Symmetric spectra - kuweb.math.ku.dk/~larsh/teaching/restricted/SymSpec.pdf · over structured ring...

Symmetric spectra

This document is a preliminary and incomplete version of a book about symmetricspectra. It probably contains many typos and inconsistencies, and hopefully not toomany actual mistakes. I intend to post updates regularly, so you may want to check myhomepage for newer versions. I am interested in feedback.

Stefan SchwedeMathematisches Institut [email protected] Bonn, Germany www.math.uni-bonn.de/people/schwede

v3.0/August 28, 2008

Contents

Introduction 3

Chapter I. Basics 91. Symmetric spectra 92. Properties of naive homotopy groups 203. Some constructions 304. Stable equivalences 465. Smash product 766. Products on homotopy groups 1007. True versus naive homotopy groups 1168. Relation to other kinds of spectra 135Exercises 155History and credits 167

Chapter II. The stable homotopy category 1711. The stable homotopy category 1712. Triangulated structure 1813. Derived smash product 1914. Generators 2025. Spectrum (co-)homology 2136. Connective covers and Postnikov sections 2197. Moore spectra 2238. Finite spectra 2289. Bousfield localization 23310. The Adams spectral sequence 239Exercises 245History and credits 248

Chapter III. Model structures 2511. Level model structures 2512. Stable model structures 2623. Operads and their algebras 2634. Model structures for algebras over an operads 2675. Connective covers of structured spectra 2746. Bousfield localization of structured spectra 278Exercises 278History and credits 279

Chapter IV. Module spectra 2811. Model structures for modules 2812. Toda brackets in triangulated categories 286History and credits 288

1

2 CONTENTS

Appendix A. Miscellaneous tools 2891. Model category theory 2892. Compactly generated spaces 2953. Simplicial sets 2994. Equivariant homotopy theory 3025. Enriched category theory 305

Bibliography 311

Index 315

INTRODUCTION 3

Introduction

This textbook is an introduction to the modern foundations of stable homotopy theory and ‘algebra’over structured ring spectra, based on symmetric spectra. We begin with a quick historical review andattempt at motivation.

A crucial prerequisite for spectral algebra is an associative and commutative smash product on agood point-set level category of spectra, which lifts the well-known smash product pairing on the homotopycategory. The first construction of what is now called ‘the stable homotopy category’, including its symmetricmonoidal smash product, is due to Boardman [5] (unpublished); accounts of Boardman’s constructionappear in [81], [78] and [2, Part III] (Adams has to devote more than 30 pages to the construction andformal properties of the smash product).

To illustrate the drastic simplification that occurred in the foundations in the mid-90s, let us drawan analogy with the algebraic context. Let R be a commutative ring and imagine for a moment that thenotion of a chain complex (of R-modules) has not been discovered, but nevertheless various complicatedconstructions of the unbounded derived category D(R) of the ring R exist. Moreover, constructions ofthe derived tensor product on the derived category exist, but they are complicated and the proof that thederived tensor product is associative and commutative occupies 30 pages. In this situation, you could talkabout objects A in the derived category together with morphisms A⊗LR A −→ A, in the derived category,which are associative and unital, and possibly commutative, again in the derived category. This notion maybe useful for some purposes, but it suffers from many defects – as one example, the category of modules(under derived tensor product in the derived category), does not in general form a triangulated category.

Now imagine that someone proposes the definition of a chain complex of R-modules and shows thatby formally inverting the quasi-isomorphisms, one can construct the derived category. She also definesthe tensor product of chain complexes and proves that tensoring with suitably nice (i.e., homotopicallyprojective) complexes preserves quasi-isomorphisms. It immediately follows that the tensor product descendsto an associative and commutative product on the derived category. What is even better, now one cansuddenly consider differential graded algebras, a ‘rigidified’ version of the crude multiplication ‘up-to-chainhomotopy’. We would quickly discover that this notion is much more powerful and that differential gradedalgebras arise all over the place (while chain complexes with a multiplication which is merely associativeup to chain homotopy seldom come up in nature).

Fortunately, this is not the historical course of development in homological algebra, but the developmentin stable homotopy theory was, in several aspects, as indicated above. In the stable homotopy categorypeople could consider ring spectra ‘up to homotopy’, which are closely related to multiplicative cohomologytheories. However, the need and usefulness of ring spectra with rigidified multiplications soon becameapparent, and topologists developed different ways of dealing with them. One line of approach uses operadsfor the bookkeeping of the homotopies which encode all higher forms of associativity and commutativity,and this led to the notions of A∞- respectively E∞-ring spectra. Various notions of point-set level ringspectra had been used (which were only later recognized as the monoids in a symmetric monoidal modelcategory). For example, the orthogonal ring spectra had appeared as I∗-prefunctors in [54], the functorswith smash product were introduced in [7] and symmetric ring spectra appeared as strictly associative ringspectra in [29, Def. 6.1] or as FSPs defined on spheres in [31, 2.7].

At this point it had become clear that many technicalities could be avoided if one had a smash producton a good point-set category of spectra which was associative and unital before passage to the homotopycategory. For a long time no such category was known, and there was even evidence that it might notexist [43]. In retrospect, the modern spectra categories could maybe have been found earlier if Quillen’sformalism of model categories [59] had been taken more seriously; from the model category perspective, oneshould not expect an intrinsically ‘left adjoint’ construction like a smash product to have a good homotopicalbehavior in general, and along with the search for a smash product, one should look for a compatible notionof cofibrations.

In the mid-90s, several categories of spectra with nice smash products were discovered, and simultane-ously, model categories experienced a major renaissance. Around 1993, Elmendorf, Kriz, Mandell and May

4 CONTENTS

introduced the S-modules [24] and Jeff Smith gave the first talks about symmetric spectra; the details ofthe model structure were later worked out and written up by Hovey, Shipley and Smith [33]. In 1995, Ly-dakis [45] independently discovered and studied the smash product for Γ-spaces (in the sense of Segal [70]),and a little later he developed model structures and smash product for simplicial functors [46]. Except forthe S-modules of Elmendorf, Kriz, Mandell and May, all other known models for spectra with nice smashproduct have a very similar flavor; they all arise as categories of continuous (or simplicial), space-valuedfunctors from a symmetric monoidal indexing category, and the smash product is a convolution product(defined as a left Kan extension), which had much earlier been studied by the category theorist Day [18].This unifying context was made explicit by Mandell, May, Schwede and Shipley in [51], where another ex-ample, the orthogonal spectra were first worked out in detail. The different approaches to spectra categorieswith smash product have been generalized and adapted to equivariant homotopy theory [20, 49, 50] andmotivic homotopy theory [21, 34, 35].

Why symmetric spectra? The author is a big fan of symmetric spectra; two important reasons arethat symmetric spectra are easy to define and require the least amount of symmetry among the models of thestable homotopy category with smash product. A consequence of the second point is that many interestinghomotopy types can be written down explicitly and in closed form. We give examples of this in Section I.??,right after the basic definitions, among these are the sphere spectrum, suspension spectra, Eilenberg-MacLane spectra, Thom spectra such as MO,MSO and MU , topological K-theory and algebraic K-theoryspectra.

Another consequence of ‘minimal symmetry’ requirements is that whenever someone writes down orconstructs a model for a homotopy type in one of the other worlds of spectra, then we immediately geta model as a symmetric spectrum by applying one of the ‘forgetful’ functors from spectra with moresymmetries which we recall in Section I.??. In fact, symmetric spectra have a certain universal property(see Shipley’s paper [72]), making them ‘initial’ among stable model categories with a compatible smashproduct.

There are already good sources available which explain the stable homotopy category, and there aremany research papers and at least one book devoted to structured ring spectra. However, my experience isthat for students learning the subject it is hard to reconcile the treatment of the stable homotopy categoryas given, for example, in Adams’ notes [2], with the more recent model category approaches to, say, S-modules or symmetric spectra. So one aim of this book is to provide a source where one can learn aboutthe triangulated stable homotopy category and about stable model categories and a good point-set levelsmash product with just one notion of what a spectrum is.

The monograph [24] by Elmendorf, Kriz, Mandell and May develops the theory of one of the competingframeworks, the S-modules, in detail. It has had a big impact and is widely used, for example becausemany standard tools can simply be quoted from that book. The theory of symmetric spectra is by nowhighly developed, but the results are spread over many research papers. The aim of this book is to collectbasic facts in one place, thus providing an alternative to [24].

Prerequisites. As a general principle, I assume knowledge of basic algebraic topology and unstablehomotopy theory. I will develop in parallel the theory of symmetric spectra based on topological spaces(compactly generated and weak Hausdorff) and simplicial sets. Whenever simplicial sets are used, I assumebasic knowledge of simplicial homotopy theory, as found for example in [28] or [53]. However, the useof simplicial sets is often convenient but hardly ever essential, so not much understanding is lost by justthinking about topological spaces throughout.

On the other hand, no prior knowledge of stable homotopy theory is assumed. In particular, we definethe stable homotopy category using symmetric spectra and develop its basic properties from scratch.

From Chapter III on I will freely use the language of Quillen’s model categories and basic results ofhomotopical algebra. The original source is [59], a good introduction is [22], and [32] is a thorough,extensive treatment.

Organization. We organize the book into chapters, each chapter into sections and some sections intosubsections. The numbering scheme for referring to definitions, theorems, examples etc. is as follows. Ifwe refer to something in the same chapter, then the reference number consists only of the arabic section

INTRODUCTION 5

number and then a running number for all kinds of environments. If the reference is to another chapter,then we add the roman chapter number in front. So ‘Lemma 3.14’ refers to a Lemma in Section 3 of thesame chapter, with running number 14, while ‘Example I.2.21’ is an example from the second section of thefirst chapter, with running number 21.

In the first chapter we introduce the basic concepts of a symmetric spectrum and symmetric ringspectrum and then, before developing any extensive theory, discuss lots of examples. [stable equivalences]There is a section on the smash product where we develop its basic formal and homotopical properties. Oneof the few points where symmetric spectra are more complicated than other frameworks is that the usualhomotopy groups can be somewhat pathological. So we spend the last section of the first chapter on thestructure of homotopy groups and the notion of semistable symmetric spectra.

The second chapter is devoted to the stable homotopy category. We develop some basic theory aroundthe stable homotopy category, such as the triangulated structure, derived smash product, homotopy (co-)limits, Postnikov sections, localization and completion, and discuss the Spanier-Whitehead category, Moorespectra and finite spectra and (Bousfield) localization and completion. In Section II.10, which is less self-contained than the rest, we give a glimpse at the mod-p Adams spectral sequence.

In Chapter III model structures enter the scene. We start by establishing the various level model struc-tures (projective, flat, injective, and their positive versions) for symmetric spectra of spaces and simplicialsets, and then discuss the associated, more important, stable model structures. We also develop the modelstructures for modules over a fixed symmetric ring spectrum and for algebras over an operad of simpli-cial sets. The latter includes the stable model structures for symmetric ring spectra and for commutativesymmetric ring spectra.

Each chapter has a section containing exercises.As a general rule, I do not attribute credit for definitions and theorems in the body of the text. Instead,

there is a section ‘History and credits’ at the end of each chapter, where I summarize, to the best of myknowledge, who contributed what. Additions and corrections are welcome.

A philosophical point: I tend to make isomorphisms explicit, thus avoiding phrases like ‘the canonicalisomorphism’ unless the isomorphism we have in mind has previously been defined. The main reasonfor this is the author’s experience that what seems canonical to the expert may often not be clear to anewcomer. Another reason is that here and there, one can get sign trouble if one is not careful aboutchoices of isomorphisms. A disadvantage is that we have to introduce lots of symbols or numbers to referto the isomorphisms.

Some conventions. Let us fix some terminology and enact several useful conventions. We think thatsome slight abuse of language and notation can often make statements more transparent, but when we allowourselves such imprecision we feel obliged to state them clearly here, at the risk of being picky.

We denote by T the category of pointed, compactly generated topological spaces. For us a compactlygenerated space is a Kelley space which is also weak Hausdorff; we review the definitions and collect variousproperties of compactly generated spaces in Section A.2. A map between topological spaces always refersto a continuous map, unless explicitly stated otherwise. Similarly, an action of a group on a space refersto a continuous action. We denote by sS the category of pointed simplicial sets. We review the definitionsand collect various properties of simplicial sets in Section A.3.

It will be convenient to define the n-sphere Sn as the one-point compactification of n-dimensionaleuclidian space Rn, with the point at infinity as the basepoint. We will sometimes need to identify the1-sphere with the space [0, 1]/0, 1, the quotient space of the unit interval with identified endpoints, andwith the space |∆[1]/∂∆[1]|, the geometric realization of the simplicial 1-simplex ∆[1] modulo its boundary.The precise identifications do not matter, but for definiteness we fix homeomorphisms now. We agree toalways use the homeomorphism

(0.1) t : [0, 1]/0, 1∼=−→ S1 , x 7→ 2x− 1

x(1− x)

with the understanding that the formula describes the restriction of t from the open interval (0, 1) to Rand that t sends the identified endpoints to the point at infinity. This homeomorphism (like many others)

6 CONTENTS

has the convenient property that it takes the ‘inversion’ of the interval to (the one-point compactificationof) reflection of R at the origin, i.e., it satisfies t(1 − x) = −t(x). The homeomorphism [0, 1] −→ ∆[1] tothe topological 1-simplex sending x to (x, 1− x) gives a homeomorphism

(0.2) [0, 1]/0, 1 ∼= |∆[1]/∂∆[1]| .The composite homeomorphism ∆[1] −→ S1 then sends the open simplex to R by (x, y) 7→ x/y − y/x.

For n ≥ 0, the symmetric group Σn is the group of bijections of the set 1, 2, . . . , n; in particular,Σ0 consists only of the identity of the empty set. It will often be convenient to identify the productgroup Σn × Σm with the subgroup of Σn+m of those permutations which take the sets 1, . . . , n andn+ 1, . . . , n+m to themselves. Whenever we do so, we implicitly use the monomorphism

+ : Σn × Σm −→ Σn+m , (τ, κ) 7→ τ + κ

given by

(τ + κ)(i) =

τ(i) for 1 ≤ i ≤ n,

κ(i− n) + n for n+ 1 ≤ i ≤ n+m.We let the symmetric group Σn act from the left on Rn by permuting the coordinates, i.e.,

γ(x1, . . . , xn) = (xγ−1(1), . . . , xγ−1(n)). This action compactifies to an action on Sn which fixes the base-point. The ‘canonical’ linear isomorphism

Rn × Rm −→ Rn+m , ((x1, . . . , xn), (y1, . . . , xm)) 7→ (x1, . . . , xn, y1, . . . , xm)

induces a homeomorphism Sn ∧ Sm −→ Sn+m which is equivariant with respect to the action of the groupΣn × Σm, acting on the target by restriction from Σn+m.

The topological spaces we consider are usually pointed, and we use the notation πnX for the n-thhomotopy group with respect to the distinguished basepoint, which we do not record in the notation.

We will often use the ‘exponential law’, i.e., the adjunction between smash product and mapping spaces.Let us fix a few conventions, in particular when we adjoint spheres. For based spaces (or simplicial sets)K,X and Z we define the adjunction bijection

(0.3) a : map(X ∧K,Z) ∼= map(X,map(K,Z))

by (a(f)(x))(k) = f(x ∧ k), where f : X ∧K −→ Z is a based continuous (or simplicial) map, x ∈ X andk ∈ K.

The m-fold loop space of a based space (or simplicial set) Z is the space (respectively simplicial set)ΩmZ = map(Sm, Z). We use the adjunction bijection and the homeomorphism Sm∧Sn ∼= Sm+n to identifyΩm(ΩnZ) = map(Sm,map(Sn, Z)) with Ωm+nZ = map(Sm+n, Z) without further notice. In particular, wecan, and will, identify Ωm with the m-fold iterate of Ω. The m-th homotopy group of a based space Z is theset of based homotopy classes of based maps Sm −→ Z or, equivalently, the set of path components of themapping space ΩmZ = map(Sm, Z). This has a natural group structure for m ≥ 1, which is commutativefor m ≥ 2. Looping a space shifts its homotopy groups; more precisely, the adjunction bijection passes toa bijection

(0.4) πk(ΩmZ) = [Sk,map(Sm, Z)] a−−→ [Sk+m, Z] = πk+mZ

on homotopy classes. So the map a takes the homotopy class of a based map f : Sk −→ ΩmZ to thehomotopy class of a(f) : Sk+m −→ Z given by a(f)(t ∧ s) = f(t)(s) for t ∈ Sk, s ∈ Sm.

Limits and colimits. Limits and colimits in a category are hardly ever unique, but the universalproperty which they enjoy makes then ‘unique up to canonical isomorphism’. We want to fix our languagefor talking about this unambiguously. We recall that a colimit of a functor F : I −→ C is a pair (F , κ)consisting of an object F of C and a natural transformation κ : F −→ cF from F to the constant functorwith value F which is initial among all natural transformations from F to constant functors. We oftenfollow the standard abuse of language and call the object F a colimit, or even the colimit, of the functor Fand denote it by colimI F . When we need to refer to the natural transformation κ which is part of the dataof a colimit, we refer to the component κi : F (i) −→ colimI F at an object i ∈ I as the canonical morphismfrom the object F (i) to the colimit. Dually for limits. [end, coends]

INTRODUCTION 7

[use of naive homotopy groups π∗ versus π∗ for true homotopy groups. The more important conceptdeserves the simpler name.]

Remark 0.5 (Manipulations rules for coordinates). Natural numbers occurring as levels of a symmetricspectrum or as dimensions of homotopy groups are really placeholders for sphere coordinates. The role ofthe symmetric group actions on the spaces of a symmetric spectrum is to keep track of how such coordinatesare shuffled. Permutations will come up over and over again in constructions and results about symmetricspectra, and there is a very useful small set of rules which predict when to expect permutations. I recommendbeing very picky about the order in which dimensions or levels occur when performing constructions withsymmetric spectra, as this predicts necessary permutations and helps to prevent mistakes. Sometimesmissing a permutation just means missing a sign; in particular missing an even permutation may not haveany visible effect. But in general the issue is more serious; for symmetric spectra which are not semistable,missing a permutation typically misses a nontrivial operation.

A first example of this are the centrality and commutativity conditions for symmetric ring spectra,which use shuffle permutations χn,1 and χn,m. A good way of remembering when to expect a shuffle is tocarefully distinguish between indices such as n + m and m + n. Of course these two numbers are equal,but the fact that one arises naturally instead of the other reminds us that a shuffle permutation should beinserted. A shuffle required whenever identifying n + m with m + n is just one rule, and here are somemore.

Main rule: When manipulating expressions which occur as levels of symmetric spectra or dimensionsof spheres, be very attentive for how these expressions arise naturally and when you use the basic rulesof arithmetic of natural numbers. When using the basic laws of addition and multiplication of naturalnumbers in such a context, add permutations according to the following rules (i)-(v).

(i) Do not worry about associativity of addition or multiplication, or the fact that 0 respectively 1are units for those operations. No permutations are required.

(ii) Whenever using commutativity of addition as in n + m = m + n, add a shuffle permutationχn,m ∈ Σn+m.

(iii) Whenever using commutativity of multiplication as in nm = mn, add a multiplicative shuffleχ×n,m ∈ Σnm defined by

χ×n,m(j + (i− 1)n) = i+ (j − 1)m

for 1 ≤ j ≤ n and 1 ≤ i ≤ m.(iv) Do not worry about left distributivity as in p(n+m) = pn+ pm. No permutation is required.(v) Whenever using right distributivity as in (n+m)q = nq +mq, add the permutation

(χ×q,n × χ×q,m) χ×n+m,q ∈ Σ(n+m)q .

Rule (v) also requires us to throw in permutations whenever we identify a product nq with an iterated sumq+ · · ·+ q (n copies) since we use right distributivity in the process. However, no permutations are neededwhen instead identifying nq with a sum of q copies of n, since that only uses left distributivity.

The heuristic rules (i) through (v) above are a great help in guessing when to expect coordinate orlevel permutations when working with symmetric spectra. But the rules are more than heuristics, and arebased on the following rigorous mathematics. Typically, there are ‘coordinate free’ constructions in thebackground (compare Exercise 8.9) which are indexed by finite sets A which are not identified with any ofthe standard finite sets n = 1, . . . , n. The outcome of such constructions may naturally be indexed bysets which are built by forming disjoint unions or products. The permutations arise because in contrastto the arithmetic rules for + and ·, their analogues for disjoint union and cartesian product of sets onlyholds up to isomorphism, and one can arrange to make some, but not all, of the required isomorphisms beidentity maps.

In more detail, when we want to restrict a ‘coordinate free’ construction to symmetric spectra, wespecialize to standard finite sets n; however, if the coordinate free construction involves disjoint union orcartesian product, we need to identify the unions or products of standard finite sets in a consistent waywith the standard finite set of the same cardinality. A consistent way to do that amounts to what is called

8 CONTENTS

a structure of bipermutative category on the category of standard finite sets. So we define binary functors+ and · on standard finite sets resembling addition and multiplication of natural numbers as closely aspossible.

We let Fin denote the category of standard finite sets whose objects are the sets n for n ≥ 0 and whosemorphisms are all set maps. We define the sum functor + : Fin×Fin −→ Fin by addition on objects andby ‘disjoint union’ on morphisms. More precisely, for morphisms f : n −→ n′ and g : m −→ m′ we definef + g : n + m −→ n′ + m′ by

(f + g)(i) =

f(i) if 1 ≤ i ≤ n, and

g(i− n) + n′ if n+ 1 ≤ i ≤ n+m.

This operation is strictly associative and the empty set 0 is a strict unit. The symmetry isomorphism isthe shuffle map χn,m : n + m −→m + n.

We define the product functor · : Fin × Fin −→ Fin by multiplication on objects and by ‘cartesianproduct’ on morphisms. To make sense of this we have to linearly order the product of the sets n and m.There are two choices which are more obvious than others, namely lexicographically with either the first orthe second coordinate defined as the more important one. Both choices work fine, and we will prefer the firstcoordinate. More precisely, for morphisms f : n −→ n′ and g : m −→m′ we define f · g : n ·m −→ n′ ·m′

by(f · g)(j + (i− 1)n) = f(j) + (g(i)− 1)n′

for 1 ≤ j ≤ n and 1 ≤ i ≤ m. The product · is also strictly associative and the set 1 is a strict unit. Thecommutativity isomorphism is the multiplicative shuffle χ×nm : n ·m −→m · n.

This choice of ordering the product of n and m has the effect of making n ·m ‘naturally’ the same asn + · · ·+ n (m copies), because we have

f · Idm = f + · · ·+ f (m copies).

Since p·k ‘is’ p+· · ·+p (k times), we can take the left distributivity isomorphism p·(n+m) = (p·n)+(p·m)as the identity (compare rule (iv)).

In contrast, Idn ·g is in general not equal to g + · · ·+ g (n copies), but rather we have

Idn ·g = χ×m′,n(g + · · ·+ g)χ×n,mfor a morphism g : m −→ m′. However, then right distributivity isomorphism cannot be taken as theidentity; since the coherence diagram

q · (n + m)χ×q,n+m //

left dist.

(n + m) · q

right dist.

q · n + q ·m

χ×q,n+χ×q,m

// n · q + m · q

is supposed to commute, we are forced to define the right distributivity isomorphism (n+m) ·q ∼= (n ·q)+(m · q) as (χ×q,n × χ×q,m) χ×n+m,q, which explains rule (v) above.

Acknowledgments. A substantial part of this book was written during a sabbatical semester taken bythe author at the Massachusetts Institute of Technology in the fall 2006, where I could also try out some ofthe contents of this book in a graduate course. I am grateful to Haynes Miller for the invitation that madethis possible. I would like to thank Mark Hovey, Brooke Shipley and Jeff Smith for the permission to use thetitle of their paper [33] also for this book. I thank the following people for helpful comments, correctionsand improvements: Daniel Davis, Moritz Groth, Lars Hesselholt, Jens Hornbostel, Katja Hutschenreuter,Tyler Lawson, Mike Mandell, Steffen Sagave and Brooke Shipley.

CHAPTER I

Basics

1. Symmetric spectra

Definition 1.1. A symmetric spectrum consists of the following data:• a sequence of pointed spaces Xn for n ≥ 0• a basepoint preserving continuous left action of the symmetric group Σn on Xn for each n ≥ 0• based maps σn : Xn ∧ S1 −→ Xn+1 for n ≥ 0.

This data is subject to the following condition: for all n,m ≥ 0, the composite

(1.2) Xn ∧ Smσn ∧ Id // Xn+1 ∧ Sm−1

σn+1∧Id // · · ·σn+m−2∧Id // Xn+m−1 ∧ S1

σn+m−1 // Xn+m

is Σn × Σm-equivariant. Here the symmetric group Σm acts by permuting the coordinates of Sm, andΣn × Σm acts on the target by restriction of the Σn+m-action. We often denote the composite map (1.2)by σm, with the understanding that σ0 is the identity map. We refer to the space Xn as the n-th level ofthe symmetric spectrum X.

A morphism f : X −→ Y of symmetric spectra consists of Σn-equivariant based maps fn : Xn −→ Ynfor n ≥ 0, which are compatible with the structure maps in the sense that fn+1 σn = σn (fn ∧ IdS1) forall n ≥ 0.

By our standing hypothesis, ‘space’ in the previous definition means a compactly generated weak Haus-dorff space, compare Appendix A.2. The category of symmetric spectra is denoted by Sp; when we needto emphasize that we use spaces (as opposed to simplicial set), we add the index T (which denotes thecategory of compactly generated weak Hausdorff space) and use the notation SpT. Symmetric spectra ofsimplicial sets, to be defined in 3.1 below, will accordingly be denoted SpsS.

Definition 1.3. A symmetric ring spectrum R consists of the following data:• a sequence of pointed spaces Rn for n ≥ 0• a basepoint preserving continuous left action of the symmetric group Σn on Rn for each n ≥ 0• Σn × Σm-equivariant multiplication maps

µn,m : Rn ∧Rm −→ Rn+m

for n,m ≥ 0, and• two unit maps

ι0 : S0 −→ R0 and ι1 : S1 −→ R1 .

This data is subject to the following conditions:(Associativity) The square

Rn ∧Rm ∧RpId∧µm,p //

µn,m∧Id

Rn ∧Rm+p

µn,m+p

Rn+m ∧Rp µn+m,p

// Rn+m+p

commutes for all n,m, p ≥ 0.

9

10 I. BASICS

(Unit) The two composites

Rn ∼= Rn ∧ S0Id∧ι0 // Rn ∧R0

µn,0 // Rn

Rn ∼= S0 ∧Rnι0∧Id // R0 ∧Rn

µ0,n // Rn

are the identity for all n ≥ 0.(Centrality) The diagram

Rn ∧ S1Id∧ι1 //

τ

Rn ∧R1

µn,1 // Rn+1

χn,1

S1 ∧Rn ι1∧Id

// R1 ∧Rn µ1,n// R1+n

commutes for all n ≥ 0. Here χn,m ∈ Σn+mdenotes the shuffle permutation which moves the first n elementspast the last m elements, keeping each of the two blocks in order; in formulas,

(1.4) χn,m(i) =

i+m for 1 ≤ i ≤ n,i− n for n+ 1 ≤ i ≤ n+m.

A morphism f : R −→ S of symmetric ring spectra consists of Σn-equivariant based maps fn : Rn −→Sn for n ≥ 0, which are compatible with the multiplication and unit maps in the sense that fn+m µn,m =µn,m (fn ∧ fm) for all n,m ≥ 0, and f0 ι0 = ι0 and f1 ι1 = ι1.

A symmetric ring spectrum R is commutative if the square

Rn ∧Rmµn,m

twist // Rm ∧Rnµm,n

Rn+m χn,m

// Rm+n

commutes for all n,m ≥ 0. Note that this commutativity diagram implies the centrality condition above.

Definition 1.5. A right module M over a symmetric ring spectrum R consists of the following data:• a sequence of pointed spaces Mn for n ≥ 0• a basepoint preserving continuous left action of the symmetric group Σn on Mn for each n ≥ 0,

and• Σn × Σm-equivariant action maps αn,m : Mn ∧Rm −→Mn+m for n,m ≥ 0.

The action maps have to be associative and unital in the sense that the following diagrams commute

Mn ∧Rm ∧RpId∧µm,p //

αn,m∧Id

Mn ∧Rm+p

αn,m+p

Mn∼= Mn ∧ S0

Id∧ι0 //

TTTTTTTTTTTTTTTTTT

TTTTTTTTTTTTTTTTTTMn ∧R0

αn,0

Mn+m ∧Rp αn+m,p

// Mn+m+p Mn

for all n,m, p ≥ 0. A morphism f : M −→ N of right R-modules consists of Σn-equivariant based mapsfn : Mn −→ Nn for n ≥ 0, which are compatible with the action maps in the sense that fn+m αn,m =αn,m (fn ∧ Id) for all n,m ≥ 0. We denote the category of right R-modules by mod-R.

Remark 1.6. We have stated the axioms for symmetric ring spectra in terms of a minimal amountof data and conditions. Now we put these conditions into perspective. We consider a symmetric ringspectrum R.

1. SYMMETRIC SPECTRA 11

(i) It will be useful to have the following notation for iterated multiplication maps. For naturalnumbers n1, . . . , ni ≥ 0 we denote by

µn1,...,ni : Rn1 ∧ . . . ∧Rni −→ Rn1+···+ni

the map obtained by composing multiplication maps smashed with suitable identity maps; byassociativity, the parentheses in the multiplications don’t matter. More formally we can definethe iterated multiplication maps inductively, setting

µn1,...,ni = µn1,n2+···+ni (IdRn1∧µn2,...,ni) .

(ii) We can define higher-dimensional unit maps ιm : Sm −→ Rm for m ≥ 2 as the composite

Sm = S1 ∧ . . . ∧ S1 ι1∧...∧ι1−−−−−−→ R1 ∧ . . . ∧R1µ1,...,1−−−−→ Rm .

Centrality then implies that ιm is Σm-equivariant, and it implies that the diagram

Rn ∧ SmId∧ιm //

twist

Rn ∧Rmµn,m // Rn+m

χn,m

Sm ∧Rn

ιm∧Id// Rm ∧Rn µm,n

// Rm+n

commutes for all n,m ≥ 0, generalizing the original centrality condition.(iii) As the terminology suggests, the symmetric ring spectrum R has an underlying symmetric

spectrum. In fact, the multiplication maps µn,m make R into a right module over itself, andmore generally, every right R-module M has an underlying symmetric spectrum as follows.We keep the spaces Mn and symmetric group actions and define the missing structure mapsσn : Mn∧S1 −→Mn+1 as the composite αn,1 (IdMn ∧ι1). Associativity implies that the iteratedstructure map σm : Mn ∧ Sm −→Mn+m equals the composite

Mn ∧ SmId∧ιm−−−−→ Mn ∧Rm

αn,m−−−→ Mn+m .

So the iterated structure map is Σn×Σm-equivariant by part (ii) and the equivariance hypothesison αn,m, and we have in fact obtained a symmetric spectrum.

The forgetful functors which associates to a symmetric ring spectrum or module spectrumits underlying symmetric spectrum have left adjoints. We will construct the left adjoints in Ex-ample 5.19 below after introducing the smash product of symmetric spectra. The left adjointsassociate to a symmetric spectrum X the ‘free R-module’ X ∧R respectively the ‘free symmetricring spectrum’ TX generated by X, which we will refer to it as the tensor algebra.

(iv) Using the internal smash product of symmetric spectra introduced in Section 5, we can identifythe ‘explicit’ definition of a symmetric ring spectrum which we just gave with a more ‘implicit’definition of a symmetric spectrum R together with morphisms µ : R ∧ R −→ R and ι : S −→ R(where S is the sphere spectrum, see Example 1.8) which are suitably associative and unital. The‘explicit’ and ‘implicit’ definitions of symmetric ring spectra coincide in the sense that they defineisomorphic categories, see Theorem 5.17.

Primary invariants of a symmetric spectrum are its homotopy groups, which come in two flavors as‘naive’ and ‘true’ homotopy groups. The former kind is defined directly from the homotopy groups of thespaces in a spectrum: the k-th naive homotopy group of a symmetric spectrum X is defined as the colimit

πkX = colimn πk+nXn

taken over the stabilization maps ι : πk+nXn −→ πk+n+1Xn+1 defined as the composite

(1.7) πk+nXn−∧S1

−−−−−→ πk+n+1

(Xn ∧ S1

) (σn)∗−−−−−→ πk+n+1Xn+1 .

For large enough n, the set πk+nXn has a natural abelian group structure and the stabilization maps arehomomorphisms, so the colimit πkX inherits a natural abelian group structure.

12 I. BASICS

As will hopefully become clear later, these naive homotopy groups are often ‘wrong’; for example, thecategory which one obtains by localizing at the class of π∗-isomorphisms is not equivalent to the stablehomotopy category as we discuss it in Chapter II. However, we need the naive homotopy groups to definethe more important true homotopy groups (see Definition 4.29 below), and also as a calculational tool toget at the true homotopy groups.

1.1. Basic examples. Before developing any more theory, we give some examples of symmetric spec-tra and symmetric ring spectra which represent prominent stable homotopy types. We discuss the spherespectrum (1.8), suspension spectra (1.13), Eilenberg-Mac Lane spectra (1.14), Thom spectra (1.16, 1.17)and spectra representing topological K-theory (1.18). It is a nice feature of symmetric spectra that onecan explicitely write down these examples in closed form with all the required symmetries. We also give,whenever possible, the naive homotopy groups of these symmetric spectra. It will turn out that all the ex-amples in the section are in fact semistable (to be defined in Definition 4.28) and hence the naive homotopygroups coincide with the more important true homotopy groups (to be defined in Definition 4.29).

Example 1.8 (Sphere spectrum). The symmetric sphere spectrum S is given by Sn = Sn, where thesymmetric group permutes the coordinates and σn : Sn ∧ S1 −→ Sn+1 is the canonical homeomorphism.This is a commutative symmetric ring spectrum with identity as unit map and the canonical homeomorphismSn∧Sm −→ Sn+m as multiplication map. The sphere spectrum is the initial symmetric ring spectrum: if Ris any symmetric ring spectrum, then a unique morphism of symmetric ring spectra S −→ R is given by thecollection of unit maps ιn : Sn −→ Rn (compare 1.6 (ii)). Being initial, the sphere spectrum plays the sameformal role for symmetric ring spectra as the integers Z play for rings. This justifies the notation ‘S’ usingthe \mathbb font. The category of right S-modules is isomorphic to the category of symmetric spectra, viathe forgetful functor mod- S −→ Sp. Indeed, if X is a symmetric spectrum then the associativity conditionshows that there is at most one way to define action maps

αn,m : Xn ∧ Sm −→ Xn+m ,

namely as the iterated structure map σm, and these do define the structure of right S-module on X.The naive homotopy group πkS = colimn πk+nS

n is called the k-th stable homotopy group of spheres,or the k-th stable stem, and will be denoted πs

k. Since Sn is (n− 1)-connected, the group πsk is trivial for

negative values of k. The degree of a self-map of a sphere provides an isomorphism πs0∼= Z. For k ≥ 1, the

homotopy group πsk is finite. This is a direct consequence of Serre’s calculation of the homotopy groups of

spheres modulo torsion, which we recall without giving a proof, and Freundenthal’s suspensions theorem[justify].

Theorem 1.9 (Serre). Let m > n ≥ 1. Then

πmSn =

(finite group)⊕ Z if n is even and m = 2n− 1(finite group) else.

Thus for k ≥ 1, the stable stem πsk = πkS is finite.

As a concrete example, we inspect the colimit system defining πs1, the first stable stem. Since the

universal cover of S1 is the real line, which is contractible, the theory of covering spaces shows that thegroups πnS1 are trivial for n ≥ 2. The Hopf map

η : S3 ⊆ C2\0 proj.−−−→ CP 1 ∼= S2

is a locally trivial fiber bundle with fiber S1, so it gives rise to long exact sequence of homotopy groups. Sincethe fiber S1 has no homotopy above dimension one, the group π3S

2 is free abelian of rank one, generated bythe class of η. By Freudenthal’s suspension theorem the suspension homomorphism −∧S1 : π3S

2 −→ π4S3

is surjective and from π4S3 on the suspension homomorphism is an isomorphism. So the first stable stem πs

1

is cyclic, generated by the image of η, and its order equals the order of the suspension of η. On the one hand,η itself is stably essential, since the Steenrod operation Sq2 acts non-trivially on the mod-2 cohomology ofthe mapping cone of η, which is homeomorphic to CP 2.


On the other hand, twice the suspension of η is null-homotopic. To see this we consider the commutativesquare

(x, y)_

S3η //

CP 1

[x : y]_

(x, y) S3

η// CP 1 [x : y]

in which the vertical maps are induced by complex conjugation in both coordinates of C2. The left verticalmap has degree 1, so it is homotopic to the identity, whereas complex conjugation on CP 1 ∼= S2 has degree−1. So (−1) η is homotopic to η. Thus the suspension of η is homotopic to the suspension of (−1) η,which by the following lemma is homotopic to the negative of η ∧ S1.

Lemma 1.10. Let Y be a pointed space, m ≥ 0 and f : Sm −→ Sm a based map of degree k. Then forevery homotopy class x ∈ πn(Y ∧ Sm) the classes (IdY ∧f)∗(x) and k · x become equal in πn+1(Y ∧ Sm+1)after one suspension.

Proof. Let dk : S1 −→ S1 be any pointed map of degree k. Then the maps f ∧ S1, Sm ∧ dk :Sm+1 −→ Sm+1 have the same degree k, hence they are based homotopic. Suppose x is represented byϕ : Sn −→ Y ∧ Sm. Then the suspension of (Y ∧ f)∗(x) is represented by (Y ∧ f ∧ S1) (ϕ ∧ S1) whichis homotopic to (Y ∧ Sm ∧ dk) (ϕ ∧ S1) = (ϕ ∧ S1) (Sn ∧ dk). Precomposition with the degree k mapSn ∧ dk of Sn+1 induces multiplication by k, so the last map represents the suspension of k · x.

The conclusion of Lemma 1.10 does not in general hold without the extra suspension, i.e., (Y ∧ f)∗(x)need not equal k ·x in πn(Y ∧Sm): as we showed above, (−1)η is homotopic to η, which is not homotopicto −η since η generates the infinite cyclic group π3S

2.As far as we know, the stable homotopy groups of spheres don’t follow any simple pattern. Much

machinery of algebraic topology has been developed to calculate homotopy groups of spheres, both unstableand stable, but no one expects to ever get explicit formulae for all stable homotopy groups of spheres. TheAdams spectral sequence based on mod-p cohomology (see Section II.10) and the Adams-Novikov spectralsequence based on MU (complex cobordism) or BP (the Brown-Peterson spectrum at a fixed prime p) arethe most effective tools we have for explicit calculations as well as for discovering systematic phenomena.

Example 1.11 (Multiplication in the stable stems). The stable stems πs∗ = π∗S form a graded commu-

tative ring which acts on the naive and true homotopy groups of every other symmetric spectrum X. Wedenote the action simply by a ‘dot’

· : πkX × πsl −→ πk+lX respectively · : πkX × πs

l −→ πk+lX.

The definition is essentially straightforward, but there is one subtlety in showing that the product is well-defined.

We first define the action of πs∗ on the naive homotopy groups π∗X of a symmetric spectrum X. Suppose

f : Sk+n −→ Xn and g : Sl+m −→ Sm represent classes in πkX respectively πsl . Then we agree that the

composite

(1.12) Sk+l+n+m Id∧χl,n∧Id−−−−−−−→ Sk+n+l+m f∧g−−−−→ Xn ∧ Smσm−−−→ Xn+m

represents the product of [f ] and [g]. The shuffle permutation χl,n is predicted by the principle that allnatural number must occur in the ‘natural order’, compare Remark 0.5. If we simply smash f and g thedimension of the sphere of origin is naturally (k + n) + (l + m), but in order to represent an element ofπk+lX they should occur in the order (k+ l)+ (n+m), whence the shuffle permutation (which here simplyintroduces the sign (−1)ln).

We check that the multiplication is well-defined. If we replace g : Sl+m −→ Sm by its suspensiong ∧ S1, then the composite (1.12) changes to its suspension, composed with the structure map σn+m :Xn+m ∧ S1 −→ Xn+m+1. So the resulting stable class is independent of the representative g of thestable class in πs

l . Independence of the representative for πkX is slightly more subtle. If we replace

14 I. BASICS

f : Sk+n −→ Xn by the representative σn (f ∧S1) : Sk+n+1 −→ Xn+1, then the composite (1.12) changesto σ1+m(f ∧ Id∧g)(Id∧χl,n+1 ∧ Id), which is the lower left composite in the commutative diagram

Sk+l+n+1+mId∧χl,n∧χ1,m //

Id∧χl,n+1∧Id **TTTTTTTTTTTTTTTT Sk+n+l+m+1f∧g∧Id //

Id∧χl+m,1

Xn ∧ Sm+1

Id∧χm,1

Sk+n+1+l+mf∧Id∧g

// Xn ∧ S1+m

σ1+m// Xn+1+m

By Lemma 1.10 the map Id∧χm,1 induces multiplication by (−1)m on homotopy groups after one sus-pension. This cancels the sign coming from the shuffle factor χ1,m in the initial horizontal map. So thecomposite is homotopic, after one suspension, to the composite σ1+m(f ∧ g ∧ Id)(Id∧χl,n ∧ Id), whichrepresents the same stable class as (1.12).

Now we verify that the dot product is biadditive. We only show the relation (x+ x′) · y = x · y+ x′ · y,and additivity in y is similar. Suppose as before that f, f ′ : Sk+n −→ Xn and g : Sl+m −→ Sm representclasses in πkX respectively πs

l . Then the sum of f and f ′ in πk+nXn is represented by the composite

Sk+npinch−−−→ Sk+n∨Sk+n

f∨f ′−−−→ Xn .

In the square

Sk+l+n+m

pinch

1∧χl,n∧1 // Sk+n+l+m

(f+f ′)∧g

++WWWWWWWWWWWWWWWWWWWWWWWW

pinch∧Id

(Sk+n∨Sk+n) ∧ Sl+m

(f∨f ′)∧g// Xn ∧ Sm

Sk+l+n+m∨Sk+l+n+m(1∧χl,n∧1)∨(1∧χl,n∧1)

// Sk+n+l+m∨Sk+n+l+m

(f∧g)∨(f ′∧g)

33gggggggggggggggggggggggg∼=

OO

the right part commutes on the nose and the left square commutes up to homotopy. After composing withthe iterated structure map σm : Xn∧Sm −→ Xn+m, the composite around the top of the diagram becomes(f + f ′) · g, whereas the composite around the bottom represents fg + f ′g. This proves additivity of thedot product in the left variable.

If we specialize to X = S then the product provides a biadditive graded pairing · : πsk × πs

l −→ πsk+l of

the stable homotopy groups of spheres. We claim that for every symmetric spectrum X the diagram

πkX × πsl × πs

j·×Id //

Id×·

πk+lX × πsj

·

πkX × πsl+j ·

// πk+l+jX

commutes, so the product on the stable stems and the action on the homotopy groups of a symmetricspectrum are associative. After choosing representing maps Sk+n −→ Xn, Sl+m −→ Sm and Sj+q −→ Sq

and unraveling all the definitions, this associativity ultimately boils down to the equality

(k × χl,n ×m× j × q) (k × l × χj,n+m × q) = (k × n× l × χj,m × q) (k × χl+j,n ×m× q)in the symmetric group Σk+l+j+q+n+m and commutativity of the square

Xn ∧ Sm ∧ Sq

Id∧∼=

σm∧Id // Xn+m ∧ Sq

σq

Xn ∧ Sm+q

σm+q// Xn+m+q


Finally, the multiplication in the homotopy groups of spheres is commutative in the graded sense, i.e.,we have xy = (−1)klyx for x ∈ πs

k and y ∈ πsl . Indeed, for representing maps f : Sk+n −→ Sn and

g : Sl+m −→ Sm the square

Sk+l+n+mId∧χl,n∧Id //

χk,l×χn,m

Sk+n+l+mf∧g // Sn+m

χn,m

Sl+k+m+n

Id∧χk,m∧Id// Sl+m+k+n

g∧f// Sm+n

commutes. The two vertical coordinate permutations induce the signs (−1)kl+nm respectively (after onesuspension) (−1)nm on homotopy groups. Since the upper horizontal composite represents xy and the lowercomposite represents yx, this proves the relation xy = (−1)klyx.

The following table gives the stable homotopy groups of spheres through dimension 8:

n 0 1 2 3 4 5 6 7 8πsn Z Z/2 Z/2 Z/24 0 0 Z/2 Z/240 (Z/2)2

generator ι η η2 ν ν2 σ ησ, ε

Here ν and σ are the Hopf maps which arises unstably as fiber bundles S7 −→ S4 respectively S15 −→ S8.The element ε in the 8-stem can be defined using Toda brackets (see Construction 6.15) as ε = ησ+〈ν, η, ν〉.The table contains or determines all multiplicative relations in this range except for η3 = 12ν. A theorem ofNishida’s says that every homotopy element of positive dimension is nilpotent. We explain in Section II.10how this table can be obtained with the help of the Adams spectral sequence.

A word of warning: it is tempting to try to define a product on the naive homotopy groups of a symmetricring spectrum R in a similar fashion, by smashing representatives and shuffling sphere coordinates into theirnatural order. This will indeed give an associative product whenever the underlying symmetric spectrumof R is semistable (see Corollary 6.9). However, if R is not semistable, then smashing of representativesdoes not descend to a well-defined product on naive homotopy groups! In that case the algebraic structurethat the homotopy groups of R enjoy is more subtle, and we discuss it in Exercise 8.35. In any case, thetrue homotopy groups of a symmetric ring spectrum have a natural multiplication, by Proposition 6.5.

Example 1.13 (Suspension spectra). Every pointed space K gives rise to a suspension spectrum Σ∞Kvia

(Σ∞K)n = K ∧ Sn

with structure maps given by the canonical homeomorphism (K ∧Sn)∧S1∼=−→ K ∧Sn+1. For example, the

sphere spectrum S is isomorphic to the suspension spectrum Σ∞S0.The naive homotopy group

πskK = πk (Σ∞K) = colimn πk+n(K ∧ Sn)

is called the kth stable homotopy group of K. Since K ∧ Sn is (n− 1)-connected, the suspension spectrumΣ∞K is connective. The Freudenthal suspension theorem implies that for every suspension spectrum, thecolimit system for a specific homotopy group always stabilizes. A symmetric spectrum X is isomorphicto a suspension spectrum (necessarily that of its zeroth space X0) if and only if every structure mapσn : Xn ∧ S1 −→ Xn+1 is a homeomorphism.

Example 1.14 (Eilenberg-Mac Lane spectra). For an abelian group A, the Eilenberg-Mac Lane spec-trum HA is defined by

(HA)n = A[Sn] ,the reduced A-linearization of the n-sphere. Let us review the linearization construction in some detailbefore defining the rest of the structure of the Eilenberg-Mac Lane spectrum.

For a general based space K, the underlying set of the A-linearization A[K] is tensor product of A withthe reduced free abelian group generated by the points of K. In other words, points of A[K] are finite sums

16 I. BASICS

of points of K with coefficients in A, modulo the relation that all A-multiples of the basepoint are zero.The set A[K] is topologized as a quotient space of the disjoint union of the spaces An ×Kn (with discretetopology on An), via the surjection

qn≥0An ×Kn −→ A[K] , (a1, . . . , an, k1, . . . , kn) 7→

n∑i=i

ai · ki .

There is a natural map Hn(K,A) −→ πn(A[K], 0) from the reduced singular homology groups of K withcoefficients in A to the homotopy groups of the linearization: let x =

∑i ai · fi be a singular chain of K

with coefficients ai in A, i.e., every fi : ∆[n] −→ K is a continuous map from the topological n-simplex. Weuse the abelian group structure of A[K] and add the maps fj pointwise and multiply by the coefficients, toget a single map x : ∆[n] −→ A[K]. More formally, x is the composite

∆[n](a1,...,an,f1,...,fn)−−−−−−−−−−−−→ An ×Kn −→ A[K] .

If the original chain x is a cycle in the singular chain complex, then the map x sends the boundary of thesimplex to the neutral element 0 of A[K]. So x factors over a continuous based map ∆[n]/∂∆[n] −→ A[K].After composing with a homeomorphism between the n-sphere and ∆[n]/∂∆[n] this maps represents anelement in the homotopy group πn(A[K], 0). If K has the based homotopy type of a CW-complex, thenthe map Hn(K,A) −→ πn(A[K], 0) is an isomorphism [ref]. In the special case K = Sn this shows thatthe A[Sn] has only one non-trivial homotopy group in dimension n, where it is isomorphic to A. In otherwords, (HA)n = A[Sn] is an Eilenberg-Mac Lane space of type (A,n).

Now we return to the definition of the Eilenberg-Mac Lane spectrum HA. The symmetric group acts on(HA)n = A[Sn] by permuting the smash factors of Sn. The structure map σn : (HA)n ∧ S1 −→ (HA)n+1

is given byA[Sn] ∧ S1 −→ A[Sn+1] ,

(∑iai · xi

)∧ y 7−→

∑iai · (xi ∧ y) .

If A is not just an abelian group but also has a ring structure, then HA becomes a symmetric ringspectrum via the multiplication map

(HA)n ∧ (HA)m = A[Sn] ∧ A[Sm] −→ A[Sn+m] = (HA)n+m(∑iai · xi

)∧(∑

jbj · yj

)7−→

∑i,j

(ai · bj) · (xi ∧ yj) .

The unit maps Sm −→ (HA)m are given by the inclusion of generators.We shall see in Example 5.20 below that the Eilenberg-Mac Lane functor H can be made into a lax

symmetric monoidal functor with respect to the tensor product of abelian groups and the smash product ofsymmetric spectra; this also explains why H takes rings (monoids in the category of abelian with respectto tensor product) to ring spectra (monoids in the category of symmetric spectra with respect to smashproduct).

Eilenberg-Mac Lane spectra enjoy a special property: the n-th space (HA)n and the loop space of thenext space (HA)n+1 are both Eilenberg-Mac Lane space of type (A,n), and in fact the map σn : (HA)n −→Ω(HA)n+1 adjoint to the structure map is a weak equivalence for all n ≥ 0. Spectra with this propertyplay an important role in stable homotopy theory, and they deserve a special name:

Definition 1.15. A symmetric spectrum of topological spaces X is an Ω-spectrum if for all n ≥ 0 themap σn : Xn −→ ΩXn+1 which is adjoint to the structure map σn : Xn ∧ S1 → Xn+1 is a weak homotopyequivalence.

In other words, HA is an Ω-spectrum. It follows that the naive homotopy groups of the symmetricspectrum HA are concentrated in dimension zero, where we have a natural isomorphism A = π0(HA)0 ∼=π0HA.

Example 1.16 (Real Thom spectra). We define a commutative symmetric ring spectrum MO whosestable homotopy groups are isomorphic to the ring of cobordism classes of closed smooth manifolds. We set

MOn = EO(n)+ ∧O(n) Sn ,


the Thom space of the tautological vector bundle EO(n) ×O(n) Rn over BO(n) = EO(n)/O(n). HereO(n) is the n-th orthogonal group consisting of Euclidean automorphisms of Rn. The space EO(n) is thegeometric realization of the simplicial space which in dimension k is the (k + 1)-fold product of copies ofO(n), and where face maps are projections. Thus EO(n) is contractible and has a right action by O(n).The right O(n)-action is used to form the orbit space MOn, where we remember that Sn is the one-pointcompactification of Rn, so it comes with a left O(n)-action.

The symmetric group Σn acts on O(n) by conjugation with the permutation matrices. Since the ‘E’-construction is natural in topological groups, this induces an action of Σn on EO(n). If we let Σn act onthe sphere Sn by coordinate permutations and diagonally on EO(n)+∧Sn, then the action descends to thequotient space MOn.

The unit of the ring spectrum MO is given by the maps

Sn ∼= O(n)+ ∧O(n) Sn −→ EO(n)+ ∧O(n) S

n = MOn

using the ‘vertex map’ O(n) −→ EO(n). There are multiplication maps

MOn ∧MOm −→ MOn+m

which are induced from the identification Sn ∧ Sm ∼= Sn+m which is equivariant with respect to the groupO(n)×O(m), viewed as a subgroup of O(n+m) by block sum of matrices. The fact that these multiplicationmaps are associative and commutative uses that

• for topological groups G and H, the simplicial model of EG comes with a natural, associative andcommutative isomorphism E(G×H) ∼= EG× EH;

• the group monomorphisms O(n) × O(m) −→ O(n + m) by orthogonal direct sum are strictlyassociative, and the following diagram commutes

O(n)×O(m) //

twist

O(n+m)

conj. by χn,m

O(m)×O(n) // O(m+ n)

where the right vertical map is conjugation by the permutation matrix of the shuffle permuta-tion χn,m.

Essentially the same construction gives commutative symmetric ring spectra MSO representing orientedbordism and MSpin representing spin bordism. For MSO this uses that conjugation of O(n) by a per-mutation matrix restricts to an automorphism of SO(n) and the block sum of two special orthogonaltransformations is again special. For MSpin it uses that the block sum pairing and the Σn-action uniquelylift from the groups SO(n) to their universal covers Spin(n).

The Thom-Pontrjagin construction provides homomorphisms πkMO −→ ΩOk from the k-th naive ho-

motopy group of the spectrum MO to the group of cobordism classes of k-dimensional smooth closedmanifolds [direction], and similarly for the other families of classical Lie groups [structure on stable nor-mal bundle; takes product in π∗MO to cartesian product of manifolds] By a theorem of Thom’s [ref], theThom-Pontrjagin map is an isomorphism. [explicit desciption of π∗MG, whenever known]

We intend to discuss these and other examples of Thom spectra in more detail in a later chapter.

Example 1.17 (Complex cobordism spectra). The Thom ring spectra MU , MSU and MSp represent-ing unitary, special unitary or symplectic bordism have to be handled slightly differently from real Thomspectra such as MO in the previous example. The point is that MU and MSU are most naturally indexedon ‘even spheres’, i.e., one-point compactifications of complex vector spaces, and MSp is most naturallyindexed on spheres of dimensions divisible by 4. However, a small variation gives MU , MSU and MSp ascommutative symmetric ring spectra, as we shall now explain. The complex cobordism spectrum MU playsan important role in stable homotopy theory because of its relationship to the theory of formal groups laws.Thus module and algebra spectra over MU are important, and we plan to study these in some detail later.

18 I. BASICS

We first consider the collection of pointed spaces MU with

(MU)n = EU(n)+ ∧U(n) S2n ,

the Thom space of the tautological complex vector bundle EU(n) ×U(n) Cn over BU(n) = EU(n)/U(n).Here U(n) is the n-th unitary group consisting of Euclidean automorphisms of Cn. The Σn-action arisesfrom conjugation by permutation matrices and the permutation of complex coordinates, similarly as in thecase of MO above.

There are multiplication maps

(MU)p ∧ (MU)q −→ (MU)p+q

which are induced from the identification Cp ⊕ Cq ∼= Cp+q which is equivariant with respect to the groupU(p)×U(q), viewed as a subgroup of U(p+q) by direct sum of linear maps. There is a unit map ι0 : S0 −→(MU)0, but instead of a unit map from the circle S1, we only have a unit map S2 −→ (MU)1. Thus wedo not end up with a symmetric spectrum since we only get structure maps (MU)n ∧ S2 −→ (MU)n+1

involving the 2-sphere. In other words, MU has the structure of what could be called an ‘even symmetricring spectrum’ (MU is really a ‘unitary ring spectrum’, as we shall define in Example 8.8 below).

In order to get an honest symmetric ring spectrum we now use a general construction which turns acommutative monoid Φ(R) in the category of symmetric sequences into a new such monoid by appropriatelylooping all the spaces involved. We set

Φ(R)n = map(Sn, Rn)

an let the symmetric group act by conjugation. Then the product of R combined with smashing maps givesΣn × Σm-equivariant maps

Φ(R)n ∧ Φ(R)m = map(Sn, Rn) ∧map(Sm, Rm) −→ map(Sn+m, Rn+m) = Φ(R)n+m

f ∧ g 7−→ f · g = µn,m (f ∧ g) .

Now we apply this construction to MU and obtain a commutative monoid MU = Φ(MU) in thecategory of symmetric sequences. We make MU into a symmetric ring spectrum via the unit map S1 −→(MU)1 = map(S1, (MU)1) which is adjoint to

ι : S2 ∼= U(1)+ ∧U(1) S2 −→ EU(1)+ ∧U(1) S

2 = (MU)1

using the ‘vertex map’ U(1) −→ EU(1). More precisely, we use the decomposition C = R · 1⊕R · i to viewS2 as the smash product of a ‘real’ and ‘imaginary’ circle, and then we view the source of the unit mapS1 −→ (MU)1 = map(S1, (MU)1) as the real circle, and we think of the imaginary circle as parameterizingthe loop coordinate in the target (MU)1. Since the multiplication of MU is commutative, the centralitycondition is automatically satisfied. Then the iterated unit map

Sn −→ (MU)n = Ωn(MU)n

is given by(x1, . . . , xn) 7−→ ((y1, . . . , yn) 7→ µ(ι(x1, y1), . . . , ι(xn, yn)))

where µ : (MU)∧n1 −→ (MU)n is the iterated multiplication map.The naive (and true) homotopy groups of MU are given by

πkMU = colimn πk+n map(Sn, (MUn)) ∼= colimn πk+2n(EU(n)+ ∧U(n) S2n) ;

so by Thom’s theorem they are isomorphic to the ring of cobordism classes of stably almost complex k-manifolds. So even though the individual spaces MUn = map(Sn, EU(n)+∧U(n)S

2n) are not Thom spaces,the symmetric spectrum which they form altogether has the ‘correct’ homotopy groups (and in fact, thecorrect stable homotopy type). [π∗MU ; semistable since orthogonal spectrum]

Essentially the same construction gives a commutative symmetric ring spectrum MSU . The symplecticbordism and MSp can also be handled similarly: it first arises as a commutative monoid MSp in symmetricsequences with structure maps (MSp)n ∧ S4 −→ (MSp)n+1 and a unit map S4 −→ (MSp)1. If we apply


the construction Φ three times, we obtain a commutative symmetric ring spectrum MSp = Φ3(MSp)representing symplectic bordism.

Example 1.18 (Topological K-theory). We define the commutative symmetric ring spectrum KU ofcomplex topological K-theory. We set

KUn = hom(qn(C0(Rn)),K(n))

where:• C0(Rn) is the C∗-algebra of continuous complex valued functions on Rn which vanish at infinity;• qn is the n-th iterate of a functor q which associates to a C∗-algebra its Cuntz algebra; qA is the

kernel of the fold morphism A∗A −→ A, where the star is the categorical coproduct of C∗-algebras(a certain completion of the algebraic coproduct);

• K is the C∗-algebra of compact operators on a fixed separable Hilbert space, and K(n) is the spatialtensor product of n copies of K;

• hom is the space of ∗-homomorphisms between two C∗-algebras with the subspace topology of thecompact open topology on the space of all continuous maps; the basepoint is the zero map.

Here C∗-algebras are not necessarily unital, and homomorphisms need not preserve units, if they exist.There is an adjunction as follows: if K is a locally compact space and A and B are C∗-algebras, then

T(K,hom(A,B)) ∼= hom(A,C0K ⊗B)

where K is the one-point compactification.There is an action of the symmetric group on qnA, but it is not obvious from what we have said

so far [define it]. The symmetric group also acts on C0(Rn) by permutation of coordinates, on K(n) bypermutation of tensor factors, and on the mapping space KUn by conjugation.

In level 0 we have q0A = A, so q0(C0(Rn)) = C, and K(0) = C. Thus we have KU0 = hom(C,C)which consists of two elements, the zero and the identity homomorphism. So we define ι0 : S0 :−→ KU0

as the homeomorphism which sends the basepoint to the zero homomorphism and the non-basepoint tothe identity of C. We have KU1 = hom(q(C0(R)),K) [identify with infinite unitary group] and via theadjunction, the unit map ι1 : S1 −→ KU1 corresponds to a C∗-homomorphism ι1 : q(C0(R)) −→ C0(R)⊗Kwhich is defined as the composite

q(C0(R)) −→ C0(R) −→ C0(R)⊗Kwhere the first map is the restriction of the morphism 1 ∗ 0 : C0(R) ∗ C0(R) −→ C0(R) to the kernel of thefold map and the second map sends f to f ⊗ e where e ∈ K is a fixed rank 1 projection. [this should be apositive Ω-spectrum. Necessary modifications to yield KO and KT , or, even better, the real, C2-equivariantversion KR?]

The Bott periodicity theorem constructs a homotopy equivalence Ω2BU ' Z × BU . It implies thatthe homotopy ring of KU contains a unit in dimension 2, the Bott class u ∈ π2KU [define the Bott classu]. Then π∗KU = Z[u, u−1] as graded rings.

There is a variant, the symmetric spectrum of real topological K-theory KO. The real version has aBott-periodicity of order 8, i.e., there is a homotopy equivalence

Ω8BO ' Z×BO ,

which gives a unit β ∈ π8KO. [define the Bott class β]The following table (and Bott periodicity) gives the (naive and true) homotopy groups of the spectrum

KO:

n 0 1 2 3 4 5 6 7 8πnKO Z Z/2 Z/2 0 Z 0 0 0 Z

generator ι η η2 ξ β

Here η is the Hurewicz image of the Hopf map, i.e., the image of the class η ∈ π1S under the uniquehomomorphism of ring spectra S −→ KO. There is a homomorphism of ring spectra KO −→ KU , the

20 I. BASICS

‘complexification map’, which is injective on homotopy groups in dimensions divisible by 4, and bijective indimensions divisible by 8. The elements ξ and β can be defined by the property that they hit 2u2 ∈ π4KUrespectively u4 ∈ π8KU under this complexification map. Thus there is the multiplicative relation ξ2 = 4β.

The symmetric spectrumX is a positive Ω-spectrum if the map σn : Xn −→ ΩXn+1 is a weak equivalencefor all positive values of n (but not necessarily for n = 0). Examples which arise naturally as positive Ω-spectra are the spectra of topologicalK-theory (Example 1.18) and algebraicK-theoryK(C) (Example 3.29)and spectra arising from special (but not necessarily very special) Γ-spaces by evaluation on spheres.

For every Ω-spectrum X and all k, n ≥ 0, the canonical map πkXn −→ πk−nX is a bijection. Indeed,the homotopy groups of ΩXn+1 are isomorphic to the homotopy groups of Xn+1, shifted by one dimension.So the colimit system which defines πk−nX is isomorphic to the colimit system

(1.19) πkXn −→ πk (ΩXn+1) −→ πk (Ω2Xn+2) −→ · · · ,

where the maps in the system are induced by the maps σn adjoint to the structure maps. In an Ω-spectrum,the maps σn are weak equivalences, so all maps in the sequence (1.19) are bijective, hence so is the mapfrom each term to the colimit πk−nX.

2. Properties of naive homotopy groups

In Section 3.2 we will discuss various constructions which one can do to a symmetric spectrum. When-ever possible we want to say how a construction effects the naive homotopy groups. So in this sectionwe develop a few general properties of naive homotopy groups. More specifically, we construct long ex-act sequences of naive homotopy groups from a morphism of symmetric spectra and we identify the naivehomotopy groups of a wedge and a finite product of spectra.

The naive homotopy groups of a symmetric spectrum do not depend on the symmetric group actions,so they are really defined for ‘sequential spectra’, i.e., ‘symmetric spectra without symmetric group actions’.We develop the basic properties of naive homotopy groups in this more general context. This extra generalityis useful because a symmetric spectrum sometimes decomposes into simpler pieces after forgetting thesymmetric group actions, but the levelwise decomposition are not equivariant. Since the naive homotopygroups don’t care about the symmetries, such a non-equivariant splitting still give information about naivehomotopy groups. An example of this strategy is the decomposition of the naive homotopy groups of atwisted smash product, see Example 3.14.

Definition 2.1. A sequential spectrum consists of a sequence of pointed spaces Xn and based mapsσn : Xn ∧ S1 −→ Xn+1 for n ≥ 0. A morphism f : X −→ Y of sequential spectra consists of based mapsfn : Xn −→ Yn for n ≥ 0, which are compatible with the structure maps in the sense that fn+1 σn =σn (fn ∧ IdS1) for all n ≥ 0. The category of sequential spectra is denoted by SpN.

We refer to the space Xn as the n-th level of the sequential spectrum X. The k-th naive homotopygroup of a sequential spectrum X is defined as the colimit

πkX = colimn πk+nXn

taken over the stabilization maps ι : πk+nXn −→ πk+n+1Xn+1 defined as for symmetric spectra as thecomposite (1.7).

2.1. Loop and suspension. The loop spectrum ΩX of a symmetric or sequential spectrum X isdefined by

(ΩX)n = Ω(Xn) ,the based mapping space from the circle S1 to the n-th level of X. In the symmetric (as opposed to‘sequential’) case, the symmetric group Σn acts on Ω(Xn) through the given action on Xn and trivially onthe circle. The structure map is the composite

(ΩX)n ∧ S1 = Ω(Xn) ∧ S1 −→ Ω(Xn ∧ S1)Ω(σn)−−−−→ Ω(Xn+1) = (ΩX)n+1

where the second map takes l ∧ t ∈ Ω(Xn) ∧ S1 to the loop S1 −→ Xn ∧ S1 which sends s to l(s) ∧ t.

2. PROPERTIES OF NAIVE HOMOTOPY GROUPS 21

The suspension S1 ∧X of a symmetric or sequential spectrum X is defined by

(S1 ∧X)n = S1 ∧Xn ,

the smash product of the circle with the n-th level of X. In the symmetric (as opposed to ‘sequential’) case,the symmetric group Σn acts on S1 ∧Xn through the given action on Xn and trivially on the circle. Thestructure map is the composite

(S1 ∧X)n ∧ S1 = S1 ∧Xn ∧ S1 Id∧σn−−−−→ S1 ∧Xn+1 = (S1 ∧X)n+1 .

We define an adjunction

(2.2) ˆ : SpN(S1 ∧X,Y )∼=−−→ SpN(X,ΩY )

which takes a morphism f : S1∧X −→ Y to the morphism f : X −→ ΩY whose n-th level fn : Xn −→ Ω(Yn)is given by (fn(x))(t) = fn(t ∧ x). If X and Y are symmetric spectra, then the adjunction bijection (2.2)restricts to a similar adjunction bijection ˆ : Sp(S1 ∧ X,Y ) ∼= Sp(X,ΩY ) between the morphisms sets ofsymmetric spectra.

Now we show that looping and suspending a spectrum shifts the naive homotopy groups. The loophomomorphism starts from the isomorphism

(2.3) α : πk+nΩ(Xn) ∼= π1+k+nXn

defined by sending a representing continuous map f : Sk+n −→ Ω(Xn) to the class of the mapf : S1+k+n −→ Xn given by f(s ∧ t) = f(t)(s), where s ∈ S1, t ∈ Sk+n. As n varies, these particu-lar isomorphisms are compatible with stabilization maps, so they induce an isomorphism

α : πk(ΩX)∼=−−→ π1+kX

on colimits. (Note that the identification α differs from the adjunction isomorphism (0.4) by precompositionwith the coordinate permutation χk+n,1 : Sk+n+1 −→ S1+k+n; the adjunction isomorphisms themselves arenot compatible with stabilization.)

The maps S1 ∧− : πk+nXn −→ π1+k+n(S1 ∧Xn) given by smashing from the left with the identity ofthe circle are compatible with the stabilization process for the homotopy groups X respectively S1 ∧X, soupon passage to colimits they induce a natural map of naive homotopy groups

S1 ∧ − : πkX −→ π1+k(S1 ∧X) .

which we call the suspension homomorphism.

Proposition 2.4. For every sequential spectrum X and integer k the loop and suspension homomor-phisms

(2.5) α : πk(ΩX) −→ π1+kX and S1 ∧ − : πkX −→ π1+k(S1 ∧X)

are isomorphisms of naive homotopy groups. Moreover, the triangles

πkXS1∧− //

πkη &&MMMMMMMMMMM π1+k(S1 ∧X) πk(ΩX) α //

S1∧− ''OOOOOOOOOOOπ1+kX

πk(Ω(S1 ∧X))

α

66mmmmmmmmmmmmπk(S1 ∧ (ΩX))

πkε

77ppppppppppp

commute where η : X −→ Ω(S1 ∧ X) and ε : S1 ∧ (ΩX) −→ X are the unit respectively counit of theadjunction (2.2).

Proof. We already argued that the loop homomorphism α on naive homotopy groups is bijectivesince it is the colimit of compatible bijections. The case of the suspension homomorphism S1 ∧ − isslightly more involved. We show injectivity first. Let f : Sk+n −→ Xn represent an element in thekernel of the suspension homomorphism. By stabilizing, if necessecary, we can assume that the suspensionS1∧f : S1+k+n −→ S1∧Xn is nullhomotopic. Then σnτ(S1∧f) : S1+k+n −→ Xn+1 is also nullhomotopic,

22 I. BASICS

where τ : S1 ∧ X ∼= X ∧ S1 is the twist homeomorphism. The maps σnτ(S1 ∧ f) and σn(f ∧ S1), thestabilization of f , only differ by a coordinate permutation of the source sphere, hence the stabilization of f isnullhomotopic. So f represents the trivial element in πkX, which shows that the suspension homomorphismis injective.

It remains to show that the suspension homomorphism is surjective. Let g : S1+k+n −→ S1 ∧Xn be amap which represents a class in π1+k(S1 ∧X). We consider the map f : σnτg : S1+k+n −→ Xn+1 where τis again the twist homeomorphism. We claim that (−1)k+n(S1 ∧ f) : S1+1+k+n −→ S1 ∧Xn+1 representsthe same class as g in π1+k(S1 ∧X). To see this, we contemplate the diagram

S1+k+n+1g∧S1

//

S1∧χk+n,1

S1 ∧Xn ∧ S1

S1∧σn

S1+1+k+nS1∧g //

S1∧f

44S1 ∧ S1 ∧XnS1∧τ // S1 ∧Xn ∧ S1

S1∧σn // S1 ∧Xn+1

The composite through the upper right corner is the stabilization of g and the composite through the lowerleft corner represents (−1)k+n(S1∧f). However, this diagram does not commute! The two composites fromS1+k+n+1 to S1 ∧Xn ∧ S1 differ by the automorphisms of S1+k+n+1 and S1 ∧Xn ∧ S1 which interchangesthe outer two sphere coordinates in each case. This coordinate change in the source induces multiplicationby −1; the coordinate change in the target is a map of degree −1, so after a single suspension it also inducesmultiplication by −1 on homotopy groups (see Lemma 1.10). Altogether this shows that the diagram abovecommutes up to homotopy after one suspension, and so the suspension map on naive homotopy groups isalso surjective. [prove the claims about unit/counit]

2.2. Mapping cone and homotopy fiber. Now we review the mapping cone and the homotopy fibreof a map of based spaces in some detail, along with their relationships to one another and to suspension andloop space. This discussion will involve various explicit formulae and it will be convenient to parametrizeloop and suspension coordinates not by the circle S1 (the onepoint compatification of R), but ratherby the homeomorphic space [0, 1]/0, 1 for which we write S1. We will later use the homeomorphismt : [0, 1]/0, 1 ∼= S1 of (0.1) to convert S1 back into S1.

The (reduced) mapping cone C(f) of a morphism of based spaces f : A −→ B is defined by

(2.6) C(f) = ([0, 1] ∧A) ∪f B .

Here the unit interval [0, 1] is pointed by 0 ∈ [0, 1], so that [0, 1]∧A is the reduced cone of A. The mappingcone comes with an inclusion i : B −→ C(f) and a projection p : C(f) −→ S1 ∧A; the projection sends theimage of B to the basepoint and is given on the cone of A by the quotient map [0, 1]∧A −→ ([0, 1]/0, 1)∧A.

The homotopy fiber is the construction ‘dual’ to the mapping cone. The homotopy fibre of a morphismf : A −→ B of based spaces is the fiber product

F (f) = ∗ ×B B[0,1] ×B A = (λ, a) ∈ B[0,1] ×A | λ(0) = ∗, λ(1) = f(a)

i.e., the space of paths in B starting at the basepoint and equipped with a lift of the endpoint to A. Asbasepoint of the homotopy fiber we take the pair consisting of the constant path at the basepoint of B andthe basepoint of A. The homotopy fiber comes with maps

ΩB i−−→ F (f)p−−→ A

given byi(ω) = (ω, ∗) respectively p(λ, a) = a .

Here ΩB is a version of the loop space parametrized by [0, 1], namely the subspace of B[0,1] consisting ofall paths ω : [0, 1] −→ B which start and end in the basepoint of B.


For a map f : A −→ B of based spaces we define a map h : [0, 1]×F (f) −→ ([0, 1]∧A)∪f B = C(f) by

(t, λ, a) 7−→

(2t, a) for 0 ≤ t ≤ 1/2, andλ(2− 2t) for 1/2 ≤ t ≤ 1.

We note that the two formulas match at t = 1/2 because λ(1) = f(a) = (1, a) in C(f). Since h(0, λ, a) andh(1, λ, a) are the basepoint of the mapping cone for all (λ, a) in F (f), the map h factors over the quotientspace ([0, 1]/0, 1)× F (f) and descends to a based map

(2.7) h : S1 ∧ F (f) −→ C(f) .

which is natural in f .

Proposition 2.8. Let f : A −→ B be a map of based spaces. Then the composites

Af−−→ B

i−−→ C(f) and F (f)p−−→ A

f−−→ B

are naturally based null-homotopic. The diagrams

CA ∪f CBpA∪∗

zzuuuuuu

uuuu

∗∪pB

$$IIIIII

IIII ΩB

map(τ,B) //

i

ΩB

Ωi

C(f)p

!!CCCC

CCCC

S1 ∧Aτ∧f

// S1 ∧B F (f)h

// ΩC(f) S1 ∧ F (f)Id∧p

//

h

;;wwwwwwwwS1 ∧A

commute up to natural, based homotopy, where τ is the involution of S1 = [0, 1]/0, 1 induced by inversiont 7→ 1− t of the unit interval, and h : F (f) −→ ΩC(f) is the adjoint of h.

Proof. We specify natural homotopies by explicit formulae. The map if : A −→ C(f) is null-homotopic by [0, 1] × A −→ C(f), (s, a) 7→ (s, a), i.e., the composite of the canonical maps [0, 1] × A −→[0, 1]∧A and [0, 1]∧A −→ C(f). The map fp : F (f) −→ B is null-homopic by [0, 1]×F (f) −→ B, (s, λ, a) 7→λ(s).

The homotopy for the left triangle will be glued together from two pieces. We define a based homotopyH : [0, 1]× CA −→ S1 ∧B by the formula

H(s, t, a) = (2− s− t) ∧ f(a)

which has to be interpreted as the basepoint if 2− s− t ≥ 1. Another based homotopy H ′ : [0, 1]×CB −→S1 ∧B is given by the formula

H ′(s, t, b) = (t− s) ∧ b

which has to be interpreted as the basepoint if t ≤ s. The two homotopies are compatible in the sense that

H(s, 1, a) = (1− s) ∧ f(a) = H ′(s, 1, f(a))

for all s ∈ [0, 1] and a ∈ A. So H and H ′ glue together and yield a homotopy

[0, 1]× (CA ∪f CB) ∼= ([0, 1]× CA) ∪Id×f ([0, 1]× CB) H∪H′

−−−−→ [0, 1]/0, 1 ∧B .

For s = 0 this homotopy starts with the map ∗ ∪ pB , and it ends for s = 1 with the map (τ ∧ f) (pA ∪ ∗).The middle square commutes up to the based homotopy

[0, 1]× ΩB −→ ΩC(f)

(s, ω) 7−→

t 7→ ∗ for 0 ≤ t ≤ s/2, andt 7→ ω(2(1− t)/(2− s)) for s/2 ≤ t ≤ 1.

24 I. BASICS

and for the triangle on the right the homotopy

[0, 1]× (S1 ∧ F (f)) −→ S1 ∧A

(s, t ∧ (λ, a)) 7−→

2t/(2− s) ∧ a for 0 ≤ t ≤ 1− s/2, and

∗ for 1− s/2 ≤ t ≤ 1.

does the job.

Lemma 2.9. Let f : A −→ B be a morphism of based spaces and β ∈ πmB a homotopy class in thekernel of πm(i) : πmB −→ πmC(f). Then there exist a homotopy class α ∈ π1+m(S1 ∧ A) such that(S1 ∧ f)∗(α) = Id∧β in π1+m(S1 ∧B).

Proof. Let b : Sm −→ B be a representative of β and let H : C(Sm) −→ C(f) be a based null-homotopy of the composite of b with i : B −→ C(f), i.e., such that H[1, x] = i(b(x)) for all x ∈ Sm. Wecollapse 1× Sm in C(Sm) and the image of B in C(f) and get a map H : S1 ∧ Sm −→ S1 ∧A induced byH on the quotient spaces. We claim that the homotopy class α = [H] has the required property.

To prove the claim we need the homotopy equivalence p∪ ∗ : C(Sm)∪1×Sm C(Sm) −→ S1 ∧ Sm whichcollapses the second cone. We obtain a sequence of equalities and homotopies

(S1 ∧ f) H (p ∪ ∗) = (S1 ∧ f) (pA ∪ ∗) (H ∪ C(b))

' (τ ∧B) (∗ ∪ pB) (H ∪ C(b))

= (τ ∧B) (S1 ∧ b) (∗ ∪ p)= (S1 ∧ b) (τ ∧ Sm) (∗ ∪ p) ' (S1 ∧ b) (p ∪ ∗)

Here H ∪ C(b) : C(Sm) ∪1×Sm C(Sm) −→ C(f) ∪B CB ∼= CA ∪f CB. The two homotopies resultfrom Proposition 2.8 applied to f respectively the identity of Sm, and we used the naturality of variousconstructions. Since the map p∪ ∗ is a homotopy equivalence, this proves that the map (S1 ∧ f) H whichrepresents (S1 ∧ f)∗(α) is homotopic to¯S1 ∧ b.

Now we can introduce mapping cones and homotopy fibers for (symmetric and sequential) spectra. Themapping cone C(f) of a morphism of symmetric or sequential spectra f : X −→ Y is defined by

(2.10) C(f)n = C(fn) = ([0, 1] ∧Xn) ∪f Yn ,

the reduced mapping cone of fn : Xn −→ Yn. In the symmetric (as opposed to ‘sequential’) case, thesymmetric group Σn acts on C(f)n through the given action on Xn and Yn and trivially on the interval.The inclusions in : Yn −→ C(f)n and projections pn : C(f)n −→ S1 ∧ Xn assemble into morphisms ofsymmetric (respectively sequential) spectra i : Y −→ C(f) and p : C(f) −→ S1 ∧X.

We define a connecting homomorphism δ : π1+kC(f) −→ πkX as the composite

(2.11) π1+kC(f)π1+k(p)−−−−−−→ π1+k(S1 ∧X) ∼= πkX ,

where the first map is the effect of the projection p : C(f) −→ S1 ∧ X on homotopy groups, and thesecond map is the inverse of the suspension isomorphism S1 ∧ − : πkX −→ π1+k(S1 ∧X) and we use thehomeomorphism t : S1 = [0, 1]/0, 1 ∼= S1 of (0.1). If we unravel all the definition, we see that δ sends theclass represented by a based map ϕ : S1+k+n −→ C(f)n to (−1)k+n times the class of the composite

S1+k+n ϕ−−→ C(f)npn−−−→ S1 ∧Xn

∼= S1 ∧Xntwist−−−→ Xn ∧ S1 σn−−→ Xn+1 .

Let f : X −→ Y be a morphism between symmetric spectra. The homotopy fiber F (f) is the symmetricspectrum defined by

(2.12) F (f)n = F (fn) ,

the homotopy fiber of fn : Xn −→ Yn. In the symmetric (as opposed to ‘sequential’) case, the symmetricgroup Σn acts on F (f)n through the given action on Xn and Yn and trivially on the interval. The inclusions


in : Ω(Yn) −→ F (f)n and projections pn : F (f)n −→ Xn assemble into morphisms of symmetric (respec-tively sequential) spectra i : ΩY −→ F (f) and p : F (f) −→ X. Put another way, the homotopy fiber is thefibre product

F (f) = ∗ ×Y Y [0,1] ×Y Xi.e., the pullback in the cartesian square of spectra

(2.13)

F (f)

p // X

(∗,f)

Y [0,1]

(ev0,ev1)// Y × Y

Here evi : Y [0,1] −→ Y for i = 0, 1 is the ith evaluation map which takes a path ω ∈ Y [0,1] to ω(i), i.e., thestart or endpoint.

We define a connecting homomorphism δ : π1+kY −→ πkF (f) as the composite

(2.14) π1+kYα−1

−−→ πk(ΩY ) ∼= πk(ΩY )πk(i)−−−−→ πkF (f) ,

where α : πk(ΩY ) −→ π1+kY is the loop isomorphism and the second identification is via the homeomor-phism t : S1 = [0, 1]/0, 1 ∼= S1 of (0.1).

Because the map h : S1 ∧ F (f) −→ C(f) defined in (2.7) is natural, it gives rise to a morphism ofspectra, symmetric or sequential,

(2.15) h : S1 ∧ F (f) −→ C(f)

which we still denote by h.

Proposition 2.16. For every morphism f : X −→ Y of sequential spectra the long sequences of abeliangroups

· · · −→ πkXπk(f)−−−−−→ πkY

πk(i)−−−→ πkC(f) δ−−→ πk−1X −→ · · ·and

· · · −→ πkXπk(f)−−−−−→ πkY

δ−−→ πk−1F (f)πk−1(p)−−−−−−→ πk−1X −→ · · ·

are exact and the morphism h : S1 ∧ F (f) −→ C(f) is a π∗-isomorphism.

Proof. We show that the sequence

πkXπk(f)−−−−−→ πkY

πk(i)−−−→ πkC(f)πk(p)−−−→ πk(S1 ∧X)

πk(Id∧f)−−−−−−→ πk(S1 ∧ Y )

is exact; when we substitute definition (2.11) of the connecting homorphism δ, this becomes the first exactsequence.

Exactness at πkY : The composite of f : X −→ Y and the inclusion Y −→ C(f) is null-homotopic, soit induces the trivial map on πk. It remains to show that every element in the kernel of πk(incl) : πkY −→πkC(f) is in the image of πkf . Let β ∈ πk+nYn represent an element in the kernel. By increasing n, ifnecessary, we can assume that incl∗(β) is trivial in πk+nC(fn). By Lemma 2.9 (ii) there is a homotopy classα ∈ π1+k+n(S1 ∧Xn) such that (Id∧fn)∗(α) = S1 ∧ β. The homotopy class α = (−1)k+n · (τS1,Xn)∗(α) ∈πk+n+1(Xn ∧ S1) then satisfies (fn ∧ Id)∗(α) = β ∧ Id, and thus (σn)∗(α) ∈ πk+n+1Xn+1 hits ι∗(β) ∈πk+n+1Yn+1. So the class represented by β in the colimit πkY is in the image of πkf : πkX −→ πkY .

Exactness at πkC(f): If we apply the previous paragraph to the inclusion i : Y −→ C(f) instead of f ,we see that the sequence

πkYπk(i)−−−→ πkC(f)

πk(incli)−−−−−→ πkC(i)is exact. We claim that the collapse map

∗ ∪ p : C(i) ∼= CY ∪f CX −→ S1 ∧Xis a homotopy equivalence, and thus induces an isomorphism of naive homotopy groups. Since the compositeof the homotopy equivalence ∗ ∪ p : C(i) −→ S1 ∧ X with the inclusion of C(f) equals the projection

26 I. BASICS

C(f) −→ S1 ∧X, we can replace the group πkC(i) by the isomorphic group πk(S1 ∧X) and still obtain anexact sequence.

To prove the claim we define a homotopy inverse

r : S1 ∧X −→ CY ∪f CX

by the formula

r(s ∧ x) =

(2s, x) ∈ CX for 0 ≤ s ≤ 1/2, and

(2− 2s, f(x)) ∈ CY for 1/2 ≤ s ≤ 1,

which is to be interpreted levelwise. [specify the homotopies r(∗ ∪ p) ' Id and (∗ ∪ p)r ' Id]Exactness at πk(S1 ∧X): If we apply the first case to the projection p : C(f) −→ S1 ∧X instead of f ,

we see that the sequence

πkC(f)πk(p)−−−→ πk(S1 ∧X)

πk(i)−−−→ πkC(p)is exact. We claim that the collapse map

??? : C(p) ∼= C(CX ∪f Y ) ∪p (S1 ∧X) −→ S1 ∧ Y

[define; give details] is a homotopy equivalence, so induces an isomorphism of naive homotopy groups.Moreover, the composite

S1 ∧X ip−−→ C(p) ???−−→ S1 ∧ Yis homotopic to the morphism τ ∧ f : S1 ∧X −→ S1 ∧ Y , whose effect on homotopy groups is the negativeof πk(S1∧f). Since the sign has no effect on the kernel, we can replace the group πkC(p) by the isomorphicgroup πk(S1 ∧ Y ) and still obtain an exact sequence.

The second sequence is exact because it is obtained from the unstable long exact sequences for thehomotopy fiber sequences F (fn) −→ Xn −→ Yn by passage to the colimit (which is exact).

For the last claim we exploit that the diagram

π1+kYδ //

−π1+k(i)

**

πkF (f)

S1∧− πk(p)

π1+k(S1 ∧ F (f))

π1+k(h)

π1+kC(f)

δ// πkX

commutes; this uses the definitions of the connecting homomorphisms, various naturalities and the homomo-topy commutativity of Proposition 2.8 [details]. The sign on the left diagonal map is the effect of the intervalinversion in the middle square of Proposition 2.8. The morphism π1+k(h) (S1∧−) : πkF (f) −→ π1+kC(f)and the identity maps of the naive homotopy groups of X and Y thus give a natural map from the longexact sequence of the homotopy fiber to the long exact sequence of the mapping cone, with an extra sign.A sign does not affect exactness of a sequence, and so the five lemma shows that π1+k(h) (S1 ∧ −) is anisomorphism. Hence h is a π∗-isomorphism.

Remark 2.17. The long exact homotopy sequence involving the homotopy fiber also exists unstably.However, a fundamental property of stable homotopy groups of spectra which is not satisfied by (unstable)homotopy groups of spaces is that also the mapping cone fits into a long exact homotopy sequence.

We draw some consequences of Proposition 2.16. An continuous based map f : A −→ B is an h-cofibration if it has the homotopy extension property, i.e., [...][are these automatically closed inclusions ?]An equivalent condition is that the map [0, 1]+ ∧ A ∪0×A B −→ [0, 1]+ ∧ B has a retraction. For everyh-cofibration the map C(f) −→ B/A which collapses the cone of A to a point is a homotopy equivalence.[...]


Lemma 2.18 (move to appendix). For every h-cofibration of based spaces f : A −→ B the map C(f) −→B/A which collapses the cone of A to a point is a homotopy equivalence.

fix 0 and 1. The homotopy extension property allows us to extend the quotient map [0, 1]+ ∧ A ∪fB× 1 −→ C(f) to a based continuous map H : [0, 1]+ ∧B −→ C(f). The map H0 = H(−, 0) : B −→ C(f)then sends A to the basepoint, so it factors over a map g : B/A −→ C(f), i.e., which satisfies gq = H0.[Construct homotopies between the composites cg : B/A −→ B/A. and gc : C(f) −→ C(f) and therespective identities.

Corollary 2.19. Let f : X −→ Y be a morphism of sequential spectra.(i) Suppose that f is levelwise an h-cofibration and denote by q : Y −→ Y/X the quotient map. Then thelong sequence of naive homotopy groups

· · · −→ πkXπk(f)−−−→ πkY

πk(q)−−−→ πk(Y/X) δ−−→ πk−1X −→ · · ·

is exact where the connecting map δ is the composite of the inverse of the isomorphism πkC(f) −→ πk(Y/X)induced by the level equivalence C(f) −→ Y/X which collapses the cone of A and the connecting homo-morphism πkC(f) −→ πk−1X defined in (2.11). Moreover, the morphism h : S1 ∧ F (f) −→ Y/X from thesuspension of the homotopy fiber to the quotient of f is a π∗-isomorphism.(ii) Suppose that f is levelwise a Serre fibration and denote by i : F −→ X the inclusion of the fiber overthe basepoint. Then the long sequence of naive homotopy groups

· · · −→ πkFπk(i)−−−→ πkX

πk(f)−−−→ πkYδ−−→ πk−1F −→ · · ·

is exact where the connecting map δ is the composite of the connecting homomorphism πkY −→ πk−1F (f)defined in (2.14) and the inverse of the isomorphism πk−1F (f) −→ πk−1F induced by the level equivalenceF −→ F (f) which sends x ∈ F to (const∗, x). Let c : C(i) −→ Y be the morphism from the mapping coneof the inclusion i : F −→ X which is trivial on the cone of F and f on X. Then c is a π∗-isomorphism.

Proof. (i) [... ] For the second part, compare the two long exact sequences and use the five lemma.(ii) If fn : Xn −→ Yn is a Serre fibration, then the map from the strict fiber to the homotopy fiber

is a weak equivalence of fn. For the second part, compare the two long exact sequences and use the fivelemma.

Corollary 2.20. (i) For every family of sequential spectra Xii∈I and every integer k the canonicalmap ⊕

i∈IπkX

i −→ πk

(∨i∈I

Xi

)is an isomorphism of abelian groups.

(ii) For every finite indexing set I, every family Xii∈I of sequential spectra and every integer k thecanonical map

πk

(∏i∈I

Xi

)−→

∏i∈I

πkXi

is an isomorphism of abelian groups.(iii) For every family of symmetric spectra the canonical morphism from the wedge to the weak product

is a π∗-isomorphism. In particular, for every finite family of symmetric spectra the canonical morphismfrom the wedge to the product is a π∗-isomorphism.

Proof. (i) We first show the special case of two summands. If A and B are two symmetric spectra,then the wedge inclusion iA : A −→ A ∨B has a retraction. So the associated long exact homotopy groupsequence of Proposition 2.16 (i) splits into short exact sequences

0 −→ πkAπk(iA)−−−−→ πk(A ∨B) incl−−→ πk(C(iA)) −→ 0 .

28 I. BASICS

The mapping cone C(iA) is isomorphic to (CA)∨B and thus homotopy equivalent to B. So we can replaceπk(C(iA)) by πkB and conclude that πk(A ∨B) splits as the sum of πkA and πkB, via the canonical map.The case of a finite indexing set I now follows by induction, and the general case follows since homotopygroups of symmetric spectra commute with filtered colimits [more precisely, the image of every compactspace in an infinite wedge lands in a finite wedge].

(ii) Unstable homotopy groups commute with products, which for finite indexing sets are also sums,which commute with filtered colimits.

(iii) This is a direct consequence of (i) and (ii). More precisely, for finite indexing set I and everyinteger k the composite map⊕

i∈IπkX

i −→ πk(∨i∈I

Xi) −→ πk(∏i∈I

Xi) −→∏i∈I

πkXi

is an isomorphism, where the first and last maps are the canonical ones. These canonical maps are isomor-phisms by parts (i) respectively (ii), hence so is the middle map.

Remark 2.21. The restriction to finite indexing sets in parts (ii) of the previous corollary is essential,and it ultimately comes from the act that infinite products do not in general commute with sequentialcolimits. Here is an explicit example: we consider the symmetric spectra S≤i obtained by truncating thesphere spectrum above level i, i.e.,

(S≤i)n =

Sn for n ≤ i,∗ for n ≥ i+ 1

with structure maps as a quotient spectrum of S. Then S≤i has trivial homotopy groups for all i. The 0thnaive homotopy group of the product

∏i≥1 S≤i is the colimit of the sequence of maps∏

i≥n

πnSn −→

∏i≥n+1

πn+1Sn+1

which first projects away from the factor indexed by i = n and then takes a product of the suspensionshomomorphisms −∧S1 : πnSn −→ πn+1S

n+1. The colimit is thus isomorphic to the quotient of an infiniteproduct of copies of the group Z by the direct sum of the same number of copies of Z. Hence the left handside of the canonical map

π0

∏i≥1

S≤i −→

∏i≥1

π0(S≤i)

is trivial, while the right hand side is not.

In Example 4.23 below we give a different example for the fact that a product of π∗-isomorphisms neednot be a π∗-isomorphisms, namely a spectrum X all of whose naive homotopy groups trivial, but such thatπ0(XN) is non-zero. That example shows in particular that the restriction to finite K in the second part ofthe following proposition is essential.

Proposition 2.22. Let K be a based space of the based homotopy type of a CW-complex. Then thefunctor K∧− preserves π∗-isomorphisms of symmetric and sequential spectra. If K has the based homotopytype of a finite CW-complex, then the functor X 7→ XK preserves π∗-isomorphisms of symmetric andsequential spectra.

Proof. Use cell induction.

Example 2.23 (Telescope and diagonal of a sequence). We will sometimes be confronted with a sequenceof morphisms of sequential or symmetric spectra

(2.24) X0 f0

−−→ X1 f1

−−→ X2 f2

−−→ · · ·


of which we want to take a kind of colimit in a homotopy invariant way, and such that the homotopy groupsof the ‘colimit’ are the colimits of the homotopy groups. We describe two constructions which do this job,the mapping telescope and the diagonal.

The mapping telescope teliXi of the sequence (2.24) is a classical construction for spaces which we applylevelwise to symmetric spectra. It is defined as the coequalizer of two maps of (sequential or symmetric)spectra ∨

i≥0Xi // //

∨i≥0 [i, i+ 1]+ ∧Xi

Here [i, i+ 1] denotes a copy of the unit interval (when in the context of spaces) respectively the 1-simplex∆[1] (when in the context of simplicial sets). One of the morphisms takes Xi to i + 1+ ∧ Xi by theidentity, the other one takes Xi to i+ 1+ ∧Xi+1 by the morphism f i. [sequential vs symmetric]

The diagonal diagiXi of the sequence (2.24) is the spectrum given by

(diagiXi)n = Xn

n ,

i.e., we take the n-th level of the n-th (sequential or symmetric) spectrum. In the symmetric case, we usegiven Σn-action on this space as part of the symmetric spectrumXn. The structure map (diagiXi)n∧S1 −→(diagiXi)n+1 is the composite around either way in the commutative square

Xnn ∧ S1

σnn //

fnn∧Id

Xnn+1

fnn+1

Xn+1n ∧ S1

σn+1n

// Xn+1n+1

Lemma 2.25. For every sequence of sequential spectra (2.24), there is a chain of two natural π∗-isomorphisms between the diagonal diagiXi and the mapping telescope teliXi of the sequence. In particularthis gives natural isomorphisms of naive homotopy groups

πk(diagiXi) ∼= colimi πk(Xi) .

If the sequence consists of morphisms of symmetric spectra, then the chain is through morphisms of sym-metric spectra as well.

Proof. We use the ‘partial telescopes’ tel[0,n]Xi, the coequalizer of two maps of (symmetric or se-

quential) spectra ∨n−1i=0 X

i // //∨ni=0 [i, i+ 1]+ ∧Xi

defined as before. The spectrum tel[0,n]Xi includes into the next spectrum tel[0,n+1]X

i with (categorical)colimit the mapping telescope. The morphism cn : tel[0,n]X

i −→ Xn which projects each wedge summand[i, i+ 1]+ ∧Xi onto Xi and then applies the morphism fn−1 · · · f i : Xi −→ Xn is a homotopy equivalence.The commutative diagram of spectra

tel[0,0]Xi //

c0

tel[0,1]Xi //

c1

tel[0,2]Xi //

c2

· · ·

X0

f0// X1

f1// X2

f2// · · ·

induces a morphismdiagn(tel[0,n]X

i) −→ diagnXn

on diagonals which is thus levelwise a homotopy equivalence, hence a π∗-isomorphism. On the other handwe have a morphism of symmetric spectra

(2.26) diagn(tel[0,n]Xi) −→ teliXi

30 I. BASICS

which is levelwise given by the inclusion of a partial telescope in the full mapping telescope. This morphismis a π∗-isomorphism by ‘cofinality‘: we use that an (unstable) homotopy group of the mapping telescope ofa sequence of based, compactly generated spaces is the colimit of the sequence of homotopy groups [ref].So the right hand side of (2.26) is a sequential colimit of groups which are themselves sequential colimits,and it is thus the colimit over the partially ordered set N×N of the functor (n, i) 7→ πk+n(Xi

n). The groupπk(diagi tel[0,n]X

i) is isomorphic to the colimit over the diagonal terms in this system. Since the diagonalembedding N −→ N × N is cofinal, the colimit over the diagonal terms is isomorphic to the colimit overN× N, which proves the isomorphism (2.26).

Let me point out an advantages of the diagonal construction over the mapping telescope of a sequenceof spectra: the diagonal construction has nicer formal and in particular multiplicative properties, as weshall see, for example, in Examples 6.27 and 8.25.

3. Some constructions

In this section we discuss various constructions involving symmetric spectra and symmetric ring spectra.We start by introducing symmetric spectra of simplicial sets, a close relative of the category of symmetricspectra of spaces. In Section 3.2 we discuss constructions which produce new symmetric spectra from oldones. We define limits and colimits (3.3), smash product with and functions from a space (3.4), shifts (3.7),shift adjoint (3.10), free (3.11) and semifree symmetric spectra (3.12), twisted smash product with a Σm-space (3.14), mapping spaces (3.17) and internal Hom spectra (3.19).

In Section 3.3 we explain some elementary constructions involving ring spectra: endomorphism ringspectra (3.21), monoid ring spectra (3.22), matrix ring spectra (3.24), inverting an integer (3.26) or anelement in π0 (3.27) of a symmetric ring spectrum and adjoining roots of unity to a symmetric ring spec-trum (3.28) and algebraic K-theory spectra (3.29).

3.1. Symmetric spectra of simplicial sets. After the interlude about naive homotopy groups ofsequential spectra we return to symmetric spectra. We will often use a variation on the notions of symmetricspectrum and symmetric ring spectrum where topological spaces are replaced by simplicial sets. We can goback and forth between the two concepts using the adjoint functors of geometric realization and singularcomplex, as we explain below.

Definition 3.1. A symmetric spectrum of simplicial sets consists of the following data:• a sequence of pointed simplicial sets Xn for n ≥ 0• a basepoint preserving simplicial left action of the symmetric group Σn on Xn for each n ≥ 0• pointed morphisms σn : Xn ∧ S1 −→ Xn+1 for n ≥ 0,

such that for all n,m ≥ 0, the composite

Xn ∧ Smσn ∧ Id // Xn+1 ∧ Sm−1

σn+1∧Id // · · ·σn+m−2∧Id // Xn+m−1 ∧ S1

σn+m−1 // Xn+m

is Σn×Σm-equivariant. Here S1 denotes the ‘small simplicial circle’ S1 = ∆[1]/∂∆[1] and Sm = S1∧. . .∧S1

is the m-th smash power, with Σm permuting the factors.

Morphisms of symmetric spectra of simplicial sets are defined just as for symmetric spectra of spaces.We denote the category of symmetric spectra of simplicial sets by SpsS. There are many situations inwhich symmetric spectra of spaces and simplicial sets can be used interchangeably. We then often use theterm ‘symmetric spectrum’ and the the notation Sp without an index T or sS as a generic term/symbolfor either the category of symmetric spectra of spaces or simplicial sets.

We similarly define a symmetric ring spectrum of simplicial sets by replacing ‘space’ by ‘simplicial set’in Definition 1.3, while also replacing the topological circle S1 by the simplicial circle S1 = ∆[1]/∂∆[1] andreplacing Sm by the m-fold smash power Sm = S1 ∧ . . . ∧ S1.

As we already mentioned we can apply the adjoint functors ‘geometric realization’, denoted | − |, and‘singular complex’, denoted S, levelwise to go back and forth between topological and simplicial symmetricspectra. We recall the definitions and main properties of these functors in Appendix A.3. We use that

3. SOME CONSTRUCTIONS 31

geometric realization is a ‘strong symmetric monoidal’ functor, i.e., there is natural, unital, associative andcommutative homeomorphism

(3.2) rA,B : |A| ∧ |B| ∼= |A ∧B|

for pointed simplicial sets A and B. Indeed, the canonical continuous map |A × B| −→ |A| × |B| isa homeomorphism (since we work in the category of compactly generated topological spaces) and thehomeomorphism rA,B is gotten from there by passing to quotients.

We already allowed ourselves the freedom to use the same symbols for the topological and simplicialspheres. The justification is that the geometric realization of the simplicial Sm is homeomorphic to thetopological Sm. Let us be completely explicit about how we identify these two spaces. Since the simplicialset S1 = ∆[1]/∂∆[1] is generated by its non-degenerate 1-simplex, its realization |S1| is a quotient space ofthe topological 1-simplex ∆[1] = (x, y) ∈ R2 |x, y ≥ 0, x + y = 1. We agree to use the homeomorphismh : |S1| −→ S1 which sends the open simplex inside |S1| to R by (x, y) 7→ x/y − y/x. Then we obtain aΣm-equivariant homeomorphism as the composite

|Sm| = |S1 ∧ · · · ∧ S1|r−1S1,...,S1−−−−−−→ |S1| ∧ · · · ∧ |S1| h

(m)

−−−→ S1 ∧ · · · ∧ S1 ∼= Sm .

Now we can define the adjoint functors ‘geometric realization’ and ‘singular complex’ for symmetricspectra. If Y is a symmetric spectrum of simplicial sets we define a symmetric spectrum |Y | of topologicalspaces by |Y |n = |Yn| with structure maps

|Yn| ∧ S1 Id∧h−1

−−−−−→ |Yn| ∧ |S1|rYn,S1−−−−→ |Yn ∧ S1| |σn|−−→ |Yn+1| .

Commutativity of the isomorphism (3.2) guarantees that the equivariance condition for the iterated struc-ture map σm is inherited by the realization |Y |.

Adjoint to the homeomorphism (3.2) is a ‘lax symmetric monoidal’ transformation of pointed simplicialsets, i.e., a natural, unital, associative and commutative morphism rX,Y : S(X) ∧ S(Y ) −→ S(X ∧ Y ) forpointed spaces X and Y . So if X is a symmetric spectrum of topological spaces, then we get a symmetricspectrum S(X) of simplicial sets by S(X)n = S(Xn) with structure map

S(Xn) ∧ S1 Id∧h−−−→ S(Xn) ∧ S(S1)rXn,S1−−−−→ S(Xn ∧ S1)

S(σn)−−−−→ S(Xn+1) .

Here h : S1 −→ S(S1) is the morphism of pointed simplicial sets which is adjoint to the h : |S1| −→ S1. Weuse the adjunction unit and counit between |− | and S levelwise to make geometric realization and singularcomplex into adjoint functors between topological and simplicial symmetric spectra.

Geometric realization and singular complex are lax symmetric monoidal functors with respect to thesmash products of pointed spaces and pointed simplicial sets (geometric realization is even strong symmetricmonoidal, i.e., commutes with the smash product up to homeomorphism). So both constructions preservemultiplications, so they take ring spectra to ring spectra and preserve commutativity.

The homotopy groups of a symmetric spectrum based on simplicial sets Y are defined as the homotopygroups of the geometric realization |Y |.

A symmetric spectrum of simplicial sets Y is an Ω-spectrum respectively positive Ω-spectrum if thegeometric realization |Y | is an Ω-spectrum, respectively positive Ω-spectrum, of topological spaces. Asymmetric spectrum of simplicial sets Y is thus an Ω-spectrum if and only if for all n ≥ 0 the map|Yn| −→ Ω|Yn+1| which is adjoint to the composite

|Yn| ∧ S1 ∼=−−→ |Yn ∧ S1| |σn|−−→ |Yn+1|

is a weak homotopy equivalence. Our definition of ‘Ω-spectrum’ differs slightly from other sources in thatwe do not require that each simplicial set Yn has to be a Kan complex. If Y is a symmetric spectrum ofsimplicial sets in which all the Yn’s are Kan, then the natural maps |ΩYn| −→ Ω|Yn| adjoint to

|ΩYn| ∧ S1 −→ |(ΩYn) ∧ S1| |evaluate|−−−−−−→ |Yn|

32 I. BASICS

are weak equivalences, and so Y is an Ω-spectrum in our sense if and only if the morphisms of simplicialsets σn : Yn −→ ΩYn+1 adjoint to the structure maps are weak equivalences.

3.2. Constructions. We discuss various constructions which product new symmetric spectra fromold ones. Whenever possible, we describe the effect that a certain construction has on the naive homotopygroups.

Example 3.3 (Limits and colimits). The category of symmetric spectra has all limits and colimits, andthey are defined levelwise. Let us be a bit more precise and consider a functor F : J −→ Sp from a smallcategory J to the category of symmetric spectra (of spaces or simplicial sets). Then we define a symmetricspectrum colimJ F in level n by

(colimJ F )n = colimj∈J F (j)n ,the colimit being taken in the category of pointed Σn-spaces (or pointed Σn-spaces). The structure map isthe composite

(colimj∈J F (j)n) ∧ S1 ∼= colimj∈J(F (j)n ∧ S1) colimJ σn−−−−−−→ colimj∈J F (j)n+1 ;

here we exploit that smashing with S1 is a left adjoint, and thus the natural map colimj∈J(F (j)n∧S1) −→(colimj∈J F (j)n) ∧ S1 is an isomorphism, whose inverse is the first map above.

The argument for inverse limits is similar, but we have to use that structure maps can also be definedin the adjoint form. We can take

(limJF )n = limj∈JF (j)n ,and the structure map is adjoint to the composite

limj∈JF (j)nlimJ σn−−−−−→ limj∈JΩ(F (j)n+1) ∼= Ω (limj∈JF (j)n+1) .

[Same for modules]The inverse limit, calculated levelwise, of a diagram of symmetric ring spectra and homomorphisms

is again a symmetric ring spectrum. In other words, symmetric ring spectra have limits and the forgetfulfunctor to symmetric spectra preserves them. Symmetric ring spectra also have co-limits, but they are notpreserved by the forgetful functor. [ends and coends]

Example 3.4 (Smash products with and functions from spaces). If K is pointed space and X asymmetric spectrum, we can define two new symmetric spectrum K ∧X and XK by smashing with K ortaking maps from K levelwise; the structure maps and symmetric group actions do not interact with.

In more detail we set

(K ∧X)n = K ∧Xn respectively (XK)n = XKn = map(K,Xn)

for n ≥ 0. The symmetric group Σn acts through its action on Xn. The structure map is given by thecomposite

(K ∧X)n ∧ S1 = K ∧Xn ∧ S1 Id∧σn−−−−−→ K ∧Xn+1 = (K ∧X)n+1

respectively by the composite

XKn ∧ S1 −→ (Xn ∧ S1)K

σKn−−→ XKn+1

where the first map is adjoint to the evaluation mapXKn ∧S1∧K −→ Xn∧S1 and the second is application of

map(K,−) to the structure map of X. For example, the spectrum K∧S is equal to the suspension spectrumΣ∞K.

Just as the functors K ∧− and map(K,−) are adjoint on the level of based spaces (or simplicial sets),the two functors just introduced are an adjoint pair on the level of symmetric spectra. The adjunction

(3.5) ˆ : Sp(K ∧X,Y )∼=−−→ Sp(X,Y K)

takes a morphism f : K∧X −→ Y to the morphism f : X −→ Y K whose n-th level fn : Xn −→ map(K,Yn)is given by (fn(x))(k) = fn(k ∧ x).

We note that if X is an Ω-spectrum, then so is XK , provided we also assume that


• K is cofibrant (for example a CW-complex) when in the context of topological spaces, or• X is levelwise a Kan complex when in the context of simplicial sets.

Indeed, under either hypothesis, the map mapping space functor map(K,−) takes the weak equivalenceσn : Xn −→ ΩXn+1 to a weak equivalence

XKn = map(K,Xn)

map(K,σn)−−−−−−−→ map(K,ΩXn+1) ∼= Ω(XKn+1) .

Loops and suspensions are the special case K = S1 of this discussion (where S1 denotes the onepointcompactification of R in the topological context, and the simplicial set ∆[1]/∂∆[1] in the simplicial context).We obtain two adjoint constructions, the suspension S1∧A and the loop spectrum ΩA = AS

1of a symmetric

spectrum A. These spectra a levelwise given by (S1 ∧ A)n = S1 ∧ An respectively (ΩA)n = Ω(An) wherethe structure maps and symmetric groups actions do not interact with the new suspension respectively loopcoordinate. As a special case of (3.5) we have an adjunction

(3.6) ˆ : Sp(S1 ∧X,Y )∼=−−→ Sp(X,ΩY )

which takes a morphism f : S1∧X −→ Y to the morphism f : X −→ ΩY whose n-th level fn : Xn −→ Ω(Yn)is given by (fn(x))(t) = fn(t ∧ x).

Example 3.7 (Shift). The shift of a symmetric spectrum A is given by

(shA)n = A1+n

with action of Σn by restriction of the Σ1+n-action on A1+n along the monomorphism (1+−) : Σn −→ Σ1+n

which is explicitly given by (1 + γ)(1) = 1 and (1 + γ)(i) = γ(i − 1) + 1 for 2 ≤ i ≤ 1 + n. The structuremaps of shA are the reindexed structure maps for A. As an example, the shift of a suspension spectrum isanother suspension spectrum, sh(Σ∞K) = Σ∞(K ∧ S1).

For any symmetric spectrum A, integer k and large enough n we have

π(k+1)+n(shA)n = πk+(1+n)A1+n ,

and the maps in the colimit system for πk+1(shA) are the same as the maps in the colimit system for πkA.Thus we get the equality of naive homotopy groups πk+1(shA) = πkA.

We can iterate the shift construction and get (shmA)n = Am+n. In every level of the symmetricspectrum shmA the symmetric group Σm acts via the ‘inclusion’ (−+n) : Σm −→ Σm+n, and these actionsare compatible with the structure maps. So in this way shmA becomes a Σm-symmetric spectrum. Wenote that for k,m ≥ 0 we have

(shk(shmA))n = (shmA)k+n = Am+k+n = (shm+k A)n .

We observe that the symbols k and m have switched places along the way, which suggests that we shouldwrite

(3.8) shk(shmA) = shm+k A

and that we should avoid writing shk(shmA) as shk+mA. Of course, m+k = k+m, and so shm+k A equalsshk+mA; but for x ∈ An, γ ∈ Σk and κ ∈ Σm we have the relation

γ · (κ · x) = (κ+ γ) · x .So the interpretation (3.8) is better because it records the equivariance properties correctly.

We have already seen in Proposition 2.4 that the loop and suspension constructions shift the homotopygroups. The previous discussion depended only on the underlying sequential spectra. In the situationof symmetric spectra, there is a natural morphism λA : S1 ∧ A −→ shA whose level n component λn :S1 ∧An −→ A1+n is the composite

(3.9) S1 ∧An∼=−−−−→

twistAn ∧ S1 σn−−−→ An+1

(1,...,n+1)−−−−−−−→ A1+n .

One should note that the shuffle permutation is necessary to get a morphism, even of sequential spectra;for sequential spectra this is not available, and in fact there is no natural morphism from the suspension to

34 I. BASICS

the shift of a sequential spectrum. The spectra S1∧A and shA have abstractly isomorphic naive homotopygroups, but the morphism λA is not generally a π∗-isomorphism! However, the symmetric spectra for whichλA is a π∗-isomorphism play an important role, and they are called semistable (compare Theorem 7.27 forthe definition and various characterizations of semistable spectra).

Example 3.10 (Shift adjoint). [other terminology? offset, displacement?] The shift functor introducedin the previous Example has a left and a right adjoint. We use the notation ‘.’ for the left adjoint becauseit is a special case of the more general construction L .m X of the twisted smash product of a Σm-spacewith a spectrum). The spectrum .A is trivial in level 0 and is given in positive levels by

(.A)1+n = Σ+1+n ∧1×Σn An .

So ignoring the action of the symmetric group, the space (or simplicial set) (.A)1+n is simply a wedge ofn+ 1 copies of An. The structure map is obtained from the structure map of A and the ‘inclusion’ of Σ1+n

into Σ1+n+1, i.e.,

Σ+1+n ∧1×Σn An ∧ S1 −→ Σ+

1+n+1 ∧1×Σn+1 An+1 , γ ∧ a ∧ t 7−→ (γ + 1) ∧ σn(a ∧ t) .[adjunction]

We claim that the naive homotopy groups of .A are a countably infinite sum of shifted copies ofthe naive homotopy groups of A. For any m ≥ 1 we let (.(m)A)1+n denote the wedge summand of(.A)1+n = Σ+

1+n ∧1×Σn An indexed by the (1×Σn)-coset containing the transposition (1,m) if 1 + n ≥ m,and a one-point space for 1 + n < m. The structure map takes (.(m)A)1+n ∧ S1 to (.(m)A)1+n+1, so as nvaries we obtain a sequential subspectrum .(m)A of .A. The action of the symmetric groups, however, doesnot stabilize .(m)A, so .(m)A is not a symmetric subspectrum. As a sequential spectrum, .A is moreoverthe internal wedge of the spectra .(m)A, so by Corollary 2.20 (i) the natural map⊕

m≥1

πk

(.(m)A

)−→ πk

(∨m≥1

.(m) A)

= πk(.A)

is an isomorphism. The shifted sequential spectrum sh(.(m)A

)is isomorphic to A, at least from level m

on, and so the group k-th naive homotopy group of .(m)A is isomorphic to the (k + 1)-th naive homotopygroup of A. So altogether we have established an isomorphism between πk(.A) and a countably infinitesum of copies of πk+1A. This calculation is a special case of the more general result for the naive homotopygroups of a twisted smash product L .m A, see (3.16). We will review the calculation of πk(.A) in a morestructured way in (7.21) below.

Example 3.11 (Free symmetric spectra). Given a pointed space (or simplicial set ) K and m ≥ 0,we define a symmetric spectrum FmK which is ‘freely generated by K in level m’. The spectrum FmK istrivial below level m and is given by

(FmK)m+n = Σ+m+n ∧1×Σn K ∧ Sn .

in levels m and above. Here 1 × Σn is the subgroup of Σm+n of permutations which fix the first melements. The structure map σm+n : (FmK)m+n∧S1 −→ (FmK)m+n+1 is given by smashing the ‘inclusion’−+ 1 : Σm+n −→ Σm+n+1 with the identity of K and the preferred isomorphism Sn ∧ S1 ∼= Sn+1.

Free symmetric spectra generated in level zero are just suspension spectra, i.e., there is a naturalisomorphism F0K ∼= Σ∞K. Free symmetric spectra generated in adjacent levels are related by a naturalisomorphism

.(FmK) ∼= F1+mK ,

where . is the shift adjoint discussed in the previous examples. Indeed, an isomorphism which immediatelymeets the eye is given in level 1 +m+ n by

Σ+1+m+n ∧1×Σm+n (Σ+

m+n ∧1×Σn K ∧ Sn)m+n −→ Σ+1+m+n ∧1×Σn K ∧ Sn

γ ∧ (τ ∧ x) 7−→ (γ(1 + τ)) ∧ xfor γ ∈ Σ1+m+n, τ ∈ Σm+n and x ∈ K ∧ Sn.


The ‘freeness’ property of FmK is made precise by the following fact: for every based continuous mapf : K −→ Xm (of morphism of based simplicial sets) there is a unique morphism of symmetric spectrumf : FmK −→ X such that the composite

K1∧−−−−→ Σ+

m ∧K = (FmK)mfm−−→ Xm

equals f . So technically speaking this bijection makes Fm : T −→ Sp into a left adjoint of the forgetfulfunctor which takes a symmetric spectrum X to the pointed space Xm. The freeness property can beobtained as a special case of semifree symmetric spectra below [ref], or it can be established by inductionon m, using the isomorphsm .(FmK) ∼= F1+mK above and the adjunction between shift and . [ref]. In anycase, the morphism f : FmK −→ X corresponding to f : K −→ Xm is given in level m+n as the composite

Σ+m+n ∧1×Σn K ∧ Sn

Id∧σn(f∧Id)−−−−−−−−→ Σ+m+n ∧1×Σn Xm+n

act−−→ Xm+n

where σn : Xm ∧ Sn −→ Xm+n is the iterated structure map of X.Using that FmK is isomorphic to the m-fold iterate of . applied to the suspension spectrum Σ∞K

allows us to identify the naive homotopy groups of the free spectrum. Indeed, by the calculation of thenaive homotopy groups of .A in the previous example and induction, πk(FmK) is isomorphic to a countablesum of copies of πk+m(Σ∞K) = πs

k+mK, as long as m ≥ 1.We point out a special case which will be relevant later. The symmetric spectrum F1S

1 freely generatedby the circle S1 in level 1 ought to be a desuspension of the suspension spectrum of the circle. And indeed, weshall see in Example 4.17 that F1S

1 stably equivalent (to be defined in Section 4) to the sphere spectrum S.More generally, the free symmetric spectrum FmK is stably equivalent to Ωm(Σ∞K), the m-fold loopspectrum of the suspension spectrum of K, compare Example 4.25.

However, the naive homotopy groups of F1S1 are a countable sum of copies of the stable stems, so

F1S1 is not π∗-isomorphic to the sphere spectrum S, whose zeroth homotopy group is a single copy of the

integers.

Example 3.12 (Semifree symmetric spectra). There are somewhat ‘less free’ symmetric spectra whichstart from a pointed Σm-space (or Σm-simplicial set) L; we want to install L in level m, and then fill inthe remaining data of a symmetric spectrum in the freest possible way. In other words, we claim that theforgetful functor

SpT −→ Σm-T , X 7−→ Xm

(and its analog for spectra of simplicial sets) has a left adjoint which we denote Gm; we refer to GmL asthe semifree symmetric spectrum generated by L in level m. The spectrum GmL is trivial below level m,and other given by

(GmL)m+n = Σ+m+n ∧Σm×Σn L ∧ Sn .

The structure map σm+n : (GmK)m+n ∧ S1 −→ (GmK)m+n+1 is defined is defined by smashing the‘inclusion’ −+1 : Σm+n −→ Σm+n+1 with the identity of L and the preferred isomorphism Sn∧S1 ∼= Sn+1.We note that every free symmetric spectrum is semifree, i.e., there is a natural isomorphism FmK ∼=Gm(Σ+

m ∧K) by ‘cancelling Σm’.The ‘semifreeness’ property of GmL, or more technically the adjunction bijection, works as follows: we

claim that for every morphism of based Σm-spaces (or Σm-simplicial sets) f : L −→ Xm there is a uniquemorphism of symmetric spectrum f : GmL −→ X such that the composite

L1∧−−−−→ Σ+

m ∧Σm L = (GmL)mfm−−→ Xm

equals f . The requirements of equivariance and compatibility with structure maps imply that there is atmost one morphism with this property. Given f , we define the corresponding morphism f : GmL −→ X inlevel m+ n as the composite

Σ+m+n ∧Σm×Σn L ∧ Sn

Id∧σn(f∧Id)−−−−−−−−→ Σ+m+n ∧Σm×Σn Xm+n

act−−→ Xm+n

36 I. BASICS

where σn : Xm∧Sn −→ Xm+n is the iterated structure map of X. We omit the straightforward verificationthat f is indeed a morphism.

Semifree spectra are the basic building blocks in the theory of symmetric spectra. This slogan can bemade precise in various ways. The first, and almost tautological, way takes the form of a natural coequalizerdiagram

(3.13)∨n≥0Gn+1(Σ+

n+1 ∧Σn×1 Xn ∧ S1)σ //I

//∨m≥0GmXm // X .

The upper map σ to be coequalized takes the nth wedge summand to the (n+1)st wedge summand by theadjoint of

σn : Xn ∧ S1 −→ Xn+1 = (Gn+1Xn+1)n+1 .

The other map I takes the nth wedge summand to the nth wedge summand by the adjoint of the wedgesummand inclusion

Xn ∧ S1 −→ Σ+n+1 ∧Σn×Σ1 (Xn ∧ S1) = (GnXn)n+1

indexed by the identity of Σn+1. The morphism to X is the wedge over the adjoints of the identity mapsof the spaces Xm.

The fact that Gm is left adjoint to evaluation at level m directly implies that X has the universalproperty of a coequalizer of σ and I. Indeed, a morphism f :

∨m≥0GmXm −→ Z corresponds bijectively

to a family of equivariant based maps fm : Xm −→ Zm. The morphism f coequalizes the maps σ and I ifand only if the maps fm satisfy fm+1 σm = σm (fm ∧ IdS1) for all m, i.e., if they form a morphism ofsymmetric spectra from X to Z.

The coequalizer (3.13) can be used to reduce certain statements about general symmetric spectra tothe special case of semifree spectra. In Proposition 5.27 we will discuss a different way in which a generalsymmetric spectrum is ‘built up’ from semifree symmetric spectra.

One can describe the naive homotopy groups of the semifree spectrum GmL functorially in terms of thestable homotopy groups of L and the induced Σm-action. This is most naturally done with all the structurepresent in the naive homotopy groups, and so we defer this discussion to Example 7.14. As for the stablehomotopy type represented by a semifree spectrum, we refer to [...]

Example 3.14 (Twisted smash product). The twisted smash product starts from a number m ≥ 0, apointed Σm-space (or Σm-simplicial set) L and a symmetric spectrum X and produces a new symmetricspectrum which we denote L.mX. This construction is a simultaneous generalization of the smash productof a space and a spectrum (Example 3.4), the left adjoint to the shift construction (Example 3.7), andsemifree symmetric spectra (Example 3.12). Once the internal smash product of symmetric spectra isavailable, we will identify the twisted smash product L .m X with the smash product of the semifreespectrum GmL and X, see Proposition 5.5 below.

We define the twisted smash product L .m X as a point in levels smaller than m and in general by

(L .m X)m+n = Σ+m+n ∧Σm×Σn L ∧Xn .

The structure map σm+n : (L.mX)m+n∧S1 −→ (L.mX)m+n+1 is obtained from Id∧σn : L∧Xn∧S1 −→L ∧Xn+1 by inducing up.

Here are some special cases. Taking X = S gives semifree symmetric spectra as GmL = L .m S and soa free spectrum FmK is isomorohic to the twisted smash product (Σ+

m ∧K) .m S. For m = 0 we get

K .0 X = K ∧X ,

the levelwise smash product of K and X. The shift adjoint can be recovered as .X = S0 .1X. The twistedsmash product has an associativity property in the form of a natural isomorphism

L .m (L′ .n X) ∼= (Σ+m+n ∧Σm×Σn L ∧ L′) .m+n X .

The twisted smash product is related by various adjunctions to other constructions. As we noted atthe end of Example 3.7, the m-fold shift of a symmetric spectrum Z has an action of Σm through spectrumautomorphisms, i.e., shm Z is a Σm-symmetric spectrum. The levelwise smash product L ∧ X (in the


sense of Example 3.4) of the underlying space of L and X also is a Σm-symmetric spectrum through theaction on L. Given a morphism f : L .m X −→ Z of symmetric spectra, we can restrict the componentin level m+ n to the summand 1 ∧ L ∧Xn in (L .m X)m+n and obtain a Σm × Σn-equivariant based mapfn = fm+n(1 ∧ −) : L ∧ Xn −→ Zm+n = (shm Z)n. The compatibility of the fm+n’s with the structuremaps translates into the property that the maps f = fnn≥0 form a morphism of Σm-symmetric spectrafrom L∧X to shm Z. Conversely, every Σm-equivariant morphism L∧X −→ shm Z arises in this way froma morphism f : L .m X −→ Z. In other words, the assignment f 7→ f is a natural bijection of functors

(3.15) Sp(L .m X,Z) ∼= Σm-Sp(L ∧X, shm Z) .

The case m = 1 and L = S0 gives a bijection,

Sp(.X,Z) ∼= Sp(X, shZ) ,

natural in the symmetric spectra X and Z, which shows that X 7→ .X is left adjoint to shifting.Now we express the naive homotopy groups of a twisted smash product L .m X in terms of the naive

homotopy groups of the spectrum L ∧ X, the levelwise smash product of the underlying based space (orsimplicial set) of L and X. For this calculation we decompose the sequential spectrum which underliesL .m X; the wedge decomposition of the shift adjoint .X in Example 3.10 is a special case.

We denote by Om the set of order preserving injections f : m −→ ω from the set m = 1, . . . ,m tothe set ω = 1, 2, . . . of positive integers. Every such f ∈ Om gives rise to a sequential subspectrum (nota symmetric subspectrum) L .fm X of L .m X as follows. We recall that a permutation γ ∈ Σm+n is an(m,n)-shuffle if the restriction of γ to 1, . . . ,m and the restriction to m+ 1, . . . ,m+ n are monotone.For f ∈ Om we denote by |f | the maximum of the values of f . If the image of f is contained in m + n fora particular n ≥ 0, then there is a unique (m,n)-shuffle χ(f) which agrees with f on m. Conversely, every(m,n)-shuffle arises in this way from a unique f ∈ Om. The sequential spectrum L .fm X is given by

(L .fm X)m+n =

(χ(f) · (Σm × Σn)

)+ ∧Σm×Σn (L ∧Xn) if Im(f) ⊂m + n∗ else.

Note that L .fm X is also trivial below level m.Since the (m,n)-shuffles provide a set of right coset representatives for the group Σm × Σn in Σm+n,

we have an internal wedge decomposition

(L .m X)m+n =∨

f∈Om

(L .fm X)m+n

The monomorphism − + 1 : Σm+n −→ Σm+n+1 takes (m,n)-shuffles to (m,n + 1)-shuffle and satisfiesχ(f)+1 = χ(f) whenever Im(f) ⊂m + n. The structure map σm+n : (L.mX)m+n∧S1 −→ (L.mX)m+n+1

thus preserves the wedge decomposition and so the underlying sequential spectrum of L .m X decomposesas a internal wedge of the sequential spectra L .fm X for f ∈ Om.

A consequence is that the underlying sequential spectrum of L .m X, and hence its naive homotopygroups do not depend on the Σm-action on L. Since the naive homotopy groups of a wedge are the directsum of the naive homotopy groups (Corollary 2.20 (i)), the inclusions induce an isomorphism⊕

f∈Om

πk(L .fm X)∼=−→ πk(L .m X) .

For f ∈ Om let n ≥ 0 be the smallest number such that Im(f) ⊂ m + n (i.e., the difference of themaximum of f andm). Then the shifted sequential spectrum shm+n(L.fmX) is isomorphic to the underlyingsequential spectrum of shn(L ∧X), so the group πk(L .fm X) = πk+m+n(shm+n(L .fm X)) is isomorphic toπk+m+n(shn(L ∧X)) = πk+m(L ∧X). Combining these two isomorphisms gives

(3.16)⊕f∈Om

πk+m(L ∧X) ∼= πk(L .m X) .

38 I. BASICS

We emphasize that (3.16) is in general only an isomorphism of abelian group, but that it does not preservethe extra structure which is available on the naive homtopy groups and which we discuss in Section 7. Wereturn to this point in Example 7.11.

We can also identify the shift of a twisted smash product. Firstly, we have sh(L .0 X) = sh(L ∧X) =L∧ shX. In general, the shift of a twisted smash product decomposes into two pieces which are themselvestwisted smash products: in Lemma 5.9 below we construct a natural isomorphism

sh(L .1+m X) ∼= (shL) .m X ∨ L .1+m (shX)

for any pointed Σ1+m-space (or simplicial set) L, where shL denotes the restriction of L to a Σm-spacealong the monomorphism 1 + − : Σm −→ Σ1+m. A way to remember this is to say that ‘shifting is aderivation with respect to twisted smash product’. We have G1+mL = L .1+m S and sh S = S1 ∧ S, so asa special case of the previous lemma we obtain a wedge decomposition of the shift of a semifree symmetricspectrum

sh(G1+mL) ∼= Gm(shL) ∨ (S1 ∧G1+mL) .

We can specialize even further to free symmetric spectra. Using F1+mK = G1+m(Σ+1+m ∧K) and the fact

that sh(Σ1+m) is the disjoint union of (1 +m) free transitive Σm-sets we obtain

sh(F1+mK) ∼=1+m∨i=1

FmK ∨ F1+m(S1 ∧K) .

Example 3.17 (Mapping spaces). There is a whole space, respectively simplicial set, of morphismsbetween two symmetric spectra. For symmetric spectra X and Y , every morphism f : X −→ Y consistsof a family of based maps fn : Xn −→ Ynn≥0 which satisfy some conditions. So the set of morphismsfrom X to Y is a subset of the product of mapping spaces

∏n≥0 map(Xn, Yn) and we give it the subspace

topology of the (compactly generated) product topology. We denote this mapping space by map(X,Y ).[also for ssets...]

Now suppose thatX and Y are symmetric spectra of simplicial sets. Then the mapping space map(X,Y )is the simplicial set whose n-simplices are the spectrum morphisms from ∆[n]+ ∧X to Y . For a monotonemap α : [n] −→ [m] in the simplicial category ∆, the map α∗ : map(X,Y )m −→ map(X,Y )n is given byprecomposition with α∗ ∧ Id : ∆[n]+ ∧X −→ ∆[m]+ ∧X. The morphism space has a natural basepoint,namely the trivial map from ∆[0]+ ∧ X to Y . We can, and will, identify the vertices of map(X,Y ) withthe morphisms from X to Y using the natural isomorphism ∆[0]+ ∧X ∼= X.

Furthermore, for a pointed space K and topological symmetric spectra X and Y we have adjunctionhomeomorphisms

map(K,map(X,Y )) ∼= map(K ∧X,Y ) ∼= map(X,Y K) ,

where the first mapping space is taken in the category T of compactly generated spaces. In the context ofsymmetric spectra of simplicial sets, the analogous isomorphisms of mapping simplicial sets hold as well.

The topological and simplicial mapping spaces are related by various adjunctions. We list some of these.For a simplicial spectrum X and a topological spectrum Y there is a natural isomorphism of simplicial sets

map(X,S(Y )) ∼= S(map(|X|, Y ))

which on vertices specializes to the adjunction between singular complex and geometric realization. Indeed,an n-simplex of the left hand side is a morphism ∆[n]+ ∧X −→ S(Y ) of symmetric spectra of simplicialsets. We can pass to the adjoint morphism |∆[n]+ ∧ X| −→ Y of symmetric spectra of spaces, exploitthat realization commutes with smash products and then identify the realization |∆[n]| with the topologicaln-simplex ∆[n] as in (3.1) of Appendix A. The outcome is a morphism ∆[n]+ ∧ |X| −→ Y whose adjoint∆[n] −→ map(|X|, Y ), a continuous map of unbased spaces, is an n-simplex of the singular complex of themapping space map(|X|, Y ).

For free symmetric spectra we have isomorphisms

(3.18) map(FmK,Y ) ∼= map(K,Ym) .


In more detail: in the context of spectra of spaces, the adjunction bijection between mapping sets which wespecified in Example 3.11 is indeed a homeomorphism In the context of spectra of simplicial sets, we canuse the adjunction bijection for ∆[n]+ ∧K and exploit Fm(∆[n]+ ∧K) = ∆[n]+ ∧ FmK and get a naturalbijection between the n-simplices of the two mapping simplicial sets. For K = S0 isomorphism (3.18)specializes to an isomorphism map(FmS0, Y ) ∼= Ym.

We have associative and unital composition maps

map(Y, Z) ∧map(X,Y ) −→ map(X,Z) .

Indeed, for symmetric spectra of topological spaces this is just the observation that composition of mor-phisms is continuous for the mapping space topology. For symmetric spectra of simplicial sets the compo-sition maps are given on n-simplices by

Sp(∆[n]+ ∧ Y, Z) ∧ Sp(∆[n]+ ∧X,Y ) −→ Sp(∆[n]+ ∧X,Z)

g ∧ f 7−→ g (Id∧f) (diag ∧ IdX)

where diag : ∆[n]+ ∧∆[n]+ −→ ∆[n]+ is the diagonal map.

Example 3.19 (Internal Hom spectra). Symmetric spectra have internal function objects: for symmet-ric spectra X and Y we define a symmetric spectrum Hom(X,Y ) in level n by

Hom(X,Y )n = map(X, shn Y )

with Σn-action induced by the action on shn Y as described in Example 3.7. The structure map σn :Hom(X,Y )n ∧ S1 −→ Hom(X,Y )n+1 is the composite

map(X, shn Y ) ∧ S1 −−→ map(X,S1 ∧ shn Y )map(X,λshn Y )−−−−−−−−−−→ map(X, shn+1 Y )

where λshn Y : S1 ∧ shn Y −→ sh(shn Y ) = shn+1 Y is the natural morphism defined in (3.9). [first map...]In order to verify that this indeed gives a symmetric spectrum we describe the iterated structure map. Letus denote by λ(m)

Y : Sm ∧ Y −→ shm Y the morphism whose n-th level is the composite

(3.20) Sm ∧ Yntwist−−−→ Yn ∧ Sm

σm−−→ Yn+mχn,m−−−→ Ym+n = (shm Y )n .

For m = 0 this is the canonical isomorphism S0 ∧ Y ∼= Y and for m = 1 this specializes to the morphismλY : S1∧Y −→ shY of (3.9); in general λ(m)

Y is a morphism of Σm-symmetric spectra. The for all k,m ≥ 0the diagram

Sk ∧ Sm ∧ YId∧λ(m)

Y //

∼=

Sk ∧ shm Yλ

(k)shm Y // shk(shm Y )

Sk+m ∧ Yχk,m∧Id

// Sm+k ∧ Yλ

(m+k)Y

// shm+k Y

commutes. This implies that the iterated structure map of the spectrum Hom(X,Y ) equals the composite

map(X, shn Y ) ∧ Sm −−→ map(X,Sm ∧ shn Y )map(X,λ

(m)shn Y )

−−−−−−−−−−→ map(X, shn+m Y )

[first map] and is thus Σn×Σm-equivariant. A natural isomorphism of symmetric spectra Hom(FmS0, Y ) ∼=shm Y is given at level n by

Hom(FmS0, Y )n = map(FmS0, shn Y ) ∼= (shn Y )m = Yn+mχn,m−−−→ Ym+n = (shm Y )n

where the second map is the adjunction bijection describes in Example 3.11. In the special case m = 0 wehave F0S

0 = S, which gives a natural isomorphism of symmetric spectra Hom(S, Y ) ∼= Y .The internal function spectrum functor Hom(X,−) is right adjoint to the internal smash product −∧X

of symmetric spectra, to be discussed in Section 5.

40 I. BASICS

3.3. Constructions involving ring spectra.

Example 3.21 (Endomorphism ring spectra). For every symmetric spectrumX, the symmetric functionspectrum Hom(X,X) defined in Example 3.19 has the structure of a symmetric ring spectrum which we callthe endomorphism ring spectrum of X. The multiplication map µn,m : Hom(X,X)n ∧ Hom(X,X)m −→Hom(X,X)n+m is defined as the composite

map(X, shnX) ∧map(X, shmX) shm ∧ Id−−−−−→ map(shmX, shm(shnX)) ∧map(X, shmX)

=map(shmX, shn+mX) ∧map(X, shmX) −−−−→ map(X, shn+mX)

where the second map is the composition pairing of Example 3.17. We refer to (3.8) for why it is ‘right’ toidentify shm(shnX) with shn+mX (note the orders in which m and n occur), so that no shuffle permutationis needed.

While this construction always works on the pointset level, one can only expect Hom(X,X) to behomotopically meaningful under certain conditions on X. The stable model structures which we discusslater will explain which conditions are sufficient.

In much the same way as above we can define associative and unital action maps Hom(X,Z)n ∧Hom(X,X)m −→ Hom(X,Z)n+m and Hom(X,X)n ∧ Hom(Z,X)m −→ Hom(Z,X)n+m for any othersymmetric spectrum Z. This makes Hom(X,Z) and Hom(Z,X) into right respectively left modules overthe endomorphism ring spectrum of X.

Example 3.22 (Spherical monoid ring). If M is a topological or simplicial monoid (depending on thekind of symmetric spectra under consideration), we define then spherical monoid ring S[M ] by

S[M ]n = M+ ∧ Sn

with symmetric group actions and structure maps only on the spheres; here M+ denotes the underlyingspaces (or simplicial set) of M with a disjoint basepoint added. In other words, the underlying symmetricspectrum of the spherical monoid ring is the suspensions spectrum, as defined in Example 1.13, of M+.The spherical monoid ring becomes a symmetric ring spectrum as follows: the unit maps of S[M ] are themaps 1 ∧ − : Sn −→ M+ ∧ Sn = S[M ]n which include via the unit element 1 of the monoid M . Themultiplication map µn,m is given by the composite

(M+ ∧ Sn) ∧ (M+ ∧ Sm) ∼= (M ×M)+ ∧ (Sn ∧ Sm)mult.∧µn,m−−−−−−−−−→ M+ ∧ Sn+m .

A right module over the spherical monoid ring S[M ] is ‘the same as’ a symmetric spectrum with a (continuousor simplicial) right action by the monoid M through endomorphisms.

If R is a symmetric ring spectrum, then the zeroth space (or simplicial set) R0 is a topological (orsimplicial) monoid via the composite

R0 ×R0 −→ R0 ∧R0µ0,0−−→ R0 .

The monoid R0 is commutative if the ring spectrm R is. Given any monoid homomorphism f : M −→ R0

we define a homomorphism of symmetric spectra f : S[M ] −→ R in level n as the composite

M+ ∧ Sn f∧Id−−−→ R+0 ∧ Sn −→ R0 ∧ Sn

σn−−→ Rn .

This morphism is in fact unital and associative, i.e., a morphism of symmetric ring spectra. Moreover, fis uniquely determined by the property that it gives back f in level 0. We can summarize this as sayingthat the construction of the monoid ring over S is left adjoint to the functor which takes a symmetric ringspectrum R to the (topological or simplicial) monoid R0. [pointed monoids]

Example 3.23 (Monoid ring spectra). The previous construction works more generally when we startwith a symmetric ring spectrum R instead of the sphere spectrum. If R is a symmetric ring spectrum and Ma topological or simplicial monoid (depending on the kind of symmetric spectra), we can define a symmetricring spectrum R[M ] by R[M ] = M+∧R, i.e., the levelwise smash product with M with a disjoint basepoint


added. The unit map is the composite of the unit map of R and the morphism R ∼= 1+ ∧R −→M+ ∧Rinduced by the unit of M . The multiplication map µn,m is given by the composite

(M+ ∧Rn) ∧ (M+ ∧Rm) ∼= (M ×M)+ ∧ (Rn ∧Rm)mult.∧µn,m−−−−−−−−−→ M+ ∧Rn+m .

If both R and M are commutative, then so is R[M ]. A right module over the symmetric ring spectrumR[M ] amounts to the same data as an R-module together with a continuous (or simplicial) right action ofthe monoid M by R-linear endomorphisms.

As we shall see later, the homotopy groups of R[M ] are the R-homology groups of the underlying spaceof M , with the Pontryagin product as multiplication. In the special case of a discrete spherical monoidring, the homotopy groups are the monoid ring, in the ordinary sense, of the homotopy groups, i.e., thereis a natural isomorphism of graded rings

π∗R[M ] ∼= (π∗R)[M ] ,

see Example 6.11

Example 3.24 (Matrix ring spectra). If R is a symmetric ring spectrum and k ≥ 1 we define thesymmetric ring spectrum Mk(R) of k × k matrices over R by

Mk(R) = map(k+, k+ ∧R) .

Here k+ = 0, 1, . . . , k with basepoint 0, and so Mk(R) is a k-fold product of a k-fold coproduct (wedge)of copies of R. So ‘elements’ of Mk(R) are more like matrices which in each row have at most one nonzeroentry. The multiplication

µn,m : map(k+, k+ ∧Rn) ∧ map(k+, k+ ∧Rm) −→ map(k+, k+ ∧Rn+m)

sends f ∧ g to the composite

k+ g−−→ k+ ∧Rmf∧Id−−−−→ k+ ∧Rn ∧Rm

µn,m−−−→ Rn+m .

We shall see below that homotopy groups take wedges and products to direct sums; this implies a naturalisomorphism of graded rings

π∗(Mk(R)) ∼= Mk(π∗R) .We revisit this in more detail in Example 6.12

Example 3.25 (Function ring spectra). If R is a symmetric ring spectrum and L an unpointed space,then the mapping spectrum RL

+(compare Example 3.4) is again a symmetric ring spectrum. The multi-

plication maps RL+

n ∧RL+

m −→ RL+

n+m are the composites

map(L+, Rn) ∧map(L+, Rm) ∧−−→ map(L+ ∧ L+, Rn ∧Rm)map(diag,µn,m)−−−−−−−−−−→ map(L+, Rn+m)

using the diagonal map L+ −→ L+∧L+. Associativity of the multiplication on RL+

comes from associativityof R and coassociativity of the diagonal map. The unit map ιn : Sn −→ RL

+

n is the composite of the unitmap of R with the map Rn −→ RL

+

n that takes a point (or simplex) x ∈ Rn to the map that sends all ofL to x. If the multiplication of R is commutative, then so is the multiplication of RL

+, since the diagonal

map is cocommutative.A concrete example if X = HA, the Eilenberg-Mac Lane spectrum of an abelian group A. Then

π−k(HAL+) ∼= Hk(L,A) ,

a natural isomorphism of abelian groups, where the right hand side is the singular cohomology of L withcoefficients in A. Indeed, since HA is an Ω-spectrum, so is HAL+ for any cofibrant space or any simplicialset. So the canonical map [L,A[Sk]] = π0 map(L+, A[Sk]) = π0(HAL

+)k −→ π−k(HAL

+) is bijective,

where [L,A[Sk]] is the set of homotopy classes of (unbased) maps. Since A[Sk] is an Eilenberg-Mac Lanespace of type (A, k), evaluation at the fundamental cohomology class ιk ∈ Hk(A[Sk], A) is an isomorphism

[L,A[Sk]] −→ Hk(L,A) , [f ] 7−→ f∗(ιk) .

42 I. BASICS

If A is a ring, then HA becomes a ring spectrum and this isomorphism takes the product of homotopygroups to the cup product in singular cohomology.

Example 3.26 (Inverting m). We consider an integer m and define S[1/m], the sphere spectrum withm inverted by starting from the sphere spectrum (of topological spaces) and using a map ϕm : S1 −→ S1

of degree m as the new unit map ι1. Since the multiplication on S is commutative, centrality is automatic.So S[1/m] has the same spaces and symmetric group actions as S, but the n-th unit map ιn of S[1/m] isthe n-fold smash power of ϕm, which is a self map of Sn of degree mn. The unit maps form a morphismS −→ S[1/m] of symmetric ring spectra which on homotopy groups induces an isomorphism

π∗ S[1/m] ∼= π∗S⊗ Z[1/m] .

For m = 0, the homotopy groups are thus trivial and for m = 1 or m = −1 the unit morphism S −→ S[1/m]is a π∗-isomorphism.

Example 3.27 (Inverting homotopy elements). Let R be a symmetric ring spectrum and let x : S1 −→R1 be a central map of pointed spaces (or simplicial sets), i.e., such that the square

Rn ∧ S1 Id∧x //

τ

Rn ∧R1

µn,1 // Rn+1

χn,1

S1 ∧Rn x∧Id

// R1 ∧Rn µ1,n// R1+n

commutes for all n ≥ 0. We observe that if R is commutative, then any map from S1 to R1 is central.We define a new symmetric ring spectrum R[1/x] as follows. For n ≥ 0 we set

R[1/x]n = map(Sn, R2n) ,

the n-fold loop space of R2n. In order to guess the correct action and multiplication maps it is helpful tothink of 2n as n+n, and not as the n-fold sum of 2’s. The group Σn acts on Sn by coordinate permutations,on Rn+n via restriction along the diagonal embedding

∆ : Σn −→ Σ2n , ∆(γ) = γ + γ ,

and by conjugation on the whole mapping space. The multiplication µn,m : R[1/x]n ∧ R[1/x]m −→R[1/x]n+m is the map

map(Sn, R2n) ∧map(Sm, R2m) −→ map(Sn+m, R2(n+m))

f ∧ g 7−→ (1n + χn,m + 1m) µ2n,2m (f ∧ g) .Note that the interpretation 2n = n + n lets us predict the shuffle permutation: the natural target of theproduct f · g is R2n+2m, whose index expands to (n+ n) + (m+m). The visual difference to 2(n+m) =(n + m) + (n + m) suggest to insert the permutation 1n + χn,m + 1m which moves the indices into thecorrect order. The multiplication map is associative since smashing of maps and the product of R are, andbecause the relation

(1n+m + χn+m,k + 1k)(1n + χn,m + 1m + 12k) = (1n + χn,m+k + 1m+k)(12n + 1m + χm,k + 1k)

holds in the group Σ2n+2m+2k. The multiplication map is Σn ×Σm-equivariant since the original multipli-cation maps are equivariant and since the diagonal embeddings satisfy

∆(γ + τ)(1n + χn,m + 1m) = (1n + χn,m + 1m)(∆(γ) + ∆(τ))

for γ ∈ Σn and τ ∈ Σm. We have R[1/x]0 = R0 and the 0th unit map for R[1/x] is the same as for R.Next we define based maps jn : Rn −→ map(Sn, R2n) as the adjoints of the maps

Rn ∧ SnId∧x∧n−−−−−→ Rn ∧R∧n1

µn,1,...,1−−−−−→ Rn+n

Since the map µn,1,...,1 x∧n : Rn ∧ Sn −→ Rn+n is Σn × Σn-equivariant the adjoint jn is Σn-equivariant.The maps jn are multiplicative in the sense of the relation µn,m(jn ∧ jm) = jn+mµn,m holds.


We define unit maps ιn : Sn −→ R[1/x]n as the composite of the unit map of R with jn (which agreeswith the n-fold power of the map j1 : S1 −→ R[1/x]1). This finishes the definition of R[1/x] which is againa symmetric ring spectrum and comes with a morphism of symmetric ring spectra j : R −→ R[1/x].

We note that the central map x does not enter in the definition of the spaces R[1/x]n, and it is notused in defining the multiplication of R[1/x], but it enters in the definition of the morphism j and hencethe unit map of the ring spectrum R[1/x]. Since R[1/x]n is the n-fold loop space of R2n, the homotopygroup πk+n(R[1/x]n) is isomorphic to πk+2nR2n; thus the colimit system which defines the naive homotopygroup πk(R[1/x]) involves ‘half of’ the groups which define πkR, but the effect of the map x is twisted intothe morphisms in the sequence, and so the homotopy groups of R and R[1/x] are typically different. Weshow in Proposition 6.29 below that if R is semistable, then so is R[1/x] and the effect of the morphismj : R −→ R[1/x] on the graded rings of homotopy groups is precisely inverting the class in π0R representedby the map x.

Example 3.28 (Adjoining roots of unity). As an application of the localization construction of Exam-ple 3.27 we construct a commutative symmetric ring spectrum which models the ‘Gaussian integers over S’with 2 inverted. We start with the spherical group ring S[C4] of the cyclic group of order 4, a commutativesymmetric ring spectrum as in Example 3.22. We invert the element

1− t2 ∈ Z[C4] = π0 S[C4]

where t ∈ C4 is a generator, and define

S[1/2, i] = S[C4][1/(1− t2)] .

In more detail, the space S[C4]1 = C+4 ∧ S1 is a wedge of 4 circles and the map from π1S[C4]1 to the stable

group π0 S[C4] is surjective. So 1 − t2 ∈ π0 S[C4] can be represented by a based map x : S1 −→ S[C4]1 towhich we apply Example 3.27. The monoid ring spectrum S[C4] is commutative and semistable, and soCorollary 6.31 below shows that the ring of true homotopy groups of S[C4][1/(1− t2)] is obtained from thering π∗S[C4] by inverting the class C(1− t2) in π0.

Because (1 + t2)(1− t2) = 0 in the group ring Z[C4], inverting 1− t2 forces 1 + t2 = 0, so t becomes asquare root of −1. Since (1− t2)2 = 2(1− t2), inverting 1− t2 also inverts 2, and in fact

π0 S[1/2, i] ∼= Z[C4][1/(1− t2)] = Z [1/2, i] .

where i is the image of t. The ring spectrum S[1/2, i] is π∗-isomorphic as a symmetric spectrum to a wedgeof 2 copies of S[1/2], and thus deserves to be called the ‘Gaussian integers over S’ with 2 inverted. Moreover,S[1/2, i] is a Moore spectrum for the ring Z[1/2, i], i.e., its integral homology is concentrated in dimensionzero (compare Section II.7).

If p is a prime number and n ≥ 1, we can similarly adjoin a primitive pn-th root of unity to the spherespectrum, provided we are also willing to invert p in the homotopy groups. We first form the monoid ringspectrum S[Cpn ] of the cyclic group of order pn, let t ∈ Cpn denote a generator and invert the elementf = p− (tq(p−1) + tq(p−2) + · · ·+ tq + 1) in Z[Cpn ] = π0S[Cpn ], where q = pn−1. This defines

S[1/p, ζ] = S[Cpn ][1/f ] .

We have f2 = pf , so inverting f also inverts the prime p and forces the expression p− f to become 0 in thelocalized ring. If we let ζ denote the image of t in the localized ring, then the latter says that ζ is a root ofthe cyclotomic polynomial, i.e.,

ζq(p−1) + ζq(p−2) + · · ·+ ζq + 1 = 0

where again q = pn−1. In fact we have Z[Cpn ][1/f ] = Z[1/p, ζ] where ζ is a primitive pn-th root of unity;moreover the commutative symmetric ring spectrum S[1/p, ζ] is a Moore spectrum for the ring Z[1/p, ζ].

We can do the same constructions starting with any semistable commutative symmetric ring spectrumR instead of the sphere spectrum, yielding a new commutative symmetric ring spectrum R[1/p, ζ]. If p isalready invertible and the cyclotomic polynomial above is irreducible in π0R, then this adjoins a primitivepn-th root of unity to the homotopy ring of R.

44 I. BASICS

These examples are a special case of a much more general phenomenon: every number ring can be‘lifted’ to an extension of the sphere spectrum by a commutative symmetric ring spectrum, provided wealso invert the ramified primes. However, the only proofs of this general fact that I know use obstructiontheory, and so we cannot give a construction which is as explicit and simple as the one above for adjoiningroots of unity.

Example 3.29 (Algebraic K-theory). There are various formalisms which associate to a category withsuitable extra structure an algebraic K-theory space. These spaces are typically infinite loop spaces ina natural way, i.e., they arise from an Ω-spectrum. One very general framework is Waldhausen’s S·-construction which accepts categories with cofibrations and weak equivalences as input and which producessymmetric spectra which are positive Ω-spectra.

We consider a category C with cofibrations and weak equivalences in the sense of Waldhausen [82]. Forany finite set Q we denote by P(Q) the power set of Q viewed as a partially ordered set under inclusions,and thus as a category. A Q-cube in C is a functor X : P(Q) −→ C. Such a Q-cube X is a cofibration cubeif for all S ⊂ T ⊂ Q the canonical map

colimS⊂U

⊂6=T

X(U) −→ X(T )

is a cofibration in C. (The colimit on the left can be formed by iterated pushouts along cofibrations, so itexists in C.)

We view the ordered set [n] = 0 < 1 < · · · < n as a category. If n = nss∈Q is a Q-tuple ofnon-negative integers, we denote by [n] the product category of the categories [ns], s ∈ Q. For a morphismi → j in [n] and a subset U ⊂ Q we let (i → j)U be the new morphisms in [n] whose sth component isis → js if s ∈ U and the identity is → is if s 6∈ U . Then for each morphism i→ j in [n], the assignment

U 7→ (i→ j)U

defines a Q-cube in the arrow category Ar[n].For a finite set Q and a Q-indexed tuple n = nss∈Q we define a category SQn C as the full subcategory

of the category of functors from Ar[n] to C consisting of the functors

A : Ar[n] −→ C , (i→ j) 7→ Ai→j

with the following properties:(i) if some component is → js of i→ j is an identity (i.e., if is = js for some s ∈ Q), then Ai→j = ∗ is

the distinguished zero object of C;(ii) for every pair of composable morphisms i→ j → k the cube

U 7→ A(j→k)U(i→j)

is a cofibration cube(iii) for every pair of composable morphisms i→ j → k the square

colimU⊂6=Q

A(j→k)U(i→j) //

Ai→k

∗ // Aj→k

is a pushout in C.The category SQn C depends contravariantly on [n], so that as [n] varies, we get a Q-simplicial category

SQ· C. We can make SQ· C into a Q-simplicial object of categories with cofibrations and weak equivalencesas follows. A morphism f : A −→ A′ is a cofibration in SQn C if for every pair of composable morphismsi→ j → k the induced map of Q-cubes(

U 7→ A(j→k)U(i→j)

)−→

(U 7→ A′(j→k)U(i→j)

)


is a cofibration cube when viewed as a (|Q|+ 1)-cube in C. A morphism f : A −→ A′ is a weak equivalencein SQn C if for every morphism i −→ j in [n] the morphism fi−→j is a weak equivalence in C. If Q has oneelement, the SQ· C is isomorphic to S·C as defined by Waldhausen [82]. If P ⊂ Q there is an isomorphismof Q-simplicial categories with cofibrations and weak equivalences

SQ· C ∼= SQ−P· (SP· C)

[define]. So a choice of linear ordering of the set Q specifies an isomorphism of categories

SQ· C ∼= S· · · ·S·C

to the |Q|-fold iterate of the S·-construction. Note that the permutation group of the set Q acts on SQ· Cby permuting the indices.

Now we are ready to define the algebraic K-theory spectrum K(C) of the category with cofibrationsand weak equivalences C. (This is really naturally a coordinate free symmetric spectrum in the sense ofExercise 8.9.) It is the symmetric spectrum of simplicial sets with nth level given by

K(C)n = N·

(wS

1,...,n· C

),

i.e., the nerve of the subcategory of weak equivalences in SQ· C for the special case Q = 1, . . . , n. Thebasepoint is the object of S1,...,n· C given by the constant functor with values the distinguished zero object.The group Σn of permutations of the set 1, . . . , n, acts on S

1,...,n· C preserving weak equivalences, so it

acts on the simplicial set K(C)n. Note that K(C)0 is the nerve of the category wC of weak equivalencesin C.

We still have to define the structure maps

σn : K(C)n ∧ S1 −→ K(C)n+1 .

[...] By a theorem of Waldhausen, the symmetric spectrum K(C) is a positive Ω-spectrum.Pairings of exact categories give rise to pairings of K-theory spectra. Consider a biexact functor

∧ : C × D −→ E between categories with cofibrations and weak equivalences. For disjoint finite subsets Qand Q′ we obtain a biexact functor of (Q∪Q′)-simplicial categories with cofibrations and weak equivalences

∧ : SQ· C × SQ′

· D −→ SQ∪Q′

· E

by assigning

(A ∧A′)i∪i′→j∪j′ = Ai→j ∧A′i′→j .

We specialize to Q = 1, . . . , n and Q′ = n+1, . . . , n+m, restrict to weak equivalences and take nerves.This yields a Σn × Σm-equivariant map K(C)n ×K(D)m −→ K(E)n+m which factors as

K(C)n ∧K(D)m −→ K(E)n+m .

These maps are associative for strictly associative pairings [explain].An example of a category with cofibrations and weak equivalences is the category Γ of finite pointed

sets n+ = 0, 1, . . . , n with 0 as basepoint, and pointed set maps. Here the cofibrations are the injectivemaps and the weak equivalences are the bijections. The ‘smash product’ functor

∧ : Γ× Γ −→ Γ , (n+,m+) 7→ (nm)+

is biexact and strictly associative so it makes the symmetric sequence K(Γ)nn≥0 into a strict monoidof symmetric sequences. Here we identify n+ ∧ m+ with (nm)+ using the lexicographic ordering. Theobject 1+ of Γ gives a 0-simplex in K(C)0; a theorem by Barratt-Priddy-Quillen asserts that the morphismS −→ K(Γ) adjoint to this is a π∗-isomorphism.

46 I. BASICS

4. Stable equivalences

In this section we introduce and discuss the important notion of a stable equivalence of symmetricspectra, see Definition 4.8. We also introduce the true homotopy groups πkA of a symmetric spectrum A(see Definition 4.29) and prove in Theorem 4.35 that the stable equivalences are precisely those morphismf : A −→ B such that the induced map πkf : πkA −→ πkB of true homotopy groups is an isomorphism forall integers k.

Another important results is that a morphism of symmetric spectra with induces isomorphisms on naivehomotopy groups is a stable equivalence (see Theorem 4.16 [and spaces]). So for morphisms of symmetricspectra we thus have the implications

homotopy equivalence =⇒ level equivalence =⇒ π∗-isomorphism =⇒ stable equivalence.

In general, the reverse implications do not hold. However, for certain classes of spectra, one can argue inthe other direction:

• every stable equivalence between semistable spectra (defined in Definition 4.28 below) is a π∗-isomorphism;

• every π∗-isomorphism between Ω-spectra is a level equivalence [ref];• in the context of simplicial sets, every level equivalence between injective spectra is a homotopy

equivalence (compare Proposition 4.5).In Proposition 4.11 below we give a list of several equivalent characterizations of stable equivalences.

In Proposition 4.22 we prove that stable equivalences are closed under various constructions such as sus-pensions, loop, wedges, finite products [more?]. Up to stable equivalence, every symmetric spectrum canbe replaced by an Ω-spectrum (Proposition 4.27). The ultimate consequence will be that the stable homo-topy category arises as the localization of the category of symmetric spectra obtained by ‘inverting stableequivalences’, compare Theorem 1.6 of Chapter II.

4.1. Injective spectra.

Definition 4.1. A symmetric spectrum of simplicial sets X is injective if for every monomorphismi : A −→ B which is also a level equivalence and every morphism f : A −→ X there exists an extensiong : B −→ X with f = gi.

As we shall see in Proposition 4.3 below, an injective spectrum is in particular levelwise a Kan complex.Injective spectra do not arise ‘in nature’ very often, so we give some examples arising as co-free and

co-semifree symmetric spectra. Moreover, we prove in Proposition 4.6 below that injectivity can always bearranged up to level equivalence.

Example 4.2 (Co-free and co-semifree symmetric spectra). Let Pm : Σm-sS −→ Sp be right adjointto evaluation at level m, considered as a Σm-simplicial set. [also in spaces] We call PmL the co-semifreesymmetric spectrum generated by the Σm-simplicial set L in level m. The spectrum PmL can explicitelybe described as follows: it is just a point above level m and for n ≤ m we have

(PmL)n = map1×Σm−n(Sm−n, L) ,

the subspace of 1 × Σm−n-equivariant maps in map(Sm−n, L), with restricted Σn-action from L. Thestructure map σn : (PmL)n ∧ S1 −→ (PmL)n+1 is adjoint to the map

map1×Σm−n(Sm−n, L) incl.−−−→ map1×Σm−n−1(Sm−n, L) ∼= Ω(map1×Σm−n−1(Sm−n−1, L)

).

The forgetful functor Σm-sS −→ sS also has a right adjoint given by K 7→ map(Σ+m,K), the function

space from the set Σm into K (i.e., a product of m! copies of K). So the composite forgetful functorSp −→ sS which takes X to the pointed simplicial set Xm has a right adjoint Rm : sS −→ Sp given byRmK = Pm(map(Σ+

m,K)).Every Kan simplicial set has the right lifting property with respect to all injective weak equivalences of

simplicial sets. So by adjointness, the co-free symmetric spectrum RmK is injective for every Kan simplicialset K. More generally, let L be a pointed Σm-simplicial set which is mixed Σm-fibrant in the following sense:

4. STABLE EQUIVALENCES 47

for every subgroup H ≤ Σm the H-fixed simplicial set LH is a Kan complex and the map LH −→ LhH

from the H-fixed points to the H-homotopy fixed points is a weak equivalence. Then H has the right liftingproperty with respect to all injective based morphisms of Σm-simplicial sets which are weak equivalencesafter forgetting the Σm-action (see Proposition A.4.5). So again by adjointness, the co-semifree symmetricspectrum PmL is injective.

Proposition 4.3. Let X be an injective spectrum.(i) For every monomorphism i : A −→ B of symmetric spectra of simplicial sets the map map(i,X) :

map(B,X) −→ map(A,X) is a Kan fibration. If in addition i is a level equivalence, thenmap(i,X) is a weak equivalence.

(ii) For every symmetric spectrum B the function space map(B,X) is a Kan complex and the homotopyrelation for morphisms from B to X is an equivalence relation.

(iii) For every n ≥ 0 the simplicial set Xn is a Kan complex. [even mixed Σn-fibrant ?](iv) For every based simplicial set K the spectrum XK is injective; in particular the loop spectrum ΩX

is injective.(v) The shifted spectrum shX is injective.

Proof. (i) We check that map(i,X) has the right lifting property with respect to every acyclic cofi-bration j : K −→ L of simplicial sets. By the adjunction between the smash pairing and mapping spaces,a lifting problem in the form of a commutative square

K //

j

map(B,X)

map(i,X)

L // map(A,X)

corresponds to a morphism K∧B∪K∧AL∧A −→ X, and a lifting corresponds to a morphism L∧B −→ Xwhich restricts to the previous morphism along the ‘pushout product’ map j∧i : K∧B∪K∧AL∧A −→ L∧B.Since j is an acyclic cofibration and i is a level cofibration, the pushout product morphism j ∧ i is levelwisean acyclic cofibration by the pushout product property in T respectively sS. So the lifting exists since weassumed that X is injective.

The second part is very similar. If i is levelwise an acyclic cofibration, then for every cofibrationj : K −→ L (not necessarily a weak equivalence) of pointed spaces (simplicial sets), the pushout productmap j ∧ i is levelwise an acyclic cofibration. So map(i,X) has the right lifting property with respect to allcofibration pointed spaces (simplicial sets).

Part (ii) is the special case of (i) where A is the trivial spectrum so that map(A,X) is a one-pointsimplicial set. Vertices of the simplicial set map(B,X) correspond bijectively to morphisms B −→ X insuch a way that 1-simplices correspond to homotopies. So the second claim of (ii) follows since in everyKan complex, the relation x ∼ y on vertices defined by existence of a 1-simplex z with d0z = x and d1z = yis an equivalence relation.

The simplicial set Xn is naturally isomorphic to the mapping space map(FnS0, X) with source the freesymmetric spectrum generated by S0 in level n. So (iii) is a special case of (ii).

Properties (iv) and (v) can be viewed as special cases of a more general fact. Suppose that R is anendofunctor of the category of symmetric spectra of simplicial sets which has a left adjoint L. Supposemoreover that L preserves monomorphisms and level equivalence. Then RX is injective for every injectivespectrum X. Indeed, given a monomorphism i : A −→ B which is also a level equivalence and everymorphism f : A −→ RX, then Li : LA −→ LB is a monomorphism and a level equivalence, so the adjointf : LA −→ X has an extension g : LB −→ X with f = g (Li). Then adjoint g : B −→ RX is thenthe required extension of f . For part (iv) we apply this argument to the adjoint pair (K ∧ −, (−)K), forpart (v) to the adjoint pair (., sh).

We now get a criterion for level equivalence by testing against injective spectra. The criterion involveshomotopy classes of morphisms,which we define first.

48 I. BASICS

Definition 4.4 (Homotopy relation). Two morphisms of symmetric spectra f0, f1 : A→ X are calledhomotopic if there is a morphism

H : I+ ∧A −→ X ,

called a homotopy, such that f0 = H i0, and f1 = H i1. Here I is either the unit interval [0, 1] when weare in the context of symmetric spectra of spaces, or the simplicial 1-simplex ∆[1] when in the context ofsimplicial sets. The morphisms ij : A −→ I+ ∧A for j = 0, 1 are the ‘end point inclusions’ which are givenlevelwise by ij(a) = j∧a (in the context of spaces) or are induced by the face morphisms dj : ∆[0] −→ ∆[1](in the context of simplicial sets) and the identification A ∼= ∆[0]+ ∧A.

For symmetric spectra of spaces, homotopy is an equivalence relation; For symmetric spectra of simpli-cial sets, ‘homotopy’ is in general neither symmetric nor transitive. The homotopy relation is an equivalencerelation when the target is injective, see Proposition 4.3 (ii) below. In any case we denote by [A,X] the setof homotopy classes of morphisms from A to X, i.e., the classes under the equivalence relation generatedby homotopy. The natural morphisms ∆[1]+ ∧ S(X) −→ S([0, 1]+ ∧ X) and [0, 1]+ ∧ |Y | ∼= |∆[1]+ ∧ Y |,compatible with the end point inclusions, show that the singular complex and geometric realization functorpreserve the homotopy relation.

By the adjunction (3.5) a homotopy from f0 to f1 can equivalently be given in the adjoint form as amorphism H : A −→ XI+ such that ev0 H = f0 and ev1 H = f1 where evj : XI+ −→ X for j = 0, 1 isgiven levelwise by evaluation at j ∈ [0, 1] (respectively the two vertices of ∆[1])

A homotopy between spectrum morphisms is really the same as levelwise pointed homotopies between(f0)n and (f1)n : An → Xn compatible with the Σn-actions and structure maps. In particular, homotopicmorphisms induce the same map of naive homotopy groups.

A morphism f : A −→ B of symmetric spectra is a homotopy equivalence if there exists a morphismg : B −→ A such that gf and fg are homotopic to the respective identity morphisms. Hence every homotopyequivalence of symmetric spectra is in particular levelwise a homotopy equivalence of spaces or simplicialsets, thus a level equivalence, but the converse is not true in general.

Yet another way to look at homotopies is via the mapping space: a homotopy H : I+ ∧ A −→ X isadjoint to a morphism of (unbased) spaces of simplicial sets I −→ map(A,X), which is either a path or a1-simplex in the mapping space. So in the context of spectra of spaces, two morphisms are homotopy ifand only if they lie in the same path component of the mapping space map(A,X). For spectra of simplicialsets, a morphism f0 is homotopic to a morphism f1 if and only if there exists a 1-simplex H ∈ map(A,X)1satisfying d0(H) = f0 and d1(H) = f1.

Proposition 4.5. A morphism f : A −→ B of symmetric spectra of simplicial sets is a level equivalenceif and only if for every injective spectrum X the induced map [f,X] : [B,X] −→ [A,X] on homotopy classesof morphisms is bijective. Every level equivalence between injective spectra of simplicial sets is a homotopyequivalence.

Proof. Suppose first that f is a level equivalence. We replace f by the inclusion of A into the mappingcylinder of f , which is homotopy equivalent to B. This way we can assume without loss of generality that fis a monomorphism. By part (i) of Proposition 4.3 the map map(f,X) : map(B,X) −→ map(A,X) is then aweak equivalence of simplicial sets, so in particular a bijection of components. Since π0 map(B,X) ∼= [B,X],and similarly for A, this proves the claim.

Now suppose conversely that [f,X] : [B,X] −→ [A,X] is bijective for every injective spectrum X. If Kis a pointed Kan complex and m ≥ 0, then the co-free symmetric spectrum RmK of Example 4.2 is injective.The adjunction for morphisms and homotopies provides a natural bijection [A,RmK] ∼= [Am,K]sS to thebased homotopy classes of morphisms of simplicial sets. So for every Kan complex K, the induced map[fm, X] : [Bm,K] −→ [Am,K] is bijective, which is equivalent to fn being a weak equivalence of simplicialsets. Since this holds for all m, the morphism f is a level equivalence.

Now consider a level equivalence f : A −→ B with A and B injective; we obtain a homotopy inverseg : B −→ A by the following standard representability argument. Since [f,A] : [B,A] −→ [A,A] isbijective there is a morphism g : B −→ A such that gf : A −→ A is homotopic to the identity. Since


[f,B] : [B,B] −→ [B,A] is a bijection which takes fg : B −→ B and IdB to the homotopy class of f , themorphism fg is homotopic to the identity of B.

The following proposition shows that the property of being injective is not too restrictive, and that itis no restriction at all up to level equivalence.

Proposition 4.6. There exists a endofunctor (−)inj : SpsS −→ SpsS on the category of symmetricspectra of simplicial sets and a natural level equivalence A −→ Ainj such that Ainj is an injective spectrum.

We use the small object argument (see Theorem 1.7 of Appendix) with respect to a certain class ofmorphisms of symmetric spectra. As usual with small object arguments we have to limit the size of objects.We call a symmetric spectrum of simplicial sets countable if the cardinality of the disjoint union of allsimplices in all levels is countable.

Lemma 4.7. A symmetric spectrum of simplicial sets X is injective if and only if for every monomor-phism i : A −→ B which is also a level equivalence with B countable and every morphism f : A −→ X thereexists an extension g : B −→ X with f = gi.

Proof. Suppose that X has the extension property with respect to all injective level equivalences withcountable target (and hence source). Consider a general injective level equivalence i : A −→ B, with norestriction on the cardinality of B, and a morphism f : A −→ X which we want to extend to B.

We denote by P the set of ‘partial extensions’: an element of P is a pair (U, h) consisting of a symmetricsubspectrum U of B which contains the image of A and such that the inclusion U −→ B (and hence themorphism A −→ U) is a level equivalence and a morphism h : U −→ X which extends f : A −→ X.The set P can be partially ordered by declaring (U, h) ≤ (U ′, h′) if U is contained in U ′ and h′ extends h.Then every chain in P has an upper bound, namely the union of all the subspectra U with the commonextension of the morphisms h. [uses that transfinite composite of injective level equivalences is an injectivelevel equivalence] So by Zorn’s lemma, the set P has a maximal element (V, k). We show that V = B, so kprovides the required extension of f , showing that the spectrum X is injective.

We argue by contradiction and suppose that V is strictly smaller than B. Then we can find a countablesubspectrum W of B which is not contained in V and such that the inclusion V ∩W −→ W is a levelequivalence [justify; the argument is in Lemmas 5.1.6 and 5.1.7 of [33]]. Since W is countable, the restrictionof k : V −→ X to the intersection V ∩W can be extended to a morphism g : W −→ X. The morphisms gand k together then provide an extension k ∪ g : V ∪W −→ X of k, which contradicts the assumption that(V, k) is a maximal element in the set P extensions.

Proof of Proposition 4.6. Let I be a set containing one morphism from every isomorphism classof injective level equivalences i : A −→ B for which B is a countable symmetric spectrum. The class ofinjective level equivalences of symmetric spectra is closed under wedges, cobase change and composition,possibly transfinite. So every I-cell complex is an injective level equivalence.

We apply the small object argument (see Theorem 1.7 of Appendix A) to the unique morphism from agiven symmetric spectrum A to the trivial spectrum. We obtain a functor A 7→ Ainj together with a naturaltransformation j : A −→ Ainj which is an I-cell complex, hence an injective level equivalence. Moreover,the morphism from Ainj to the trivial spectrum is I-injective. Lemma 4.7 shows that Ainj is an injectivespectrum.

4.2. Stable equivalences. We first define stable equivalences for symmetric spectra of simplicial sets,and then symmetric spectra of spaces.

Definition 4.8. A morphism f : A −→ B of symmetric spectra of simplicial sets is a stable equivalenceif for every injective Ω-spectrum X the induced map

[f,X] : [B,X] −→ [A,X]

of homotopy classes of spectrum morphisms is bijective.A morphism f of symmetric spectra of topological spaces is a stable equivalence if the singular complex

S(f) : S(A) −→ S(B) is a stable equivalence in the previous sense.

50 I. BASICS

Proposition 4.5 immediately implies that every level equivalence of symmetric spectra is a stable equiv-alence. Theorem 4.16 below shows that more generally every π∗-isomorphism is a stable equivalence.

Lemma 4.9. (i) Let A be a symmetric spectrum of simplicial sets, Y a symmetric spectrum of spacesand h : |A| −→ Y a morphism. Then h is a stable equivalence if and only if its adjoint h : Y −→ S(Y ) is astable equivalence.(ii) A morphism of symmetric spectra of simplicial sets is a stable equivalence if and only if its geometricrealization is a stable equivalence of symmetric spectra of spaces.

Proof. (i) The composite of the adjunction unit A −→ S|A| and S(h) : S|A| −→ S(Y ) is the adjoint h.Since the adjunction unit is a level, hence stable, equivalence, h is a stable equivalence if and only if S(h)is. But the latter is equivalent, by definition, to h being a stable equivalence.

(ii) A morphism g : A −→ B of symmetric spectra of simplicial sets is a stable equivalence if and onlyits composite with the adjunction unit η : B −→ S|B| is a stable equivalence (since η is a level, hencestable, equivalence). By (i), the composite ηg : A −→ S|B| is a stable equivalence if and only if its adjointηg = |g| : |A| −→ |B| is.

Proposition 4.10. Every stable equivalence between Ω-spectra is a level equivalence.

Proof. A spectrum of spaces is an Ω-spectrum if and only if its singular complex is an Ω-spectrum ofsimplicial sets. So the case of spaces is a direct consequence of the case of simplicial sets.

For symmetric spectra of simplicial sets we argue as follows. Let f : X −→ Y be a stable equivalencebetween Ω-spectra. By Proposition 4.6 we can assume that X and Y are even injective Ω-spectra. Since[f,X] : [Y,X] −→ [X,X] is bijective there exists a morphism g : Y −→ X such that gf is homotopic to theidentity of X. Since fgf is homotopic to f and [f, Y ] : [Y, Y ] −→ [X,Y ] is bijective, we conclude that fgis homotopic to the identity of Y . So f is a homotopy equivalence, in particular a level equivalence.

We establish some useful criteria for stable equivalences:

Proposition 4.11. For every morphism f : A −→ B of symmetric spectra of simplicial sets thefollowing are equivalent:

(i) f is a stable equivalence;(ii) for every injective Ω-spectrum X the induced map map(f,X) : map(B,X) −→ map(A,X) is a

weak equivalence of simplicial sets;(iii) for every injective Ω-spectrum X the induced map Hom(f,X) : Hom(B,X) −→ Hom(A,X) is a

level equivalence of symmetric spectra;(iv) the mapping cone C(f) of f is stably equivalent to the trivial spectrum;

Proof. (i)⇒(ii) For every simplicial set K and every injective Ω-spectrum X the function spectrumXK is again injective by Proposition 4.3 (iv) and an Ω-spectrum by Example 3.4. We have an adjunctionbijection [K,map(A,X)] ∼= [A,XK ] where the left hand side means homotopy classes of morphisms of simpli-cial sets. So if f is a stable equivalence, then [f,XK ] is bijective, hence [K,map(f,X)] : [K,map(B,X)] −→[K,map(B,X)] is bijective. Since this holds for all simplicial sets K, map(f,X) is a weak equivalence.

(ii)⇒(iii) For every injective Ω-spectrum X and n ≥ 0 the shifted spectrum shnX is again injectiveProposition 4.3 (v)) and an Ω-spectrum. So if f : A −→ B satisfies (ii), it also satisfies (iii) since the nthlevel of the spectrum Hom(A,X) is defined as map(A, shnX).

Condition (iii) implies (i) since the set [A,X] of homotopy classes of spectrum morphisms equals theset of path components π0 map(A,X) = π0(Hom(A,X)0).

(iii)⇒(iv) Let X be an injective Ω-spectrum. The simplicial set map(C(f), X) is isomorphic tothe homotopy fibre of the morphism Hom(f,X)0 : Hom(B,X)0 −→ Hom(A,X)0 between Kan sim-plicial sets. So if condition (iii) holds, then map(C(f), X) is contractible. In particular, the set[C(f), X] = π0 (Hom(C(f), X)0) contains only one element, so the morphism from C(f) to the trivialspectrum is a stable equivalence.

(iv)⇒(i)


For this purpose we introduce a device which attempts to turn a symmetric spectrum into an Ω-spectrum while keeping the naive homotopy groups; however, the attempt is not always successful. We setRX = Ω(shX) and obtain a morphism λX : X −→ RX with n-th level the composite

Xnσn−−→ Ω(Xn+1)

Ω(χn,1)−−−−−→ Ω(X1+n) = (Ω(shX))n .

The morphism λX is in fact adjoint to the morphism λX : S1 ∧X −→ shX defined in (3.9).We iterate this construction and let R∞X be the mapping telescope (see Example 2.23) of the sequence

(4.12) XλX−−→ RX

R(λX)−−−−−→ R2XR2(λX)−−−−−−→ · · · .

This construction comes with a canonical natural morphism λ∞X : X −→ R∞X, the embedding of the initialterm into the mapping telescope.

We can identify the (unstable) homotopy groups of the levels of the spectrum R∞X with the (stable)naive homotopy groups of X. A word of warning: while the next proposition shows that the groupsπk+n(R∞X)n and πk+n+1(R∞X)n+1 are isomorphic, the spectrum R∞X is in general not an Ω-spectrumsince the isomorphism obtained from the proposition need not coincide with the stabilization map of thespectrum R∞X. In fact, R∞X is an Ω-spectrum if and only if X is semistable, see Theorem 7.27 below.

Proposition 4.13. Let X be a symmetric spectrum of simplicial sets which is levelwise Kan, k aninteger and n ≥ 0 such that k + n ≥ 0. There is a natural bijection

bnX : πkX = πk+n(shnX)∼=−−→ πk+n(R∞X)n .

Proof. Given a based space Z, we recall the adjunction bijection (0.3)

a : πk(ΩmZ) ∼= πk+mZ

which takes the homotopy class of a based map f : Sk −→ ΩmZ to the homotopy class of a(f) : Sk+m −→ Zgiven by a(f)(t ∧ s) = f(t)(s) for t ∈ Sk, s ∈ Sm [spaces vs. ssets...]. We obtain a bijection fromπk+n+mXn+m to πk+n(RmX)n as the composite

(4.14) πk+n+mXn+ma−−→ πk+n(ΩmXn+m)

Ωm(χn,m)∗−−−−−−−→ πk+n(ΩmXm+n) = πk+n(RmX)n .

For every m ≥ 0 the diagram

πk+n+mXn+ma //

ι∗

πk+n(ΩmXn+m)Ωm(χn.m)∗ // πk+n(Ωm(Xm+n)) πk+n(RmX)n

πk+nRm(λ∗X)n

πk+n+m+1Xn+m+1 a

// πk+n(Ωm+1Xn+m+1)Ωm+1(χn,m+1)

// πk+n(Ωm+1Xm+1+n) πk+n(Rm+1X)n

commutes, where ι∗ is the stabilization map. Indeed, if f : Sk+n+m −→ Xn+m represents an element of thesource πk+n+mXn+m, then the diagram

Sk+na(f) //

a(f∧S1)

a(ι∗f)

''

ΩmXn+mΩm(χn,m) // ΩmXm+n

Ωm(σm+n)

Rm(λ∗X)n

ww

Ωm+1(Xn+m ∧ S1)

Ωm+1(σn+m)

Ωm(ΩXm+n+1)

Ωm+1(m×χn,1)

Ωm+1Xn+m+1Ωm+1(χn,m+1)

//

Ωm+1(χn,m×1)

22eeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeΩm+1Xm+1+n

commutes, and the two images in πk+n(Rm+1X)n are the two outmost composites. So we can pass tocolimits over m in the vertical direction and obtain a natural bijection bnX as the composite

πkX = colimm πk+n+mXn+m −→ colimm πk+nRm(λ∗X)n −→ πk+nR

∞(λ∗X)n ;

52 I. BASICS

the second map is bijective because homotopy groups commute with filtered colimits.

Proposition 4.15. Let X be a symmetric Ω-spectrum of simplicial sets. Then the morphism λ∞X :X −→ R∞X is a level equivalence.

Proof. The n-th level of the morphism λX is the composite of the adjoint structure map σn : Xn −→ΩXn+1 and an isomorphism. So if X is an Ω-spectrum, then the morphism λX is a level equivalence. Sincethe shift and loop functors preserves level equivalences between spectra which are levelwise Kan. So allthe morphisms Rn(λX) are then level equivalences, hence so is the inclusions into the mapping telescopeλ∞X : X −→ R∞X [ref to App]

Theorem 4.16. Every π∗-isomorphism of symmetric spectra is a stable equivalence.

Proof. We treat the context of spectra of simplicial sets first. We start with an observation for aninjective Ω-spectrum X. Proposition 4.15 shows that the morphism λ∞X : X −→ R∞X is a level equivalence.Since X is an injective spectrum the map [λ∞X , X] : [R∞X,X] −→ [X,X] is bijective by Proposition 4.5.So there exists a morphism r : R∞X −→ X such that the composite rλ∞ is homotopic to the identity ofX (the other composite need not be homotopic to the identity of R∞X).

Now we establish the following special case of the theorem: let C be a symmetric spectrum of simplicialsets which is levelwise Kan and all of whose naive homotopy groups vanish. We show that then C isstably contractible. By Proposition 4.13 and the hypothesis on C all homotopy groups of the simplicial set(R∞C)n are trivial for every level n, so the symmetric spectrum R∞C is levelwise weakly contractible. ByProposition 4.5 the set [R∞C,X] has just one element for every injective Ω-spectrum X.

Shifting, looping and taking mapping telescopes are constructions which preserves the homotopy rela-tion, hence so does the functor R∞. So we can define a map

[C,X] −→ [R∞C,X] by [ϕ] 7→ [r R∞ϕ] .

There also is a map [R∞C,X] −→ [C,X] in the other direction given by precomposition with λ∞C : C −→R∞C. Since r is a retraction (up to homotopy) to λ∞X , the composite of the two natural maps is the identityon [C,X]. Since the set [R∞C,X] has only one element, the same is true for the set [C,X]. Since X is anarbitrary injective Ω-spectrum, C is indeed stably contractible.

Now we consider a symmetric spectrum C of simplicial sets, not necessarily levelwise Kan, such thatall naive homotopy groups of C vanish. We show that then C is stably contractible. We apply the functors‘geometric realization’ and ‘singular complex’ to replace C by the level equivalent spectrum S(|C|) whichis levelwise Kan and has trivial naive homotopy groups. So S(|C|) is stably contractible by the case above;since level equivalences are stable equivalences, C is stably contractible.

Now we prove the theorem. Let f : A −→ B be a π∗-isomorphism and let C(f) be its mapping cone.By the long exact of naive homotopy groups (see Proposition 2.16; taking mapping cones commutes withrealization) the naive homotopy groups of C(f) are trivial. By the previous paragraph, C(f) is stablycontractible, so f is a stable equivalence by the criterion (iv) of Proposition 4.11.

[spaces...]

Here is an important example of a stable equivalence which is not a π∗-isomorphism,

Example 4.17. Let λ : F1S1 −→ F0S

0 = S denote the morphism which is adjoint to the identity inlevel 1. For any injective Ω-spectrum X we consider the commutative square

[F0S0, X]

[λ,X] //

eval0 ∼=

[F1S1, X]

eval1∼=

π0(X0) ι// π1(X1)

The vertical maps given by evaluation at levels 0 respectively 1 are adjunction bijections. Since X is anΩ-spectrum, the stabilization map ι : π0(X0) −→ π1(X1) is bijective. So the map [λ,X] is bijective and λis indeed a stable equivalence.


In Example 3.11 we determined the 0-th naive homotopy group of F1S1 as an infinitely generated free

abelian group, whereas π0S is free abelian of rank 1. Thus the morphism λ is not a π∗-isomorphism.

The stable equivalence λ : F1S1 −→ S of the previous example is only one special case of a whole class of

stable equivalences (which are typically not π∗-isomorphisms either). For every symmetric spectrum A weintroduced a morphism λA : S1 ∧A −→ shA. Using the adjunction between suspension and loop we get anadjoint morphism λA : A −→ Ω(shA) = sh(ΩA). Even though sources and target of these two morphismshave abstractly isomorphic naive homotopy groups, λA and λA are not in general π∗-isomorphisms.

Using the adjunction (., sh) we can adjoin λA and λ to get two more natural morphisms

λA : .(S1 ∧A) −→ A and λA : .A −→ ΩA .

[make explicit?]. For the sphere spectrum, the morphism λS specializes to the morphism λ : F1S1 −→ S,

under the identification F1S1 ∼= .(S1 ∧ S).

Proposition 4.18. For every symmetric spectrum A the morphism λA : .(S1 ∧ A) −→ A is a stableequivalence. In the context of topological spaces, or if A is levelwise Kan, then the morphism λA : .A −→ ΩAis a stable equivalence.

Proof. We first treat the case of symmetric spectra of simplicial sets. For every symmetric spectrumX the adjunction bijections for the pairs (., sh) and (S1 ∧ −,Ω), applied to morphisms and homotopies,provide natural bijections

[.(S1 ∧A), X] ∼= [A,Ω(shX)] .

Under this correspondence the map [λA, X] : [A,X] −→ [.(S1∧A), X] becomes the map [A, λX ] : [A,X] −→[A,Ω(shX)]. If X is an injective Ω-spectrum, then λX : X −→ Ω(shX) is a level equivalence betweeninjective spectra, hence a homotopy equivalence. So [A, λX ], and consequently also [λA, X] is bijective,which proves that λA is a stable equivalence.

For symmetric spectra of spaces A we reduce to the previous case as follows. By the above the morphismλS(A) : .(S1 ∧ S(A)) −→ S(A) is a stable equivalence, where S(A) is the singular complex of A. In thecommutative diagram

.(S1 ∧A)λA //

A

.(S1 ∧ |S(A)|)

∼= // | . (S1 ∧ S(A))||λS(A)|

// |S(A)|

both vertical maps are stable equivalences (since suspensions and . preserve stable equivalences byLemma ??).

For the second statement we suppose that we are in the topological context or A is levelwise Kan. Theadjunction counit ε : S1 ∧ ΩA −→ A is then a π∗-isomorphism (Proposition 2.4). Since the shift adjoint. preserves π∗-isomorphisms (compare Example 3.10), the morphism .(εA) is a π∗-isomorphism, hence astable equivalence. The morphism λΩA is a stable equivalence by the first part, so the commutative triangle

.(S1 ∧ ΩA)λΩA //

.(εA) %%KKKKKKKKKK ΩA

.A

λA

==

shows that λA is a stable equivalence.

Our next task is to develop various equivalent characterizations for stable equivalences.

Proposition 4.19. For every morphism f : A −→ B of symmetric spectra the following are equivalent:(i) f is a stable equivalence;

54 I. BASICS

(ii) the mapping cone C(f) of f is stably equivalent to the trivial spectrum;(iii) the suspension S1 ∧ f : S1 ∧A −→ S1 ∧B is a stable equivalence;(iv) the shift adjoint .f : .A −→ .B is a stable equivalence.

In the context of spaces, or if A and B are levelwise Kan, then conditions (i)-(iv) are also equivalent to thefollowing two conditions:

(v) the homotopy fiber F (f) of f is stably equivalent to the trivial spectrum;(vi) the loop Ωf : ΩA −→ ΩB is a stable equivalence.

Proof. We start by showing the equivalence of conditions (i) through (iv) for symmetric spectra ofsimplicial sets. The equivalence of conditions (i) and (ii) is contained in Proposition 4.11.

(ii)⇒(iii) If the mapping cone C(f) is stable contractible, then for every injective Ω-spectrum X thesimplicial set map(C(f), X) is contractible by Proposition 4.11. The simplicial set map(C(f), X) is isomor-phic to the homotopy fibre of the morphism map(f,X) : map(B,X) −→ map(A,X) between Kan simplicialsets, so map(f,X) induces isomorphisms on positive dimensional homotopy groups and an injection on pathcomponents. If we loop once, we get a weak equivalence Ω map(f,X) : Ω map(B,X) −→ Ω map(A,X).Since Ω map(A,X) is naturally isomorphic to map(S1 ∧A,X), we conclude that map(S1 ∧ f,X) is a weakequivalence. So S1 ∧ f is a stable equivalence by Proposition 4.11.

(vi)⇒(i) If .f is a stable equivalence, so is the suspension S1 ∧ (.f) since we already know thatcondition (i) implies (iii). The spectra S1 ∧ (.A) and .(S1 ∧A) are naturally isomorphic, so the morphism.(S1∧f) is a stable equivalence. Because the natural morphism λA : .(S1∧A) −→ A is a stable equivalence(Proposition 4.18), the morphism f is then a stable equivalence.

(i)⇒(iv) Let X be an injective Ω-spectrum. Then the shifted spectrum shX is another injective Ω-spectrum (Proposition 4.3 (v)), and so [f, shX] : [B, shX] −→ [A, shX] is bijective. The set [A, shX]is naturally bijective to [.A,X], so we conclude that [.f,X] : [.B,X] −→ [.A,X] is bijective for everyΩ-spectrum X. So .f is a stable equivalence.

(iii)⇒(i) Suppose that S1 ∧ f is a stable equivalence. By the already established implication (i)⇒(iv)we know that then .(S1 ∧ f) is a stable equivalence. So f is a stable equivalence by Proposition 4.18.

For the rest of the proof we assume that we are in the context of spaces or A and B are levelwise Kan.(ii)⇔(v) By the already established equivalence between conditions (ii) and (iii), the homotopy fiber

F (f) is stably contractible if and only if its suspensions S1∧F (f) is stably contractible. By Proposition 2.16the spectrum S1 ∧ F (f) is π∗-isomorphic to the mapping C(f). By Theorem 4.16 the spectra S1 ∧ F (f)and C(f) are then stably equivalent. Altogether this proves that F (f) is a stable contractible if and onlyif C(f) is.

Conditions (iv) and (vi) are equivalent because the natural morphism λA : .A −→ ΩA is a stableequivalence by Proposition 4.18.

Remark 4.20. As far as I can see, S1 ∧ − does not in general preserve weak equivalences of spaces,hence not level equivalences of symmetric spectra of spaces [give example]. However, S1 ∧ − preserves allstable equivalences in the topological context.

For every pair of symmetric spectra of simplicial sets A and B the canonical morphism A∨B −→ A×Bis a π∗-isomorphism by Corollary 2.20. So Theorem 4.16 immediately implies

Corollary 4.21. For every pair of symmetric spectra A and B the canonical morphism A∨B −→ A×Bis a stable equivalence.

For the mapping telescope and the diagonal of a sequence of symmetric spectra, see Example 2.23.

Proposition 4.22. (i) A wedge of stable equivalences is a stable equivalence.(ii) A finite product of stable equivalences is a stable equivalence.


(iii) Consider a commutative square of morphisms of symmetric spectra

Af //

ϕA

B

ϕB

A′

f ′// B′

and let ϕC : C(f) −→ C(f ′) be the map induced by ϕA and ϕB on mapping cones. Then if two ofthe three morphisms ϕA, ϕB and ϕC are stable equivalences, so is the third.

(iv) Consider a commutative of symmetric spectra of simplicial sets

Ai //

f

B

g

C

j// D

Suppose that one of the following conditions holds:(a) the square is a pushout and i or f is injective.(b) the square is a pullback and j or g is levelwise a (Serre respectively Kan) fibration.Then f is a stable equivalence if and only if g is.

(v) For every m ≥ 0 and every based Σm-simplicial set L the twisted smash product functor L .m −preserves stable equivalences. In particular, for every simplicial set K the functor K ∧− preservesa stable equivalences.

(vi) Let K be a finite based CW-complex respectively a finite based simplicial set, and f : A −→ Ba stable equivalence. Suppose that A and B are levelwise Kan complexes when in the simplicialcontext. Then the morphism fK : AK −→ BK is a stable equivalence.

(vii) We consider a commutative diagram of symmetric spectra

A0f0

//

ϕ0

A1f1

//

ϕ1

A2f2

//

ϕ2

A3 · · ·

ϕ3

B0

g0// B1

g1// B2

g2// B3 · · ·

in which all vertical morphisms ϕn : An −→ Bn are stable equivalences. Then the map teln ϕn :telnAn −→ telnBn induced on mapping telescopes and the map diagn ϕn : diagnAn −→ diagnBn

induced on the diagonal symmetric spectrum are also stable equivalences. In the context of simpli-cial sets, the map colimn≥0 ϕ

n : colimn≥0An −→ colimn≥0B

n induced on colimits is also a stableequivalence.

(viii) Let fn : An −→ An+1 for n ≥ 0 be a sequence of composable stable equivalences of symmetricspectra of simplicial sets. Then canonical morphism A0 −→ telnAn to the mapping telescope is astable equivalence. In the context of simplicial sets, the canonical morphism A0 −→ colimn≥0A

n

to the colimit is a stable equivalence.(ix) Let I be a filtered category and let A,B : I −→ Sp be functors which take all morphisms in I

to monomorphisms of symmetric spectra. If τ : A −→ B is a natural transformation such thatτ(i) : A(i) −→ B(i) is a stable equivalence for every object i of I, then the induced morphismcolimI τ : colimI A −→ colimI B on colimits is a stable equivalence.

(x) Let G be a group and f : A −→ B an G-equivariant morphism of G-symmetric spectra. If theunderlying morphism of symmetric spectra is a stable equivalence, then so is the induced morphismfhG : AhG −→ BhG on homotopy orbit spectra. equivalence.

56 I. BASICS

Proof. (i) For symmetric spectra of simplicial sets, we argue from the definition: for every familyAii∈I of symmetric spectra and every injective Ω-spectrum X the natural map

[∨i∈I

Ai, X] −→∏i∈I

[Ai, X]

is bijective by the universal property of the wedge, applied to morphisms and homotopies. A wedge ofstable equivalences is a stable equivalence.

For symmetric spectra of spaces we note that for every family Aii∈I of symmetric spectra the canonicalmap

∨i∈I S(Ai) −→ S(

∨i∈I A

i) is a π∗-isomorphism because naive homotopy groups take wedges to sums(Corollary 2.20) and taking singular complex does not chance the naive homotopy groups. This reducesthe claim for spaces to the claim for simplicial sets.

(ii) Finite products are stably equivalent to finite products by Corollary 4.21, so the claim follows frompart (i).

(iii) We start with the special case where A′ and B′ are trivial spectra. In other words, we show firstthat given any morphism of symmetric spectra f : A −→ B, then if two of the spectra A, B and themapping cone C(f) are stably contractible, then so is the third.

[simplicial sets] We will use Proposition 4.11 repeatedly, namely that a morphism f of symmetricspectra of simplicial sets is a stable equivalence if and only if the map map(f,X) is a weak equivalence ofsimplicial sets for every for every injective Ω-spectrum X. For every injective Ω-spectrum X the mappingspace map(C(f), X) is isomorphic to the homotopy fiber of map(f,X) : map(B,X) −→ map(A,X). If Ais stably contractible, then map(A,X) is contractibly, and so map(C(f), X) −→ map(B,X) induced bythe inclusion i : B −→ C(f) is a stable equivalence. So B is stably contractible if and only if C(f) isstably contractible. One case remains, namely when B amd C(f) are assumed to be stably contractible.Since the inclusion i : B −→ C(f) is a monomorphism and X is injective, it induces a Kan fibrationmap(i,X) : map(C(f), X) −→ map(B,X) (by Proposition 4.3 (i)). Since the quotient spectrum C(f)/i(B)is isomorphic to S1 ∧A, the fibre of map(i,X) is isomorphic to map(S1 ∧A,X), which is contractible sincemap(C(f), X) and map(B,X) are. Then the suspension S1 ∧ A is stably contractible, hence so is A byProposition 4.19.

In the general case we exploit that the order in which we take iterated mapping cones in a commutativesquare does not matter. More precisely, the mapping cone of the morphism ϕC : C(f) −→ C(f ′) isisomorphic to the mapping cone of the morphism f ′′ : C(ϕA) −→ C(ϕB) induced by f and f ′. [...] ByProposition 4.19, a morphism is a stable equivalence if and only if its mapping cone if stably contractible.So the general case follows by applying the special case to the morphism f ′′ : C(ϕA) −→ C(ϕB).

(iv) We start with case (a) of a pushout square. If f (and hence g) is injective, then the symmetricspectrum of strict cofibers C/f(A) is level equivalent to the mapping cone F (f), and similarly for g. Sincethe square is a pushout, the strict cofibers are isomorphic. So the mapping cones F (f) and F (g) arelevel equivalent. The criterion (ii) of Proposition 4.19 shows that f is a stable equivalence if and onlyif g is. If i (and hence j) is injective, we use ‘properness’ to reduce to the previous case. We choose afactorization [mapping cylinder?] g = ψf ′ where f ′ : A −→ C ′ is injective and ψ : C ′ −→ C is a levelequivalence. The morphism f ′ is then a stable equivalence if and only if f is, so by the above the pushouti∗(f ′) : B −→ C ′∪AB is a stable equivalence if and only if f is. Now g factors as the composite of the levelequivalence (by properness) jψ ∪ g : C ′ ∪A B −→ D and the morphism i∗(f ′), so g is a stable equivalenceif and only if f is.

The case (b) of a pullback square is strictly dual. If g (and hence f) is a level fibration, then thesymmetric spectrum of strict fibers is level equivalent to the homotopy fiber F (g), and similarly for f .Since the square is a pullback, the strict fibers are isomorphic. So the homotopy fibers F (g) and F (f)are level equivalent. The criterion (v) of Proposition 4.19 shows that g is a stable equivalence if and onlyif f is. If j (and hence i) is a level fibration, we use ‘properness’ to reduce to the previous case. Wechoose a factorization g = g′ϕ where ϕ : B −→ B′ is a level equivalence and g′ : B′ −→ D is a levelfibration. The morphism g′ is then a stable equivalence if and only if g is, so by the above the pullbackj∗(g′) : C×DB′ −→ C is a stable equivalence if and only if g is. Now f factors as the composite of the level


equivalence (by properness) (f, ϕi) : A −→ C ×D B′ and the morphism j∗(g′), so f is a stable equivalenceif and only if g is.

(v) [simplicial sets] The twisted smash product functor L .m − preserves monomorphisms and levelequivalences, so its right adjoint X 7→ Hom(L, shmX)Σm preserves injective spectra. [also Ω-spectra???]

If f : A −→ B is a stable equivalence, then for every injective Ω-spectrum the map[f,Hom(L, shmX)Σm ] : [B,Hom(L, shmX)Σm ] −→ [A,Hom(L, shmX)Σm ] is bijective. By adjointnessfor morphisms and homotopies, this means that the map [L .m f,X] : [L .m B,X] −→ [L .m A,X] isbijective. So L .m f is a stable equivalence. [The first statement follows from the adjunction bijection[K ∧A,X] ∼= [A,XK ] and the fact that XK is an injective Ω-spectrum whenever K is.]

[spaces] For every symmetric spectrum of simplicial sets A the adjunction counit |S(A)| −→ A is a levelequivalence and so f : A −→ B is a π∗-isomorphism if and only if S(f) : S(A) −→ S(B) is a π∗-isomorphism.For symmetric spectra of spaces, stable equivalence were defined after taking singular complexes, so thecase of symmetric spectra of spaces follows.

For a cofibrant based Σm-space L and symmetric spectrum of spaces A, we claim that the morphism

|S(L) .m S(A)| ∼= |S(L)| .m |S(A)| ε.mε−−−→ L .m A

is a stable equivalence. [...] By Lemma 4.9 (i) the adjoint morphism ε . ε : S(L) .m S(A) −→ S(L .m A) isthen also a stable equivalence. If f : A −→ B is a stable equivalence of symmetric spectra of spaces, thenboth vertical morphisms in the commutative square

S(L) .m S(A)S(L).mS(f) //

cε.ε

S(L) .m S(B)

cε.ε

S(L .m A)S(L.mf)

// S(L .m B)

are stable equivalences. Moreover, the upper horizontal morphism is a stable equivalence by the firstparagraph, so the lower morphism is a stable equivalence. Thus L .m f is a stable equivalence, by the verydefinition of that notion for symmetric spectra of spaces.

(vi) For the second statement we observe that the functor A 7→ AK commutes with homotopy fibersand preserves the property of being levelwise Kan. So passage to homotopy fibers and the equivalence ofconditions (i) and (v) in Proposition 4.19 reduces the proof to showing that if A is levelwise Kan and stablycontractible, then so is AK . We prove this by induction over the dimension of K. If K is 0-dimensional,then AK is a finite product of copies of A, this stably contractible by part (ii). Now suppose that K haspositive dimension n. We do another induction on the number of non-degenerate n-simplices of K. Wewrite K = L ∪∂∆[n]+ ∆[n]+ for a simplicial subset L with one non-degenerate n-simplex less than K. Weobtain pullback square of symmetric function spectra

AK //

AL

A∆[n]+ // A∂∆[n]+

in which the horizontal morphisms are restrictions, thus levelwise Kan fibrations. The spectrum A∂∆[n]+

is stably contractible since ∂∆[n]+ has smaller dimension. Since ∆[n]+ is weakly equivalent to S0 thespectrum A∆[n]+ is level equivalent to AS

0 ∼= A, thus stably contractible. In particular, the lower horizontalmorphism is a stable equivalence, hence so is the upper one by part (iv). Since AL is stably contractible byinduction, so is AK .

(vii) [Simplicial case] For every injective Ω-spectrum X the simplicial set map(telnAn, X) is isomorphicto the ‘mapping microscope’ of the tower of Kan [ref] simplicial sets map(fn, X) : map(An+1, X) −→map(An, X). The microscope construction takes sequences of weak equivalences between Kan simplicialsets to weak equivalences. So the map map(teln ϕn, X) is a weak equivalence, and so teln ϕn is a stable

58 I. BASICS

equivalence by Proposition 4.11. Lemma 2.25 relates the mapping telescope telnAn to the diagonal spectrumdiagnAn through a chain of two natural π∗-isomorphisms. Since every π∗-isomorphism is also a stableequivalence, the result for mapping telescopes implies the one for diagonals.

In the simplicial world, the colimit of a sequence is level equivalent to the mapping telescope [ref], hencealso stably equivalent.

(viii) We can reduce statement (viii) to (vi) by comparing the given sequence with the constant sequenceconsist of the spectra A0 and its identity map. The canonical map A0 −→ telnA0 is a level equivalence.

(ix) For every injective Ω-spectrum X the simplicial set map(colimI A,X) is isomorphic to the inverselimit of the functor map(A,X) : Iop −→ sS, and similarly for the functor B. Since A and B consists ofinjective morphisms, all morphisms in the inverse systems map(A,X) and map(B,X) are Kan fibrations(by Proposition 4.3 (i)). Filtered inverse limits of weak equivalences along Kan fibrations are again weakequivalences, so the map map(colimI B,X) −→ map(colimI A,X) is a weak equivalence of simplicial set,which means that colimI A −→ colimI B is a stable equivalence.

(x) [simplicial sets] Use the isomorphism

map(AhG, X) ∼= mapG(EG+,map(A,X)) = map(A,X)hG

and the fact that homotopy fixed points takes underlying weak equivalences between Kan simplicial G-setsto weak equivalences.

Example 4.23. We give another example which shows that stable equivalences are in general badlybehaved with respect to infinite products: we present a stably contractible symmetric spectrum H(A/A•)such that the countably infinite product of copies of X is not stably contractible. This shows that therestriction to finite products in part (ii) of Proposition 4.22 is essential. Since the product

∏n≥1X is the

same a s mapping spectrum from a infinite discreted space (or an infinite constant simplicial set) to X, theexample also shows that the restriction to finite CW-complexes (respectively simplicial sets) is essential inpart (iv) of Proposition 4.22.

For the example we pick an abelian group A and an exhaustive filtration 0 = A0 ⊂ A1 ⊂ A2 ⊂ . . . suchthat An is a proper subgroup of An+1 for all n. We define a modified Eilenberg-Mac Lane spectrum by

Xn = (A/An)[Sn] ,

the linearization, with coefficient group A/An, of the n-sphere (either topological or simplicial). Thesymmetric group acts on Xn through the permutation action on the sphere. The structure map is thecomposite

(A/An)[Sn] ∧ S1 −→ (A/An)[Sn+1] −→ (A/An+1)[Sn] ,where the first is the structure map of the Eilenberg-Mac Lane spectrum for the group A/An (see Exam-ple 1.14) and the second map is induced by the quotient map A/An −→ A/An+1 of coefficient groups. (Thiscan be seen as an example of the more general construction of the Eilenberg-Mac Lane spectrum associatedto an I-functor, compare Exercise 8.27).

As we argued in Example 1.14, the space Xn is an Eilenberg-Mac Lane space of type (A/An, n) (butthe spectrum X is not an Ω-spectrum). So the naive homotopy groups of X are trivial in all non-zerodimensions. The naive homotopy group π0X is isomorphic to the colimit of the sequence of projectionmaps

A = A/A0 −→ A/A1 −→ A/A2 −→ · · ·so π0X is trivial since the groups An exhaust A. Thus X is stably contractible by Theorem 4.16.

Now we calculate the naive homotopy groups of the infinite product XN of copies of X. Again, thenspace (or simplicial set) in level n is an Eilenberg-Mac Lane space of dimension n, so the naive homotopygroups are trivial in all non-zero dimensions. The naive homotopy group π0(XN) is isomorphic to the colimitof the sequence maps

AN = (A/A0)N −→ (A/A1)N −→ (A/A2)N −→ · · ·each of which is in infinite product of projection maps. If we choose a sequence of elements an ∈ An−An−1,for n ≥ 1, then the tuple (an)n ∈ AN = π0X0 does not become zero at any finite stage of the colimit


system, hence it represents a non-trivial element in π0(XN). Since the spectrum X underlies an orthogonalspectrum, it is semistable (compare Proposition 7.28). Since the product XN is semistable and has anon-trivial homotopy group, it is not stably contractible.

Example 4.24. We have mentioned many constructions which preserves stable equivalences, and nowwe also mention one which does not, namely shifting; this should be contrasted with the fact that shiftingdoes preserve π∗-isomorphisms because πk+1(shX) equals πkX as abelian groups. An example is thefundamental stable equivalence λ : F1S

1 −→ S of Example 3.11 which is adjoint to the identity of S1. Thesymmetric spectrum sh(F1S

1) is isomorphic to the wedge of F0S1 = Σ∞S1 and F1S

2, while sh S ∼= F0S1;

the map shλ : sh(F1S1) −→ sh S is the projection to the wedge summand. The complementary summand

F1S2 ∼= S1 ∧ F1S

1 is stably equivalent, via the suspension of λ, to S1 ∧ S ∼= Σ∞S1, and is thus not stablycontractible (for example since π1(Σ∞S1) ∼= π0S is infinite cyclic).

Example 4.25 (Stable homotopy type of free spectra). We show that the free symmetric spectrum FmKgenerated by a based space K in level m is stably equivalent to the m-fold loop of the suspension spectrumof K. To produce a stable equivalence we start with the adjunction unit K −→ Ωm(K∧Sm) = Ωm(Σ∞K)mand form the adjoint morphism

ϕm : FmK −→ Ωm(Σ∞K) .We claim that ϕm is a stable equivalence.

We argue by induction on m, the case m = 0 being clear since ϕ0 is an isomorphism. In general, thesquare

.(FmK)λFmK //

∼=

Ω(FmK)

Ω(ϕm)

F1+mK

ϕ1+m// Ω1+m(Σ∞K)

commutes, where the left morphism was constructed in [...] The morphism λFmK is a stable equivalence byProposition 4.18, the morphism Ω(ϕm) is a stable equivalence by Proposition 4.19 and induction. So ϕ1+m

is a stable equivalence.

Example 4.26 (Stable homotopy type of semifree spectra). We show that the semifree symmetricspectrum GmL generated by a based Σm-space L in level m is stably equivalent to a Σm-homotopy orbitspectrum of the m-fold loop of the suspension spectrum of L.

To produce a stable equivalence we start from the stable equivalence

ϕm : FmL −→ Ωm(Σ∞L)

constructed in the previous Example 4.25. The morphism is Σm-equivariant with respect to the action ofthe symmetric group on the source [...] and on the target by conjugation, using the given action on L andthe coordinate permutations of the sphere Sm. Proposition 4.22 (x) allows us to deduce a stable equivalenceof homotopy orbit spectra

(ϕm)hΣm : (FmL)hΣm −→ (Ωm(Σ∞L))hΣm .

The action of Σm on the free spectrum FmL is free [...], so the natural map

(FmL)hΣm −→ (FmL)Σmfrom homotopy orbits to strict orbits is a level equivalence [ref]. The ortbit spectrum (FmL)Σm , finally, isisomorphic to the semifree spectrum GmL. So altogether we have obtained a chain of two natural stableequivalences linking the semifree spectrum GmL to the homotopy orbit spectrum (Ωm(Σ∞L))hΣm .

We close this section with a construction which shows that up to stable equivalence, every symmetricspectrum can be replaced by an Ω-spectrum. We construct a functor Q : Sp −→ Sp with values in Ω-spectratogether with a natural stable equivalence ηA : A −→ QA. The main construction is with symmetric spectraof spaces. We use a special case of the small object argument with respect to a certain set K of stable

60 I. BASICS

equivalences. First we let λn : Fn+1S1 −→ FnS

0 denote the morphism which is adjoint to the wedgesummand inclusion S1 −→ (FnS0)n+1 = Σ+

n+1 ∧ S1 indexed by the identity element. The morphism λnfactors through the mapping cylinder as λn = rncn where cn : Fn+1S

1 −→ Z(λn) = ([0, 1]+ ∧ Fn+1S1) ∪λ

FnS0 is the ‘front’ mapping cylinder inclusion and rn : Z(λn) −→ FnS

0 is the projection, which is ahomotopy equivalence. We then define K as the set of all pushout product maps

im ∧ cn : Fn+1(Dm)+ ∧ S1 ∪Fn+1(Sm−1)+∧S1 (Sm−1)+ ∧ Z(λn) −→ (Dm)+ ∧ Z(λn)

for n,m ≥ 0, where Dm = C(Sm−1) is the cone of the (m− 1)-sphere, so homeomorphic to a m-disc.Now we apply the small object argument (see Theorem 1.7 of Appendix A) for the set K to the unique

morphism from a given symmetric spectrum A to the trivial spectrum [spell this out instead of ref to App].We obtain a functor A 7→ QA together with a natural transformation j : A −→ QA which is a K-cellcomplex and such that QA is K-injective.

To get a ‘Q’-functor for symmetric spectra of simplicial set we could mimic the above construction usingthe small object argument in the world of simplicial sets. But instead we reduce to the case of spaces bysetting QA = S(Q(|A|)) when A is a symmetric spectrum of simplicial sets. The natural stable equivalenceis then obtained as the composite

ηA : A unit−−→ S|A|S(η|A|)−−−−−→ S(Q(|A|)) .

Proposition 4.27. For every symmetric spectrum A the spectrum QA is an Ω-spectrum and the mor-phism ηA : A −→ QA is a stable equivalence.

Proof. The singular complex of the Ω-spectrum of spaces Q|B| is an Ω-spectrum of simplicial sets.Every level equivalence is a stable equivalence and the singular complex functor preserves stable equivalences(by definition). Hence the case of simplicial sets follows from the case of spaces.

Hence the real content lies in the case of spectra of spaces. We start by showing that QA is alwaysan Ω-spectrum. By construction QA has the right lifting property with respect to K, so we show thatthis right lifting property characterizes Ω-spectra. By adjointness, a symmetric spectrum X has the rightlifting property with respect to the set K if and only if for all n ≥ 0 the map of spaces map(cn, X) :map(Z(λn), X) −→ map(Fn+1S

1, X) ∼= ΩXn+1 has the right lifting property for all inclusions of spheresinto discs; this is equivalent to map(cn, X) being simultaneously a Serre fibration and weak equivalence.Since the mapping cylinder Z(λn) is homotopy equivalent to FnS0, the space map(Z(λn), X) is homotopyequivalent to map(FnS0, X) ∼= Xn. [spell out] So altogether the right lifting property with respect to theset K implies that the map σn : Xn −→ ΩXn+1 is a weak equivalence, i.e., X is an Ω-spectrum.

The morphism j : A −→ QA belongs to the class of K-cell complexes, the closure of the set Kunder wedges, cobase change and countable composition. Each morphism in K is simultaneously a stableequivalence and h-cofibration and the class of stable equivalences which are h-cofibrations is closed underunder wedges, cobase and countable composition by Proposition 4.22 [fix]. So j is a stable equivalence.

4.3. True homotopy groups. In this section we introduce the true homotopy groups of a symmetricspectrum and use them to characterize stable equivalences: by Theorem 4.35 below a morphism of symmetricspectra is a stable equivalence if and only it it induces isomorphisms of all true homotopy groups. Thedefinition involves the notion of a semistable symmetric spectrum:

Definition 4.28. A symmetric spectrum of spaces or of simplicial sets is semistable if it admits aπ∗-isomorphism to an Ω-spectrum.

Semistable spectra should be thought of as the ones where the naive homotopy groups are ‘correct’:within this class, the naive and true homotopy groups coincide and stable equivalences coincide with π∗-isomorphisms. Clearly, every Ω-spectrum is semistable. More generally, suppose that X is a symmetricspectrum which is ultimately an Ω-spectrum, i.e., there exists an m ≥ 0 such that the adjoint structure mapσn : Xn −→ ΩXn+1 is a weak equivalence for all n ≥ m. Then the m-fold shift of X is an Ω-spectrum, andhence so is Ωm(shmX), them-fold loop spectrum of them-fold shift ofX. The morphismX −→ Ωm(shmX)adjoint to λ(m)

X : Sm ∧ X −→ shmX of (3.20) is a weak equivalence in levels m and above, and hence a


π∗-isomorphism. Thus X is semistable. In particular, Eilenberg-Mac Lane spectra (Example 1.14) and thespectra KU and KO of complex respectively real topological K-theory (Example 1.18) are semistable sincethey are Ω-spectra respectively positive Ω-spectra.

All the basic examples of Section 1.1 are semistable, although we need more tools to show that. Inparticular the sphere spectrum (Example 1.8), suspension spectra (Example 1.13), and the various Thomspectra of Examples 1.16 and 1.17 are semistable, see Example 7.29 below. Free and semifree symmetricspectra generated in positive levels are typically not semistable, compare Example 7.30 for more details.

We will investigate semistable symmetric spectra in more detail in Section 7.4, where we provide a longlist of characterizations of semistable spectra in Theorem 7.27. Among these is the fact that a symmetricspectrum X is semistable if and only if the morphism λX : S1 ∧X −→ shX is a π∗-isomorphism. Sufficientconditions for semistability are that a symmetric spectrum underlies an orthogonal spectrum and that allnaive homotopy groups are finitely generated (see Proposition 7.28 below).

Now comes a key definition in the theory of symmetric spectra.

Definition 4.29. Let A be a symmetric spectrum of spaces and k an integer. The k-th true homotopygroups of A is given by

πkA = NatSpssT→set(Sp(A,−), πk) ,

the set of natural transformations, of functors from semistable symmetric spectra of spaces to sets, fromthe restriction of the representable functor Sp(A,−) to the restriction of the functor πk of k-th naivehomotopy groups. We define the true homotopy groups of a symmetric spectrum of simplicial sets as thetrue homotopy groups of its geometric realization.

In the definition of true homotopy groups, the class of semistable spectra could be replaced by variousother classes to yield isomorphic groups, see Exercise 8.15 for details. For us, the class of semistable spectrais the most convenient one.

One caveat in the above definition is that it is not a priori clear that the collection of natural trans-formations from the restriction of Sp(A,−) to πk actually forms a set (as opposed to a proper class); afterall, the category of semistable symmetric spectra is not small, nor even equivalent to a small category. Wejustify that πkA is indeed a set in Remark 4.36 below.

The set πkA becomes an abelian group by pointwise addition of natural transformations. Moreover,quite formally πk becomes a functor from Sp to the category of abelian groups by precomposition. In moredetail, if g : A −→ B is a homomorphism of symmetric spectra and τ = τX : Sp(A,X) −→ πkXX∈Spss anatural transformation, then we define g∗(τ) ∈ πkB by

(g∗(τ))X : Sp(B,X) −→ πkX , ϕ 7→ τX(ϕg)

where X runs through all semistable symmetric spectra.Admittedly, the definition of true homotopy groups is quite abstract, and it is not immediately clear

if and how the true homotopy groups of a symmetric spectrum A are determined by the spaces An, theirhomotopy groups and the rest of the available structure. We have chosen this definition of true homotopygroups because it can be easily stated and some important properties follow tautologically (for examplethat naive and true homotopy groups coincide for semistable spectra). We will later develop various otherviewpoints on true homotopy groups, some of which have a more ‘concrete’ feeling to them. For example,in Proposition II.1.15 we will identify πkA with the group of maps from the k-dimensional sphere spectrumSk to A in the stable homotopy category (to be introduced in Chapter II). In Section 8 we will discussvarious detection functors, by which we mean a functor D : Sp −→ Sp with values in semistable spectraand such that DA is naturally stably equivalent to A. It follows that for every detection functor D, thetrue homotopy groups of A can be calculated as the naive homotopy groups of the spectrum DA. InExercise 8.15 we discuss that one can use other classes in place of ‘semistable’ to define πk, for example theclasses of all Ω-spectra or all injective Ω-spectra.

The more important true homotopy groups are in general different from the naive homotopy groups;when this happens, the naive groups can be thought of as ‘pathological’. The two invariants are related by

62 I. BASICS

a (quite tautological) natural map c : πkA −→ πkA. For a class x ∈ πkA and a semistable spectrum X, theclass c(x) is defined at X by

(4.30) c(x)X : Sp(A,X) −→ πkX , f 7→ (πkf)(x) .

It is straightforward and formal to verify that c is indeed natural in A and additive.For a general symmetric spectrum A which is not semistable, the restriction of the functor Sp(A,−) to

the subcategory Spss is not representable, so one cannot identify natural transformations out of it purelyformally. However, when A is itself semistable, then the restriction of Sp(A,−) is a representable functoron the category of semistable spectra. So as a special case of the Yoneda lemma we obtain:

Lemma 4.31. For every semistable symmetric spectrum A and integer k the natural map c : πkA −→πkA from the naive to true homotopy group is an isomorphism.

As we will see in Section 7.4, the class of semistable spectra contains many spectra of interest, forexample suspensions spectra, Ω-spectra, restriction of orthogonal spectra and Thom spectra. A symmetricspectrum of simplicial sets is semistable if and only if its geometric realization is. Conversely, a symmetricspectrum of spaces is semistable if and only if its singular complex is (since the adjunction unit is a levelequivalence).

By definition the true homotopy groups of a symmetric spectrum of simplicial sets are the true homotopygroups of its realization. We need to know analogous statement for symmetric spectra of spaces, whichrequires us to prove out front that level equivalences preserve true homotopy groups.

Lemma 4.32. Every level equivalence induces an isomorphism of true homotopy groups. Hence for everysymmetric spectrum A of spaces the adjunction counit ε : |S(A)| −→ A induces a natural isomorphism

πkS(A) = πk|S(A)| πkε−−−→ πkA .

Proof.

Construction 4.33. In Example 1.11 we discussed an action of the stable stems πs∗ (also known as

the naive homotopy groups of the sphere spectrum) on the naive homotopy groups π∗X of a symmetricspectrum X. We can promote this to an action of the stable stems on the true homotopy groups π∗X ofX in a rather formal way. Indeed, if y ∈ πs

l is a stable homotopy class, then right multiplication with yis a natural transformation − · y : πk −→ πk+l of abelian group valued functors on symmetric spectra. Sopostcomposition with the transformation − · y (or rather its restriction to semistable spectra) provides amap

− · y : πkA −→ πk+lA , τ 7→ (− · y) τof true homotopy groups for every symmetric spectrum A, for which we use the same symbol. Biadditivity,unitality and associativity of the action of πs

l of naive homotopy groups directly imply the respectiveproperties of the action on true homotopy groups. The fact that the natural map c : π∗X −→ π∗X fromtrue to naive homotopy groups is πs

∗-linear is also straightforward and formal from the definitions.

The original definition of stable equivalences involves homotopy classes of morphisms into injectiveΩ-spectra. The next proposition links true homotopy groups to such homotopy classes. Given a symmetricspectrum of simplicial sets A, a true homotopy class τ ∈ πkA and an injective Ω-spectrum X of simplicialsets we define a map e(τ)X : [A,X] −→ πk|X| as the composite

[A,X]|−|−−→ [|A|, |X|]

τ|X|−−→ πk|X| .

We have to check that the second map is well-defined, i.e., that τ|X| : SpT(|A|, |X|) −→ πk|X| sendshomotopic morphisms to the same naive homotopy class [...]. The maps e(τ)X are natural in X, so theyconstitute a natural transformation

e(τ) : [A,−] −→ πk

of functors on the category of injective Ω-spectra of simplicial sets.


Proposition 4.34. For every symmetric spectrum of simplicial sets A the map

e : πkA −→ NatSpinj.ΩsS →set([A,−], πk)

is a bijection.

fix this. We have to prove that elements of a true homotopy group (i.e., natural transformations) canbe recovered from their values on injective Ω-spectra. More precisely, for every natural transformation

τ : Sp(A,−)|Spinj.Ω −→ πk

defined on the full subcategory Sp(A,−)|Spinj.Ω of injective Ω-spectra there is a unique element τ ∈ πkAwhose restriction to injective Ω-spectra equals τ .

Indeed, if X is any semistable spectrum, then by definition there is a π∗-isomorphism ϕ : X −→ Zwhose target Z is an Ω-spectrum. If we compose with the level equivalence of Proposition 4.6 we get aninjective Ω-spectrum Z ′ = Z inj and a natural π∗-isomorphism ϕ′ : X −→ Z ′. The diagram

Sp(A,X)τX //

Sp(A,ϕ′)

πkX

πk(ϕ′)

Sp(A,Z ′)

τZ′// πkZ ′

commutes by the naturality of τ . Since the map πk(ϕ′) is an isomorphism, the transformation τ is deter-mined by its restriction to injective Ω-spectra. Conversely, if a natural transformation τ , defined only oninjective Ω-spectra, is given, then the commutative square leaves us no choice as to define τ ∈ πkA at asemistable spectrum X by

τX = πk(ϕ′)−1 τZ′ Sp(A,ϕ′) ,

where ϕ′ : X −→ Z ′ is a π∗-isomorphism to an injective Ω-spectrum. If ϕ′′ : X −→ Z ′′ is another π∗-isomorphism to an injective Ω-spectrum, then there exists a morphism ψ : Z ′ −→ Z ′′ such that ψϕ′ ishomotopic to ϕ′′ (since ϕ′ is a stable equivalence by [...]). Since ϕ′ and ϕ′′ are π∗-isomorphisms, so is ψ,and we deduce the relation [homotopic...]

πk(ϕ′′)−1 τZ′′ Sp(A,ϕ′′) = πk(ϕ′)−1 πk(ψ)−1 τZ′′ Sp(A,ψ) Sp(A,ϕ′)= πk(ϕ′)−1 πk(ψ)−1 πk(ψ) τZ′ Sp(A,ϕ′) = πk(ϕ′)−1 τZ′ Sp(A,ϕ′)

where the second equalitiy is naturality of τ for morphisms between injective Ω-spectra. So the definitionof τX is independent of the chosen π∗-isomorphism to an injective Ω-spectrum. [natural]

Now we can proof a key result about true homotopy groups, namely that they characterize stableequivalences.

Theorem 4.35. For every morphism f : A −→ B of symmetric spectra the following are equivalent

(i) f is a stable equivalence;(ii) the map πkf : πkA −→ πkB of true homotopy groups is an isomorphism for all integers k.

Proof. We treat the case of spectra of simplicial sets first.(i)=⇒(ii): By definition, the stable equivalence f induces a bijection [f,X] : [B,X] −→ [A,X] of sets of

homotopy classes for every injective Ω-spectrum X. Thus for every natural transformation τ : [A,−] −→ πkof set valued functors on injective Ω-spectra there is a unique such transformation τ : [B,−] −→ πk satisfyingτ [f,−] = τ . By Proposition 4.34, such natural transformations are in natural bijection with elements ofthe true homotopy group πk. So f induces an isomorphism of true homotopy groups.

64 I. BASICS

(ii)=⇒(i): Proposition 4.27 provides a commutative square

Af //

ηA

B

ηB

QA

Qf// QB

in which the vertical morphisms are stable equivalences and the spectra QA and QB are Ω-spectra, so inparticular semistable. By the already established implication the vertical morphisms induce isomorphismsof true homotopy groups; since f also induces isomorphisms of true homotopy groups, so does Qf . SinceQA and QB are semistable, true and naive homotopy groups are naturally isomorphic (Lemma 4.31), soQf induces isomorphisms on naive homotopy groups. So Qf is a stable equivalence by Theorem 4.16 [evenlevel equivalence]. Since the other three maps in the commutative square are stable equivalences, so if themorphism f .

For a morphism f of symmetric spectra of spaces, Lemma 4.32 shows that f induces isomorphisms oftrue homotopy groups if and only if the singular complex S(f) does. By definition, f is a stable equivalenceif and only if S(f) is, which reduces the case of spectra of spaces to the case of simplicial sets.

Remark 4.36. We can now justify that the our definition of true homotopy groups actually yields a set(as opposed to a proper class). For every symmetric spectrum A the stable equivalence ηA : A −→ QA ofProposition 4.27 has an Ω-spectrum as target. So QA is in particular semistable, and so πkQA is in bijectionwith the naive homotopy group πkQA, which is a set. Hence πkQA is itself a set. By Theorem 4.35, themorphism ηA induces a bijection between πkQA and πkA, so the latter is also a set.

Proposition 4.37. Every stable equivalence between semistable spectra is a π∗-isomorphism.

Proof. Every morphism f : A −→ B of symmetric spectra gives rise to a commutative square

πkAπkf //

c

πkB

c

πkA

πkf// πkB

If A and B are semistable, then the vertical maps are bijective. So if f is a π∗-isomorphism, it also inducesan isomorphism of true homotopy groups, and is thus a stable equivalence by Theorem 4.35.

4.4. Properties of true homotopy groups. The true homotopy groups share many of the formalproperties of naive homotopy groups. [list them here]

We start by investigating the effect of the loop and suspension constructions on true homotopy groups.We need that suspension, looping and shifting preserve semistable spectra.

Proposition 4.38. A symmetric spectrum of spaces is semistable if and only if its singular complexis. A symmetric spectrum of simplicial sets is semistable if and only if its geometric realization is. Letf : A −→ B be a π∗-isomorphism of symmetric spectra. Then A is semistable if and only if B is semistable.

Proof. Since geometric realization and singular complex both preserve π∗-isomorphisms and Ω-spectra, both preserve semistable spectra. For the other directions we argue separately.

Let Y be a symmetric spectrum of simplicial sets. If the geometric realization is semistable andϕ : |Y | −→ Z a π∗-isomorphism to an Ω-spectrum, then the adjoint ϕ : Y −→ S(Z) is a π∗-isomorphism toan Ω-spectrum, so Y is semistable.

Let X be a symmetric spectrum of spaces. If the singular complex S(X) is semistable and ϕ : S(X) −→Z a π∗-isomorphism to an Ω-spectrum, then the morphism |ϕ| : |S(X)| −→ |Z| is a π∗-isomorphism to anΩ-spectrum. We form the double mapping cylinder of |ϕ| and the adjunction counit ε : |S(X)| −→ X.


Then the resulting map from X to the mapping cylinder is the required π∗-isomorphism to an Ω-spectrum[...].

Now we treat the last statement involving a π∗-isomorphism f : A −→ B. By what we just showed itsuffices to treat the case of simplicial sets. If B is semistable, then there is a π∗-isomorphism B −→ Z to anΩ-spectrum Z. The composite with f is then the required π∗-isomorphism that shows that A is semistable.

If A is semistable, then there is a π∗-isomorphism ϕ : A −→ Z to an Ω-spectrum Z. If we composewith the level equivalence l : Z −→ Z inj of Proposition 4.6, then Z inj is an injective Ω-spectrum. Sinceevery π∗-isomorphism is a stable equivalence (Theorem 4.16), there is a morphism ψ : B −→ Z inj suchthat ψf : A −→ Z inj is homotopic to lϕ. Since all f, l and ϕ are π∗-isomorphisms, so is ψ, and hence B issemistable.

Proposition 4.39. For every symmetric spectrum A the following are equivalent:

(i) A is semistable;(ii) the suspension S1 ∧A is semistable;

In the context of spaces, or if the symmetric spectrum A is levelwise Kan, then conditions (i) and (ii) arefurthermore equivalent to

(iii) the loop spectrum ΩA is semistable.

Proof. (ii)⇒(i) If S1 ∧ A is semistable, then there exists a π∗-isomorphism ψ : S1 ∧ A −→ Z to anΩ-spectrum Z. Since looping preserves naive homotopy groups and Ω-spectra, and the adjunction counitis a π∗-isomorphism (by Proposition 2.4), the composite

Aη−→ Ω(S1 ∧A)

Ωψ−−→ ΩZ

(which is also the adjoint of ψ) is a π∗-isomorphism to an Ω-spectrum. Hence A is semistable.(i)⇒(ii) If A is semistable, then there exists a π∗-isomorphism ψ : A −→ Z to an Ω-spectrum Z. The

suspension S1 ∧ ϕ : S1 ∧ A −→ S1 ∧ Z is then a π∗-isomorphism by Proposition 2.4. Since Z is an Ω-spectrum, the morphism λZ : Z −→ Ω(shZ) is a level equivalence; its adjoint λZ : S1 ∧ Z −→ shZ is thena π∗-isomorphism by Proposition 2.4. The composite λZ(S1 ∧ ϕ) is the π∗-isomorphism to an Ω-spectrumwhich shows that S1 ∧A is semistable.

(iii)⇒(i) If ΩA is semistable, then so is S1∧ΩA (since we already showed that condition (i) implies (ii)).The adjunction unit ε : S1∧ΩA −→ A is a π∗-isomorphism (Proposition 2.4, here we use that A is levelwiseKan in the simplicial context), so A is semistable by Proposition 4.38.

(i)⇒(iii) If A is semistable, then there exists a π∗-isomorphism ϕ : A −→ Z to an Ω-spectrum Z. Sincelooping preserves naive homotopy groups (Proposition 2.4) and Ω-spectra, the morphism Ωϕ : ΩA −→ ΩZis a π∗-isomorphism to an Ω-spectrum. Hence the loop spectrum ΩA is semistable.

We define natural homomorphisms of true homotopy groups

α : πk(ΩA) −→ π1+kA and S1 ∧ − : πkA −→ π1+k(S1 ∧A) .

Let τ ∈ πk(ΩA) be any true homotopy class (i.e., natural transformation) and X a semistable symmetricspectrum. Then ΩX is also semistable by Proposition 4.39 and we can define α(τ) ∈ π1+kA at X as thecomposite

Sp(A,X) Ω−−→ Sp(ΩA,ΩX) τΩX−−−−→ πk(ΩX) α−−→ π1+kX

where αX : πk(ΩX) −→ π1+kX is the loop isomorphism for naive homotopy groups defined in (2.5).The suspension homomorphism S1 ∧ − : πkA −→ π1+k(S1 ∧ A) of true homotopy groups is defined

as follows. Let τ ∈ πkA be any homotopy class (i.e., natural transformation), X a semistable symmetricspectrum and g : S1 ∧ A −→ X a morphism. Then the target of the adjoint morphism g : A −→ ΩX isagain semistable (by Proposition 4.39) so we can set

(S1 ∧ τ)X(g) = αX(τΩX(g)) ∈ π1+kX .

66 I. BASICS

Proposition 4.40. For every symmetric spectrum A and integer k the loop and suspension homomor-phisms

α : πk(ΩA) −→ π1+kA and S1 ∧ − : πkA −→ π1+k(S1 ∧A)

are isomorphisms of true homotopy groups. Moreover, the triangles

πkAS1∧− //

πkη &&MMMMMMMMMMM π1+k(S1 ∧A) πk(ΩA) α //

S1∧− ''NNNNNNNNNNNπ1+kA

πk(Ω(S1 ∧A))

α

66mmmmmmmmmmmmπk(S1 ∧ (ΩA))

πkε

88ppppppppppp

commute where η : A −→ Ω(S1 ∧ A) and ε : A −→ S1 ∧ (ΩA) are the unit respectively counit of theadjunction. In particular, the adjunction unit and counit induce isomorphisms of true homotopy groups.

The squares

πkAc //

S1∧−

πkA

S1∧−

πk(ΩA) c //

α

πk(ΩA)

α

π1+k(S1 ∧A) c

// π1+k(S1 ∧A) π1+kA c// π1+kA

commute.

Proof. We prove the two compatibility conditions first. Let X be any symmetric spectrum, g :S1 ∧A −→ X a morphism and g : A −→ ΩX its adjoint. Then the square

πkAπkg //

S1∧−

πk(ΩX)

S1∧−

α

((RRRRRRRRRRRRR

π1+k(S1 ∧A)π1+k(S

1∧g)//

π1+kg

55π1+k(S1 ∧ ΩX)π1+k(ε)

// π1+kX

of naive homotopy groups commutes by Proposition 2.4, naturality of the suspension isomorphism for naivehomotopy groups, and because g equals the composite of S1∧g and the adjunction counit ε : S1∧ΩX −→ X.So if X is semistable and x ∈ πkA any naive homotopy class, we calculate

(S1 ∧ c(x))X(g) = α (c(x)ΩX(g)) = α ((πkg)(x)) = (π1+kg)(S1 ∧ x) = c(S1 ∧ x)X(g) ,

where the first, second and fourth equality hold by definition. This proves the first naturality claim S1 ∧c(x) = c(S1 ∧ x). The other naturality is straightforward: if x ∈ πk(ΩA) is any naive homotopy class, X asemistable symmetric spectrum and g : A −→ X any morphism, then we have

c(α(x))X(g) = g∗(α(x)) = α((Ωg)∗(x)) = α(c(x))X(g) ;

the second equality is naturality for the loop isomorphisms for naive homotopy groups, the other equalitiesare the definitions.

Now we prove that S1 ∧ − and α are isomorphisms of true homotopy groups in the special case ofa semistable symmetric spectrum A. For such A, the suspensions S1 ∧ A and the loop spectrum ΩA arealso semistable (Proposition 4.39) and hence the natural map c from naive to true homotopy groups is abijective for A, S1 ∧A and ΩA. Thus the compatibility squares and the corresponding properties for naivehomotopy groups (Proposition 2.4) show that S1 ∧ − and α are isomorphisms of true homotopy groupsfor A.

If the claim holds for B and f : A −→ B is a stable equivalence, then the claim holds for A by naturalityand because suspending and looping preserve stable equivalences (Proposition 4.11 resp. ). Proposition 4.27


provides a natural stable equivalence ηA : A −→ QA whose target QA is an Ω-spectrum, so in particularsemistable. So the claim holds for all symmetric spectra A.

The two relations α(πkη) = S1 ∧ − and πkε(S1 ∧ −) = α (i.e., commuting triangles) are again formalconsequences: the stable equivalence ηA : A −→ QA of Proposition 4.27 reduces the argument from A tothe Ω-spectrum QA, and for this the tautological map c : πk(QA) −→ πk(QA) is an isomorphism. So therelations for true homotopy groups follow from the corresponding relations for naive homotopy groups (seeProposition 2.4).

What we said about the loop spectrum works as well for symmetric spectra of simplicial sets as longas they are levelwise Kan complexes.

Now prove that the shift adjoint . always shifts the true homotopy groups. We observe that shiftingpreserves semistability: if A is semistable and ϕ : A −→ Z a π∗-isomorphism to an Ω-spectrum Z, then themorphism shϕ : shA −→ shZ is a π∗-isomorphism to an Ω-spectrum. Hence the shifted spectrum shA issemistable.

For a symmetric spectrum A we define a natural map . : π1+kA −→ πk(.A) as follows. For a homotopyclass τ ∈ π1+kA, a semistable symmetric spectrum X and a morphism f : .A −→ X we can set

(.τ)X(f) = τshX(f)

in the group π1+k(shX) = πkX, where f : A −→ shX is the adjoint of f .

Proposition 4.41. For every symmetric spectrum A and integer k the composite map

πk(ΩA) α−−→ π1+kA.−−→ πk(.A)

is inverse to the negative of the effect of the morphism λA : .A −→ ΩA on πk. Thus the map

. : π1+kA −→ πk(.A)

is an isomorphism of true homotopy groups.

Proof. After unraveling all definitions this boils down to the fact that the two morphism

λΩA , Ω(λA) : ΩA −→ Ω2(shA)

differ by the permutation of the two loop coordinates in the target. So these morphisms induce the negativeof each other on naive homotopy groups. [...]

The morphism λA is a stable equivalence by Proposition 4.18, so λA induces isomorphisms of truehomotopy groups by Theorem 4.35. Since α : πk(ΩA) −→ π1+kA is an isomorphism by Proposition 4.40,the map . : π1+kA −→ πk(.A) is an isomorphism as well.

Corollary 2.20 says that naive homotopy groups commute with finite products and arbitrary coproducts.Now we show that analogous result for true homotopy groups.

Lemma 4.42. Let Aii∈I be a family of semistable symmetric spectra. Then the coproduct∨i∈I A

i

and the weak product∏′

i∈I Ai are again semistable.

Proof. By assumption there exist π∗-isomorphisms ϕi : Ai −→ Zi for all i ∈ I such that all Zi areΩ-spectra. By Corollary 2.20 (iii) the weak product∏

i∈I

′ϕi :

∏i∈I

′Ai −→

∏i∈I

′Zi

is again a π∗-isomorphism, and the target is another Ω-spectrum. So the weak product is semistable.By Corollary 2.20 (iii) the canonical morphism from the coproduct to the weak product is a π∗-

isomorphism. So the composite of this morphism with the π∗-isomorphism above shows that the wedge∨i∈I A

i is semistable.

68 I. BASICS

Proposition 4.43. (i) For every family of symmetric spectra Aii∈I and every integer k the canonicalmap ⊕

i∈IπkA

i −→ πk

(∨i∈I

Ai

)is an isomorphism of abelian groups.

(ii) For every finite indexing set I, every family Aii∈I of symmetric spectra and every integer k thecanonical map

πk

(∏i∈I

Ai

)−→

∏i∈I

πkAi

is an isomorphism of abelian groups.(iii) For every family of symmetric spectra the canonical morphism from the wedge to the weak product

is a stable equivalence. In particular, for every finite family of symmetric spectra the canonical morphismfrom the wedge to the product is a stable equivalence.

Proof. (i) For every i ∈ I we can choose a stable equivalence ϕi : Ai −→ Bi whose target is semistable,for example as in Proposition 4.27. In the commutative diagram⊕

i∈I πkAi

⊕i∈I πkB

iLπkϕ

i

ooLI c //

⊕i∈I πkB

i

πk(∨

i∈IAi)

πk(∨

i∈IBi)

πk(WI ϕ

i)

ooc

// πk(∨

i∈IBi)

the right vertical map of naive homotopy groups is an isomorphism by Corollary 2.20 (i). The right upperand lower maps are isomorphisms by Proposition 4.31 since the spectra Bi and their wedge are semistable.The wedge

∨I ϕ

i :∨I A

i −→∨I B

i is a stable equivalence by Proposition 4.22 (i), so the lower left map isan isomorphism. Altogether it follows that the canonical map from the direct sum of the groups πkAi tothe homotopy group of the wedge

∨i∈I A

i is an isomorphism.(iii) The canonical morphism is a π∗-isomorphism by Corollary 2.20 (iii), hence a stable equivalence by

Theorem 4.16.(ii) For finite indexing sets I the canonical map

∨I A

i −→∏I A

i from the wedge to the product is astable equivalence by part (iii), so it induces an isomorphism of true homotopy groups. So the claim followsfrom part (i) and the fact that finite sums and finite products of abelian groups are isomorphic.

Given a morphism of symmetric spectra f : X −→ Y , the mapping cone C(f) and homotopy fiber F (f)were defined in (2.6) respectively (2.13). We can mimick the definitions of the connecting homomorphismsfor naive homotopy groups (compare (2.11) respectively (2.14)) with true homotopy groups. We thenobtain long exact sequences of true homotopy groups as follows. We define a connecting homomorphismδ : π1+kC(f) −→ πkX as the composite

(4.44) π1+kC(f)π1+k(p)−−−−−−→ π1+k(S1 ∧X) ∼= πkX ,

where the first map is the effect of the projection p : C(f) −→ S1 ∧X on true homotopy groups, and thesecond map is the inverse of the suspension isomorphism S1 ∧ − : πkX −→ π1+k(S1 ∧ X). We define aconnecting homomorphism δ : π1+kY −→ πkF (f) as the composite

(4.45) π1+kYα−1

−−→ πk(ΩY )πk(i)−−−−→ πkF (f) ,

where α : πk(ΩY ) −→ π1+kY is the loop isomorphism and i : ΩX −→ F (f) the injection of the loopspectrum into the homotopy fiber. Since loop and suspension isomorphisms for naive and true homotopygroups are compatible with the tautological map c : πk −→ πk, both connecting homomorphisms arecompatible with the tautological map c : πk −→ πk.


Proposition 4.46. Let f : X −→ Y be a morphism of symmetric spectra.(i) The long sequence of true homotopy groups

· · · −→ πkXπk(f)−−−−−→ πkY

πk(i)−−−→ πkC(f) δ−−→ πk−1X −→ · · ·is exact.

(ii) In the simplicial context, suppose also that X and Y are levelwise Kan complexes. Then the longsequence of true homotopy groups

· · · −→ πkXπk(f)−−−−−→ πkY

δ−−→ πk−1F (f)πk−1(p)−−−−−−→ πk−1X −→ · · ·

is exact.(iii) Suppose that f is an h-cofibration of symmetric spectra of topological spaces or an injective mor-

phism of symmetric spectra of simplicial sets. Denote by q : Y −→ Y/X the quotient map. Then the naturalsequence of true homotopy groups

· · · −→ πkXπk(f)−−−→ πkY

πk(q)−−−→ πk(Y/X) δ−−→ πk−1X −→ · · ·is exact, where the connecting map δ is the composite of the inverse of the isomorphism πkC(f) −→πk(Y/X) induced by the level equivalence C(f) −→ Y/X which collapses the cone of X and the connectinghomomorphism πkC(f) −→ πk−1X defined in (4.44).

(iv) Suppose that f is levelwise a Serre fibration of spaces respectively Kan fibration of simplicial sets.Denote by i : F −→ X the inclusion of the fiber over the basepoint. Then the natural sequence of truehomotopy groups

· · · −→ πkFπk(i)−−−→ πkX

πk(f)−−−→ πkYδ−−→ πk−1F −→ · · ·

is exact, where the connecting map δ is the composite of the connecting homomorphism πkY −→ πk−1F (f)defined in (4.45) and the inverse of the isomorphism πkF (f) −→ πkF induced by the level equivalenceF −→ F (f) which send x ∈ F to (const∗, x).

(v) True homotopy groups commute with filtered colimits over closed embeddings.

Proof. (ii) Proposition 4.27 provides stable equivalences ηX : X −→ QX and ηY : Y −→ QY and amorphism Qf : QX −→ QY satisfying Qf ηX = ηY f . By [...] the induced morphism on homotopyfibers η : F (f) −→ F (Qf) is then again a stable equivalence. We compare the sequence in question withthe long homotopy sequence of Qf : QX −→ QY :

πkXπkf //

πkηX

πkYδ //

πkηY

πk−1F (f)πkp //

πkη

πk−1Xπk−1f //

πk−1ηX

πk−1Y

πk−1ηX

πkQX

πkQf // πkQYδ // πk−1F (Qf)

πkQp // πk−1QXπk−1Qf // πk−1QY

πkQXπkQf

//

c ∼=

OO

πkQYδ

//

c ∼=

OO

πk−1F (Qf)πkQp

//

c ∼=

OO

πk−1QXπk−1Qf

//

c∼=

OO

πk−1QY

c∼=

OO

The lower row of naive homotopy groups is exact by Proposition 2.16. Since the spectra QX and QY areΩ-spectra, so is the homotopy fibre F (Qf). The lower vertical maps are instances of the natural map fromnaive homotopy groups, and they are isomorphisms since QX, QY and F (Qf) are semistable.

So the middle row is exact. The upper vertical morphisms of true homotopy groups are all induced bystable equivalences, so they are bijective by Theorem 4.35, and thus the upper row is exact, as we had toshow.

We deduce statement (i) from (ii) by using the π∗-isomorphism S1∧F (f) −→ C(f) which is compatiblewith the connecting homomorphisms [...].

(iii) Since f is an h-cofibration (in the topological context) respectively levelwise injective (in thesimplicial context), the quotient spectrum Y/X is level equivalent to the mapping cone C(f).

70 I. BASICS

(iv) If f : X −→ Y is a morphism of symmetric spectra which is levelwise a Serre fibration of spacesrespectively Kan fibration of simplicial sets, the strict fiber F is level equivalent to the homotopy fiber.

(v)

Remark 4.47. Since shifting does not in general preserve stable equivalences (compare Example 4.24),the true homotopy groups of the spectrum shX are not in general an invariant of the true homotopy groupsof X.

Now we calculate the true homotopy groups in some examples. Most of the time this ‘calculation’will consist in a reduction of the problem to the calculation of naive homotopy groups of other symmetricspectra.

In Example 4.17 we considered the stable equivalence λ : F1S1 −→ S which is adjoint to the identity

in level 1. This morphism is not a π∗-isomorphism, but a consequence of Theorem 4.35 is that the stableequivalence λ also induces isomorphisms of true homotopy groups. Since the sphere spectrum S is semistable[do we know this?], its naive and true homotopy groups ‘coincide’ and so the homotopy groups of thespectrum F1S

1 are isomorphic to the stable stems. However, we can also calculate the true homotopygroups π∗(FmSn) directly with the tools developed so far. The naive fundamental class

(4.48) ιmn ∈ πn−m(FmSn)

is the class represented by the wedge summand inclusion

Sn1∧−−−−→ Σ+

m ∧ Sn = (FmSn)m .

[was this defined earlier?] The true fundamental class

(4.49) ιmn ∈ πn−m(FmSn)

is the image c(ιmn ) of the naive fundamental class under the map c : πm−n(FmSn) −→ πm−n(FmSn). As anatural transformation, the value of ιmn at a semistable spectrum X sends a morphism f : FmSn −→ X tothe class in πn−mX represented by the composite

Sn1∧−−−−→ Σ+

m ∧ Sn = (FmSn)mfm−−−→ Xm .

The next proposition uses the action of the stable stems on the true homotopy groups of a spectrumwhich was introduced in Construction 4.33.

Proposition 4.50. The graded homotopy group groups π∗(FmSn) are freely generated as a πs∗-module

by the fundamental class ιmn .

Proof. We have isomorphisms [specify] S1 ∧FmSn ∼= FmS1+n and .(FmSn) ∼= F1+mS

n such that thecomposites

πk(FmSn)S1∧−−−−−−→ π1+k(S1 ∧ FmSn)

π1+k(...)−−−−−→ π1+k(FmS1+n)

and

πk+1(FmSn).−−→ πk(.(FmSn))

πk(...)−−−−→ πk(F1+mSn)

take the fundamental classes to the fundamental classes [check...]. The suspension homomorphism and themap . are πs

∗-linear isomorphisms by Proposition 4.40 respectively Proposition 4.41 [πs∗-linear...]. So theclaim holds for (m,n) if and only if it holds for (m,n + 1) if and only if it holds for (m + 1, n). Thus itsuffices to prove the statement for m = n = 0. The spectrum F0S

0 is isomorphic to the sphere spectrumand thus semistable, so

πsk = πkS

c−−→ πkS ∼= πk(F0S0)

is bijective. Since the map c is πs∗-linear, this proves the claim.


Example 4.51. The true homotopy groups of a free symmetric spectrum FmK generated by a basedspace (or simplicial set) K in level m are isomorphic to the stable homotopy groups of K, shifted mdimensions. One way to see this is to exploit that FmK is isomorphic to the m-fold iterate of the shiftadjoint applied to the suspensions spectrum of K,

FmK ∼= .m(Σ∞K)

(compare Example 3.11). So an m-fold application of Proposition 4.41 provides an isomorphism of truehomotopy groups

.m : πsm+kK = πm+k(Σ∞K)

∼=−→ πk .m (Σ∞K) ∼= πk(FmK) .

We have exploited that suspension spectra are semistable, and hence true and naive homotopy groupscoincide.

In the context of spaces, another way to derive this calculation is to use the stable equivalence

FmK −→ Ωm(Σ∞K)

which we derived in Example 4.25 and which is adjoint to the adjunction unit K −→ Ωm(K ∧ Sm) =Ωm(Σ∞K)m. This stable equivalence induces an isomorphism between the true homotopy groups of FKand those of Ωm(Σ∞K), which by Proposition 4.40 and semistability of suspension spectra are isomorphicto the naive homotopy groups πm+k(Σ∞K) = πs

m+kK.

Example 4.52. There does not seem to be a functorial description of the true homotopy groups of asemifree spectrum GmL in terms of elementary invariants of the Σm-space L. However, as we will see inExample 7.45, there is a spectral sequence

E2p,q = Hp(Σm, (πs

q+mL)(sgn)) =⇒ πp+q(GmL)

which starts from the homology of Σm with coefficients in the stable homotopy of L, twisted by the signrepresentation, and converges to the true homotopy groups of the semifree spectrum GmL.

As we showed in Example 4.26, the semifree spectrum GmL is stably equivalent to the homotopyorbit spectrum (Ωm(Σ∞L))hΣm with respect to the conjugation action of the symmetric group. In thisinterpretation, the spectral sequence becomes the homotopy orbit spectral sequence [ref].

Example 4.53 (Realization). By a simplicial spectrum we mean a contravariant functor X from thesimplex category ∆ of standard partially ordered sets [n] to the category of symmetric spectra (of spaces orof simplicial sets). We usually denote the value X([n]) of X at the object [n] by Xn, and refer to it as thespectrum of n-simplices of X. We sometimes use the notation X• when we wish to emphasize the simplicialdirection, where the bullet is meant as a placeholder for objects from the category ∆.

We will sometimes encounter simplicial spectra X•, often from some kind of bar construction, and wantto ‘realize’ them into a single symmetric spectrum. So the realization of X• is the symmetric spectrum |X•|is the symmetric spectrum defined as the coend [reference]

|X•| = ∆[•]+ ∧∆ X• = coequalizer( ∨

α:[k]−→[n]∆[k]+ ∧Xnα∗∧Id //Id∧α∗

//∨n∆[n]+ ∧Xn

)of the functor ∆[•]+ ∧X• : ∆×∆op −→ Sp. Here ∆[n] should be interpreted as the topological n-simplexwhen in the context of symmetric spectra of spaces, and as the simplicial n-simplex when working withsimplicial sets.

Realizations of simplicial spectra are given ‘levelwise’. Indeed, since coequalizers of symmetric spectraand smashing with a based spaces (or simplicial set) are give levelwise, the realization |X•| can be obtainedby calculating the realization of the simplicial space (or bisimplicial set) (X•)m for every level m ≥ 0, andthen using that the resulting spaces (simplicial set) naturally inherit symmetric group actions and structuremaps which make them into a symmetric spectrum. In the simplicial context this gives another way to lookat |X•|: since the realization of a bisimplicial set is naturally isomorphic to the diagonal simplicial set [ref],

72 I. BASICS

the realization |X•| can be defined by diagonalizing the two simplicial directions. In other words, we candefine the k-simplices of the mth level |X•|k as

(|X•|m)([k]) = (X([k])m)([k]) .

[realization commutes with limits and colimits]The realization of a simplicial spectrum X• comes with a natural morphism

|X•| −→ coequalizer(X1

d0 //d1

// X0

)to the coequalizer of the two face morphisms from X1 to X0; this morphism is adjoint to the collection ofmorphisms ∆[n]+ ∧Xn

proj−−→ Xnd∗−→ X0 where d : [0] −→ [n] is any morphism in ∆.

[Construct the spectral sequence of a relaization

E2p,q = Hp(Nπq(X•)) =⇒ πp+q|X•|

Apply to homotopy colimits]

4.5. A calculus of fractions. For symmetric spectra X and Y we define the fraction category X,Y as follows. An object of X,Y is a a triple (Z, f, g) consisting of a symmetric spectrum Z, a stableequivalence f : Z −→ X and a morphism g : Z −→ Y . A morphism in X,Y from (Z, f, g) to (Z ′, f ′, g′)is a morphism ϕ : Z −→ Z ′ such that f ′ϕ = f and g′ϕ = g. Note that such a morphism ϕ is necessarily astable equivalence since f and f ′ are. We sometimes visualize objects of X,Y as diagrams X ∼←− Z −→ Yand morphisms as commutative diagrams

Z∼uukkkkkkkkkk

∼

))SSSSSSSSSS

X Y

Z ′∼

iiSSSSSSSSSS

55kkkkkkkkkk

Note that X,Y is a covariant functor in the variable Y and a covariant functor for stable equivalences inthe variable X, by composition of morphisms.

We define a natural map l : [X,Y ] −→ π0X,Y where the left hand side is the set of homotopy classesof spectrum morphisms Sp(X,Y ) [in simplicial sets: the equivalence relation generated by ‘homotopy’] Amorphism g : X −→ Y gives rise to an object (X, Id, g) of the category X,Y ; if g′ : X −→ Y is homotopicto g via a homotopy H : [0, 1]+ ∧X −→ Y then the commutative diagram

X

=

vvmmmmmmmmmmmmmmmmm

i0

r

((QQQQQQQQQQQQQQQQQ

X [0, 1]+ ∧Xproj.oo H // Y

X

=

hhQQQQQQQQQQQQQQQQQi1

OO

g′

66mmmmmmmmmmmmmmmmm

shows that (X, Id, g) and (X, Id, g) are in the same path component of the category X,Y . So we get awell-defined map l : [X,Y ] −→ π0X,Y which is natural in Y .

Proposition 4.54. [careful about spaces vs. simplicial sets...] If Y is an injective Ω-spectrum, thenthe map

l : [X,Y ] −→ π0X,Y is bijective. For every pair of stable equivalences α : X −→ X ′ and β : Y −→ Y ′ the map

π0α, β : π0X,Y −→ π0X ′, Y ′is bijective.


Proof. If Y is an injective Ω-spectrum we can define a map

π0X,Y −→ [X,Y ]

in the other direction as follows. Given an object (Z, f, g) of X,Y the map [f, Y ] : [X,Y ] −→ [Z, Y ] isbijective. So there is a morphism g : X −→ Y , unique up to homotopy, such that gf is homotopic to g. Wesend the path component of the object (Z, f, g) to the homotopy class of the morphism g. [well-defined...]

For a morphism g : X −→ Y the object l[f ] is given by (X, Id, g), and g itself is a possible lift for g.So the composite [X,Y ] l−→ π0X,Y −→ [X,Y ] is the identity. To prove our claim it remains to show thatl : [X,Y ] −→ π0X,Y is surjective. Given any object (Z, f, g) there is a morphism g : X −→ Y and ahomotopy H : [0.1]+ ∧ Z −→ Y from g to gf . Then the commutative diagram

Z

f

wwwwwwwwwwwwwwwwwwwwwww

i0

g

##GGGGGGGGGGGGGGGGGGGGGGG

[0, 1]+ ∧ Zfproj.

uukkkkkkkkkkkkkkkkH

))SSSSSSSSSSSSSSSS

X Zf

oo

i1

OO

gf//

f

Y

X

=

iiTTTTTTTTTTTTTTTTTTTg

55jjjjjjjjjjjjjjjjjjj

shows that (Z, f, g) is in the same path component as l[g]Now we show that π0X,Y takes stable equivalences in the second variable to bijections. Suppose

β : Y −→ Y ′ is a stable equivalence; we construct a map β! : π0X,Y ′ −→ π0X,Y in the other direction.Given an object (Z, f, g) of X,Y ′ we choose a factorization

Zγ−−→∼

W(fW ,gW )−−−−−−→ (X,Y ′)

of the morphism (f, g) : Z −→ X × Y ′ as a level equivalence followed by a level fibration [make specificfunctorial choice, for example by the homotopy mapping space]. We note that the component fW : W −→ Xis a stable equivalence since γ and fW γ = f are. We define V as the pullback of the two morphisms(fW , gW ) : W −→ (X,Y ′) and Id×β : X × Y −→ X × Y ′. We get a commutative square

V(fV ,gV ) //

κ ∼

X × Y

∼ Id×β

W(fW ,gW )

// X × Y ′

Since the morphism (fW , gW ) is a level fibration and Id×β a stable equivalence [ref], the morphism κ :V −→ W is a stable equivalence [ref]. Since κ and fW are stable equivalence, so is fV = fWκ. Thus thetriple (V, fV , gV ) is an object of the category X,Y . The various ingredients of the construction (homotopymapping space, pullback) are functorial, so a morphism (Z, f, g) −→ (Z ′, f ′, g′) in the category X,Y ′gives rise to a morphism (V, fV , gV ) −→ (V, fV , gV ) in the category X,Y . The construction thus passesto a well-defined map on path components and we can set β![Z, f, g] = [V, fV , gV ].

In the construction above, the image X,β(V, fV , gV ) = (V, fV , βgV ) of the object (V, fV , gV ) isin the same component as the original object (Z, f, g) because κ : (V, fV , βgV ) −→ (W, fW , gW ) andγ : (Z, f, g) −→ (W, fW , gW ) are morphisms in X,Y ′. So the composite X,β β! is the identity ofπ0X,Y ′. [notation...] In the other direction we start with an object (Z, f, g) of the category X,Y , takethe image (Z, f, βg) in X,Y ′ and apply the construction β!. For this we consider the factorization

Zγ−−→∼

W(fW ,gW )−−−−−−→ (X,Y ′)

74 I. BASICS

of the morphism (f, βg) : Z −→ X × Y ′ as a level equivalence followed by a level fibration [for example bythe homotopy mapping space]. We define V as the pullback of the two morphisms (fW , gW ) : W −→ (X,Y ′)and Id×β : X × Y −→ X × Y ′ and get a commutative square

V(fV ,gV ) //

κ ∼

X × Y

∼ Id×β

W(fW ,gW )

// X × Y ′

The universal property of pullbacks yields a unique morphism γ : Z −→ V satisfying κγ = γ : Z −→ Wand (fV , gV ) γ = (f, g) : Z −→ (X,Y ). But then γ is a morphism in X,Y from (Z, f, g) to (V, fV , gV ) =β!(Z, f, βg). So the composite β! X,β is the identity of π0X,Y . This finished the proof that the mapX,β is bijective.

It remains to show that π0X,Y takes stable equivalences in the first variable to bijections. But this isa formal consequence of the things we already proved. We choose a stable equivalence β : Y −→ Y ′ whosetarget Y ′ is an injective Ω-spectrum. We get a commutative diagram

π0X,Y π0X,β //

π0α,Y

π0X,Y ′

π0α,Y ′

[X,Y ′]loo

[α,Y ′]

π0X ′, Y

π0X′,β// π0X ′, Y ′ [X,Y ′]

loo

The two right horizontal maps are bijections since Y ′ is an injective Ω-spectrum, by the first paragraphabove. The two left horizontal maps are bijective by the second paragraph above. The right vertical mapis bijective by the definition of stable equivalences. So π0α, Y is also bijective.

A symmetric spectrum A is stably contractible if the unique morphism from the trivial spectrum toA is a stable equivalence. Equivalently, the unique morphism from A to the trivial spectrum is a stableequivalence or the true homotopy groups are trivial.

Proposition 4.55. Let X and Y be symmetric spectra. An object (Z, f, g) of the fraction categoryX,Y lies in the basepoint component if and only if the morphism g : Z −→ Y factors through some stablycontractible spectrum.

Proof. Suppose that there is a stably contractible spectrum C and morphisms i : Z −→ C andp : C −→ Y such that g = pi. Then the following is a path from (Z, f, g) to the basepoint object in X,Y

Zf

uukkkkkkkkkkkkkkkkkk

(f,i)

g

))SSSSSSSSSSSSSSSSSS

X X × Cproj.oo pproj. // Y

X

=

iiSSSSSSSSSSSSSSSSSS(Id,∗)

OO

∗

55kkkkkkkkkkkkkkkkkk

Now suppose convesely that (Z, g, f) lies in the basepoint component of X,Y . We choose a stableequivalence j : Y −→ Y ′ with target an injective Ω-spectrum (for example as in [...]). By Proposition 4.54the composite jg : Z −→ Y ′ represents the basepoint in [Z, Y ′], so there is a nullhomotopy H : CZ −→ Y ′

of jg. We choose a factorization CZ −→ C ′ −→ Y ′ as a stable equivalence followed by a level fibration andlet P denote the pullback of the diagram

C ′ −→ Y ′j←−− Y .


By ‘right properness’ the projection P −→ C ′ is a stable equivalence, so the spectrum P is stably con-tractible. By the universal property of the pullback, the morphism g : Z −→ Y has a lift to P whosepropjection to C ′ is the composite of the cone inclusion Z −→ CZ and the stable equivalence CZ −→ C ′.This provides the required factorization.

Example 4.56. The special case X = S of the above ‘calculus of fractions’ will be important later whenwe discuss products on true homotopy groups. So we take some time to spell out this example and relateit to true homotopy groups. Suppose (Z, f, g) is an object of the category S, Y . The stable equivalencef : Z −→ S induces a bijection π0f : π0S −→ π0Z of true homotopy groups, so we can define a homotopyclass 〈Z, f, g〉 in π0Y by

〈Z, f, g〉 = (π0g)((π0f)−1(1)

)where 1 ∈ π0S is the fundamental class. Two objects in the same path component of S, Y give the samehomotopy class. We claim that the resulting natural map

(4.57) 〈−〉 : π0S, Y −→ π0Y

is bijective. [exercise: sum in this picture] Since both sides of the map take stable equivalences in Y tobijections, we can assume without loss of generality that Y is an injective Ω-spectrum. Then we considerthe composite

[S, Y ] l−−→ π0S, Y 〈−〉−−−→ π0Y

with the map l which is a bijection by Proposition 4.54. The composite map is a bijection since Y is anΩ-spectrum (compare ???), which proves the claim.

We also have the tautological map c : π0Y −→ π0Y from naive to true homotopy groups, and here ishow that fits into the picture. Suppose x : Sn −→ Yn is a based continuous map which represents a naivehomotopy class [x] ∈ π0Y . We let x : FnSn −→ Y denote the adjoint of x, a morphism symmetric spectra.Then the triple (FnSn, λ, x) is an object of the category S, Y , and we claim that it represents the truehomotopy class c[x] ∈ π0Y . Put another way, we claim that the assignment

π0Y −→ π0S, Y , [x : Sn → Yn] 7→ (FnSn, λ, x)

when composed with the map (4.57) yields the tautological map c : π0Y −→ π0Y (this also implies, aposteriori, that the assignment is well-defined, since c is). The proof of this is, one more time, a simplenaturality argument: we have a commutative diagram

π0S

c ∼=

π0(FnSn)π0(λ)oo

c

π0x // π0Y

c

π0S π0(FnSn)

π0(λ)

∼=ooπ0x

// π0Y

and a fundamental class ι ∈ π0(FnSn) [ref/define] which maps to the fundamental class in π0S under theupper left map and to the class [x] under the upper right map. Thus

c[x] = (π0x)(ι) = (π0x)(c(ι)) = (π0x)((π0λ)−1(1)

)= 〈FnSn, λ, x〉 ,

as claimed.

=============leftover========

Construction 4.58. Let J be a category and F : J −→ Sp a functor. We define the homotopy colimitby taking homotopy colimits levelwise in T or sS as explained in Appendix A.?. [structure maps]

We define the homotopy limit by taking homotopy limits levelwise in T or sS as explained in Appen-dix A.?. [structure maps]

The category J-Ab of functors from J to abelian groups has enough projective and enough injectiveobjects [say more] and so derive additive functors. In particular, the colimit functor colim : J-Ab −→ Ab

76 I. BASICS

has left derived functors colimnJ , the derived colimits, and the limit functor lim : J-Ab −→ Ab has right

derived functors limnJ , derived limits.

[Appendix: derived (co-)limits can be calculated by the standard normalized complex, and thus vanishabove the dimension of the nerve of J ]

In the context of spaces we assume in addition that for every object i ∈ J the spectrum F (i) is levelwisecofibrant. There is a spectral sequence

(4.59) E2p,q = colimp

J(πqF ) =⇒ πp+q(hocolimJ F )

[convergence; edge morphism]In the context of simplicial sets we assume in addition that for every object i ∈ J the spectrum F (i) is

levelwise Kan. There is a spectral sequence

(4.60) Ep,q2 = limpJ(π−qF ) =⇒ π−p−q(holimJ F )

[convergence; edge morphism]

Example 4.61. J = • −→ • leads to mapping cone (homotopy fiber) and the spectral sequencedegenerates to the short exact sequences.

Example 4.62. Let M be a monoid and X a symmetric spectrum equipped with a left action of M . Wecan view this data as a functor X : BM −→ Sp from the category BM with one object whose morphismmonoid is M . We denote by XhM the homotopy colimit of the functor X and call it the homotopy orbitspectrum. By inspection of the construction, XhM is isomorphic to the spectrum EM+ ∧M X [explain].In the special case of the category BM , the left derived functors of the colimit are precisely the homologyfunctors, i.e., colimn

BM F = Hn(M,F (∗)) = TorZ[M ]n (Z, F (∗)). So the homotopy colimit spectral sequence

specializes to a spectral sequence

E2p,q = Hp(M, πqX) =⇒ πp+q(XhM ) .

We denote by XhM the homotopy limit of the functor X : BM −→ Sp and call it the homotopy fixedpoint spectrum. By inspection of the construction, XhM is isomorphic to the spectrum mapM (EM+, X)[explain]. In the special case of the category BM , the left derived functors of the limit are precisely thecohomology functors, i.e., limn

BMF = Hn(M,F (∗)) = ExtnZ[M ](Z, F (∗)). So the homotopy limit spectralsequence specializes to a spectral sequence

Ep,q2 = Hp(M, π−qX) =⇒ π−p−q(XhM ) .

5. Smash product

5.1. Construction of the smash product. One of the main features which distinguishes symmetricspectra from the more classical spectra without symmetric group actions is the internal smash product. Thesmash product of symmetric spectra is very much like the tensor product of modules over a commutativering. To stress that analogy, we recall three different ways to look at the classical tensor product and thengive analogies involving the smash product of symmetric spectra.

In the following, R is a commutative ring and M,N and W are right R-modules.

(A) Tensor product via bilinear maps. A bilinear map from M and N to another right R-moduleW is a map b : M ×N −→ W such that for each m ∈ M the map b(m,−) : N −→ W is R-linear and foreach n ∈ N the map b(−, n) : M −→W is R-linear. The tensor product M ⊗R N is the universal exampleof a right R-module together with a bilinear map from M ×N . In other words, there is a specified bilinearmap i : M ×N −→M ⊗R N such that for every R-module W the map

HomR(M ⊗R N,W ) −→ BilinR(M ×N,W ) , f 7→ f iis bijective. As usual, the universal property characterizes the pair (M ⊗R N, i) uniquely up to preferredisomorphism.

5. SMASH PRODUCT 77

(B) Tensor product as an adjoint to internal Hom. The category of right R-modules has‘internal Hom-objects’: the set HomR(N,W ) of R-linear maps between two right R-modules N and W isnaturally an R-module by pointwise addition and scalar multiplication. For fixed right R-modules M andN , the functor HomR(M,HomR(N,−)) : mod-R −→ mod-R is representable and tensor product M ⊗R Ncan be defined as a representing R-module. This point of view is closely related to the first approach sincethe R-modules HomR(M,HomR(N,W )) and BilinR(M ×N,W ) are naturally isomorphic.

(C) Tensor product as a construction. Often the tensor product M ⊗R N is introduced as aspecific construction, usually the following: M ⊗RN is the free R-module generated by symbols of the formm⊗ n for all m ∈M and n ∈ N subject to the following set of relations

• (m+m′)⊗ n = m⊗ n+m′ ⊗ n , m⊗ (n+ n′) = m⊗ n+m⊗ n′• (mr)⊗ n = (m⊗ n) · r = m⊗ (nr)

for all m,m′ ∈ M , n, n′ ∈ N and r ∈ R. Since this is a minimal set of relations which make the mapM ×N −→M ⊗R N given by (m,n) 7→ m⊗ n into a bilinear map, the tensor product is constructed as tohave the universal property (A).

Now we introduce the smash product of symmetric spectra in three ways, analogous to the ones above.

(A) Smash product via bilinear maps. We define a bimorphism b : (X,Y ) −→ Z from a pair ofsymmetric spectra (X,Y ) to a symmetric spectrum Z as a collection of Σp×Σq-equivariant maps of pointedspaces or simplicial sets, depending on the context,

bp,q : Xp ∧ Yq −→ Zp+q

for p, q ≥ 0, such that the diagram

(5.1) Xp ∧ Yq ∧ S1

Id∧σq

vvnnnnnnnnnnnnnnn

bp,q∧Id

Id∧twist // Xp ∧ S1 ∧ Yq

σp∧Id

Xp ∧ Yq+1

bp,q+1((QQQQQQQQQQQQQQQZp+q ∧ S1

σp+q

Xp+1 ∧ Yq

bp+1,q

Zp+q+1 Zp+1+q

1×χ1,q

oo

commutes for all p, q ≥ 0. In Exercise 8.16 we give a justification for calling this notion ‘bimorphism’.The smash product X ∧ Y can now we introduced as the universal example of a symmetric spectrum

with a bimorphism fromX and Y . More precisely, we will show in (C) below that for every pair of symmetricspectra (X,Y ) the functor which assign to Z ∈ Sp the set of bimorphism from (X,Y ) to Z is representable.A smash product of X and Y is then a representing object, i.e., a pair consisting of a symmetric spectrumX ∧ Y and a bimorphism ι : (X,Y ) −→ X ∧ Y such that for every symmetric spectrum Z the map

(5.2) SpΣ(X ∧ Y, Z) −→ Bimor((X,Y ), Z) , f 7−→ fi = fp+q ip,qp,qis bijective. Very often only the object X ∧ Y will be referred to as the smash product, but one shouldkeep in mind that it comes equipped with a specific, universal bimorphism. We will often refer to thebijection (5.2) as the universal property of the smash product of symmetric spectra.

(B) Smash product as an adjoint to internal Hom. In Example 3.19 we introduced ‘internalHom objects’ in the category of symmetric spectra. For every pair of symmetric spectra (X,Y ) we definedanother symmetric spectrum Hom(X,Y ) such that the morphism from X to Y are (in natural bijectionto) the vertices of the 0th level of Hom(X,Y ). We claim that for fixed symmetric spectra X and Y , the

78 I. BASICS

functor Hom(X,Hom(Y,−)) : Sp −→ Sp is representable. The smash product X ∧ Y can then be definedas a representing symmetric spectrum. This point of view can be reduced to perspective (A) since thesets Sp(X,Hom(Y, Z)) and Bimor((X,Y ), Z) are in natural bijection (see Exercise 8.16). In particular,since the functor Bimor((X,Y ),−) is representable, so is the functor Sp(X,Hom(Y,−)). [extend this to anisomorphism of spectra Hom(X,Hom(Y, Z)) ∼= Bimor((X,Y ), Z)]

(C) Smash product as a construction. Now we construct a symmetric spectrum X ∧ Y from twogiven symmetric spectra X and Y . We want X ∧ Y to be the universal recipient of a bimorphism from(X,Y ), and this pretty much tells us what we have to do. For n ≥ 0 we define the nth level (X ∧Y )n as thecoequalizer, in the category of pointed Σn-spaces or pointed Σn-simplicial sets (depending on the context),of two maps

αX , αY :∨

p+1+q=n

Σ+n ∧Σp×Σ1×Σq Xp ∧ S1 ∧ Yq −→

∨p+q=n

Σ+n ∧Σp×Σq Xp ∧ Yq .

The wedges run over all non-negative values of p and q which satisfy the indicated relations. The map αXtakes the wedge summand indexed by (p, 1, q) to the wedge summand indexed by (p+ 1, q) using the map

σXp ∧ Id : Xp ∧ S1 ∧ Yq −→ Xp+1 ∧ Yqand inducing up. The other map αY takes the wedge summand indexed by (p, 1, q) to the wedge summandindexed by (p, 1 + q) using the composite

Xp ∧ S1 ∧ YqId∧twist−−−−−→ Xp ∧ Yq ∧ S1

Id∧σYq−−−−→ Xp ∧ Yq+1Id∧χq,1−−−−−→ Xp ∧ Y1+q

and inducing up.The structure map (X ∧Y )n ∧S1 −→ (X ∧Y )n+1 is induced on coequalizers by the wedge of the maps

Σ+n ∧Σp×Σq Xp ∧ Yq ∧ S1 −→ Σ+

n+1 ∧Σp×Σq+1 Xp ∧ Yq+1

induced from Id∧σYq : Xp∧Yq∧S1 −→ Xp∧Yq+1. One should check that this indeed passes to a well-definedmap on coequalizers. Equivalently we could have defined the structure map by moving the circle past Yq,using the structure map of X (instead of that of Y ) and then shuffling back with the permutation χ1,q; thedefinition of (X ∧Y )n+1 as a coequalizer precisely ensures that these two possible structure maps coincide,and that the collection of maps

Xp ∧ Yqx∧y 7→1∧x∧y−−−−−−−−→

∨p+q=n

Σ+n ∧Σp×Σq Xp ∧ Yq

projection−−−−−−→ (X ∧ Y )p+q

form a bimorphism – and in fact a universal one.

The smash product X ∧ Y is a functor in both variables. This is fairly evident from the construction(C), but it can also be deduced from the universal property (A) or the adjunction (B) as follows. If we usethe universal property (A) the contravariant functoriality of the set Bimor((X,Y ), Z) in X and Y turns intofunctoriality of the representing objects. In more detail, if f : X −→ X ′ and g : Y −→ Y ′ are morphismsof symmetric spectra, then the collection of pointed maps

Xp ∧ Yqfp∧gq−−−−→ X ′

p ∧ Y ′qi′p,q−−→ (X ′ ∧ Y ′)p+q

p,q≥0

forms a bimorphism (X,Y ) −→ X ′ ∧ Y ′. So there is a unique morphism of symmetric spectra f ∧ g :X ∧ Y −→ X ′ ∧ Y ′ such that (f ∧ g)p+q ip,q = i′p,q (fp ∧ gq) for all p, q ≥ 0. The uniqueness part of theuniversal property implies that this is compatible with identities and composition in both variables.

If we define the smash product as a representing object for the functor Hom(X,Hom(Y,−)), thenfunctoriality in X and Y follows from functoriality of the latter functor in X and Y .

5. SMASH PRODUCT 79

5.2. Coherence isomorphisms. Now that we have constructed a smash product functor we caninvestigate its formal and homotopical properties. The formal properties will be discussed in the rest ofthis section, but we postpone the homotopical analysis until Section 3 of Chapter II.

The first thing to show is that the smash product is symmetric monoidal. Since ‘symmetric monoidal’is extra data, and not a property, we are obliged to construct associativity isomorphisms

αX,Y,Z : (X ∧ Y ) ∧ Z −→ X ∧ (Y ∧ Z) ,

symmetry isomorphismsτX,Y : X ∧ Y −→ Y ∧X

and right unit isomorphismsrX : X ∧ S −→ X

which satisfy a certain list of coherence conditions. We then define left unit isomorphisms lX : S∧X −→ Xas the composite of the symmetry isomorphism τS,X and the right unit rX .

First construction. We can obtain all the isomorphisms of the symmetric monoidal structure justfrom the universal property. So suppose that for each pair of symmetric spectra (X,Y ) a smash productX ∧ Y and a universal bimorphism i = ip,q : (X,Y ) −→ X ∧ Y have been chosen.

For construction the associativity isomorphism we notice that the familyXp ∧ Yq ∧ Zr

ip,q∧Id−−−−→ (X ∧ Y )p+q ∧ Zrip+q,r−−−−→ ((X ∧ Y ) ∧ Z)p+q+r

p,q,r≥0

and the familyXp ∧ Yq ∧ Zr

Id∧iq,r−−−−−→ Xp ∧ (Y ∧ Z)q+rip,q+r−−−−→ (X ∧ (Y ∧ Z))p+q+r

p,q,r≥0

both have the universal property of a trimorphism (whose definition is hopefully clear) out of X, Y and Z.The uniqueness of representing objects gives a unique isomorphism of symmetric spectra

αX,Y,Z : (X ∧ Y ) ∧ Z ∼= X ∧ (Y ∧ Z)

such that (αX,Y,Z)p,q,r ip+q,r (ip+q ∧ Id) = ip,q+r (Id∧iq,r).The symmetry isomorphism τX,Y : X ∧ Y −→ Y ∧X corresponds to the bimorphism

(5.3)Xp ∧ Yq

twist−−−→ Yq ∧ Xpιq,p−−→ (Y ∧X)q+p

χq,p−−−→ (Y ∧X)p+qp,q≥0

.

The block permutation χq,p is crucial here: without it the diagram (5.1) would not commute and wewould not have a bimorphism. If we restrict the composite τY,X τX,Y in level p + q along the mapip,q : Xp ∧ Yq −→ (X ∧ Y )p+q we get ip,q again. Thus τY,X τX,Y = IdX∧Y and τY,X is inverse to τX,Y .

In much the same spirit, the universal properties can be used to provide a right unit isomorphism.Because of the commuting left part of the diagram (5.1) a bimorphism b : (X, S) −→ Z is completelydetermined by the components bp,0 : Xp ∧ S0 −→ Zp, which constitute a morphism b•,0 : X −→ Z;moreover, every morphism from X to Z arises in this way from a unique bimorphism out of (X, S). Hencethe morphism rX : X ∧S −→ X corresponding to the bimorphism consisting of the iterated structure mapsσm : Xn ∧ Sm −→ Xn+m is an isomorphism of symmetric spectra.

Second construction. The coherence isomorphisms can also be obtained from the construction ofthe smash product in (C) above, as opposed to the universal property. In level n the spectra (X ∧ Y ) ∧ Zand X ∧ (Y ∧ Z) are quotients of the spaces∨

p+q+r=n

Σ+n ∧Σp+q×Σr

(Σ+p+q ∧Σp×Σq Xp ∧ Yq

)∧ Zr

80 I. BASICS

respectively ∨p+q+r=n

Σ+n ∧Σp×Σq+r Xp ∧

(Σ+q+r ∧Σq×Σr Yq ∧ Zr

).

The wedges run over all non-negative values of p, q and r which sum up to n. We get a well-defined mapsbetween these two wedges by wedging over the maps

Σ+n ∧Σp+q×Σr

(Σ+p+q ∧Σp×Σq Xp ∧ Yq

)∧ Zr ←→ Σ+

n ∧Σp×Σq+r Xp ∧(Σ+q+r ∧Σq×Σr Yq ∧ Zr

)σ ∧ ((τ ∧ x ∧ y) ∧ z) 7−→ (σ(τ × 1)) ∧ (x ∧ (1 ∧ y ∧ z))

σ(1× γ) ∧ ((1 ∧ x ∧ y) ∧ z) ←− σ ∧ (x ∧ (γ ∧ y ∧ z))

where σ ∈ Σn, τ ∈ Σp+q, γ ∈ Σq+r, x ∈ Xp, y ∈ Yq and z ∈ Zr.The symmetry isomorphism τX,Y : X ∧X −→ Y ∧X is obtained by wedging over the maps

Σ+n ∧Σp×Σq Xp ∧ Yq −→ Σ+

n ∧Σq×Σp Yq ∧Xp

σ ∧ x ∧ y 7−→ (σχp,q) ∧ y ∧ x

where σ ∈ Σn, x ∈ Xp and y ∈ Yq and passing to quotient spaces. The shuffle permutation χp,q is neededto make this map well-defined on quotients.

Theorem 5.4. The associativity, symmetry and unit isomorphisms make the smash product of sym-metric spectra into a symmetric monoidal product with unit object the sphere spectrum S. This product isclosed symmetric monoidal in the sense that the smash product is adjoint to the internal Hom spectrum,i.e., there is an adjunction isomorphism

Hom(X ∧ Y, Z) ∼= Hom(X,Hom(Y, Z)) .

Proof. We have to verify that several coherence diagrams commute. We start with the pentagoncondition for associativity. Given a fourth symmetric spectrum W we consider the pentagon

((W ∧X) ∧ Y ) ∧ ZαW,X,Y ∧Id

ssgggggggggggggggggggαW∧X,Y,Z

++WWWWWWWWWWWWWWWWWWWW

(W ∧ (X ∧ Y )) ∧ Z

αW,X∧Y,Z ''NNNNNNNNNNN(W ∧X) ∧ (Y ∧ Z)

αW,X,Y∧Zwwooooooooooo

W ∧ ((X ∧ Y ) ∧ Z)Id∧αX,Y,Z

// W ∧ (X ∧ (Y ∧ Z)))

If we evaluate either composite at level o+ p+ q + r and precompose with

Wo ∧Xp ∧ Yq ∧ Zrio,p∧Id∧ Id−−−−−−−→ (W ∧X)o+p ∧ Yq ∧ Zrio+p,q∧Id−−−−−−→ ((W ∧X) ∧ Y )o+p+q ∧ Zr

io+p+q,r−−−−−→ (((W ∧X) ∧ Y ) ∧ Z)o+p+q+r

then both ways around the pentagon yield the composite

Wo ∧Xp ∧ Yq ∧ ZrId∧ Id∧iq,r−−−−−−−→Wo ∧Xp ∧ (Y ∧ Z)q+rId∧ip,q+r−−−−−−→Wo ∧ (X ∧ (Y ∧ Z))p+q+r

io,p+q+r−−−−−→ (W ∧ (X ∧ (Y ∧ Z)))o+p+q+r .

So the uniqueness part of the universal property shows that the pentagon commutes.

5. SMASH PRODUCT 81

Coherence between associativity and symmetry isomorphisms means that the two composites from theupper left to the lower right corner of the diagram

(X ∧ Y ) ∧ ZαX,Y,Z //

τX,Y ∧Id

X ∧ (Y ∧ Z)τX,Y∧Z // (Y ∧ Z) ∧X

αY,Z,X

(Y ∧X) ∧ Z

αY,X,Z// Y ∧ (X ∧ Z)

Id∧τX,Z// Y ∧ (Z ∧X)

should be equal, and the same kind of argument as for the pentagon relation for associativity works.It remains to check the coherence conditions relating associativity and symmetry isomorphisms to the

unit morphisms. We define the left unit isomorphism lX : S ∧X ∼= X as the composite lX = rXτS,X of theright unit and the symmetry isomorphism. Then the unit isomorphism are compatible with symmetry, andfurthermore we have lS = rS : S ∧ S −→ S since both arise from the bimorphism Sn ∧ Sm

∼=−→ Sn+m madeup from the canonical isomorphisms. Finally, the triangle

(X ∧ S) ∧ YαX,S,Y //

rX∧Id &&NNNNNNNNNNX ∧ (S ∧ Y )

Id∧lYxxpppppppppp

X ∧ Y

should commute, which is true since it holds after restriction with the maps

(Xp ∧ Sq) ∧ Yrip,q∧Id−−−−→ (X ∧ S)p+q ∧ Yr

ip+q,r−−−−→ ((X ∧ S) ∧ Y )p+q+r .

5.3. Smash product and various other constructions. Now we identify the smash products ofcertain kinds of symmetric spectra and relate it by natural maps to other constructions. We start bydescribing the smash product with a semifree spectrum. In Example 3.14 we introduced the twisted smashproduct L .m X of a pointed Σm-space L and a symmetric spectrum X.

Let X be a symmetric spectrum and L be a pointed Σm-space (or Σm-simplicial set) for some m ≥ 0.The twisted smash product L .m X was defined in Example 3.14 and consists of a point in levels smallerthan m is given in general by

(L .m X)m+n = Σ+m+n ∧Σm×Σn L ∧Xn .

In order to link L .m X to GmL ∧X we note that as n varies, the (m,n)-components

L ∧Xn = (GmL)m ∧Xnιm,n−−−→ (GmL ∧X)m+n = (shm(GmL ∧X))n

of the universal bimorphism ι : (GmL,X) −→ GmL∧X in fact define a morphism of Σm-symmetric spectrab : L ∧ X −→ shm(GmL ∧ X). By the adjunction (3.15) this morphism corresponds to a morphism ofsymmetric spectra b : L .m X −→ GmL ∧X.

Proposition 5.5. Let L be a pointed Σm-space (or Σm-simplicial set) for some m ≥ 0 and X asymmetric spectrum. Then the morphism of symmetric spectra

b : L .m X −→ GmL ∧X

is a natural isomorphism.

Proof. In (3.15) we constructed a natural bijection

Sp(L .m X,Z) ∼= Σm-Sp(L ∧X, shm Z) .

82 I. BASICS

The adjunctions between the smash product and mapping spaces, the definition of the homomorphismspectrum and the fact that Gm is adjoint to evaluation at level m yield a natural bijection

Σm-Sp(L ∧X, shm Z)) ∼= Σm-sS(L,map(X, shm Z))

= Σm-sS(L,Hom(X,Z)m) ∼= Sp(GmL,Hom(X,Z)) .

Combining all these isomorphisms gives a representation of the functor Sp(GmL,Hom(X,−)) by the sym-metric spectrum L.mX. Since the smash product GmL∧X represents the same functor, we get a preferredisomorphism L .m X ∼= GmL ∧X, which in fact equals the morphism b.

We specialize the previous proposition in several steps. The special case m = 0 provides a naturalisomorphism

K ∧X = K .0 X ∼= (Σ∞K) ∧X

for pointed spaces (or simplicial sets) K and symmetric spectra X. We can also consider a Σm-space Land a Σn-space L′. If we spell out all definitions we see that L .m (GnL′) is isomorphic to the semifreesymmetric spectrum Gm+n(Σ+

m+n ∧Σm×Σn L∧L′). So Proposition 5.5 specializes to a natural isomorphism

(5.6) Gm+n(Σ+m+n ∧Σm×Σn L ∧ L′) ∼= GmL ∧GnL′ .

The isomorphism is adjoint to the Σm+n-equivariant map

Σ+m+n ∧Σm×Σn L ∧ L′ −→ (GmL ∧GnL′)m+n

which in turn is adjoint to the Σm × Σn-equivariant map

L ∧ L′ = (GmL)m ∧ (GnL′)nim,n−−−→ (GmL ∧GnL′)m+n

given by the universal bimorphism. So the isomorphism (5.6) rephrases the fact that a bimorphism from(GmL,GnL′) to Z is uniquely determined by its (m,n)-component, which can be any Σm×Σn-equivariantmap L ∧ L′ −→ Zm+n. The isomorphism (5.6), and the ones which follow below, are suitably associative,commutative and unital.

As a special case we can consider smash products of free symmetric spectra. If K and K ′ are pointedspaces or simplicial set then we have FmK = Gm(Σ+

m ∧ K) and FnK′ = Gn(Σ+

n ∧ K ′), so the isomor-phism (5.6) specializes to an associative, commutative and unital isomorphism

Fm+n(K ∧K ′) ∼= FmK ∧ FnK ′ .

As the even more special case for m = n = 0 we obtain a natural isomorphism of suspension spectra

(Σ∞K) ∧ (Σ∞L) ∼= Σ∞(K ∧ L)

for all pairs of pointed spaces (or pointed simplicial sets) K and L.The bimorphism

(shX)p ∧ Yq = X1+p ∧ Yqi1+p,q−−−−→ (X ∧ Y )1+p+q = sh(X ∧ Y )p+q

corresponds to a natural homomorphism of symmetric spectra

(5.7) ξX,Y : (shX) ∧ Y −→ sh(X ∧ Y ) .

5. SMASH PRODUCT 83

The homomorphism ξX,Y is compatible with the unit and associativity isomorphisms in the sense that thefollowing diagrams commute

((shX) ∧ Y ) ∧ ZαshX,Y,Z //

ξX,Y ∧Id

(shX) ∧ (Y ∧ Z)

ξX,Y∧Z

(shX) ∧ SrshX

''OOOOOOOOOOOO

ξX,S

sh(X ∧ Y ) ∧ Z

ξX∧Y,Z

shX

sh((X ∧ Y ) ∧ Z)sh(αX,Y,Z)

// sh(X ∧ (Y ∧ Z)) sh(X ∧ S)sh(rX)

77oooooooooooo

Moreover, the map ξS,Y ‘is’ the morphism λY : S1∧Y −→ shY is the following sense: we have sh S = Σ∞S1

and so (sh S) ∧ Y ∼= S1 ∧ Y . With this identification, the following square commutes

(sh S) ∧ Y

∼=

ξS,Y // sh(S ∧ Y )

sh(lY )

S1 ∧ Y

λY

// shY

As a special case of the above, or by direct verification from the definitions, we obtain that the square

(5.8) (S1 ∧X) ∧ YαS1,X,Y //

λX∧Id

S1 ∧ (X ∧ Y )

λX∧Y

(shX) ∧ Y

ξX,Y

// sh(X ∧ Y )

commutes.We also need a version of the morphism ξX,Y where in the second factor in the source is shifted. So we

define the morphism ξX,Y : X ∧ (shY ) −→ sh(X ∧ Y ) as the composite ξX,Y = sh(τY,X) ξY,X τX,shY .Put another way, ξ is defined so as to make the square

(shY ) ∧XτshY,X //

ξY,X

X ∧ (shY )

ξX,Y

sh(Y ∧X)

sh(τY,X)// sh(X ∧ Y )

commute. [compare ξ with the map L.1+m (shY ) −→ sh(L.1+mY ) if X = G1+mL] The natural morphismξX,Y satisfies analogous compatibilities with the unit and associativity isomorphisms as ξX,Y , which we donot spell out.

For a symmetric spectrum X we define two natural morphisms

ζL,X : (shL) .m X −→ sh(L .1+m X) and ξL,X : L .1+m (shX) −→ sh(L .1+m X)

as follows. In level m+ n the morphism ζL,X is given by

(shL) .m X = Σ+m+n ∧Σm×Σn (shL) ∧Xn −→ Σ+

1+m+n ∧Σ1+m×Σn L ∧Xn = sh(L .1+m X)

obtained by smashing the monomorphism +− : Σm+n −→ Σ1+m+n with the identity of L and Xn. [laterwe have to identify them with instances of the shearing morphism ξ]

Lemma 5.9. For m ≥ 0, every pointed Σ1+m-space (or simplicial set) L and symmetric spectrum Xthe morphism

ζL,X ∨ ξL,X : (shL) .m X ∨ L .1+m (shX) −→ sh(L .1+m X)

84 I. BASICS

is an isomorphism. Moreover the morphism λL.1+mX : S1 ∧ (L .1+m X) −→ sh(L .1+m X) equals thecomposite of the map

S1 ∧ (L .1+m X) ∼= L .1+m (S1 ∧X)L.1+mλX−−−−−−→ L .1+m (shX)

ξL,X−−−→ sh(L .1+m X) .

Proof. We note that Σ1+m+k consists of exactly two double cosets with respect to the subgroups1× Σm+k (acting from the left) and Σ1+m × Σk (acting from the right), namely

A = γ ∈ Σ1+m+k | γ−1(1) ≤ 1 +m and B = γ ∈ Σ1+m+k | γ−1(1) > 1 +m .These two cosets are represented by the identity and the cycle (1, 2, . . . , 1 +m+ 1). We view Σ1+m+k as aleft-Σm+k, right-(Σ1+m ×Σk)-biset, where the left action is through the monomorphism 1 +− : Σm+k −→Σ1+m+k. As a biset, Σ1+m+k = A q B then decomposes as the disjoint union of two transitive doublecosets, which we identify as follows. The maps

Σm+k ×(Σm×Σk) (Σ1+m × Σk) −→ (1× Σm+k) · 1 · (Σ1+m × Σk) = A

(σ, (α, β)) 7−→ (1 + σ) · (α+ β)

respectively

Σm+k ×(Σ1+m×Σ−1+k) (Σ1+m × Σk) −→ (1× Σm+k) · (1, 2, . . . , 1 +m+ 1) · (Σ1+m × Σk) = B

(σ, (α, β)) 7−→ (1 + σ) · (1, 2, . . . , 1 +m+ 1) · (α+ β)

are isomorphisms of Σm+k-(Σ1+m×Σk)-bisets since the targets are transitive and both sides have the samenumber of elements. In the source of the first map we use the monomorphism 1+− : Σm −→ Σ1+m; in thesource of the second map, we use the monomorphism 1 +− : Σ−1+k −→ Σk.

Now we can identify level m+ k of the spectrum sh(L .1+m X) as

(sh(L .1+m X))m+k = (L .1+m X)1+m+k = Σ+1+m+k ∧Σ1+m×Σk (L ∧Xk)

∼= (Σm+k ×(Σm×Σk) (Σ1+m × Σk))+ ∧Σ1+m×Σk (L ∧Xk)

∨ (Σm+k ×(Σ1+m×Σ−1+k) (Σ1+m × Σk))+ ∧Σ1+m×Σk (L ∧Xk)∼= Σ+

m+k ∧(Σm×Σk) ((shL) ∧Xk) ∨ Σ+m+k ∧(Σ1+m×Σ−1+k) (L ∧ (shX)−1+k)

= ((shL) .m X)m+k ∨ (L .1+m (shX))m+k

All these isomorphisms are Σm+k-equivariant, and they are compatible with the structure maps because thedouble coset decomposition and the representatives for the double cosets which were use are stable whenincreasing k along the ‘inclusion’ −+ 1 : Σ1+m+k −→ Σ1+m+k+1.

Proof. We note that Σ1+m+k consists of exactly two double cosets with respect to the subgroups1× Σm+k (acting from the left) and Σ1+m × Σk (acting from the right), namely

A = γ ∈ Σ1+m+k | γ−1(1) ≤ 1 +m and B = γ ∈ Σ1+m+k | γ−1(1) > 1 +m .These two cosets contain the identity respectively and the cycle (1, 2, . . . , 1 +m+ 1).

We can thus decompose level m+ k of the spectrum sh(L .1+m X) as

(sh(L .1+m X))m+k = (L .1+m X)1+m+k = Σ+1+m+k ∧Σ1+m×Σk (L ∧Xk)

A+ ∧Σ1+m×Σk (L ∧Xk) ∨ B+ ∧Σ1+m×Σk (L ∧Xk)∼= Σ+

m+k ∧(Σm×Σk) ((shL) ∧Xk) ∨ Σ+m+k ∧(Σ1+m×Σ−1+k) (L ∧ (shX)−1+k)

= ((shL) .m X)m+k ∨ (L .1+m (shX))m+k

All these isomorphisms are Σ1+m+k-equivariant, and thus equivariant for the restricted action of Σm+k,and they are compatible with the structure maps because the double coset decomposition is preserved whenincreasing k along the ‘inclusion’ − + 1 : Σ1+m+k −→ Σ1+m+k+1. So altogether we get an internal wedgedecomposition of the symmetric spectrum sh(L.1+mX) into two summands which we denote shA(L.1+mX)respectively shB(L .1+m X).

5. SMASH PRODUCT 85

Now we show that the morphisms ζL,X : (shL) .m X −→ sh(L .1+m X) and ξL,X : L .1+m (shX) −→sh(L .1+m X) are isomorphisms onto the summands shA(L .1+m X) respectively shB(L .1+m X).

We view Σ1+m+k as a left-Σm+k, right-(Σ1+m×Σk)-biset, where the left action is through the monomor-phism 1 + − : Σm+k −→ Σ1+m+k. As a biset, Σ1+m+k = A q B then decomposes as the disjoint union oftwo transitive double cosets, which we identify as follows. The maps

Σm+k ×(Σm×Σk) (Σ1+m × Σk) −→ (1× Σm+k) · 1 · (Σ1+m × Σk) = A

(σ, (α, β)) 7−→ (1 + σ) · (α+ β)

respectively

Σm+k ×(Σ1+m×Σ−1+k) (Σ1+m × Σk) −→ (1× Σm+k) · (1, 2, . . . , 1 +m+ 1) · (Σ1+m × Σk) = B

(σ, (α, β)) 7−→ (1 + σ) · (1, 2, . . . , 1 +m+ 1) · (α+ β)

are isomorphisms of Σm+k-(Σ1+m×Σk)-bisets since the targets are transitive and both sides have the samenumber of elements. In the source of the first map we use the monomorphism 1+− : Σm −→ Σ1+m; in thesource of the second map, we use the monomorphism 1 +− : Σ−1+k −→ Σk.

We have G1+mL = L .1+m S and sh S = S1 ∧ S, so as a special case of the previous lemma we obtain awedge decomposition of the shift of a semifree symmetric spectrum

(5.10) sh(G1+mL) ∼= Gm(shL) ∨ (S1 ∧G1+mL)

under which the morphism λG1+mL : S1 ∧ G1+mL −→ sh(G1+mL) becomes the inclusion of the secondwedge summand.

We can specialize even further to free symmetric spectra. Using F1+mK = G1+m(Σ+1+m ∧K) and the

fact that sh(Σ1+m) is the disjoint union of (1 +m) free transitive Σm-sets we obtain

(5.11) sh(F1+mK) ∼=1+m∨i=1

FmK ∨ F1+m(S1 ∧K) .

[naive homotopy groups of L .mX can be described functorially from the naive homotopy groups of L∧Xand the Σm-action on that... is the stable homotopy type that of (Ωm(L ∧X))hΣm ??]

We can use the morphisms ξX,Y and ξX,Y to express the shift of a smash product X ∧ Y as anamalgamated union of the spectra (shX) ∧ Y and X ∧ (shY ). By direct verification from the definitionswe see that the diagram

(5.12) (S1 ∧X) ∧ Y twist //

λX∧Id

X ∧ (S1 ∧ Y )Id∧λY // X ∧ (shY )

ξX,Y

(shX) ∧ Y

ξX,Y

// sh(X ∧ Y )

commutes [check]. Moreover, up to the associativity isomorphism, the composite morphism from the initialto the terminal vertex agrees with the morphism λX∧Y : S1 ∧ (X ∧ Y ) −→ sh(X ∧ Y ) [check].

Proposition 5.13. For every pair of symmetric spectra X and Y the square (5.12) is a pushout.

Proof. All four corners of the square as well as the pushout commute with colimits. Since everysymmetric spectrum is naturally a coequalizer of two morphisms between wedges of semifree spectra (com-pare (3.13)), it it suffices to verify the special case where X = GmL is the semifree symmetric spectrumgenerated by a pointed Σm-space (or simplicial set) L.

In this special case the spectrum sh(GmL) is naturally isomorphic to the wedge of Gm−1(shL) andS1 ∧ GmL (see (5.10)) and under this decomposition, the morphism λGmL : S1 ∧ GmL −→ sh(GmL)corresponds to the wedge summand inclusion. If we smash sh(GmL) with Y we get an induced wedgesplitting and so it suffices to show that sh(GmL ∧ Y ) is the wedge of GmL ∧ (shY ) and the remaining

86 I. BASICS

summand Gm−1(shL) ∧ Y , via the morphisms ξGmL,Y and ??? : Gm−1(shL) ∧ Y −→ sh(GmL ∧ Y ). If weuse the natural isomorphisms between the functors GmL∧− and L.m−, then this is precisely the contentof Lemma 5.9. Here we omit the verification that the morphisms which provide the splittings of Lemma 5.9and (5.10) are instances of the morphism ξ and ξ.

There are natural composition morphisms

(5.14) : Hom(Y, Z) ∧ Hom(X,Y ) −→ Hom(X,Z)

which are associative and unital with respect to a unit map S −→ Hom(X,X) adjoint to the identity of X(which is a vertex in level 0 of the spectrum Hom(X,X)). The composition morphism is obtained, by theuniversal property of the smash product, from the bimorphism consisting of the maps

map(Y, shn Z) ∧map(X, shm Y ) shm ∧ Id−−−−−→ map(shm Y, shm(shn Z)) ∧map(X, shm Y )

=map(shm Y, shn+m Z) ∧map(X, shm Y ) −−−−→ map(X, shn+m Z)

where the second map is the composition pairing of Example 3.17. We refer to (3.8) for why it is ‘right’ toidentify shm(shn Z) with shn+m Z (note the orders in which m and n occur), so that no shuffle permutationis needed. If we specialize to X = Y = Z, we recover the multiplication of the endomorphism ring spectrumas defined in Example 3.21.

If X and Y are symmetric spectra we can also define natural coherent morphisms

XK ∧ Y L −→ (X ∧ Y )K∧L

for pointed spaces (simplicial sets) K and L and morphisms

map(A,X) ∧map(B, Y ) −→ map(A ∧B,X ∧ Y ) and

Hom(A,X) ∧Hom(B, Y ) −→ Hom(A ∧B,X ∧ Y )

for symmetric spectra A and B.The functors of geometric realization and singular complex (compare Section 1) which relate symmetric

spectra of spaces respectively simplicial sets are nicely compatible with the smash products for symmetricspectra. As for the unstable objects themselves, geometric realization is strong symmetric monoidal, i.e.,commutes with the smash product up to coherent isomorphism. The singular complex is at least laxsymmetric monoidal, i.e., allows an associative and commutative natural transformation.

Here are some more details. We let Y and Y ′ be symmetric spectra of simplicial sets. Then |Y | is thegeometric realization, a symmetric spectrum of topological spaces, and similarly for |Y ′|. We can considerthe composite maps

|Yp| ∧ |Y ′q |rYp,Y ′q−−−−→ |Yp ∧ Y ′q |

|ip,q|−−−→ |(Y ∧ Y ′)p+q|where the isomorphism rA,B : |A| ∧ |B| ∼= |A ∧ B| was discussed in (3.2) and ip,q is a component of theuniversal bimorphism. As p and q vary the collection of these maps constitute a bimorphism from (|Y |, |Y ′|)to |Y ∧ Y ′|, which gives rise to a preferred morphism of symmetric spectra of spaces

(5.15) rY,Y ′ : |Y | ∧ |Y ′| −→ |Y ∧ Y ′| .

This morphism is a natural isomorphism [justify?], and associative, unital and commutative.Now we let X and X ′ be two symmetric spectra of topological spaces. Similarly as above, the composite

maps

S(Xp) ∧ S(X ′q) −→ S(Xp ∧X ′

q)S(ip,q)−−−−→ S((X ∧X ′)p+q)

constitute a bimorphism from (S(X),S(X ′)) to S(X ∧ X ′), which gives rise to a preferred morphism ofsymmetric spectra of simplicial sets

S(X) ∧ S(X ′) −→ S(X ∧X ′) .

This natural morphism is also associative, unital and commutative, but in general not an isomorphism.

5. SMASH PRODUCT 87

Now we can make precise the idea that symmetric ring spectra are the same as monoid objects in thesymmetric monoidal category of symmetric spectra with respect to the smash product.

Construction 5.16. Let us define an implicit symmetric ring spectrum as a symmetric spectrum Rtogether with morphisms µ : R ∧ R −→ R and ι : S −→ R which are associative and unital in the sensethat the following diagrams commute

(R ∧R) ∧RαR,R,R //

µ∧Id

R ∧ (R ∧R)Id∧µ // R ∧R

µ

S ∧R

lR''NNNNNNNNNNNNN

ι∧Id // R ∧Rµ

R ∧ SId∧ιoo

rRwwppppppppppppp

R ∧R µ// R R

We say that the implicit symmetric ring spectrum (R,µ, ι) is commutative if the multiplication is unchangedwhen composed with the symmetric isomorphism, i.e., if the relation µ τR,R = µ holds.

Given an implicit symmetric ring spectrum (R,µ, ι) we can make the collection of Σn-spaces Rn into asymmetric ring spectrum in the sense of the original Definition 1.3 as follows. As unit maps we simply takethe components of ι : S −→ R in levels 0 and 1. We define the multiplication map µn,m : Rn∧Rm −→ Rn+m

as the composite

Rn ∧Rmin,m−−−→ (R ∧R)n+m

µn+m−−−−→ Rn+m .

Then the associativity condition for µ above directly translates into the associativity condition of Defini-tion 1.3 for the maps µn,m. Evaluating the two commuting unit triangles in level 0 gives the unit conditionof Definition 1.3. Spelling out the condition µ(ι ∧ Id) = lR = rR τS,R in level 1 + n and composing withthe map i1,n : S1∧Rn −→ (S∧R)1+n gives the centrality condition of Definition 1.3. Finally, the conditionµ(Id∧ι) = rR in level n+ 1 composed with ιn,1 : Rn ∧ S1 −→ (R ∧ S)n+1 shows that µn,1 (Id∧ι1) equalsthe structure map σn : Rn ∧ S1 −→ Rn+1 of the underlying symmetric spectrum of R. So the conceivablydifferent meaning of ‘underlying symmetric spectrum’ in the sense of Remark 1.6 (iii) in fact coincides withthe underlying spectrum R [commutativity]. Altogether this proves:

Theorem 5.17. The construction 5.16 which turns an implicit symmetric ring spectrum into a symmet-ric ring spectrum in sense of the original Definition 1.3 is an isomorphism between the category of implicitsymmetric ring spectra and the category of symmetric ring spectra. The functor restricts to an isomorphismfrom the category of commutative implicit symmetric ring spectra to the category of commutative symmetricring spectra.

Now that we have carefully stated and proved Theorem 5.17 we will start to systematically blur the dis-tinction between implicit and explicit symmetric ring spectra. Whenever convenient we use the isomorphismof categories to go back and forth between the two notions without further mentioning.

Example 5.18 (Smash product of ring spectra). Here is a construction of a new symmetric ringspectrum from old ones for which the possibility to define ring spectra ‘implicitly’ is crucial. If R and S aresymmetric ring spectra, then the smash product R∧S has a natural structure as symmetric ring spectrumas follows. The unit map is defined from the unit maps of R and S as the composite

Sr−1

S,S =l−1S,S←−−−−−−− S ∧ S ι∧ι−−−−−→ R ∧ S .

The multiplication map of R ∧ S is defined from the multiplications of R and S as the composite

(R ∧ S) ∧ (R ∧ S)Id∧τS,R∧Id−−−−−−−−→ (R ∧R) ∧ (S ∧ S)

µ∧µ−−−→ R ∧ S ,

where we have suppressed some associativity isomorphisms. It is a good exercise to insert these associativityisomorphisms and observe how the hexagon condition for associativity and symmetry isomorphisms entersthe verification that the product of R ∧ S is in fact associative.

88 I. BASICS

Example 5.19 (Tensor and symmetric algebra). Another class of examples which can only be givenas implicit symmetric ring spectra are symmetric ring spectra ‘freely generated’ by a symmetric spectrum.These come in two flavors, an associative and a commutative (and associative) version.

Given a symmetric spectrum X we define the tensor algebra as the symmetric spectrum

TX =∨n≥0

X ∧ · · · ∧X︸︷︷︸n

with the convention that a 0-fold smash product is the unit object S. The unit morphism ι : S −→ TXis the inclusion of the wedge summand for n = 0. The multiplication is given by ‘concatenation’, i.e., therestriction of µ : TX ∧ TX −→ TX to the (n,m) wedge summand is the canonical isomorphism

X∧n ∧X∧m ∼=−→ X∧(n+m)

followed by the inclusion of the wedge summand indexed by n+m. In order to be completely honest here weshould throw in several associativity isomorphisms; strictly speaking already the definition of TX requireschoices of how to associate expressions such as X ∧X ∧X and higher smash powers. However, all of this istaken care of by the coherence conditions of the associativity (and later the symmetry) isomorphisms, andwe will not belabor this point any further.

Given any symmetric ring spectrum R and a morphism of symmetric spectra f : X −→ R we can definea new morphism f : TX −→ R which on the nth wedge summand is the composite

X∧n f∧n−−→ R∧nµn−−→ R .

Here µn is the iterated multiplication map, which for n = 0 has to be interpreted as the unit morphismι : S −→ R. This extension f : TX −→ R is in fact a homomorphism of (implicit) symmetric ring spectra.Moreover, if g : TX −→ R is any homomorphism of symmetric ring spectra then g = g1 for g1 : X −→ Rthe restriction of g to the wedge summand indexed by 1. Another way to say this is that

Homring spectra(TX,R) −→ Sp(X,R) , g 7→ g1

is a natural bijection. In fact, this bijection makes the tensor algebra functor into a left adjoint of theforgetful functor from symmetric ring spectra to symmetric spectra.

The construction has a commutative variant. We define the symmetric algebra generated by a symmetricspectrum X as

PX =∨n≥0

(X∧n)/Σn .

Here Σn permutes the smash factors [elaborate] of X∧n using the symmetry isomorphisms, and we takethe quotient symmetric spectrum. This symmetric spectrum has unique unit and multiplication maps suchthat the quotient morphism TX −→ PX becomes a homomorphism of symmetric ring spectra. So the unitmorphism ι : S −→ PX is again the inclusion of the wedge summand for n = 0 and the multiplication isthe wedge of the morphisms

(X∧n)/Σn ∧ (X∧m)/Σm −→ (X∧(n+m))/Σn+m

induced on quotients by X∧n ∧X∧m ∼= X∧(n+m).

Example 5.20. For two abelian groups A and B, a natural morphism of symmetric spectra

mA,B : HA ∧HB −→ H(A⊗B)

is obtained, by the universal property (5.2), from the bilinear morphism

(HA)n ∧ (HB)m = (A⊗ Z[Sn]) ∧ (B ⊗ Z[Sm])

−→ (A⊗B)⊗ Z[Sn+m] = (H(A⊗B))n+m

5. SMASH PRODUCT 89

given by (∑i

ai · xi

)∧

∑j

bj · yj

7−→∑i,j

(ai ⊗ bj) · (xi ∧ yj) .

A unit map S −→ HZ is given by the inclusion of generators. With respect to these maps, H becomes alax symmetric monoidal functor from the category of abelian groups to the category of symmetric spectra.As a formal consequence, H turns a ring A into a symmetric ring spectrum with multiplication map

HA ∧HA mA,A−−−−→ H(A⊗A)Hµ−−→ HA .

This is the ‘implicit’ construction of an Eilenberg-Mac Lane ring spectrum whose explicit variant appearedin Example 1.14. Similarly, an A-module structure on B gives rise to an HA-module structure on HB.

The definition of the symmetric spectrum HA makes just as much sense when A is a simplicial abeliangroup; thus the Eilenberg-Mac Lane functor makes simplicial rings into symmetric ring spectra, respectingpossible commutativity of the multiplications.

5.4. Latching spaces.

Construction 5.21 (Skeleton filtration). There is a functorial way to write a symmetric spectrum asa sequential colimit of spectra which are made from the information below of fixed level. This is somewhatanalogous to the skeleton filtration of a simplicial set, which ultimately arises from filtering the category ∆of finite ordered sets by cardinality. We thus refer to this as the skeleton filtration of a symmetric spectrum.The word ‘filtration’ should maybe be set in quotes here because the maps from the skeleta to the symmetricspectrum need not be injective.

A k-truncated symmetric spectrum a sequence of pointed spaces (or simplicial sets) Xn for 0 ≤ n ≤ k, abasepoint preserving continuous left action of the symmetric group Σn on Xn for each 0 ≤ n ≤ k and basedstructure maps σn : Xn ∧ S1 −→ Xn+1 for 0 ≥ n ≤ k − 1. This data is subject to the same equivariancecondition as for symmetric spectrum on the iterated structure maps σm : Xn ∧ Sm −→ Xn+m, but onlywhere it makes sense, i.e., as long as n + m ≤ k. A morphism f : X −→ Y of k-truncated symmetricspectra consists of Σn-equivariant based maps fn : Xn −→ Yn for 0 ≤ n ≤ k which are compatible withthe structure maps in the sense that fn+1 σn = σn (fn ∧ IdS1) for all 0 ≤ n ≤ k − 1. The category ofsymmetric spectra is denoted by SpΣ

≤k. There is a forgetful functor τ≤k : SpΣ −→ SpΣ≤k which forgets all

information above level k, and which we call truncation at level k.We claim that the truncation functor has both a left and right adjoint. The easiest way to see this is to

interpret k-truncated symmetric spectra as a brand of enriched functors and then use the general machine ofenriched Kan extensions. We let Σ≤k be the full topological (or simplicial) subcategory of Σ with objectsthe natural numbers at most k. In Proposition 8.1 we provided an isomorphism of categories betweensymmetric spectra and enriched functors Σ −→ T; this isomorphism restricts to an isomorphism betweenthe category of k-truncated symmetric spectra and enriched functors Σ≤k −→ T. Under this isomorphism,the truncation functor τ≤k : SpΣ −→ SpΣ

≤k corresponds to restriction to the subcategory Σ≤k. The generalmachine [ref to Appendix] provides left and right enriched Kan extensions l, r : SpΣ

≤k −→ SpΣ [notation?]resulting in two adjunctions

Sp≤k

l''

r

77Sp

τ≤koo .

We define a functorF k : Sp −→ Sp

by F kX = l(τ≤kX) and refer to F kX as the k-th skeleton of the symmetric spectrum X. We use theconvention that F−1X is the trivial spectrum. The adjunction unit provides a natural morphism κk :F kX −→ X. Since the inclusion Σ≤k −→ Σ is fully faithful, the left Kan extension functor l does notchange the values in levels below and including k; in other words, the adjunction unit κk : F kX −→ X is

90 I. BASICS

an isomorphism in levels 0, . . . , k. A word of warning: although the functor F k behaves in many ways likea skeleton, then morphism κk : F kX −→ X in not generally injective! Factorization jk : F k−1X −→ F kX;in every given level n, the sequence of morphisms stabilizes to X in the strong sense that almost all maps(κk)n : (F kX)n −→ Xn are isomorphisms. So X is a colimit of the sequence, with respect to the morphismsκk.

We can analyze to some extent what happens from the (m − 1)th filtration to the mth filtration. Forthis we need the notion of latching space.

Definition 5.22. The m-th latching space LmX of a symmetric spectrum X is the based Σm-space (orΣm-simplicial set) (Fm−1X)m. The mth level of the canonical morphism κm−1 : Fm−1X −→ X providesa natural map of pointed Σm-spaces (resp. Σm-simplicial sets) νm : LmX −→ Xm.

In the previous definition, the 0th latching space L0X has to be interpreted as the one-point space(or simplicial set). In Proposition 5.30 below we will recognize the latching spaces as the spaces in thesymmetric spectrum X ∧ S, the smash product of the truncated sphere spectrum with X.

Since the latching spaces play important roles in what follows, we make their definition more explicit. Ifwe unravel the definition of left Kan extension in this situation and specialize to the appropriate values weobtain the latching space LmX as a coequalizer, in the category of pointed Σm-spaces (resp. Σm-simplicialsets), of two maps ∨

p≤q≤m−1 Σ(q,m) ∧Σ(p, q) ∧Xp∧Id //Id∧

//∨p≤m−1 Σ(p,m) ∧Xp .

The upper map is induced by composition in the category Σ, the lower map by the structure maps of X.[rewrite this as a coequalizer∨n−2

p=0 Σ+n ∧Σp×Σ1×Σn−p−1 Ap ∧ S1 ∧ Sn−p−1 ////

∨n−1p=0 Σ+

n ∧Σp×Σn−p Ap ∧ Sn−p .

One of the maps takes the wedge summand indexed by p to the wedge summand indexed by p + 1 usingthe map

σp ∧ Id : Ap ∧ S1 ∧ Sn−p−1 −→ Ap+1 ∧ Sn−p−1

and inducing up. The other map takes the wedge summand indexed by p to the wedge summand indexedby p using the canonical isomorphism

Ap ∧ S1 ∧ Sn−p−1 ∼=−→ Ap ∧ Sn−p

and inducing up. Maybe have this as exercise ?]For example, L0A is a point (by definition) and L1A = A0 ∧ S1. The second latching space L2A is the

pushout of the diagram

A0 ∧ S2 act on S2

←−−−−−−− Σ+2 ∧A0 ∧ S2 Id∧σ0∧Id−−−−−−→ Σ+

2 ∧A1 ∧ S1 .

Thus L2A is the quotient of Σ+2 ∧A1 ∧ S1 by the equivalence relation generated by

γ ∧ σ0(a ∧ x) ∧ y ∼ (γ(1, 2)) ∧ σ0(a ∧ y) ∧ x

for a ∈ A0 and x, y ∈ S1. In general, LnA is a quotient of Σ+n ∧Σn−1 An−1 ∧ S1 by a suitable equivalence

relation.

Example 5.23. As a first concrete example we calculate the skeleton filtration and the latching spacesof semifree spectra. We consider a based Σm-space (or Σm-simplicial set) L and identify the canonicalfiltration of the semifree spectrum GmL. Since GmK is trivial below level m, the truncation τ≤i(GmL)is trivial for i < k, hence so is the spectrum F i(GmL). In sharp contrast, we claim that for i ≥ m thecanonical morphism

F i(GmL) −→ GmL

5. SMASH PRODUCT 91

is an isomorphism. To see this, we first check that for i ≥ m the truncation τ≤i(GmL) is also ‘semifree’ inthe category of k-truncated symmetric spectra, i.e., the map

Sp≤i(τ≤i(GmL), Z) −→ TΣm(L,Zm)

is bijective for all m-truncated spectra Z. The argument for this is the same as in ??? where the anal-ogous property of GmL in the category of all symmetric spectra. But then the canonical morphismκi : F i(GmL) −→ GmL must be an isomorphism for i ≥ m since source and target represent the samefunctor on the category of symmetric spectra [spell out ?]. As a corollary we can deduce that the latchingobjects of semifree spectra are given by

Ln(GmK) = (Fn−1(GmK))n ∼=

∗ for n ≤ m(GmK)n for n > m,

where the isomorphism in the second case is given by the map νn : Ln(GmK) −→ (GmK)n.Now we characterize the latching space by the functor that it represents. The structure map σm−1 :

Xm−1 ∧ S1 −→ Xm of the symmetric spectrum X lifts to a Σm−1-equivariant map σm−1 : Xm−1 ∧ S1 −→LmX [...], i.e., such that νmσm−1 = σm−1. For every n = 0, . . . ,m − 1 the composite of σm−1 withσm−n−1 ∧ Id : Xn ∧ Sm−n −→ Xn−1 is Σn ×Σm−n-equivariant [...]. In fact, σm−1 is the universal exampleof a map with these properties:

Proposition 5.24. For every symmetric spectrum X and based Σm-space (or simplicial set) Z the map

mapΣm(LmX,Z) −→ map(Xm−1 ∧ S1, Z)

given by precomposition with σm−1 : Xm−1 ∧ S1 −→ LmX is a bijection from the space of based Σm-mapsLmX −→ Z onto the subspace of those maps f : Xm−1 ∧ S1 −→ Z such that the composite

f (σm−n−1 ∧ Id) : Xn ∧ Sm−n −→ Z

is Σn × Σm−n-equivariant for all n = 0, . . . ,m− 1.

Proof.

For example, the latching map νm : LmX −→ Xm corresponds, under the bijection of Proposition 5.24,to the structure map σm−1 : Xm−1 ∧ S1 −→ Xm.

Example 5.25. We identify the latching spaces of twisted smash products. Let K be a pointedΣm-space (or simplicial set) for some m ≥ 0, X a symmetric spectrum and k ≥ 0; we now construct aΣm+k-equivariant isomorphism

(5.26) q : Lm+k(K .m X)∼=−−→ Σ+

m+k ∧Σm×Σk (K ∧ LkX) .

The construction starts from the map

K ∧Xk−1 ∧ S1 Id∧σk−1−−−−−−→ K ∧ LkX1∧−−−−→ Σ+

m+k ∧Σm×Σk K ∧ LkXwhich is Σm × Σk−1-equivariant, so it can be extended uniquely to a Σm+k−1-equivariant map

q : (K .m X)m+k−1 ∧ S1 =(Σ+m+k−1 ∧Σm×Σk−1 K ∧Xk−1

)∧ S1 −→ Σ+

m+k ∧Σm×Σk K ∧ LkX .

The map q has the property that its composite with

σm+k−n−1 ∧ Id : (K .m X)n ∧ Sm+k−n −→ (K .m X)m+k−1 ∧ S1

is Σn×Σm+k−n-equivariant for all n = 0, . . . ,m+k. So Proposition 5.24 provides a unique Σm+k-equivariantmap q (5.26) whose composite with σm+k−1 : (K .m X)m+k−1 ∧ S1 −→ Lm+k(K .m X) is q. [since sourceand target of q commute with colimits, it suffices to check the case of semifree spectra; for X = GnL wehave K .m (GnL) ∼= Gm+m(Σ+

m+n ∧Σm×Σn K ∧ L)]Under the isomorphism q, the latching map νm+k : Lm+k(K .m X) −→ (K .m X)m+k corresponds to

Σ+m+k ∧Σm×Σk (K ∧ νk) : Σ+

m+k ∧Σm×Σk (K ∧ LkX) −→ Σ+m+k ∧Σm×Σk (K ∧Xk) .

92 I. BASICS

In ??? we explained how a symmetric spectrum is built from free symmetric spectra in the form of acoend; in (3.13) we explained a way in which every symmetric spectrum is built from semifree spectra as acoequalizer. Here is another way in which all symmetric spectra are built from semifree spectra.

Proposition 5.27. For every symmetric spectrum X and every n ≥ 0 the commutative square

(5.28) GnLnXGnνn //

GnXn

Fn−1X

jn// FnX

is a pushout square, where the vertical morphisms are adjoint to the identity map of LnX = (Fn−1X)nrespectively (FnX)n = Xn.

Proof. We start with the special case X = GmK of a semifree symmetric spectrum generated bya pointed Σm-space (or simplicial set) K. The latching objects and filtration of a semifree symmetricspectrum were calculated in Example 5.23 above. Inspection of the answer shows that for m 6= n bothhorizontal morphism in the square (5.28) are isomorphism, thus the square is a pushout. For m = n incontrast, both vertical morphism in (5.28) are isomorphisms, so again the square is a pushout. This provesthe special case of a semifree symmetric spectrum.

For the general case we use that the square (5.28) is natural in the spectrum X and all four cornerscommute with colimits in X. Since moreover pushouts commute with colimits and X is naturally a co-equalizer of two morphisms between wedges of semifree spectra (compare (3.13)), the general case followsfrom the special case.

We can reinterpret the latching space using the smash product with a truncated sphere spectrum.The latching space was originally defined in 5.22 using the canonical filtration of a spectrum as LmX =(Fm−1X)m. We denote by S the truncated sphere spectrum i.e., the symmetric subspectrum of S withlevels

(5.29) Sn =

∗ for n = 0Sn for n ≥ 1.

On other words, S differs from the sphere spectrum S only in one missing point in level 0. Now let X beany symmetric spectrum. The map im−1,1 : Xm−1 ∧ S1 −→ (X ∧ S)m from the universal bimorphism from(X, S) to X ∧ S is Σm−1-equivariant. Moreover, for every n = 0, . . . ,m − 1 the composite of im−1,1 withσm−n−1 ∧ Id : Xn ∧ Sm−n −→ Xm−1 ∧ S1 equals in,m−n : Xn ∧ Sm−n −→ (X ∧ S)m, so it is Σn × Σm−n-equivariant. Proposition 5.24 show that there is thus a unique Σm-equivariant map im−1,1 : LmX −→(X ∧ S)m whose composite with σm−1 : Xm−1 ∧ S1 −→ LmX is im−1,1. Since the map im−1,1 is natural inX so is im−1,1.

Proposition 5.30. For every symmetric spectrum X and m ≥ 0 the natural map im−1,1 : LmX −→(X ∧ S)m is an isomorphism. Moreover, the composite

LmXim−1,1−−−−→ (X ∧ S)m

(Id∧incl.)m−−−−−−−→ (X ∧ S)m(rX)m−−−−→ Xm

equals the latching map νn : LmX −→ Xm.

Proof. We first show that claim about the latching map. If we precompose with σm−1 : Xm−1∧S1 −→LmX we get

(rX)m (Id∧incl.)m im−1,1 σm−1 = (rX)m (Id∧incl.)m i(X,S)m−1,1 = (rX)m i(X,S)

m−1,1 = σm−1

where we use, in that order, the definition of im−1,1, naturality of the universal bimorphism and thedefinition of the right unit morphism. The composite of the latching map νn : LmX −→ Xm with σm−1

5. SMASH PRODUCT 93

also equals the structure map σm−1 : Xm−1∧S1. By Proposition 5.24 a Σm-map out of LmX is determinedby its composite with σm−1, which proves (rX)m (Id∧incl.)m im−1,1 = νm.

Now we show that im−1,1 : LmX −→ (X ∧ S)m is an isomorphism. We first check the special caseX = GnL of a semifree symmetric spectrum generated by based Σn-space (or simplicial set) L. If m ≤ n,then Lm(GnL) and (GnL∧ S)m ∼= L .n S are trivial, so im−1,1 is an isomorphism. If m > n then Id∧incl. :GnL∧ S −→ GnL∧ S is an isomorphism in level m since GnL∧A ∼= L .n A which in level m depends onlyon L and Am−n. The latching map νm (by Example 5.23) and the unit isomorphism are also isomorphisms,hence so is im−1,1

For general X we exploit that every symmetric spectrum can be written as a coequalizer (3.13) of a mapbetween wedges of semifree spectra. Since the map im−1,1 is natural and its source and target commutewith colimits, the special case of semifree spectra thus implies the general case.

A generalization of the previous proposition to the ‘generalized latching spaces’ (F iX)m for i < m isgiven in Exercise 8.20.

5.5. Flat symmetric spectra. We recall that a morphism of pointed simplicial sets is a cofibrationif it is a categorical monomorphism, i.e., dimensionwise injective. A morphism of pointed spaces is acofibration if it is a retract of a pointed cell complex. These are the cofibrations in the standard modelstructures on pointed simplicial sets respectively pointed spaces.

Definition 5.31. A morphism f : X −→ Y of symmetric spectra is a level cofibration if for every n ≥ 0the morphism fn : Xn −→ Yn is a cofibration of the underlying pointed spaces or pointed simplicial sets(depending on the context). A symmetric spectrum A is flat if the functor A∧− preserves level cofibrations.

To motivate the terminology we recall that a module over a commutative ring is called flat if tensoringwith it preserves monomorphisms. Level cofibrations of symmetric spectra of simplicial sets are just thecategorical monomorphisms, i.e., those morphisms which are injective in every spectrum level and everysimplicial dimensions. So in the context of simplicial sets, a symmetric spectrum A is flat if and only if A∧−preserves monomorphisms. In the context of symmetric spectra of topological spaces, simply requiring thatA ∧ − preserves monomorphisms is not the right condition, so the analogy with flatness in algebra is lesstight.

We will show in Chapter III that flat symmetric spectra are the cofibrant objects in various ‘flat modelstructures’.

Example 5.32. Let L be a pointed Σm-space (or Σm-simplicial set), for some m ≥ 0, whose underlyingpointed space is cofibrant (this is automatic in the context of simplicial sets). Then smashing with thesemifree symmetric spectrum preserves level cofibrations and level acyclic cofibrations. In particular, thesemifree spectrum GmL is flat. As special cases, this applies to free symmetric spectra FnK and suspensionspectra Σ∞K, for every based cofibrant space K (respectively every based simplicial set).

Indeed, if X is another symmetric spectrum then GmL∧X is isomorphic to the twisted smash productL .m X (see Proposition 5.5) and so it is trivial in levels below m and we have a natural isomorphism

(5.33) (GmL ∧X)m+n∼= Σ+

m+n ∧Σm×Σn L ∧Xn

for n ≥ 0. If f : X −→ Y is a level cofibration respectively a level acyclic cofibration then Id∧fn :L∧Xn −→ L∧Yn is a cofibration respectively acyclic cofibration by the pushout product property [ref]. Asa pointed space (simplicial set), the right hand side of (5.33) is a wedge of

(m+nm

)copies of L ∧Xn. Hence

the morphism Id∧f : GmL ∧X −→ GmL ∧ Y is levelwise a cofibration respectively acyclic cofibration.

Some other properties of flat spectra are fairly straightforward from the definition:

Proposition 5.34. (i) A wedge of flat symmetric spectra is flat.(ii) For symmetric spectra of simplicial sets, a filtered colimit of flat symmetric spectra is flat.(iii) The smash product of two flat symmetric spectra is flat.(iv) If A is a flat symmetric spectrum and K a cofibrant space (respectively simplicial set), then K ∧A

is flat.

94 I. BASICS

Proof. Properties (i) and (ii) follow from the two facts that the smash product commutes with colimitsand that a filtered colimit (for simplicial sets) or a wedge of level cofibrations is a level cofibration.

(iii) Let A and be B flat symmetric spectra. If f : X −→ Y is a level cofibration, then so is Id∧f :B ∧X −→ B ∧ Y since B is flat; then Id∧ Id∧f : A∧B ∧X −→ A∧B ∧ Y is also a level cofibration sinceA is flat (where we have implicitly used the associativity isomorphisms). Thus A ∧B is flat.

(iv) The spectrum K ∧ A is isomorphic to the smash product (Σ∞K) ∧ A. Since A is flat and thesuspension spectrum if flat by Example 5.32, (iv) follows from (iii).

An example of a non-flat symmetric spectrum is S, the subspectrum of the sphere spectrum given byS0 = ∗ and Sn = Sn for n ≥ 1. So the difference between S and S is only one missing point in level 0, butthat point makes a huge difference for the flatness property. Indeed, since S is trivial in level 0 we have

(S ∧ S)2 = Σ+2 ∧ S1 ∧ S1 = Σ+

2 ∧ S2

while (S∧ S)2 ∼= S2 = S2. So S∧− does not take the inclusion S −→ S to a monomorphism, hence S is notflat.

Now we develop a convenient criterion for recognizing flat symmetric spectra which involves latchingspaces LmA which were defined in 5.22 via the canonical filtration of A.

Proposition 5.35. (i) Let A be a symmetric spectrum such that for every m ≥ 0 the latching morphismνm : LmA −→ Am is cofibration of underlying pointed spaces (respectively simplicial sets). Then the functorA ∧ − preserves level cofibrations and level acyclic cofibrations.(ii) A symmetric spectrum is flat if and only for every m ≥ 0 the latching morphism νm : LmA −→ Am iscofibration of underlying pointed spaces, respectively simplicial sets.

Thus for symmetric spectra of simplicial set, flatness is equivalent to the condition that the morphismνm : LmA −→ Am is injective for all m ≥ 1 (since L0A = ∗, ν0 is automatically a cofibration in thesimplicial context).

Proof of Proposition 5.35. (i) Let f : X −→ Y be a level cofibration (respectively level acycliccofibration) of symmetric spectra. We use the skeleton filtration of A by the spectra FmA (see Construc-tion 5.21) and show inductively that the map Id∧f : FmA∧X −→ FmA∧Y is a level cofibration (resp. levelacyclic cofibration). Since FmA agrees with A up to level m, the smash product FmA ∧ X agrees withA ∧X up to level m, and similarly for Y . Since m can be arbitrarily large, this proves the claim.

We can start the induction with m = −1, where there is nothing to show. For the inductive step weuse the pushout square of Proposition 5.27. Since smashing is a left adjoint the spectrum FmA ∧ X is apushout of the upper row in the commutative diagram

(5.36)

Fm−1A ∧XId∧f

LmA .m Xνm.mId //

Id .mf

oo Am .m X

Id .mf

Fm−1A ∧ Y LmA .m Y

νm.mId//oo Am .m Y

and FmA ∧ Y is a pushout of the lower row; we have used the identification GmL ∧X ∼= L .m X providedby Proposition 5.5. The left vertical morphism is a level cofibration (resp. acyclic cofibration) by inductionhypothesis, and we claim that in addition the pushout product map of the right square in (5.36)

νm .m f : LmA .m Y ∪LmA.mX Am .m X −→ Am .m Y

is a level cofibration (resp. acyclic cofibration). It is then a general model category fact (see Lemma A.1.10)that induced map on pushouts is levelwise a cofibration (resp. acyclic cofibration).

It remains to justify the claim that νm .m f is a level cofibration (resp. acyclic cofibration). There isnothing to show below level m since both sides are trivial. For n ≥ 0 a we have

(L .m X)m+n = Σ+m+n ∧Σm×Σn L ∧Xn ,

5. SMASH PRODUCT 95

which non-equivariantly is a wedge of(n+mn

)copies of L∧Xn. Since νm : LmA −→ Am is a cofibration and

fn : Xn −→ Yn is a cofibrations (resp. acyclic cofibration), the pushout product property [ref] shows thatthe map

νm ∧ fn : LmA ∧ Yn ∪LmA∧Xn Am ∧Xn −→ Am ∧ Ynis a cofibration (resp. acyclic cofibration). So (νm .m f)m+n is a cofibration (resp. acyclic cofibration).

(ii) If the latching maps for A are cofibrations, then A ∧ − preserves level cofibrations by (i), so A isflat. If conversely A is flat, then Id∧i : A∧ S −→ A∧S is a level cofibration because the inclusion i : S −→ Sof the truncated sphere spectrum (5.29) is a level cofibration. By Proposition 5.30 the m-th level of thismorphism is isomorphic to the latching map νm : LmA −→ Am, which is thus a cofibration.

Proposition 5.37. Let A be a flat symmetric spectrum.(i) The shifted spectrum shA is again flat.(ii) All structure maps σn : An ∧ S1 −→ An+1 are cofibrations. Moreover, in the context of topologicalspaces, the pointed spaces LnA and An are cofibrant for all n ≥ 0.(iii) The morphism λA : S1 ∧A −→ shA is a level cofibration.(iv) If A is a flat symmetric spectrum of simplicial sets, then the geometric realization |A| is a flat symmetricspectrum of topological spaces.(v) For every injective spectrum X the internal function spectrum Hom(A,X) is injective.

Proof. (i) As a special case of Proposition 5.13 we obtain a pushout square

S1 ∧A ∧ S(Id∧λS)twist //

λA∧Id

A ∧ (sh S)

ξA,S

(shA) ∧ SξA,S

// sh(A ∧ S)

Since the morphism λS : S1 ∧ S −→ sh S is a level cofibration (it is even an isomorphism except in level 0)and A is flat, the upper horizontal morphism is a level cofibration. Since cofibrations are stable undercobase change, the lower horizontal morphism ξA,S is also a level cofibration.

The morphism Id∧i : A ∧ S −→ A ∧ S ∼= A is a level cofibration since A is flat, hence its shift is also alevel cofibration. Thus the composite

(shA) ∧ SξA,S−−→ sh(A ∧ S)

sh(Id∧i)−−−−−→ shA

is a level cofibration. Since this composite coincides with Id∧i : (shA) ∧ S −→ shA, the spectrum shA isflat by Proposition 5.35. [plus the identification Lm(shA) ∼= ((shA) ∧ S)m]

(ii) For n = 0 the claim holds since L0A is a point and ν0 : L0A −→ A0 is a cofibration whenever A isflat. For n ≥ 1 we use part (i) to deduce that the n-fold shift of A is flat. Thus the map

An ∧ S1 = L1(shnA) −→ shn(A)1 = An+1

is a cofibration, which proves the first claim. In the context of topological spaces the second claim about Anfollows by induction on n, using that suspension preserves cofibrancy. To see that the latching space LnA iscofibrant we note that by flatness, the smash product A∧ S is level cofibrant since S is. So LnA ∼= (A∧ S)nis cofibrant.

(iii) The nth level of λA agrees with the structure map σn : An ∧ S1 −→ An+1 up to composition withtwo isomorphisms. By (ii) the structure map σn is a cofibration, hence so is (λA)n for all n ≥ 0.

(iv) The latching space Lm|A| is homeomorphic to |LmA| in a way compatible with the maps ν|A|m and|νAm| to |Am|. [add this to the section on the filtration] Since geometric realization takes cofibrations ofsimplicial sets to cofibrations of spaces, the claim follows from the flatness criterion of Proposition 5.35.

(v) Given a level acyclic cofibration of symmetric spectra i : K −→ L and a morphism g : K −→Hom(A,X), we have to produce an extension g : L −→ Hom(A,X) satisfying g f = g. The morphismi ∧ Id : K ∧ A −→ L ∧ A is again a level acyclic cofibration by Proposition 5.35. Since X is injective, the

96 I. BASICS

adjoint G : K ∧ A −→ X of g has an extension G : L ∧ A −→ X satisfying G(f ∧ Id) = G. The adjointL −→ Hom(A,X) of G is then the required extension of g.

We recall that every based simplicial set is cofibrant, hence every symmetric spectrum of simplicial setis level cofibrant. So the condition ‘level cofibrant’ in the next proposition is vacuous in the context ofsimplicial sets.

Proposition 5.38. Smashing with a flat symmetric spectrum preserves level equivalences between levelcofibrant spectra and it preserves all π∗-isomorphisms and stable equivalences.

Proof. Let A be a flat symmetric spectrum. By Proposition 5.35 the functor A ∧ − preserves levelacyclic cofibrations. We use a version of ‘Ken Brown’s lemma’ [ref] to deduce that A ∧ − takes a levelequivalence f : X −→ Y between level cofibrant spectra to a level equivalence. The morphism f factors asa composite

XiX−−→ Z(f)

p−−→ Y

where Z(f) = [0, 1]+ ∧X ∪f Y is the mapping cylinder of f [ref?]. Since the projection p : Z(f) −→ Y isa homotopy equivalence and f is a level equivalence, the ‘front inclusion’ iX : X −→ Z(f) and the ‘backinclusion’ are level equivalences. Since X and Y are cofibrant, these two inclusions are also level cofibrations.By Proposition 5.35 the morphisms A∧ iX : A∧X −→ A∧Z(f) and A∧ iY : A∧Y −→ A∧Z(f) are levelacyclic cofibrations. Since A∧ p : A∧Z(f) −→ A∧ Y is a retraction to A∧ iY , it is then level equivalence.But then A ∧ f is a level equivalence as the composite of the two level equivalences A ∧ iX and A ∧ p.

We now prove the statement which refers to π∗-isomorphisms. Smash product withA commutes with themapping cone construction, so using the long exact sequence of naive homotopy groups of Proposition 2.16 (i)it suffices to show that if a symmetric spectrum C has trivial stable homotopy groups, then so does A∧C.By [...] the symmetric spectrum L ∧ C has trivial stable homotopy groups for every pointed simplicial setL [cofibrant space]. The isomorphism

πk(L .m X) ∼=⊕f∈Om

πk+m(L ∧X)

(see (3.16)) combined with the isomorphism L.mC ∼= GmL∧C of Proposition 5.5 shows that the homotopygroups of GmL ∧ C are trivial. In other words, the claim holds for semifree symmetric spectra.

If A is an arbitrary flat symmetric spectrum we again use induction and show that FmA ∧ C hastrivial homotopy groups for all m ≥ 0. Since homotopy groups commute with filtered colimits [spaces:overh-cofibrations...] this show that the groups π∗(A∧C) vanish and it finishes the argument. In the inductivestep we use the pushout square of Proposition 5.27. Since A is flat the morphism νm : LmA −→ Am isinjective, hence jm ∧ Id : Fm−1A ∧ C −→ FmA ∧ C is a monomorphism. We know by induction that thehomotopy groups of Fm−1A ∧ C vanish, and we know that the homotopy groups of

(FmA ∧ C)/(Fm−1A ∧ C) ∼= Gm(Am/LmA) ∧ Cvanish by the special case. So the long exact sequence of homotopy groups shows that the homotopy groupsof FmA ∧ C are trivial.

Now we show that A ∧ − preserves stable equivalences. We treat the case of spectra of simplicial setsfirst. If Z is an injective Ω-spectrum, then Hom(A,Z) is injective by part (v) of Proposition 5.37 andan Ω-spectrum by [ref]. So for every stable equivalence f : X −→ Y the induced map [f,Hom(A,Z)] :[Y,Hom(A,Z)] −→ [X,Hom(A,Z)] is bijective. By adjunction, [f ∧ A,Z] : [Y ∧ A,Z] −→ [X ∧ A,Z] isbijective. Since this holds for all injective Ω-spectra Z, the morphism f ∧ Id : X ∧A −→ Y ∧A is a stableequivalence. [In the context of spectra of spaces ...]

Example 5.39. Here is an example which shows that smashing with an arbitrary symmetric spectrumdoes not preserve level equivalences. Let X be the symmetric spectrum with X0 = S0, X1 = CS1 andXn = ∗ for n ≥ 2. Here CS1 = [0, 1] ∧ S1 is the cone on S1, where the unit internal [0, 1] is pointed by 0.The only nontrivial structure map σ0 : X0 ∧ S1 −→ X1 is the cone inclusion S1 −→ CS1. Let Y be thesymmetric spectrum with Y0 = S0 and Yn = ∗ for n ≥ 1. Then the unique morphism f : X −→ Y which

5. SMASH PRODUCT 97

is the identity in level 0 is a level equivalence, but we claim that f ∧ S : X ∧ S −→ Y ∧ S is not a levelequivalence. Indeed, in level 2 we have

(X ∧ S)2 = L2X ∼= pushout(S2 act←−− Σ+2 ∧ S2 i∧S1

−−−→ Σ+2 ∧ (CS1) ∧ S1)

which is the suspension of the double cone on S1, i.e., a simplicial 3-sphere. In contrast,

(Y ∧ S)2 = L2Y ∼= pushout(S2 act←−− Σ+2 ∧ S2 −→ ∗)

is a point, so f∧S is not a weak equivalence in level 2. [give an examples where π∗-isos or stable equivalencesare not preserved]

In (7.39) we defined the standard resolution X\ of a symmetric spectrum X and proved various usefulfacts about it. This resolution comes with a natural level equivalence ε : X\ −→ X and a natural isomor-phism (K ∧ X)\ ∼= K ∧ (X\) for pointed spaces (or simplicial sets) K. In the context of simplicial sets,the spectrum X\ is projective, thus flat. In the context of symmetric spectra of spaces we use modify theconstruction to obtain a flat (in fact, projective) resolution as follows. For a spectrum of spaces B we formthe singular complex S(B), take the standard resolution S(B)\ and geometrically realize. Since S(B)\ isprojective, so is the realization |S(B)\| by Proposition 5.37 (iv). The composite

|S(B)\||rS(B)|−−−−→ |S(B)|| counit−−−−→ B

is then a natural level equivalence with flat source.

Example 5.40. We identify the standard resolution in some particular cases. As before we denoteby I the injection category with objects 0, 1, 2, . . . and morphisms the injective set maps. The spacesΣ(m,n) and I(m,n)+ ∧ Sm−n are both wedges of m!/(m − n)! spheres of dimension m − n, so they arehomeomorphic; however, to construct a homeomorphism we have to choose a set of coset representatives ofthe group Σm−n in Σn?, so there is no canonical such homeomorphism. In contrast, the spaces Σ(m,n)∧Smand Sn ∧ I(m,n)+ are canonically homeomorphic, via

Σ(m,n) ∧ Sm −→ Sn ∧ I(m,n)+

[σ ∧ x] ∧ y 7→ σ∗(y ∧ x) ∧ ρ(σ) .

Here we use the canonical homeomorphism between Sm ∧ Sn−m and Sn, and ρ(σ) : m −→ n denotes therestriction of the bijection σ ∈ Σn to m ⊂ n. This isomorphism is compatible with composition in thecategories Σ and I in the sense that the square commutes for

Σ(m,n) ∧Σ(k,m) ∧ Sk //

∧Id

Σ(m,n) ∧ Sm ∧ I(k,m)+ // Sn ∧ I(m,n)+ ∧ I(k,m)+

Id∧

Σ(k, n) ∧ Sk // Sm ∧ I(k, n)+

all k,m, n ≥ 0. Moreover, the composite

Σ(m,n) ∧ Sm −→ Sn ∧ I(m,n)+proj−−→ Sn

is the action of the Σ-functor corresponding to the symmetric sphere spectrum S.The simplicial space B•(S)n has as its space of k-simplices the wedge∨

i0≤···≤ik

Σ(ik, n) ∧Σ(ik−1, ik) ∧ · · · ∧Σ(i0, i1) ∧ Si0

(where we used that (Fik)n = Σ(ik, n)). If we use the homeomorphism k + 1 times we can identify thisspace with the space ∨

i0≤···≤ik

Sn ∧ I(ik, n)+ ∧ I(ik−1, ik)+ ∧ · · · ∧ I(i0, i1)+ .

98 I. BASICS

By the naturality properties of the map, this is in fact a homeomorphism of simplicial spaces B•(S)n ∼=Sn ∧ B•(I(−, n), I, S0), where S0 is the constant I-functor with value S0. After geometric realization wethus obtain a homeomorphism

S\n = |B•(S)n| = |Sn∧B•(I(−, n), I, S0)| ∼= Sn∧|B•(I(−, n)+, I, S0)| = Sn∧(I(−, n)+∧hIS0) ∼= Sn∧(hocolimIop I(−, n))+ .

Under this isomorphism, the n-th level of the resolution morphism S] −→ S is induced by the projectionto the homotopy colimit to the one-point space (which is a homotopy equivalence...I has initial object 0,so Iop has terminal object...). More generally, (FmSm)]n = Sn ∧ (I(−, n)×hI I(m,−))+.

In Section 4.5 we have defined the fraction category X,Y for symmetric spectra X and Y . Wenow define a variation, the flat fraction category X,Y [. This is simply the full subcategory of thefraction category X,Y with objects those triples (Z, f, g) consisting of a symmetric spectrum Z, a stableequivalence f : Z −→ X and a morphism g : Z −→ Y , such that the spectrum Z is flat. Like the fullfraction category, X,Y [ is a covariant functor in Y and a contravariant functor for stable equivalences inX.

Corollary 5.41. For every pair of symmetric spectra X and Y , the inclusion X,Y [ −→ X,Y induces a bijection

π0X,Y [ −→ π0X,Y .

Proof. We obtain a functor [ : X,Y −→ X,Y by sending (Z, f, g) to (Z[, fr, gr), and similarlyon morphisms, where r : Z[ −→ Z is the functorial flat resolution constructed in [...]. The resolutionmorphism r : Z[ −→ Z is a natural morphism from (Z[, fr, gr) to (Z, f, g) in X,Y , so it give a naturaltransformation from [ to the identity functor. Since the functor [ takes values in the full subcategoryX,Y [, it induced a map π0X,Y −→ π0X,Y [ on path components which is inverse to the mapinduced by the inclusions.

In Example 4.56 we discussed the fraction category in the special case X = S. An important result isthat the map natural map (4.57)

〈−〉 : π0S, Y −→ π0Y , 〈Z, f, g〉 = (π0g)((π0f)−1(1)

)is bijective. If we restrict this construction to the flat fraction category, we obtain another bijectionπ0S, Y [ −→ π0Y . We also factored the tautological map c : π0Y −→ π0Y through π0S, Y by sending[x] ∈ π0Y represented by x : Sn −→ Yn to the class of the object. (FnSn, λ, x) where x : FnSn −→ Y isadjoint to x. Since the free spectrum FnS

n is flat, the map actually factors through π0S, Y [.[later: smash product of two semistable spectra is semistable if at least one of the factors is flat]The maps in,n : Xn ∧ Yn −→ (X ∧ Y )n+n from the universal bimorphism make the following diagram

commute induce maps

πk+2n(Xn ∧ Yn)πk+2nin,n−−−−−−→ πk+2n(X ∧ Y )2n −−−→ πk(X ∧ Y )

of homotopy groups, where the second in the canonical map to the colimit. We want to let n grow tostabilize the groups on the left, but the maps to use have to be chosen with care. We define ’stabilizationmaps’

πk+2n(Xn ∧ Yn)−∧S2

−−−−→ πk+2n+2(Xn ∧ Yn ∧ S1 ∧ S1)(−1)n·πk+2n+2((σ

Xn ∧σ

Yn )twist)−−−−−−−−−−−−−−−−−−−−→ πk+2n+2(Xn+1 ∧ Yn+1)

Proposition 5.42. Consider symmetric spectra X and Y such that Y is flat and semistable Then themaps πk+2n(Xn ∧ Yn) −→ πk(X ∧ Y ) induce a well-defined map

(5.43) colimn πk+2n(Xn ∧ Yn) −→ πk(X ∧ Y ) .

which is an isomorphism of abelian groups.

[warning: if one is flat, but not semistable, the map is not even well-defined...]

5. SMASH PRODUCT 99

Proof. The diagram

Xn ∧ Yn ∧ S1 ∧ S1

in,n∧Id

twist // Xn ∧ S1 ∧ Yn ∧ S1σXn ∧σ

Yn // Xn+1 ∧ Yn+1

in+1.n+1 // (X ∧ Y )n+1+n+1

1×χ1,n×1

(X ∧ Y )n+n ∧ S1 ∧ S1

σ2// (X ∧ Y )n+n+1+1

commutes by the defining property of a bimorphism. If X and Y are semistable and at least one of themis flat, then X ∧ Y is again semistable[ref] and so the effect of the permutation 1 × χn,1 × 1 becomesmultiplication by the sign (−1)n in the colimit πk(X ∧ Y ) [ref].

So on the level on homotopy groups, the diagram

πk+2n(Xn ∧ Yn)

πk+2nin,n

−∧S2// πk+2n+2(Xn ∧ Yn ∧ S1 ∧ S1)

πk+2n+2((σXn ∧σ

Yn )twist) // πk+2n+2(Xn+1 ∧ Yn+1)

πk+2n+2in+1.n+1

πk+2n(X ∧ Y )n+n can.

// πk(X ∧ Y ) πk+2n+2(X ∧ Y )2n+2(−1)ncan.

oo

commutes. This proves that we can a well-defined map from the colimit over the groups πk+2n(Xn ∧ Yn).We seem to have [check] natural morphisms

X4Y −→ Φ(X ∧ Y ) ←− X ∧ Y

where (X4Y )n = map(Sn−, Xn ∧ Yn) with diagonal, conjugation Σn-action. Is the adjunction isoπk+n(X4Y )n ∼= πk+2n(Xn ∧ Yn) compatible with stabilization ? The right map should have been shownto be a π∗-iso for semistable Y .

Since X is semistable, the two morphisms

Xλ∞X−−→ R∞ = teln(Ωn(shnX)) ?←− teln Ωn(Σ∞Xn)

are π∗-isomorphisms [ref]. Since Y is flat, smashing with it preserves π∗-isomorphisms (compareProposition5.38), which provides the first two of a chain of π∗-isomorphisms

X ∧ Y λ∞X ∧Id−−−−→ (R∞X) ∧ Y ?←− teln Ωn(Σ∞Xn) ∧ Y?−→ teln Ωn(Σ∞Xn ∧ Y ) ∼= teln Ωn(Xn ∧ Y ) .

The sequence of isomorphisms

πk(teln Ωn(Xn ∧ Y )) ∼= colimn≥0 πk(Ωn(Xn ∧ Y )) ∼= colimn≥0 πk+n(Xn ∧ Y ) ∼= colimn,m≥0 πk+n+m(Xn ∧ Ym)

finishes the argument.

Proposition 5.44 (check !!!). X ∧ Y is stably equivalent to

hocolim(n,m)∈I×I Ωn+m(Σ∞(Xn ∧ Ym))

[spaces: level cofibrant] When is it π∗-isomorphic ?

Proof. Use the diagonal trick: hocolim is stably equivalent to the total diagonal

(diag Ω•+•(Σ∞(X• ∧ Y•)))n = Ωn+n(Sn ∧Xn ∧ Yn)

Study ‘Φ’ for a bi-spectrum such as X∧Y ?

100 I. BASICS

6. Products on homotopy groups

The true homotopy groups are nicely compatible with the multiplicative structure given by the smashproduct, as we explain in this section. We construct a biadditive pairing of true homotopy groups

· : πkX × πlY −→ πk+l(X ∧ Y )

for pairs of integers k, l, with certain desirable properties which are listed in the next theorem. The theoremcan be summarized in fancy language by saying that the pairing makes the graded true homotopy groupsinto a lax symmetric monoidal functor from the category of symmetric spectra (under smash product) tothe category of graded abelian groups (under graded tensor product, with Koszul sign convention for thesymmetry isomorphism). Later we show that if X is (k − 1)-connected, Y is (l− 1)-connected and at leastone of them is flat, then X ∧ Y is (k + l − 1)-connected and the map (6.2) is an isomorphism, compareRemark II.4.19.

A formal consequence of the compatibility properties is that the true homotopy groups of a symmetricring spectrum naturally form a graded ring, which is commutative in the graded sense if the ring spectrumis, compare Proposition 6.5.

Theorem 6.1. There is a unique family of natural pairings

(6.2) · : πkX × πlY −→ πk+l(X ∧ Y )

of true homotopy groups for k, l ∈ Z subject to the following two conditions.

(Normalization) The unit isomorphism S ∧ S ∼= S takes the product 1 · 1 ∈ π0(S ∧ S) to 1, where 1 ∈ π0S isthe fundamental class.

(Suspension) The pairing is compatible with the suspensions isomorphisms in the following sense. For allintegers k, l and homotopy classes x ∈ πkX and y ∈ πlY the isomorphism

π1+k+l((S1 ∧X) ∧ Y ) −→ π1+k+l(S1 ∧ (X ∧ Y ))

takes (S1 ∧ x) · y to S1 ∧ (x · y) and the isomorphism

π1+k+l(X ∧ (S1 ∧ Y )) −→ π1+k+l(S1 ∧ (X ∧ Y ))

takes x · (S1 ∧ y) to S1 ∧ (x · y) [sign (−1)k???] in the sense that the diagrams

πkX × πlY

(S1∧−)×Id

· // πk+l(X ∧ Y )

S1∧−

πkX × πlY

Id×(S1∧−)

·// πk+l(X ∧ Y )

S1∧−

π1+k(S1 ∧X)× πlY

·

πkX × π1+l(S1 ∧ Y )

·

π1+k+l(S1 ∧ (X ∧ Y )) ∼=// π1+k+l(X ∧ (S1 ∧ Y )) πk+1+l(S1 ∧ (X ∧ Y )) ∼=

// π1+k+l(S1 ∧ (X ∧ Y ))

commute.Moreover, the pairing has the following properties:

(Associativity) For every triple of symmetric spectra X,Y and Z the diagram

πkX × πlY × πjZ·×Id //

Id×·

πk+l(X ∧ Y )× πjZ

·

πkX × πl+j(Y ∧ Z) ·// πk+l+j(X ∧ Y ∧ Z)

6. PRODUCTS ON HOMOTOPY GROUPS 101

commutes.(Commutativity) For every pair of symmetric spectra X and Y the square

πkX × πlY· //

twist

πk+l(X ∧ Y )

πk+l(τX,Y )

πlY × πkX ·

// πl+k(Y ∧X)

commutes up to the sign (−1)kl.(Unitality) For every symmetric spectrum X both composites around the square

πkX·1 //

1·

πk(X ∧ S)

πk(rX)

πk(S ∧X)

πk(lX)// πkX

are identities.(Action of stable stems) The composite

πkX × πsl

Id×c−−−→ πkX × πlS·−−→ πk+l(X ∧ S)

πk+l(rX)−−−−−−→ πk+lX

equals the action of the stable stems on the true homotopy groups constructed in 4.33.

The proof of Theorem 6.1 will make repeated use of the fact that we can easily identify natural trans-formations out of the product of homotopy groups of a finite number of symmetric spectra.

Proposition 6.3. For n ≥ 1, let F : Spn −→ Ab be a functor of n variables to the category of abeliangroups which takes stable equivalences in each variable to isomorphisms when all variables are flat. Thenevaluation at (1, . . . , 1) ∈ (π0S)n is a bijection from the abelian group of natural transformations

π0X1 × · · · × π0X

n −→ F (X1, . . . , Xn)

to the group F (S, . . . ,S).

Proof. For ease of exposition we only treat the case n = 2 which shows all relevant features of thegeneral case. The proof of the general case mainly differs in the more complex notation.

Let τ : π0X × π0Y −→ F (X,Y ) be a natural transformation. For every true homotopy class x ∈ π0Xthere exists a triple (Z, f, g) where Z is a flat symmetric spectrum, f : Z −→ S a stable equivalence andg : Z −→ X a morphism such that the condition (π0g)((π0f)−1(1)) = x holds (see Example 4.56). Similarly,for every y ∈ π0Y there exists (Z ′, f ′ : Z ′ −→ S, g′ : Z ′ −→ Y ) with Z ′ flat, f ′ a stable equivalence and(π0g

′)((π0f′)−1(1)) = y. Naturality gives a commutative diagram

π0S× π0S

τ

π0Z × π0Z′

τ

π0f×π0f′

∼=oo π0g×π0g

′// π0X × π0Y

τ

F (S,S) F (Z,Z ′)

F (f,f ′)

∼=ooF (g,g′)

// F (X,Y )

Since the sphere spectrum S and the spectra Z and Z ′ are flat the map F (f, f ′) : F (Z,Z ′) −→ F (S,S) isan isomorphism by the hypothesis on F . So the two left horizontal maps are bijective, and we have therelation

(6.4) τX,Y (x, y) = F (g, g′)(F (f, f ′)−1(τS,S(1, 1)

).

Thus the element τS,S(1, 1) of F (S,S) determines the transformation τ .On the other hand, the above analysis tells us how we have to define the transformation τ associated

to a given element z ∈ F (S,S). Given x ∈ π0X and y ∈ π0Y we choose triples (Z, f : Z −→ S, g : Z −→ X)

102 I. BASICS

and (Z ′, f ′ : Z ′ −→ S, g′ : Z ′ −→ Y ) with Z and Z ′ flat, f and f ′ stable equivalences and such that therelations

(π0g)((π0f)−1(1)) = x and (π0g′)((π0f

′)−1(1)) = y

hold. As above, the map F (f, f ′) :) : F (Z,Z ′) −→ F (S,S) is an isomorphism and we can define τX,Y (x, y)by τX,Y (x, y) = F (g, g′)

(F (f, f ′)−1(z)

).

The key point is now to show that this definition is independent of the choices. We show this for thevariable X, the argument in Y is analogous. So suppose that (Z, f , g) is above triple which represents x.By [...] this triple is related to (Z, f, g) by a path in the category S, X[. We can assume without loss ofgenerality that the is a morphism ϕ : Z −→ Z from (Z, f, g) to (Z, f , g). The data gives a commutativediagram

F (Z,Z ′)

F (f,f ′)

F (g,g′) //

F (ϕ,Id)

((PPPPPPPPPPPPF (X,Y )

F (S,S) F (Z, Z ′)F (f ,f ′)

oo

F (g,g′)

OO

in which the three maps which do not end in F (X,Y ) are isomorphisms. So we get

F (g, g′) F (f, f ′)−1 = F (g, g′) F (ϕ, Id) F (f, f ′)−1 = F (g, g′) F (f , f ′)−1

If we evaluate this on the pair (1, 1) we obtain the independence of the definition of τX,Y .We have τS,S(1, 1) = z because we can represent the fundamental class 1 ∈ π0S by the triple (S, Id, Id).

This finishes the proof.

Proof of Theorem 6.1. We start by showing that the normalization and suspension propertiesuniquely determine the pairing. Since the suspension isomorphisms are bijective (Proposition 4.40), thesuspension property shows that the pair for dimension (0, 0) determine the pairing in general dimensions(k, l). If we apply Proposition 6.3 in the case n = 2 to the functor F (X,Y ) = π0(X ∧ Y ) we see that thenormalization condition determines the product x · y.

On the other hand, the Proposition 6.3 also tells that there exists a natural pairing

· : π0X × π0Y −→ π0(X ∧ Y )

satisfying the normalization condition. [extend to general dimensions] [unitality, action]For the associativity condition we observe that the two maps

π0X × π0Y × π0Z −→ π0(X ∧ Y ∧ Z)

which take a triple (x, y, z) to x · (y · z) respectively (x · y) · z are both natural transformations of functorsin three variable which take the same values on the triple (1, 1, 1) ∈ (π0S)3, by the normalization condition.The uniqueness part of Proposition 6.3 in the case n = 3 and the functor F (X,Y, Z) = π0(X ∧ Y ∧ Z)proves that x · (y · z) = (x · y) · z. [general dimensions (k, l);remark that we suppress associativity isos...]

The commutativity property follows from the characterization of the homotopy group pairing. Indeed,we can define a (potentially) new natural pairing of true homotopy groups as the composite

πkX × πlYtwist−−−→ πlY × πkX

·−→ πl+k(Y ∧X)(−1)kl·πl+k(τY,X)−−−−−−−−−−−−→ πk+l(X ∧ Y ) .

This pairing is also normalized and compatible with suspension; for example, we have

(S1 ∧ x) y = (−1)(1+k)l · πl+1+k(τY,S1∧X)(y · (S1 ∧ x))

= (−1)(1+k)l · πl+1+k(Id∧τY,X)(πl+1+k(τY,S1 ∧ Id)(y · (S1 ∧ x))

)= (−1)kl · πl+1+k(Id∧τY,X)

(S1 ∧ (y · x)

)= (−1)kl · (S1 ∧ (πl+k(τY,X)(y · x))) = S1 ∧ (x y) .


The second equation uses that the composite

Y ∧ S1 ∧XτY,S1∧Id−−−−−−−→ S1 ∧ Y ∧X Id∧τY,X−−−−−→ S1 ∧X ∧ Y

equals τY,S1∧X (where we allow ourselves to suppress the associativity isomorphisms; this is the hexagoncondition). The third equation uses the suspension property for the original pairing and the fourth equalityis the naturality of the suspension isomorphism. The proof of the other suspension condition is similar.So by the uniqueness property, the product has to coincide with the original pairing. Since τY,X is theinverse of τX,Y , this proves that the commutativity square commutes.

For unitality we apply Proposition 6.3 in the case n = 1 to the functor F (X) = π0X. The naturaltransformation π0X −→ π0X which takes x to π0(rX)(x · 1) takes the fundamental class 1 ∈ π0S to itself,by the normalization; by the uniqueness property, we must have π0(rX)(x · 1) = x in general, and similarlyfor left multiplication by 1. [general dimension]

It remains to show that we recover the action of the stable stems on the true homotopy groups. Forany given class y ∈ πs

l the two maps π0X −→ πlX which send x to πk(rX)(x · cy) respectively x ·y agree forX = S on the fundamental class. If we apply Proposition 6.3 for n = 1 and F (X) = πlX, the uniquenesspart shows that πk(rX)(x · cy) = x · y in general. [general dimension k]

As a formal consequence of Theorem 6.1 we obtain

Proposition 6.5. Let R be a symmetric ring spectrum with multiplication (in internal form) µ :R ∧R −→ R. Then the composite maps

πkR× πlR·−−→ πk+l(R ∧R)

πk+l−−−→ πk+lR

make the true homotopy groups of R into a graded ring. If R is commutative, then this product on π∗R isgraded commutative. The true homotopy groups of a right R-module naturally form a graded right moduleover the graded ring π∗R via the composite map

πkR× πlM·−−→ πk+l(R ∧M)

πk+l(α)−−−−−→ πk+lM ,

where α : R ∧M −→M is the action morphism in internal form.

Of course, there are analogous statements for left modules and bimodules.

6.1. A product on naive homotopy groups. The construction of the pairing (6.2) was about asabstract as the definition of true homotopy groups. An important tool for calculating the latter in examplesare the more concrete naive homotopy groups. We now explain that the naive homotopy groups can also beused to determine the product on true homotopy groups, at least for semistable spectra. For this purposewe first construct a biadditive pairing of naive homotopy groups

(6.6) · : πkX × πlY −→ πk+l(X ∧ Y )

for all integers k, l under the additional hypothesis that the M-action [not yet defined] on πlY is trivial.The condition on theM-action comes up naturally when checking that an unstable product is well-definedon stable homotopy classes. We then verify that whenever the product · on naive homotopy groups isdefined, the square

πkX × πlY· //

c×c

πk+l(X ∧ Y )

c

πkX × πlY ·

// πk+l(X ∧ Y )

commutes, where c is the tautological map from naive to true homotopy groups. In the special case whereX and Y are semistable and at least one of them is flat, the vertical maps are all bijections, and so theproduct on naive homotopy groups effectively calculated the product of true homotopy groups.

104 I. BASICS

Given two homotopy classes f ∈ πk+nXn and g ∈ πl+mYm we denote by f · g the homotopy class inπk+l+n+m(X ∧ Y )n+m given by the composite

(6.7) Sk+l+n+m Id∧χl,n∧Id−−−−−−−→ Sk+n+l+m f∧g−−−−→ Xn ∧ Ymin,m−−−−→ (X ∧ Y )n+m

where in,m is a component of the universal bimorphism. This dot operation is associative, i.e., if Z isanother symmetric spectrum and h ∈ πj+qZq is a homotopy class, then we have (f · g) · h = f · (g · h)in πk+l+j+n+m+q(X ∧ Y ∧ Z)n+m+q (where we omit associativity isomorphisms). After spelling out thedefinitions, this associativity ultimately boils down to the equality

(k × χl,n ×m× j × q) (k × l × χj,n+m × q) = (k × n× l × χj,m × q) (k × χl+j,n ×m× q)

in the symmetric group Σk+l+j+q+n+m and the associativity of the smash product.Using the dot product we can rewrite the stabilization map ι∗ : πk+nXn −→ πk+n+1Xn+1 as ι∗(f) =

f · ι1, where Y = S and ι1 ∈ π1S1 is the class of the unit map of the sphere spectrum S, i.e., the identity

map of S1. In the special case X = S we can also multiply with the unit map ι1 from the left; if we identifyS ∧ Y with Y via the left unit isomorphism, and then ι1 · g is the composite of the top row in the diagram

Sl+1+mχl,1×m //

l×χ1,m ''PPPPPPPPPPPP S1+l+mS1∧g //

χ1,l+m

S1 ∧ Ym(λY )m //

twist

Y1+m

Sl+m+1g∧S1

// Ym ∧ S1σm

// Ym+1

χm,1

OO

The diagram commutes, using centrality of ι1, so that we have the relation

ι1 · g = (−1)m(χm,1)∗(g · ι1)

in πl+1+mY1+m. The image of the right hand side in the stable group πlY is precisely d · [g], where d· is theaction of the special monoid element d ∈M, see Example 7.4. So we deduce the relation [ι1 · g] = d · [g] inπlY .

Now we investigate to what extent the dot product passes to stable homotopy groups. If we replacethe right factor g ∈ πl+mYm by the next representative ι∗(g) then associativity gives

f · ι∗(g) = f · (g · ι1) = (f · g) · ι1 = ι∗(f · g) ,

so the class [f · g] in the stable group πk+l(X ∧ Y ) only depends on the class [g] in πlY , and not on theunstable representative. [so we have a well defined map

πk+nXn × πlY −→ πk+l(X ∧ Y ) .

We can also get this map as the effect of the morphism bn.• : Xn∧Y −→ shn(X ∧Y ) on homotopy groups...

Xn ∧ S1 ∧ YId∧λY //

σn∧Id

Xn ∧ shYbn,• // sh(shn Y )

Xn+1 ∧ Ybn+1,•

// shn+1(X ∧ Y )

πk+n(Xn)× πlY −→ πk+l+n(Xn ∧ Y )πk+l+nbn,•−−−−−−−→ πk+l+n(shn(X ∧ Y ))

The following square commutes

πk+nXn × πlYId×d //

ι×Id

πk+nXn × πlY

·

πk+n+1Xn+1 × πlY ·// πk+l(X ∧ Y )


However, the dot product interacts differently with stabilization in the left factor, for we have

(6.8) ι∗(f) · g = (f · ι1) · g = f · (ι1 · g)

which in general will not have the same image in πk+l(X ∧ Y ) as f · g. However, under the hypothesis thatthe monoid M acts trivially on the group πlY we have in particular that [ι1 · g] = d · [g] = [g] in πlY . Byequation (6.8) we then get that ι∗(f) · g and f · g do have the same image in πk+l(X ∧ Y ). So if πlY isM-fixed we get a well-defined paring

· : πkX × πlY −→ πk+l(X ∧ Y ) , [f ] · [g] = [f · g] .

The proof that the dot operation is biadditive is the same as in the special case X = Y = S, compareExample 1.11. We claim that the product ifM-linear in the left variable, i.e., we have (α ·x) · y = α · (x · y)for α ∈ M, x ∈ πkX and y ∈ πlY . To see this we can assume that x = [f ] for some class f ∈ πk+nXn andα ∈ Σn with sgn(α) = +1. Then we have

(α · [f ]) · [g] = [α∗f ] · [g] = [(α∗f) · g] = [(α× 1)∗(f · g)] = α · ([f ] · [g]) .

The third equation uses that the map in,m : Xn ∧ Ym −→ (X ∧ Y )n+m is Σn × Σm-equivariant.Now we claim that the product is associative, graded commutative and unital, whenever it is defined.

Associativity for the unstable dot product immediately implies associativity for its stabilized version, i.e.,for a triple of symmetric spectra X,Y and Z such that the monoid M acts trivially on πlY and πjZ thediagram

πkX × πlY × πjZ

·×Id

Id×· // πkX × πl+j(Y ∧ Z)

·

πk+l(X ∧ Y )× πjZ ·// πk+l+j(X ∧ Y ∧ Z)

commutes. Finally, the product is graded-commutative in the following sense. For every pair of symmetricspectra X and Y and integers k, l such thatM acts trivially on πkX and πlY the square

πkX × πlY· //

twist

πk+l(X ∧ Y )

πk+l(τX,Y )

πlY × πkX ·

// πl+k(Y ∧X)

commutes up to the sign (−1)kl. To see this we consider two homotopy classes f ∈ πk+nXn and g ∈ πl+mYmwhich represents classes in πkX respectively πlY . We have a commutative diagram

Sk+l+n+mk×χl,n×m //

χk,l×χn,m

Sk+n+l+mf∧g //

χk+n,l+m

Xn ∧ Ymin,m //

twist

(X ∧ Y )n+m

(τX,Y )n+m

Sl+k+m+n

l×χk,m×n// Sl+m+k+n

g∧f// Ym ∧Xn im,n

// (Y ∧X)m+n χm,n// (Y ∧X)n+m

in which the composite through the upper right corner represents πk+l(τX,Y )([f ] · [g]). So we get the relation

πk+l(τX,Y )(f · g) = (−1)kl+nm(χm,n)∗(g · f)

as elements of πk+l+n+m(X ∧ Y )n+m. By M-linearity in the left variable and since πlY is M-fixed wededuce

[(−1)nm(χm,n)∗(g · f)] = χm,n · [g · f ] = (χm,n · [g]) · [f ] = [g] · [f ]

in πk+l(Y ∧X). Thus πk+l(τX,Y )([f ] · [g]) = (−1)kl[g] · [f ].

106 I. BASICS

For every symmetric spectrum X both composites around the square

πkX·1 //

1·

πl(X ∧ S)

πl(rX)

πl(S ∧X)

πl(lX)// πlX

are identities, where 1 ∈ π0S is the class of the identity maps of spheres. Indeed, the class 1 is representedby the class of the identity in π0S

0, and we have f · Id = f = Id ·f even before we pass to the stablehomotopy groups.

Let us show that the products on naive and true homotopy groups are compatible with the tautologicalmorphism c : π∗X −→ π∗X, i.e., that we have c(x·y) = cx·cy whenever the left hand side is defined. [reduceto bidegree (0, 0)...] Let [x] and [y] be represented by based maps x : Sn −→ Xn respectively y : Sm −→ Ym,and let x : FnSn −→ X respectively y : FmSm −→ Y be the adjoint morphisms of symmetric spectra. Thisdata gives rise to a commutative diagram

S ∧ S

lS

FnSn ∧ FmSm

λ∧λoo x∧y //

∼=

X ∧ Y

S Fn+mSn+m

λoo

fx·y77ppppppppppp

[specify the middle iso; justify commutativity] As we explained in Example 4.56, the triple (FnSn, λ, x)then represents c[x] ∈ π0X and the triple (FmSm, λ, y) represents c[y] ∈ π0Y . The product c[x] · c[y] is thusrepresented by the triple (FnSn ∧ FmSm, λ ∧ λ, x ∧ y), and the commutative diagram is an isomorphismfrom this to the triple (Fn+mS

n+m, λ, x · y), which represents c([x] · [y]), in the category S, X ∧ Y . Sinceobjects in the same path component of S, X ∧Y represent the same homotopy classes in π0X, this provesthe relation c(x · y) = cx · cy.

Corollary 6.9. Let R be a semistable symmetric ring spectrum. Then the dot product defines thestructure of graded ring on the naive homotopy groups of R such that the tautological map c : π∗R −→ π∗Ris an isomorphism of graded rings. If M is a semistable right R-module, then the dot product gives thenaive homotopy groups of M the structure of a right π∗R-module.

Again, there are analogous graded actions on the naive homotopy groups of semistable left modulesand bimodules.

For a general symmetric ring spectrum R which is not semistable, the naive homotopy groups π∗Rshould be regarded as pathological, and then the true homotopy groups are what one really cares about. Inthis situation, the naive homotopy groups do not have a preferred structure of graded ring (while the truehomotopy groups do, compare Proposition 6.5). Instead, the natural algebraic structure present in π∗R isthat of an algebra over the injection operad. While this is an interesting piece of algebra, it is not relevantfor stable homotopy theory, and so we defer this discussion to Exercise 8.36.

Example 6.10 (Eilenberg-Mac Lane ring spectra). In Example 1.14 we associates to every abeliangroup A a semistable symmetric Eilenberg-Mac Lane spectrum HA whose homotopy groups (naive andtrue) are concentrated in dimension 0, where they are isomorphic to the group A. We explained how anadditional ring structure on A can be used to make HA into a symmetric ring spectrum. Now we do thenecessary reality check: for a ring A the isomorphism jA [ref] between A and π0HA is indeed multiplicative.We recall from [...] that the isomorphism jA : A ∼= π0HA sends an element a ∈ A to the class representedby the based map S0 −→ A = (HA)0 which sends the non-basepoint element of S0 to a.

As we explained in Example 5.20, the multiplication of HA is composed of the a special case of themonoidal transformation mA,B : HA ∧HB −→ H(A⊗B) and the multiplication map A⊗A −→ A of thering A. So multiplicativity of the isomorphism jA is a formal consequence of the fact that for all abelian


groups A and B the composite

A⊗B jA⊗jB−−−−→ π0HA⊗ π0HB·−→ π0(HA ∧HB)

π0(mA,B)−−−−−−→ H(A⊗B)

equals jA⊗B . [This is (?!) straightforward to verify from the definitions.]

Example 6.11 (Monoid ring spectra). revisit Example 3.22

Example 6.12 (Matrix ring spectra). revisit Example 3.24

Example 6.13 (Opposite ring spectrum). For every symmetric ring spectrum R we can define theopposite ring spectrum Rop by keeping the same spaces (or simplicial sets), symmetric group actions andunit maps, but with new multiplication µop

n,m on Rop given by the composite

Ropn ∧Rop

m = Rn ∧Rmtwist−−−→ Rm ∧Rn

µm,n−−−→ Rm+nχm,n−−−→ Rn+m = Rop

n+m .

As a consequence of centrality of ι1, the higher unit maps for Rop agree with the higher unit maps for R.By definition, a symmetric ring spectrum R is commutative if and only if Rop = R. In the internal form,the multiplication µop is obtained from the multiplication µ : R ∧R −→ R as the composite

R ∧R τR,R−−−→ R ∧R µ−→ R .

For example, we have (HA)op = H(Aop) for the Eilenberg-Mac Lane ring spectra (Example 1.14) of anordinary ring A and its opposite, and (R[M ])op = (Rop)[Mop] for the monoid ring spectra (Example 3.22)of a simplicial or topological monoid M and its opposite.

By the centrality of the unit, the underlying symmetric spectra of R and Rop are equal (not justisomorphic), hence R and Rop have the same (not just isomorphic) naive and true homotopy groups. Thegraded commutativity of the external product implies that we have

(6.14) π∗(Rop) = (π∗R)op

(again equality) as graded rings [expand], where the right hand side is the graded-opposite ring, i.e., thegraded abelian group π∗R with new product x ·op y = (−1)kly · x for x ∈ πkR and y ∈ πlR. If R issemistable, then we have can similarly conclude for the naive homotopy groups that π∗(Rop) = (π∗R)op asgraded rings. [also follows formally since c : π∗R −→ π∗R is isomorphism]

Construction 6.15 (Toda brackets). The homotopy groups of a symmetric ring spectrum have evenmore structure than that of a graded ring, they also have ‘secondary’ (and higher. . . ) forms of multiplica-tions, called Toda brackets. Toda brackets are the homotopical analogues of Massey products in differentialgraded algebras, and they satisfy similar relations. We will restrict ourselves to the simplest kind of suchbrackets, namely triple brackets (as opposed to four-fold, five-fold,. . . ) with single entries (as opposed to‘matric’ Toda brackets).

So we let R be a symmetric ring spectrum and M a right R-module. The Toda bracket 〈x, y, z〉 isdefined for every triple of homogeneous elements x ∈ π∗M and y, z ∈ π∗R which satisfies the relationsxy = 0 = yz. If the dimensions of x, y and z are k, l and j respectively, then the bracket is not a singlehomotopy class, but a subset of the homotopy group πk+l+j+1M .

For the construction we represent the classes x, y and z by ‘fractions’ as explained in Example 4.56.So we choose an object (Z, f, g) of the flat fraction category Sk,M[ such that x = 〈Z, f, g〉. Here Z is aflat symmetric spectrum, f : Z −→ Sk a stable equivalence and f : Z −→ M any morphism of symmetricspectra, and the right hand side was defined in Example 4.56 (for flat, see Corollary 5.41) by

〈Z, f, g〉 = (πkg)((πkf)−1(ιk)) ∈ πkM

for ιk ∈ πkSk the fundamental class; this is possible by (4.57). Similarly we choose objects (Z ′, f ′, g′)and (Z ′′, f ′′, g′′) of the flat fraction categories Sl, R[ respectively Sj , R[ such that y = 〈Z ′, f ′, g′〉 andz = 〈Z ′′, f ′′, g′′〉. The product xy ∈ πk+lM is then represented by the fraction (Z ∧ Z ′, f ∧ f ′, g · g′) inSk ∧ Sl,M where g · g′ is the composite of

Z ∧ Z ′ g∧g′−−−−→ M ∧R α−−→ M .

108 I. BASICS

We note that f ∧ f ′ : Z ∧ Z ′ −→ Sk ∧ Sl is again a stable equivalence since all the spectra involved areflat. We have assumed that xy = 0, so Proposition 4.55 allows us to choose a stably contractible spectrumC and a factorization g · g′ = pi for suitable morphisms i : Z ∧ Z ′ −→ C and p : C −→ M . Since wealso have yz = 0 in πl+jR we can choose another stably contractible spectrum C ′ and a factorization ofg · g′ : Z ′ ∧ Z ′′ −→ R as the composite of two morphisms i′ : Z ′ ∧ Z ′′ −→ C ′ and p′ : C ′ −→ R. From thisdata we cook up a morphism of symmetric spectra

p · g′′ ∪ g · p′ : C ∧ Z ′′ ∪Z∧Z′∧Z′′ Z ∧ C ′ −→ M ;

the source of this morphism is the pushout of the diagram

C ∧ Z ′′ i∧Id←−−− Z ∧ Z ′ ∧ Z ′′ Id∧i′−−−→ Z ∧ C ′

and we now construct a canonical homotopy class ι in πk+l+j+1(C ∧ Z ′′ ∪Z∧Z′∧Z′′ Z ∧ C ′). Now we canfinally define the Toda bracket 〈x, y, z〉 as the set of all elements of the form

(p · g′′ ∪ g · p′)∗(ι) ∈ πk+l+j+1M

for all possible representations (in the above sense) of x, y and z as flat fractions, and all possible factor-izations of g · g′ and g′ · g′′ through stably contractible spectra.

Later we will discuss two different constructions of the Toda bracket in slightly different contexts.In II.2.1 we define Toda brackets for composable morphisms in triangulated categories. [... and in ...]The context of triangulated categories is the most convenient one for proving various properties of Todabrackets, such as the indeterminacy and certain juggling formulas. A priori, the Toda bracket 〈x, y, z〉 issome subset of the group πk+l+j+1M , but in fact one can say more:

Proposition 6.16. Whenever defined, the Toda bracket 〈x, y, z〉 ⊂ πk+l+j+1M is a coset for the sub-group (x · πl+j+1R) + (πk+l+1M · z).

The subgroup x · πl+j+1R+ πk+l+1M · z is called the indeterminacy of the Toda bracket 〈x, y, z〉.Toda brackets satisfy a number of relations. The first of these is a kind of ‘higher form of associativity’

and is often referred to as a juggling formula. For the juggling formula we need another homogeneous classw ∈ π∗R satisfying zw = 0. Then we have

(6.17) x · 〈y, z, w〉 = (−1)k+1〈x, y, z〉 · w

modulo the common indeterminacy x · πs|yz|+1 · w. [check the sign]Toda bracket are functorial under homomorphisms of symmetric ring spectra [more generally in pair

morphisms (f, ϕ) : (S,N) −→ (R,M)]. More precisely, it is straightforward to check that for every homo-morphism f : S −→ R we have

f∗ (〈x, y, z〉) ⊆ 〈f∗(x), f∗(y), f∗(z)〉

whenever the bracket on the left is defined. The indeterminacy of the right hand side may be larger thanthe image of the indeterminacy of the bracket 〈x, y, z〉, which is why in general we only have containment,not necessarily equality, as subsets on πk+l+j+1R.

As we exaplain in Exercise 8.22, the Toda brackets of the form 〈x, y, z〉 for varying y and z containsignificant information about the structure of π∗(M/xR) as a graded π∗R-module, where M/xR is themapping cone of a morphism of R-modules which realizes left multiplication by x in homotopy.

Example 6.18. Here are some examples of non-trivial Toda bracket. In the stable stems, i.e., thehomotopy groups of the sphere spectrum (compare the table in Example 1.11) we have

η2 ∈ 〈2, η, 2〉 mod (0) 6ν ∈ 〈η, 2, η〉 mod (12ν)ν2 ∈ 〈η, ν, η〉 mod (0) 40σ ∈ 〈ν, 24, ν〉 mod (0)

ησ + ε ∈ 〈ν, η, ν〉 mod (0) ε ∈ 〈η, ν, 2ν〉 mod (ησ)


With the help of juggling formulas, we can use Toda brackets to deduce relations which themselves don’trefer to brackets. For example, the juggling formula (6.17)

η · 〈2, η, 2〉 = 〈η, 2, η〉 · 2

holds in πs3 with zero indeterminacy. By the table above, the first bracket contains η3 while the second

bracket contains 12ν. So we get the multiplicative relation η3 = 12ν as a consequence of the Toda bracketsinvolving 2 and η.

The first bracket η2 ∈ 〈2, η, 2〉 in the table above is a special case of Toda’s relation

ηx ∈ 〈2, x, 2〉

which holds for all 2-torsion classes x in the homotopy of every commutative [?] symmetric ring spectrum.This in turn is a special case of relations which hold between Toda brackets and power operations in thehomotopy ring of commutative symmetric ring spectra. We plan to get back to this later.

Another example of a non-trivial Toda bracket is u ∈ 〈1, η, 2〉 (modulo 2u) in π2KU . Here the complextopological K-theory spectrum KU (compare Example 1.18) is viewed just as a symmetric spectrum (andnot as a ring spectrum). So the Bott class u and 1 are homotopy classes of KU , while η and 2 have to beviewed as elements in the stable stems.

Since the relation η2 ∈ 〈2, η, 2〉 holds in πs2, it also holds in the second homotopy group of every

ring spectrum (with possibly bigger indeterminacy, and possibly with η2 = 0); In the homotopy of realtopological K-theory KO the class η2 is non-zero (compare the table in Example 1.18), so we get a non-trivial bracket η2 ∈ 〈2, η, 2〉 (modulo 0) in π2KO. We also have ξ ∈ 〈2, η, η2〉 (modulo 2ξ) in π4KO and2β ∈ 〈ξ, η, η2〉 (modulo 4β) in π8KO. [prove. Also η2β = Sq1(ξ) ∈ 〈ξ, η, ξ〉 (modulo 0) in π10KO]

We consider a symmetric ring spectrum R and a based map x : Sl+m −→ Rm. We let x : FmSl+m −→ Rdenote the adjoint morphism of symmetric spectra. For every right R-module M we can define a morphismof symmetric spectra

(6.19) ρx : FmSl+m ∧M −→ M

as the composite

FmSl+m ∧M

τFmSl+m,M−−−−−−−−→ M ∧ FmSl+m

Id∧x−−−→ M ∧R act−−→ M.

The name ρx stands for ‘right multiplication by x’. [relate the composite of ρy and ρx to ρx·y]

Proposition 6.20. Let R be a symmetric ring spectrum, x : Sl+m −→ Rm a based map and M a rightR-module.

(i) The morphism ρx : FmSl+m∧M −→M realizes right multiplication by the class 〈x〉 = c[x] of πlRin homotopy. More precisely, the composite

πkM·ιml+m∼=

// πk+l(M ∧ FmSl+m)πk+l(ρx) // πk+lM

is equal to − · 〈x〉 (up to a sign ???), where ιml+m ∈ πl(FmSl+m) is the fundamental class definedin (4.49).

(ii) Suppose that the map x is central, i.e., the square

Rn ∧ Sl+mId∧x //

τ

Rn ∧Rmµn,m // Rn+m

χn,m

Sl+m ∧Rn x∧Id

// Rm ∧Rn µm,n// Rm+n

commutes for all n ≥ 0. Then the morphism of symmetric spectra ρx : FmSl+m ∧M −→ M is ahomomorphism of right R-modules.

110 I. BASICS

Proof. Part (i) is an exercise in naturality properties. We start from the observation that [x] ∈ πlRis the image of the naive fundamental class ιml+m ∈ πl(FmSl+m) (see (4.48)) under the morphism x :FmS

l+m −→ R. By naturality of the morphism c from naive to true homotopy groups the class 〈x〉 = c[x]in πlR is the image of the true fundamental class ιml+m ∈ πl(FmSl+m). We have

πk+l(ρx)(a · ιml+m) = πk+l(α)(πk+l(IdM ∧x)(a · ιml+m))

= πk+l(α)(a · (πl(x)(ιml+m))) = πk+l(α)(a · 〈x〉)

where the first equality uses the definition of ρx as α(IdM ∧x) [mind the twist], and the second is naturalityof the external dot product.

Part (ii): [fill in]

Remark 6.21. For a based map x : Sl+m −→ Rm and a right R-module M we can adjoin the morphismρx : FmSl+m ∧M −→ M to a morphism ρx : Sl+m ∧M −→ shmM . For example, if x = ι1 : S1 −→ R1 isthe unit map, then ρι1 : S1 ∧M −→ shM equals the morphism λM defined in (3.9). While the morphismρx makes sense for any map x and R-module M , we can only analyze it homotopically if M is semistable.In that case shmM is stably equivalent to the m-fold suspension of M and the morphism ρx is stablyequivalent to the m-fold suspension of ρx. So if M is semistable, then also ρx realizes right multiplicationby c[x] ∈ πlR in the sense that the diagram

πkM·[x] //

Sl+m∧− ∼=

πk+lM

∼=

πl+m+k(Sl+m ∧M)πl+m+k(ρx)

// πl+m+k(shmM)

commutes up to the sign (−1)k(l+m). [specify an iso on RHS][is this still OK ?: As usual, the sign in part (ii)above can be predicted by remembering that the group πk+lM is ‘naturally’ equal to πk+l+m(shmM)(compare Example 3.7), whereas the ‘natural’ target of the lower vertical map is πl+m+k(shmM); sosecretly, k sphere coordinates move past l +m other coordinates, hence the sign.]

[rk: only the classes in πkR which are in the image of the morphism c : πkR −→ πkR can be realizedba morphisms x : Sl+m −→ Rm. The spectral sequence gives obstructions]

Example 6.22 (Killing a homotopy class). We describe a construction that can be used to ‘kill’ theaction of a homotopy class in a ring spectrum on a given module. Now consider a symmetric ring spectrumR, a right R-module M and a central map x : Sl+m −→ Rm. We let M/x denote the mapping cone of themorphism ρx : FmSl+m ∧M −→ M (see (6.19)). Then by Proposition 6.20 (i) the morphism ρx realizesmultiplication by 〈x〉 ∈ πlR on true homotopy groups. The long exact true homotopy group sequence of amapping cone (Proposition 4.46 (i)) breaks up into a short exact sequence of π∗R-modules [the connectingmorphism is π∗R-linear !]

0 −→ π∗M/(π∗−lM)〈x〉 −→ π∗(M/x) −→ π∗−l−1M〈x〉 −→ 0

where the first map is induced by the mapping cone inclusion M −→ M/x and −〈x〉 denotes the π∗R-submodule of homotopy classes annihilated by 〈x〉. If 〈x〉 acts injectively on the true homotopy groups of M ,then we can conclude that the morphism M −→M/x realizes the quotient map π∗M −→ π∗M/(π∗−lM)〈x〉on true homotopy groups.

Example 6.23 (Killing a regular sequence). We can iteratively kill homotopy classes as in the previousexample and thereby kill the action of certain ideals in the homotopy groups of a symmetric ring spectrum.We just saw that we can only control the homotopy groups of M/x if the homotopy class 〈x〉 which is killedis not a zero divisor on π∗M . So iterating the construction naturally leads us to consider regular sequences.

Recall that a sequence, finite or countably infinite, of homogeneous elements yi in a graded commutativering R∗ is a regular sequence for a graded R∗-module M∗ if y1 acts injectively on M∗ and for all i ≥ 2 the


element yi acts injectively on M∗/M∗ · (y1, . . . , yi−1). A homogeneous ideal I of R∗ is a regular ideal for M∗if it can be generated by a regular sequence, finite or countably infinite, for M∗.

To simplify the exposition we now assume that the ring spectrum R we work over is semistable andcommutative. Semistability guarantees that ever class in the true homotopy group πlR is realizable by amap Sl+m −→ Rm for some m ≥ 0; commutativity guarantees that any such map is automatically central.[semistable is not loss of generality since any R is stably equivalent, as a symmetric ring spectrum, to asemistable one; if R is commutative, then the replacement can be chosen commutative as well]

Proposition 6.24. Let R be a commutative semistable symmetric ring spectrum, M a right R-moduleand I a homogeneous ideal of π∗R. If I is a regular ideal for the module π∗M then there exists a semistableR-module M/I and a homomorphism q : M −→ M/I of R-modules such that the induced homomorphismof true homotopy group

π∗(q) : π∗M −→ π∗(M/I)

is surjective and has kernel equal to (π∗M)I.

Proof. We choose a sequence y1, y2, . . . of homogeneous elements of π∗R which generate the ideal Iand form a regular sequence for π∗M . We construct inductively a sequence of semistable R-modules M i

and homomorphismsM = M0 q1−→ M1 q2−→ M2 q3−→ · · ·

such that the composite morphism M −→ M i is surjective on homotopy groups and has kernel equal to(π∗M) · (y1, . . . , yi).

The induction starts with i = 0, where there is nothing to show. In the ith step we let l be thedimension of the homotopy class yi; since R is semistable the map c : πlR −→ πlR is bijective and we canchoose a based map x : Sl+m −→ Rm such that yi = c[x] = 〈x〉 in πlR. By induction the true homotopygroups of M i−1 realize the π∗R-module π∗M/(π∗M) · (y1, . . . , yi−1). Since we have a regular sequence forπ∗M the class yi = 〈x〉 acts injectively on the homotopy of M i−1, so the morphism qi : M i−1 −→M i−1/xconstructed in Example 6.22 realizes the projection π∗M i−1 −→ π∗M

i−1/(π∗M i−1) · yi. We can thus takeM i = M i−1/x; then the composite morphism M −→ M i is again surjective on homotopy groups and itskernel is

(π∗M) · (y1, . . . , yi−1) + (π∗M) · yi = (π∗M) · (y1, . . . , yi) .This finishes the argument if I is generated by a finite regular sequence.

If the generating sequence is countably infinite we define M/I as the diagonal (see ???; Example 2.23is this for naive groups) of the above sequence of R-modules M i. Then the natural map

colimi π∗Mi −→ π∗(M/I)

is an isomorphism (see (??)), and the left hand side is isomorphic to

colimi (π∗M/(π∗M) · (x1, . . . , xi)) ∼= π∗M/(π∗M) · (x1, x2, . . . ) = π∗M/(π∗M) · I .

Let R be a symmetric ring spectrum and x ∈ πkR a zero divisor in the homotopy ring of R. Then wemay not be able realize the module π∗R/(π∗−lR)〈x〉 as the homotopy of an R-module, and Toda bracketsgive the first obstructions to such a realization.

Proposition 6.25. Let R be a symmetric ring spectrum and x ∈ πkR a homotopy class. If there isa right R-module whose homotopy groups are isomorphic to π∗R/x · π∗−kR as a graded right π∗R-module,then for all homogeneous homotopy classes y, z with xy = 0 = yz the Toda bracket 〈x, y, z〉 contains 0.

Proof. We choose a right R-module N and a π∗-linear epimorphism p : π∗R −→ π∗N with kernelx · (π∗−kR). The bracket 〈p(1), x, y〉 is then defined with indeterminacy p(1) · πk+l+1R + πk+1N · y. Sincep(1) generates π∗N as a graded π∗R-module, the indeterminacy of the bracket 〈p(1), x, y〉 is the entire groupπk+l+1N , and so we have 〈p(1), x, y〉 = πk+l+1N .

112 I. BASICS

We choose an element α ∈ πk+l+j+1R of the bracket 〈x, y, z〉. Using the juggling formula (6.17) we get

p(α) = p(1) · α ∈ p(1) · 〈x, y, z〉 = −〈p(1), x, y〉 · z = πl+1N · z = p(πl+1R · z) .

Since the kernel of p equals x · π∗−kR we deduce that α lies in x · πl+j+1R+ πk+l+1R · z. This is preciselythe indetermincy of the bracket 〈x, y, z〉, so the bracket also contains 0.

Example 6.26. Here are some non-realizability results which can be obtained with the help of Propo-sition 6.25. We have the relation η2 ∈ 〈2, η, 2〉 in πs

2 with zero indeterminacy. So the bracket 〈2, η, 2〉 doesnot contain zero and hence there is no symmetric spectrum whose homotopy groups realize Z/2 ⊗ πs

∗ asa graded πs

∗-module. For an odd prime p, the bracket 〈p, α1, α1〉 in πs4p−5 does not contain zero, hence

Z/p⊗ πs∗ is not realizable as the homotopy of a spectrum.

In Example 6.18 we also exhibited various nonzero triple Toda brackets in the homotopy of the realtopological K-theory spectrum KO, such as

η2 ∈ 〈2, η, 2〉 , ξ ∈ 〈2, η, η2〉 and 2β ∈ 〈ξ, η, η2〉 .

Moreover, in all three cases, the class on the left is not in the indeterminacy group, so the three brackets donot contain zero. Proposition 6.25 lets us conclude that the π∗KO-modules Z/2⊗π∗KO(2), π∗KO/η2·π∗KOand π∗KO/ξ · π∗KO are not realizable as the homotopy of any KO-module spectrum. In Exercise 8.23we show more generally that the only cyclic π∗KO-modules which are realizable as the homotopy of aKO-module spectrum are the free module, the trivial module and Z/n⊗ π∗KO for n an odd integer.

[show that any KO-morphism KO/η −→ KU which extends the complexification map KO −→ KU isa stable equivalence]

Example 6.27 (Inverting a homotopy class). In Example 3.27 we defined a new symmetric ring spec-trum R[1/x] from a given symmetric ring spectrum R and a central map x : S1 −→ R1. We now generalizethis construction to central maps x : Sl+m −→ Rm and also analyze it homotopically.

First we extend the localization construction to right R-modules M . We define a right R[1/x]-moduleM [1/x] by

M [1/x]p = map(S(l+m)p,M(1+m)p) .

The symmetric group Σp acts on S(l+m)p and M(1+m)p by permuting the p blocks of l + m respectively1 +m coordinates, i.e., by restriction along the diagonal embeddings

∆ : Σp −→ Σ(l+m)p respectively ∆ : Σp −→ Σ(1+m)p .

More precisely, the diagonal embedding ∆ : Σp −→ Σnp is defined by ∆(γ) = Idn ·γ (using the notation ofRemark 0.5) which unravels to

∆(γ)(j + (i− 1)n) = j + (γ(i)− 1)

for i ≤ j ≤ n and 1 ≤ i ≤ p. The action of Σp on the whole mapping space M [1/x]p is then by conjugation.A special case of this is M = R and we now define Σp × Σq-equivariant action maps

αp,q : M [1/x]p ∧R[1/x]q −→ M [1/x]p+q

as the composite

map(S(l+m)p,M(1+m)p) ∧map(S(l+m)q, R(1+m)q) −→ map(S(l+m)(p+q),M(1+m)(p+q))

f ∧ g 7−→ µ(1+m)p,(1+m)q (f ∧ g) .

The action maps are associative because smashing and the original action maps are. The verification themaps αp,q are Σp × Σq-equivariant ultimately boils down to the equivariant for the original action mapsand the fact that the diagonal embeddings ∆p : Σp −→ Σnp, ∆q : Σq −→ Σnq and ∆p+q : Σp+q −→ Σn(p+q)

satisfy∆p(γ)×∆q(τ) = ∆p+q(γ × τ) .


The next piece of data we define are maps jp : Mp −→ M [1/x]p which we will later recognize as ahomomorphism of R-modules. We denote by xp : S(l+m)p −→ Rmp the composite

S(l+m)p x(p)

−−→ R(p)m

µm,...,m−−−−−→ Rmp ;

because x is central the map xp is Σp-equivariant if we let Σp act on source and target through the diagonalembeddings. We define

jp : Mp −→ map(S(l+m)p,M(1+m)p) = M [1/x]p

as the adjoint to

Mp ∧ S(l+m)p Id∧xp−−−−→ Mp ∧Rmpαp,mp−−−−→ Mp+mp

ξ∗−→ M(1+m)p

where ξ ∈ Σ(1+m)p is given by

ξ(i) =

1 + (i− 1)(1 +m) if 1 ≤ i ≤ p(j + 1) + (k − 1)(1 +m) if i = p+mk + j for 1 ≤ k ≤ p, 1 ≤ j ≤ m.

The map jp is Σp-equivariant. In terms of the adjoint of jp this means that for every permutationγ ∈ Σp the outer composites in the diagram

Mp ∧ S(l+m)p Id∧xp //

γ∧∆(γ)

Mp ∧Rmpαp,mp //

γ∧∆(γ)

Mp+mp

γ×∆(γ)

ξ // M(1+m)p

∆(γ)

Mp ∧ S(l+m)p

Id∧xp// Mp ∧Rmp αp,mp

// Mp+mpξ

// M(1+m)p

agree. The left square commutes by equivariance of xp, the middle square by the equivariance of the actionsmaps of M . The right square commutes because the relation

ξ (γ ×∆(γ)) = ∆(γ) ξ

holds in the symmetric group Σ(1+m)p.The collection of maps jp is multiplicative in the sense that the squares

Mp ∧Rqαp,q //

jp∧jq

Mp+q

jp+q

M [1/x]p ∧R[1/x]q αp,q

// M [1/x]p+q

commute [elaborate].ForM = R we define the unit maps ιn : Sn −→ R[1/x]n as the composite of the unit map ιn : Sn −→ Rn

for R with jn : Rn −→ R[1/x]n. [central] Then j : R −→ R[1/x] is a homomorphism of symmetric ringspectra and j : M −→M [1/x] is a morphism of R-modules if we view M [1/x] as an R-module by restrictionof scalars along j : R −→ R[1/x]. In other words, we have constructed a functor

mod-R −→ mod-R[1/x] , M 7−→ M [1/x]

and a natural transformation of R-modules j : M −→ j∗(M [1/x]).[leftover: Both mapping telescope and diagonal preserve module structures. Suppose that each symmet-

ric spectrum Xi in the sequence (2.24) has the structure of right module over a symmetric ring spectrumR and that all morphisms f i are R-linear. Then the mapping telescope is naturally an R-module sinceall constructions used to build it preserves the action by the ring spectrum. The diagonal is naturally an

114 I. BASICS

R-module as well, with action maps (diagiXi)n ∧ Rm −→ (diagiXi)n+m equal to the composite aroundeither way in the commutative square

Xnn ∧Rm

αnn,m //

(fn+m−1···fn)n∧Id

Xnn+m

(fn+m−1···fn)n+m

Xn+mn ∧Rm

αn+mn,m

// Xn+mn+m

]

Remark 6.28. The permutation ξ can be predicted as follows by the general rules which we introducedin Remark 0.5. The natural ‘coordinate free’ target of the map αp,m,...,m is indexed by the set

p + m + · · ·+ m︸︷︷︸p

= 1 · p + m · p .

This set is equal to the set (1 + m) · p, but the way the parenthesis arise naturally reminds us that weshould use the right distributivity isomorphism

p + m · p = 1 · p + m · p∼=−→ (1 + m) · p

to identify the two sets. The right distributivity isomorphism is defined using the multiplicative shufflesas χ×p,1+m (χ×1,p × χ×m,p), which is precisely the permutation ξ. Note that in contrast, the definition ofthe actions map αp,q does not need any permutations. Indeed, the natural coordinate free target of αp,q isindexed by the set (1 + m) ·p + (1 + m) · q, which is equal to the set (1 + m) · (p + q). Here, however, theparenthesis suggest using the left distributivity isomorphism, which is the identity permutation.

The relation ξ (γ ×∆(γ)) = ∆(γ) ξ which came up in the proof of the equivariance on the map jpjust expresses the fact that the multiplicative shuffle χ×n,m : n ·m −→m · n is a natural isomorphism

1 · p + m · pχ×1,p+χ

×m,p //

ξ

++

Id1 ·γ+Idm ·γγ+∆(γ)

p · 1 + p ·m = p · (1 + m)χ×p,1+m //

γ·Id1 +γ·Idm γ·Id1+m

(1 + m) · p

∆(γ)γ·Id1

1 · p + m · p

χ×1,p+χ×m,p

//

ξ

33p · 1 + p ·m = p · (1 + m)χ×p,1+m

// (1 + m) · p

Now we analyze the construction homotopically, under the additional assumption that the class of x inπlR is M-fixed. The centrality condition on the map x implies that then the stable homotopy class [x] iscentral in the graded ring (π∗R)(0). [when is R[1/x] equivalent to just changing the unit map ?]

Proposition 6.29. Let R be a symmetric ring spectrum and x : Sl+m −→ Rm a central map whoseclass [x] in πlR isM-fixed. Then for every semistable right R-module M [levelwise Kan] the R[1/x]-moduleM [1/x] is again semistable [and levelwise Kan] and the morphism j : M −→M [1/x] of R-modules inducesa natural isomorphism

(6.30) (π∗M)[1/[x]] ∼= π∗(M [1/x]

)of graded modules over (π∗R)(0). In the special case M = R the morphism of symmetric ring spectraj : R −→ R[1/x] induces an isomorphism of graded rings

(π∗R)(0)[1/[x]] ∼= π∗(R[1/x]

)(0).


Proof. As an R-module M [1/x] is equal to the diagonal [No: not correct Σn- action] (see Exam-ple 2.23) of the sequence of R-module homomorphisms

Mρx−−−−−→ Ωl+m(shmM)

Ωl+m shm(ρx)−−−−−−−−−→ Ω(l+m)2(shm2M) · · ·

· · · Ω(l+m)p(shmpM)Ω(l+m)p shmp(ρx)−−−−−−−−−−−→ · · ·

where ρx is adjoint to the ‘right multiplication’ homomorphism defined in Example 6.22. The map j :M −→M [1/x] is the canonical morphism from the initial term of a sequence to the diagonal spectrum.

By Proposition 6.20 (ii) the effect of ρx on homotopy groups is right multiplication by the class [x] ∈πlR. So since the homotopy groups of the diagonal are isomorphic to the colimit of the homotopy groups(see (??)), we deduce that j induces the isomorphism

π∗(M [1/x]) = π∗ diagp(Ω(l+m)p(shmpM)

)∼= colimp π∗

(Ω(l+m)p(shmpM)

)∼= colimp π∗+lpM

with the last colimit being taken over iterated multiplication by [x]. Since the right hand side is the resultof inverting [x] in π∗M , this proves the first claim.

In the special case M = R we now know that the morphism of symmetric ring spectra j : R −→ R[1/x]induces an isomorphism of graded (π∗R)(0)-modules

(π∗R)(0)[1/[x]] ∼= π∗(R[1/x]

)(0).

This is necessarily a multiplicative isomorphism.

An important special case of the above is when the symmetric ring spectrum R is commutative (whichmakes centrality of the map x is automatic) and semistable (so that all of π∗R is M-fixed). For easierreference we spell out Proposition 6.29 in this special case.

Corollary 6.31. Let R be a commutative semistable symmetric ring spectrum [levelwise Kan] andx : Sl+m −→ Rm a based map. Then R[1/x] is a commutative symmetric ring spectrum, the homomorphismof symmetric ring spectra j : R −→ R[1/x] sends the class [x] ∈ πlR to a unit in the l-th homotopy groupof R[1/x] and the induced morphism of graded commutative rings

(π∗R)[1/[x]] ∼= π∗(R[1/x]

)is an isomorphism.

Example 6.32 (Brown-Peterson, Johnson-Wilson spectra and Morava K-theory). If we apply themethod of ‘killing a regular sequence’ to the Thom spectrum MU we can construct a whole collection ofimportant spectra. In Example 1.17 we constructed MU as a commutative symmetric ring spectrum, andMU is semistable because it underlies an orthogonal spectrum (compare Proposition 7.28). As input for thefollowing construction we need the knowledge of the homotopy ring of MU . The standard way to performthis calculation is in the following sequence of steps:

• calculate, for each prime p, the mod-p cohomology of the spaces BU(n) and BU ,• use the Thom isomorphism to calculate the mod-p cohomology of the Thom spectrum MU as a

module over the mod-p Steenrod algebra,• use the Adams spectral sequence, which for MU collapses at the E2-term, to calculate the p-

completion of the homotopy groups of MU ,• and finally assemble the p-local calculations into the integral answer.

When the dust settles, the result is that π∗MU is a polynomial algebra generated by infinitely manyhomogeneous elements xi of dimension 2i for i ≥ 1. The details of this calculation can be found in [77]and [64] [check this; other sources?]. A very different geometric approach to this calculation was describedby Quillen [61], who determines the ring of cobordism classes of stably almost complex manifolds, whichby Thom’s theorem is isomorphic to π∗MU . (Quillen’s argument, however, needs as an input the a prioriknowledge that the homotopy groups of MU are finitely generated in each dimension.)

116 I. BASICS

Now fix a prime number p. Using the close connection between the ring spectrum MU and the theoryof formal groups laws one can make a particular choices for the (pn − 1)-th generator xpn−1, the so-calledHazewinkel generator, which is then denoted vn. Killing all polynomial generators except those of the formxpn−1 produces a semistable MU -module BP with homotopy groups π∗(BP ) = Z[v1, v2, v3, . . . ] where thedegree of vn is 2pn − 2. Localizing at p produces a semistable MU -module BP , called the Brown-Petersonspectrum, with homotopy groups

π∗BP ∼= Z(p)[v1, v2, v3, . . . ] .The original construction of this spectrum by Brown and Peterson was quite different, and we say more

about the history of BP in the ‘History and credits’ section at the end of this chapter.Now we can keep going and kill more of the polynomial generators vi in the homotopy of BP , and possi-

bly also invert another generator. In this way we can produce various MU -modules BP/I and (BP/I)[v−1n ]

together with MU -homomorphisms from BP whose underlying stable homotopy types play important rolesin stable homotopy theory. Some examples of spectra which we can obtained in this way are given in thefollowing table, along with their homotopy groups:

BP 〈n〉 Z(p)[v1, v2, . . . , vn] E(n) Z(p)[v1, v2, . . . , vn, v−1n ]

P (n) Fp[vn, vn+1, . . . ] B(n) Fp[v−1n , vn, vn+1, . . . ]

k(n) Fp[vn] K(n) Fp[vn, v−1n ]

[discuss uniqueness] The spectrum E(n) is referred to as the Johnson-Wilson spectrum and k(n) respectivelyK(n) are the connective and periodic Morava K-theory spectra.

We have so far only constructed the spectra above as MU -modules. The way we have presented thehomotopy groups of the various spectra above does not only give graded modules over the homotopy ringof MU , but in fact graded commutative algebras. This already hints that the spectra have more structure.In fact, all the spectra above can be constructed as MU -algebra spectra, so in particular as symmetric ringspectra. We may or may not get back to this later.

7. True versus naive homotopy groups

The naive homotopy groups of a symmetric spectrum do not take the action of the symmetric groupsinto account; this has the effect that there is extra structure on πk which we will discuss now. In order tounderstand the relationship between π∗-isomorphisms and stable equivalences, it is useful to exploit all thealgebraic structure available on the naive homotopy groups of a symmetric spectrum. This extra structureis an action of the injection monoidM the monoid of injective self-maps of the set of natural numbers undercomposition. The cycle operator, which we used to define semistable spectra, is subsumed in this picture asthe action of a particular injection. TheM-modules that come up, however, have a special property whichwe call tameness, see Definition 7.1. Tameness has strong algebraic consequences and severely restricts thekinds of M-modules which can arise as homotopy groups of symmetric spectra.

7.1. M-action on homotopy groups. The definition of homotopy groups does not take the sym-metric group actions into account; using these actions we will now see that the homotopy groups of asymmetric spectrum have more structure.

Definition 7.1. The injection monoid is the monoidM of injective self-maps of the set ω = 1, 2, 3, . . . of natural numbers under composition. AnM-module is a left module over the monoid ring Z[M]. We callanM-module W tame if for every element x ∈W there exists a number n ≥ 0 with the following property:for every element f ∈M which fixes the set n = 1, . . . , n elementwise we have fx = x.

As we shall soon see, the homotopy groups of a symmetric spectrum have a natural tame M-action.An example of anM-module which is not tame is the free module of rank 1. Tameness has many algebraicconsequences which we discuss in the next section. We will see in Remark 7.24 that the M-action givesall natural operations on the homotopy groups of a symmetric spectrum; more precisely, we show that thering of natural operations on π0X is a completion of the monoid ring ofM, so that an arbitrary operationis a sum, possibly infinite, of operations by elements from M.

7. TRUE VERSUS NAIVE HOMOTOPY GROUPS 117

Construction 7.2. [Omit I-functors and define in one go; move I-functors to exercises] Wedefine an action of the injection monoidM on the homotopy groups of a symmetric spectrum X.

Step 1: from symmetric spectra to I-functors. For every integer k we assign an I-functor πkX to thesymmetric spectrum X. On objects, this I-functor is given by

(πkX)(n) = πk+nXn

if k + n ≥ 2 and (πkX)(n) = 0 for k + n < 2. If α : n −→ m is an injective map and k + n ≥ 2, thenα∗ : (πkX)(n) −→ (πkX)(m) is given as follows. We choose a permutation γ ∈ Σm such that γ(i) = α(i)for all i = 1, . . . , n and set

α∗(x) = sgn(γ) · γ(ιm−n(x))where ι : πk+nXn −→ πk+n+1Xn+1 is the stabilization map (1.7).

We have to justify that this definition is independent of the choice of permutation γ. Suppose γ′ ∈ Σmis another permutation which agrees with α on n. Then γ−1γ′ is a permutation of m which fixed thenumbers 1, . . . , n, so it is of the form γ−1γ′ = 1 × τ for some τ ∈ Σm−n, where 1 is the unit of Σn. Itsuffices to show that for such permutations the induced action on πk+mXm via the action on Xm satisfiesthe relation

(7.3) (1× τ) · (ιm−n(x)) = sgn(τ) · (ιm−n(x))

for all x ∈ πk+nXn. To justify this we let f : Sk+n −→ Xn represent x. Since the iterated structure mapσm−n : Xn ∧ Sm−n −→ Xm is Σn × Σm−n-equivariant, we have a commutative diagram

Sk+mf∧Id //

Id∧τ

Xn ∧ Sm−nσm−n //

Id∧τ

Xm

1×τ

Sk+mf∧Id

// Xn ∧ Sm−nσm−n

// Xm

The composite through the upper right corner represents (1 × τ)(ιm−n(x)). Since the effect on homo-topy groups of precomposing with a coordinate permutation of the sphere is multiplication by the sign ofthe permutation, the composite through the lower left corner represents sgn(τ) · (ιm−n(x)). This provesformula (7.3) and completes the definition of α∗ : (πkX)(n) −→ (πkX)(m).

The inclusion n −→ n + 1 induces the stabilization map ι over which the colimit πkX is formed, so ifwe denote the inclusion by ι, then two meanings are consistent. We let N denote the subcategory of I whichcontains all objects but only the inclusions as morphisms, and then we have

πkX = colimN πkX .

Step 2: from I-functors to tame M-modules. The next observation is that for any I-functor F thecolimit of F , formed over the subcategory N of inclusions, has a natural left action by the injection monoidM. Applied to the I-functor πkX coming from a symmetric spectrum X, this yields the M-action on thestable homotopy group πkX.

We let Iω denote the category with objects the sets n for n ≥ 0 and the set ω and with all injective mapsas morphisms. So Iω contains I as a full subcategory and contains one more object ω whose endomorphismmonoid is M. We will now extend an I-functor F to a functor from the category Iω in such a way thatthe value of the extension at the object ω is the colimit of F , formed over the subcategory N of inclusions.It will thus be convenient, and suggestive, to denote the colimit of F , formed over the subcategory N ofinclusions, by F (ω) and not introduce new notation for the extended functor. TheM-action on the colimitof F is then the action of the endomorphisms of ω in Iω on F (ω).

So we set F (ω) = colimN F and first define β∗ : F (n) −→ F (ω) for every injection β : n −→ ω asfollows. We set m = maxβ(n), denote by β|n : n −→ m the restriction of β and take β∗(x) to be theclass in the colimit represented by the image of x under

(β|n)∗ : F (n) −→ F (m) .

118 I. BASICS

It is straightforward to check that this is a functorial extension of F , i.e., for every morphism α : k −→ nin I we have (βα)∗(x) = β∗(α∗(x)).

Now we let f : ω −→ ω be an injective self-map of ω, and we want to define f∗ : F (ω) −→ F (ω). If[x] ∈ F (ω) is an element in the colimit represented by a class x ∈ F (n), then we set f∗[x] = [(f |n)∗(x)]where f |n : n −→ ω is the restriction of f and f∗ : F (n) −→ F (ω) was defined in the previous paragraph.Again it is straightforward to check that this definition does not depend on the representative x of the class[x] in the colimit and that the extension is functorial, i.e., we have (fα)∗(x) = f∗(α∗(x)) for injectionsα : n −→ ω as well as (fg)∗[x] = f∗(g∗[x]) when g is another injective self-map of ω. As an example, if wealso write ι : n −→ ω for the inclusion, then we have ι∗(x) = [x] for x ∈ F (n).

The definition just given is in fact the universal way to extend an I-functor F to a functor on thecategory Iω, i.e., we have just constructed a left Kan extension of F : I −→ Ab along the inclusion I −→ Iω.However, we do not need this fact, so we omit the proof.

A trivial but important observation straight from the definition is that the action of the injection monoidM on πkX is tame in the sense of Definition 7.1: every element [x] ∈ πkX in the colimit is represented bya class x ∈ πk+nXn for some n ≥ 0; then for every element f ∈ M which fixes the numbers 1, . . . , n, wehave f · [x] = [x].

Example 7.4. To illustrate the action of the injection monoid M on the homotopy groups of a sym-metric spectrum X we make it explicit for the injection d : ω −→ ω given by d(i) = i+ 1. For every n ≥ 1,the map d and the cycle (1, 2, . . . , n, n + 1) of Σn+1 agree on n, so d acts on πkX as the colimit of thesystem

πkX0ι //

ι

πk+1X1ι //

−(1,2)·ι

πk+2X2ι //

(1,2,3)·ι

· · · ι // πk+nXnι //

(−1)n(1,2,...,n,n+1)·ι

πk+1X1 ι// πk+2X2 ι

// πk+3X3 ι// · · ·

ι// πk+n+1Xn+1 ι

//

(at least for k ≥ 0; for negative values of k only a later portion of the system makes sense). Because theaction of the injection d on πkX is determined by the actions of the cycles on unstable homotopy groups,we refer to d the cycle operator.

Lemma 7.5. (i) The injection monoidM acts trivially on the naive homotopy groups of every semistablesymmetric spectrum.

(ii) For every symmetric spectrum A the morphism c : πkA −→ πkA from the naive to the true homotopygroups coequalizes the M-action, i.e., we have c(fx) = cx for all f ∈M and x ∈ πkA.

Proof. (i)(ii) If X is semistable [show], then the injection monoidM acts trivially on the naive homotopy groups

of X. So for x ∈ πkA, X a semistable symmetric spectrum, f ∈M and ϕ : A −→ X a morphism we have

c(fx)X(ϕ) = (πkϕ)(fx) = f((πkϕ)(x)) = (πkϕ)(x) = c(x)X(ϕ)

in πkX by the naturality of the M-action.

7.2. Algebraic properties of tame M-modules. In this section we discuss some algebraic prop-erties of tame M-modules. It turns out that tameness is a rather restrictive condition. We start with aresult which controls a lot of the homological algebra of the monoid ring ofM.

Lemma 7.6. The classifying space BM of the injection monoidM is contractible. More generally, thesimplicial set EM×M Pn is weakly contractible.

Proof. The classifying space BM is the geometric realization of the nerve of the category BM withone object whose monoid of endomorphisms isM. Let t ∈M be given by t(i) = 2i. We define an injectiveendomorphism ct :M−→M as follows. For f ∈M and i ∈ ω we set

ct(f)(i) =

i if i is odd, and

2 · f(i/2) if i is even.


Even though t is not bijective, the endomorphism ct behaves like conjugation by t in the sense that theformula ct(f) · t = t · f holds. Thus t provides a natural transformation from the identity functor of BMto B(ct). On the other hand, if s ∈ M is given by s(i) = 2i − 1, then ct(f) · s = s for all f ∈ M, so sprovides a natural transformation from the constant functor of BM with values 1 ∈ M to B(ct). Thusvia the homotopies induced by t and s, the identity of BM is homotopic to a constant map, so BM iscontractible.

The simplicial set EM×M Pnis isomorphic to the nerve of the translation category T (M,Pn) whoseobjects are the elements of Pn and whose morphism from x to y are those monoid elements f which satisfyfx = y. We consider the functor n + − : BM −→ T (M,Pn) which sends the unique object of BM tothe element (1, 2, . . . , n) of Pn and whose behavior on morphisms is given by f 7→ n + f . This functorsends the monoid M isomorphically onto the endomorphism monoid of (1, 2, . . . , n) in T (M,Pn), whichmeans that n + − is fully faithful as a functor. For every element x ∈ Pn there exists a bijection σ ∈ Msuch that σ · (1, . . . , n) = x, so every object of T (M,Pn) is isomorphic to the object (1, . . . , n). Thus thefunctor n + − : BM −→ T (M,Pn) is an equivalence of categories, so it induces a weak equivalence ofnerves. Altogether, EM×M Pn is weakly equivalent to the classifying space BM, which we showed isweakly contractible.

We introduce some useful notation and terminology. For an injective map f : ω −→ ω we write |f |for the smallest number i ≥ 0 such that f(i + 1) 6= i + 1. So in particular, f restricts to the identity on1, . . . , |f |. An element x of an M-module W has filtration n if for every f ∈ M with |f | ≥ n we havefx = x. We denote by W (n) the subgroup of W of elements of filtration n; for example, W (0) is the set ofelements fixed by all f ∈ M. We say that x has filtration exactly n if it lies in W (n) but not in W (n−1).By definition, anM-module W is tame if and only if every element has a finite filtration, i.e., if the groupsW (n) exhaust W .

The following lemmas collect some elementary observations, first for arbitraryM-modules and then fortame M-modules.

Lemma 7.7. Let W be any M-module.(i) If two elements f and g of M coincide on n = 1, . . . , n, then fx = gx for all x ∈ W of

filtration n.(ii) For n ≥ 0 and f ∈M set m = maxf(n). Then f ·W (n) ⊆W (m).(iii) We denote by d ∈M the map given by d(i) = i+ 1. If x ∈W has filtration exactly n with n ≥ 1,

then dx has filtration exactly n+ 1.(iv) Let V ⊆ W be an M-submodule such that the action of M on V and W/V is trivial. Then the

action of M on W is also trivial.

Proof. (i) We can choose a bijection γ ∈ M which agrees with f and g on n, and then γ−1f andγ−1g fix n elementwise. So for x of filtration n we have (γ−1f)x = x = (γ−1g)x. Multiplying by γ givesfx = gx.

(ii) If g ∈M satisfies |g| ≥ m, then gf and f agree on n. So for all x ∈W (n) we have gfx = fx by (i),which proves that fx ∈W (m).

(iii) We have d ·W (n) ⊆W (n+1) by part (ii). To prove that d increases the exact filtration we considerx ∈W (n) with n ≥ 1 and show that dx ∈W (n) implies x ∈W (n−1).

For f ∈ M with |f | = n − 1 we define g ∈ M by g(1) = 1 and g(i) = f(i − 1) + 1 for i ≥ 2.Then we have gd = df and |g| = n. We let h be the cycle h = (f(n) + 1, f(n), . . . , 2, 1) so that we have|hd| = f(n) = maxf(n). Then fx ∈W (f(n)) by part (ii) and so

fx = (hd)(fx) = h(g(dx)) = (hd)x = x .

Altogether this proves that x ∈W (n−1).(iv) Since the M-action is trivial on V and W/V , every f ∈ M determines an additive map δf :

W/V −→ V such that x − fx = δf (x + V ) for all x ∈ W . These maps satisfy δfg(x) = δf (x) + δg(x) andso δ is a homomorphism from the monoid M to the abelian group of additive maps from W/V to V . By

120 I. BASICS

Lemma 7.6 the classifying space BM of the monoid M is contractible, so H1(BM, A) = Hom(M, A) istrivial for every abelian group A. Thus δf = 0 for all f ∈M, i.e.,M acts trivially on W .

Lemma 7.8. Let W be a tame M-module.(i) Every element of M acts injectively on W .(ii) If the filtration of elements of W is bounded, then W is a trivial M-module.(iii) If the map d given by d(i) = i+ 1 acts surjectively on W , then W is a trivial M-module.(iv) If W is finitely generated as an abelian group, then W is a trivial M-module.

Proof. (i) Consider f ∈ M and x ∈ W (n) with fx = 0. Since f is injective, we can choose h ∈ Mwith |hf | ≥ n. Then x = (hf)x = h(fx) = 0, so f acts injectively.

(ii) Lemma 7.7 (iii) implies that if W = W (n) for some n ≥ 0, then n = 0, so the M-action is trivial.(iii) Suppose M does not act trivially, so that W (0) 6= W . Let n be the smallest positive integer such

that W (0) 6= W (n). Then by part (iii) of Lemma 7.7, any x ∈ W (n) −W (0) is not in the image of d, so ddoes not act surjectively.

(iv) The union of the nested sequence of subgroups W (0) ⊆ W (1) ⊆ W (2) ⊆ · · · is W . Since finitelygenerated abelian groups are Noetherian, we have W (n) = W for all large enough n. By part (ii), themonoidM must act trivially.

Parts (i), (iii) and (iv) of Lemma 7.8 can fail for non-tame M-modules: we can let f ∈ M act on theabelian group Z as the identity if the image of f : ω −→ ω has finite complement, and we let f acts as 0 ifits image has infinite complement.

Example 7.9. We introduce some important tame M-modules Pn for n ≥ 0. The module Pn is thefree abelian group with basis the set of ordered n-tuples of pairwise distinct elements of ω (or equivalentlythe set of injective maps from n to ω). The monoid M acts from the left on this basis by componentwiseevaluation, i.e., f(x1, . . . , xn) = (f(x1), . . . , f(xn)), and it acts on Pn by additive extension. For n = 0,there is only one basis element, the empty tuple, and so P0 is isomorphic to Z with trivial M-action. Forn ≥ 1, the basis is countably infinite and theM-action is non-trivial. The module Pn is tame: the filtrationof a basis element (x1, . . . , xn) is the maximum of the components. So the filtration subgroup P(m)

n isgenerated by the n-tuples all of whose components are less than or equal to m. An equivalent way of sayingthis is that P(m)

n = Z[I(n,m)], the free abelian group generated by all injections from n to m; in particular,P(m)n is trivial for m < n.

The module Pn represents the functor of taking elements of filtration n: for every M-module W , themap

HomM-mod(Pn,W ) −→ W (n) , ϕ 7→ ϕ(1, . . . , n)is bijective.

7.3. Examples. We discuss several classes of symmetric spectra with a view towards the M-actionon the stable homotopy groups.

Example 7.10 (Eilenberg-Mac Lane spectra). Every tame M-module W can be realized as the ho-motopy group of a symmetric spectrum. For this purpose we modify the construction of the symmetricEilenberg-Mac Lane spectrum of an abelian group. We define a symmetric spectrum HW of simplicial setsby

(HW )n = W (n) ⊗ Z[Sn] ,whereW (n) is the filtration n subgroup ofW and Z[Sn] refers to the simplicial abelian group freely generatedby the simplicial set Sn = S1∧. . .∧S1, divided by the subgroup generated by the basepoint. The symmetricgroup Σn takes W (n) to itself and we let it act diagonally on (HW )n, i.e., on Sn by permuting the smashfactors. If M acts trivially on W , then this is just the ordinary Eilenberg-Mac Lane spectrum introducedin Example 1.14. Note that HW is an Ω-spectrum if and only if the M-action on W is trivial.

Since (HW )n is an Eilenberg-Mac Lane space of type (W (n), n) the homotopy groups of the symmetricspectrum HW are concentrated in dimension zero where we have π0HW ∼=

⋃n≥0W

(n) = W asM-modules.


Example 7.11 (Twisted smash products). We describe the homotopy groups of a twisted smash productL .m X (see Example 3.14) as a functor of the homotopy groups of L ∧X, using all available structure onthose. This is essentially a reinterpretation of the additive isomorphism (3.16) paying close attention to theaction on the injection monoid. Since L.mX is isomorphic to GmL∧X this gives a description of the naivehomotopy groups of smash products with semifree spectra. Since free and semifree symmetric spectra arespecial cases of twisted smash products, this will specialize to formulas for the naive homotopy groups offree and semifree symmetric spectra.

We define a homomorphism of monoids + : Σm ×M −→M by

(γ + f)(i) =

γ(i) for 1 ≤ i ≤ m, and

f(i−m) +m for m+ 1 ≤ i.

We denote by Z[M]〈m〉 the monoid ring ofM with its usual left multiplication action, but with right actionby the monoid Σm ×M via restriction along the homomorphism + : Σm ×M −→ M. So we consider asymmetric spectrum X and a based Σm-space (or any pointed Σm-simplicial set) L for some m ≥ 0. In thenext proposition we use the left Σm-action on πk+m(L ∧X) through the action on L, twisted by the signrepresentation sgnm of Σm.

Proposition 7.12. For every m ≥ 0, every based Σm-space (or Σm-simplcial set) L and every sym-metric spectrum X the natural map

Z[M]〈m〉 ⊗Σm×M (sgnm⊗πk+m(L ∧X)) −→ πk(L .m X)(7.13)

f ⊗ (1⊗ [α]) 7−→ f · [(1 ∧ −)α]

is an isomorphism ofM-modules. Here f is any element of the injection monoidM, α : Sk+m+n −→ L∧Xn

represents a naive homotopy class in πk+m(L ∧X) and (1 ∧ −)α is the composite of α with the map

1 ∧ − : L ∧Xn −→ Σ+m+n ∧Σm×Σn L ∧Xn = (L .m X)m+n ,

which represents an element of naive homotopy group πk(L .m X).

We remark that as a right Σm ×M-module, Z[M]〈m〉 is free of countably infinite rank. One possiblebasis is given by the ‘(m,∞)-shuffles’, i.e., by those bijections f ∈ M which satisfy f(i) < f(i + 1) for alli 6= m. In other words, all bijective f which keep the sets m = 1, . . . ,m and m+ 1,m+ 2, . . . in theirnatural order. So Proposition 7.12 in particular implies that the underlying abelian group of πk(L .m X)is a countably infinite sum of copies of the underlying abelian group of πk+m(L ∧X), i.e., this generalizesthe additive calculation of (3.16).

Proof. We start by arguing that the map (7.13) is well-defined; naturality and M-linearity are thenclear from the definition. As n varies, the maps 1 ∧− : L ∧Xn −→ Σ+

m+n ∧Σm×Σn L ∧Xn = (L .m X)m+n

form a morphism of sequential spectra 1 ∧ − : L ∧X −→ shm(L .m X) (which is not compatible with theactions of the symmetric groups). So the induced map on naive homotopy groups

ψ : πk+m(L ∧X) −→ πk+m(shm(L .m X)) = πk(L .m X) , [α] 7−→ [1 ∧ α]

is well-defined and additive (but typically notM-linear). The map ψ satisfies the relations

ψ(γ∗(x)) = sgn(γ) · ψ(x) and ψ(fx) = (1m + f) · ψ(x)

for γ ∈ Σm, f ∈ M and x ∈ πk+m(L ∧ X) [justify]. Hence ψ extends to well-defined and M-linear fromZ[M]〈m〉 ⊗Σm×M (sgnm⊗πk+m(L ∧ X)) by sending f ⊗ (1 ⊗ x) to f · ψ(x). So the map (7.13) is indeedwell-defined (and hence natural andM-linear).

We note that the composite⊕f∈Om

πk+m(L ∧X) −→ Z[M]〈m〉 ⊗Σm×M (sgnm⊗πk+m(L ∧X))(7.13)−−−−−→ πk(L .m X) .

is the bijection (3.16), where the first map sends a class x ∈ πk+m(L ∧ X) in the summand indexed byf ∈ Om to f ⊗ x ∈ Z[M]〈m〉 ⊗Σm×M (sgnm⊗πk+m(L ∧ X)) where f ∈ M is the unique (m,∞)-shuffle

122 I. BASICS

which restricts to f on m. Since the (m,∞)-shuffles form a right Σm×M-basis of Z[M]〈m〉, the first map,and hence the map (7.13), are bijective.

Example 7.14 (Free and semifree symmetric spectra). We saw in Example 3.11 that the zeroth stablehomotopy group of the free symmetric spectrum F1S

1 is free abelian of countably infinite rank. We nowrefine this calculation to an isomorphism of M-modules π0(FmSm) ∼= Pm, see (7.16) below; here Pm isthe tame M-module which represents taking filtration m elements, see Example 7.9. So while the groupsπ0(FmSm) are all additively isomorphic for different positive m, the M-action distinguishes them. Inparticular, there cannot be a chain of π∗-isomorphisms between FmS

m and FnSn for n 6= m. [specify the

generators ιmn ]The calculation of the M-action on free and semifree symmetric spectra is a special case of the very

general formula (7.13) for the naive homotopy groups of a twisted smash product. Let L be a pointed space(or simplicial set) with a left action by the symmetric group Σm, for some m ≥ 0. Recall that GmL denotesthe semifree symmetric spectrum generated by L in level m, defined in Example 3.12, which is also equalto the twisted smash product L .m S of L with the sphere spectrum. The functor Gm is left adjoint toevaluating a symmetric spectrum at level m, viewed as a functor with values in pointed Σm-spaces. Whenwe specialize (7.13) to GmL = L .m S we obtain an isomorphism ofM-modules

Z[M]〈m〉 ⊗Σm×M (sgnm⊗πk+m(L ∧ S)) ∼= πk(L .m S) = πk(GmL) .

The homotopy groups of the spectrum L ∧ S = Σ∞L are the stable homotopy groups of L. Since theM-action on πk+m(L ∧ S) is trivial [justify] we get

Z[M]〈m〉 ⊗Σm×M (sgnm⊗πk+m(L ∧ S)) ∼= (Z[M]〈m〉 ⊗1×M Z)⊗Σm (sgnm⊗πsk+mL) .

The tame M-module Pm has a compatible right Σm-action which is given on the basis by permuting thecomponents of an m-tuple, i.e., (x1, . . . , xm)γ = (xγ(1), . . . , xγ(m)). The map

Z[M]〈m〉 ⊗1×M Z −→ Pm , f ⊗ 1 7−→ (f(1), . . . , f(m))

is an isomorphism of M-Σm-bimodules; so combining all these isomorphisms we finally get a naturalisomorphism ofM-modules

(7.15) Pm ⊗Σm (πsk+mL)(sgn)

∼=−−→ πk(GmL) , f ⊗ [α] 7−→ f · [(1 ∧ −)α] .

As above, f is any element of the injection monoidM, α : Sk+m+n −→ L∧Sn represents a stable homotopyclass in πs

k+mL and (1 ∧ −)α is the composite of α with the map

1 ∧ − : L ∧ Sn −→ Σ+m+n ∧Σm×Σn L ∧ Sn = (GmL)m+n .

On the left of the tensor symbol, the group Σm acts by what is induced on stable homotopy groups by theaction on L, twisted by sign.

Free symmetric spectra are special cases of semifree symmetric spectra. For a pointed space K (withoutany group action) we have FmK ∼= Gm(Σ+

m ∧K) and πsk+m(Σ+

m ∧K) ∼= Z[Σm] ⊗ πsk+mK as Σm-modules.

So (7.15) specializes to a natural isomorphism ofM-modules

(7.16) Pm ⊗ πsk+mK

∼= πk(FmK) .

Here πsk+mK is the (k +m)th stable homotopy group of K; the monoidM acts only on Pm.

Example 7.17 (Loop and suspension). The loop ΩX and suspension S1∧X of a symmetric spectrum Xare defined by applying the functors Ω respectively S1∧− levelwise, where the structure maps do not interactwith the new loop or suspension coordinates, compare Example ??. We already saw in Proposition 2.4 thatloop and suspension simply shift the homotopy groups, and we shall now prove that the M-action isunchanged in this process.

For every symmetric spectrum X the map S1 ∧ − : πk+nXn −→ π1+k+n(S1 ∧ Xn) is Σn-equivariantand a natural transformations of I-functors as n varies. [remove the I-functor]So the induced morphismS1 ∧− : πkX −→ π1+k(S1 ∧X) on colimits isM-linear, and hence, by Proposition 2.4 an isomorphismof M-modules. Also by Proposition 2.4 the loop isomorphism α : πk(ΩX) −→ π1+kX is the composite


of the suspension isomorphism S1 ∧ − : πk(ΩX) −→ π1+k(S1 ∧ (ΩX)), which is M-linear by the above,and the map induced by the adjunction counit ε : S1 ∧ (ΩX) −→ X, which is M-linear. Hence the loopisomorphism α is M-linear.

Example 7.18 (Shift). The shift is another construction for symmetric spectra which reindexes thehomotopy groups, but unlike the suspension, this construction changes the M-action in a systematic way.The shift of a symmetric spectrum X was defined in Example 3.7 by (shX)n = X1+n with action of Σn viathe monomorphism (1 +−) : Σn −→ Σ1+n. The structure maps of shX are the reindexed structure mapsfor X.

If we view Σn as the subgroup ofM of maps which fix all numbers bigger than n, then the homomor-phism (1 + −) : Σn −→ Σ1+n has a natural extension to a monomorphism (1 + −) : M −→ M given by(1 × f)(1) = 1 and (1 + f)(i) = f(i − 1) + 1 for i ≥ 2. The image of the monomorphism 1 + − is thesubmonoid of those g ∈ M with g(1) = 1. If W is an M-module, we denote by W (1) the M-module withthe same underlying abelian group, but with M-action through the endomorphism 1 × −. We call W (1)the shift of W . Since |1 + f | = 1 + |f |, shifting an M-module shifts the filtration subgroups, i.e., we haveW (1)(n) = W (1+n) for all n ≥ 0. Thus the M-module W (1) is tame if and only if W is.

For any symmetric spectrum X, integer k and large enough n we have

π(k+1)+n(shX)n = πk+(1+n)X1+n ,

and the maps in the colimit system for πk+1(shX) are the same as the maps in the colimit system for πkX.Thus we get πk+1(shX) = πkX as abelian groups. However, the action of a permutation on πk+1+n(shX)nis shifted by the homomorphism 1 +−, so we have

(7.19) π∗+1(shX) = (π∗X)(1)

as M-modules.

Example 7.20 (Shift adjoint). The shift functor has a left adjoint . given by (.X)0 = ∗ and

(.X)1+n = Σ+1+n ∧1×Σn Xn

for n ≥ 0. Here Σn acts from the right on Σ1+n via the monomorphism (1 + −) : Σn −→ Σ1+n. Thestructure map (Σ+

1+n ∧Σn Xn) ∧ S1 −→ Σ+1+n+1 ∧Σn+1 Xn+1 is induced by (−+ 1) : Σ1+n −→ Σ1+n+1 (the

‘inclusion’) and the structure map of X.The effect on homotopy groups of the functor . is given as a special case of the general formula (7.13) for

the homotopy groups of a twisted smash product. Indeed, that formula specializes to a natural isomorphismofM-modules

(7.21) Z[M]+ ⊗M πk+1X ∼= πk(.X) .

Here Z[M]+ denotes the monoid ring of M with its usual left action, but with right action through themonomorphism (1 + −) : M −→ M given by (1 + f)(1) = 1 and (1 + f)(i) = f(i − 1) + 1 for i ≥ 2 (inProposition 7.12 we used the notation Z[M]〈1〉). As a rightM-module, Z[M]+ is free of countably infiniterank (one possible basis is given by the transpositions (1, n) for n ≥ 1). So the isomorphism (7.21) inparticular implies that the underlying abelian group of πk(.X) is a countably infinite sum of copies of theunderlying abelian group of πk+1X, a fact which we already observed in Example 3.10.

The functor Z[M]+ ⊗M − is left adjoint to HomM(Z[M]+,−), which is a fancy way of writing thealgebraic shift functor W 7→W (1). Under the isomorphism (7.21) and the identification (7.19), the adjunc-tion between shift and S0. as functors of symmetric spectra corresponds exactly to the adjunction betweenW 7→W (1) and Z[M]+ ⊗Z[M] − as functors of tame M-modules.

Example 7.22 (Infinite products). Finite products of symmetric spectra are π∗-isomorphic to finitewedges, so stable homotopy groups commute with finite products. But homotopy groups do not in generalcommute with infinite products. This should not be surprising because stable homotopy groups involves asequential colimit, and these generally do not preserve infinite products.

There are even two different ways in which commutation with products can fail. First we note that aninfinite product of a family Wii∈I of tame M-modules is only tame if almost all the modules Wi have

124 I. BASICS

trivialM-action. Indeed, if there are infinitely many Wi with non-trivialM-action, then by Lemma 7.8 (ii)the product

∏i∈IWi contains tuples of elements whose filtrations are not bounded. We define the tame

product of the family Wii∈I bytame∏i∈I

Wi =⋃n≥0

(∏i∈I

W(n)i

),

which is the largest tame submodule of the product and thus the categorical product in the category oftame M-modules.

Now we consider a family Xii∈I of symmetric spectra. Since the monoid M acts tamely on thehomotopy groups of any symmetric spectrum, the natural map from the homotopy groups of the productspectrum to the product of the homotopy groups always lands in the tame product. But in general, thisnatural map

(7.23) πk

(∏i∈I

Xi

)−→

tame∏i∈I

πkXi

need not be an isomorphism.In Remark 2.21 we exhibit a countable family of symmetric spectra with trivial naive homotopy groups

whose product has non-trivial (even infinitely generated) naive homotopy groups. Now we modify thatexample slightly [...] As an example we consider the symmetric spectra (F1S

1)≤i obtained by truncatingthe free symmetric spectrum F1S

1 above level i, i.e.,

((F1S1)≤i)n =

(F1S

1)n for n ≤ i,∗ for n ≥ i+ 1

with structure maps as a quotient spectrum of F1S1. Then (F1S

1)≤i has trivial homotopy groups for all i.The 0th homotopy group of the product

∏i≥1(F1S

1)≤i is the colimit of the sequence of maps∏i≥n

P(n)1 −→

∏i≥n+1

P(n+1)1

which first projects away from the factor indexed by i = n and then takes a product of inclusions P(n)1 −→

P(n+1)1 . The colimit is the quotient of the tame product

∏tamei≥1 P1 by the sum

⊕i≥1 P1; so π0 of the

product is non-zero and even has a non-trivial M-action. [check whether this is stably contractible; is theproduct of semistable spectra again semistable; is the product of π∗-isos between semistable spectra]

In Exercise 8.29 we show that the naive homotopy groups of the spectrum R∞X depend functoriallyon the naive homotopy groups of a symmetric spectrum X, if we take the M-actions into account.

Remark 7.24. The injection monoidM gives essentially all natural operations on the homotopy groupsof symmetric spectra. More precisely, we now identify the ring of natural operations π0X −→ π0X witha completion of the monoid ring Z[M]. Moreover, tame M-modules can equivalently be described as thediscrete modules over the ring of operations. We will not need this information later, so we will be brief.

We define the ring Z[[M]] as the endomorphism ring of the functor π0 : Sp −→ Ab. So an elementof Z[[M]] is a natural self-transformation of the functor π0, and composition of transformations gives theproduct. The following calculation of this ring depends on the fact that the homotopy group functor π0

is pro-represented, in the level homotopy category of symmetric spectra, by the inverse system of freesymmetric spectra FnSn, and that we know π0(FnSn) by Example 7.14.

In more detail: for every n ≥ 0 we let jn ∈ πn(FnSn)n be the wedge summand inclusion Sn −→Σ+n ∧ Sn = (FnSn)n indexed by the unit element of Σn. Then evaluation at jn is a bijection

[FnSn, X] −→ πnXn , [f ] 7→ f∗(jn)

where the left hand side means homotopy classes of morphisms of symmetric spectra. We write λ :Fn+1S

n+1 −→ FnSn for the morphism adjoint the wedge summand inclusion Sn+1 −→ Σ+

n+1 ∧ (Sn ∧S1) =


(FnSn)n+1 indexed by the unit element of Σn+1. Then we have

λ∗(jn+1) = ι∗(jn)

in the group πn+1(FnSn)n+1 which implies that the squares

[FnSn, X]∼= //

[λ,X]

πnXn

ι∗

[Fn+1S

n+1, X] ∼=// πn+1Xn+1

commute. Passage to colimits give a natural isomorphism

colimn [FnSn, X] −→ π0X .

From here the Yoneda lemma shows that we get an isomorphism of abelian groups

(7.25) β : Z[[M]] −→ limn

π0(FnSn) ,

(where the limit is taken over the maps π0λ) by sending a natural transformation τ : π0 −→ π0 to the tupleτFnSn [jn]n.

It remains to exhibit the ring Z[[M]] as a completion of the monoid ring Z[M]. The natural action ofM on the 0th homotopy group of a symmetric spectrum provides a ring homomorphism Z[M] −→ Z[[M]].We define a left ideal In of Z[M] as the subgroup generated by all differences of the form f − g for allf, g ∈ M such that f and g agree on n. If W is a tame M-module and if x ∈ W (n) has filtration n, thenIn ·x = 0. So the action of the monoid ring Z[M] on any tame module automatically extends to an additivemap

(limn

Z[M]/In)⊗W −→ W .

(Warning: In is not a right ideal for n ≥ 1, so the completion does not a priori have a ring structure). Sincethe homotopy groups of every symmetric spectrum form tame M-modules, this gives a map of abeliangroups

α : limn

Z[M]/In −→ Z[[M]]

which extends the map from the monoid ring Z[M].To prove that α is a bijection we show that the composite βα : limn Z[M]/In −→ limn π0(FnSn)

with the isomorphism (7.25) is bijective. But this holds because the composite arise from compatibleisomorphisms

Z[M]/In −→ π0(FnSn) , f + In 7−→ f · [jn] ,

which in turn uses the isomorphisms Pn ∼= π0(FnSn) from Example 7.14.

7.4. Semistable symmetric spectra. The semistable spectra form an important class of symmetricspectra since for these, the naively defined homotopy groups of (1.7) coincide with the ‘true’ homotopygroups. As a slogan, for semistable spectra the homotopy groups are ‘correct’, and they are pathologicalotherwise. Many symmetric spectra which arise naturally are semistable, compare Example 7.29.

As we have seen in Example 3.7 and Proposition 4.40, the suspension and shift construction bothshift the naive homotopy groups. However, there is in general no morphism of symmetric spectra whichrealizes an isomorphism on homotopy groups. The two constructions are related by the natural morphismλX : S1∧X −→ shX which was defined in (3.9). Since every element of the injection monoid acts injectivelyon a tameM-module, the next lemma in particular implies that the morphism λX induces a monomorphismon naive homotopy groups.

In Theorem 7.27 below we collect various equivalent characterizations of semistable symmetric spectra.

126 I. BASICS

Lemma 7.26. For every symmetric spectrum X the effect on naive homotopy groups of the morphismλX : S1 ∧X −→ shX coincides with the action of the cycle operator d ∈M in the sense that the square

πkXd· //

S1∧−

(πkX)(1)

π1+k(S1 ∧X)πk(λX)

// π1+k(shX)

commutes up to the sign (−1)k.

Proof. The level n component λn : S1 ∧Xn −→ (shX)n = X1+n is the composite

S1 ∧Xn

∼=−−−−−→twist

Xn ∧ S1 σn−−−→ Xn+1(1,...,n,n+1)−−−−−−−−→ X1+n

(using only the structure map σn without the twist isomorphism and cycle permutation (1, . . . , n, n + 1)does not yield a morphism of symmetric spectra !) So the square

πk+nXn

S1∧−

ι∗ // πk+n+1Xn+1

(1,...,n,n+1)∗

π1+k+n(S1 ∧Xn)

π1+k+n(λX)// π1+k+nX1+n

does not in general commute since both ways around the square differ by the coordinate permutation ofS1+k+n which moves the first coordinate past the other ones. So the square commutes up to the sign ofthis permutation, which is (−1)k+n. As n increases, the maps (−1)n(1, . . . , n, n + 1)∗ ι∗ : πk+nXn −→πk+1+nX1+n stabilize to the left multiplication of the cycle operator d, see Example 7.4, which completesthe proof.

In Section 4.2 we used a spectrum R∞X as a tool for showing that π∗-isomorphisms are stable equiv-alences. We had set RX = Ω(shX), which comes with a morphism λX : X −→ RX adjoint to λX . Thespectrum R∞X was defined as the mapping telescope of the sequence of spectra RnX, see (4.12). Thisconstruction comes with a canonical natural morphism λ∞X : X −→ R∞X. For a description of the naivehomotopy groups of R∞X as a functor of π∗X see Exercise 8.29.

Theorem 7.27. For every symmetric spectrum X of spaces the following conditions are equivalent.(i) There exists a π∗-isomorphism from X to an Ω-spectrum, i.e., X is semistable.(ii) The tautological map c : πkX −→ πkX from the naive to true homotopy group is an isomorphism

for all integers k.(iii) The action of the injection monoid M is trivial on all naive homotopy groups of X.(iv) The cycle operator d acts trivially on all naive homotopy groups of X.(v) The morphism λX : S1 ∧X −→ shX is a π∗-isomorphism.(vi) The morphism λX : X −→ Ω(shX) is a π∗-isomorphism.(vii) The morphism λ∞X : X −→ R∞X is a π∗-isomorphism.(viii) The symmetric spectrum R∞X is an Ω-spectrum.

[I suspect that we can add to the list: λX and/or λX and/or λ∞X are stable equivalences.]

A symmetric spectrum A of simplicial sets is semistable if and only if its geometric realization |A| issemistable, and naive homotopy groups are defined after geometric realization. Moreover, suspension andshift commute with geometric realization on the nose, and looping commutes with geometric realization upto weak equivalence for spectra which are levelwise Kan. So Theorem 7.27 holds for spectra of simplicialsets with the only modification that in conditions (ii), (iii) and (v) the spectrum X should be levelwiseKan.


Proof of Theorem 7.27. (i)⇒ (ii): since the true homotopy groups were defined as transformationsout of a representable functor, this implication is an instance of the Yoneda lemma (compare Lemma 4.31)

(ii) ⇒ (iii): for any spectrum X, the tautological map c : πkX −→ πkX factors over the coinvariantsof the M-action, compare Lemma 7.5 (ii). So if c is bijective, then the M-action on πkX is trivial.

Condition (iv) is a special case of condition (iii).Conditions (iv) and (v) are equivalent because the effect of λX on naive homotopy groups and the

action of the cycle operator only differ by an isomorphism (see Lemma 7.26).Conditions (v) and (vi) are equivalent because counit and unit of the adjunction between loop and

suspension are π∗-isomorphisms. More precisely, we have the relation λX = Ω(λX) η where η : X −→Ω(S1 ∧X) is the adjunction unit. So Proposition 2.4 implies commutativity of the square

πkXπkλX //

S1∧− ∼=

πk(Ω(shX))

α∼=

π1+k(S1 ∧X)π1+kλX

// π1+k(shX)

for all integers k. Since the vertical suspension and loop morphisms are isomorphisms, conditions (v)and (vi) are equivalent.

(vi) ⇒ (vii): If λX is a π∗-isomorphism, then so is Rn(λX) for all n since looping and shifting reindexthe naive homotopy groups. The spectrum R∞X is the mapping telescope of the sequence of morphismsRn(λX) : RnX −→ Rn+1X, hence the naive homotopy groups of R∞X are isomorphic to the colimit of thenaive homotopy groups of RnX [ref]; under the isomorphism, the map πk(λ∞X ) corresponds to the canonicalmap from πkX to the colimit, which is bijective. Hence λ∞X is a π∗-isomorphism.

(iii) ⇒ (viii): In Proposition 4.13 we constructed a natural bijection bnX : πkX −→ πk+n(R∞X)n forn ≥ 0 and integers K with k + n ≥ 0. We now claim that the square

πkX

(1n+d)·

bnX // πk+n(R∞X)n

ι∗

πkX

bn+1X

// πk+n+1(R∞X)n+1

commutes, where 1n + d ∈ M is the injection which fixes the numbers 1 through n and increases allother numbers by 1 [prove...]. If the injection monoid acts trivially on πkX for all k, we deduce thatthe stabilization maps of the spectrum R∞X are all bijections. Thus the adjoint structure maps σn :(R∞X)n −→ Ω(R∞X)n+1 induce bijections of all homotopy groups based at the distinguished basepoint;since these maps are loop maps [ref...], they are weak equivalences, so that R∞X is an Ω-spectrum.

Finally, conditions (vii) and (vii) (which by all the previous steps both follow from (i)) together have (i)as a special case.

The characterization of semistable spectra by the triviality of theM-action on naive homotopy groupsis convenient for proving several sufficient conditions for semistability.

Proposition 7.28. Let X be a symmetric spectrum which satisfies one of the following conditions.(i) For every k ∈ Z there is an n ≥ 0 such that the canonical map πk+nXn −→ πkX is surjective.(ii) Every even permutation γ ∈ Σn acts as the identity on the homotopy groups of Xn.(iii) The symmetric spectrum X is underlying an orthogonal spectrum.(iv) The naive homotopy groups of X are dimensionwise finitely generated as abelian groups.

Then the M-action on all homotopy groups of X is trivial and so X is semistable.

Proof. (i) Under the assumption every element of πkX has filtration n. But tame M-modules withbounded filtration necessarily have trivial M-action by Lemma 7.8 (ii).

128 I. BASICS

(ii) We consider more generally any I-functor F which takes all even permutations to identity maps.Given f ∈ M and an element [x] ∈ F (ω) in the colimit represented by x ∈ F (n), then we can findm ≥ maxf(n) and an even permutation γ ∈ Σm such that γ agrees with f on n. Since γ is even, wethen have f∗[x] = [(f |n)∗(x)] = [(γ|n)∗(x)] = [γ∗(ιm−n∗ (x))] = [ιm−n∗ (x)] = [x]. Thus the monoid M actstrivially on the colimit F (ω).

(iii) The inclusion Σn −→ O(n) as permutation matrices sends all even permutations to the pathcomponent of the unit in O(n). So if the Σn-action on a pointed space Xn extends to an O(n)-action, thenall even permutations act as the identity on the homotopy groups of Xn. So part (ii) applies.

(iv) If πkX is finitely generated as an abelian group, then tameness forces the M-action to be trivialon πkX (Lemma 7.8 (iv)).

Example 7.29. An important special case where condition (i) in Proposition 7.28 above holds is whenthe homotopy groups of a symmetric spectrum X stabilize, i.e., for each k ∈ Z there exists an n ≥ 0 suchthat from the group πk+nXn on, all maps in the sequence (1.7) defining πkX are isomorphisms.

Examples of symmetric spectra with stabilizing homotopy groups include all suspension spectra, Ω-spectra, or Ω-spectra from some point Xn on. So it includes Eilenberg-Mac Lane spectra HA associatedto an abelian group (see Example 1.14) as well as spectra of topological K-theory (Example 1.18) andalgebraic K-theory (Example 3.29). So all these kinds of symmetric spectra are semistable.

All suspension spectra Σ∞K for based spaces (or simplicial sets) and the various Thom spectra MO,MSO and MSpin of Example 1.16 or MU , MSU and MSp of Example 1.17 are underlying orthogonalspectra, so they are all semistable.

Example 7.30. We collect some examples of symmetric spectra which are not semistable. Example 7.14identifies the homotopy groups of free and semifree symmetric spectra as

πk(FmK) ∼= Pm ⊗ πsk+mK respectively πk(GmL) ∼= Pm ⊗Σm (πs

k+mL)(sgn) .

Since Pm is free of countably infinite rank as a right Σm-module, the free or semifree symmetric spectragenerated in positive level m are never semistable unless K respectively L has trivial stable homotopygroups. For a semifree symmetric generated in positive level m we have sh(Gm) ∼= Gm−1(shL)∨(S1∧GmL)by (5.10), and the morphism λGmL : S1 ∧GmL −→ sh(GmL) is the inclusion of a wedge summand.

If W is a tame M-module with non-trivial M-action, then π0HW ∼= W as M-modules and so thegeneralized Eilenberg-Mac Lane spectrum HW as defined in Example 7.10 is not semistable.

Example 7.22 shows that an infinite product of symmetric spectra with trivial homotopy groups canhave homotopy groups with non-trivial M-action. In particular, infinite products of semistable symmetricspectra need not be semistable.

IfX has at least one non-trivial homotopy group, then .X ∼= F1S0∧X is not semistable by Example 7.20.

The ‘trivial M-action’ criterion is often handy for showing that semistability is preserved by certainconstructions. We give a few examples of this in the following proposition.

Example 7.31. If f : X −→ Y is any morphism of symmetric spectra, then the homotopy groups ofthe spectra X, Y and the mapping cone C(f) = [0, 1]+ ∧X ∪f Y are related by a long exact sequence oftame M-modules (we use that the M-action does not change under loop and suspension). Trivial tameM-modules are closed under taking submodules, quotient modules and extensions (Lemma 7.7 (iv)); so iftwo out of three graded M-modules π∗X, π∗Y and π∗C(f) have trivial M-action, then so does the third.Thus the mapping cone of any morphism between semistable symmetric spectra is semistable.

If f : X −→ Y is an h-cofibration [define] of symmetric spectra, or simply an injective morphism whenin the simplicial context, then the mapping cone C(f) is π∗-isomorphic to the quotient Y/X. Thus if twoof the spectra X, Y and Y/X are semistable, then so is the third.

Example 7.32. Semistability is preserved under suspension, loop, wedges, shift and sequential colimitsalong h-cofibrations (or injective morphisms when in the simplicial context) since these operations preservethe property ofM acting trivially on homotopy groups.


We shall see in Proposition 7.36 below that the smash product of two semistable symmetric spectra issemistable, at least if one of the factors is level cofibrant (which is automatic in the context of simplicialsets) and the other flat. Moreover, if X is a semistable symmetric spectrum and A is a Γ-space of simplicialsets, then A(X) is semistable (see Proposition II.8.16 (iii)) [also true for simplicial functors?]

Example 7.33. For a symmetric spectrum X and a pointed space K we let K ∧X be the symmetricspectrum obtained by smashing K levelwise with X (compare Example 3.4). For example, when K = S1 isthe circle, this specializes to the suspension of X. We claim that if X is semistable and K is a CW-complex,then the symmetric spectrum K ∧X is again semistable.

We first prove the claim for finite dimensional CW-complexes by induction over the dimension. If K is0-dimensional, then K ∧X is a wedge of copies of X, thus semistable. If K has positive dimension n andK(n−1) is its (n− 1)-skeleton, then K/K(n−1) is a wedge of n-spheres and so the quotient of K ∧X by thesubspectrum K(n−1)∧X is a wedge of n-fold suspension of X. By induction the subspectrum K(n−1)∧X issemistable; since the inclusion is an h-cofibration and the quotient spectrum is also semistable, so is K ∧X.For a general CW-complex K the symmetric spectrum K ∧X is the sequential colimit, over h-cofibrations,of the smash product of X with the skeleta of K. So K ∧X is semistable.

The geometric realization of any simplicial set is a CW-complex, so in the simplicial context we concludethat for any pointed simplicial set K and any semistable symmetric spectrum X the symmetric spectrumK ∧X is again semistable.

Example 7.34. Let F : J −→ Sp be a functor from a small category J to the category of symmetricspectra. If F (j) is semistable for each object j of J , then the homotopy colimit of F over J is semistable.

Indeed, the homotopy colimit is the geometric realization of the simplicial replacement q∗F in thesense of Bousfield and Kan [11, Ch. XII, 5.1], a simplicial object of symmetric spectra. The spectrum ofn-simplices of q∗F is a wedge, indexed over the n-simplices of the nerve of J , of spectra which occur asvalues of F . The geometric realization |q∗F | is the sequential colimit, over h-cofibrations, of the realizationsof the skeleta sknq∗F in the simplicial direction, so it suffices to show that each of these is semistable.The skeleton inclusion realizes to an h-cofibration | skn−1q∗F | −→ | sknq∗F | whose quotient symmetricspectrum is a wedge, indexed over the non-degenerate n-simplices of the nerve of J , of n-fold suspensionsof spectra which occur as values of F . So the quotient spectra are semistable, and so by induction thesymmetric spectra | sknq∗F | are semistable.

Now we show that the smash product of two semistable symmetric spectra is again semistable, undersome mild cofibrancy hypothesis. In (5.7) we defined a natural map ξX,Y : (shX) ∧ Y −→ sh(X ∧ Y ) forsymmetric spectra X and Y . This map is not always a π∗-isomorphism. For example if X = S is the spherespectrum, then shX = Σ∞S1 and the map ξS,Y is isomorphic to the map λ : Y : S1 ∧ Y −→ shY . Thismap is a π∗-isomorphism if and only if Y is semistable. For semistable Y , however, we have: [recall thatfor simplicial sets, ‘level cofibrant’ is automatic]

Proposition 7.35. Let Y be a flat, semistable symmetric spectrum. Then for every level cofibrantsymmetric spectrum X, the map

ξX,Y : (shX) ∧ Y −→ sh(X ∧ Y )

is a π∗-isomorphism.

Proof. We start with the special case where both X and Y are flat. Proposition 5.13 provides apushout square

S1 ∧X ∧ Y

λX∧Id

(Id∧λY )twist // X ∧ (shY )

ξX,Y

(shX) ∧ Y

ξX,Y

// sh(X ∧ Y ).

Since Y is semistable the morphism λY : S1 ∧ Y −→ shY is a π∗-isomorphism. Since Y is flat, thismorphism is also a level cofibration by Proposition 5.37 (iii). Since X is flat, smashing with it preserves

130 I. BASICS

π∗-isomorphism between level cofibrant spectra (by Proposition 5.38) and level cofibrations (by definition).Since both S1 ∧ Y and shY are flat, thus level cofibrant, the upper horizontal morphism in the pushoutsquare is thus a level cofibration and a π∗-isomorphism. But then the lower horizontal morphism is also alevel cofibration, thus π∗-isomorphism by the long exact sequences of homotopy groups and the five lemma.

If X is level cofibrant (but not necessarily flat) we choose a level equivalence X ′ −→ X with flat source,for example as in Proposition ??. Then in the commutative square

(shX ′) ∧ YξX′,Y //

sh(X ′ ∧ Y )

(shX) ∧ Y

ξX,Y

// sh(X ∧ Y )

the upper horizontal map is a π∗-isomorphism by the above. Since the four spectra X,X ′, shX and sh(X ′)are level cofibrant and Y is flat, both vertical maps are level equivalences by Proposition 5.38. Thus thelower map is a π∗-isomorphism, which finishes the proof.

Proposition 7.36. Let X and Y be two semistable spectra one of which is flat and the other levelcofibrant. Then the smash product X ∧ Y is semistable.

Proof. Suppose that Y is flat. By Theorem 7.27 a symmetric spectrum A is semistable if and only ifthe map λA : S1 ∧A −→ shA is a π∗-isomorphism. The map λX∧Y : S1 ∧X ∧ Y −→ sh(X ∧ Y ) factors asthe composition

S1 ∧X ∧ Y λX∧Id−−−−→ (shX) ∧ Y ξX,Y−−−→ sh(X ∧ Y )

(where we suppress an associativity isomorphism). Since X is semistable, the map λX is a π∗-isomorphism,and hence so is λX∧Id, by Proposition 5.38. The second map ξX,Y is a π∗-isomorphism by Proposition 7.35.

7.5. A spectral sequence for true homotopy groups. For semistable spectra the naive and truehomotopy groups agree (essentially by definition of the latter, compare Lemma 4.31). If X is not semistable,then there must be at least one dimension k for which the injection monoid M acts non-trivially on πkX;since c factors over the M-coinvariants, the map c : πkX −→ πkX is then not injective.

For spectra which are not semistable it would thus be interesting to describe the true homotopy groups interms of the naive homotopy groups, which are often more readily computable from an explicit presentationof the symmetric spectrum. The bad news is that the true homotopy groups are not a functor of the classicalhomotopy groups, not even if one takes the M-action into account [give an example as exercise]. But thenext best thing is true: the naive and true homotopy groups are only a spectral sequence apart from eachother.

In this section we construct a spectral sequence (see Theorem 7.41)

E2p,q = Hp(M, πqX) =⇒ πp+qX

which converges strongly to the true homotopy groups of a symmetric spectrum X and whose E2-term isgiven by the homology of the naive homotopy groups, viewed as modules over the injection monoid M.The homology groups above are defined as Tor groups over the monoid ring of M, i.e., Hp(M,W ) =TorZ[M]

p (Z,W ). We refer to this spectral sequence as the naive-to-true spectral sequence.We will see below that the naive-to-true spectral sequence collapses in many cases, for example for

semistable symmetric spectra (Example 7.43) and for free symmetric spectra (Example 7.44), and it alwayscollapses rationally (Example 7.47). naive-to-true spectral sequence typically does not collapse for semifreesymmetric spectra, see Example 7.45.

We need to develop more homological algebra of tame M-modules.


Proposition 7.37. (i) Let Z[M]+ denote the monoid ring ofM with its usual left action, but with rightaction through the monomorphism (1×−) :M−→M given by (1× f)(1) = 1 and (1× f)(i) = f(i− 1)+1for i ≥ 2. Then for every n ≥ 0 the map

κ : P1+n −→ Z[M]+ ⊗M Pnwhich sends the generator (1, . . . , n+ 1) to the element 1⊗ (1, . . . , n) of filtration n+ 1 in Z[M]+ ⊗M Pnis an isomorphism of M-modules.

(ii) For every n ≥ 0 and every abelian group A, the homology groups Hp(M,Pn⊗A) vanish in positivedimensions.

(iii) For every n ≥ 0 and every Σn-module B we have a natural isomorphism

H∗(M,Pn ⊗Σn B) ∼= H∗(Σn;B) .

Proof. (i) For any n-tuple (x1, . . . , xn) of pairwise distinct natural numbers we can choose g ∈ Mwith g(i) = xi for 1 ≤ i ≤ n. Because of

f ⊗ (x1, . . . , xn) = f ⊗ g(1, . . . , n) = f(1× g) · (1⊗ (1, . . . , n))

the element 1⊗ (1, . . . , n) generates Z[M]+⊗MPn, so the map κ is surjective. The map Z[M]+⊗MPn −→P1+n which sends f⊗(x1, . . . , xn) to (f(1), f(x1 +1), . . . , f(xn+1)) is right inverse to κ since the compositesends the generator (1, . . . , n+ 1) to itself. So κ is also injective.

(ii) The groups TorZ[M]p (Z, A) are isomorphic to the singular homology groups with coefficients in A of

the classifying space BM of the monoid M. This classifying space is contractible by Lemma 7.6, so thegroups TorZ[M]

p (Z, A) vanish for p ≥ 1, which proves the case n = 0.For n ≥ 1, the M-modules P1+n ⊗ A and Z[M]+ ⊗M Pn ⊗ A are isomorphic by part (i). Since the

M-bimodule Z[M]+ is free as a left and right module separately, the balancing property of Tor groupsyields

TorZ[M]∗ (Z,P1+n ⊗A) ∼= TorZ[M]

∗ (Z,Z[M]+ ⊗M Pn ⊗A)∼= TorZ[M]

∗ (Z⊗M Z[M]+,Pn ⊗A) ∼= TorZ[M]∗ (Z,Pn ⊗A)

since Z ⊗M Z[M]+ is again the trivial right M-module Z. So induction on n shows that the groupsTorZ[M]

p (Z,Pn ⊗A) vanish in positive dimensions.(iii) Since Pn is free as a right Σn-module, the functor Pn ⊗Σn − is exact. The functor takes the free

Σn-module of rank 1 to Pn, so by part (ii) it takes projective Σn-modules to tame M-modules which areacyclic for the functor Z⊗M −.

Thus if P• −→ B is a projective resolution of B by Σn-modules, then Pn ⊗Σn P• is a resolution ofPn ⊗Σn B which can be used to calculate the desired Tor groups. Thus we have isomorphisms

TorZ[M]∗ (Z,Pn ⊗Σn B) = H∗(Z⊗M Pn ⊗Σn P•) ∼= H∗(Z⊗Σn P•) = H∗(Σn;B) .

To construct the spectral sequence from naive to true homotopy groups we introduce a constructionthat will also be useful at various other places, the standard resolution X] of a symmetric spectrum. Thestandard resolution is the cotriple resolution obtained from the adjunctions between free symmetric spectraand evaluation.

For a symmetric spectrum X we define PX =∨n≥0 FnXn. This is a functor which comes with a

natural augmentationε : PX −→ X

which takes the n-th summand to X by the adjoint of the identity of Xn.

Lemma 7.38. For every symmetric spectrum X the following properties hold.(i) the induced map π∗ε : π∗(PX) −→ π∗X of naive homotopy groups is surjective;(ii) for every p ≥ 1 and all integers q the group Hp(M, πq(PX)) is trivial;(iii) the map c : Z⊗M (π∗(PX)) −→ π∗(PX) induced by c : π∗PX −→ π∗PX is bijective.

132 I. BASICS

Proof. (i) Let y : Sk+n −→ Yn represent an element in the naive homotopy group πkY . The composite

FnSk+n Fny−−→ FnYn −→ Y

is the morphism of symmetric spectra adjoint to y, and it takes the naive fundamental class ιnk+n ∈πk(FnSk+n) [ref] to [y] ∈ πkY . This proves the claim since this morphism factors through ε : PY −→ Y .

(ii) We haveπk(PX) ∼=

⊕n

πk(FnXn) ∼=⊕n

Pn ⊗ πsk+nXn .

as M-modules, compare (7.16). The M-homology of the right hand side vanishes in positive dimensionsby Proposition 7.37 (ii).

Part (iii) follows from the stable equivalence FnK −→ Ωn(Σ∞K) with semistable target and theisomorphism (7.16).

We can iterate the functor P and make the iterations into a simplicial spectrum. In more detail, we canconsider the underlying N-graded based space (or simplicial set) UX = Xnn≥0 of a symmetric spectrum.This is a functor U : Sp −→ TN with left adjoint

F• : TN −→ Sp ,∨n≥0

FnXn .

We have PX = F•(UX) and the augmentation morphism ε : PX −→ X defined above is the counit of theadjunction. As the composite of right and left adjoint, the functor P has the structure of a cotriple withaugmentation ε and coaction

F•(ηUX) : PX −→ PPX

obtained by applying the left adjoint to the adjunction unit ηUX : UX −→ U(F•(UX)).We obtain an augmented simplicial object B•(X) in the category of symmetric spectra whose object of

k-simplices is given byBm(X) = Pm+1(X) ,

the (m+ 1)-th iterate of the functor P [ref to App]. We define the standard resolution X\ as the realizationof this simplicial spectrum, i.e.,

(7.39) X\ = |B•(X)| .The augmentation ε : PX = B0(X) −→ X provides a natural morphism X\ −→ X. [We have add ?:

B0(X) = PY =∨n≥0

FnYn =∨n≥0

Σ(n,−) ∧ Yn .

Inductively we conclude that

Bm(Y ) = Pm+1Y =∨

im≥···≥i0

Σ(im,−) ∧Σ(im−1, im) ∧ . . . ∧Σ(i0, i1) ∧ Yi0 .

] The standard resolution has various useful properties which we collect in the following proposition.

Proposition 7.40. For every symmetric spectrum X the standard resolution X\ has the followingproperties.(i) The augmentation ε : X] −→ X is a level equivalence.(ii) For every integer k the chain complex

· · · πkBm+1(X) −→ πkBm(X) −→ πkBm−1(X) −→ · · · −→ πkB1(X) −→ πkB0(X)

is a resolution of πkX by Z-flat tame M-modules.(iii) If X is levelwise cofibrant (which is no restriction in the context of simplicial sets), then the spectrumX] is projective [not yet defined...].(iv) There is an isomorphism (K ∧X)] ∼= K ∧ (X]) which is natural in pointed spaces (or simplicial sets)K and symmetric spectra A. [strongly associative, commutative and unital in K; lax associative and unitalin A]


Proof. (i) If we apply the forgetful functor U : Sp −→ TN dimensionwise to the simplicial spectrumB•(X) we obtain a N-graded augmented based space (or simplicial set) with ‘extra degeneracies’ [ref]. Sincegeometric realization of simplicial symmetric spectra are taken levelwise, in every level n the augmentationX]n = |B•(X)n| −→ Xn is a homotopy equivalence, compare Proposition A.3.4.

(ii) For fixed n ≥ 0 the augmented based space B•(X)n has extra degeneracies. If in addition k + n ≥2 then the augmented simplicial abelian group πk+n(B•(X)n) has extra degeneracies, and so the chaincomplex

· · ·πk+n(Bm+1(X)n) −→ πk+n(Bm(X)n) −→ · · · −→ πk+n(B1(X)n) −→ πk+n(B0(X)n) −→ πk+nXn −→ 0

is acyclic by Proposition A.3.5. Since filtered colimits are exact, we can pass to the colimit over n andobtain that the augmented chain complex πkB•(X) is acyclic. In other words, the complex in question isa resolution of πkX. For every m ≥ 0 the spectrum Bm(X) = Pm+1(X) has Z-flat naive homotopy groupsby Lemma 7.38 (ii).

(iii)(iv) Smash products of a based spaces or simplicial set with a symmetric spectrum are defined levelwise,

commute with colimits and realization [sic!]. This gives the natural isomorphism (K∧X)] = |B•(K∧X)| ∼=|K ∧B•(X)| ∼= K ∧ |B•(X)| = K ∧ (X]).

Theorem 7.41 (Naive-to-true spectral sequence). There is a strongly convergent half-plane spectralsequence

E2p,q = Hp(M, πqX) =⇒ πp+qX

with dr-differential of bidegree (−r, r − 1). The E2-term is given by the homology of the monoid M withcoefficients in the M-module π∗X. The spectral sequence is natural in the symmetric spectrum X withdr-differential of bidegree (−r, r − 1). The edge homomorphism

Z⊗M πqX = E20,q −→ πqX

is induced by the natural transformation c : πqX −→ πqX.

Proof. The spectral sequence for the true homotopy groups of the simplicial spectrum B•(X) [ref]has the form

E2p,q = Hp(πq(B•(X))) =⇒ πp+q|B•(X)|

with differentials of the correct bidegree. The E2p,q-term of the spectral sequence consists of the q-th

homology group of the simplicial abelian group πq(B•(X)). The augmentation ε : |B•(X)| = X] −→ X is alevel equivalence, hence stable equivalence, so the abutment is isomorphic to the (p+ q)-th true homotopygroup of X.

By Proposition 7.40 (ii) the complex of πq(B•(X)) of naive homotopy groups is a resolution of πqXby Z-flat M-modules. The homology of the complex Z ⊗M πq(B•(X)) thus calculates the Tor groupsTorZ[M]

p (Z, πqX) = Hp(M, πqX). Lemma 7.38 (iii) identifies the chain complex Z ⊗M πq(B•(X)). withthe complex of πq(B•(X)), and thus the homology group Hp(M, πqX) with the E2

p,q-term of the spectralsequence.

[convergence should be dealt with earlier; edge homomorphism] [is the following the filtration of πlX ?Let Πi be the group of all true homotopy classes x ∈ π∗X with the following property: for every morphismf : X −→ Y which can be written as the composite of i morphisms each of which is trivial on naivehomotopy groups π∗, we have (πnf)(x) = 0] Then Π0 = 0, Π1 = Im(c) and Πi ⊆ Πi+1.]

Remark 7.42. We are making a big point about the fact that the action of the injection monoid Mon the naive homotopy groups of a symmetric spectrum is tame. So one could expect that the E2-termof the naive-to-true spectral sequence should be given by homological algebra in the abelian category oftame M-modules. In other words, it may be a little surprising that the E2-term is given by Tor groupsover the monoid ring Z[M], which are the absolute derived functors of W 7→ Z⊗MW (as opposed to some‘tamely derived’ or relative derived functors). The explanation is the following: while the abelian categoryof tameM-modules has no nonzero projective objects, the modules Pn play a role analogous to projective

134 I. BASICS

generators. Proposition 7.37 (which ultimately is a consequence of the contractibility of the classifying spaceBM) says that the particular Tor groups TorZ[M]

∗ (Z,Pn) vanish in positive dimension (even though Pn isnot flat as a left M-module). So to calculate TorZ[M]

∗ (Z,W ) for tame M-modules we can use resolutionsby sum of modules of the form Pn, as opposed to projective resolutions.

Example 7.43 (Semistable spectra). WhenX is a semistable symmetric spectrum, the injection monoidacts trivially on the naive homotopy groups of X, so H0(M, πkX) is isomorphic to πkX. and the higherhomology groups vanish by part (ii) of Proposition 7.37. Thus E2

p,q = 0 for p 6= 0 in the naive-to-truespectral sequence and the edge homomorphism

π∗X −→ π∗X

is an isomorphism. We recover the (tautological) fact that the natural map c : πkX −→ πkX is anisomorphism for semistable X.

Example 7.44 (Free spectra). For the free symmetric spectrum generated by a pointed space (orsimplicial set) K in level m, (7.16) provides an isomorphism ofM-modules

πk(FmK) ∼= Pm ⊗ πsk+mK .

The higher Tor groups for suchM-modules vanish by part (ii) of Proposition 7.37. Thus E2p,q = 0 for p 6= 0

in the naive-to-true spectral sequence and the edge homomorphism

Z⊗M π∗(FmK) −→ π∗(FmK)

is an isomorphism. The left hand side is isomorphic to Z⊗MPm⊗πsk+mK

∼= πsk+mK, the (k+m)th stable

homotopy group of K.We can also use Example ?? instead of the spectral sequence of Theorem 7.41 to calculate the true

homotopy groups of the free spectrum FmK. Indeed, there we introduced a stable equivalence from FmKto Ωm(Σ∞K). The spectrum Ωm(Σ∞K) is semistable, so its naive and true homotopy groups coincide.

Example 7.45 (Semifree spectra). For semifree symmetric spectra (see Example 3.12) the naive-to-truespectral sequence typically does not degenerate. If L is a cofibrant based Σm-space (or simplicial set), theisomorphism (7.15) and Proposition 7.37 (iii) allows us to rewrite the E2-term of the spectral sequence as

Hq(M, πq(GmL)) ∼= Hq(M,Pm ⊗Σm (πsk+mL)(sgn)) ∼= Hq(Σm, (πs

k+mL)(sgn)) .

[In general GmL is stably equivalent to the semistable symmetric spectrum (Ωm(Σ∞L))hΣm ]As an example we consider the semifree symmetric spectrumG2S

2, where S2 is a Σ2-space by coordinatepermutations. We first identify the stable equivalence type of G2S

2. The spectrum G2S2 is isomorphic to

the quotient spectrum of Σ2 permuting the smash factors of (F1S1)∧2. Since the Σ2-action on (F1S

1)∧2 isfree [not yet shown], the map

EΣ+2 ∧Σ2 (F1S

1)(2) −→ (F1S1)∧2/Σ2 = G2S

2

which collapses EΣ2 to a point is a level equivalence. On the other hand, the stable equivalence λ(2) :(F1S

1)(2) −→ S is Σ2-equivariant, so it induces a stable equivalence

EΣ+2 ∧Σ2 (F1S

1)(2) −→ EΣ+2 ∧Σ2 S = Σ∞BΣ+

2

on homotopy orbit spectra. Altogether we conclude that G2S2 is stably equivalent to Σ∞BΣ+

2 .The naive-to-true spectral sequence for G2S

2 has as E2-term the Tor groups of π∗(G2S2). According

to (7.15) these homotopy groups are isomorphic to P2 ⊗Σ2 (πs∗+2S

2)(sgn). The sign representation cancelsthe sign action induced by the coordinate flip of S2, so we have an isomorphism ofM-modules πq(G2S

2) ∼=P2 ⊗Σ2 π

sqS

0, this time with trivial action on the stable homotopy groups of spheres. Using part (iii) ofProposition 7.37, the naive-to-true spectral sequence for G2S

2 takes the form

E2p,q∼= Hp(Σ2;πs

qS0) =⇒ πs

p+q(BΣ+2 ) .

This spectral sequence has non-trivial differentials and it seems likely that it coincides with the Atiyah-Hirzebruch spectral sequence for the stable homotopy of the space BΣ+

2 .

8. RELATION TO OTHER KINDS OF SPECTRA 135

Example 7.46 (Eilenberg-Mac Lane spectra). In Example 7.10 we associate an Eilenberg-Mac Lanespectrum HW to every tameM-module W . The homotopy groups of HW are concentrated in dimension 0,where we get the module W back. So the naive-to-true spectral sequence for HW collapses onto the axisq = 0 to isomorphisms

πp(HW ) ∼= Hp(M,W ) .

In particular, the true homotopy groups of HW need not be concentrated in dimension 0. One can showthat HW is in fact stably equivalent to the product of the Eilenberg-Mac Lane spectra associated to thegroups Hp(M,W ), shifted up p dimensions. [make exercise]

Here is an example which shows that for non-trivial W the Eilenberg-Mac Lane spectrum HW can bestably contractible: we let W be the kernel of a surjection Pn −→ Z. Proposition 7.37 and the long exactsequence of Tor groups show that the groups Hp(M,W ) vanish for all p ≥ 0. Thus the true homotopygroups of HW are trivial, i.e., HW is stably contractible.

Example 7.47 (Rational collapse). We claim that for every tameM-module W and all p ≥ 1, we haveQ ⊗ Hp(M,W ) = 0. So the spectral sequence of Theorem 7.41 always collapses rationally and the edgehomomorphism is a rational isomorphism

Q⊗M π∗X −→ Q⊗ π∗X .

In particular, for every symmetric spectrumX the tautological map c : π∗X −→ π∗X is rationally surjective.The rational vanishing of higher Tor groups is special for tameM-modules.

To prove the claim we consider a monomorphism i : V −→ W of tame M-modules and show that thekernel of the map Z ⊗M i : Z ⊗M V −→ Z ⊗M W is a torsion group. The inclusions W (n) −→ W inducean isomorphism

colimn Z⊗Σn W(n) ∼=−−→ Z⊗MW .

For every n ≥ 0, the kernel of Z ⊗Σn i(n) : Z ⊗Σn V

(n) −→ Z ⊗Σn W(n) is annihilated by the order of the

group Σn. Since the kernel of Z ⊗M i is the colimit of the kernels of the maps Z ⊗Σn i(n), it is torsion.

Thus the functor Q⊗M− is exact on short exact sequences of tameM-modules and the higher Tor groupsvanish as claimed.

Example 7.48 (Connective spectra). Let A be a symmetric spectrum which is ‘naively (k − 1)-connected’ for some integer k in the sense that the naive homotopy groups below dimension k are trivial.Then the homology group Hp(M, πqA) is trivial whenever p+ q < k, and so the true homotopy groups alsovanish below dimension k, by the naive-to-true spectral sequence. Moreover, the edge homomorphism ofthe spectral sequence is an isomorphism

Z⊗M πkA ∼= πkA

for the k-th true homotopy group of A.Somewhat more generally, we can deduce that, roughly speaking, the true homotopy groups up to

certain dimension only depend on the naive homotopy groups up to that dimension (as long as inducedby a morphism). More precisely: let f : A −→ B be a morphism of symmetric spectrum which inducesan isomorphism on πk for k < n and an epimorphism on πn. Then f also induces an isomorphism on πkfor k < n and an epimorphism on πn. Indeed, under this hypothesis the mapping cone C(f) is naivelyn-connected (by the long exact sequence of naive homotopy groups, Proposition 2.16) and thus the truehomotopy groups of C(f) vanish in dimensions n and below (by the above). So the long exact sequence oftrue homotopy groups (Proposition 4.46) shows the claim about f .

8. Relation to other kinds of spectra

In this section we discuss some other kinds of spectra and how they relate to symmetric spectra:orthogonal spectra SpO, unitary spectra SpU, Γ-spaces, simplicial and continuous functors and S-modules

136 I. BASICS

MS . The following diagram of categories and functors provides an overview:

SpU

Φ

''OOOOOOOOOOOOO MS

Γ-spaces/T //

continuous functors //

OO

SpO // SpT

Γ-spaces/sS // (simpl. functors) // SpsS

The functors pointing to the right are ‘forgetful’ or evaluation functors; functors pointing down fromthe second ‘topological’ row to the third ‘simplicial’ row are given by taking singular complexes. Allfunctors in the diagram have left adjoints. The lower rectangle commutes up to natural isomorphism. Thetriangle only commutes up to natural π∗-isomorphism: for every continuous functor F there is a naturalπ∗-isomorphism whose V th term is the map F (SV ) −→ map(SiV , F (SC⊗RV )) adjoint to the assembly mapF (SV )∧ SiV −→ F (SC⊗RV ). We did not say how to make an orthogonal spectrum from an S-module, butthis construction can be found in [50].

We also develop some theory around functors from the category I.

8.1. Symmetric spectra as continuous functors. We will now see various other kinds of spectraand how they relate to symmetric spectra.

The original definition of symmetric spectra is the most ‘down to earth’ way to introduce these objects.In a sense, Definition 1.1 describes the structure which the space Xn of a symmetric spectrum have by ‘gen-erators’ (the actions of the symmetric groups and the structure maps σn) and ‘relations’ (the equivarianceproperties for the iterated structure maps). It will often be convenient to reinterpret symmetric spectra as‘enriched functors’ defined on a based topological (or simplicial) category which encodes all possible naturaloperations between the spaces in a symmetric spectrum. We will now explain this different point of viewin more detail; we focus on the case of symmetric spectra of spaces and then say which changes are neces-sary to encompass symmetric spectra of simplicial sets. [special case of enriched category theory over thesymmetric monoidal category T; for generalities on this kind of enriched category theory see Section A.5]We define a based topological category Σ as follows. The objects of Σ are the natural numbers 0, 1, 2, . . .and the based space of morphisms from n to m is given by Σ(n,m) = Σ+

m ∧1×Σm−n Sm−n, which is to be

interpreted as a one-point space if m < n. Composition is defined by : Σ(m, k) ∧Σ(n,m) −→ Σ(n, k) isdefined by

[τ ∧ z] [γ ∧ y] = [τ(γ × 1) ∧ (y ∧ z)]where τ ∈ Σk, γ ∈ Σm, z ∈ Sk−m and y ∈ Sm−n. The identity in Σ(n, n) = Σ+

n ∧ S0 is the identity of Σn(smashed with the non-basepoint of S0). An enriched functor from Σ to the category T of based compactlygenerated spaces consists of

• based spaces X(n) for every n ≥ 0• a based, continuous action map : Σ(n,m) ∧X(n) −→ X(m) for all n,m ≥ 0

such that Idn ∈ Σ(n, n) acts as the identity and the square

Σ(m, k) ∧Σ(n,m) ∧X(n) Id∧ //

∧Id

Σ(m, k) ∧X(m)

Σ(n, k) ∧X(n) // X(k)

commutes for all n,m, k ≥ 0. [morphisms]Given a symmetric spectrum X we define a continuous based functor X : Σ −→ T on objects by

X(n) = Xn. The action map Σ(n,m) ∧Xn −→ Xm is defined by

[τ ∧ z] ∧ x 7−→ τ∗(σm−n(x ∧ z)) .


It is straightforward to check that this indeed defines a based continuous functor [where is the equivariancecondition used?].

Proposition 8.1. The assignment X 7→ X defines an isomorphism between the category of symmetricspectra and the category of based continuous functors from Σ to the category T of based compactly generatedspaces.

Proof. We describe the inverse isomorphism. Given an enriched functor Y : Σ −→ T we define asymmetric spectrum Y by Yn = Y (n) with Σn-action given by the composite

Σm × Y (n) −→ Σ(n, n) ∧ Y (n) Y−−→ Y (n) .

The structure map σn : Y (n) ∧ S1 −→ Y (n+ 1) is the adjoint of the composite

Y (n) ∧ S1 ∼= S1 ∧ Y (n) 1∧−−−−→ Σ+n+1 ∧ S1 ∧ Y (n) = Σ(n, n+ 1) ∧ Y (n) Y−−→ Y (n+ 1) .

[σn is Σn-equivariant, hence so are all iterated structure maps] With this definition, the iterated structuremap σm : Y (n) ∧ Sm −→ Y (n+m) comes out as the adjoint of the composite

Sm1∧−−−−→ Σ+

n+m ∧1×Σm Sm = Σ(n, n+m) Y−−→ T(Y (n), Y (n+m))

(this uses the functoriality of Y ). The map 1 ∧ − : Sm −→ Σ(n, n+m) is Σm-equivariant with respect tothe permutation action on Sm and the action on Σ(n, n + m) by γ · [τ ∧ z] = [(1 × γ)τ ∧ z]. Then mapY : Σ(n, n + m) −→ T(Y (n), Y (n + m)) is Σm-equivariant by functoriality of Y . Hence altogether theiterated structure map σm is Σn × Σm-equivariant.

[on morphisms ?]It is now straightforward to check that (X)X and (Y ) = Y on objects and morphisms, so we have an

isomorphism of categories.[spell out for simplicial sets]

8.2. Orthogonal spectra.

Definition 8.2. An orthogonal spectrum consists of the following data:• a sequence of pointed spaces Xn for n ≥ 0• a base-point preserving continuous left action of the orthogonal group O(n) on Xn for each n ≥ 0• based maps σn : Xn ∧ S1 −→ Xn+1 for n ≥ 0.

This data is subject to the following condition: for all n,m ≥ 0, the iterated structure map

σm : Xn ∧ Sm −→ Xn+m

is O(n)×O(m)-equivariant. The orthogonal group acts on Sm since this is the one-point compactificationof Rn and O(n)×O(m) acts on the target by restriction, along orthogonal sum, of the O(n+m)-action.

A morphism f : X −→ Y of orthogonal spectra consists of O(n)-equivariant based maps fn : Xn −→ Ynfor n ≥ 0, which are compatible with the structure maps in the sense that fn+1 σn = σn (fn ∧ IdS1) forall n ≥ 0. We denote the category of orthogonal spectra by SpO.

An orthogonal ring spectrum R consists of the following data:• a sequence of pointed spaces Rn for n ≥ 0• a base-point preserving continuous left action of the orthogonal group O(n) on Rn for each n ≥ 0• O(n)×O(m)-equivariant multiplication maps µn,m : Rn ∧Rm −→ Rn+m for n,m ≥ 0, and• O(n)-equivariant unit maps ιn : Sn −→ Rn for all n ≥ 0.

This data is subject to the same associativity and unit conditions as a symmetric ring spectrum (seeDefinition 1.3) and a centrality condition for every unit map ιn. In the unit condition, permutations suchas χn,m ∈ Σn+m have to be interpreted as permutation matrices in O(n+m). An orthogonal ring spectrumR is commutative if for all n,m ≥ 0 the relation χn,mµn,m = µm,ntwist holds as maps Rn∧Rm −→ Rm+n.

138 I. BASICS

A morphism f : R −→ S of orthogonal ring spectra consists of O(n)-equivariant based maps fn :Rn −→ Sn for n ≥ 0, which are compatible with the multiplication and unit maps (in the same sense as forsymmetric ring spectra).

Orthogonal spectra are ‘symmetric spectra with extra symmetry’ in the sense that every orthogonalspectrum X has an underlying symmetric spectrum UX. Here (UX)n = Xn and the symmetric group actsby restriction along the monomorphism Σn −→ O(n) given by permutation matrices. The structure mapsof UX are the structure maps of X. Many spectra that we have discussed already have this extra symmetry,i.e., they are underlying orthogonal spectra. Examples are the sphere spectrum, suspension spectra or thevarious Thom spectra such as MO and MU arise from orthogonal spectra by forgetting symmetry, butthey do not extend to continuous functors. [all semistable]

Definition 8.3. For m ≥ 0 the free orthogonal spectrum FOmK generated by a based space K in level m

is trivial below level m and is otherwise given by

(FOmK)m+n = O(m+ n)+ ∧1×O(n) K ∧ Sn .

Structure map.

Just as for symmetric spectra, the free functor for orthogonal spectra is left adjoint to evaluation at alevel. [semifree orthogonal spectra]

A morphism of symmetric spectrum FmS0 −→ FO

m is given by the adjoint of S0 −→ (FOm )m = O(m)+∧

S0 which sends the non-basepoint of S0 to the neutral element of O(m), smashed with the non-basepointof S0. In the special case n = 0 the morphism S ∼= F0S

0 −→ FO0 is in fact an isomorphism.

As the number m varies the spectra FOm form a contravariant enriched functor from the category

Σ which parameterizes symmetric spectra. This means that we have to specify associative and unitalhomomorphisms of symmetric spectra

FOm ∧Σ(k,m) −→ FO

k .

In level m+ n, we define

O(m+ n)+ ∧1×O(n) Sn ∧Σ(k,m) −→ O(m+ n)+ ∧1×O(m+n−k) S

m+n−k

[...]The naive homotopy groups of the free symmetric spectra FmS

m are ‘too big’ in the sense that themorphism λ : FmSm −→ S is not injective on naive homotopy groups as soon as m ≥ 1. Replacing the freesymmetric spectrum by the free orthogonal spectra fixes this, as the following proposition shows.

Proposition 8.4. (i) The morphism of othogonal spectra λO : Sm ∧ FOm −→ S adjoint to the identity

of Sm induces an isomorphism of naive homotopy groups.(ii) The morphism η : FΣ

m −→ U(FOm ) adjoint to the map S0 −→ O(m)+ = U(FO

m )m which sends thenon-basepoint to the unit element of O(n) is a stable equivalence of symmetric spectra.

Proof. (i) We use the space L(Rm,Rm+n) of linear isometries from Rm to Rm+n. Precomposition withi : Rm−1 −→ Rm, i(x) = (x, 0) is a locally trivial fiber bundle i∗ : L(Rm,Rm+n) −→ L(Rm−1,Rm+n) withfiber an n-sphere, so i∗ induces isomorphisms of homotopy groups below dimension n. Since L(R0,Rm+n)is a one-point space, we conclude by induction that L(Rm,Rm+n) is (n− 1)-connected.

A homeomorphism

(Sm ∧ FOm )m+n = Sm ∧

(O(m+ n)+ ∧1×O(n) S

n) ∼= Sm+n ∧ L(Rm,Rm+n)+

is given by sending x ∧ [A ∧ y] to A · (x ∧ y) ∧ ρ(A) where ρ(A) is the restriction of a linear isometryA ∈ O(m+n) = L(Rm+n,Rm+n) to Rm. Under this homeomorphisms the (m+n)-th level of the morphismλO corresponds to the map

Sm+n ∧ L(Rm,Rm+n)+ Id∧i∗−−−−→ Sm+n

which is induced by the unique map L(Rm,Rm+n) −→ ∗. The space L(Rm,Rm+n) is (n− 1)-connected, sothe map λO

m+n induces an isomorphism of homotopy groups below dimension 2n. As n goes to infinity, weconclude that λO induces an isomorphism on naive homotopy groups.


(ii) Since suspension preserves and detects stable equivalences [ref], it suffices to show that the morphismSm ∧ η : Sm ∧ FΣ

m −→ Sm ∧ U(FOm ) is a stable equivalence. In the commutative triangle

FΣmS

mη∧Id //

λ &&LLLLLLLLLLU(FO

mSm)

U(λO)wwppppppppppp

S = U(FO0 )

the right vertical morphism induces an isomorphism on naive homotopy groups by part (i), and is thus astable equivalence. So the left vertical morphism is a stable equivalence by [...]. So the morphism η ∧ Id,and thus η itself, are stable equivalences.

Proposition 8.5. The forgetful functor U : SpO −→ SpT from orthogonal spectra to symmetric spectraof spaces has a left adjoint P and a right adjoint R [P and R already used...]. For every flat symmetricspectrum A the adjunction unit η : A −→ U(PA) is a stable equivalence of symmetric spectra. For everymixed fibrant Ω-spectrum X [not yet defined] the adjunction counit η : U(RX) −→ X is a level equivalenceof symmetric spectra.

Proof. Construct left adjoint. Define right adjoint by (UY )n = map(U(FOn ), Y ). Show that PU

preserves flat spectra.In the special case of the free symmetric spectrum FΣ

m the adjunction unit FΣn −→ U(P (FΣ

n )) = U(FOn )

is a stable equivalence by Proposition 8.4 (ii). Moreover, FΣn and U(P (FΣ

n )) are both flat [show.] If X is amixed fibrant Ω-spectrum the induced map

map(−, X) : U(RX)n = map(U(FOn ), X) −→ map(Fn, X)

is thus a weak equivalence. Up to isomorphism, this map is the n-th level of the adjunction counitU(RX) −→ X, which is thus a level equivalence.

If X is a mixed fibrant Ω-spectra, then so is U(RX) by the above, and ε : U(RX) −→ X is a levelequivalence. So for every flat symmetric spectrum A the map [A, ε] : [A,U(RX)] −→ [A,X] is bijective.But this map is isomorphic to [η,X] : [U(PA), X] −→ [A,X], which is thus bijective. Since A and U(PA)are flat, the adjunction unit A −→ U(PA) is a stable equivalence by the criterion [...]

It follows immediate [in simplicial sets] that the composite functor P(X) = U(P (X])) is a detectionfunctor, where X] is the standard resolution of X defined in [...]. Indeed, since P(X) is underlying anorthogonal spectrum, it is semistable by Proposition 7.28 (c). In the chain of natural morphisms

Xε←−− X] unit−−→ U(P (X])) = PX

the left map is a level equivalence by Proposition 7.40 (i), and the right map is a stable equivalence byProposition 8.5 since X] is flat. [Exercise: show that PX = FO

• ∧hΣ X]The following proposition essentially says that the prolongation functor P is a detection functor.

Proposition 8.6. A morphism f : A −→ B of symmetric spectra of simplicial sets is a stable equiva-lence if and only if the morphism Pf : P|A| −→ P|B| is a π∗-isomorphism.

For a simplicial functor G : Σop −→ Sp and a symmetric spectrum X define G ∧hΣ X. Note theisomorphism G ∧hΣ X ∼= G ∧Σ (X]) [or define this way...]

Proposition 8.7. (i) If G is objectwise semistable, then the spectrum G ∧hΣ X is semistable for everysymmetric spectrum X.

(ii) If G is objectwise flat, then G ∧hΣ X is flat.(iii) Let ϕ : G −→ G′ is a natural transformation of contravariant simplicial functors from Σ to

symmetric spectra. Suppose that for every n ≥ 0 the morphism ϕ(n) : G(n) −→ G′(n) is a level equivalence(respectively π∗-isomorphism, respectively stable equivalence). Then the induce morphism of homotopycoends

ϕ ∧hΣ X : G ∧hΣ X −→ G′ ∧hΣ X

140 I. BASICS

is a level equivalence (respectively π∗-isomorphism, respectively stable equivalence).

Examples are the natural maps

FΣn −→ U(FO

n ) −→ Ωn(S(S)) .

The first map is a stable equivalence, the second a π∗-isomorphism. Another example is

X ∧ Y ∼= X ∧ (F• ∧Σ Y ) ∼= (F• ∧X) ∧Σ Y ∼= (.•X) ∧Σ Y

If Y is flat, then the map(.•X) ∧hΣ Y −→ (.•X) ∧Σ Y ∼= X ∧ Y

is a level equivalence. The functor X 7→ .nX preserves π∗-isomorphisms and stable equivalences, so X ∧ Ypreserves π∗-isomorphisms and stable equivalences in X for flat Y .

Example 8.8 (Unitary spectra). Unitary spectra are the complex analogues of orthogonal spectra; aunitary spectrum consists of pointed spaces Xn for n ≥ 0 endowed with a based, continuous left action ofthe unitary group U(n) and structure maps σn : Xn ∧ S2 −→ Xn+1 subject to the equivariance conditionthat the composites

σm : Xn ∧ S2m −→ Xn+m

are U(n)× U(m)-equivariant where U(m) acts of S2m via the isomorphism with the one-point compactifi-cation of Cm [make the isomorphism explicit in the R-basis 1, i of C]. We denote the category of unitaryspectra by SpU.

Given a unitary spectrum X we can produce an orthogonal spectrum Φ(X) as follows. We set

Φ(X)n = map(Sn, Xn) ,

The orthogonal group acts on Sn, on Xn via the complexification map O(n) −→ U(n) and on the mappingspace by conjugation. The structure map σn : Φ(X)n ∧ S1 −→ Φ(X)n+1 is adjoint to the map

map(Sn, Xn) ∧ S1 ∧ Sn+1 ∼= map(Sn, Xn) ∧ Sn ∧ S2

eval∧Id−−−−−→ Xn ∧ S2 σn−−→ Xn+1

where we have used a shuffle isomorphism S1 ∧ Sn+1 ∼= Sn ∧ S2 [specify].The functor Φ : SpU −→ SpO turns unitary ring spectra into orthogonal ring spectra. An example

of this is the complex cobordism spectrum MU of Example 1.17 which arises naturally as a unitary ringspectrum, made into an orthogonal spectrum via the functor Φ. More precisely, the symmetric sequencedenoted MU in Example 1.17 comes from a unitary spectrum with nth space

MUn = EU(n)+ ∧U(n) S2n ,

the Thom space of the vector bundle over BU(n) with total space EU(n)×U(n) Cn.

Example 8.9 (Periodic complex cobordism). We define the periodic complex cobordism spectrumMUP ,a unitary spectrum, as follows. [make coordinate free] For a complex inner product space V we considerthe ‘full Grassmannian’ Gr(V ⊕ V ) of V ⊕ V . A point in Gr(V ⊕ V ) is any complex sub-vectorspace ofV ⊕V , and this space is topologized as the disjoint union of the Grassmannians of k-dimensional subspacesof V ⊕ V for k = 0, . . . , 2 dim(V ). Over the full Grassmannian Gr(V ⊕ V ) sits a tautological hermitianvector bundle (of non-constant rank!): the total space of this bundle consist of pairs (U, x) where U is acomplex sub-vectorspace of V ⊕ V and x ∈ U .

We define (MUP )(V ) as the Thom space of this tautological vector bundle, i.e., the quotient space ofthe unit disc bundle by the sphere bundle. The multiplication

(8.10) (MUP )(V ) ∧ (MUP )(W ) −→ (MUP )(V ⊕W )

sends (U, x) ∧ (U ′x′) to (U + U ′, (x, x′)) where U + U ′ is the image of U ⊕ U ′ under the isometryId∧τ ∧ Id : (V ⊕ V )⊕ (W ⊕W ) ∼= (V ⊕W )⊕ (V ⊕W ). The unit map SV −→ (MUP )(V ) sends x ∈ V to(∆(V ), (v, v)) where ∆(V ) is the diagonal copy of V in V ⊕ V .


As the name suggests, MUP is a periodic version of the Thom spectrum MU . More precisely, weclaim that MUP is Z-graded unitary ring spectrum whose piece of degree k is π∗-isomorphic to a 2k-foldsuspension of MU . For every integer k and complex inner product space V we let Gr(k)(V ⊕ V ) be thesubspace of Gr(V ⊕ V ) consisting of subspaces of dimension dim(V ) + k. So the full Grassmannian is thedisjoint union of the spaces Gr(k)(V ⊕ V ) for k = −dim(V ), . . . ,dim(V ). We let (MUP (k))(V ) be theThom space of the tautological (dimV +k)-plane bundle over Gr(k)(V ⊕V ), so that (MUP )(V ) is the one-point union of the Thom spaces (MUP (k))(V ) for k = −dim(V ), . . . ,dim(V ). We note that the unit mapSV −→ (MUP )(V ) has image in the degree 0 summand (MUP (0))(V ). The multiplication map (8.10) is‘graded’ in the sense that its retriction to (MUP (k))(V )∧(MUP (l))(W ) has image in (MUP (k+l))(V ⊕W ).Together this implies that MUP (k) is a unitary subspectrum of MUP and altogether the periodic spectrumdecomposes as

MUP =∨k∈Z

MUP (k) .

Now we explain why MUP (k) is, up to π∗-isomorphism, a 2k-fold suspension of MUP (0). We observethat the Grassmannian Gr(−1)(C⊕C) has only one point (the zero subspace of C⊕C), and so MUP (−1)(C)is a 0-sphere. So a special case of the multiplication map is

MUP (1+k)(V ) ∼= MUP (−1)(C) ∧MUP (1+k)(V ) −→ MUP (k)(C⊕ V ) .

If we let V vary, these maps form a morphism of unitary spectra

MUP (1+k) −→ shC MUP (k) ,

where the right hand side is the shift of a unitary spectrum by the inner product space C. Since the mapMUP (1+k)(V ) −→ MUP (k)(C⊕ V ) is highly connected [prove] the previous morphism of unitary spectrais in fact a π∗-isomorphism.

It remains to relate the spectrum MUP (0) to MU through morphisms of unitary spectra which areπ∗-isomorphisms. [...]

If we replace complex inner product spaces by real inner product spaces throughout, we obtain ancommutative orthogonal ring spectrum MOP which is a periodic version of the unoriented Thom spectrumMO in much the same way.

8.3. Continuous and simplicial functors. By a continuous functor we mean a functor F : T −→ Tfrom the category of pointed compactly generated spaces to itself which is pointed in that it takes one-pointspaces to one-point spaces and continuous in the sense that for all pointed spaces K and L the map

F : T(K,L) −→ T(F (K), F (L))

is continuous with respect to the compact open topology on the mapping spaces. The (continuous !) map

Ll 7→(k 7→k∧l)−−−−−−−−→ T(K,K ∧ L) F−−−−−→ T(F (K), F (K ∧ L)) .

then has an adjointF (K) ∧ L −→ F (K ∧ L)

which we call the assembly map. The assembly map is natural in K and L, it is unital in the sense that thecomposite

F (K) ∼= F (K) ∧ S0 assembly−−−−−→ F (K ∧ S0) ∼= F (K)is the identity and it is associative in the sense that the diagram

(F (K) ∧ L) ∧M ass.∧Id //

∼=

F (K ∧ L) ∧M ass. // F ((K ∧ L) ∧M)

F (∼=)

F (K) ∧ (L ∧M)

assembly// F (K ∧ (L ∧M))

commutes for allK,L andM , where the vertical maps are associativity isomorphisms for the smash product.

142 I. BASICS

As usual, there is also a simplicial version. A simplicial functor is an enriched, pointed functor F :sS −→ sS from the category of pointed simplicial sets to itself. So F assigns to each pointed simplicial setK a pointed simplicial set F (K) and to each pair K,L of pointed simplicial sets a morphism of pointedsimplicial sets

F : map(K,L) −→ map(F (K), F (L))which is associative and unital and such that F (∗) ∼= ∗. The restriction of F to vertices is then a functorin the usual sense. The same kind of adjunctions as for continuous functors provides a simplicial functorwith an assembly map F (K) ∧ L −→ F (K ∧ L), again unital and associative.

To every continuous (respectively simplicial) functor F we can associate a symmetric spectrum of spaces(respectively of simplicial sets) F (S) by

F (S)n = F (Sn)where Σn permutes the coordinates of Sn. The structure map σn : F (Sn)∧ S1 −→ F (Sn+1) is an instanceof the assembly map. More generally, we can evaluate a continuous (simplicial) functor F on a symmetricspectrum X and get a new symmetric spectrum F (X) by defining F (X)n = F (Xn) with structure map thecomposite

F (Xn) ∧ S1 assembly−−−−−→ F (Xn ∧ S1)F (σn)−−−−→ F (Xn+1) .

In the context of topological spaces, the space F (Sn) even has an action of the orthogonal group O(n), andin fact the functor (cont.funct) −→ SpT naturally lifts to orthogonal spectra.

Some of the symmetric spectra which we described are restrictions of continuous functors to spheres,for example suspension spectra and Eilenberg-Mac Lane spectra. Free symmetric spectra FmK or semifreesymmetric spectra GmK do not arise this way (unless m = 0 or K = ∗) and cobordism spectra like MOand MU don’t either.

A consequence of the formal properties of the assembly map is that the structure of a triple on acontinuous or simplicial functor T yields a multiplication on the symmetric spectrum T (S). Indeed, usingthe assembly map twice and the triple structure map produces multiplication maps

T (K) ∧ T (L) −→ T (K ∧ T (L)) −→ T (T (K ∧ L)) −→ T (K ∧ L) ;

here K and L are pointed spaces. If we apply this to spheres, we get Σp × Σq-equivariant maps

T (Sp) ∧ T (Sq) −→ T (Sp+q)

which provide the multiplication. The unit maps come from the natural transformation Id −→ T byevaluating on spheres. In the context of topological spaces, evaluating a triple at spheres as above evengives an orthogonal ring spectra.

Here are some examples.• The identity triple gives the sphere spectrum as a symmetric ring spectrum.• Let Gr be the reduced free group triple, i.e., it sends a pointed set K to the free group generated

by K modulo the normal subgroup generated by the basepoint. Since Gr(Sn) is weakly equivalentto ΩSn+1, which in the stable range is equivalent to Sn, the unit maps form a π∗-isomorphismS −→ Gr(S). The same conclusion would hold with the free reduced monoid functor, also knownas the ‘James construction’ J , since J(Sn) is also weakly equivalence to ΩSn+1 as soon as n ≥ 1.

• Let M be a topological monoid and consider the pointed continuous functor K 7→M+ ∧K. Themultiplication and unit of M make this into a triple whose algebras are pointed sets with leftM -action. The associated symmetric ring spectrum is the spherical monoid ring S[M ].

• Let A be a ring and consider the free reduced A-module triple A[K] = A[K]/A[∗]. Then A[S] =HA, the Eilenberg-Mac Lane ring spectrum. We shall see later [ref] that for every symmetricspectrum of simplicial sets X the symmetric spectrum A[X] is π∗-isomorphic to the smash productHA ∧X.• Let B be a commutative ring and consider the triple X 7→ I(B(X)), the augmentation ideal of

the reduced polynomial algebra over B, generated by the pointed set X. The algebras over thistriple are non-unital commutative B-algebras, or augmented commutative B-algebras (which are


equivalent categories). The ring spectrum associated to this triple is denoted DB, and it is closelyrelated to topological Andre-Quillen homology for commutative B-algebras. The ring spectrumDB is rationally equivalent to the Eilenberg-Mac Lane ring spectrum HB, but DB has torsion inhigher homotopy groups.

More generally, if we evaluate a triple T on a symmetric ring spectrum R, then the resulting spectrum T (R)is naturally a ring spectrum with multiplication maps

T (Rn) ∧ T (Rm) −→ T (Rn ∧Rm)T (µn,m)−−−−−→ T (Rn+m) .

Example 8.11 (S-modules). We describe a functor Φ :MS −→ SpT from the category of S-modulesin the sense of Elmendorf, Kriz, Mandell and May [24] to the category of symmetric spectra of spaces.The functor Φ preserves homotopy groups and multiplicative structures. For this we need the S-moduleS ∧L LS -1 = S ∧L LΣ∞1 S

0 defined in [24, II 1.7], which we abbreviate to S -1c . What matters is not the

precise form of S -1c , but that it is a cofibrant desuspension of the sphere S-module, i.e., it comes with a

weak equivalence S -1c ∧ S1 −→ S, where S1 denotes the circle. For n > 0 we define S -n

c to be the n-foldsmash power of the S-module S -1

c , endowed with the permutation action of the symmetric group on nletters. We set S0

c = S, the unit of the smash product; here the notation is slightly misleading since S0c is

not cofibrant. The functor Φ is then given by

Φ(X)n = MS(S -nc , X)

where the right hand side is the topological mapping space in the category of S-modules. The symmetricgroup acts on the mapping space through the permutation action of the source. The desuspension mapS -1c ∧ S1 −→ S induces a map

MS(S -nc , X) −→ MS(S -(n+1)

c ∧ S1, X) ∼= T(S1,MS(S -(n+1)c , X))

whose adjointMS(S -n

c , X) ∧ S1 −→MS(S -(n+1)c , X)

makes Φ(X) into a symmetric spectrum. For n ≥ 1, the S-module S -nc is a cofibrant model of the (-n)-sphere

spectrum. So the functor Φ takes weak equivalences of S-modules to maps which are level equivalencesabove level 0, and the i-th homotopy group of the space Φ(X)n is isomorphic to the (i − n)-th homotopygroup of the S-module X by [24, II 1.8]. In particular there is a natural isomorphism of stable homotopygroups π∗Φ(X) ∼= π∗X.

If R is an S-algebra with multiplication µ : R ∧ R −→ R and unit i : S −→ R, then Φ(R) becomes asymmetric ring spectrum with multiplication maps

MS(S -mc , R) ∧ MS(S -n

c , R) ∧−−→ MS(S -(m+n)c , R ∧ R)

µ−−→MS(S -(m+n)c , R) .

The unit maps i0 : S0 −→ Φ(S)0 =MS(S,R) is the unit i and the unit map S1 −→ Φ(S)1 =MS(S -1c , R) is

adjoint to the composite S -1c ∧ S1 −→ S

i−→ R. If R is a commutative S-algebra, then Φ(R) is a commutativesymmetric ring spectrum.

8.4. Γ-spaces. Many continuous or simplicial functors arise from so called Γ-spaces, and then theassociated symmetric spectra have special properties. The category Γ is a skeletal category of the categoryof finite pointed sets: there is one object n+ = 0, 1, . . . , n for every non-negative integer n, and morphismsare the maps of sets which send 0 to 0. A Γ-space is a covariant functor from Γ to the category of spacesor simplicial sets taking 0+ to a one point space (simplicial set). A morphism of Γ-spaces is a naturaltransformation of functors. We follow the established terminology to speak of Γ-spaces even if the valuesare simplicial sets.

A Γ-spaceX can be extended to a continuous (respectively simplicial, depending on the context) functorby a coend construction. If X is a Γ-space and K a pointed space or simplicial set, the value of the extendedfunctor on K is given by ∫ n+∈Γ

Kn ∧ X(n+) ,

144 I. BASICS

where we use thatKn = map(n+,K) is contravariantly functorial in n+. We will not distinguish notationallybetween the original Γ-space and its extension. The extended functor is continuous respectively simplicial.

In the simplicial context, the extension of a Γ-space admits the following different (but naturallyisomorphic) description. First, X can be prolonged, by direct limit, to a functor from the category ofpointed sets, not necessarily finite, to pointed simplicial sets. Then if K is a pointed simplicial set we geta bisimplicial set [k] 7→ X(Kk) by evaluating the (prolonged) Γ-space degreewise. The simplicial set X(K)defined by the coend above is naturally isomorphic to the diagonal of this bisimplicial set.

Symmetric spectra which arise from Γ-spaces have special properties. Here we restrict to Γ-spaces ofsimplicial sets, where things are easier to state. First, every simplicial functor which arises from a Γ-spaceX preserves weak equivalences of simplicial sets, see [12, Prop. 4.9]. So if f : A −→ B is a level equivalenceof symmetric spectra of simplicial sets, then X(f) : X(A) −→ X(B) is again a level equivalence. We shallsee later that X(−) also preserves π∗-isomorphisms and stable equivalences [ref]. Another special propertyis that symmetric spectra of the form X(S) for Γ-spaces of simplicial sets X are connective and the colimitsystems for the stable homotopy groups stabilize in a uniform way. This is because for every Γ-space X,the simplicial set X(Sn) is always (n−1)-connected [12] and the structure map X(Sn)∧S1 −→ X(Sn+1) is2n-connected [45, prop. 5.21]. In particular, the symmetric spectrum X(S) is semistable for every Γ-spaceof simplicial sets X. For Γ-spaces of topological spaces Y , the fact that Y (S) is semistable can also bededuced from the fact that Y (S) underlyies an orthogonal spectrum. Moreover, up to π∗-isomorphisms,Γ-spaces model all connective spectra (see Theorem 5.8 of [12] [also reference to [70]?])

A Γ-space X is called special if the map X((k + l)+) −→ X(k+) ×X(l+) induced by the projectionsfrom (k + l)+ ∼= k+ ∨ l+ to k+ and l+ is a weak equivalence for all k and l. In this case, the weak map

X(1+)×X(1+) ∼←− X(2+)X(∇)−−−→ X(1+)

induces an abelian monoid structure on π0 (X(1+)). Here ∇ : 2+ −→ 1+ is defined by ∇(1) = 1 = ∇(2).The Γ-space X is called very special if it is special and the monoid π0 (X(1+)) is a group. By Segal’s theorem([70, Prop. 1.4] or [12, Thm. 4.2]), the spectrum X(S) associated to a special Γ-space X by evaluation onspheres is a positive Ω-spectrum. If X is very special, then X(S) is even an Ω-spectrum (i.e., from the 0thlevel on). In particular, the homotopy groups of a very special Γ-space X are naturally isomorphic to thehomotopy groups of the simplicial set X(1+).

Example 8.12. Let A be an abelian group. We consider the Γ-space HA which assigns to a finitepointed set k+ the simplicial abelian group A⊗Z[k+]. ThenHA(S) equals the Eilenberg-Mac Lane spectrumHA as defined in Example 1.14. Proposition 8.16 shows that the Eilenberg-Mac Lane spectrum HA is flatand the assembly morphism

HA ∧X −→ HA(X)

is a π∗-isomorphism for every symmetric spectrum of simplicial sets X. The nth level of the target spectrumis A ⊗ Z[Xn] whose homotopy groups are the reduced homology groups of the pointed simplicial set Xn

with coefficients in A. So we get natural isomorphisms of abelian groups

(8.13) πk(HA ∧X) ∼= colimn Hk+n(X;A) .

Proposition 8.14. Let A be a Γ-space of simplicial sets and K and L pointed simplicial sets.• The assembly map A(K) ∧ L −→ A(K ∧ L) is injective.• The commutative square

A(S0) ∧K ∧ LassS0,L∧Id

//

assS0,K∧Id

A(L) ∧K

assL,K

A(K) ∧ L

assK,L// A(K ∧ L)

is a pullback.


Proof. (i) To verify that the assembly morphism is injective we can inspect it in each simplicialdimension separately, which means we can assume that K and L are pointed sets (i.e., constant in thesimplicial direction). For l ∈ L be denote by pl : K ∧ L −→ K the map of pointed sets given by

pl(k ∧ l′) =

k if l′ = l,∗ else.

Here we use the reduction to set, since this assignment is not compatible with a simplicial direction in l.Now consider two elements y ∧ l, y′ ∧ l′ ∈ A(K)∧L with the same image under the assembly map. We

have A(pl)(ass(y ∧ l)) = y and

A(pl)(ass(y′ ∧ l′)) =

y′ if l′ = l,∗ else.

So if l′ = l then we also have y′ = y. If l 6= l′ then y′ = ∗. Reversing the roles of the two elements wealso get y = ∗, which means that both elements are the basepoint in A(K) ∧ L. So in any case we havey ∧ l = y′ ∧ l′ in A(K) ∧ L, and so the assembly map is injective.

(ii) As in part (i) we can assume thatK and L are pointed sets (since the pullback property for simplicialsets can be tested dimensionwise). We consider two elements y ∧ l ∈ A(K)∧L and x∧ k ∈ A(L)∧K whichhave the same image in A(K ∧ L). We set z = A(pl)(x) where pl : L −→ S0 sends all points of L except lto the basepoint in S0.

Then we have

assS0,K(z ∧ k) = A(ik)(z) = A(ik)(A(pl)(x)) = A(pl)(A(ik)(x)) = A(pl)(assL,K(x ∧ k))= A(pl)(assK,L(y ∧ l)) = y

where pl : K ∧ L −→ K is defined as in part (i) and where we exploited that the square of pointed sets

Lpl //

ik

S0

ik

K ∧ L pl

// K

commutes. Thus z ∧ k ∧ l ∈ A(S0) ∧K ∧ L is a preimage of y ∧ l. Since the square commutes we have

assL,K((assS0,L ∧ Id)(z ∧ k ∧ l)) = assK,L(assS0,K ∧ Id)(z ∧ k ∧ l)) = assK,L(y ∧ l) = assL,K(x ∧ k) .

By part (i) every assembly map is injective, thus

(assS0,L ∧ Id)(z ∧ k ∧ l) = x ∧ k

and so z ∧ k ∧ l is a common preimage of the two points we started with. Again since assembly maps areinjective, such a preimage is unique, which proves the pullback property.

Evaluating a Γ-space A on simplicial spheres give symmetric spectrum A(S). More generally, we canevaluate a Γ-space A on any symmetric spectrum of simplicial sets X and obtain a new symmetric spectrumA(X) by A(X)n = A(Xn) with structure maps

A(Xn) ∧ S1 assembly−−−−−→ A(Xn ∧ S1)A(σn)−−−−→ A(Xn+1) .

Note the relationssh(A(X)) = A(shX) and A(K ∧X) = AK∧−(X) .

In particular, sh(A(S)) = A(Σ∞S1) = AS1∧−(S).If Y is another symmetric spectrum, then the composite maps

A(Xn) ∧ Ymassembly−−−−−→ A(Xn ∧ Ym)

A(in,m)−−−−−→ A((X ∧ Y )n+m)

146 I. BASICS

form a bimorphism from (A(X), Y ) to A(X ∧ Y ) and thus assemble into a natural morphism of symmetricspectra

(8.15) A(X) ∧ Y −→ A(X ∧ Y )

which we also refer to as the assembly map.[Are the following true:

• The assembly map A(X)∧Y −→ A(X ∧Y ) is injective? Maybe only for flat Y ? True if Y = GmLis semistable...

This would imply that if X is flat, so is A(X): let f : Y −→ Z be an injective morphism andconsider the commutative square

A(X) ∧ Yassembly //

Id∧f

A(X ∧ Y )

A(Id∧f)

A(X) ∧ Z

assembly// A(X ∧ Z)

The right vertical morphism is injective since X is flat and A(−) preserves monomorphisms. Thehorizontal assembly maps are injective. So the composite from A(X)∧Y to A(X ∧Z) is injective,hence so the left vertical map. Thus A(X) is flat.• If X or Y is flat, then assembly map A(X) ∧ Y −→ A(X ∧ Y ) is a π∗-isomorphism. This is a

consequence of (ii) below.

]

Proposition 8.16. Let A be is a Γ-space of simplicial sets and X a symmetric spectrum of simplicialsets.

(i) The symmetric spectrum A(S) obtained by evaluating A on spheres is flat.[is A(X) flat if X is flat?]

(ii) The assembly morphism A(S) ∧X −→ A(X) is a π∗-isomorphism.(iii) If X is semistable then the symmetric spectrum spectrum A(X) is again semistable.

Proof. (i) For a finite set B we denote by SB the smash product of copies of S1 indexed by the setB. More formally, we can define SB as the quotient of the function simplicial set map(B,S1) by the wedge.We consider the diagram

∨C⊆D(nA(SC) ∧ SD−C ∧ Sn−D assem.∧Id //

Id∧assoc.//∨B(nA(SB) ∧ Sn−B assem. // A(Sn)

in which the right map coequalizes the two left maps. The coequalizer of the left two maps is isomorphicto the latching object Ln(A(S)) and the right object is A(Sn). So we need to show that the morphism fromthe coequalizer to A(Sn) is injective.

We consider two elements x ∧ k ∈ A(SB) ∧ Sn−B and x′ ∧ k′ ∈ A(SB′) ∧ Sn−B′ which represent two

classes in the coequalizer whose images in A(Sn) agree. We first consider the special case that one of thetwo indexing sets is contained in the other, so we assume without loss of generality that B′ ⊆ B. If B = B′

we have x ∧ k = x′ ∧ k′ since the assembly map A(SB) ∧ Sn−B −→ A(Xn) is injective. OtherwiseNow we treat the general case where the two elements x∧k ∈ A(SB)∧Sn−B and x′∧k′ ∈ A(SB

′)∧Sn−B′

are indexed by subsets B and B′ of n which are not comparable. Since the assembly map A(SB∪B′) ∧

Sn−B∪B′ −→ A(Sn) is injective, the images of the two coequalizer classes already agree in A(SB∪B′) ∧

Sn−B∪B′ . Smashing with a pointed simplicial set preserves pullbacks, so by Proposition ?? (ii) we have a


pullback square

A(SB∩B′) ∧ SB−B∩B′ ∧ SB′−B∩B′ ∧ Sn−B∪B′

ass?,?∧Id //

ass?,?∧Id

A(SB′) ∧ SB−B∩B′ ∧ Sn−B∪B′

ass?,?

A(SB) ∧ SB′−B∩B′ ∧ Sn−B∪B′

ass?,?// A(SB∪B

′) ∧Bn−(B∪B′)

So there is a (unique) class z∧l in A(SB∩B′)∧SB−B∩B′∧SB′−B∩B′∧Sn−B∪B′ which is a common preimage

of both x∧k and x′∧k′. The class z∧l ∈ A(SB∩B′)∧Sn−B∩B′ also represents an element in the coequalizer.

Since B ∩B′ is contained in B as well as in B′ the special case above shows that the class z ∧ l representsthe same element as x ∧ k and the same element as x′ ∧ k′, which finishes the proof.

(ii) We choose a strict (i.e., objectwise) weak equivalence of Γ-spaces Ac −→ A such that Ac is cofibrantin the strict Quillen model structure [ref.] Then for every simplicial set K, the map Ac(K) −→ A(K) is aweak equivalence and thus the morphisms of symmetric spectra Ac(S) −→ A(S) and Ac(X) −→ A(X) arelevel equivalences. In the commutative square

Ac(S) ∧X //

Ac(X)

A(S) ∧X // A(X)

the left vertical morphism is a level equivalence since Ac(S) and A(S) are flat. So in order to show that thelower assembly map is a π∗-isomorphism we can show that the upper assembly map is one; in other words,we can assume without loss of generality that the Γ-space A is Q-cofibrant.

Now use induction over the skeleta of the Γ-space. This reduced to the special case A = Γn of arepresentable Γ-space where the assembly map becomes (S× · · · × S) ∧X = Γn(S) ∧X −→ Γn(X) = Xn

which is a π∗-isomorphism.(iii) The symmetric spectrum A(S) is flat (by Proposition 5.34 (ii)) and semistable, so by Proposi-

tion 7.36 the smash product A(S) ∧X is semistable. By Proposition 8.16 the assembly map A(S) ∧X −→A(X) is a π∗-isomorphism, hence its target is also semistable.

The functor(Γ-spaces) −→ Sp , A 7→ A(S)

has a right adjoint which we now discuss, and together these adjoint functors provide a pointset levelconstruction of connective covers for injective [flat fibrant] Ω-spectra. We define a functor Λ : Sp −→ Γspby setting

(ΛX)(n+) = map(Sn, X) .

Here X is any symmetric spectrum, n+ an object of the category Γ and Sn is the n-fold cartesian productof the sphere spectrum. We can view Sn as the function object, in the sense of Example 3.19, of mapsfrom the discrete pointed simplicial set n+ to the sphere spectrum; this makes clear how n+ 7→ Sn is acontravariant functor of n+, so that ΛX is indeed a covariant functor of n+.

The functor Λ is right adjoint to evaluating a Γ-space at spheres via the bijection (...) The adjunctionunit ε : A −→ Λ(A(S)) is given at n+ by the map

A(n+) −→ map(Sn, A(S))

using A(n+) ∼= Γsp(Γn, A) and Sn ∼= Γn(S). The adjunction counit η : (ΛX)(S) −→ X ???.The following proposition says that for injective Ω-spectra the adjunction counit η : (ΛX)(S) −→ X is

a connective cover. So

Proposition 8.17. For every injective Ω-spectrum X the Γ-space ΛX is very special, and thus thesymmetric spectrum (ΛX)(S) is a connective Ω-spectrum. Moreover, the adjunction counit η : (ΛX)(S) −→

148 I. BASICS

X induces isomorphisms of homotopy groups in all non-negative dimensions. A symmetric spectrum isconnective if and only if it is stable equivalence to A(S) for some Γ-space A.

Proof. For all k+, l+ ∈ Γ, the canonical morphism of symmetric spectra Sk ∨ Sl −→ Sk+l is a π∗-isomorphism by Theorem ??. So the induced map on mapping spaces

(ΛX)(k+ l+) = map(Sk+l, X) −→ map(Sk∨Sl, X) ∼= map(Sk, X)×map(Sl, X) = (ΛX)(k+)×(ΛX)(l+)

is a weak equivalence by???. This means that the Γ-space ΛX is special. The induced monoid structureon π0(ΛX(1+)) = π0 map(S, X) = π0X agrees with the addition on the 0th homotopy group (justify), soit has inverses. Altogether we conclude that the Γ-space ΛX is very special whenever X is an injectiveΩ-spectrum.

The adjunction counit η : (ΛX)(S) −→ X is an isomorphism at level 0. Since source and target of ηare Ω-spectra, η induces isomorphisms of homotopy groups in all non-negative dimensions.

Now we show that a symmetric spectrum X is connective if and only if it is stably equivalent to A(S) fora Γ-space A. One direct is clear by now, namely that A(S) is connective for every Γ-space A. Now supposeconversely that X is a connective spectrum. We can choose a stable equivalence X −→ ωX with target aninjective Ω-spectrum [ref]. Since ωX and (Λ(ωX))(S) are both connective, the adjunction counit is then astable equivalence from (Λ(ωX))(S) to ωX. So altogether, X is stably equivalent to (Λ(ωX))(S).

We shall see later [...] that evaluation at S and Λ are actually a Quillen adjoint functor pair with re-spect to the flat stable model structure on symmetric spectra and the (BF or Q) model structure on Γ-spaces.

8.5. I-spaces. Symmetric spectra are intimately related to the category I of (standard) finite sets andinjective maps. As before we denote by I the category with objects the sets n = 1, . . . , n for n ≥ 0 (where0 is the empty set) and with morphisms all injective maps. In other words, I is the subcategory of thecategory Fin of standard finite sets (compare Remark 0.5) with only injective maps as morphisms. Wedenote by TI the category of I-spaces, i.e., covariant functors from I to the category of pointed spaces.

Example 8.18. Here is an alternative perspective on the I-functor πkX associated to a symmetricspectrum X. In Example 8.19 we associated to every I-space T : I −→ T a symmetric spectrum T ∧ S.This construction has a right adjoint Ω• : Sp −→ TI defined as follows. If X is a symmetric spectrum, weset

(Ω•X)(n) = map(Sn, Xn)

on objects, where the symmetric group Σn acts by conjugation, i.e., (γ∗f)(x) = γf(γ−1x) for f : Sn −→ Xn

and γ ∈ Σn.If α : n −→ m is an injective map then α∗ : map(Sn, Xn) −→ map(Sm, Xm) is given as follows. We

choose a permutation γ ∈ Σm such that γ(i) = α(i) for all i = 1, . . . , n and set

α∗(f) = γ∗(σm−n(f ∧ Sm−n)) ,

i.e., we let γ acts as just defined on the composite

Sm ∼= Sn ∧ Sm−n f∧Sm−n−−−−−−→ Xn ∧ Sm−nσm−n−−−−→ Xm .

The proof of the relation (7.3) above in fact shows that this definition is independent of the choice ofpermutation γ. Functoriality of the assignment α 7→ α∗ is then straightforward.

The isomorphism πk map(Sn, Xn) ∼= πk+nXn (adjoint the loop coordinates to the right) gives an iso-morphism of abelian groups between πk(Ω•X)(n) and (πkX)(n). The Σn-action on the source sphere inmap(Sn, Xn) induces the sign action on homotopy groups, so the above isomorphism is Σn-equivariant.Since the stabilization map ι∗ : πk+nXn −→ πk+n+1Xn+1 corresponds precisely to the effect of ι∗ :map(Sn, Xn) −→ map(Sn+1, Xn+1) on πk, we in fact have an isomorphism of I-functors πk(Ω•X) ∼= (πkX).


Example 8.19 (Smash product with I-spaces). Given an I-space T : I −→ T and a symmetric spectrumA, we can form a new symmetric spectrum T ∧A by setting

(T ∧A)n = T (n) ∧An

with diagonal action of Σn (which equals the monoid of endomorphism of the object n of I). The structuremap is given by

(T ∧A)n ∧ S1 = T (n) ∧An ∧ S1 T (ι)∧σn−−−−−→ T (n + 1) ∧An+1 = (T ∧A)n+1

where ι : n −→ n + 1 is the inclusion. If K is a pointed space and T = cK the constant functor with valuesK, then (cK) ∧A is equal to K ∧A, i.e., this construction reduces to the pairing of Example 3.4. Anotherexample is I(n,−)+ ∧ S ∼= FnS

n [expand]. We prove in Proposition 8.24 (i) below that the symmetricspectrum T ∧ A is stably equivalent to (hocolimI T ) ∧ A, the levelwise smash product of the homotopycolimit of the I-space T with A.

We have an adjunction

Sp(T ∧ S, X) ∼= TI(T,Ω•X)

which takes a morphism ϕ : T ∧ S −→ X to the natural transformation ϕ : T −→ Ω•X whose value at theobject n is the adjoint T (n) −→ map(Sn, Xn) of ϕn : T (n) ∧ Sn −→ Xn.

Example 8.20 (Diagonal of a I-spectrum). Suppose we are given a functor H : I −→ Sp from thecategory I to the category of symmetric spectra. We define a new symmetric spectrum diagH, the diagonalof H. The levels of the diagonal are given by

(diagH)n = H(n)n ,

i.e., we take the n-th level of the symmetric spectrum H(n) with the diagonal Σn-action. The structuremap (diagH)n ∧ S1 −→ (diagH)n+1 is the composite around either way in the commutative square

H(n)n ∧ S1σnn //

H(ι)n∧Id

H(n)n+1

H(ι)n+1

H(n + 1)n ∧ S1

σn+1n

// H(n + 1)n+1

where ι : n −→ n + 1 is the inclusion. This constructions generalizes the diagonal of a sequence ofsymmetric spectra as discussed in Example 2.23. Indeed, given a sequence (Xi, f i) we can define a I-spectrum by H(n) = Xn and α : n −→ m induces fm−1 · · · fn : Xn −→ Xm. Then the diagonal diagiXi

as defined in Example 2.23 equals the diagonal diagH as defined here.The pairing of an I-space T and a symmetric spectrum A of Example 8.19 can be viewed as the

diagonal an ‘external smash product’ T ∧A, which is the I-spectrum defined by (T ∧A)(n) = T (n) ∧ A.Then T ∧A = diag(T ∧A).

Our next aim is to prove that for suitable I-spectra H the diagonal is stably equivalent to the homotopycolimit hocolimIH, compare Proposition 8.24. For this purpose we construct a natural morphism

(8.21) (diagH)] −→ hocolimIH

from the standard resolution (7.39) of the diagonal spectrum (which is level equivalent to diagH by Propo-sition 7.40 (i)) to the suspension spectrum of the homotopy colimit of H.

The spaces Σ(n,m) = Σ+m ∧1×Σm−n S

m−n and I(n,m)+ ∧ Sm−n are both wedges of m!/(m − n)!spheres of dimension m− n, so they are homeomorphic; however, to construct a homeomorphism we haveto choose coset representatives for the group Σm−n in Σm, so there is no preferred such homeomorphism.The situation improves if we suspend n times: the spaces Sn ∧Σ(n,m) and I(n,m)+ ∧ Sm are canonically

150 I. BASICS

homeomorphic, via

Sn ∧Σ(n,m) −→ I(n,m)+ ∧ Sm(8.22)

x ∧ [γ ∧ y] 7→ ρ(γ) ∧ γ · (x ∧ y) .

Here we use the canonical homeomorphism between Sn ∧ Sm−n and Sm, and ρ(σ) : n −→ m denotes therestriction of the bijection σ ∈ Σm to n ⊂m. More generally we can define a based continuous map

An ∧Σ(n,m) −→ I(n,m)+ ∧Am(8.23)

x ∧ [γ ∧ y] 7→ ρ(γ) ∧ γ · σm−n(x ∧ y)

which is natural in the symmetric spectrum A. This map is compatible with composition in the categoriesΣ and I in the sense that the square commutes for

An ∧Σ(n,m) ∧Σ(m, k) //

Id∧

Σ(n,m) ∧Am ∧ I(m, k)+ // I(n,m)+ ∧ I(m, k)+ ∧Ak

∧Id

An ∧Σ(n, k) // I(n, k)+ ∧Ak

all k,m, n ≥ 0. The upshot is that a m varies, the maps (8.23) constitute a morphism of symmetric spectra

An ∧ Fn = An ∧Σ(n,−) −→ I(n,−)+ ∧A .

For A = S the sphere spectrum this morphism is an isomorphism. We map the unbased I-space I(n,−) tothe constant I-space with value a point, add disjoint basepoints and smash with the sphere spectrum to geta morphism p : I(n,−)+ ∧A −→ A. The composite

An ∧Σ(n,m) −→ I(n,m)+ ∧Amproj∧Id−−−−−→ Am

is the action of the Σ-functor corresponding to the symmetric sphere spectrum A, which means that thecomposite

An ∧ Fn −→ I(n,−)+ ∧A p−−→ A

is the evaluation morphism [for A = S this is the stable equivalence λ : FnSn −→ S (up to the twistingisomorphism Sn ∧ Fn ∼= FnS

n)].We fix m ≥ 0 and 0 ≤ i0 ≤ · · · ≤ im and consider the continuous map of based spaces

H(i0)i0 ∧Σ(i0, i1) ∧ . . . ∧Σ(im−1, im) −→ I(i0, i1)+ ∧ . . . ∧ I(im−1, im)+ ∧H(i0)immade up form iterates of the map (8.23) for the spectrum H(i0). The adjoint is a morphism of symmetricspectra

(diagH)i0 ∧Σ(i0, i1) ∧ . . . ∧Σ(im−1, im) ∧ Fim −→ I(i0, i1)+ ∧ . . . ∧ I(im−1, im)+ ∧H(i0) .

If we sum over all tuples (i0, . . . , im) as above, we obtain a morphism Inductively we deduce morphisms ofsymmetric spectra

Bm(diagH) = Pm+1(diagH) ∼=∨

i0≤···≤im

(diagH)i0 ∧Σ(i0, i1) ∧ . . . ∧Σ(im−1, im) ∧ Fim

−→∨

i0≤···≤im

I(i0, i1)+ ∧ . . . ∧ I(im−1, im)+ ∧H(i0) = Bm(S, I,H)

[check compatibility with the simplicial operators] These morphisms are compatible with the simplicialstructure as m varies in the simplicial category ∆, so they induce a morphism

(diagH)] = |B•(diagH)| −→ |Bk(S, I,H)| = hocolimIH

of realizations.We point out a special case. If H = T ∧A is the external smash product of an I-space T and a symmetric

spectrum A, then diagH = T ∧ H. Moreover, smashing with A commutes with homotopy colimits, so


the homotopy colimit of T ∧A is isomorphic to (hocolimI T ) ∧ A. So in the special case H = T ∧A themorphism (8.21) becomes a natural morphism

(T ∧A)] −→ (hocolimI T ) ∧A .

By part (ii) of the next proposition, this map is a stable equivalence whenever A is semistable.

Proposition 8.24. (i) Let T be an I-space. [If we work in the context of simplicial sets or if all valuesof the I-space are cofibrant,] The morphism (T ∧S)] −→ (hocolimI T )∧S is a stable equivalence. Hence thesymmetric spectrum T ∧ S is stably equivalent to (hocolimI T ) ∧ S = Σ∞(hocolimI T ).(ii) Let H : I −→ SpsS be a functor to symmetric spectra of simplicial sets such that for every n the spectrumH(n) is semistable. Then the morphism (diagH)] −→ hocolimIH a stable equivalence. Thus the homotopycolimit hocolimIH and the diagonal diagH of the functor H are stably equivalent.

[also true if externalM-action on true/naive homotopy groups of the telescope is trivial ?]

Proof. (i) This part is a special of (ii) where H is given by H = Σ∞T . But the proof of (ii) will relyon this special case. For every k ≥ 0 the projection gives a stable equivalence

Bm(T ∧ S) ∼=∨

i0≤···≤im

T (i0) ∧ I(i0, i1)+ ∧ . . . ∧ I(im−1, im)+ ∧ I(im,−)+ ∧ S

WId∧p−−−−−→

∨i0≤···≤im

T (i0) ∧ I(i0, i1)+ ∧ . . . ∧ I(im−1, im)+ ∧ S ∼= Bk(T, I, S0) ∧ S .

The projection morphism p : I(im,−)+ ∧ S −→ S is isomorphic to the stable equivalence λ : FimSim . So

we are taking the realization of a morphism of simplicial spectra which is a stable equivalence in everysimplicial dimensions. Thus the realization (T ∧ S)] −→ (hocolimI T ) ∧ S is again a stable equivalence by[...].

(ii) Now we show that if the Proposition holds for a I-spectrum H, then it also holds for the loopedI-spectrum ΩH (where loops are taken object- and levelwise). In the commutative diagram

(Ω(diagH))]

εΩ(diagH)

(diag(ΩH))] //

hocolimI(ΩH)

Ω(diagH) Ω((diagH)])

Ω(εdiagH)oo

Ω(−)// Ω (hocolimIH)

the left vertical and lower left horizontal morphisms are level equivalences by Proposition 7.40 (i). Thusthe middle vertical morphism is a level equivalence. Since homotopy colimit commutes with suspension,it commutes with loops up to π∗-isomorphism; so the right vertical morphism is a π∗-isomorphism, thusa stable equivalence. Since looping preserves stable equivalence, the lower right morphism is a stableequivalence. Hence the upper right morphism is a stable equivalence, i.e., the proposition holds for ΩH. Soaltogether the claim is true whenever the I-spectrum is of the form H = Ωm(T ∧ S) for some I-space T andm ≥ 0.

Now suppose we have a sequence H0 −→ Hi −→ · · · of I-spectra and natural transformations suchthat the proposition holds for all Hi. We form the mapping telescope teliHi object- and levelwise, andobtain another I-spectrum. Since homotopy colimit and diagonal commute with mapping telescopes (up toisomorphism) and since a telescope of stable equivalences is a stable equivalence [ref], the proposition thenalso holds for the mapping telescope teliHi.

Now we prove the general case of an I-spectrum H which is objectwise semistable. Since H(n) issemistable, we have the chain of π∗-isomorphisms

H(n) λ∞−−→ R∞(H(n)) ←−− telm Ωm(Σ∞H(n)m) .

We let n vary through the objects of I and get natural transformations of I-spectra

Hλ∞−−→ R∞H ←−− telm Ωm(H(•)m ∧ S)

152 I. BASICS

which are objectwise π∗-isomorphisms. By the above, the proposition holds for telm Ωm(H(•)m∧S). Becauseof the isomorphisms

πk(diagH)] ∼= πk(diagH) ∼= colimn πkH(n)

the construction H 7→ (diagH)] takes objectwise π∗-isomorphisms of I-spectra to π∗-isomorphisms. Ahomotopy colimit of an objectwise π∗-isomorphism is also a π∗-isomorphism. So the proposition holdsfor H.

[exercise ? The simplicial spectrum B•(S)n has as its spectrum of k-simplices the wedge∨i0≤···≤ik

Si0 ∧Σ(i0, i1) ∧ . . . ∧Σ(ik−1, ik) ∧ Fik

(where we used that Fik = Σ(ik,−)). If we use the homeomorphism (8.22) k+ 1 times we can identify thisspace with the space ∨

i0≤···≤ik

I(i0, i1)+ ∧ . . . ∧ I(ik−1, ik)+ ∧ I(ik,−)+ ∧ S.

By the naturality properties of the map, this is in fact a homeomorphism of simplicial spectra B•(S) ∼=B•(I(−, n), I, S0)∧S, where S0 is the constant I-functor with value S0. After geometric realization we thusobtain a homeomorphism

S]n = |B•(S)n| = ∼= (hocolimIop I(n,−))+ ∧ Sn .

In other words, S] ∼= (hocolimn∈I I(n,−))+ ∧ S; under this isomorphism, the n-th level of the resolu-tion morphism S] −→ S is induced by the projection to the homotopy colimit to the one-point space(which is a homotopy equivalence...I has initial object 0, so Iop has terminal object...). Moreover, then-th level of the morphism ??? S] −→ (hocolimI cS

0) ∧mS becomes the map induced by the projectionhocolimn∈I I(n,−) −→ NI which is a weak equivalence by ???.]

Example 8.25 (Ring spectra from multiplicative I-spaces). We can use the construction which pairs anI-space with a symmetric spectrum (see Example 8.19) to produce symmetric ring spectra which model thesuspension spectra of certain infinite loop spaces such as BO, the classifying space of the infinite orthogonalgroup, even if these do not have a strictly associative multiplication. This works for infinite loop spaceswhich can be represented as ‘monoids of I-spaces’, as we now explain.

The symmetric monoidal sum operation restricts from the category Fin of standard finite sets to thecategory I. Thus I has a symmetric monoidal product ‘+’ given by addition on objects and defined formorphisms f : n −→ n′ and g : m −→m′ we define f + g : n + m −→ n′ + m′ by

(f + g)(i) =

f(i) if 1 ≤ i ≤ n, and

g(i− n) + n′ if n+ 1 ≤ i ≤ n+m.

The product + is strictly associative and has the object 0 as a strict unit. The symmetry isomorphism isthe shuffle map χn,m : n + m −→m + n.

Consider an I-space T : I −→ T with a pairing, i.e., an associative and unital natural transformationµn,m : T (n) ∧ T (m) −→ T (n + m). If R is a symmetric ring spectrum, then the smash product T ∧R (seeExample 8.19) becomes a symmetric ring spectrum with respect to the multiplication map

(T ∧R)n ∧ (T ∧R)m −→ (T ∧R)n+m

defined as the composite

T (n) ∧Rn ∧ T (m) ∧RmId∧twist∧Id−−−−−−−−→ T (n) ∧ T (m) ∧Rn ∧Rm

µn,m∧µn,m−−−−−−−→ T (n + m) ∧Rn+m .


If the transformation µ is commutative in the sense that the square

T (n) ∧ T (m)µn,m //

twist

T (n + m)

T (χn,m)

T (m) ∧ T (n)

µm,n

// T (m + n)

commutes for all n,m ≥ 0 and if the multiplication of R is commutative, then the product of T ∧R is alsocommutative. This construction generalizes monoid ring spectra (see Example 3.22): if M is a topological(respectively simplicial) monoid, then the constant I-functor with values M+ inherits an associative andunital product from M which is commutative if M is. The smash product of a ring spectrum R with sucha constant multiplicative functor equals the monoid ring spectrum R[M ].

We have (T ∧ R)op = T op ∧ Rop for the smash product of an I-space with multiplication and a ringspectrum.

A more interesting instance of this construction is a commutative symmetric ring spectrum which modelsthe suspension spectrum of the space BO+, the classifying space of the infinite orthogonal group, suppliedwith a disjoint basepoint. Here we start with the ‘I-topological group’ O, a functor from I to topologicalgroups whose value at n is O(n), the n-th orthogonal group. The behavior on morphisms is determined byrequiring that a permutation γ ∈ Σn acts as conjugation by the permutation matrix associated to γ andthe inclusion ι : n −→ n + 1 induces

ι∗ : O(n) −→ O(n+ 1) , A 7−→(A 00 1

).

A general injective set map α : n −→m then induces the group homomorphism α∗ : O(n) −→ O(m) givenby

(α∗A)i,j =

Aα−1(i),α−1(j) if i, j ∈Im(α),

1 if i = j and i 6∈Im(α),0 if i 6= j and i or j is not contained in Im(α).

Orthogonal sum of matrices gives a natural transformation of group valued functors

O(n)×O(m) −→ O(n + m) , (A,B) 7−→(A 00 B

).

This transformation is unital, associative and commutative, in a sense which by now is hopefully clear. Theclassifying space functor B takes topological groups to topological spaces and commutes with products up tounital, associative and commutative homeomorphism. So by taking classifying spaces objectwise we obtainan I-space BO with values BO(n) = BO(n). This I-space inherits a unital, associative and commutativeproduct in the sense discussed above, but with respect to the cartesian product, as opposed to the smashproduct, of spaces. So if we add disjoint basepoints and perform the construction above, we obtain asymmetric spectrum BO+ ∧ S whose value in level n is the space BO(n)+ ∧ Sn. By our discussion above,this is a commutative symmetric ring spectrum. Since the action of the symmetric group on BO(n)+ ∧ Snextends to an action of the n-th orthogonal group (BO+ ∧S is in fact underlying an orthogonal spectrum),the symmetric spectrum BO+ ∧ S is semistable.

Proposition 8.24 (i) shows that the symmetric spectrum BO+∧S is stably equivalent to the suspensionspectrum of the space hocolimI BO+. The space is homeomorphic to the homotopy colimit, in the categoryof unbased spaces, of the I-space BO. The inclusion of the mapping telescope telnBO(n) into hocolimI BOis a stable homotopy equivalence [even unstable equivalence ?], so hocolimI BO is stably equivalent to BO,the classifying space of the infinite orthogonal group BO. The upshot of all of this is that BO+ ∧ S is acommutative symmetric ring spectrum which is stably equivalent to the suspension spectrum of the spaceBO+. Since both spectra are semistable, they are even π∗-isomorphic.

This construction can be adapted to yield commutative symmetric ring spectra which model the sus-pension spectra of BSO+, BSpin+, BU+, BSU+ and BSp+. In each case, the respective family of classical

154 I. BASICS

groups fits into an ‘I-topological group’ with commutative product, and from there we proceed as for theorthogonal groups. More examples of the same kind are obtained from families of discrete groups which fitinto ‘I-groups’ with commutative product, for example symmetric groups, alternating groups or general orspecial linear groups over some ring. [is commutativity of the ring needed? check BSp]

The smash product of I-spaces [define] and the smash product of symmetric spectra are compatiblewith paring of Example 8.19 in the following sense. For I-spaces T, T ′ and symmetric spectra X and X ′ wecan consider the maps

(T ∧X)n ∧ (T ′ ∧X ′)m = T (n) ∧Xn ∧ T ′(m) ∧X ′m∼= T (n) ∧ T (m) ∧Xn ∧X ′

m

−→ (T ∧ T ′)(n + m) ∧ (X ∧X ′)n+m = ((T ∧ T ′) ∧ (X ∧X ′))n+m

for n,m ≥ 0. As n and m vary, these maps form a bimorphism from ((T ∧X,T ′∧X ′) to (T ∧T ′)∧(X∧X ′),so the universal property of the smash product provides a morphism of symmetric spectra

(T ∧X) ∧ (T ′ ∧X ′) −→ (T ∧ T ′) ∧ (X ∧X ′) .

Note that here the various smash product signs have three different meanings: three smash products arepairings between an I-space and a symmetric spectrum, two smash products are internal smash products ofsymmetric spectra and one is the internal smash product of I-spaces. We omit the proof that this morphismis natural and has appropriate associativity, commutativity and unitality properties.

8.6. Another detection functor. By a detection functor we mean an endofunctor D : Sp −→ Sp onthe category of symmetric spectra with values in semistable spectra and such that D is related by a chain ofnatural stable equivalences to the identity functor. For any such detection functor and symmetric spectrumX, the naive homotopy groups of DX are then naturally isomorphic to the true homotopy groups of X,

πkX ∼= πk(DX) .

Thus a morphism f : X −→ Y is a stable equivalence if and only if the morphism Df : DX −→ DY isa π∗-isomorphism. In this sense the naive homotopy groups of DX ‘detect’ stable equivalences, hence thename.

We have already seen two detection functors, namely the functor Q of Proposition 4.27 which takesvalues in Ω-spectra and comes with a single natural stable equivalence ηX : A −→ QA. Since the functorQ is constructed with the help of the small object argument, it is not so very explicit. Another examplethe prolongation functor PX = UP (X]), which is a detection functor by Proposition 8.6 [spaces vs ssets].

In this section we introduce another detection functor D due to Shipley [ref]. The main result of thissection then is:

Theorem 8.26. The following are equivalent for a morphism f : A −→ B of symmetric spectra.(i) f : A −→ B is a stable equivalence.(ii) The morphism Df : DA −→ DB is a π∗-isomorphism.(iii) The morphism D2f : D2A −→ D2B is a level equivalence.

For a symmetric spectrum of simplicial sets X we define a functor DX : I −→ SpT. On objects, thefunctor is given by

(DX)(n) = Ωn(Σ∞|Xn|) .For an injection α : n −→ m the morphism α∗ : (DX)(n) −→ (DX)(m) is given by [...] We define thesymmetric spectrum DX by

DX = hocolimI DX ,

the homotopy colimit [reference] of the functor DX. For a symmetric spectrum of spaces Y we define thedetection functor by first taking singular complex, i.e., we set DY = D(SY ). We hope that using the samesymbol for two different (but closely related) detection functors causes no trouble.

Proposition 8.27. For every symmetric spectrum X the spectrum DX is semistable. There is achain of two natural stable equivalences between D and the identity functor (when in the context of spaces)respectively the geometric realization functor (when in the context of simplicial sets).

EXERCISES 155

Proof. For every n ≥ 0 the symmetric spectrum Ωn(Σ∞|Xn|) is underlying an orthogonal spectrum,and for every injection α, the morphism α∗ preserves the orthogonal group actions. Hence the homotopycolimits DX is underlying an orthogonal spectrum, and so it is semistable by Proposition ??.

We can define two natural morphisms

Xλ−→ hocolimIRX ←− DX ,

the second one arising from the morphism of I-spectra DX −→ RX. We claim that these two morphismsare stable equivalences whenever X is semistable. Granted this, we obtain a chain of two natural stableequivalences

X −→ hocolimIR(QX) ←− DX .

The left morphism is the composite of the stable equivalence η : X −→ QX whose target is an Ω-spectrum(compare Proposition 4.27) with QX −→ hocolimIR(QX). The right morphism is the composite of thestable equivalence Dη : DX −→ D(QX) with D(QX) −→ hocolimIR(QX).

It remains to prove the claim. The I-spectrum DX is always semistable. The I-spectrumRX and theconstant I-spectrum with values X are objectwise semistable whenever X is. So if X is semistable, thenProposition 8.24 (ii) reduces the claim to the verification that the maps of diagonal spectra

X = diag(cX) −→ diag(RX) ←− diag(DX)

are stable equivalences. In level n, these morphisms are given by

Xn?−→ ΩnXn+n

Ωn(σn)←−−−−− Ωn(Xn ∧ Sn) .

Both are even π∗-isomorphisms whenever X is semistable[prove].

For the equivalence of properties (ii) and (iii) in Theorem 8.26 we have to prove more properties aboutthe detection functors D.

Proposition 8.28. Let X be a symmetric spectrum of spaces.(i) If X is semistable, then DX is an Ω-spectrum.(ii) If f : A −→ B is a level equivalence, then so is Df : DA −→ DB.

Proof. (ii) Since f is a level equivalence, the natural transformation Df : DA −→ DB of functorsI −→ Sp is a level equivalence at every object n ∈ I. Since homotopy colimits of symmetric spectra aformed levelwise, the induced map on homotopy colimits Df : DA −→ DB is also a level equivalence [needthat DX is levelwise and objectwise cofibrant space...]

Exercises

Exercise 8.1. The definition of a symmetric spectrum contains some redundancy. Show that theequivariance condition for the iterated structure map is already satisfied if for every n ≥ 0 the followingtwo conditions hold:(i) the structure map σn : Xn ∧ S1 −→ Xn+1 is Σn-equivariant where Σn acts on the target by restrictionfrom Σn+1 to the subgroup Σn.(ii) the composite

Xn ∧ S2 σn ∧ Id−−−−→ Xn+1 ∧ S1 σn+1−−−→ Xn+2

is Σ2-equivariant.

Exercise 8.2. Let X be a symmetric spectrum such that for infinitely many n the action of Σn on Xn

is trivial. Show that all naive homotopy groups of X are trivial. (Hint: identify the quotient space of theΣ2-action on S2.) What can be said if infinitely many of the alternating groups act trivially?

156 I. BASICS

Exercise 8.3. Let S[k] denote the symmetric subspectrum of the sphere spectrum obtained by trun-cating below level k, i.e.,

(S[k])n =

∗ for n < k

Sn for n ≥ k.

Show that for every symmetric spectrum X and all k ≥ 0 the inclusion S[k] −→ S induces a π∗-isomorphismS[k] ∧X −→ S ∧X ∼= X.

Exercise 8.4. For an abelian group A we define a symmetric spectrum KA of simplicial sets as follows.We set KAn = A[∆[n]/∂∆[n]], the A-linearization of the simplicial n-simplex modulo its boundary. Thesymmetric group acts on KAn by multiplication by sign. The structure map

σn : A[∆[n]/∂∆[n]] ∧ S1 −→ A[∆[n+ 1]/∂∆[n+ 1]]

is the ‘A-linear extension’ of the map (∆[n]/∂∆[n]) ∧ S1 −→ A[∆[n + 1]/∂∆[n + 1]] that sends the ithgenerating (n+ 1)-simplex of the source to (−1)i times the generating (n+ 1)-simplex of the target.

Show that the above really defines a symmetric spectrum and calculate the naive homotopy groups ofKA in terms of the group A. For which A is KA an Ω-spectrum?

Exercise 8.5. In this exercise we identify the skeleton filtration (see Construction 5.21). Let L be apointed Σm-space (or simplicial set) and X a symmetric spectrum. Show that the filtration F i(L .m X) istrivial for 0 ≤ i < m and that construct a natural isomorphism

Fm+n(L .m X) = L .m (FnX)

for n ≥ 0.

[homotopy mapping spaces maph(A,X) is homotopy invariant for X levelwise Kan [in sS] respectivelyA level cofibrant [in T]. Interpret points in maph(A,X) as ‘infinitely homotopy coherent morphisms’. Showmaph(A,X) ∼= map(A[, X) If A is flat or X injective, then maph(A,X) −→ map(A,X) is a weak equiva-lence. f is stable equivalence iff for every Ω-spectrum X the induced map maph(f,X) : maph(B,X) −→maph(A,X) of homotopy mapping spaces is a weak equivalence of simplicial sets]

[ For the next proposition we recall that a commutative square of simplicial sets

Vα //

ϕ

W

ψ

X

β// Y

is called homotopy cartesian if for some (hence any) factorization of the morphism g as the composite of aweak equivalence w : W −→ Z followed by a Kan fibration f : Z −→ Y the induced morphism

V(ϕ,wα)−−−−→ X ×Y Z

is a weak equivalence. The definition is in fact symmetric in the sense that the square is homotopy cartesianif and only if the square obtained by interchanging X and W (and the morphisms) is homotopy cartesian.So if the square is homotopy cartesian and ψ (respectively β) is a weak equivalence, then so is ϕ (respectivelyα).

Proposition 8.6. Consider a pullback square of symmetric spectra of simplicial sets

Ai //

f

B

g

C

j// D

EXERCISES 157

in which the morphism g is levelwise a Kan fibration. Then for every injective Ω-spectrum X the commu-tative square of simplicial sets

map(D,X)map(g,X) //

map(j,X)

map(B,X)

map(i,X)

map(C,X)

map(f,X)// map(A,X)

is homotopy cartesian.

Proof. In a first step we prove the proposition under the additional assumption that the morphismj : C −→ D is a monomorphism. This implies that its basechange i : A −→ B is also a monomorphism. Inthis situation the morphism j ∪ g : C ∪A B −→ D is a π∗-isomorphism [ref], thus a stable equivalence byTheorem 4.16. We fix an injective Ω-spectrum X and consider the commutative diagram of simplicial sets

map(D,X)map(g,X) //

map(j,X)

map(f∪i,X)

))SSSSSSSSSmap(B,X)

map(i,X)

map(C ∪A B,X)

33ggggggggggggg

yyssssssssssssss

map(C,X)map(f,X)

// map(A,X)

The right vertical map map(i,X) is a Kan fibration by Proposition 4.3 (i) and the lower right part of thediagram is a pullback. The morphism map(f ∪ i,X) is a weak equivalence, so the outer commutative squareis homotopy cartesian.

Now we prove the general case. We factor the morphism j as the mapping cylinder inclusion C −→ Z(j)followed by the projection p : Z(j) −→ D which is a homotopy equivalence. Then the square decomposesas the composite of two pullback squares

Ai //

f

Z(j)×D Bp //

B

g

C // Z(j)

p// D

For every injective Ω-spectrum X the functor map(−, X) takes the left pullback square to a homotopycartesian square of simplicial sets by the special case above. The projection p is a level equivalence, henceso is its basechange p. So both p and p become weak equivalences after applying map(−, X) and thefunctor map(−, X) also takes the right pullback square to a homotopy cartesian square. The composite oftwo homotopy cartesian squares is homotopy cartesian, which proves the claim.

]

Exercise 8.7. Let A be a flat symmetric spectrum and X a symmetric spectrum such that X0 iscofibrant and all structure morphisms σn : Xn ∧ S1 −→ Xn+1 are cofibrations. Show that then themorphism

ξA,X : (shA) ∧X −→ sh(A ∧X)

defined in (5.7) is a level cofibration.Hint: the special case X = S featured in the proof of Proposition 5.37.

158 I. BASICS

Exercise 8.8. (i) We define a functor + : Σ×Σ −→ Σ on objects by addition of natural numbers andon morphisms by

[τ ∧ z] + [γ ∧ y] = [(τ × γ) ∧ (1× χn,m × 1)∗(z ∧ y)] ∈ Σ(n+ n,m+ m)

for [τ ∧ z] ∈ Σ(n,m) and [γ ∧ y] ∈ Σ(n, m). Show that ‘+’ is strictly associative and unital, i.e., a strictmonoidal product on the category Σ. Define a symmetry isomorphism to make this into a symmetricmonoidal product. Show that under the correspondence between symmetric spectra and based continuousfunctors Σ −→ T, shifting of spectra corresponds to precomposition with the functor 1 +− : Σ −→ Σ.

(ii) Show that the construction in (i) can be extended to an isomorphism between the categories ofsymmetric ring spectra and strong monoidal functors from Σ to T such that it takes commutative symmetricring spectra isomorphically onto the full subcategory of symmetric monoidal functors.

(iii) Present a continuous functor X : Σ −→ T as a continuous coend,

X ∼=∫ top

n∈Σ

Xn ∧ FnS0 .

This leads to a presentation of a symmetric spectrum X as a coequalizer∨n≤m Xn ∧Σ(n,m) ∧ FmS0 act∧Id //

Id∧coact//∨n≥0 Xn ∧ FnS0 // X

Exercise 8.9 (Coordinate free symmetric spectra). There is an equivalent definition of symmetricspectra which is, in a certain sense, ‘coordinate free’. If A is a finite set we denote by RA the set offunctions from A to R with pointwise structure as a R-vector space. We let SA denote the one-pointcompactification of RA, a sphere of dimension equal to the cardinality of A. A coordinate free symmetricspectrum consists of the following data:

• a pointed space XA for every finite set A• a based continuous map α∗ : XA ∧ SB−α(A) −→ XB for every injective map α : A −→ B of finite

sets, where B − α(A) is the complement of the image of α.This data is subject to the following conditions:

• (Unitality) For every finite set A, the composite

XA∼= XA ∧ S∅

(IdA)∗−−−−→ XA

is the identity.• (Associativity) For every pair of composable injections α : A −→ B and β : B −→ C the diagram

XA ∧ SB−α(A) ∧ SC−β(B)Id∧β!

//

α∗∧Id

XA ∧ SC−β(α(A))

(βα)∗

XB ∧ SC−β(B)

β∗

// XC

commutes. In the top vertical map we use the homeomorphism β! : SB−α(A) ∧ SC−β(B) ∼=SC−β(α(A)) which is one-point compactified from the linear isomorphism RB−α(A) × RC−β(B) ∼=RC−β(α(A)) which uses β on the basis elements indexed by B−α(A) and the identity on the basiselements indexed by C − β(B).

A coordinate free symmetric spectrum X gives rise to a symmetric spectrum as follows. For this we letn = 1, . . . , n denote the ‘standard’ set with n elements and identify Rn with Rn and Sn with Sn. We setXn = Xn. A permutation γ ∈ Σn acts on Xn as the composite

Xn∼= Xn ∧ S∅

γ∗−→ Xn .

(i) Show that the above construction indeed defines a symmetric spectrum in the sense of Definition 1.1.(ii) Show that this ‘forgetful’ functor from coordinate free symmetric spectra to symmetric spectra is

an equivalence of categories.

EXERCISES 159

(iii) Work our how symmetric ring spectra are formulated in this language.

Exercise 8.10. Here is an interpretation of orthogonal spectra analogous to the perspective on sym-metric spectra of Exercise 8.1 We define a based topological category O with objects the natural numbers0, 1, 2, . . . and the based space of morphisms from n to m is given by O(n,m) = O(m)+ ∧1×O(m−n) S

m−n,[notation O already used in Example 8.25 ...] which is to be interpreted as a one-point space if m < n.The identity in O(n, n) = O(n)+ ∧ S0 is the identity of O(n) (smashed with the non-basepoint of S0) andcomposition : O(m, k) ∧O(n,m) −→ O(n, k) is defined by

[τ ∧ z] [γ ∧ y] = [τ(γ × 1) ∧ (y ∧ z)]

where τ ∈ O(k), γ ∈ O(m), z ∈ Sk−m and y ∈ Sm−n.(i) Given an orthogonal spectrum X we define a map X : O(n,m) ∧Xn −→ Xm by

[τ ∧ z] ∧ x 7−→ γ∗(σm−n(x ∧ z)) .

Verify that this assignment defines a based continuous functor X : O −→ T from the category O to thecategory of pointed spaces. Show that X 7→ X is an isomorphism between the category of orthogonalspectra and the category of based continuous functors O −→ T.

(ii) We define a functor + : O×O −→ O on objects by addition of natural numbers and on morphismsby

[τ ∧ z] + [γ ∧ y] = [(τ × γ) ∧ (1× χn,m × 1)∗(z ∧ y)] ∈ O(n+ n,m+ m)

for [τ ∧ z] ∈ O(n,m) and [γ ∧ y] ∈ O(n, m). Show that ‘+’ is strictly associative and unital, i.e., a strictmonoidal product on the category O. Define a symmetry isomorphism to make this into a symmetricmonoidal product.

(iii) Show that the construction in (i) can be extended to an isomorphism between the categoriesof orthogonal ring spectra and strong monoidal functors from O to T such that it takes commutativeorthogonal ring spectra isomorphically onto the full subcategory of symmetric monoidal functors.

(iv) Define a continuous based functor Σ −→ O such that the forgetful functor from orthogonal tosymmetric spectra corresponds to precomposition with the functor Σ −→ O. Interpret the prolongationfunctor P : Sp −→ SpO and the right adjoint ? : Sp −→ SpO of the forgetful functor as topological left andright Kan extension along Σ −→ O.

(v) Define a based topological category OR from an orthogonal ring spectrum R such that R-modulescorrespond to continuous functors OR −→ T.

(vi) Formulate unitary spectra as continuous functors from a topological category U, etc.

Exercise 8.11 (Coordinate free orthogonal spectra). [fix this... needs Thom spectra] There is a morenatural notion where we use vector spaces with inner product to index the spaces in an orthogonal spectrum.A coordinate free orthogonal spectrum X consists of the following data:

• a pointed space X(V ) for each inner product space V , i.e., finite dimensional real vector spacewith a euclidian scalar product,

• a base-point preserving continuous left action of the orthogonal group O(V ) of V on X(V ), foreach inner product space V ,

• a structure map σV,W : X(V ) ∧ SW −→ X(V ⊕W ) for each pair of inner product spaces V andW which is O(V )×O(W )-equivariant. Here SW is the one-point compactification of W on whichthe group O(W ) acts by extension of the action on W , fixing the basepoint at infinity. The groupO(V )×O(W ) acts on the target by restriction of the O(V ⊕W )-action.

This data should satisfy two conditions: the composite

X(V ) ∼= X(V ) ∧ S0 σV,0−−−→ X(V ⊕ 0) ∼= X(V )

160 I. BASICS

should be the identity [isometries V ∼= W should also act] and the square

X(V ) ∧ SW ∧ SZId∧µW,Z //

σV,W∧Id

X(V ) ∧ SW⊕Z

σV,W⊕Z

X(V ⊕W ) ∧ SZ σV⊕W,Z

// X(V ⊕W ⊕ Z)

commutes for all inner product spaces V,W and Z.A morphism f : X −→ Y of coordinate free orthogonal spectra consists of O(V )-equivariant pointed

maps f(V ) : X(V ) −→ Y (V ) for all V which are compatible with the structure maps in the sense thatf(V ⊕W ) σV,W = σV,W (f(V ) ∧ Id) for all V and W .

A coordinate free orthogonal spectrum X gives rise to a coordinate free symmetric spectrum UX(see Exercise 8.9) by forgetting symmetry. For a finite set A the space (UX)A is X(RA), the value ofX at the inner product space RA which has A as orthonormal basis. [define the structure maps α∗ :(UX)A ∧ SB−α(A) −→ (UX)B ]

Exercise 8.12. Let R be a symmetric ring spectrum. Define mapping spaces (simplicial sets) andfunction symmetric spectra of homomorphisms between two given R-modules. Check that for all k ≥ 0 theendomorphism ring spectrum HomR(k+∧R, k+∧R) of the R-module k+∧R is isomorphic, as a symmetricring spectrum, to the matrix ring spectrum Mk(R) (see Example 3.24).

Exercise 8.13. In [...] we defined a functor R∞ : Sp −→ Sp; for a symmetric spectrum X, R∞X is acertain mapping telescope of spectra RnX where RX = Ω(shX).

Let f : A −→ B be a π∗-isomorphism of symmetric spectra of spaces. Show that then the morphismR∞f : R∞A −→ R∞B is a level equivalence.

Exercise 8.14. The true homotopy groups of a symmetric spectrum A of simplicial sets were definedin 4.29 via the geometric realization. This exercise shows that instead we could have defined πkA as a groupof natural transformations, analogous to Definition 4.29, but within the category of semistable symmetricspectra of simplicial sets.

We define a group πsSk A by

πsSk A = NatSpsssS→set(Sp(A,−), πk) ,

the set of natural transformations, of functors from semistable symmetric spectra of simplicial sets to sets,from the restriction of the representable functor to the restriction of k-th naive homotopy group functor.We construct a morphism of abelian groups

πsSk A −→ πk|A|

as follows. Given a ‘simplicial’ true homotopy class τ of dimension k and a semistable symmetric spectrumof spaces X, we define the image of τ at X as the composite

SpT(|A|, X) ∼= SpsS(A,S(X))τS(X)−−−→ πkS(X) = πk|S(X)| πkε−−−→ πkX .

The first map is the adjunction bijection, and the last map is induced by the counit ε : |S(X)| −→ X. Showthat this morphism is a natural isomorphism.

Exercise 8.15 (Variations on true homotopy groups). Let C be a class of symmetric spectra and definea ‘homotopy group relative to C’ by

πCkA = NatC→set(Sp(A,−)|C , πk) ,

[...] For C′ ⊂ C we get a homomorphism πCk −→ πC′

k by restricting a natural transformation to the smallercategory.

(i) Let C′ ⊂ C be two classes of symmetric spectra such that there is a functor F : C −→ C with valuesin C′ and a natural π∗-isomorphism X −→ FX. Show that the restriction homomorphism πCk −→ πC

′

k is anisomorphism.

EXERCISES 161

(ii) Show that the true homotopy group πkA is naturally isomorphic to the homotopy group relativeany of the following classes of symmetric spectra: (a) Ω-spectra; (b) flat fibrant Ω-spectra; (c) injectiveΩ-spectra; (d) positive Ω-spectra; (e) flat fibrant positive Ω-spectra; (f) injective positive Ω-spectra; (g) flatfibrant semistable spectra; (h) injective semistable spectra. Can you think of other classes of symmetricspectra that can be used here?

(iii) Suppose that the class C contains a symmetric spectrum which is not semistable. Show that thenπCkA is not naturally isomorphic to the true homotopy group functor.

(iv) Let C be the class of symmetric spectra X such that for all n ≥ 0 the alternativ group An actstrivially on Xn. Show that every spectrum in C is semistable and that the restriction map πkA = πss

k A −→πCkA extends to an isomorphism Q⊗ πkA ∼= πCkA

Exercise 8.16. This exercise is supposed to motivate the term ‘bimorphism’ which we used in the firstway to introduce the smash product of symmetric spectra. Let X,Y and Z be symmetric spectra.

(i) Let bp,q : Xp∧Yq −→ Zp+q be a collection of Σp×Σq-equivariant maps. Show that the commutativityof the left part of (5.1) is equivalent to the condition that for every p ≥ 0 the maps bp,q : Xp ∧ Yq −→ Zp+qform a morphism bp.• : Xp ∧ Y −→ shp Z of symmetric spectra as q varies. Show that the commutativityof the right part of (5.1) is equivalent to the condition that for every q ≥ 0 the composite maps

Yq ∧Xptwist−−−→ Xp ∧ Yq

bp,q−−→ Zp+qχp,q−−−→ Zq+p

form a morphism Yq ∧X −→ shq Z of symmetric spectra as p varies.

(ii) Let b = bp,q : Xp∧Yq −→ Zp+q be a bimorphism. Define bp : Xp −→ map(Y, shp Z) as the adjointof the morphism of symmetric spectra bp.• : Xp ∧ Y −→ shp Z (compare part (i)). Show that as p varies,the maps bp form a morphism of symmetric spectra b : X −→ Hom(Y, Z). Show then that the assignment

Bimor((X,Y ), Z) −→ Sp(X,Hom(Y, Z)) , b 7→ b

is bijective and natural in all three variables.

Exercise 8.17. The way Hovey, Shipley and Smith introduce the smash product in their originalpaper [33] is quite different from our exposition, and this exercise makes the link. Thus the paper [33]has the solutions to this exercise. A symmetric sequence consists of pointed spaces (or simplicial set) Xn,for n ≥ 0, with based, continuous (respectively simplicial) Σn-action on Xn. Morphisms f : X −→ Y aresequences of equivariant based maps fn : Xn −→ Yn. The tensor product X⊗Y of two symmetric sequencesX and Y is the symmetric sequence with nth term

(X ⊗ Y )n =∨

p+q=n

Σ+n ∧Σp×Σq Xp ∧ Yq .

(i) Make the tensor product into a closed symmetric monoidal product on the category of symmetricsequences.

(ii) Show that the sequence of spheres S = Snn≥0 forms a commutative monoid in the category ofsymmetric sequences. Show that the category of symmetric spectra is isomorphic to the category of rightS-modules in the monoidal category of symmetric sequences.

(iii) Given a commutative monoid R in the monoidal category of symmetric sequences and two rightR-modules M and N , show that the coequalizer M ∧R N of the two morphisms

αM ⊗ Id, Id⊗(αN τR,N ) : M ⊗R⊗N −→ M ⊗Nis naturally a right R-module. Show that the smash product over R is a closed symmetric monoidal producton the category of right R-modules.

(iv) Show that the smash product over S corresponds to the smash product of symmetric spectra underthe isomorphism of categories of part (ii).

Exercise 8.18. Define a notion of ‘commuting homomorphisms’ between symmetric ring spectra suchthat homomorphism of symmetric ring spectra R∧S −→ T are in natural bijection with pairs of commuting

162 I. BASICS

homomorphisms (R −→ T, S −→ T ). Deduce that the smash product is the categorical coproduct forcommutative symmetric ring spectra.

Exercise 8.19. In this exercise we discuss an operator on symmetric spectra which has formal prop-erties very similar to differentiation of functions. We define a functor

∂ : Sp −→ Spby

∂X = cokernel(λX : S1 ∧X −→ shX)where the map λX was defined in (3.9). Show that the functor ∂ has the following properties.

(i) the ‘derivative’ is additive in the sense that ∂ commutes with colimits and satisfies ∂(K ∧X) ∼=K ∧ ∂X for a pointed space (or simplicial set) K.

(ii) for the smash product of two symmetric spectra X and Y we have the ‘Leibniz rule’ in the formof a natural isomorphism

∂(X ∧ Y ) ∼= (∂X) ∧ Y ∨ X ∧ (∂X) .

(iii) for the semifree symmetric spectrum generated by a pointed Σn-space (or simplicial set) L wehave ∂(GnL) = Gn−1(shL) where shL is the restriction of L along the homomorphism 1 × − :Σn−1 −→ Σn.

(iv) the free symmetric spectrum FnS0 ‘differentiates’ formally like the function xn because we have

∂(FnS0) = n+ ∧ Fn−1S0.

(v) the n-fold iterated derivative ∂(n)X has a natural action of the symmetric group Σn in such a waythat its 0th level admits a natural isomorphism of Σn-spaces

(∂(n)X)0 ∼= cokernel(νn : LnX −→ Xn) .

(vi) there is an analogue of the Taylor expansion f(x) =∑n≥0 f

(n)(0)/n! · xn, but only ‘up to exten-sions’ in the following sense: the graded symmetric spectrum associated to the filtration of X bythe spectra FnX, see (??), is isomorphic to∨

n≥0

(∂(n)X)0 ∧Σn (F1S0)∧n .

(vii) suppose that X is flat. Then ∂X is flat and X is semistable if and only if all naive homotopygroups of ∂X are trivial.

Exercise 8.20. In Proposition 5.30 we identified the latching space LmX = (Fm−1X)m of a symmetricspectrum X with the m-th level of the spectrum X ∧ S. In this exercise we show that also the other levelsof the spectra in the canonical filtration of X can be obtained from smash products with truncated spherespectra.

(i) Describe the functor which the Σm-space (F iX)m represents in a way which generalized Proposi-tion 5.24.

(ii) We denote by S[k] the sphere spectrum truncated below level k, i.e., the symmetric subspectrum ofS with levels

(S[k])n =

∗ for n < k

Sn for n ≥ k.

For example we have S[0] = S and S[1] = S.Construct a natural Σm-equivariant isomorphism between (F iX)m and (X ∧ S[m−i]) for i ≤ m.

Exercise 8.21. In Example 3.12 we saw that every symmetric spectrum is naturally a coequalizer ofsemifree symmetric spectra. This fact has a dual version: every symmetric spectrum X is naturally anequalizer made up of co-semifree symmetric spectra. More precisely, we claim that the diagram

X //∏m≥0 PmXm

σ //I

//∏n≥0 Pn(ΩXn+1)

EXERCISES 163

is an equalizer diagram. Here the upper morphism is the product of the morphisms Pmσm : PmXm −→Pm(ΩXm+1); the lower map is the product of the morphisms Pm+1Xm+1 −→ Pm(ΩXm+1) which are adjointto the identity of (Pm+1Xm+1)m = ΩXm+1. [Morphisms to X are the same as morphisms to the equalizer]

Since the morphisms in the symmetric category Σ are ‘directed’ (in the sense that for n > m the spaceof morphisms from n to m has just one point), the ‘opposite’ of the skeleton filtration of a spectrum by thesubspectra FmX is rather boring: we define the truncation of X at level m τ≤mX

(τ≤mX)n =

∗ for n > m,

Xn for n ≤ m.

Show that the commutative square

τ≤mX //

τ≤m−1X

PmXm

// τ≤m−1(PmXm)

is a pullback.

Exercise 8.22. Let R be a symmetric ring spectrum, M a right R-module and x ∈ πkM a homogeneoushomotopy class. Denote by M/xR the mapping cone of a R-homomorphism λx : FnSk+n ∧R −→M whichtakes the unit class to x. [so x must be in the image of c : π∗M −→ π∗M ] The morphism λx should bethought of as ‘left multiplication by x’ (and indeed, this is its effect in homotopy). Then the Toda bracketsof the form 〈x, y, z〉 for varying y and z contain significant information about the structure of π∗(M/xR)as a graded π∗R-module.

(i) Show that there is a short exact sequence of graded right π∗R-modules [the connecting morphismis π∗R-linear !]

0 −→ π∗M/x · π∗−kRj−−→ π∗(M/xR) δ−−→ ann∗−k−1(x) −→ 0

where ann∗(x) = y ∈ π∗R | x · y = 0 is the annihilator (right) ideal of x.(ii) Let y ∈ πlR and z ∈ πjR be homogeneous classes which satisfy xy = 0 = yz. Show that the Toda

bracket 〈x, y, z〉 equals the set of all elements α of πk+l+j+1M for which there exists y ∈ πk+l+1 with

δ(y) = y and j(α+ x · π∗−kR) = y · z .

Exercise 8.23. Recall the commutative symmetric ring spectra KU and KO of complex and realtopological K-theory from Example 1.18.

(i) Show that every graded π∗KU -module is realizable as the homotopy of a KU -module spectrum.(ii) Show that the only cyclic graded π∗KO-modules which are realizable as the homotopy of a KO-

module spectrum are the free module, the trivial module and Z/n ⊗ π∗KO for n an odd integer. (Hint:produce enough nonzero Toda brackets and use Proposition 6.25)

(iii) Recall from Example 6.22 how to ‘mod out’ a homotopy class x of a symmetric ring spectrum Ron an R-module M , producing an R-module M/x. Calculate the homotopy groups of KO/2 as a π∗KO-module and show that the double suspension of KO/ξ is stably equivalent, as a KO-module spectrum, toKO/2.

Exercise 8.24. One can break the construction of the M-action on the naive homotopy groups of asymmetric spectrum up into two steps and pass through the intermediate category of I-functors. As before,the category I has an object n = 1, . . . , n for every non-negative integer n, including 0 = ∅. Morphismsin I all injective maps. An I-functor is a covariant functor from the category I to the category of abeliangroups.

(i)From symmetric spectra to I-functors. Given a symmetric spectrum X and an integer k we assignan I-functor πkX to the symmetric spectrum X. On objects, this I-functor is given by

(πkX)(n) = πk+nXn

164 I. BASICS

if k + n ≥ 2 and (πkX)(n) = 0 for k + n < 2. If α : n −→ m is an injective map and k + n ≥ 2, thenα∗ : (πkX)(n) −→ (πkX)(m) is given as follows. We choose a permutation γ ∈ Σm such that γ(i) = α(i)for all i = 1, . . . , n and set

α∗(x) = sgn(γ) · γ(ιm−n(x))where ι : πk+nXn −→ πk+n+1Xn+1 is the stabilization map (1.7). Justify that this definition is independentof the choice of permutation γ and really defines a functor on the category I.

(ii) Show that every I-functor arises as the I-functor π0 of a symmetric spectrum. (Hint: revisit theconstruction of the Eilenberg-Mac Lane spectrum a second time.)

(iii) From I-functors to tame M-modules. Let F be any I-functor F ; construct a natural tame leftaction by the injection monoidM on the colimit of F , formed over the subcategory N of inclusions, in sucha way that for the I-functor πkX coming from a symmetric spectrum X as in (i) , this yields theM-actionon the stable homotopy group πkX.

(iv) Show that the tame M-module Pn which represents the functor of taking filtration n (see Exam-ple 7.9) is isomorphic, as anM-module, to the colimit of the representable I-functor I(n,−).

Exercise 8.25. Let W be anM-module. Show that the assignment n 7→W (n) extends to an I-functorW (•) in such a way that W 7→ W (•) is right adjoint to the functor which assigns to an I-functor F theM-module F (ω). The counit of the adjunction (W (•))(ω) −→ W is injective with image the subgroupof elements of finite filtration, which is also the largest tame submodule of W . The assignment W 7→(W (•))(ω) =

⋃nW

(n) is right adjoint to the inclusion of tame M-modules into all M-modules. Therestriction of W 7→W (•) to tame M-modules is fully faithful.

Exercise 8.26. We recall that the m-th stable homotopy group πsmK of a based space (or simplicial

set) is defined as the m-th maive homotopy groups πm(Σ∞K), i.e., as the colimit of the sequence of abeliangroups

πmK−∧S1

−−−−→ πm+1(K ∧ S1) −∧S1

−−−−→ πm+2(K ∧ S2) −∧S1

−−−−→ · · · .Show that the naive homotopy group πkX of a symmetric spectrum X can also be calculated from thesystem of stable as opposed to unstable homotopy groups of the individual spaces Xn. where we stabilizefrom the left. Smashing with the identity of S1 from the left provides a map S1∧− : πs

mK −→ πs1+m(S1∧K)

which is a special case of the suspension isomorphism (compare Proposition ??) for the suspension spectrumof K.

For a symmetric spectrum we can then define an I-functor πskX by setting (πs

kX)(n) = πsk+nXn on

objects (with no restriction on k + n). We let the inclusion ι : n −→ n + 1 act as the composite

πsk+nXn

S1∧−−−−−→ πs1+k+n(S

1 ∧Xn)(−1)k+ntwist−−−−−−−−→ πs

k+n+1(Xn ∧ S1)πsk+n+1σn−−−−−−→ πs

k+n+1Xn+1 .

[check that 1 × Σm acts trivially on the image of ιm.] and defining the action of a morphism n −→ m inthe same way as for the I-functor πkX of unstable homotopy groups.

Exercise 8.27. Let F : I −→ Ab be any I-functor. Define a natural ‘generalized Eilenberg-Mac Lanespectrum’ HF to F so the I-functor π0(HF ) is isomorphic to F . Hint: the various Eilenberg-Mac Lanespectra of Examples 1.14, 4.23 and 7.10 should become special cases.

Exercise 8.28. Show that the extended action of the completed monoid ring Z[[M]] on a tame M-module W (see Remark 7.24) makes it a discrete module in the sense that the action map

Z[[M]]×W −→ W

is continuous with respect to the discrete topology on W and the filtration topology on Z[[M]]. Showthat conversely, if W is discrete module over Z[[M]], then its underlying M-module is tame. Show thatthis establishes an isomorphism between the category of tame M-modules and the category of discreteZ[[M]]-modules.

EXERCISES 165

Exercise 8.29. Let W be anM-module.(i) Show that the map d· : W −→ W (1) given by the left multiplication by the cycle operator d is

M-linear with respect to the shiftedM-action on the target.(ii) Denote by W (∞) the colimit of the sequence

Wd·−−→ W (1) d·−−→ W (2) d·−−→ · · ·

of M-modules and homomorphisms. For a symmetric spectrum X, construct a natural isomorphism ofM-modules

πk(R∞X) ∼= (πkX)(∞) .

Exercise 8.30. Show that for every I-functor F there are natural isomorphisms of abelian groups

colimpI F

∼= Hp(M, colimN F ) .

for all p ≥ 0.

Exercise 8.31. In Remark 7.24 we introduced the ring Z[[M]] of natural operations on the 0-thhomotopy group of symmetric spectra and identified it with a certain completion of the monoid ring Z[M].

(i) An I-set is a functor from the category I of finite sets and injections to the category of sets. Showthat the endomorphism monoid of the ‘colimit over inclusions’ functor

(I-sets) −→ (sets) , F 7→ F (ω)

is isomorphic to M.(ii) Show that the endomorphism ring of the ‘colimit over inclusions’ functor

(I-functors) −→ (ab. groups) , F 7→ F (ω)

is isomorphic to the ring Z[[M]].

Exercise 8.32. Readers familiar with the linear isometries operad and the theory of S-modules inthe sense of Elmendorf, Kriz, Mandell and May [24] will notice the similarity between the -product ofM-spectra and the smash product of L-spectra defined in [24, I.5]. Indeed, lurking behind the -productis a certain discrete analog M of the linear isometries operad, which we now discuss.

The injection operadM is the operad of sets defined by lettingM(n) be the set of injections from theset ω × n into ω, for n ≥ 0. Note that for n = 0 the source is the empty set, so M(0) has exactly oneelement, and M(1) is the monoid M. The symmetric groups permute the second coordinates in ω × n.The operad structure is via disjoint union and composition, i.e., M is a suboperad of the endomorphismoperad of the set ω in the symmetric monoidal category of sets under disjoint union. More precisely, theoperad structure morphism

γ : M(n)×M(i1)× · · · ×M(in) −→ M(i1 + · · ·+ in)

sends (ϕ, f1, . . . , fn) to ϕ (f1 + · · ·+ fn).

(i) Show that the collection M(n)n≥0 is an operad with respect to the structure maps γ definedabove.

(ii) For all n,m ≥ 1 the map

a : M(2)×M2 (M(n)×M(m)) −→ M(n+m)

given by ϕ(ψ, λ) 7→ ϕ (ψ + λ) is an isomorphism ofM-Mn+m-bisets.(ii’) The injection operad of sets has similar formal properties as the linear isometries operad. Show

that the operadic composition map

γ : M(1)×M(n) −→ M(n)

166 I. BASICS

is a free and transitive action of the monoidM on the setM(n) for n ≥ 1. Show that the injection operadhas ‘Hopkins’ property’ that the map

M(2)×M(1)×M(1)×M(i)×M(j)γ×Id //Id×γ2

//M(2)×M(i)×M(j)γ //M(i+ j)

is a split coequalizer for all i, j ≥ 1.(iii) Show that although the monoid M×M does not act transitively on the set M(2), the orbit set

M(2)/M(1)×M(1) =M(2)/M(1)2 has only one element.

Exercise 8.33. We now give a construction which associates to an I-functor with Σm-action F (i.e.,a covariant functor F : I −→ Z[Σm]-mod) a new I-functor .mF and give a formula for the M-module(.mF )(ω).

Given F : I −→ Z[Σm]-mod we define a new I-functor .mF by (.mF )(k) = 0 for k < m and

(8.34) (.mF )(m + n) = Z[Σm+n]⊗Σm×Σn F (n) .

We define .mF on morphisms α : m + n −→ m + k in I as follows. We choose a permutation γ ∈ Σm+k

which agrees with α on m + n and define

α∗ : (.mF )(m + n) = Z[Σm+n]⊗Σm×Σn F (n) −→ Z[Σm+k]⊗Σm×Σk F (k) = (.mF )(m + k)

by α∗(τ ⊗ x) = γ(τ × 1k−n)⊗ ι∗(x) where ι : n −→ k is the inclusion.(i) Check that this defined a functor(ii) In (7.11) we defined a homomorphism of monoids × : Σm ×M −→M by

(γ × f)(i) =

γ(i) for 1 ≤ i ≤ m, and

f(i−m) +m for m+ 1 ≤ i.

As before we denote by Z[M]〈m〉 the monoid ring of M with its usual left multiplication action, but withaction by the monoid Σm×M via restriction along the homomorphism × : Σm×M −→M. Since F takesvalues in Σm-modules, the colimit F (ω) not only has an action of M, but also a compatible left action bythe group Σm. So we can form

Z[M]〈m〉 ⊗Σm×M F (ω)which is a leftM-module via the left multiplication action of Z[M] on itself. Show that for every I-functorwith Σm-action F the natural map

Z[M]〈m〉 ⊗Σm×M F (ω) −→ (.mF )(ω)

f ⊗ [x] 7−→ f · [1⊗ x]is an isomorphism ofM-modules.

Relate this to the I-functor of stable homotopy groups of a twisted smash product.

Exercise 8.35. Let R be a symmetric ring spectrum. Show that there is a natural structure of agraded ring on the graded subgroup (π∗R)(0) of M-fixed elements in the homotopy groups of R. If R iscommutative, then the product on (π∗R)(0) is graded-commutative. The homotopy groups of every rightR-module naturally form a graded right module over the graded ring (π∗R)(0).

Thus if R is semistable then the homotopy groups π∗R form a graded ring which acts naturally on thehomotopy groups of every right R-module.

Exercise 8.36. As we have seen in Proposition 6.5 the true homotopy groups of any symmetric ringspectrum form a graded ring. However, the naive homotopy groups of a symmetric ring spectrum R doin general not form a graded ring in any natural way (unless R is semistable, when the naive and truehomotopy groups coincide). As we mentioned in [...] the problem is that the smash product pairing

πk+nRn × πl+mRm −→ πk+n+l+mRn+m

induced by the multiplication Rn ∧ Rm −→ Rn+m is generally not compatible with the stabilization inthe left factor and so does not usually induce a map πkR × πlR −→ πk+lR on colimits. In this exercise

HISTORY AND CREDITS 167

we reveal the natural structure on the naive homotopy groups of a symmetric ring spectrum, namely agraded algebra structure over the injection operad. For semistable symmetric spectra, the naive and truehomotopy groups are isomorphic. By part (iii) of this exercise, in that case the action of the injectionoperad reduces to the multiplication on the homotopy groups.

(i) Let R be a symmetric ring spectrum. For every n ≥ 1, every ϕ ∈M(n) and all integers ki, constructa natural multilinear map

ϕ∗ : πk1R× · · · × πknR −→ πk1+···knR .

[...] Show that these maps make the naive homotopy groups π∗R into a graded (non-symmetric) algebraover the (non-symmetric) operadM.

(ii) Show that if the multiplication of R is commutative, then the algebra structure of part (i) makesπ∗R into a graded algebra over the operadM, i.e., the symmetric property also holds.

(iii) Suppose A is an algebra over the injection operad in the monoidal category of abelian groupsunder tensor product. Show that if the action of the injection monoid M = M(1) on A is trivial, thenall injections ϕ ∈ M(n) induced the same map A⊗n −→ A. Conclude that the full subcategory of thoseM-algebras for which M =M(1) acts trivially is equivalent to the category of rings.

Exercise 8.37. In Example 7.10 we associated to every tame M-module W an Eilenberg-Mac Lanespectrum HW and an M-linear isomorphism jW : W ∼= π0(HW ). Show that this isomorphism is multi-plicative, i.e., that for every pair of tame M-modules W and V the composite map

WVjWjV−−−−−→ π0HWπ0HV

·−→ π0(HW ∧HV )π0(mW,V )−−−−−−→ H(WV )

equals jW⊗V . (Note that ifM acts trivially on W and V , then this specializes to Example 6.10.

Exercise 8.38. We claim that the quotient spectrum Q = A× B/A ∨ B is stably contractible. Sincethe canonical morphism is injective, this quotient spectrum Q is level equivalent to the mapping cone,[how to do this for spaces ?; need that S(A) ∨ S(B) −→ S(A ∨B) is π∗-isomorphism] which is thus stablecontractible. So the morphism is a stable equivalence by Proposition 4.19 (iii).

In every level n the morphism λQn : S1 ∧Qn −→ Q1+n is homotopic to the trivial morphism [spell out].So the morphism λQ induces the trivial map on naive homotopy groups. By Proposition 4.39 the cycleoperator d : πkQ −→ πkQ is then trivial in all dimensions. Since this operator is also injective, all naivehomotopy groups of Q are trivial. So Q is stably contractible by Theorem 4.16.

History and credits

I now summarize the history of symmetric spectra and symmetric ring spectra, and the genesis of theexamples which were discussed above, to the best of my knowledge. My point with respect to the examplesis not when certain spectra first appeared as homotopy types or ring spectra ‘up to homotopy’, but ratherwhen a ‘highly structured’ multiplication was first noticed in one form or another. Additions, correctionsand further references are welcome.

Symmetric spectra and symmetric ring spectra were first introduced under this name in the article [33]by Hovey, Shipley and Smith. However, these mathematical concepts had been used before, in particular inseveral papers related to topological Hochschild homology and algebraic K-theory. For example, symmetricring spectra appeared as strictly associative ring spectra in [29, Def. 6.1] and as FSPs defined on spheresin [31, 2.7].

There is a key observation, however, which is due to Jeff Smith and which was essential for the devel-opment of symmetric spectra and related spectra categories. Smith noticed that symmetric ring spectraare the monoids in a category of symmetric spectra which has a smash product and a compatible stablemodel structure. Smith gave the first talks on this subject in 1993. In the fall of 1995, Hovey, Shipleyand Smith started a collaboration in which many remaining issues and in particular the model structureswere worked out. The results first appeared in a joint preprint on the Hopf algebraic topology server(at hopf.math.purdue.edu), the K-theory preprint server (at www.math.uiuc.edu/K-theory/) and the

168 I. BASICS

ArXiv (under math.AT/9801077) in January 1998. This preprint version has a section about symmetricspectra based on topological spaces which did not make it into the published version [33] because the refereerequested that the paper be shortened.

Several of the examples which we gave in Section 1.1 had been around with enough symmetries beforesymmetric spectra were formally introduced. For example, Bokstedt and Waldhausen introduced functorswith smash product, or FSPs for short, in [7], from which symmetric ring spectra are obtained by restrictingto spheres. Eilenberg-Mac Lane spectra (Example 1.14) and monoid ring spectra (Example 3.22) arise inthis way from FSPs and seem to have first appeared in [7] (or already in Gunnarson’s preprint [29] ?).Matrix ring spectra (Example 3.24) were also treated as FSP in [7] [first published reference ?].

Cobordism spectra first appeared as highly structured ring spectra in the form of as ‘I∗-prefunctors’in [54]. I∗-prefunctors are the same as [commutative ?] orthogonal ring spectra, and the underlyingsymmetric ring spectra are what we present in Example 1.16. The construction in Example 8.8 which turnsunitary spectra into orthogonal spectra by looping with the imaginary spheres appears to be new.

The model for the complex topological K-theory spectrum in Example 1.18 is a specialization of a moregeneral construction for C∗-algebras by Joachim and Stolz [37]. Earlier, Joachim had given a different modelfor real topological K-theory as a commutative symmetric ring spectrum in [36].

Waldhausen notes on p. 330 of [82] that the iterated S·-construction defines a (sequential) spectrumwhich is an Ω-spectrum from level 1 upwards. Waldhausen’s construction predates symmetric spectra,and it was later noticed by Hesselholt [26, Appendix] that iterating the S·-construction in fact has allthe symmetries needed to form a symmetric spectrum. Moreover, bi-exact pairings of input data yieldsmultiplications of associated K-theory spectra. Our treatment of the algebraic K-theory spectrum onExample 3.29 follows very closely the Appendix of [26].

Free and semifree symmetric spectra, suspensions, loop and shifts of symmetric spectra were firstdiscussed in the original paper [33] of Hovey, Shipley and Smith.

The particular method for inverting a homotopy elements in a symmetric ring spectrum described inExamples 3.26, 3.27 and 6.27 seems to be new. The construction of Example 3.28 for adjoining roots ofunity to a symmetric ring spectrum is due to Schwanzl, Vogt and Waldhausen [65]. They originally wroteup the construction in the context of S-modules, but their argument only needs that one can form monoidrings and invert homotopy elements within the given framework of commutative ring spectra. So as soonas these constructions are available, their argument carries over to symmetric ring spectra.

I learned the model of the periodic complex cobordism spectrum MUP given in Example 8.9 fromMorten Brun, who adapted a construction of Strickland [76, Appendix] from ‘complex S-modules’ tounitary spectra.

The smash product of symmetric spectra was defined by Hovey, Shipley and Smith in their originalpaper [33]. However, their exposition of the smash product differs substantially from ours. Hovey, Shipleyand Smith use the category of symmetric sequences (sequences of pointed spaces Xn, for n ≥ 0, with Σn-action on Xn) as an intermediate step towards symmetric spectra and in the construction of the smashproduct, compare Exercise 8.17. I chose to present the smash product of symmetric spectra in a differentway because I want to highlight its property as the universal target for bimorphisms.

The M-action on the homotopy groups of symmetric spectra was first studied systematically by theauthor in [67]. However, several results related to theM-action on homotopy groups are already contained,mostly implicitly, in the papers [33] and [71]. The definition of semistable symmetric spectra and thecharacterizations (ii)-(v) of Theorem 7.27 appear in Section of [33]; the criterion of trivial M-action onhomotopy groups (Theorem 7.27 (i)) first appears in [67]. I owe the proof of Lemma 7.6 to Neil Strickland.

Multiplications on the ‘classical’ homotopy groups of a symmetric ring spectrum have not previouslybeen discussed in the literature. One reason may be that the naive approach to defining a product on thehomotopy groups of a symmetric ring spectrum does not always work because it is not in general compatiblewith stabilization. The preferred way to bypass this issue has so far been to consider ‘derived homotopygroups’, i.e., to redefine homotopy groups as the classical homotopy groups of a stably fibrant replacement(which we will discuss in the next chapter). In this approach, a symmetric spectrum is replaced by anΩ-spectrum, which is in particular semistable, and thus has a well-defined multiplication.


The idea to construct various spectra from the Thom spectrum MU by killing a regular sequence andpossibly inverting an element (see Example 6.32) is taken from Chapter V of [24] where this process iscarried out in the world of S-modules. This strategy had previously been adapted to symmetric spectra inWeiner’s Diplomarbeit [83].

The original construction of the Brown-Peterson spectrum in the paper [13] by Brown and Petersonwas quite different. They constructed a spectrum whose mod-p homology realizes a certain polynomialsubalgebra of the dual Steenrod algebra. Later Quillen gave a construction of the spectrum BP using thetheory of formal groups, and Quillen’s approach is still at the heart of most current applications of BP .Quillen used p-typical formal groups to produce an idempotent endomorphism e : MU(p) −→ MU(p) ofthe p-localization of MU in the stable homotopy category (see Section II) which is even a homomorphismof homotopy ring spectra (see Section II.3 below). The ‘image’ of this idempotent is isomorphic, in thestable homotopy category, to the spectrum BP , and Quillen’s construction produces it as a homotopy ringspectrum. Part II of Adams’ notes [2] are a good exposition of Quillen’s results in this area. [originalpaper?]

The spectral sequence of Theorem 7.41 was first constructed by Shipley in [71], by completely differentmethod; more precisely, Shipley obtains a spectral sequence with isomorphic E2-term and isomorphicabutment, and so it seems very likely that the spectral sequences are isomorphic. Shipley constructs aspectral sequence of the form

E2p,q = colimp

I (πsqX) =⇒ πp+qDX

which also converges to the true homotopy groups of a symmetric spectrum X and whose E2-term dependson the I-functor πs

qX of stable homotopy groups of the spectrum X. Here D is Shipley’s detection func-tor [71, Def. 3.1.1]. The proof that Shipley’s E2-term (the derived functors of colimit) is isomorphic to theTor groups which arise as the E2-term in Theorem 7.41 was given by the author in [67].

The argument used in Lemma 1.24 to reduce the lifting property to a set of morphisms with boundedcardinality is taken from [33, Lemma 5.1.4 (6)] and ultimately goes back to Bousfield, who used it in [9] toestablish a ‘local’ model structure for simplicial sets with respect to a homology theory.

The category of Γ-spaces was introduced by Segal [70], who showed that it has a homotopy categoryequivalent to the usual homotopy category of connective spectra. The category we denote Γ is reallyequivalent to the opposite of Segal’s category Γ, so that covariant functors from Γ are ‘the same’ ascontravariant from Γ. Bousfield and Friedlander [12] considered a bigger category of Γ-spaces in which theones introduced by Segal appeared as the special Γ-spaces. Their category admits a closed simplicial modelcategory structure with a notion of stable weak equivalences giving rise again to the homotopy categoryof connective spectra. Then Lydakis [45] showed that Γ-spaces admit internal function objects and asymmetric monoidal smash product with good homotopical properties.

After the discovery of smash products and compatible model structures for Γ-spaces and symmetricspectra it became obvious that variations of this theme are possible. Simplicial functors were first used forthe purposes of describing stable homotopy types by Bokstedt and Waldhausen when they introduced ‘FSPs’in [7]. Various model structures and the smash product of simplicial functors were systematically studiedby Lydakis in [46]. The paper [51] contains a systematic study of ‘diagram spectra’, their model structuresand smash products, which includes symmetric spectra, Γ-spaces and simplicial functors. Here orthogonalspectra and continuous functors (defined on finite CW-complexes) make their first explicit appearance. Thecategory of S-modules is very different in flavor from the categories diagram spectra, and it is defined andstudied in the monograph [24].

CHAPTER II

The stable homotopy category

1. The stable homotopy category

Now we come to the main definition of this chapter, introducing the stable homotopy category. Forthis definition we choose for each symmetric spectrum A a stable equivalence pA : A −→ ωA with targetan injective Ω-spectrum, which is possible by combining Propositions I.4.27 and I.4.6. We insist that if Ais already an injective Ω-spectrum, then ωA = A and pA is the identity. This is not really necessary, butwill implify some arguments. The following definition depends on these choices, but only very slighty, aswe explain in Remark 1.2 below.

Definition 1.1. The stable homotopy category SHC has as objects all symmetric spectra of simplicialsets. For two such spectra, the morphisms from X to Y in SHC are given by [X,ωY ], the set of homotopyclasses of spectrum morphisms from X to the chosen replacement ωY . If f : X −→ ωY is a homomorphismof symmetric spectra we denote by [f ] : X −→ Y its homotopy class, considered as a morphism in SHC.

Composition in the stable homotopy category is defined as follows. Let f : X −→ ωY and g : Y −→ ωZbe morphism of symmetric spectra which represent morphism from X to Y respectively from Y to Z inSHC. Then there is a morphism g : ωY −→ ωZ, of symmetric spectra, unique up to homotopy, such thatg pY is homotopic to g. The composite of [f ] ∈ SHC(X,Y ) and [g] ∈ SHC(Y, Z) is then defined by

[g] [f ] = [g f ] ∈ SHC(X,Z) .

There are a few things to check so that Definition 1.1 makes sense. To see that composition in thestable homotopy category is associative we consider four symmetric spectra X,Y, Z and W and threehomomorphisms f : X −→ ωY , g : Y −→ ωZ and h : Z −→ ωW of symmetric spectra. We also pickhomomorphisms g : ωY −→ ωZ and h : ωZ −→ ωW such that g pY ' g and h pZ ' h. Then we have

([h][g])[f ] = [h g][f ] = [(h g) f ] = [h (g f)] = [h][g f ] = [h]([g][f ])

where the second equality uses that (h g)pY is homotopic to hg. It is straightforward to check that [pX ],the homotopy class of the chosen stable equivalence pX : X −→ ωX, is a two-sided unit for composition,so pX represents the identity of X in SHC.

Remark 1.2. The definition of the stable homotopy category depends on the unspecified choices ofstable equivalences pX : X −→ ωX with targets injective Ω-spectra. However, if p′X : X −→ ω′X is anothersuch choice, then there is a unique homotopy class of morphisms of symmetric spectra κ : ωX −→ ω′Xsuch that κ pX is homotopic to p′X . By symmetry and uniqueness, κ is a homotopy equivalence. So if weuse p′X instead of pX in the definition of the stable homotopy category, the resulting morphism sets are incanonical bijection. Altogether, the stable homotopy category is independent of the choices up to preferredismorphism of categories which is the identity on objects. This strong uniqueness property is also reflectedin the universal property of the stable homotopy category, see Theorem 1.6 below.

Remark 1.3. The choice pA : A −→ ωA of stable equivalence to an injective Ω-spectrum can infact be made functorially at the pointset level (and not just up to homotopy), since the level equivalentinjective replacement A 7→ Ainj of Proposition I.4.6. and the stably equivalent Ω-spectrum replacement Q ofProposition I.4.27 are both functorial. However, if we want this extra functoriality, we cannot simultaneouslyarrange things so that ωA = A if A is already an injective Ω-spectra. The pointset level functoriality of ωis irrelevant for the current discussion, and so we continue without it.

171

172 II. THE STABLE HOMOTOPY CATEGORY

In Chapter III we will show that the stable equivalences can be complemented by various useful choicesof cofibrations and fibrations, thus arriving at different stable model category structures for symmetricspectra. For one particular choice (the injective stable model structure), every symmetric spectrum iscofibrant and the fibrant objects are precisely the injective Ω-spectra. Moreover, the ‘concrete’ homotopyrelation using homotopies defined on ∆[1]+ ∧ A coincides with the model category theoretic homotopyrelation using abstract cylinder objects. Thus the stable homotopy category as introduced above turns outto be the homotopy category, in the sense of model category theory, with respect to the injective stablemodel structure.

The stable homotopy category comes with a functor γ : Sp −→ SHC from the category of symmetricspectra of simplicial sets which is the identity on objects. For a morphism ϕ : X −→ Y of symmetricspectra we set

γ(ϕ) = [pY ϕ] in SHC(X,Y ) ,

where pY : Y −→ ωY is the chosen stable equivalence. Note that we have γ(pY ) = [pY ] since pωY = Id byconvention. Thus for every morphism f : X −→ ωY of symmetric spectra we have the relation γ(pY ) [f ] =γ(f) as morphisms from X to ωY in the stable homotopy category. Since γ(pY ) = [pY ] is an isomorphismwith inverse [IdωY ], this can also be rewritten as

(1.4) [f ] = γ(pY )−1 γ(f) ∈ SHC(X,Y ) .

In other words, every morphism in the stable homotopy category can be written as a ‘fraction’, i.e., thecomposite of a morphism of symmetric spectra with the inverse of a stable equivalence.

We also note that for morphisms α : W −→ X and f : X −→ ωY we have the relation

(1.5) [f ] γ(α) = [fα] ∈ SHC(W,ωY ) .

Indeed, if f : ωX −→ ωY is such that fpX is homotopic to f , then fpXα is homotopic to fα and so

[f ] γ(α) = [f ] [pX α] = [f pX α] = [fα] .

We now show that γ is indeed a functor and in fact the universal functor which takes stable equivalencesto isomorphisms.

Theorem 1.6. The functor γ : Sp −→ SHC is a localization of the category of symmetric spectra atthe class of stable equivalences. More precisely, we have

(i) The assignment γ : Sp −→ SHC is a functor which takes stable equivalences to isomorphisms.Moreover, a morphism of symmetric spectra is a stable equivalence if and only if the functor γtakes it to an isomorphism in the stable homotopy category.

(ii) For every functor F : Sp −→ C which takes stable equivalences to isomorphisms, there exists aunique functor F : SHC −→ C such that F γ = ε.

(iii) Let τ : F −→ G be a natural transformation between functors F,G : Sp −→ C which takes stableequivalences to isomorphisms. Then there exists a unique natural transformation τ : F −→ Gbetween the induced functors F , G : SHC −→ C such that τ γ = τ . If τ is a natural isomorphism,so is τ .

Proof. (i) First we check functoriality. Since the homotopy class of pY is the identity of Y in SHC,γ preserves identities. For composable morphism of symmetric spectra ϕ : X −→ Y and ψ : Y −→ Z wehave

γ(ψ)γ(ϕ) = [pZ ψ]γ(ϕ) = [pZ (ψϕ)] = γ(ψϕ)

by (1.5). So γ is indeed functorial.Now we show that γ takes stable equivalences between symmetric spectra to isomorphisms in the stable

homotopy category. If ϕ : X −→ Y is any morphism of symmetric spectra, then there exists a morphism

1. THE STABLE HOMOTOPY CATEGORY 173

ω(ϕ) : ωX −→ ωY , unique up to homotopy, such that the square

Xϕ //

pX

Y

pY

ωX

ω(ϕ)// ωY

commutes up to homotopy. In particular, we have γ(ϕ) = [pY ϕ] = [ω(ϕ)pX ]. If ϕ is a stable equivalence,then so is ω(ϕ) (since both pX and pY are stable equivalences). Since ωX and ωY are injective Ω-spectra, thestable equivalence ω(ϕ) is then even a homotopy equivalence [ref], so there exists a morphism of symmetricspectra ψ : ωY −→ ωX such that ψω(ϕ) and ω(ϕ)ψ are homotopic to the respective identity morphisms.

The composite ψpY : Y −→ ωX represents a morphism from Y to X in the stable homotopy category,and we claim that [ψ pY ] is inverse to γ(ϕ). Indeed, we have

γ(ϕ)[ψ pY ] = [pY ϕ][ψ pY ] = [ω(ϕ) pX ][ψ pY ] = [ω(ϕ) ψ pY ] = [pY ] = IdY

and

[ψ pY ]γ(ϕ) = [ψ pY ][ω(ϕ) pX ] = [ψ ω(ϕ) pX ] = [pX ] = IdX .

[show that ϕ is a stable equivalence if and only if γ(ϕ) is an isomorphism in the stable homotopy category](ii) We consider a functor G : SHC −→ C and prove the uniqueness property by showing that G is

completely determined by the composite functor G γ : Sp −→ C. This is clear on objects since γ is theidentity on objects. If f : X −→ ωY is a morphism of symmetric spectra which represents a morphism[f ] : X −→ Y in SHC, then we can apply G to the equation (1.4) and obtain

G([f ]) = (G γ)(pY )−1 (G γ)(f) .

Thus also the behavior of G on morphisms is determined by the composite G γ.Now we tackle the existence property. Given a functor F : Sp −→ C which takes stable equivalences

to isomorphisms we set F (X) = F (X) on objects. Given a homomorphism f : X −→ ωY , the uniquenessargument tells us that we have to define the value of F on [f ] by

(1.7) F ([f ]) = F (pY )−1 F (f) .

We have to check that this is well-defined and functorial.To see that the assignment (1.7) is well-defined we have to show that the C-morphism F (f) only

depends on the homotopy class of f : X −→ ωY . Indeed, the morphism c : ωY −→ (ωY )∆[1]+ which sendsa point to the ‘constant path’ is a homotopy equivalence, hence a stable equivalence. So by hypothesis,F (c) : F (ωY ) −→ F ((ωY )∆[1]+) is an isomorphism in C. The restriction maps to the two end pointsr0, r1 : (ωY )∆[1]+ −→ ωY satisfy r0 c = IdωY = r1 c, so we have

F (r0) F (c) = IdF (ωY ) = F (r1) F (c) .

Since F (c) is an isomorphism, we deduce F (r0) = F (r1).Now suppose that f, f ′ : X −→ ωY are homotopic morphisms via some homotopyH : X −→ (ωY )∆[1]+ .

Then we have

F (f) = F (r0) F (H) = F (r1) F (H) = F (g) ,

which proves that the formula (1.7) is well-defined.By the various definitions we have

F (IdX) = F ([pX ]) = F (pX)−1 F (pX) = IdF (X)


so F is unital. For associativity we consider two homomorphisms f : X −→ ωY and g : Y −→ ωZ as wellas a homomorphism g : ωY −→ ωZ such that g pY is homotopic to g. Then we have

F ([g] [f ]) = F ([g f ]) = F (pZ)−1 F (g f)

= F (pZ)−1 F (g pY ) F (pY )−1 F (f)

= (F (pZ)−1 F (g)) (F (pY )−1 F (f)) = F (g) F (f)

where we used functoriality of F and homotopy invariance of F . Thus F is a functor.Finally, we have to check the relation F γ = F . On objects this holds by definition. For a homomor-

phism ϕ : X −→ Y of symmetric spectra we have

F (γ(ϕ)) = F ([pY ϕ]) = F (pY )−1 F (pY ϕ) = F (ϕ) ,

which finishes the proof.(iii)

As we say in (1.4), every morphism from X to Y in the stable homotopy category allows a presentationas a ‘left fraction’

[f ] = γ(pY )−1 γ(f) .We now explain that we can similarly express morphisms in SHC as ‘right fractions’. Suppose we are givena triple (Z, f, g) where Z is a symmetric spectrum, f : Z −→ X a stable equivalence and g : Z −→ Y anymorphism. We get a morphism from X to Y in the stable homotopy category as

γ(g) γ(f)−1 ∈ SHC(X,Y ) .

The triples (Z, f, g) as above are the objects of a category X,Y in which a morphism ϕ : (Z, f, g) −→(Z ′, f ′, g′) is a spectrum morphism ϕ : Z −→ Z ′ satisfying f ′ϕ = f and g′ϕ = g. The path componentsπ0X,Y of this category were studied in Proposition I.4.54. [well-defined on components]

Proposition 1.8. For every pair of symmetric spectra X and Y the map

π0X,Y −→ SHC(X,Y ) , [Z, f, g] 7→ γ(g) γ(f)−1

is bijective.

Proof. Both sides of the map takes stable equivalences in the variable Y to bijections, so we canassume without loss of generality that Y is an injective Ω-spectrum. In that case the set SHC(X,Y ) equalsthe set [X,Y ] of homotopy classes of spectrum morphisms and we claim that the map π0X,Y −→ [X,Y ]above is the inverse of the bijection l : [X,Y ] −→ π0X,Y discussed in Proposition I.4.54. Indeed, themap l sends the homotopy class of a morphism g : X −→ Y to the path component of the triple (X, Id, g),and we have γ(g) = [g].

The stable homotopy category does not have general limits and colimits, but it has coproducts andproducts of arbitrary size:

Proposition 1.9. (i) The stable homotopy category has coproducts and the functor γ : Sp −→SHC preserves coproducts.

(ii) The stable homotopy category has products. The functor γ : Sp −→ SHC preserves finite products.(iii) For every finite family of symmetric spectra the canonical morphism in the stable homotopy cate-

gory from the coproduct to the product is an isomorphism.

Proof. (i) We show that γ preserves arbitrary coproducts. Since every object of SHC is in the imageof γ, this in particular shows that coproducts exist in the stable homotopy category.

Let Xii∈I be a family of symmetric spectra. Let αj : Xj −→∨i∈I X

i denote the inclusion of the jthwedge summand. We then have to show that for every symmetric spectrum Y the map

SHC(∨i∈I

Xi, Y ) −→∏i∈ISHC(Xi, Y ) , [f ] 7−→

([f ] γ(αi)

)i∈I


is a bijection. By definition of morphisms in SHC this amounts to verifying that the map

[∨i∈I

Xi, ωY ] −→∏i∈I

[Xi, ωY ] , [f ] 7−→([fαi]

)i∈I

between sets of homotopy classes of morphisms is a bijection, where we used (1.5). This last claim is clearsince a morphism out of a wedge corresponds bijectively to a familiy of morphisms from each summand,and similarly for homotopies.

(ii) We start by constructing products in the stable homotopy category, which are in general not givenby the product as symmetra spectra. Let Y ii∈I be a family of symmetric spectra with chosen stableequivalences pY i : Y i −→ ω(Y i). We form the product of the injective Ω-spectra ω(Y i); for each indexj ∈ I this symmetric spectrum comes with a projection

πj :∏i∈I

ω(Y i) −→ ω(Yj)

which represent a morphism [πj ] in SHC from∏i∈I ω(Y i) to Yj . We claim that these morphisms make∏

i∈I ω(Y i) into a product, in the stable homotopy category, of the family Y ii∈I .To see this we have to show that for every symmetric spectrum X the map

SHC(X,∏i∈I

ω(Y i)) −→∏i∈ISHC(X,Y i) , [f ] 7−→

([πi] [f ]

)i∈I

is a bijection. Since ω(Y i) is an injective Ω-spectrum for all i ∈ I, so is the product. So by our convention,ω(∏

i∈I ω(Y i))

=∏i∈I ω(Y i) and the morphism pQ

i∈I ω(Y i) is the identity. By definition of morphisms inSHC we thus have to verifying that the map

[X,∏i∈I

ω(Y i)] −→∏i∈I

[X,ω(Y i)] , [f ] 7−→([πi f ]

)i∈I

between sets of homotopy classes of morphisms is a bijection, where we used (1.5). This is again clear sincea morphism to a product corresponds bijectively to a familiy of morphisms to each factor, and similarly forhomotopies.

It remains to show that γ preserves finite products. For ease of notation we treat the case of two factorsY and Y ′; the general case then follows by induction on the number of factors. Since ωY and ωY ′ areinjective Ω-spectra, so is their product. There is thus a morphism ψ : ω (Y × Y ′) −→ (ωY )× (ωY ′), uniqueup to homotopy, such that ψ pY×Y ′ : Y ×Y ′ −→ (ωY )× (ωY ′) is homotopic to pY × pY ′ . The morphismspY × pY ′ : Y × Y ′ −→ (ωY )× (ωY ′) (by Proposition I.4.22 (ii)) and pY×Y ′ are stable equivalences, henceso is ψ. As a stable equivalence between injective Ω-spectra, ψ is thus a homotopy equivalence. The map

SHC(X,Y × Y ′) −→ SHC(X,Y )× SHC(X,Y ′) , [f ] 7−→([πY ] [f ], [πY ′ ] [f ]

)equals the composite

[X,ω(Y × Y ′)] [X,ψ]−−−→ [X, (ωY )× (ωY ′)]([X,πωY ],[X,πωY ′ ])−−−−−−−−−−−−→ [X,ωY ]× [X,ωY ′]

The first map is a bijection since ψ is a homotopy equivalence. The second map is a bijection since amorphism to a product corresponds bijectively to a pair of morphisms to each factor, and similarly forhomotopies.

(iii) It suffices to consider two factors. The canonical morphism κ : X ∨ Y −→ X × Y of symmetricspectra is a π∗-isomorphism by part (iii) of Corollary I.2.20, thus a stable equivalence, and so the morphismγ(κ) : γ(X ∨ Y ) −→ γ(X × Y ) is an isomorphism is the stable homotopy category. Since γ preservescoproducts and finite product, γ(κ) is the canonical morphism, in the stable homotopy category, from acoproduct of γ(X) and γ(Y ) to a product. This proves the claim since every object in SHC is in the imageof γ.

The next thing we show is that the stable homotopy category is additive, i.e., there is a natural com-mutative groups structure on the homomorphism sets such that composition is biadditive. This definitionmakes it sound as if ‘additive category’ is extra structure on a category (namely the addition on morphism


sets), but in fact, ‘additive category’ is really a property of a category (namely having finite sums whichare isomorphic to products). So we present the construction of the addition on hom-sets in this generality;in Remark 1.13 below we make the addition on the stable homotopy category a bit more explicit.

Construction 1.10. Let C be a category which has finite products and a zero object. Suppose furtherthat ‘finite products are coproducts’; more precisely, assume that for every pair of objects A and B themorphisms i1 = (Id, 0) : A −→ A × B and i2 = (0, Id) : B −→ A × B make A × B into a co-product of Aand B, where ‘0’ is the unique morphism which factors through a zero object. In other words, we demandthat for every object X the map

C(A×B,X) −→ C(A,X)× C(B,X) , f 7→ (fi1, fi2)

is a bijection.In this situation we can define a binary operation on the morphism set C(A,X) for every pair of objects

A and X. Given morphisms a, b : A −→ X we let a⊥b : A × A −→ X be the unique morphism such that(a⊥b)i1 = a and (a⊥b)i2 = b. Then we define a+ b : A −→ X as (a⊥b)∆ where ∆ = (Id, Id) : A −→ A×Ais the diagonal morphism.

Proposition 1.11. Let C be a category which has finite products and a zero object, and in which ‘finiteproducts are coproducts’ in the sense of Construction 1.10. Then for every pair of objects A and X of C thebinary operation + makes the set C(A,X) of morphisms into an abelian monoid. The neutral element is theclass of the zero morphism. Moreover, the group structure is natural for all morphisms in both variables,or, equivalently, composition is biadditive.

If, moreover, the morphism i1⊥∆ : A× A −→ A× A is an isomorphism, then the abelian monoid hasadditive inverse, i.e., is an abelian group.

First proof. The proof is lengthy, but quite formal. For the associativity of ‘+’ we consider threemorphisms a, b, c : A −→ X. Then a+ (b+ c) respectively (a+ b) + c are the two outer composites aroundthe diagram

A∆

vvmmmmmmmmm∆

((QQQQQQQQQ

A×AId×∆

A×A∆×Id

A× (A×A)

a⊥(b⊥c) ((PPPPPPPPP(A×A)×A

(a⊥b)⊥cvvnnnnnnnnn

X

If we fill in the canonical associativity isomorphism A × (A × A) ∼= (A × A) × A then the upper part ofthe diagram commutes because the diagonal morphism is coassociative. The lower triangle then commutessince the two morphisms a⊥(b⊥c), (a⊥b)⊥c : A× (A× A) −→ X have the same ‘restrictions’, namely a, brespectively c.

The commutativity is a consequence of two elementary facts: first, b⊥a = (a⊥b)τ where τ : A×A −→A × A is the automorphism which interchanges the two factors; this follows from τi1 = i2 and τi2 = i1.Second, the diagonal morphism is cocommutative, i.e., τ∆ = ∆ : A −→ A×A. Altogether we get

a+ b = (a⊥b)∆ = (a⊥b)τ∆ = (b⊥a)∆ = b+ a .

As before we denote by 0 ∈ C(A,X) the unique morphism which factors through a zero object. Then wehave a⊥0 = ap1 in C(A × A,X) where p1 : A × A −→ A is the projection onto the first factor. Hencea+ 0 = (a⊥0)∆ = ap1∆ = a; by commutativity we also have 0 + a = a.


Now we verify naturality of the addition on C(A,X) in A and X. To check (a + b)c = ac + bc fora, b : A −→ X and c : A′ −→ A we consider the commutative diagram

A′

∆

c // A∆ //

∆

A×A

a⊥b

A′ ×A′

ac⊥bc

33c×c // A×A a⊥b // X

in which the composite through the upper right corner is (a+ b)c. We have (a⊥b)(c× c)i1 = (a⊥b)(c, 0) =ac = (ac⊥bc)ı1 and similarly for i2 instead i1. So (a⊥b)(c × c) = ac⊥bc since both sides have the same‘restrictions’ to the two factors of A′ × A′. Since the composite through the lower left corner is ac + bc,we have shown (a + b)c = ac + bc. Naturality in X is even easier. For a morphism d : X −→ Y we haved(a⊥b) = da⊥db : A × A −→ Y since both sides have the same ‘restrictions’ da respectively db to the twofactors of A×A. Thus d(a+ b) = da+ db by the definition of ‘+’.

Now we know that the addition makes the set C(A,X) into an abelian monoid, and it remains to showthat additive inverses exist. An arbitrary abelian monoid M has additive inverses if and only if the map

M2 −→ M2 , (a, b) 7−→ (a, a+ b)

is bijective. Indeed, the inverse of a ∈ A is the second component of the preimage of (a, 0). For the abelianmonoid C(A,X) the square

C(A×A,X)(i1⊥∆)∗ //

(i∗1 ,i∗2) ∼=

C(A×A,X)

(i∗1 ,i∗2)∼=

C(A,X)2

(a,b) 7→(a,a+b)// C(A,X)2

by definition and both vertical maps are bijective. Since i1⊥∆ is an isomorphism, the upper map is bijective,hence so is the lower map, and so the monoid C(A,X) has inverses.

Corollary 1.12. The stable homotopy category SHC is an additive category.

Proof. We apply Proposition 1.11 to the stable homotopy category. Finite coproducts and productsexist and are isomorphic by Proposition 1.9. So we have to verify that for every symmetric spectrum A themorphism i1⊥∆ : A× A −→ A× A is an isomorphism in the stable homotopy category. For every integerk the composite map

πkA⊕ πkA −→ πk(A×A)πk(i1⊥∆)−−−−−−→ πk(A×A) −→ πkA× πkA

it sends (a, b) to (a+ b, b) where the first and last maps are the canonical ones. The composite map is anisomorphism since πkA is a group, i.e., has additive inverses. Since canonical maps are isomorphisms, so isthe middle map, and so i1⊥∆ is π∗-isomorphisms, thus a stable equivalence.

Remark 1.13. As we have just seen, the stable homotopy category is additive. However, the binaryoperation ‘+’ on the morphism sets in SHC was defined in a rather abstract fashion, and we want to makeit more explicit.

We let X be an injective Ω-spectrum. The wedge inclusion ι : X∨X −→ X×X (which is the canonicalmorphism in Sp from coproduct to product) is a stable equivalence (see Proposition I.4.22 (ii)). Thus theinduced map

[ι,X] : [X ×X,X] −→ [X ∨X,X]on homotopy classes of morphisms is a bijection. So there exists a morphism m : X × X −→ X, uniqueup to homotopy, such that mι : X ∨ X −→ X equals the fold map (which is the identity on each wedgesummand). Given any other symmetric spectrum A and two morphisms f, g : A −→ X we define a newmorphism f+g : A −→ X as f+g = m(f, g). This construction is well-defined on homotopy classes and we


claim that induced construction on the set [A,X] of homotopy classes of morphisms from A to X coincideswith the operation ‘+’ defined in Construction 1.10 in the case of the stable homotopy category.

Indeed, if A and B are symmetric spectra, then morphisms from A to B in the stable homotopycategory are defined as [A,ωB], the set of homotopy classes of morphisms from A to the chosen stablyequivalent replacement. Given morphisms of symmetric spectra f, g : A −→ ωB, then the compositem(f × g) : A×A −→ ωB satisfies

(m(f × g)) i1 = m(f, 0) ' f

and (m(f × g)) i2 is similarly homotopic to g. So the homotopy class [m(f × g)] ∈ [A × A,ωB] equals[f ]⊥[g], and thus

[f ] + [g] = ([f ]⊥[g])∆ = [m(f × g)∆] = [m(f, g)] .

The operation which sends a pair of morphisms f, g : A −→ X to m(f, g) is is not associative orcommutative on the pointset level, but by Proposition 1.11 the induced operation ‘+’ on the set [A,X] ofhomotopy classes of morphisms is an abelian group structure. An equivalent way of saying this is that themorphism m is associative, commutative and unital up to homotopy, and has a homotopy inverse.

The additivity of the stable homotopy category is a fundamental result which deserves two differentproofs.

Second proof. If X is an injective Ω-spectrum then λ∗ : X −→ Ω(shX) is a natural level equivalence,homotopy equivalence (by Proposition ?? (i)) between injective Ω-spectra. So λ∗ induces a homotopyequivalence

map(A, λ∗) : map(A,X) −→ map(A,Ω(shX)) ∼= Ω map(A, shX)

on mapping spaces. Since the target is the simplicial loop space, the loop addition defines a group structureon the set of components π0 map(A,Ω(shX)) which we pull back along the bijection induced by map(A, λ∗)to a natural group structure on π0 map(A,X). Now we show that the natural bijection

[A,X] ∼= π0 map(A,X)

takes the operation ‘+’ to the loop product in the components of the mapping space map(A,X) and we showsimultaneously that the product on the right hand side is abelian. For this we consider the commutativediagram

[A,X]2∼= //

+

(π0 map(A,X))2

loop product

[A×A,X]

(i∗0 ,i∗1)∼=

OO

∼= //

[∆,X]

π0 map(A×A,X)

(i∗0 ,i∗1) ∼=

OO

π0 map(∆,X)

[A,X] ∼=

// π0 map(A,X)

of sets in which all horizontal and the left upper vertical map are bijections. The left vertical compositedefines ‘+’. The right vertical composite coincides with the loop product in π0 map(A,X) since it is ahomomorphism which sends (f, ∗) and (∗, f) to f . Since the group multiplication (π0 map(A,X))2 −→π0 map(A,X) is a homomorphism of groups, the group π0 map(A,X), and thus [A,X], is abelian.

The circle S1 has a self-map of degree -1 which we can smash with a symmetric spectrum of spaces Xto realize a self-map of S1 ∧X which becomes the additive inverse of the identity morphism in the stablehomotopy category. It will be convenient later to have something analogous for symmetric spectra based onsimplicial sets, but the simplicial circle S1 = ∆[1]/∂∆[1] is too rigid and has no endomorphism of degree -1.So we need to use a simplicial model of the circle which is slightly larger. We define a simplicial set S1 as


a quotient of the horn Λ2[2] (the simplicial subset of ∆[2] generated by d0∆[2] and d1∆[2]) by identifyingthe two ‘outer’ vertices, compare the picture.

•2 • 2

0 •

d1∆[2]AA

identify• 1

d0∆[2]]];;;;;;;

•

e1

??

e0

__

0 ∼ 1

Λ2[2] S1

We use the common image of the two outer vertices in S1 as the basepoint and denote the two non-degenerate 1-simplices of S1 by e0 and e1. There is a unique involution τ : S1 −→ S1 which interchangese0 and e1. The geometric realization of S1 is homeomorphic to a circle and τ realizes a map of degree −1.

Proposition 1.14. Let X be a symmetric spectrum of simplicial sets. Then the localization functorγ : Sp −→ SHC takes the morphism τ ∧ Id : S1 ∧X −→ S1 ∧X to the additive inverse of the identity inthe group SHC(S1 ∧X, S1 ∧X).

Proof. We define three based maps p0, p1,∇ : S1 −→ S1 = ∆[1]/∂∆[1] as follows. The map pi sendsei to the generating 1-simplex of S1 and it sends e1−i to the basepoint, for i = 0, 1. The map ∇ sends e0and e1 to the generating 1-simplex of S1.

We have p1 = p0τ . Both p0 and p1 are weak equivalences, so for every symmetric spectrum of simplicialsets X the morphisms p0 ∧ Id, p1 ∧ Id : S1 ∧X −→ S1 ∧X are level equivalences. We have ∇ = fold pinchfor the fold map S1 ∨ S1 −→ S1 and the ‘pinch’ map S1 −→ S1 ∨ S1 which sends e0 respectively e1 tothe generating 1-simplex of the first respectively second copy of S1. The spectrum (S1 ∨ S1) ∧ X is acoproduct, thus a product, of two copies of S1 ∧X in the stable homotopy category. The pinch morphismS1 ∧X −→ (S1 ∨ S1) ∧X thus represents the diagonal morphism of in SHC. Hence we have the relation

γ(∇∧X) = γ((fold ∧X) (pinch ∧X) = γ(p0) + γ(p1)

in the group SHC(S1 ∧X,S1 ∧X).The map ∇ factors as the composite

S1 = Λ2[2]/(0 ∼ 1) incl−−→ ∆[2]/d2∆[2] s0−→ ∆[1]/∂∆[1] = S1

where the second map is induced by the morphism s0 : ∆[2] −→ ∆[1] on quotients. Since the simplicial set∆[2]/d2∆[2] is contractible, the spectrum ∆[2]/d2∆[2]∧X is a zero object in the stable homotopy category.Thus the morphism γ(∇∧X) : S1 ∧X −→ S1 ∧X is the zero morphism in the stable homotopy category.This proves γ(p1 ∧X) + γ(p0 ∧X) = 0 in SHC(S1 ∧X,S1 ∧X).

Since we have

γ(p0 ∧X) γ(τ ∧X) = γ((p0τ) ∧X) = γ(p1 ∧X) = −γ(p0 ∧X)

and p0 ∧X is a homotopy equivalence, we conclude that γ(τ∧) = − Id.

Now we discuss some examples in which we can identify morphisms in the stable homotopy categorywith other possibly more familiar expressions. We start by identifying morphisms in SHC out of free spectraFmS

n.The functors of true homotopy groups take stable equivalences to isomorphism (compare Theo-

rem I.4.35), so they factor uniquely through the localization functor γ : Sp −→ SHC. We use the samenotation πk for the induced functor which is defined on the stable homotopy category. In (4.49) of Chapter Iwe have defined the true fundamental class ιmn ∈ πm−n(FnSm) as the image of the naive homotopy class inπm−n(FnSm).

Proposition 1.15. For every symmetric spectrum X and all m,n ≥ 0 the map

(1.16) SHC(FmSn, X) −→ πm−nX , α 7−→ (πm−nα)(ιmn )

is an isomorphism of abelian groups.


Proof. Every symmetric spectrum is isomorphic, in the stable homotopy category, to an injec-tive Ω-spectrum, so we can assume that X is injective Ω-spectrum. In that case the morphism groupSHC(FmSn, X) is the set of homotopy classes of morphisms from FmS

n to X. The free-forgetful adjunc-tion gives a bijection

SHC(FmSn, X) ∼= [Sn, Xm] = πnXm .

For every Ω-spectrum X the canonical map πnXm −→ πm−nX is bijective [ref.]; since the composite ofthe adjunction and the stabilization map equals evaluation at the fundamental class, this prove that theevaluation map (1.16) is bijective. [check additivity]

Example 1.17. Let K be a based simplicial set and X and Ω-spectrum. We claim that then the map

γ : [K,X0]sS ∼= [Σ∞K,X]Sp −→ SHC(Σ∞K,X)

given by the localization functor γ : Sp −→ SHC is bijective. Ever stable equivalence between Ω-spectra isa level equivalence, so both sides of the map take stable equivalence to bijection. We can thus assume thatX is an injective Ω-spectrum, in which case the map γ : [Σ∞K,X]Sp −→ SHC(Σ∞, X) is the identity bydefinition of morphisms in SHC.

Example 1.18. If K is a finite pointed simplicial set and X a semistable symmetric spectrum which islevelwise Kan, we define a natural bijection between SHC(Σ∞K,X) and the colimit of the homotopy sets[K ∧ Sn, Xn]sS formed over the maps

[K ∧ Sn, Xn]−∧S1

−−−−→ [K ∧ Sn+1, Xn ∧ S1][K,σn]−−−−→ [K ∧ Sn+1, Xn+1] .

The map is defined without restrictions on X as follows. A based map of simplicial sets f : K ∧Sn −→Xn is adjoint to a morphism of symmetric spectra f : Fn(K ∧ Sn) −→ X; homotopic maps of simplicialsets are adjoint to homotopic morphisms of spectra, so we can define a map

[K ∧ Sn, Xn]sS −→ SHC(Σ∞K,X)

by sending [f ] to γ(f) γ(Id)−1 where Id : Fn(K ∧ Sn) −→ Σ∞K is adjoint to the identity of (Σ∞K)n =K∧Sn and a stable equivalence by [...] We omit the verification that the maps are compatible as n increases;we then obtain a well-defined natural map

(1.19) colimn [K ∧ Sn, Xn] −→ SHC(Σ∞K,X) .

Now we show that the map (1.19) is bijective whenever X is semistable and levelwise Kan. By Proposi-tion ?? (i) the morphism λ∞X : X −→ R∞X is a π∗-isomorphism, thus a stable equivalence, with target anΩ-spectrum. In the commutative square

colimn [K ∧ Sn, Xn]sS //

colimn [K,ΩnXn]− sS // [K, (R∞X)0]sS

γ

SHC(Σ∞K,X)

SHC(Id,λ∞X )// SHC(Σ∞K,R∞X)

the right vertical map is bijective by Example 1.17, and the lower map is bijective since λ∞X is a stableequivalence. The upper two maps are the adjunction bijection respectively induced from the canonical mapsΩnXn = (RnX)0 −→ (R∞X)0. That right upper map is bijective since K is finite [ref]. This proves thatthe map (1.19) is also bijective.

Example 1.20. We specialize the previous Example 1.18 to various classes of semistable symmetricspectra which we have see in Section I.??. For example, we consider morphisms in SHC from a suspensionsspectrum into shn(HA), then n-fold shift of the Eilenberg-Mac Lane spectrum of an abelian group A,compare Example I.1.14. The pointed simplicial set (shnHA)0 = (HA)n = A⊗ Z[Sn] is an Eilenberg-MacLane space of type (A,n), so it represents reduced simplicial cohomology. More precisely, (HA)n has a

2. TRIANGULATED STRUCTURE 181

fundamental class ι ∈ Hn((HA)n, A) in the n-cohomology group with coefficients in A [define it] such thatfor every other based simplicial set K the evaluation map

[K, (HA)n]sS −→ Hn(K,A) , [f ] 7→ f∗(ι)

is bijective. We can thus get bijections

SHC(Σ∞K, shnHA)γ←−∼= [K, (HA)n]

∼=−−→ Hn(K,A) .

Another example is shk(S(S)) = Σ∞S(Sk), the n-fold shift of the singular complex of the topologicalsphere spectrum. As a special case of the bijection (1.19) we obtain an isomorphism

πks (K) = colimn [K ∧ Sn, Sk+n] −→ SHC(Σ∞K,Σ∞S(Sk))

from the k-th stable cohomotopy group of K to the set of morphisms from the suspensions spectrum of Kto the shifted sphere spectrum.

[also SHC(Σ∞K,KU) ∼= K0(|K|) and SHC(Σ∞K,KO) ∼= KO0(|K|) — the symbol K is unfortunatehere...]

2. Triangulated structure

We have seen that the stable homotopy category is an additive category with products and coproducts.In this section we make the stable homotopy category into a triangulated category. First we recall thedefinition.

Let T be a category equipped with an endofunctor Σ : T −→ T. A triangle in T (with respect to thefunctor Σ) is a triple (f, g, h) of composable morphisms in T such that the target of h is equal to Σ appliedto the source of f . We will often display a triangle in the form

Af−→ B

g−→ Ch−→ ΣA .

A morphism from a triangle (f, g, h) to a triangle (f ′, g′, h′) is a triple of morphisms α : A −→ A′, β : B −→B′ and γ : C −→ C ′ in T such that the diagram

Af //

α

Bg //

β

C

γ

h // ΣA

Σ(α)

A′

f ′// B′

g′// C ′

h′// Σ(A′)

commutes. A morphism of triangles is an isomorphism (i.e., has an inverse morphism) if and only all threecomponents are isomorphisms in T.

Definition 2.1. A triangulated category is an additive category T together with a self-equivalenceΣ : T −→ T and a collection of triangles, called distinguished triangles, which satisfy the following axioms(T1), (T2), (T3) and (T4).

We refer to the equivalence Σ of a triangulated category as the suspension, since that is what it willbe in our main example. In algebraic contexts, this equivalence is often denoted X 7→ X[1] and called the‘shift’.

(T1) For every object X and every zero object 0 the triangle X Id−→ X −→ 0 −→ ΣX is distinguished.Every morphism f is part of a distinguished triangle (f, g, h). Any triangle which is isomorphic to adistinguished triangle is itself distinguished.

(T2) Distinguished triangles can be rotated: if a triangle (f, g, h) is distinguished if and only if thetriangle (g, h,−Σf) is.


(T3) Consider two distinguished triangles (f, g, h) and (f ′, g′, h′). Any pair (α, β) of morphisms sat-isfying βf = f ′α can be extended to a morphism of triangles, i.e., there exists a morphism γ making thefollowing diagram commute

Af //

α

Bg //

β

C

γ

h // ΣA

Σ(α)

A′

f ′// B′

g′// C ′

h′// Σ(A′) .

(T4) [Octahedral axiom] Consider distinguished triangles (f1, g1, h1), (f2, g2, h2) and (f3, g3, h3) suchthat f1 and f2 are composable and f3 = f2f1. Then there exist morphisms x and y such that (x, y,Σ(g1)h2)is a distinguished triangle and the following diagram commutes

(2.2) Af1 // B

f2

g1 // C

x

h1 // ΣA

Af3

// D g3//

g2

Eh3

//

y

ΣA

Σ(f1)

F

h2

F

Σ(g1)h2

h2

// ΣB

ΣBΣ(g1)

// ΣC

These are Verdier’s original axioms [79] of a triangulated category. The axioms above contain someredundancy, as was observed by May [56].

Now we define the triangulated structure for the stable homotopy category. The suspension functorin the stable homotopy category is essentially given by suspension of symmetric spectra. In more detail,we recall that the functor S1 ∧ − of symmetric spectra preserves stable equivalences of symmetric spectraof simplicial sets, and so the composite functor γ (S1 ∧ −) : Sp −→ SHC takes stable equivalences toisomorphism. So by the universal property of the functor γ : Sp −→ SHC (see Theorem 1.6 (ii)), there isa unique functor

Σ : SHC −→ SHC

which satisfies Σ γ = γ (S1 ∧ −). Then Σ is given on objects by ΣX = S1 ∧X, and we claim that thebehavior on morphisms is as follows. If f : X −→ ωY represents a morphism [f ] from X to Y in SHC, then

Σ[f ] = [κ (S1 ∧ f)] ∈ SHC(ΣX,ΣY ),

where κ : S1∧ωY −→ ω(S1∧Y ) is any morphism (uniquely determined up to homotopy) whose compositewith the stable equivalence S1 ∧ pY : S1 ∧ Y −→ S1 ∧ ωY is homotopic to the chosen stable equivalencepS1∧Y : S1 ∧ Y −→ ω(S1 ∧ Y ). Indeed, we have

[κ (S1 ∧ f)] = γ(pS1∧Y )−1 γ(κ) γ(S1 ∧ f) = γ(S1 ∧ pY )−1 γ(S1 ∧ f)

= Σ(γ(pY ))−1 Σ(γ(f)) = Σ(γ(pY )−1γ(f)) = Σ[f ]

where we used the relation (1.4) twice, as well as γ(pS1∧Y ) = γ(κ (S1 ∧ pY )).

Proposition 2.3. The suspension functor Σ : SHC −→ SHC is a self-equivalence of the stable homo-topy category.


Proof. We choose a morphism κ : S1∧ωY −→ ω(S1∧Y ) whose composite with the stable equivalenceS1 ∧ pY : S1 ∧ Y −→ S1 ∧ωY is homotopic to the chosen stable equivalence pS1∧Y : S1 ∧ Y −→ ω(S1 ∧ Y ).Suspension of morphisms from X to Y can be factored as the composite

SHC(X,Y ) = [X,ωY ][X,κ]−−−→ [X,Ω(ω(S1 ∧ Y ))] ∼= [S1 ∧X,ω(S1 ∧ Y )] = SHC(ΣX,ΣY )

where κ : ωY −→ Ω(ω(S1 ∧ Y )) is the adjoint of κ and the second map is the bijection induced onhomotopy classes by the adjunction between S1 ∧ − and Ω. The morphism κ : ωY −→ Ω(ω(S1 ∧ Y )) isa stable equivalence [justify] between injective Ω-spectra, thus a homotopy equivalence, so the first map isbijective. Hence the composite above is bijective, which means that the suspension functor is fully faithful.

For every symmetric spectrum X the injective Ω-spectrum ωX is in particular levelwise Kan, so theadjunction counit S1∧Ω(ωX) −→ ωX is a π∗-isomorphism by Proposition I.??. The homotopy class of thisadjunction counit is thus an isomorphism, in SHC, from Σ(Ω(ωX)) to X. So suspension is also essentiallysurjective on objects of the stable homotopy category.

[Fix the rest] If X is an Ω-spectrum, then so is the shifted spectrum shX. The left adjoint S0.0 toshifting (see Example I.7.20) preserves level equivalences and level injections, so shifting also preserves theproperty of being injective. Moreover, shifting preserves homotopies since sh(∆[1]+ ∧X) = ∆[1]+ ∧ shX.If X is an Ω-spectrum and levelwise Kan, then so is the loop spectrum ΩX, and the functor Ω preservesinjective spectra and the homotopy relation. Moreover, for every Ω-spectrum X the natural map

λ∗ : X −→ Ω(shX)

is a level equivalence, thus a homotopy equivalence by part (i) of Proposition ??. And so on the level ofthe stable homotopy category, Ω and shift are inverse to each other. So we have shown

Proposition 2.4. The shift functor and the functor Ω are quasi-inverse self-equivalences of the stablehomotopy category.

Loops of symmetric spectra are defined levelwise, so we have Ω(shX) = sh(ΩX). We thus have thetwo morphisms λ∗ΩY and Ω(λ∗Y ) from ΩY to Ω2(shY ), and they differ by the involution of Ω2(shY ) = Y S

2

which flips the two coordinates in S2. So we have shown

Lemma 2.5. The relations

λ∗ΩY = −Ω(λ∗Y ) and λ∗shY = − sh(λ∗Y )

hold in the groups [ΩY,Ω2(shY )] respectively [shY,Ω(sh2 Y )].

Now we define the distinguished triangles in the stable homotopy category; they arise from mappingcone sequences as follows. Recall from (2.6) of Chapter I that the mapping cone of a morphism ϕ : X −→ Ybetween symmetric spectra of simplicial sets is defined as

C(ϕ) = (∆[1] ∧X) ∪ϕ Y ,

where the simplicial 1-simplex ∆[1] is pointed by 0, so that ∆[1] ∧X is the simplicial cone of X. The coneconstruction comes with a canonical monomorphism i : Y −→ C(ϕ) and a projection p : C(ϕ) −→ S1 ∧X;the latter sends the image of Y to the basepoint and is the projection ∆[1] −→ S1, smashed with theidentity of X, on the cone.

We recall that the localization functor γ : Sp −→ SHC is the identity on objects and the suspensionfunctor Σ : SHC −→ SHC is given on objects by S1 ∧ −. So by applying γ we obtain a diagram in thestable homotopy category

Xγ(ϕ)−−−→ Y

γ(i)−−→ C(ϕ)γ(p)−−−→ ΣX

which we call the elementary distinguished triangle associated to the morphism ϕ : X −→ Y . A distinguishedtriangle is any diagram

Af−−→ B

g−−→ Ch−−→ ΣA


in the stable homotopy category which is isomorphic to an elementary distinguished triangle, i.e., such thatthere is a morphism ϕ : X −→ Y of symmetric spectra and isomorphisms α : A −→ X, β : B −→ Y andγ : C −→ C(ϕ) in SHC such that the diagram

Af //

α

Bg //

β

C

γ

h // ΣA

Σ(α)

X

γ(ϕ)// Y

γ(i)// C(ϕ)

γ(p)// ΣX

commutes.Now we can state and prove the main result of this section.

Theorem 2.6. The suspension functor and the class of distinguished triangles make the stable homotopycategory into a triangulated category.

Proof. (T1) For every symmetric spectrum X the mapping cone of the identity morphism is sim-plicially contractible, so in particular a zero object in the stable homotopy category. So the elementarydistinguished triangle associated to the identity morphism has the form the triangle X Id−→ X −→ 0 −→ ΣX.

A morphism [f ] from X to Y in the stable homotopy category is represented by a morphism f : X −→ωY of symmetric spectra, where pY : Y −→ ωY is the chosen stable equivalence with target an injectiveΩ-spectrum. We have γ(f) = γ(pY ) [f ] (compare (1.4)) and γ(pY ) is an isomorphism in SHC. The lowerrow of the commutative diagram

X[f ] // Y

γ(pY )

γ(i)γ(pY ) // C(f)γ(p) // ΣX

Xγ(f)

// ωYγ(i)

// C(f)γ(p)

// ΣX

is an elementary distinguished triangle, so it shows that [f ] occurs in a distinguished triangle.It is worth noting that the triangle which we get this way depends on the choice of representing

morphism f , and is thus not natural in the homotopy class [f ]. In fact, if f ′ : X −→ ωY is homotopicto f , then mapping cone C(f) is homotopy equivalent to C(f ′), but any construction of such a homotopyequivalence involves a choice of homotopy between f and f ′. Different choices of homotopies will in generallead to different homotopy classes of homotopy equivalences.

By definition, every triangle isomorphic to a distinguished triangle is itself distinguished.

(T2) We first show that if (f, g, h) is a distinguished triangle, then the triangle (g, h,−Σf) is alsodistinguished. The other implication will be shown in Proposition 2.8 (iii) below.

It suffices to show this for the elementary distinguished triangle associated to a morphism ϕ : X −→ Yof symmetric spectra. The projection p : C(ϕ) −→ S1∧X takes the entire image of Y (under i : Y −→ C(ϕ))to the basepoint, so we can extend it to a morphism

ε : C(iϕ) = CY ∪iϕ C(ϕ) −→ S1 ∧X

by sending the cone of Y to the basepoint. Since ε collapses a contractible spectrum, it is a level equivalence,thus also a stable equivalence, and it is taken by γ to an isomorphism in the stable homotopy category. Wehave a diagram in SHC

Yγ(iϕ) // C(ϕ)

γ(iiϕ )// C(iϕ)

γ(ε)

γ(piϕ )// ΣY

Yγ(iϕ)

// C(ϕ)γ(pϕ)

// ΣX−Σγ(ϕ)

// ΣY


in which the upper row is the elementary distinguished triangle associated to the morphism iϕ : Y −→ C(ϕ)and the lower row is what we are interested in. All vertical morphisms are isomorphisms and the left andmiddle square commute (since we already have ε iiϕ = iϕ as morphisms of symmetric spectra). So if wecan show that the right square commutes, then we can conclude that the triangle (γ(iϕ), γ(pϕ),−Σγ(ϕ)) isdistinguished.

So it remains to show that the morphism piϕ : CY ∪ϕ CX −→ S1 ∧ Y and the composite

CY ∪ϕ CXε−→ S1 ∧X S1∧ϕ−−−→ S1 ∧ Y

become additive inverses in the group SHC(CY ∪ϕ CX,S1 ∧ Y ) after applying the localization functorγ.[...] Both factor through Id∪C(ϕ) : CY ∪ϕ CX −→ CY ∪Y CY , but one then collapses the left, and theother the right copy of CY . By Proposition 1.14, γ takes the cone interchange automorphism τ ∧ IdY ofS ∧ Y = CY ∪Y CY to the additive inverse of the identity.

Before we prove axiom (T3) we supply a lemma that well be needed there.

Lemma 2.7. Consider a square of morphisms of symmetric spectra

Xϕ //

a

Y

b

X ϕ

// Y

which commutes up to homotopy. Then there exists a morphism δ : C(ϕ) −→ C(ϕ) in the stable homotopycategory such that the following diagram of distinguished triangles commutes

Xγ(ϕ) //

γ(a)

Yγ(iϕ) //

γ(b)

C(ϕ)

δ

γ(pϕ) // ΣX

Σ(γ(a))

X

γ(ϕ)// Y

γ(iϕ)// C(ϕ)

γ(pϕ)// ΣX

If moreover a and b are stable equivalences, then the morphism δ is an isomorphism in SHC.

The cone construction is natural enough that if the composite ϕa : X −→ Y equals bϕ ‘on the nose’(and not just up to homotopy), then there is a morphism from C(ϕ) to C(ϕ) which makes the correspondingdiagram commute in the category of symmetric spectra, i.e., before applying the localization functor. If thesquare only commutes up to homotopy, then the morphism δ in SHC will depend on a choice of homotopy.

Proof. We choose a homotopy H : ∆[1]+ ∧ X −→ Y from ϕa to bϕ. We consider the morphism ofsymmetric spectra

Ca ∪H ∪ b : C long(ϕ) = (∆[1] ∧X) ∪X (∆[1]+ ∧X) ∪ϕ Y −→ CX ∪ϕ Y = C(ϕ) .

Note that the source of this morphism is not the mapping cone C(ϕ) but rather a modification C long(ϕ)which is ‘twice as long’; the ‘extra space’ is needed to accommodate the homotopy. A level equivalence(thus stable equivalence) j : C long(ϕ) −→ C(ϕ) is defined as the identity on the cone of X and Y ands+0 ∧ Id : ∆[1]+ ∧X −→ ∆[0]+ ∧X ∼= X on the middle part.


The various morphisms fit into a diagram of symmetric spectra

Yiϕ //

b

ilongϕ &&MMMMMMMMMMMM C(ϕ)

pϕ // S1 ∧X

S1∧a

C long(ϕ)

j'

OO

Ca∪H∪b

Yiϕ

// C(ϕ)pϕ

// S1 ∧ X

which commutes strictly, where ilongϕ is the inclusion of Y into the ‘long cone’. Since j becomes an isomor-

phism in the stable homotopy category, the diagram

X

γ(a)

γ(ϕ) // Y

γ(b)

γ(iϕ) // C(ϕ)γ(pϕ) //

γ(Ca∪H∪b)γ(j)−1

ΣX

Σ(γ(a))

X

γ(ϕ)// Y

γ(iϕ)// C(ϕ)

γ(pϕ)// ΣX

commutes in SHC. In other words, δ = γ(Ca ∪H ∪ b) γ(j)−1 is the required morphism.[δ is iso if a, b are stable equivalences]

(T3) We are given two distinguished triangles (f, g, h) and (f , g, h) and two morphisms α and β in thestable homotopy category satisfying βf = fα. We have to extend this data to a morphism of triangles.If we can solve the problem for isomorphic triangles, then we can also solve it for the original triangles.So we can assume that the triangle (f, g, h) and (f , g, h) are the elementary distinguished triangle arisingfrom morphisms of symmetric spectra ϕ : X −→ Y and ϕ : X −→ Y . We then have to find a morphismγ : C(ϕ) −→ C(ϕ) in the stable homotopy category such that the diagram

Xγ(ϕ) //

α

Yγ(iϕ) //

β

C(ϕ)

γ

γ(pϕ) // ΣX

Σ(α)

X

γ(ϕ)// Y

γ(iϕ)// C(ϕ)

γ(pϕ)// ΣX .

commutes in SHC. We let a : X −→ ωX and b : Y −→ ωY be morphisms of symmetric spectra whichrepresent α and β. We choose a morphism of symmetric spectra Φ : ωX −→ ωY such that Φ pX ishomotopic to pY ϕ. Then Φa represents γ(ϕ)α and bϕ represents βγ(ϕ). Thus in the diagram in Sp

Xϕ //

a

Y

b

ωX

Φ

// ωY

X ϕ//

pX

OO

Y

pY

OO


both squares commute up to homotopy. We apply Lemma 2.7 to both squares and obtain a commutativediagram in the stable homotopy category

Xγ(ϕ) //

γ(a)

Yγ(iϕ) //

γ(b)

C(ϕ)

δ1

γ(pϕ) // ΣX

Σ(γ(a))

ωX

γ(Φ)

// ωYγ(iΦ)

// C(Φ)γ(pΦ)

// Σ(ωX)

Xγ(ϕ)

//

γ(pX)

OO

Yγ(iϕ)

//

γ(pY )

OO

C(ϕ)γ(pϕ)

//

δ2

OO

ΣX

Σ(γ(pX))

OO

The morphisms γ(pX), γ(pY ) and Σ(γ(pX)) are stable equivalences, hence so is δ2 by the last sentenceof Lemma 2.7. Moreover, we have α = γ(pX)−1γ(a) and β = γ(pY )−1γ(b), so the morphism δ−1

2 δ1 :C(ϕ) −→ C(ϕ) is the required morphism.

Before we verify the octahedral axiom (T4) we draw some consequences of the other axioms which willbe used many times in the sequel. Part (iii) also proves the remaining half of axiom (T2).

Proposition 2.8. Let T be an additive category equipped with a self-equivalence Σ : T −→ T and aclass of distinguished triangles for which axioms (T1), one half of (T2) and (T3) hold. Then the followingproperties hold.

(i) For every distinguished triangle (f, g, h) and every object X of T , the two sequences of abeliangroups

T (X,A)T (X,f)−−−−−→ T (X,B)

T (X,g)−−−−→ T (X,C)T (X,h)−−−−−→ T (X,ΣA)

and

T (ΣA,X)T (h,X)−−−−−→ T (C,X)

T (g,X)−−−−−→ T (B,X)T (f,X)−−−−−→ T (A,X)

are exact.(ii) Let (α, β, γ) be a morphism of distinguished triangles. If two out of the three morphisms are

isomorphisms, then so is the third.(iii) Let (f, g, h) be a triangle such that the triangle (g, h,−Σf) is distinguished. Then the triangle

(f, g, h) is distinguished.

Proof. (i) We start by showing exactness of the first sequence at T (X,B). By (T3) applied to thepair (Id, f) there is a (necessarily unique) morphism from any zero object to C such that the diagram

AId // A //

f

0

// A[1]

Af

// B g// C

h// A[1]

commutes (the top row is distinguished by (T1 a)). So gf = 0 and thus the image of T (X, f) is containedin the kernel of T (X, g) for every object X.

Conversely, let ψ : X −→ B be a morphism in the kernel of T (X, g), i.e., such that gψ = 0. Applying(T3) to the pair (ψ, 0) gives a morphism ϕ : ΣX −→ ΣA such that the diagram

X //

ψ

0 //

ΣX

ϕ

− Id // ΣX

Σψ

B g

// Ch

// ΣA −Σf// ΣB


commutes (both rows are distinguished by (T1 a) and the part of (T2) which we assume). Since shiftingis full, there exists a morphism ϕ : X −→ A such that ϕ = Σ(ϕ), and since shifting is faithful we havefϕ = ψ, so ψ is in the image of T (X, f). Altogether, the first sequence is exact at T (X,B). If we applythis to the triangle (g, h,−Σf) (which is distinguished by the part of (T2) which we assume), we deducethat the first sequence is also exact at T (X,C). Exactness of the second sequence is similar, using part (i)for suitable maps out of the distinguished triangle (f, g, h) and its rotations.

(ii) We first treat the case where α and β are isomorphisms. If X is any object of T we have acommutative diagram

T (X,A)f∗ //

α∗

T (X,B)g∗ //

β∗

T (X,C)h∗ //

γ∗

T (X,ΣA)(−Σf)∗ //

Σ(α)∗

T (X,ΣB)

(Σβ)∗

T (X,A′)

f ′∗

// T (X,B′)g′∗

// T (X,C ′)h′∗

// T (X,ΣA′)(−Σ(f ′))∗

// T (X,Σ(B′))

where we write f∗ for T (X, f), etc. The top row is exact by part (ii) applied to the triangles (f, g, h) and(g, h,−Σf), which are distinguished by hypothesis, respectively axiom (T2). Similarly, the bottom row isexact. Since α and β (and hence Σ(α) and Σ(β)) are isomorphisms, all vertical maps except possibly themiddle one are isomorphisms of abelian groups. So the five lemma says that γ∗ is an isomorphisms. Sincethis holds for all objects X, the morphism γ : C −→ C ′ is an isomorphism.

If β and γ are isomorphisms, we apply the same argument to the triple (β, γ,Σ(α)). This is a morphismfrom the distinguished (by (T2)) triangle (g, h,−Σ(f)) to the distinguished triangle (g′, h′,−Σ(f ′)). By theabove, Σ(α) is an isomorphism, hence so is α since shifting is an equivalence of categories. The third caseis similar.

(iii) If the triangle (g, h,−Σf) is distinguished, then so is (−Σf,−Σg,−Σh) by two applications of (T2).Axiom (T1) lets us choose a distinguished triangle

Af−→ B

g−→ Ch−→ ΣA

and by three applications of (T2), the triangle (−Σf,−Σg,−Σh) is distinguished. By (T3) there is amorphism γ : ΣC −→ ΣC such that the diagram

ΣA−Σf // ΣB

−Σg // ΣC

γ

−Σh // Σ2A

ΣA −Σf// ΣB −Σg

// ΣC −Σh

// Σ2A

commutes. By part (ii), γ is an isomorphism. Since shifting is an equivalence of categories, we have γ = Σγfor a unique isomorphism γ : C −→ C. Thus (f, g, h) is isomorphic to the distinguished triangle (f, g, h),so it is itself distinguished by axiom (T1).

Now we can approach the last axiom for the stable homotopy category.(T4) We have to consider distinguished triangles (f1, g1, h1), (f2, g2, h2) and (f3, g3, h3) such that f1

and f2 are composable and f3 = f2f1, where the notation is as in diagram (2.2). We have to constructmorphisms x and y such that (x, y,Σ(g1) h2) is a distinguished triangle and the diagram (2.2) commutes.We first reduce to the special case where the three distinguished triangles are elementary distinguished andarise from a pair of composable morphism of symmetric spectra. This reduction requires some care anduses the previously established axioms.

We can assume without loss of generality that the triangle (f1, g1, h1) is the elementary distinguishedtriangle (γ(ϕ), γ(iϕ), γ(pϕ)) associated to a morphism of symmetric spectra ϕ : X −→ Y . The source of thefirst morphism f2 : Y −→ D in the second distinguished triangle (f2, g2, h2) is then the object Y , and wecannot simply replace it by an isomorphic object. Suppose that ψ : Y −→ ωD is a morphism of symmetric


spectra which represents f2. By axiom (T3) we can find a morphism δ : F −→ C(ψ) which makes thediagram

Yf2 // D

g2 //

γ(pD)

F

δ

h2 // ΣY

Yγ(ψ)

// ωDγ(iψ)

// C(ψ)γ(pψ)

// ΣY

commutes, and δ is then an isomorphism by part (ii) of Proposition 2.8. We can thus replace (f2, g2, h2) bythe isomorphic elementary distinguished triangle associated to the morphism ψ : Y −→ ωD. To simplifynotation we rename the target of ψ to Z = ωD.

We then have f3 = γ(ψ) γ(ϕ) = γ(ψϕ) and the triangle (f3, g3, h3) is isomorphic to the elementarydistinguished triangle of the composite ψϕ : X −→ Z, where the isomorphisms atX and Z are the identities;this uses (T3) and Proposition (2.8) (ii) one more time. So we can finally assume that the triangle (f3, g3, h3)is the elementary distinguished triangle of ψϕ.

We are now in the situation of the following commutative diagram, where the two dotted morphismshave to be constructed:

Xγ(ϕ) // Y

γ(ψ)

γ(iϕ) // C(ϕ)

γ(Id∪ψ)

γ(pϕ) // ΣX

Xγ(ψϕ)

// Zγ(iψϕ)

//

γ(iψ)

C(ψϕ)γ(pψϕ)

//

γ(Cϕ∪Id)

ΣX

Σγ(ϕ)

C(ψ)

γ(pψ)

C(ψ)

Σγ(iϕ)γ(pψ)

γ(pψ)// ΣY

ΣYΣγ(iϕ)

// ΣC(ϕ)

The morphism

Id∪ψ : C(ϕ) = CX ∪ϕ Y −→ CX ∪ψϕ Z = C(ψϕ)

which gives rise to one of the fillers above satisfies (Id∪ψ)iϕ = iψϕψ : Y −→ C(ψϕ) as well as pψϕ(Id∪ψ) =pϕ, so the two squares which involve γ(Id∪ψ) commute.

The morphism

Cϕ ∪ Id : C(ψϕ) = CX ∪ψϕ Z −→ CY ∪ψ Z = C(ψ)

which gives rise to the other filler satisfies (Cϕ∪ Id)iψϕ = iψ as well as pψ(Cϕ∪ Id) = (S1∧ϕ)pψϕ. Since wehave γ(S1∧ϕ) = Σγ(ϕ) in the stable homotopy category, the two squares which involve γ(Cϕ∪Id) commuteas well. So it remains to show that the second column is a distinguished triangle. Since Σ(γ(iϕ)) γ(pψ) =γ(S1 ∧ iϕ) γ(pψ) = γ((S1 ∧ iϕ) pψ) this follows from the following Lemma 2.9.

Lemma 2.9. For every pair of composable morphisms of symmetric spectra ϕ : X −→ Y and ψ : Y −→ Zthe image of the sequence

C(ϕ)Id∪ψ−−−→ C(ψϕ)

Cϕ∪Id−−−−→ C(ψ)(S1∧iϕ)pψ−−−−−−−→ S1 ∧ C(ϕ)

under the localization functor γ : Sp −→ SHC is a distinguished triangle.


Proof. The cone construction is natural for morphisms of symmetric spectra, so the commutativesquare in Sp

Yψ //

iϕ

Z

iψϕ

C(ϕ)

Id∪ψ// C(ψϕ)

gives rise to a commutative diagram in the stable homotopy category

Y

γ(iϕ)

γ(ψ) // Z

γ(iψϕ)

γ(iψ) // C(ψ)

∼= γ(C(iϕ)∪iψϕ)

γ(pψ) // ΣY

Σγ(iϕ)

C(ϕ)

γ(Id∪ψ)// C(ψϕ)

γ(iId∪ψ)// C(Id∪ψ)

γ(pId∪ψ)// ΣC(ϕ)

both of whose rows are elementary distinguished triangles. The third vertical morphism is an isomorphism[...], so we can replace the object C(Id∪ψ) in the lower distinguished triangle by C(ψ) while also changingthe morphisms. This give the

C(ϕ)Id∪ψ−−−→ C(ψϕ)

γ((Ciϕ)∪iψϕ)−1γ(iId∪ψ)−−−−−−−−−−−−−−−−−→ C(ψ)(S1∧iϕ)pψ−−−−−−−→ S1 ∧ C(ϕ)

distinguished triangle, which is almost what we are after, except that we need to identify the secondmorphism with the morphism γ(Cϕ ∪ Id) : C(ψϕ) −→ C(ψ).

There is a homotopyH : CX∧J+ −→ C(CX), relative toX, from the morphism iCX : CX −→ C(CX)to C(iX) [...] We glue this homotopy with the constant homotopy on Z to a homotopy

H : C(ψϕ) ∧ J+ = (CX ∪ψϕ Z) ∧ J+ −→ C(CX ∪ϕ Y ) ∪Id∪ψ (CX ∪ψϕ Z) = C(Id∪ψ)

from the morphism iId∪ψ : C(ψϕ) −→ C(Id∪ψ) to the composite (C(iϕ) ∪ iψϕ) (Cϕ ∪ Id). So afterlocalizing we obtain the relation

γ(iId∪ψ) = γ (C(iϕ) ∪ iψϕ) γ(Cϕ ∪ Id)

in the stable homotopy category. Composition with the inverse of γ(C(iϕ) ∪ iψϕ) gives the remainingrelation.

The universal functor γ : Sp −→ SHC commutes with suspensions and takes mapping cone sequencesto distinguished triangles, by definition. The next proposition shows that homotopy fiber sequences alsogive rise to distinguished triangles in the stable homotopy category.

We recall from (2.13) of Chapter I that the homotopy fiber F (ϕ) of a morphism ϕ : X −→ Y betweensymmetric spectra is the pullback in the cartesian square

F (ϕ)

p // X

(∗,ϕ)

Y ∆[1]

(ev0,ev1)// Y × Y

We write ‘elements’ of F (ϕ) as pairs (ω, x) where ω is a path in Y starting at the basepoint and x is a pointin X such that ϕ(x) is the endpoint of ω. There are morphisms

ΩY i−→ F (ϕ)prX−−→ X

ϕ−→ Y

the first two being given by

i(ω) = (ω, ∗) respectively prX(ω, x) = x .

3. DERIVED SMASH PRODUCT 191

The composite pi is the trivial map and the composite ϕp comes with a preferred null-homotopy [specifyit].

Proposition 2.10. (i) Let ϕ : X −→ Y be a level fibration of symmetric spectra of simplicial sets andlet F denote the fiber of ϕ over the basepoint. Denote by c : C(i) −→ Y the morphism from the mappingcone of the inclusion i : F −→ X which is trivial on the cone of F and ϕ on X. Then c is a stableequivalence and the sequence

Fγ(i)−−−−→ X

γ(ϕ)−−−−→ Yγ(p)γ(c)−1

−−−−−−−→ ΣFis a distinguished triangle in the stable homotopy category.(ii) Let ϕ : X −→ Y be a morphism of symmetric spectra. In the context of simplicial sets, suppose alsothat X and Y are levelwise Kan complexes. Then the sequence

F (ϕ)γ(prX)−−−−→ X

γ(ϕ)−−−→ Yγ(p)γ(c)−1

−−−−−−−→ ΣF (ϕ)

to a distinguished triangle in the stable homotopy category.

Proof. (i) The morphism c is a π∗-isomorphism by Corollary 2.19 (iv), thus a stable equivalence. Wehave ci = ϕ as morphisms of symmetric spectra, so the diagram

Fγ(i) // X

γ(i) // C(i)

γ(c)

γ(p) // ΣF (ϕ)

Fγ(i)

// Xγ(ϕ)

// Yγ(p)γ(c)−1

// ΣF (ϕ)

commutes in the stable homotopy category. The upper row is the elementary distinguished triangle associ-ated to ϕ; since all vertical morphisms are isomorphisms, the lower triangle is distinguished.

(ii) This is a special case of (i). Indeed, the morphism ev0 : Y ∆[1] ×Y X −→ Y which takes (ω, x) tothe starting point of ω is a level fibration. Moreover, the strict fiber of ev0 is precisely the homotopy fiberof ϕ. So by (i) the lower row in the diagram

F (ϕ)γ(i) // Y ∆[1] ×Y X

γ(ev0) //

γ(prX)

Yγ(p)γ(c)−1

// ΣF (ϕ)

F (ϕ)γ(prX)

// Xγ(ϕ)

// Yγ(p)γ(c)−1

// ΣF (ϕ)

in SHC is a distinguished triangle. The left square commutes already on the level of morphisms of symmetricspectra, then middle square commutes because ϕ prX : Y ∆[1] ×Y X −→ Y is homotopic, via the pathω, to ev0. Since the vertical morphisms in this diagram are isomorphisms, the upper triangle is alsodistinguished.

3. Derived smash product

The main result of this section is that the pointset level smash product of symmetric spectrum (seeSection I.5) descends to a closed symmetric monoidal product on the stable homotopy category. Recall thatγ : Sp −→ SHC denotes the universal functor from symmetric spectra to the stable homotopy categorywhich inverts stable equivalences (see Theorem 1.6).

Theorem 3.1. The stable homotopy category admits a closed symmetric monoidal product

∧L : SHC × SHC −→ SHC ,called the derived smash product with the sphere spectrum S as unit object. Moreover, there is a naturaltransformation ψA,B : γ(A) ∧L γ(B) −→ γ(A ∧ B) which makes the localization functor γ : Sp −→ SHCinto a lax symmetric monoidal functor and which is an isomorphism whenever A or B is flat.


Before we go into the actual construction of the derived smash product we want to make some commentson what is important here. The construction of the derived smash product is really a largely formalconsequence of the following facts which we have already established:

• there is a symmetric monoidal smash product for symmetric spectra,• the smash product is homotopically well-behaved whenever at least one factor is flat,• every symmetric spectrum is stably equivalent (even level equivalent) to a flat symmetric spectrum.

To make the construction of the derived smash product more transparent we first define it on the fullsubcategory SHC[ of the stable homotopy category whose objects are the flat symmetric spectra. On thissubcategory we can define ∧L on objects by the pointset level smash product, and there is a canonical wayto extend ∧L to morphisms in SHC[. Since there are no choices involved, the coherence properties are thenfairly formal.

Every symmetric spectrum is level equivalent, thus isomorphic in SHC, to a flat symmetric spectrum(compare Proposition I.??), and so the inclusion SHC[ −→ SHC is an equivalence of categories. A choiceof inverse equivalence (which amounts to choices of ‘flat resolutions’) gives us an extension of the derivedsmash product from the category SHC[ to all of SHC, with all necessary coherence for free.

The proof of Proposition 3.4 uses that we can perform a certain ‘calculus of fractions’ entirely insidethe category of flat symmetric spectra. Indeed, Propositions I.4.54 and 1.8 together imply that the map

(3.2) π0X,Y [ −→ SHC(X,Y ) , [Z, f, g] 7→ γ(g) γ(f)−1

is bijective. In particular, every morphism in SHC(X,Y ) is of the form γ(g)γ(ε′)−1 for some flat symmetricspectrum Z, stable equivalence ε : Z −→ X and morphism g : Z −→ Y .

Proposition 3.3. Consider two functors

F,G : (SHC[)n −→ Cof n variables for some n ≥ 1. Then for every natural transformation τ : F (γ[)n −→ G (γ[)n of functors(SpΣ,[)n −→ C there is a unique natural transformation τ : F −→ G such that τ (γ[)n = τ . If τ is anatural isomorphism, so is τ .

Proof. Since the functor γ[ : SpΣ,[ −→ SHC[ is the identity on objects, there can be at most onenatural transformation τ : F −→ G such that τ (γ[)n = τ ; more precisely, at every n-tupel of flatsymmetric spectra X1, X2, . . . , Xn the C-morphism τX1,...,Xn : F (X1, . . . , Xn) −→ G(X1, . . . , Xn) has to beequal to τX1,...,Xn .

The substance of the proposition is in the proof that naturality of τ implies naturality of τ . To shownaturality of τ as a transformation of n variables it suffices to show naturality in each variable separately,while the other n − 1 variables are fixed. We show naturality in the first variable. So we consider flatsymmetric spectra X2, . . . , Xn−1, Y and Y ′ and a morphism α : Y −→ Y ′ in SHC[. If α is of the formα = γ(ϕ) for some morphism of symmetric spectra ϕ : Y −→ Y ′, then the naturality square

F (Y,X2, . . . , Xn)τY,X2,...,Xn //

F (γ(ϕ),Id,...,Id)

G(Y,X2, . . . , Xn)

G(γ(ϕ),Id,...,Id)

F (Y ′, X2, . . . , Xn) τY,X2,...,Xn

// G(Y ′, X2, . . . , Xn)

commutes since τ = τ objectwise and τ is natural. If τ is natural for a particular morphism α in the firstvariable and α is invertible, then τ is also natural for the inverse α−1 in the first variable.

By (3.2) an arbitrary morphism α from Y to Y ′ in SHC[ can be written in the form α = γ(ψ)γ(ϕ)−1

for suitable morphisms ϕ : Z −→ Y and ψ : Z −→ Y between flat symmetric spectra such that ϕ is astable equivalence. By the above, τ is natural for both γ(ψ) and γ(ϕ)−1, so τ is natural for an arbitrarymorphism in the first variable, hence natural in general.

If τ is a natural isomorphism, then we apply the previous argument to the inverse transformation τ−1

and obtain a natural transformation τ−1 satisfying τ−1 (γ[)n = τ−1. Both composites of τ−1 with τ


restrict to identity transformations along (γ[)n, so by the uniqueness, the transformations τ−1 and τ areinverse to each other.

We denote by Sp[ the full subcategory of Sp consisting of flat symmetric spectra; since this category isclosed under smash product and contains the sphere spectrum, it is symmetric monoidal by restriction of allthe structure from the full category of symmetric spectra. We denote by γ[ : Sp[ −→ SHC[ the restrictionof the localization functor to flat symmetric spectra.

Proposition 3.4. There is a functor

∧L : SHC[ × SHC[ −→ SHC[

on the stable homotopy category of flat symmetric spectra which satisfies

∧L (γ[ × γ[) = γ[ ∧

as functors Sp[ × Sp[ −→ SHC[. Moreover, the functor ∧L can be extended to a symmetric monoidalstructure on the category SHC[ in such a way that the functor γ[ : Sp[ −→ SHC[ is strong symmetricmonoidal (with respect to the identity transformation).

Remark 3.5. We leave it as Exercise 11.3 to show that there is only one functor ∧L : SHC[×SHC[ −→SHC[ which satisfies ∧L (γ[×γ[) = γ[ ∧ and there is only one way to extend this functor to a symmetricmonoidal structure on SHC[ for which γ[ : Sp[ −→ SHC[ is strong symmetric monoidal. In other words,on the subcategory SHC[, the derived smash product constructed in Proposition 3.4 is very canonical.

Proof of Proposition 3.4. With start some preparations. If X is a flat symmetric spectrum, thensmashing with X preserves stable equivalences [ref], and so the composite functor γ (X∧−) : Sp −→ SHCtakes stable equivalence to isomorphisms. By the universal property of the functor γ (see Theorem 1.6 (ii))there is a unique functor

lX : SHC −→ SHCsatisfying (lX) γ = γ (X ∧ −). On objects, this functor is then given by (X ∧L −)(Y ) = X ∧ Y , thepointset level smash product of symmetric spectra, and the proof of Theorem 1.6 (ii) reveals how to extendthis definition to morphisms. If Y is another flat symmetric spectrum, then by the same argument there isa unique functor rY : SHC −→ SHC satisfying (rY ) γ = γ (− ∧ Y ).

We show that these two constructions have a certain compatibility in the range where both are defined.More precisely, suppose we are given four flat symmetric spectra X,X ′, Y and Y ′ and morphisms α : X −→X ′ and β : Y −→ Y ′ in SHC[, then we claim the relation

(3.6) (rY ′)(α) (lX)(β) = (lX ′)(β) (rY )(α)

as morphisms from X ∧ Y to X ′ ∧ Y ′ in SHC[. We will see below that this relations is exactly what isneeded to combine the various individual functors into a two-variable functor.

To prove relation (3.6) we go through a sequence of four steps. In Step 1 we suppose that α = γ(a)and β = γ(b) are images of morphisms of symmetric spectra a : X −→ X ′ and b : Y −→ Y ′ under thelocalization functor. Then we have

(rY ′)(α) (lX)(β) = (rY ′)(γ(a)) (lX)(γ(b)) = γ(a ∧ Y ′) γ(X ∧ b) = γ(a ∧ b)= γ(X ′ ∧ b) γ(a ∧ Y ) = (lX ′)(γ(b)) (rY )(γ(a)) = (lX ′)(β) (rY )(α) .

Step 2: if the relation (3.6) holds for a pair of morphisms (α, β) in the stable homotopy category and α(respectively β) is an isomorphism, then the relation also holds for the pair (α−1, β) (respectively (α, β−1)).For example, in the first case we simply take the equation (3.6) for (α, β), composite it from the left with(rY ′)(α)−1 and from the right with (rY )(α)−1 and obtain the relation for (α−1, β).

Step 3: clearly, if α : X −→ X ′ and α′ : X ′ −→ X ′′ are composable morphisms in the stable homotopycategory and the relation (3.6) holds for the pairs (α, β) and (α′, β), then it also holds for the pair (α′α, β),and similarly for composable morphisms in the second variable.


Step 4: by (3.2) the morphisms α and β can be written as α = γ(ψ) γ(ϕ)−1 respectively β =γ(ψ′) γ(ϕ′)−1 where ϕ : Z −→ X, ψ : Z −→ X ′, ϕ′ : Z −→ Y and ψ′ : Z ′ −→ Y ′ are morphisms ofsymmetric spectra, ϕ and ϕ′ are stable equivalences and Z and Z ′ are flat. So steps 1-3 combine to provethe general case.

Now we are ready to define the derived smash product on the stable homotopy category of flat symmetricspectra. The requirement ∧L (γ[ × γ[) = γ[ ∧ forces us to define the derived smash product on pairs ofobjects by the pointset level smash product. For morphisms α : X −→ X ′ and β : Y −→ Y ′ in SHC[ wedefine α ∧L β : X ∧L Y −→ X ′ ∧L Y ′ as the common value of the the equation (3.6). To see that this isindeed a functor we consider further morphisms α′ : X ′ −→ X ′′ and β′ : Y ′ −→ Y ′′ in SHC[ and calculate

(α′α) ∧L (β′β) = (rY ′′)(α′α) (lX)(β′β)

= (rY ′′)(α′) (rY ′′)(α) (lX)(β′β)

= (rY ′′)(α′) (lX ′)(β′β) (rY )(α)

= (rY ′′)(α′) (lX ′)(β′) (lX ′)(β) (rY )(α)

= (α′ ∧L β′) (α ∧L β)

Seeing that the derived smash product preserves identities is clear.For morphisms ϕ : X −→ X ′ and ψ : Y −→ Y ′ of symmetric spectra we have

γ(ϕ) ∧L γ(ψ) = (rY ′)(γ(ϕ)) (lX)(γ(ψ)) = γ(ϕ ∧ Y ′) γ(X ∧ ψ)(3.7)

= γ((ϕ ∧ Y ′) (X ∧ ψ)) = γ(ϕ ∧ ψ) ,

so we have the equality ∧L (γ[ × γ[) = γ[ ∧ as functors.Now we define coherence isomorphisms for the functor ∧L which make it into a symmetric monoidal

structure on the category SHC[. For the unit isomorphism we apply Proposition 3.3 to the functor S∧L− :SHC[ −→ SHC[ and the identity functor of SHC[. The pointset level unit isomorphism

lY : S ∧ Y −→ Y

restricts to a natural isomorphism between the functors (S ∧L −)γ[ and γ[, so there is a unique naturalisomorphism l : (S∧L −)S −→ Id such that lγ = l. We define l as the left unit isomorphism for the derivedsmash product on SHC[. The right unit isomorphism r : (−∧L S) −→ Id is obtained in the same way fromthe right unit isomorphism r : X ∧ S −→ X.

To obtain the derived associativity isomorphism we apply Proposition 3.3 to the functors

∧L (∧L × Id) , ∧L (Id×∧L) : SHC[ × SHC[ × SHC[ −→ SHC[ .

The pointset level associativity isomorphism αX,Y,Z : (X ∧ Y ) ∧ Z −→ X ∧ (Y ∧ Z) restricts to a naturalisomorphism between the functors ∧L (∧L × Id) (γ[)3 and ∧L (Id×∧L) (γ[)3, so there is a uniquenatural isomorphism α : ∧L (∧L × Id) −→ ∧L (Id×∧L), which we define to be the derived associativityisomorphism.

The construction of the derived symmetry isomorphism should now be clear, but we still give it. Thepointset level symmetry isomorphism τX,Y : X ∧ Y −→ Y ∧X restricts to a natural isomorphism betweenthe functors ∧L (γ[)2 and ∧L T (γ[)2, where T : SHC[×SHC[ −→ SHC[×SHC[ is the automorphismwhich interchanges the two factors. So there is a unique natural isomorphism τ : ∧L −→ ∧L T satisfyingτ (γf lat)2 = τ , the derived symmetry isomorphism.

The various coherence conditions at the level of the stable homotopy category SHC[ are now a directconsequence of the corresponding coherence conditions at the level of symmetric spectra and the uniquenessstatement in Proposition 3.3. We treat one case in detail and omit the other cases, which are very similar.


The two composites around the diagram

(3.8)

(X ∧L Y ) ∧L ZαX,Y,Z //

τX,Y ∧LId

X ∧L (Y ∧L Z)τX,Y∧LZ // (Y ∧L Z) ∧L X

αY,Z,X

(Y ∧L X) ∧L Z

αY,X,Z// Y ∧L (X ∧L Z)

Id∧LτX,Z// Y ∧L (Z ∧L X)

are two natural transformations from the functor

∧L (∧L × Id) : SHC[ × SHC[ × SHC[ −→ SHC[

to the functor ∧L (Id×∧L) C where C is the automorphism of SHC[ × SHC[ × SHC[ which cycli-cally permutes the factors. After composition with (γ[)3 the two natural transformations become equalsince the corresponding diagram for the smash product in Sp commutes. So the uniqueness statement inProposition 3.3 guarantees that the diagram (3.8) commutes, and we have verified the coherence betweenassociativity and symmetry isomorphisms.

There is one final claim which we have to check, namely that the identity transformation makes thefunctor γ[ : Sp[ −→ SHC[ into a strong symmetric monoidal functor. However, when we unravel what thismeans, we see that all conditions hold by construction. Indeed, if X and Y are flat symmetric spectra, thenγ(X) ∧L γ(Y ) and γ(X ∧ Y ) are the same object (namely the smash product X ∧ Y ) and for morphismϕ : X −→ X ′ and ψ : Y −→ Y ′ of symmetric spectra the two morphisms γ(ϕ) ∧L γ(ψ) and γ(ϕ ∧ ψ) areequal, see (3.7) above. [what else to check ?]

Now we can extend the derived smash product from the category SHC[ to the entire stable homotopycategory and prove Theorem 3.1.

Proof of Theorem 3.1. For every symmetric spectrum X we choose a flat symmetric spectrum X[

and a stable equivalence X[ −→ X. [fill in details]We have to show that the monoidal structure given by the smash product is closed. For symmetric

spectra Y and Z we define the derived function spectrum by

F (Y, Z) = Hom(Y, ωZ) ,

the symmetric internal function spectrum (see Example I.3.19) from Y to the chosen stably equivalent‘resolution’ ωZ of Z. This spectrum comes with an evaluation morphism

εY,Z : F (Y, Z) ∧ Y = Hom(Y, ωZ) ∧ Y −→ ωZ

of symmetric spectra which is adjoint to the identity of Hom(Y, ωZ); so εY,Z is the morphism correspondingto the bimorphism from (Hom(Y, ωZ), Y ) to ωZ with (p, q)-component

map(Y, shp(ωZ)) ∧ Yq −→ (ωZ)p+q , f ∧ y 7→ fq(y) .

We define a morphism εY,Z : F (Y, Z) ∧L Y −→ Z in the stable homotopy category as the composite

F (Y, Z) ∧L YψF (Y,Z),Y−−−−−−−→ F (Y, Z) ∧ Y [εY,Z ]−−−−−→ Z

where the second morphism is the the homotopy class of εY,Z . We also call εY,Z an evaluation morphism,possibly with adjective ‘derived’ if we need to distinguish it from εY,Z .

Now we show that the derived function spectrum construction is right adjoint to the derived smashproduct. We start by showing that for every triple of objects Y and Z of the stable homotopy category themap

(3.9) SHC(X,F (Y, Z)) −→ SHC(X ∧L Y, Z) , α 7→ εY,Z (α ∧L Y )

is bijective. In order to show this, we first make a reduction argument. The symmetric spectrum F (Y, Z)is already a contravariant functor in Y (but not in Z) on the pointset level, and the map is natural in Y .Moreover, the functor F (−, Z) = Hom(−, ωZ) takes stable equivalences to level equivalences, and thus to


isomorphism in SHC, since ωZ is an injective Ω-spectrum (compare Proposition I.4.19). So we may replaceY by any stably equivalent object and can thus assume without loss of generality that Y is flat.

If Y is flat then by Proposition I.5.37 (v) the function spectrum Hom(Y, ωZ) is injective and by [...] itis an Ω-spectrum. We have a commutative diagram

SHC(X,F (Y, Z)) // SHC(X ∧L Y, Z) [X ∧L Y, ωZ]

[X,Hom(Y, ωZ)]

adjunction// [X ∧ Y, ωZ]

in which all other maps are bijections (some even identities), which proves the (3.9) is bijective.Now still have to justify that F (Y, Z) is a functor in the second variable. A morphism ζ : Z −→ Z ′

gives rise to a map

SHC(X ∧L Y, ζ) : SHC(X ∧L Y, Z) −→ SHC(X ∧L Y, Z ′)

which is natural in X and Y . Combining this with the adjunction bijections (3.9) gives a map

SHC(X,F (Y, Z)) −→ SHC(X,F (Y, Z ′)) ,

still natual in X. By the Yoneda lemma, this transformation is induced by a unique morphism from F (Y, Z)to F (Y, Z ′) in the stable homotopy category, which we define to be F (Y, ζ). By the very construction,F (Y,−) is right adjoint to −∧L Y with respect to the bijection (3.9). A similar representability argumentsshows that altogether we obtain a functor

F : SHCop × SHC −→ SHC .

In Section 5.3 we proved various relations between the pointset level smash product and other con-structions with symmetric spectra. We see now that many of these relation descend to the stable homotopycategory and have analogues for the derived smash product. For a Σm-simplicial set L and a Σn-simplicialset L′ we have a natural isomorphism between semifree symmetric spectrum

(3.10) Gm+n(Σ+m+n ∧Σm×Σn L ∧ L′) ∼= GmL ∧L GnL′ .

Since semifree symmetric spectra are flat, the derived smash product is given by the pointset level smashproduct. Thus the isomorphism is obtained from the corresponding pointset level isomorphism (3.10)of Chapter I by applying the localization functor γ : Sp −→ SHC. As a special case we can considersmash products of free symmetric spectra. If K and K ′ are pointed spaces or simplicial set then we haveFmK = Gm(Σ+

m ∧ K) and FnK′ = Gn(Σ+

n ∧ K ′), so the isomorphism (5.6) specializes to an associative,commutative and unital isomorphism

Fm+n(K ∧K ′) ∼= FmK ∧L FnK ′ .

As the even more special case for m = n = 0 we obtain a a natural isomorphism of suspension spectra

(Σ∞K) ∧L (Σ∞L) ∼= Σ∞(K ∧ L)

for all pairs of pointed spaces (or pointed simplicial sets) K and L.The special case X = Sn of the adjunction bijection (3.9) yields a natural isomorphism between

SHC(Sn, F (Y, Z)) and SHC(Sn ∧L Y, Z). The former group is in turn isomorphic, via evaluation at thefundamental class, to πnF (Y, Z) (see Proposition 1.17). Combining this gives a natural isomorphism

πnF (Y, Z) ∼= SHC(Sn ∧L Y, Z) .

The pointset level composition morphisms (compare I.(5.14))

: Hom(Y, Z) ∧ Hom(X,Y ) −→ Hom(X,Z)


‘pass’ to the stable homotopy category and admit a derived version. Indeed, the composite

(F (Y, Z) ∧L F (X,Y )) ∧L X α−−→ F (Y, Z) ∧L (F (X,Y ) ∧L X)Id∧εX,Y−−−−−→ F (Y, Z) ∧L Y εY,Z−−−→ Z

has an adjoint

(3.11) : F (Y, Z) ∧L F (X,Y ) −→ F (X,Z)

We omit the verification that this ‘derived composition map’ is associative and unital and that it becomescomposition in the category SHC on homotopy groups in the sense that the diagram

π0F (Y, Z) × π0F (X,Y ) · //

∼=

π0

(F (Y, Z) ∧L F (X,Y )

) π0() // π0F (X,Z)

∼=

SHC(Y, Z) × SHC(X,Y ) // SHC(X,Z)

commutes. [is the product · already defined ?]Another useful fact about the derived smash product and its adjoint derived function spectrum is a

compatibility with the triangulated structure. In fact, if we fix an symmetric spectrum X then the functorsX ∧L −, −∧LX, F (X,−) and F (−, X) all all exact functors on the stable homotopy category, as the nextproposition shows.

Definition 3.12. Let T and T ′ be triangulated categories. An exact functor is a pair (F, τ) consistingof an additive functor F : T −→ T ′ and a natural isomorphism τ : F Σ ∼= Σ F such that for everydistinguished triangle (f, g, h) in T the diagram

FAFf−−−→ FB

Fg−−−→ FCτFh−−−→ Σ(FA)

is a distinguished triangle in T ′.

[contravariant functor is exact it exact with respect to the opposite triangulation on T op] [automaticallyadditive ? F (0) = 0 seems to suffice] When the natural isomorphism τ : FΣ ∼= ΣF is understood, we oftenrefers to the functor F an exact functor. One should keep in mind, though, that ‘exact’ is not just aproperty of a functor, but extra structure. A useful fact is that adjoints of exact functors are again exact:

Proposition 3.13. Let T and T ′ be triangulated categories F : T −→ T ′ a functor and τ : F Σ ∼= ΣFa natural isomorphism. Let G : T ′ −→ T be a right adjoint of F with adjunction counit ε : FGX −→ X.Then the adjoint of the natural isomorphism

F (Σ(GX)) τGX−−−→ Σ(FGX) Σε−→ ΣX

is an isomorphism ψ : Σ(GX) −→ G(ΣX) and the pair (G,ψ−1) is an exact functor if and only if the pair(F, τ) is an exact functor.

For symmetric spectra A and Z the associativity isomorphism (S1 ∧ Z) ∧A ∼= S1 ∧ (Z ∧A) gives rise,by applying the localization functor γ : Sp −→ SHC to a natural isomorphism [on flat spectra; derive...]

κZ,A : (ΣZ) ∧L A −→ Σ(Z ∧L A)

in the stable homotopy category. A similar isomorphism

κZ,A : Z ∧L (ΣA) −→ Σ(Z ∧L A)

can then be defined as the composite

Z ∧L (ΣA)τZ,ΣA−−−−→ (ΣA) ∧L Z κA,Z−−−→ Σ(A ∧L Z)

Σ(τA,Z)−−−−−→ Σ(Z ∧L A) .

[isos for F (Z, Y ): the composite

(ΣF (Z,A)) ∧L Z ∼= Σ(F (Z,A) ∧L Z)ΣεZ,A−−−−→ ΣZ


has an adjoint ΣF (A,Z) −→ F (A,ΣZ) which is an isomorphism [...] and the composite

(Σ(F (ΣA,Z)) ∧L A ∼= F (ΣA,Z) ∧L (ΣA)εΣA,Z−−−−→ Z

has an adjoint Σ(F (ΣA,Z)) −→ F (A,Z) which is an isomorphism [...] ...]

Proposition 3.14. Let Z be a symmetric spectrum of simplicial sets and

Af−−→ B

g−−→ Ch−−→ ΣA

a distinguished triangle in the stable homotopy category. Then the four diagrams

Z ∧L A Id∧Lf−−−−−→ Z ∧L B Id∧Lg−−−−−→ Z ∧L C κZ,A(Id∧Lh)−−−−−−−−−→ Σ(Z ∧L A)

A ∧L Z f∧LId−−−−−→ B ∧L Z g∧LId−−−−−→ C ∧L Z κA,Z(h∧LId)−−−−−−−−−→ Σ(A ∧L Z)

F (Z,A)F (Id,f)−−−−−−→ F (Z,B)

F (Id,g)−−−−−−→ F (Z,C)?−1F (Id,h)−−−−−−−−→ ΣF (Z,A)

F (ΣA,Z)F (h,Id)−−−−−→ F (C,Z)

F (g,Id)−−−−−−→ F (B,Z)?−1F (f,Id)−−−−−−−−→ ΣF (ΣA,Z)

are also distinguished triangles.

Proof. We can assume without loss of generality that Z is flat (since every object of SHC is isomor-phic to a flat spectrum) and that the given distinguished triangle is the elementary distinguished triangleassociated to a morphism ϕ : X −→ Y of symmetric spectra. Using the functorial flat resolution ofProposition I.?? we can moreover assume that X and Y are flat. Then the mapping cone C(ϕ) is flatby ???.

We have a diagram in the stable homotopy category

Z ∧L XId∧Lγ(ϕ) // Z ∧L Y

Id∧Lγ(i) // Z ∧L C(ϕ)

γ(j)

κZ,X(Id∧Lγ(p))// Σ(Z ∧L X)

Z ∧Xγ(Id∧ϕ)

// Z ∧ Yγ(i)

// C(Id∧ϕ)γ(p)

// Σ(Z ∧X)

in which the lower row is the elementary distinguished triangle associated to the morphism Id∧ϕ : Z∧X −→Z∧Y . In the upper row, all derived smash products are between pairs of flat spectra, so they are representedby the pointset level smash products. Finally

j : Z ∧ C(ϕ) = Z ∧ (∆[1] ∧X ∪ϕ Y ) −→ ∆[1] ∧ (Z ∧X) ∪Id∧ϕ Z ∧ Y = C(Id∧ϕ)

is the isomorphism which moves Z past the cone coordinate. Since all vertical morphisms in the diagramare isomorphisms and the lower row is distinguished, so is the upper row.

The second case is proved in essentially the same way as the first. Alternatively, the derived twistisomorphisms provide an isomorphism between the first sequence and the second.

Proposition 3.13 makes the third case a formal consequence of the second since the functors F (Z,−) isright adjoint to Z ∧L −.

In the fourth case we argue as follows. Again is suffices to consider an elementary distinguished triangleassociated a morphism ϕ : X −→ Y of symmetric spectra. As usual, ωZ is the stably equivalent ‘resolution’of Z by an injective Ω-spectrum. We consider the homotopy fibre sequences associated to the morphismF (ϕ,Z) = Hom(ϕ, ωZ) from F (Y, Z) = Hom(Y, ωZ) to F (X,Z) = Hom(X,ωZ) and get a distinguishedtriangle [symbol F overused...]

F (F (ϕ,Z))γ(prX)−−−−→ F (Y, Z)

γ(Hom(varphi,ωZ))−−−−−−−−−−−−−→ F (X,Z)γ(Hom(varphi,ωZ))−−−−−−−−−−−−−→ ΣF (F (Y, Z))

by Proposition 2.10 (ii). We apply the functor Hom(−, ωZ) to the mapping cone sequence of ϕ and obtaina homotopy fibre sequence

Hom(X,ωZ)Hom(ϕωZ)←−−−−−−− Hom(Y, ωZ) ←− Hom(C(ϕ), ωZ) ←− Hom(S1 ∧X,ωZ)


Fγ(i) // X

γ(i) // C(i)

γ(c)

γ(p) // ΣF (ϕ)

Fγ(i)

// Xγ(ϕ)

// Yγ(p)γ(c)−1

// ΣF (ϕ)

Remark 3.15. In a triangulated category with a symmetric monoidal product one can formulatemore compatibility conditions between the two structures, as has been done by May in [56, Sec. 4] [andKeller-Neeman?] The stable homotopy category satisfies all ‘reasonable’ conditions of this kind since thetriangulated structure and the derived smash product arise from compatible structures on the level ofsymmetric spectra. We will not be any more precise than this, but refer the reader to Section 5 of [56] forfurther explanations.

3.1. Homotopy ring spectra. Now that we constructed the derived smash product we can considermonoid objects in the stable homotopy category with respect to the derived smash product. For us a homo-topy ring spectrum or ring spectrum up to homotopy is a symmetric spectrum S together with morphismsµ : S ∧L S −→ S and ι : S −→ S in the stable homotopy category which are associative and unital in thesense that the following diagrams commute

(S ∧L S) ∧L SαS,S,S //

µ∧LId

S ∧L (S ∧L S)Id∧Lµ // S ∧L S

µ

S ∧L S

lS&&MMMMMMMMMMMM

ι∧LId // S ∧L Sµ

S ∧L SId∧Lιoo

rSxxqqqqqqqqqqqq

S ∧L S µ// S S

A homotopy ring spectrum (S, µ, ι) is homotopy commutative if the multiplication is unchanged whencomposed with the derived symmetry isomorphism, i.e., if the relation µ τS,S = µ holds in the stablehomotopy category.

The definition of the derived smash product was such that the universal functor γ : Sp −→ SHC is laxsymmetric monoidal (with respect to the universal transformation ψ : ∧L (γ × γ) −→ γ ∧). A formalconsequence is that γ takes symmetric ring spectra to homotopy ring spectra. Indeed, if (R,µ : R ∧R −→R, ι : S −→ R) is a symmetric ring spectrum, then R becomes a ring spectrum up to homotopy with respectto the multiplication map

γ(R) ∧L γ(R)ψR,R−−−→ γ(R ∧R)

γ(µ)−−−→ γ(R)

and the unit map γ(ι) : S −→ R.The converse is far from being true. More precisely, given a ring spectrum up to homotopy S one

can ask if there is a symmetric ring spectrum R such that γ(R) is isomorphic to S as a homotopy ringspectrum. There is an infinite sequence of coherence obstructions for the associativity to get a positiveanswer. [Illustrate the pentagon coherence condition?] The question of when a homotopy commutativehomotopy ring spectrum is represented by a commutative symmetric ring spectrum is even more subtle.We hope to get back to this later, and discuss some of the obstruction theories available to tackle such‘rigidification’ questions.

A concrete example is the mod-pMoore spectrum SZ/p for a prime p ≥ 5. Indeed, SZ/p has a homotopyassociative and homotopy commutative multiplication in the stable homotopy category for p ≥ 5, but thereis no symmetric ring spectrum whose underlying spectrum is a mod-p Moore spectrum.

If we specialize the derived composition pairing (3.11) to X = Y = Z, we see that the derived functionspectrum F (X,X) is a homotopy ring spectrum. However, these homotopy ring spectra always a rise fromsymmetric ring spectra. Indeed, the endomorphism ring spectrum as defined in Example 3.21 [...].


The spaces Sn represent the unstable homotopy groups and are related by isomorphisms Sm ∧ Sn ∼=Sm+n [...] which are obviously associative and unital. In the stable homotopy category, objects can bedesuspended, so we have ‘stable spheres’ Sk for all integers k with the analogous properties. However, thesymmetric spectra Sk can not be chosen so that Sm ∧ Sn and Sm+n are isomorphic as symmetric spectrafor all m,n ∈ Z. However, when we pass to the stable homotopy category we can consistently choose suchisomorphisms. We want these isomorphisms in SHC to have suitable associativity properties, which is a bitless obvious than in the case of spaces.

In the (unstable) homotopy category of based spaces the sphere Sn represents the homotopy groupsfunctor πn, by definition. There is an analogous statement in the stable homotopy category, as follows. Foran integer n we define the n-dimensional sphere spectrum Sn by

(3.16) Sn =

Σ∞Sn for n ≥ 1,

S for n = 0, andF−nS

0 for n ≤ −1.

Let us recall the definition of the fundamental classes ιn ∈ πnSn (originally introduced in (4.49) ofChapter I). For n ≥ 0 the identity of the sphere Sn is a based map Sn −→ (Σ∞Sn)0 whose homotopyclass is the naive fundamental class ιn ∈ πnSn (we used the notation ι0n earlier). For n < 0 the mapS0 −→ Σ+

−n ∧ S0 = (F−nS0)−n indexed by the identity element of the symmetric group has a homotopyclass in π0(Sn−n) which represents the naive fundamental class ιn ∈ πnSn (we used the notation ι−n0 earlier).The ‘true’ fundamental class is the image this naive fundamental class ιn ∈ πnS under the tautological mapc : πnSn −→ πnSn. By Proposition 1.15 the evaluation map

SHC(Sk, X) −→ πkX , α 7−→ (πkα)(ιk)

is an isomorphism of abelian groups for every symmetric spectrum X and all integers k.Since the spectrum S1+n represents the homotopy group functor π1+n, there is a unique morphism

(3.17) βn : S1+n −→ S1 ∧ Sn

in the stable homotopy category such that π1+n(βn) : π1+nS1+n −→ π1+n(S1 ∧ Sn) takes the fundamentalclass of S1+n to S1 ∧ ιn, the suspension of the previous fundamental class. The true homotopy groups ofSn are freely generated as a graded module over the stable stems πs

∗ by the fundamental class ιn[ref]. Sincethe suspensions isomorphism is πs

∗-linear, [ref...], the homotopy groups of S1 ∧ Sn are freely generated overπs∗ by the class S1∧ ιn. Since the morphism βn takes one generator to the other, it induces isomorphisms of

all true homotopy groups, so βn is an isomorphism in the stable homotopy category. [Exercise: representedby certain morphisms of symmetric spectra]

If m and n are both positive or both negative, then there are isomorphisms of symmetric spectraSn ∧ Sm ∼= Sn+m, but if n and m have opposite signs, the spectra Sn ∧ Sm and Sn+m are not isomorphicin Sp. However, the next propositions shows that in the stable homotopy category, we can consistentlychoose such isomorphisms for all integers n and m. The next proposition in particular shows that thesphere spectra Sn are invertible for all integers n, i.e., they have inverses in SHC (up to isomorphism) forthe derived smash product. We shall see in Proposition 8.6 that conversely, every invertible object of thestable homotopy category is isomorphic to Sn for an integer n.

In Theorem I.6.1 we constructed a natural pairing of true homotopy groups, which in the special caseof sphere spectra is a biadditive map

· : πmSm × πnSn −→ πm+n(Sm ∧ Sn) .The sphere spectra are flat, so the pointset smash product which appears in the target is also the derivedsmash product. Since the spectrum Sm+n represents the homotopy group functor πm+n, there is a uniquemorphism

(3.18) αm,n : Sm+n −→ Sm ∧L Sn

in the stable homotopy category such that πm+n(αm+n) : πm+nSm+n −→ πm+n(Sm ∧ Sn) takes thefundamental class of Sm+n to ιm · ιn.


Proposition 3.19. For all m,n ∈ Z the morphism αm,n : Sm+n −→ Sm ∧L Sn is an isomorphism inthe stable homotopy category. Moreover, these isomorphisms satisfy the following properties:

• (normalization) αm,0 and α0,n are inverses to the right respectively left unit isomorphisms;• (associativity) the square

(Sl ∧L Sm) ∧L SnαSl,Sm,Sn //

αl,m∧Id

Sl ∧L (Sm ∧L Sn)Id∧αm,n // Sl+m ∧L Sm

αl+m,n

Sl ∧L Sm,n αl.m+n

// Sl+m+n

commutes in the stable homotopy category for all integers l,m and n.• (commutativity) the square

Sm+n(−1)mn //

αm,n

Sn+m

αn,m

Sm ∧L Sn τSm,Sn

// Sn ∧L Sm

commutes in the stable homotopy category for all integers m and n .

Proof. The sources of the associativity and commutativity squares are sphere spectra; since theserepresent homotopy groups, the diagrams commute if we can show that the fundamental classes have thesame image both ways around the squares. But this amounts to the associativity respectively commutativityproperties of the smash product pairing and the defining properties of the morphisms αm,n.

Now we argue that the morphism αm,n are isomorphisms. Since S1 = Σ∞S1 we have a naturalisomorphism S1 ∧L X = (Σ∞S1) ∧L X ∼= S1 ∧X [ref]. In the special case X = Sn, the composite of thisisomorphism with α1,n : S1+n −→ S1 ∧L Sn equals the morphism βn : S1+n −→ S1 ∧Sn since both have thesame effect on the fundamental class ι1+n. So α1,n is an isomorphism in SHC since the other two morphismsare. The commutativity property then implies that αm,1 is also an isomorphism. The associativity propertyfor m = 1 then shows that αl+1,n is an isomorphism if and only if αl,1+n is. Since α0,n is an isomorphismfor all integers n, an induction on the absolute values of m shows that αm,n is an isomorphism for all mand n.

Remark 3.20. One cannot define associative stable equivalences Sm ∧ Sn −→ Sm+n in the category ofsymmetric spectra. For example, A(1,−1, 1) is a problem. In SHC we can add a sign to compensate forthe twist permutation of S2.

We now define graded homomorphism groups in the stable homotopy category by

[X,Y ]n = [Sn ∧L X,Y ]

for n ∈ Z. Note that the left unit isomorphism lX : S0 ∧LX −→ X can be used to identify [X,Y ]0 with thehomomorphism group from X to Y in SHC and for n ≥ 0, the object Sn ∧L X is isomorphic to the n-foldsuspension of X in SHC. We can also define a graded composition

: [Y, Z]m ⊗ [X,Y ]n −→ [X,Z]m+n

(f, g) 7−→ f (Sm ∧L g) (αm,n ∧L X)

which is associative by Proposition 3.19. Composition is also unital with unit the left unit isomorphismlX : S0 ∧L X −→ X. [compare S1 ∧L X with ΣX = S1 ∧X] [iso πkF (X,Y ) ∼= [X,Y ]k]

Recall from Theorem 1.6 that γ : Sp −→ SHC denotes the universal localization functor. Suppose that(S, µ, ι) is a homotopy ring spectrum, i.e., a symmetric spectrum which is a monoid in the stable homotopycategory under the derived smash product. Then the derived smash product of morphisms can be used to


make the graded abelian group [S, S]∗ into a graded ring. The unit is given by the unit map ι : S −→ Swhich is an element of [S, S]0, and the multiplication

[S, S]m ⊗ [S, S]n −→ [S, S]m+n

takes f ⊗ g to the composite

Sm+nα−1m,n−−−→ Sm ∧L Sn f∧Lg−−−−→ S ∧L S µ−−→ S .

If the homotopy ring spectrum S is commutative, then this multiplication is graded commutative. [define[S, X]m ⊗ [S, Y ]n −→ [S, X ∧L Y ]m+n first ?] [if this comes from a symmetric ring spectrum R thenevaluation at the fundamental classes is an isomorphism of graded rings [S, R]∗ ∼= π∗R ]

In the special case of a homotopy ring spectrum the map SHC(Sk, R) −→ πkR which evaluates at thefundamental class is also multiplicative and unital, i.e., and isomorphism of graded rings.

4. Generators

Let T be a triangulated category which has infinite sums. An object C of a triangulated category T iscalled compact (sometimes called finite or small ) if for every family Xii∈I of objects the canonical map⊕

i∈I[C, Xi] −→ [C,

⊕i∈I

Xi]

is an isomorphism.An object G of T is called a weak generator if the following condition holds: if X is an object such that

the groups [ΣkG,X] are trivial for all k ∈ Z, then X is a zero object. So weak generators detect whetherobjects are trivial. But weak generators also detect isomorphism [expand]

Proposition 4.1. The sphere spectrum S is compact and a weak generator of the stable homotopycategory.

Proof. If X is a symmetric spectrum for which the graded abelian group [S, X]∗ is trivial, then thetrue homotopy groups of X are trivial by Proposition 1.15. Thus X is stably equivalent to the trivialspectrum, hence a zero object in SHC. This proves that the sphere spectrum is a graded weak generatorof the stable homotopy category.

According to Proposition 1.9 (i), the coproduct in SHC of a family Xii∈I of symmetric spectra isgiven by the wedge. We have a commutative square⊕

i∈I [S, Xi] //

[S,⊕

i∈I Xi]

⊕i∈Iπ

t0(X

i) // πt0(∨

i∈I Xi)

in which the vertical maps are evaluation at the fundamental class, which are isomorphisms by the above.[difference between true and classical homotopy groups...] The lower horizontal maps is an isomorphism by[...], hence so is the upper horizontal map, which shows that the sphere spectrum is compact.

Let T be a triangulated category with sums. A localizing subcategory of T is a full subcategory X whichis closed under sums and under extensions in the following sense: if

A −→ B −→ C −→ ΣA

if a distinguished triangle in T such that two of the objects A, B or C belong to X , then so does the third.As we just saw, the sphere spectrum S is a compact weak generator of the stable homotopy category. Butis also generates the stable homotopy category in the sense that the whole stable homotopy category is thesmallest localizing subcategory containing S.

Proposition 4.2. Every localizing subcategory of the stable homotopy category which contains thesphere spectrum S is all of SHC.

4. GENERATORS 203

This proposition is really a special case of a general fact about triangulated categories, Proposition 4.11below. For the proof we need the notion of homotopy colimits in triangulated categories. Triangulatedcategories typically do not have limits or colimits (except for sums and products). For a sequence ofcomposable morphisms, a certain construction called ‘homotopy colimit’ exist and has some of the propertiesof a categorical colimit, but it is not a colimit in the homotopy category.

Definition 4.3 (Homotopy colimit). Let T be a triangulated category with infinite sums. We considera countably infinite sequence

X0f0−→ X1

f1−→ X2 · · ·of morphisms in T . A homotopy colimit of the sequence consists of an object X together with morphismsϕn : Xn −→ X satisfying ϕn+1fn = ϕn such that there exists a distinguished triangle⊕

n≥0

Xn1−f−−−→

⊕n≥0

Xn

Lϕn−−−−→ X −→ Σ(

⊕n≥0

Xn) .

Here we denote by 1− f :⊕

n≥0Xn −→⊕

n≥0Xn the morphism whose restriction to the ith summand isthe difference of the canonical morphism Xi −→

⊕n≥0Xn and the composition of fi : Xi −→ Xi+1 with

the canonical morphism Xi+1 −→⊕

n≥0Xn.

We repeat the earlier warning that homotopy colimits are not colimits in the triangulated category, andthey are less functorial and canonical that categorical colimits since the don’t enjoy a universal property.We will see in Exercise 11.4 below that the homotopy colimit is unique up to isomorphism; however, incontrast to ordinary colimits there is usually no preferred isomorphism between two homotopy colimits.

Our prime example of a triangulated category is the stable homotopy category, and we now relatedthe abstract notion of homotopy colimit in a triangulated category to sequential colimits of symmetricspectra. The following lemma say that a homotopy colimit in the stable homotopy category can essentiallybe calculated as the colimit, in the category of symmetric spectra, over arbitrary choices of morphismswhich represent the given homotopy classes.

Proposition 4.4. Let fn : Xn −→ Xn+1 be morphisms of symmetric spectra of simplicial sets forn ≥ 0. Then the colimit, in the category of symmetric spectra, of the sequence of morphisms fn representsthe homotopy colimit of the sequence of morphisms γ(fn) in the stable homotopy category.

Proof. Let C be a colimit of the sequence fn and let ϕn : Xn −→ C be the canonical morphisms.We construct a distinguished triangle in the stable homotopy category

⊕n≥0

XnId−f−−−→

⊕n≥0

Xn⊕γ(ϕn)−−−−−→ C −→ Σ

⊕n≥0

Xn

.

[fix this] Every filtered colimit of Kan simplicial sets is again Kan, a filtered colimit of weak equivalences isa weak equivalence and the simplicial loop space functor map(S1,−) commutes with filtered colimits. Thisproves that the colimit of the sequence of morphisms fn is again levelwise Kan and an Ω-spectrum. (...)

We now see how one can calculate maps from and to a homotopy colimit, in the latter case from compactobjects.

Definition 4.5. Let T be a triangulated category. A contravariant functor E from T to the categoryof abelian groups is called cohomological if it takes sums in T to products of abelian groups and if for everydistinguished triangle (f, g, h) in T the sequence of abelian groups

E(ΣA)E(h)−−−→ E(C)

E(g)−−−→ E(B)E(f)−−−→ E(A)

is exact.


Since distinguished triangles can be rotated, the 4-term exact sequence produced by a cohomologicalfunctor can be extended indefinitely in both directions resulting in a long exact sequence. The mainexamples of cohomological functors are the contravariant representable functors T (−, X) for an object Xof T .

Lemma 4.6. Let T be a triangulated category, fn : Xn −→ Xn+1 a sequence of composable morphismsand (X, ϕn) a homotopy colimit of the sequence fn.(i) For every cohomological functor E : T op −→ Ab the short sequence of abelian groups

(4.7) 0 −→ lim1nE(ΣXn) −→ E(X) −→ limnE(Xn) −→ 0

is exact, where the right map arises from the system of compatible homomorphisms E(ϕn) : E(X) −→E(Xn). In particular, for every object Y of T the short sequence of abelian groups

(4.8) 0 −→ lim1n [ΣXn, Y ] −→ [X, Y ] −→ limn[Xn, Y ] −→ 0

is exact.(ii) For every compact object Y of T the maps [Y, ϕn] : [Y,Xn] −→ [Y, X] express the [Y, X] as the colimitof the abelian groups [Y,Xn]. [state this for ‘homological’ functors, using that [G,−] is homological forcompact G?]

Proof. (i) We apply E to the defining triangle of the homotopy colimit and use the universal propertyof sums we get an exact sequence∏

nE(ΣXn)

1−E(f)−−−−−→∏

nE(ΣXn) −−−→ E(X)

QE(ϕn)−−−−−−→

∏nE(Xn)

1−E(f)−−−−−→∏

nE(Xn) .

Kernel respectively cokernel of the self-map 1 − E(f) of the product∏nE(Xn) are the limit respectively

derived limit of the sequence of maps E(fn) : E(Xn+1) −→ E(Xn), which proves that (4.8) is exact.(ii) By definition of ‘compact’, the functor [Y,−] takes sums to sums. So applying [Y,−] to the defining

triangle of the homotopy colimit gives an exact sequence⊕n≥0

[Y,Xn]1−[Y,f ]−−−−−→

⊕n≥0

[Y,Xn]L

[Y,ϕn]−−−−−→ [Y, X] −−−→⊕n≥0

[Y,ΣXn]1−[Y,f ]−−−−−→

⊕n≥0

[Y,Xn[1]] .

The map 1−[Y, f ] is always injective and its cokernel is a colimit of the sequence of maps [Y, fn] : [Y,Xn] −→[Y,Xn+1], which proves the claim.

Remark 4.9. Among topologists, the short exact sequence (4.8) is often called the Milnor exact se-quence [since Milnor introduced it in ...?]. The surjectivity of the second map in the Milnor sequencesays that the data (X, ϕn) has ‘half‘ of the universal property of a categorical colimit: given morphismsgn : Xn −→ Y in the triangulated category T which are compatible in the sense that we have gn+1fn = gn,then the tuple gnn is an element in the limit of the groups [Xn, Y ]. So by surjectivity the exists a mor-phism g : X −→ Y restricting to gn on each Xn. However, when the lim1 term is non-trivial, there ismore than one such morphism g. So the lim1 term measures to what extent the homotopy colimit lacks theuniqueness part of the universal property. In ??? we give an example of a Milnor sequence with non-triviallim1 term.

Proposition 4.10. Let T be a triangulated category with infinite sums and let C be a set of compactobject of T . Let 〈C〉+ denote the smallest class of objects of T which contains C and is closed under sums(possibly infinite) and ‘extensions to the right’ in the following sense: if

A −→ B −→ C −→ ΣA

is a distinguished triangle such that A and B belong to the class, then so does C. Then for every cohomo-logical functor E : T op −→ Ab there exists an object R in class 〈C〉+ and an element u ∈ E(R) such thatfor every object G of C the evaluation map

evu : [G,R] −→ E(G) , f 7→ E(f)(u)

is bijective.

4. GENERATORS 205

Proof. By induction on n we now construct objects Rn in 〈C〉+ and elements un : E(Rn) andin : Rn −→ Rn+1 such that E(in)(un+1) = un. We start with

R0 =⊕G∈C

⊕x∈E(G)

G ,

which comes with a tautological element u0 ∈ E(C(X)) which is determined by the property that it restrictsto x ∈ E(G) on the summand indexed by x. Note that R0 belongs to 〈C〉+ and evu0 : [G,R0] −→ E(G) issurjective for all G ∈ C.

In the inductive step we suppose that the pair (Rn, un) has already been constructed. We let In(G)denote the kernel of the evaluation morphism evun : [G,Rn] −→ E(G) and consider

Cn =⊕G∈C

⊕x∈In(G)

G ,

which comes with a tautological morphism τ : C(X) −→ Rn which is given by x on the summand indexedby x. We choose a distinguished triangle

Cnτ−−→ Rn

in−→ Rn+1 −→ ΣCn .

Since E is cohomological the sequence

E(ΣCn) −→ E(Rn+1)E(in)−−−−−→ E(Rn)

E(τ)−−−−→ E(Cn)

is exact. Under the isomorphism E(Cn) ∼=∏G∈C

∏x∈In(G)E(G) the map E(τ) takes un ∈ E(Rn) to

the family evun(x) which is zero by definition. So there exists an element un+1 ∈ E(Rn+1) satisfyingE(in)(un+1) = un.

Now we choose a homotopy colimit (R, ϕn : Rn −→ Rn), in the sense of Definition 4.3, of thesequence of morphisms in : Rn −→ Rn+1. Since all the objects Rn are in 〈C〉+, so is R. The short exactsequence (4.7) shows that we can choose an element u ∈ E(R) satisfying E(ϕn)(u) = un in E(Rn) for alln ≥ 0. We claim that any such r has the property that evu : [G,R] −→ E(G) is bijective.

Since E(ϕ0)(u) = u0 in E(R0) the composite [G,R0] −→ [G,R] evu−−→ E(G) is evaluation at u0, which issurjective. Hence evu : [G,R] −→ R(G) is also surjective. To show that evu is injective we let α : G −→ R bean element in its kernel, i.e., such that E(α)(u) = 0. Since G is compact, there is an n ≥ 0 and a morphismα′ : G −→ Rn such that α = ϕnα

′, by Lemma 4.6 (ii). Then E(α′)(un) = E(α′)(E(ϕn)(u)) = E(α)(u) = 0.So α′ lies in In(G) and indexes one of the summands of Cn. So α′ factors through the tautological morphismτ : Cn −→ Rn, and hence

α = ϕnα′ = ϕn+1inα

′ = 0since the morphisms in and τ are adjacent in a distinguished triangle, and so have trivial composite. Henceevu : [G,R] −→ E(G) is also injective, which finishes the proof.

Now we show that for compact objects, the meanings of ‘generator’ for a triangulated category coincide.For arbitrary objects, condition (ii) in the next proposition implies condition (i), but not necessarily theother way around. In the special case of the stable homotopy category and the weak generator D = S, thenext proposition implies Proposition 4.2.

Proposition 4.11. Let T be a triangulated category with infinite sums and let G be a set of compactobjects of T . Then the following conditions are equivalent.

(i) G is a set of weak generators.(ii) Every localizing subcategory of T which contains G is all of T .

Proof. Let us first assume that G is a set of weak generators and let X be a localizing subcategory ofT which contains G. We let X be any object of T .

We apply Proposition 4.10 to the set C = ΣkGk∈Z,G∈G of all positive and negative suspensions ofobjects in G. Since a localizing subcategory is closed under shifts in both directions, the set C is containedin X , and hence so is the class 〈C〉+. Proposition 4.10 applied to the representable functor [−, X] provides


a morphism u : R −→ X such that [ΣkG, u] : [ΣkG,R] −→ [ΣkG,X] is bijective for all k ∈ Z and G ∈ G.Since G weakly generates, u must be an isomorphism. Since R is contained in 〈C〉+ ⊂ X , so is X. So thelocalizing subcategory X contains all objects of T .

Now we assume condition (ii) and show that G is then a set of weak generators. This implication doesnot need the assumption that the object in G are compact. We let X be an object of T such that the gradedabelian group [G,X]∗ is trivial for every G ∈ G. We let X be the class of all those objects A of T for whichthe graded abelian group [A,X]∗ is trivial. The X is a localizing subcategory of T and contains G, henceit contains all objects, in particular the object X itself. Thus the group [X,X] is trivial, so X must be azero object.

Remark 4.12. For a set of not necessarily compact objects the two conditions are not generally equiv-alent.

Definition 4.13. A triangulated category T is called compactly generated if T has sums and a set ofcompact objects which are weak generators.

By Proposition 4.11 we could replace the condition ‘weak generators’ by ‘generators’ in the sense of[...] (as long as we insist of compact objects.) The stable homotopy category is our main example of acompactly generated triangulated category, where we can take the sphere spectrum S as a single compactgenerator. More generally we will show in ??? that the triangulated derived category of a symmetric ringspectrum is compactly generated, where the free module of rank one can be taken as a single compactgenerator. [examples with ‘many generators’]

Proposition 4.14. Every cohomological functor defined on a compactly generated triangulated categoryis representable.

Proof. Let T be compactly generated and let G be a set of compact generators. Given a cohomologicalfunctor E : T op −→ Ab we apply Proposition 4.10 with C = ΣkGk∈Z,G∈G the set of all positive andnegative suspensions of all objects in G. We obtain an object R of T and an element u ∈ E(R) such thatevu : [ΣkG,R] −→ E(ΣkG) is bijective for all integers k and G ∈ G.

We let X be the class of all objects X of T for which the evaluation morphism evu : [X,R] −→ E(X)is bijective. Since [−, R] and E are both cohomological functors, the class X is localizing. By the above, italso contains the set of generators. Proposition 4.11 shows that X = T , so the pair (R, u) represents thefunctor E.

Example 4.15. The representability result given by Proposition 4.14 can sometimes be used to con-struct spectra with prescribed homotopy groups. One example of this is the Brown-Comenetz dual (of thesphere spectrum). The construction uses the contravariant endofunctor of abelian groups

A 7−→ A∨ = Ab(A,Q/Z)

which is sometimes called the Pontryagin dual of A. There is a natural evaluation homomorphism

A −→ (A∨)∨ , a 7−→ (ϕ 7→ ϕ(a))

which injects A into its double dual. If A is finite, then the evaluation is an isomorphism A ∼= (A∨)∨; inthat case A is also isomorphic to its (single) dual A∨, but this isomorphism is not natural. The group Q/Zis injective as an abelian group, i.e., the Pontryagin duality functor Ab(−,Q/Z) is exact.

We consider the contravariant functor

SHC −→ Ab , X 7−→ (π0X)∨ .

For every family Xii∈I the natural map⊕i∈I

πn(Xi) −→ πn

(∨i∈I

Xi

)

4. GENERATORS 207

is an isomorphism [ref] and Pontryagin duality takes sums to products, so the functor takes sums toproducts. Every distinguished triangle (f, g, h) in the stable homotopy category gives rise to a long exactsequence of homotopy groups

π0Aπ0f−−→ π0B

π0g−−→ π0Cπ0h−−→ π0(ΣA)

[ref] Since Pontryagin duality is exact, we get an exact sequence

(π0A)∨(π0f)∨←−−−− (π0B)∨

(π0g)∨

←−−−− (π0C)∨(π0h)

∨

←−−−− (π0(ΣA))∨ .

So we have constructed a cohomological functor. The Brown-Comenetz dual IS is a representing spectrumfor this cohomological functor; it comes with a universal element u ∈ (π0IS)∨, i.e., a homomorphismu : π0IS −→ Q/Z. We can calculate the homotopy groups of IS as follows. For any integer k we composethe action of the stable stems with the universal homomorphism to a homomorphism of abelian groups

πkIS ⊗ πs−k

·−→ π0ISu−−→ Q/Z .

This map is a perfect pairing in the sense that its adjoint

u : πkIS −→ Ab(πs−k,Q/Z) = (πs

−k)∨

is an isomorphism. Indeed, we have a commutative square

[Sk, IS]evu //

evιk

[S,Sk]∨

πkIS

u// (πs

−k)∨

in which the other three maps are isomorphisms. Since the stable homotopy groups of spheres are finiteexcept in dimension zero (compare Theorem I.1.9), and thus self-dual (in a non-canonical way), one could saythat the homotopy groups of the Brown-Comenetz dual IS are the stable homotopy groups of spheres ‘turnedupside down’. There are non-trivial stable homotopy groups of spheres in arbitrarily high dimensions, sothe spectrum IS has non-trivial homotopy groups in arbitrarily low dimensions.

A symmetric spectrum X is n-connected if the true homotopy groups πkX are trivial for k ≤ n. So‘connective’ and ‘(-1)-connected’ have the same meaning.

Proposition 4.16. For an integer n, let 〈Sn〉+ denote the smallest class of symmetric spectra whichcontains the n-dimensional sphere spectrum Sn and is closed under sums (possibly infinite) and ‘extensionsto the right’ in the following sense: if

A −→ B −→ C −→ ΣA

is a distinguished triangle such that A and B belong to the class, then so does C. Then 〈Sn〉+ equals theclass of (n− 1)-connected spectra.

Proof. Since the sphere spectrum S is connective and suspensions shifts homotopy groups, the spherespectrum Sn is (n−1)-connected. The class of (n−1)-connected spectra is closed under sums (since homotopygroups commute with sums) and extensions to the right (by the long exact sequence of homotopy groupsassociated to a distinguished triangle), so every spectrum in the class 〈Sn〉+ is (n− 1)-connected.

For the converse we let X be any (n − 1)-connected symmetric spectrum. We let C = Skk≥n bethe set of sphere spectra of dimension at least n. Every class closed under extensions to the right is inparticular closed under suspensions, so we in fact have 〈C〉+ = 〈Sn〉+. We apply Proposition 4.10 tothe representable functor [−, X]. We obtain a symmetric spectrum R belonging to the class 〈Sn〉+ and amorphism u : R −→ X such that [Sk, u] : [Sk, R] −→ [Sk, X] is bijective for all k ≥ n. Since Sk representsthe k-th homotopy group this means that u induces isomorphisms of homotopy groups in dimensions n andabove. Since R (by the previous paragraph) and X are (n − 1)-connected, the morphism u also inducesisomorphisms of homotopy groups below dimension n, so u is an isomorphism in the stable homotopycategory. Thus X belongs to 〈C〉+, which finishes the proof.


Proposition 4.17. For a symmetric spectrum X the following conditions are equivalent.(i) X is connective.(ii) X belongs to 〈S〉+, the smallest class of objects of the stable homotopy category which contains the

sphere spectrum S and is closed under sums (possibly infinite) and extensions to the right.

Proposition I.?? lets us add another condition which is equivalent to X being connective, namely thatX stably equivalent to a symmetric spectrum of the form A(S) for a Γ-space A.

The characterization of connective (i.e., (-1)-connected) spectra as being generated by the sphere spec-trum S under sums and extensions to the right (see Proposition 4.16) can be useful for reducing claimsabout connective spectra to the special case of the sphere spectrum. The following result is an examplewhere we use this strategy in the proof.

Proposition 4.18. Let X and Y be symmetric spectra such that X is (k − 1)-connected and Y is(l − 1)-connected. Then the derived smash product X ∧L Y is (k + l − 1)-connected and the pairing

· : πkX ⊗ πlY −→ πk+l(X ∧L Y )

of of Theorem I.6.1 is an isomorphism of abelian groups.

Proof. We fix an (l− 1)-connected spectrum Y and let X be the class of all (k− 1)-connected spectraX for which the theorem is true. Since πkSk ∼= Z and πk+l(Sk ∧L Y ) ∼= πlY , the class X contains the spherespectrum [more precise]. The class X is also closed under sums since both sides of the map commute withsums in X. Finally, X is closed under extensions to the right. Indeed, suppose that

A −→ B −→ C −→ ΣA

is a distinguished triangle such that A and B belong to X . Then the triangle

A ∧L Y f∧LId−−−−−→ B ∧L Y g∧LId−−−−−→ C ∧L Y κA,Y (h∧LId)−−−−−−−−−→ Σ(A ∧L Y )

is distinguished by Proposition 3.14. We apply homotopy groups and get a commutative diagram

πkA⊗ πlYπkf⊗Id //

·

πkB ⊗ πlYπkg⊗Id //

·

πkC ⊗ πlY //

·

0

πk+l(A ∧L Y )πk+l(f∧Id)

// πk+l(B ∧L Y )πk+l(g∧Id)

// πk+l(C ∧L Y ) // 0

The upper row is exact since A is (k − 1)-connected, thus πk(ΣA) ∼= πk−1A = 0, and tensoring with πlYis right exact. The lower row is exact since πk+l(Σ(A ∧L Y )) is isomorphic to πk+l(A ∧L (ΣY )), which istrivial since ΣY is l-connected and A belongs to X . Since both rows are exact and the left and middlevertical map are isomorphisms, the right vertical map is an isomorphism and thus C ∈ X . Proposition 4.17now applies and shows that every (k − 1)-connected spectrum belongs to the class X , which is what weclaimed.

[alternative proof earlier: wlog Ω-spectra, then use πk(X ∧L Y ) ∼= colimn πk+2n(Xn ∧ Yn)]

Remark 4.19. The previous theorem about the lowest potentially non-trivial homotopy group of asmash product immediately implies a similar result for the pointset level smash product whenever at leastone factor is flat. Indeed, if X and Y are symmetric spectra at least one of which is flat, then by Theorem 3.1the natural map ψX,Y : X ∧L Y −→ X ∧ Y from the derived to the pointset level smash product is anisomorphism in SHC. So if X is (k− 1)-connected and Y is (l− 1)-connected and one of them is flat, thenX ∧ Y is (k + l − 1)-connected and the pairing

· : πkX ⊗ πlY −→ πk+l(X ∧ Y )


4. GENERATORS 209

Proposition 4.20. Let X be a coconnective symmetric spectrum, i.e., the homotopy group πnX istrivial for all n ≥ 1, and let A be a connective spectrum. Then the map

π0 : [A,X] −→ HomAb(π0A, π0X)


Proof. We consider the class X of all connective spectra A such that for all coconnective X the mapπ0 : [A,X] −→ HomAb(π0A, π0X) is an isomorphism. The map

π0 : [S, X] −→ HomAb(π0S, π0X)

is an isomorphism of abelian groups because π0S is free abelian of rank 1, generated by the unit 1 ∈ π0S.So the sphere spectrum S belongs to X .

Now consider a family Aii∈I of objects from X . We have a commutative square

[⊕

I Ai, X]

π0 //

∼=

HomAb(π0

(⊕I A

i), π0X)

∼=

HomAb(⊕

I π0(Ai), π0X)

∼=∏

I [Ai, X] QIπ0

// ∏I HomAb(π0(Ai), π0X)

in which the vertical maps are isomorphisms by the universal property of sums and because homotopygroups commute with sums. The lower horizontal map is an isomorphism by the assumption on the objectsAi, hence the upper map is an isomorphism; this proves that the class X is closed under sums.

Now consider a distinguished triangle

A −→ B −→ C −→ ΣA

such that A and B belong to X . We consider the commutative diagram

0 // [C,X] //

π0

[B,X] //

π0 ∼=

[A,X]

π0 ∼=

0 // HomAb(π0C, π0X) // HomAb(π0B, π0X) // HomAb(π0A, π0X)

The upper row is exact since Σ−1X is coconnective, thus π0(Σ−1X) = 0 and so the group [ΣA,X] ∼=[A,Σ−1X] ∼= HomAb(π0A, π0(Σ−1X)) is trivial. The lower row is exact since π0A −→ π0B −→ π0C −→π−1A = 0 is. Since the vertical maps for B and A are isomorphisms, so is the one for C. Thus C alsobelongs to X which means that the class X is closed under extensions to the right. So by part (ii) ofProposition 4.16 the class X contains all connective spectra.

Theorem 4.21 (Uniqueness of Eilenberg-Mac Lane spectra). (i) Let X be a connective symmet-ric spectrum and A an abelian group. Then the map

π0 : SHC(X,HA) −→ HomAb(π0X,A)

is an isomorphism of abelian groups.(ii) Let X be a symmetric spectrum whose homotopy groups are trivial in dimensions different from 0.

Then there is a unique morphism in the stable homotopy category from the Eilenberg-Mac Lanespectrum H(π0X) to X which induces the isomorphism π0(Hπ0X) ∼= π0X on homotopy.

(iii) The restriction of the functor π0 : SHC −→ Ab to the full subcategory of spectra with homotopyconcentrated in dimension 0 is an equivalence of categories.


Proof. (i) Since Eilenberg-Mac Lane spectra are coconnective this is a special case of Proposition 4.20.(ii) Part (i) gives a morphism f : X −→ HA, unique in the stable homotopy category, which induces

induces the isomorphism π0(Hπ0X) ∼= π0X on homotopy. Since source and target of f have no homotopyin dimensions other than 0, f is an isomorphism in SHC.

(iii) The restriction of π0 to the full subcategory of spectra with homotopy concentrated in dimension 0is fully faithful by (i) and essentially surjective since every abelian group has an Eilenberg-Mac Lanespectrum.

[alternative proof earlier: [X,HA] = map(π0X,A) ??, the wlog X an Ω-spectrum]

Remark 4.22. In the theory of triangulated categories, the notion of a t-structure formalizes thebehavior of ‘connective’ and ‘co-connective’ objects. What we have show can be summarized in this languageas saying that in the situation of the classes of connective and co-connective spectra provide a t-structureon the stable homotopy category. Moreover, the objects with homotopy groups concentrated in dimension 0form the so-called heart of the t-structure. So Theorem 4.21 can be rephrased in a fancy way as saying thatthe functor π0 is an equivalence of abelian categories from the heart of this t-structure to the category ofabelian groups.

Lemma 4.23. Let n be an integer and f : X −→ Y and g : Y −→ Z composable morphisms in thestable homotopy category such that X is (n− 1)-connected, Z is (n+ 1)-coconnected and

0 −→ πkXπkf−−−−→ πkY

πkg−−−−→ πkZ −→ 0

is exact for all k ∈ Z. Then there is a unique morphism h : Z −→ ΣX such that (f, g, h) is a distinguishedtriangle.

Proof. Since Z is (n+1)-coconnected, the exactness of the above sequence implies that πkf is bijectivefor k > n. Similarly, πkg is bijective for k < n.

We choose a distinguished triangle

Xf−−→ Y

g′−−→ Z ′h′−−→ ΣX

which extends f . We contemplate the long exact sequence of homotopy groups of this triangle: since πkf isbijective for k > n and injective for k = n, the spectrum Z ′ has trivial homotopy groups above dimensionk. Since X is (n − 1)-connected, the map πkg

′ : πkY −→ πkZ′ is an isomorphism for k < n and passes to

an isomorphism from the cokernel of πkf : πkX −→ πkY to πkZ ′.The composite of gf : X −→ Z is trivial in SHC by Proposition 4.20 (or rather its n-fold shifted version).

So there exists a morphism ϕ : Z ′ −→ Z satisfying ϕg′ = g. By the assumptions on the homotopy groupsof Z and the above calculation of the homotopy groups of Z ′, the morphism ϕ induces an isomorphism ofall homotopy groups. So ϕ is an isomorphism in the stable homotopy category and we can replace Z ′ bythe isomorphism spectrum to obtain a distinguished triangle of the desired form with h = h′ϕ−1.

For the uniqueness statement we let h : Z −→ ΣX be another morphism which extends (f, g) to adistinguished triangle. Axiom (T3) of the triangulated category allows us to choose an endomorphismϕ : Z −→ Z which makes the diagram

Xf // Y

g // Zh //

ϕ

ΣX

Xf

// Y g// Z

h

// ΣX

commute. We have (IdZ −ϕ)g = 0, so there exists a morphism ψ : ΣX −→ Z such that ψh = (Id−ϕ) byexactness of (f, g, h). Since ΣX is n-connected and Z is (n+ 1)-coconnected, the morphism ψ is trivial byProposition 4.20 (or rather its n-fold shifted version). So ϕ equals the identity of Z and thus h = h.

4. GENERATORS 211

By an extension of abelian groups we means a pair of homomorphisms i : A −→ B and j : B −→ C ofabelian groups such that i is injective, j is surjective and the image of i equals the kernel of B. [i.e., shortexact sequence]

Proposition 4.24. (i) For every extension of abelian groups A i−→ Bj−→ C there is a unique morphism

δ : HC −→ Σ(HA) in the stable homotopy category such that the diagram

HAHi−−−→ HB

Hj−−−→ HCδ−−→ Σ(HA)

is a distinguished triangle. This morphism δ will be called the Bockstein operator associated to the exten-sion.(ii) The Bockstein operator is natural for morphisms of extensions in the following sense. Given a commu-tative diagram of abelian groups

Ai //

α

B

β

j // C

γ

A′

i′// B′

j′// C ′

in which both rows are extensions and two distinguished triangles (Hi,Hj, δ) and (Hi′,Hj′, δ′), then therelation

Σ(Hα) δ = δ Hγholds as morphism HC −→ Σ(HA′) in the stable homotopy category.(iii) The Bockstein operator only depends on the Yoneda class of the extensions and this construction definesa group isomorphism

Ext(C,A) −−→ [HC,Σ(HA)]for all abelian groups A and C.(iv) The composite of every composable pair of Bockstein operations is zero.

Proof. (i) This is the special case of Lemma 4.23 with n = 0 for the composable morphisms Hi :HA −→ HB and Hj : HB −→ HC.

(ii) Suppose we are given extensions (i, j) and (i′, j′), a morphism (α, β, γ) from (i, j) to (i′, j′) as wellas morphisms δ : HC −→ Σ(HA) and δ′ : HC ′ −→ Σ(HA′) such that (Hi,Hj, δ) and (Hi′,Hj′, δ′) aredistinguished triangles. Axiom (T3) of the triangulated category let us choose a morphism ϕ : HC −→ HC ′

which makes the diagram

HAHi //

Hα

HBHj //

Hβ

HCδ //

ϕ

Σ(HA)

Σ(Hα)

HA′

Hi′// HB

Hj′// HC

δ′// Σ(HA)

commute. The relation π0(ϕ) π0(Hj) = π0(Hj′) π0(Hβ) = π0(Hγ) π0(Hj) holds since j′β = γj. SinceHj is surjective on π0, we deduce that ϕ and Hγ induce the same map π0. Then ϕ = Hj by Theorem 4.21,and thus Σ(Hα) δ = δ′ ϕ = δ′ Hγ, as we claimed.

(iii) Two extensions (i, j) and (i′, j′) represent the same Yoneda class if and only if there is a homo-morphism f : B −→ B′ (necessarily an isomorphism) satisfying fi = i′ and j′f = j. We can apply (ii) tothe morphism (IdA, f, IdC) from (i, j) to (i′, j′) and the distinguished triangles (i, j, δ) and (i′, j′, δ′). Thenaturality statement then boils down to the equation δ = δ′. So the Bockstein operator only depends onthe Yoneda class of the extensions.

For the additivity of the Bockstein operator construction we first show that the Bockstein respectsexternal direct sums. If we are given extensions (i, j) and (i′, j′). we can form the direct sum extension(i⊕ i′, j⊕ j′). The Eilenberg-Mac Lane functor commutes with sums [show...], i.e., the canonical morphismHA ⊕ HA′ −→ H(A ⊕ A′) is an isomorphism. Since the direct sum of distinguished triangles is also a


distinguished triangle [ref] we deduce that the Bockstein operator for the direct sum extensions (i⊕i′, j⊕j′)equals the composite

H(C ⊕ C ′)∼=−→ HC ⊕HC ′ δ⊕δ′−−−→ Σ(HA)⊕ Σ(HA′)

∼=−→ Σ(H(A⊕A′)) .

Now we specialize to two extensions (i, j) and (i′, j′) which d start with the same group A and endwith the same group C. The addition in the Yoneda Ext-group is induced by the Baer sum of extensions(ϕ : A −→ ∆∗(j, j′)/(i,−i′)(A), ψ : ∆∗(j, j′)/(i, i′)(A) −→ C), defined as follows. We have nested subgroups

(i, i′)(A) ⊆ ∆∗(j, j′) ⊆ B ⊕B′

where ∆∗(j, j′) consists of the pairs (b, b′) satisfying j(b) = j′(b′), and (i,−i′)(A) consists of pairs of theform (i(a),−i′(a)) for a ∈ A. The middle group in the Baer sum is the factor group ∆∗(j, j′)/(i,−i′)(A).We define morphisms ϕ and ψ by ϕ(a) = (i(a), 0) + (i,−i′)(A) = (0, i′(a)) + (i,−i′)(A) respectivelyψ ((b, b′) + (i,−i′)(A)) = j(b) = j′(b′). The Baer sum comes with a commutative diagram of extensions

A⊕Ai⊕i′ // B ⊕B′

j⊕j′ // C ⊕ C

A⊕Ai⊕i′ //

+

∆∗(j, j′) //

incl

OO

C

∆

OO

A ϕ// ∆∗(j, j′)/(i,−i′)(A)

ψ// C

By two applications of naturality we obtain that the Bockstein of the Baer sum extensions equals the sumof the individual Bockstein operators. [check... possible sign ?]

To prove injectivity we consider an extension (i, j) whose Bockstein operator is zero. Then the triangle... splits, i.e., there is a morphism ψ : HC −→ HB in the stable homotopy category such thatHjψ = IdHC .This implies that π0(Hj) and thus j : B −→ C is split surjective, so (i, j) represents the zero element inthe group Ext(C,A).

For surjectivity we consider any morphism δ : HC −→ Σ(HA) and embed it in a distinguished triangle

HAf−−→ Y

g−−→ HCδ−−→ Σ(HA) .

The long exact sequence of homotopy groups associated to this triangle reduces shows that the homotopygroups of Y are concentrated in dimension zero and gives an extension

(4.25) A = π0HAπ0f−−−→ π0Y

π0g−−−→ π0HC = C .

Theorem 4.21 (ii) constructs a preferred isomorphism in SHC from the spectrum Y to the Eilenberg-MacLane spectrum H(π0Y ). So we can replace Y in the above distinguished triangle by the isomorphism objectH(π0Y ) in a way which turns f into H(π0f) and g into H(π0g). The uniqueness of Bockstein operatorsthen shows that the original morphism δ is the Bockstein associated to the extension (4.25)

(iii) The reason behind property (iii) is the fact that the category of abelian groups has no non-trivialExt-groups beyond dimension one. In more detail we argue as follows. Let

Ai−−→ B

j−−→ C and Ci′−−→ D

j′−−→ E

be two ‘composable’ extensions. We wish to show that the composite

HEδ′−−→ Σ(HC)

Σ(δ)−−−−→ Σ2(HA)

5. SPECTRUM (CO-)HOMOLOGY 213

is trivial. We choose a free abelian group F , an epimorphism ε : F −→ E and a lift ε : F −→ D of ε. Thisdata yields a morphism of extensions

Kincl //

F

ε

ε // E

Ci′

// Dj′

// E

where K denotes the kernel of ε. Now the kernel K is again a free abelian group, so we can choose a liftλ : K −→ B and get another morphism of extensions

A(1,0) // A⊕K

(iλ)

(01) // K

A

i// B

j// C

By naturality of the Bockstein operator, the composite Σ(δ(i,j)) δ(i′,j′) equals the composite Σ(δ(ε,incl)) δ((1,0),(0

1)). Since the Bockstein of a split extension is zero, this proves Σ(δ(i,j)) δ(i′,j′) = 0.

An important example is the mod-p Bockstein β : HZ/p −→ Σ(HZ/p) associated to the extension

Z/p ·p−−→ Z/p2 proj−−−−→ Z/p

where the first map sends n+ pZ to np+ p2Z and the second map sends n+ p2Z to n+ pZ.[introduce homotopy limit ? in Exercise ?]

5. Spectrum (co-)homology

We define the ‘ordinary’ homology and cohomology groups of a symmetric spectrum with coefficientsin an abelian group as generalized cohomology groups in the special case of Eilenberg-Mac Lane spectra.

Definition 5.1. Let A be an abelian group, k an integer and X a symmetric spectrum. The k-thhomology group of X with coefficients in A is define as

Hk(X,A) = πk(HA ∧X) .

The k-th cohomology group of the symmetric spectrum X with coefficients in A is defined as

Hk(X,A) = [X,HA]k = [X, Sk ∧HA] ,

the group of degree k maps in the stable homotopy category from X to the Eilenberg-Mac Lane spectrumof A.

[remark that HA is flat, so no derived smash product necessary][Exercise 11.7: (co-)homology can be defined from a chain complex; every stable equivalence induces

isomorphisms on (co-)homology groups]The ‘ordinary’ spectrum homology groups admit other interpretations. For example, we can replace

the smash product HA ∧X by the ‘linearization’ of X.

Definition 5.2. For a symmetric spectrum of simplicial sets X the free abelian group spectrum Z[X]is obtained by applying the reduced free abelian group functor to X levelwise and dimensionswise. Moreprecisely, Z[X] is the symmetric spectrum of abelian groups which in level n is given by Z[X]n = Z[Xn],the dimensionwise reduced simplicial free abelian group generated by the simplices of Xn. The Σn-actionis induced by the action on Xn and the structure map is the composite

Z[Xn]⊗ Z[S1]∼=−→ Z[Xn ∧ S1]

Z[σn]−−−→ Z[Xn+1] .


If A is abelian group, then the A-linearization A[X] of X is the symmetric spectrum of abelian groupsdefined as A[X] = A ⊗ Z[X], where the tensor product is taken levelwise in the spectrum direction anddimensionwise in the simplicial direction.

If A is a ring, then the linearization A[X] is given, levelwise and dimensionwise, by the reduced freeA-module generated by the symmetric spectrum X. If M is an A-modules, then M [X] is naturally asymmetric spectrum of A-modules.

As an example of linearization which we have already seen, the Eilenberg-Mac Lane spectrum HA ofExample 1.14 equals (the underlying symmetric spectrum of) the linearization A[S] of the sphere spectrum.Because HA arises from a Γ-space in this fashion, there is an assembly morphism

HA ∧X = (HA)(S) ∧X −→ A[S ∧X] ∼= A[X] ,

(compare I.(8.15)), which is a π∗-isomorphism (thus a stable equivalence) by Proposition 8.16 (ii). Thusthe A-homology of X can be calculated as the true homotopy groups of the linearization A[X].

Proposition 5.3. Let X be a semistable symmetric spectrum and A an abelian group. Then the k-thhomology group Hk(X,A) is naturally isomorphic to each of the following groups:

• the naive homotopy groups π∗(HA ∧X) of the smash product,• the naive homotopy groups π∗(A[X]) of the linearization,• the colimit colimn Hk+n(Xn, A) of the reduced homology groups of the levels, taken over the system

Hk+n(Xn, A) ∼= Hk+n+1(Xn ∧ S1, A)Hk+n+1(σn,A)−−−−−−−−−→ Hk+n+1(Xn+1, A)

with the first map being the suspension isomorphism [define suspension isomorphism consistently].

Proof. Since HA is flat and both HA and X are semistable, the smash product HA∧X is semistableby Proposition I.7.36. So the natural map c : πK(HA∧X) −→ πk(HA∧X) = Hk(X,A) is an isomorphism.

If X is semistable, then A[X] is semistable by Proposition 8.16 (iii) and so [...](i) We have isomorphisms

πk+n(HA)n = πk+nA[Xn] ∼= Hk+n(Xn, A)

compatible with stabilization. Passing to the colimit in n provides an isomorphism πk(HA) ∼=colimn Hk+n(Xn, A).

[refer to evaluation of simplicial functor on a spectrum; If B is a Γ-space then (Z[B])(S) = Z[B(S)].]

Remark 5.4. Let us denote the colimit by Hk(X,A) the colimit of the groups Hk+n(Xn, A) formedas in Proposition 5.3 and refer to it as the k-th naive homology group of X with coefficients in A. Forspectra which are not semistable, one cannot generally hope that the naive homology group calculatesthe spectrum homology group Hk(X,A). For example, the naive group H0(F1S

1,Z) is free abelian ofcountably infinite rank, by essentially the same argument as for π0(F1S

1) in Example I.3.11. However,the spectrum F1S

1 is stably equivalent to the semistable sphere spectrum, so H0(F1S1,Z) is isomorphic to

H0(S,Z) = π0(HZ ∧ S) ∼= Z.We can say more about the relationship between the naive and true homology groups also for spectra

which are not necessarily semistable. Since Hk(X,A) is the colimit, over the category N of inclusions, ofan I-functor, it has a natural structure of a tame module over the injection monoidsM (compare step 2 inConstruction I.7.2). Moreover, the proof of Proposition 5.3 really show that for any X, semistable or not,the naive homology group Hk(X,A) is isomorphic, as anM-module, to the naive homotopy group πkA[X]of the linearization. Since the assembly map HA ∧X −→ A[X] is a π∗-isomorphisms, the naive and truehomotopy groups of the linearization are isomorphic to the naive respectively true homotopy groups of thespectrum HA ∧X. The naive-to-true spectral sequence for the linearization HA ∧X (see Theorem I.7.41)becomes a spectral sequence

E2p,q = Hp(M, Hq(X,A)) =⇒ Hq+p(X,A) .


If X is semistable, then so is HA ∧ X, and thus the naive homology groups have trivial M-action. Thespectral sequence then degenerates to the isomorphism Hk(A,X) ∼= Hk(A,X) between naive and truehomology groups, which recovers part of Proposition 5.3.

The spectrum cohomology groups of X can also be related to the cohomology groups of the spaces (orsimplicial sets) which make up the symmetric spectrum X. The relationship is slightly more complicatedthan for homology. The precise statements are as follows:

Proposition 5.5. For a semistable symmetric spectrum X, an abelian group A and an integer k thereis a natural short exact sequence

0 −→ lim1nH

k+n−1(Xn, A) −→ Hk(X,A) −→ limnHk+n(Xn, A) −→ 0

where the limit is taken over the inverse system of reduced cohomology groups

Hk+n+1(Xn+1, A)Hk+n+1(σn,A)−−−−−−−−−−→ Hk+n+1(Xn ∧ S1, A) ∼= Hk+n(Xn, A)

and the derived limit is taken of the analogous sequence with dimensions shifted by 1.

Proof. Since X is semistable, it is stably equivalent to the mapping telescope teln Ωn(Σ∞Xn) [ref].The mapping telescope is a homotopy colimit in the triangulated stable homotopy category, so the Milnorsequence (4.8) of Chapter II becomes a short exact sequences

0 −→ lim1n [ΣΩn(Σ∞Xn),HA] −→ [teln Ωn(Σ∞Xn),HA] −→ limn[Ωn(Σ∞Xn),HA] −→ 0

Now replace [Ωn(Σ∞Xn),HA] by the isomorphic group [Σ∞Xn,Sn ∧HA] and finally by Hn(Xn, A).

Proposition 5.6. (i) For suspension spectra there are natural isomorphism of (co)homologygroups

Hk(Σ∞K,A) ∼= Hk(K,A) and Hk(Σ∞K,A) ∼= Hk(K,A)for k ∈ Z and pointed spaces (or simplicial sets) K. [specify the maps]

(ii) There are natural short exact sequences of abelian groups

0 −→ A⊗Hk(X,Z) −→ Hk(X,A) −→ Tor(A,Hk−1(X,Z)) −→ 0

and0 −→ Ext(Hk−1(X,Z), A) −→ Hk(X,A) −→ Hom(Hk(X,Z), A) −→ 0

which (non-naturally) split.(iii) Given a short exact sequence of abelian groups

0 −→ Ai−−→ B

j−−→ C −→ 0

there is a unique morphism HC −→ Σ(HA) in the stable homotopy category such that the diagram

HAHi−−−→ HB

Hj−−−→ HCδ−−→ Σ(HA)

is a distinguished triangle. The long sequences which fit into a long exact sequence

· · · −→ Hk(X,A)Hk(X,i)−−−−−→ Hk(X,B)

Hk(X,j)−−−−−→ Hk(X,C) δ−−→ Hk−1(X,A) −→ · · ·and

· · · −→ Hk(X,A)Hk(X,i)−−−−−→ Hk(X,B)

Hk(X,j)−−−−−→ Hk(X,C) δ−−→ Hk+1(X,A) −→ · · ·are exact, where the connecting homomorphisms are defined as

πk(HC ∧X)πk(δ∧Id)−−−−−−→ πk(Σ(HA) ∧X) ∼= πk−1(HA ∧X)

in the homological case and as the composite

[X, Sk ∧HC][X,Sk∧δ]−−−−−→ [X, Sk ∧ Σ(HA)] ∼= [X, Sk+1(HA)]

in the cohomological case.


(vi) [Suspension isomorphism, long exact sequence for mapping cone]

Proof. (i) Suspension spectra are semistable, so by Proposition 5.3 the spectrum homology groupHk(Σ∞K,A) can be calculated as the colimit of the reduced cohomology groups Hk+n(K ∧ Sn, A). In thisparticular case, all the maps in the colimit system are isomorphisms, hence so is the canonical map fromHk(K,A) = Hk((Σ∞K)0, A) to the colimit. The cohomological case is similar, but it uses Proposition 5.5instead of Proposition 5.3 and the fact that the lim1-term over a system of isomorphisms is trivial.

(ii) [Adapt the proof for the new definition] It is a purely algebraic fact [ref] that every chain complexC of free abelian groups gives rise to natural short exact sequences

0 −→ A⊗HkC −→ Hk(A⊗ C) −→ Tor(A,Hk−1C) −→ 0

and0 −→ Ext(Hk−1C,A) −→ Hk(Hom(C,A)) −→ Hom(HkC,A) −→ 0

which split non-naturally. For a ∈ A and a k-cycle x of C the first map takes an elementary tensor a⊗ [x]to [a⊗ x]. [second map] So (ii) is the special case for the singular chain complex CX.

(iii) Associated to the short exact sequence of abelian groups there is a The long cohomological sequenceis exact because it is obtained by applying SHC(X,−) to the triangle. By Proposition 3.14 the diagram

HA ∧X Hi∧Id−−−−−→ HB ∧X Hj∧Id−−−−−→ HC ∧X κHA,X(δ∧Id)−−−−−−−−−→ Σ(HA ∧X)

is also a distinguished triangle. [Because Eilenberg-Mac Lane spectra are flat we can use pointset levelsmash products].

There are various relations between homotopy and homology groups of spectra. The Hurewicz andWhitehead theorems for singular homology of spaces immediately imply Hurewicz and Whitehead theoremsfor spectrum homology. The only caveat is that the Whitehead theorem only applies to spectra whosehomotopy groups are bounded below. The unit morphism S −→ HZ can be smashed with a symmetricspectrum to yield a morphism X ∼= S ∧ X −→ HZ ∧ X. The Hurewicz homomorphism is the naturalmorphism of abelian groups

h : πkX −→ Hk(X,Z)

induced by this morphism on homotopy groups. This is closely related to the classical Hurewicz homo-morphism for topological spaces: if X is semistable, then the source of the Hurewicz homomorphism isisomorphic to the colimit of the homotopy groups πk+nXn, whereas the target group is isomorphic tothe colimit of the reduced homology groups Hk+n(Xn,Z) (by Proposition 5.3). Exercise 11.12 shows theHurewicz homomorphism which we just defined corresponds to the map induced by the classical Hurewiczhomomorphisms for the spaces Xn which by suitable passage to colimits.

[induced by the homomorphism of symmetric spectra X −→ Z[X] given by the inclusion of generators.]

Proposition 5.7. (i) (Stable Hurewicz theorem) Let X be a (k − 1)-connected symmetric spec-trum for some integer k. Then the homology groups of X are trivial below dimension k and theHurewicz homomorphism πkX −→ Hk(X,Z) is an isomorphism.

(ii) (Stable Whitehead theorem) Let f : X −→ Y be a morphism between symmetric spectrum whosehomotopy groups are bounded below. Then f is a stable equivalence if and only if it inducesisomorphisms on all integral homology groups.

(iii) Let A be a uniquely divisible abelian group (i.e., a Q-vector space). Then the smash product pairing

A⊗ πlX = π0(HA)⊗ πlX·−−→ πl(HA ∧X) = Hl(X,A)

s an isomorphism for every integer k. In particular, the Hurewicz homomorphism induces anisomorphisms

Q⊗ πkX ∼= Q⊗Hk(X,Z) ∼= Hk(X,Q)

for all symmetric spectra X and integers k.


Proof. (i) This is a special case of Proposition 4.18, or rather its corollary Remark 4.19. Since theEilenberg-Mac Lane spectrum HZ is flat and (−1)-connected, the natural map

· : π0(HZ) ⊗ πkX −→ πk(HZ ∧X)

is an isomorphism of abelian groups. The group π0(HZ) is isomorphic to Z and the Hurewicz map is givenby 1 · − : πkX −→ πk(HZ ∧X), so it is an isomorphism.

(ii) One direction does not need the hypothesis that X and Y are bounded below: if f : X −→ Y is astable equivalence, then so is HZ ∧ f : HZ ∧X −→ HZ ∧ Y since the Eilenberg-Mac Lane spectrum is flat[ref].

Suppose conversely that f is an integral homology isomorphism. Then [level cofibrant] the mappingcone C(f) has trivial integral homology by the long exact sequence of homology groups [ref]. Since thehomotopy groups of X and Y are bounded below, so are the homotopy groups of the cone, by the longexact sequence of homotopy groups [ref]. So if C(f) had a non-trivial homotopy group, there would be aminimal one, and by part (i) there would also be a non-trivial homology group in that minimal dimension.This would contract what we concluded before, so all homotopy groups of the mapping cone C(f) vanish,and so f is a stable equivalence, one more time by the long exact sequence of homotopy groups.

(iii) We let X be the smallest class of symmetric spectra X for which that map A⊗πkX −→ Hk(X,A) isan isomorphism for every integer k. This class is closed under stable equivalences, thus under isomorphismin the stable homotopy category. The sphere spectrum S belongs to X by Serre’s calculation of homotopygroups of spheres modulo torsion. Both sides of the map commute with sums, so the class X is closed undersums in SHC. Finally, both sides of the map take distinguished triangles in the stable homotopy categoryto long exact sequences, so the class X is closed under extensions by the 5-lemma. In other words, X is alocalizing subcategory of the stable homotopy category which contains the sphere spectrum, so it containsall symmetric spectra by Proposition 4.2

Example 5.8. The hypothesis in the Whitehead theorem (Proposition 5.7 (ii)) that homotopy groupsare bounded below is essential. In general, H∗(−; Z)-isomorphisms need not be π∗-isomorphism (and alsonot stable equivalences). As an example let Mn(p) = Sn ∪p Dn+1 be the mod-p Moore space of dimensionn+ 1, for a prime number p. For n ≥ 2, this space is characterized up to weak equivalence by the propertythat its reduced integral homology is concentrated in dimension n, and Hn(Mn(p); Z) is cyclic of order p.

Since Mn(p) is the mapping cone of the degree p map on the sphere Sn, every generalized cohomologytheory E gives rise to a long exact sequence of reduced cohomology groups

· · · −→ Em−1(Sn)×p−−−→ Em−1(Sn) δ−−→ Em(Mn(p)) −→ Em(Sn)

×p−−−→ Em(Sn) δ−−→ · · ·

which splits up into short exact sequences

(5.9) 0 −→ Z/p⊗ Em−1(Sn) −→ Em(Mn(p)) −→ pEm(Sn) −→ 0

(pA denotes the subgroup of an abelian group A consisting of those elements a which satisfy pa = 0).In particular, this applies to complex topological K-theory KU∗ (compare Example I.1.18). The

coefficients of complex K-theory are 2-periodic and πnKU = KU−n = KU0(Sn) is a free abelian group of

rank one when n is even and trivial when n is odd. So for KU -theory, the short exact sequence (5.9) showsthat

KUm(Mn(p)) ∼=

Z/p if m− n is odd, and0 if m− n is even.

Adams showed that for every odd prime p and sufficiently large n there exists a map

v : Σ2p−2Mn(p) −→ Mn(p)

which induces an isomorphism in reduced complex K-theory. This implies that every iterated compositionof (suspended copies of) the map v induces an isomorphism in K-theory, hence every such composite isstably essential. A map with this property is called a periodic self-map. The Moore space at the prime 2


also admits a periodic self-map, but the period and some other details are different from the situation atodd primes.

The mod-p Moore spectrum is the spectrum Ω2(Σ∞M2(p)). Note that the spectrum homologyH∗(M(p); Z) is isomorphic to Z/p, concentrated in dimension 0 (by [...]). The Adams map v :Σ2p−2Mn(p) −→Mn(p) can be used to define a morphism of spectra

v1 : M(p) −→ S−2p+2 ∧M(p)

which also induces an isomorphism on complex topological K-theory; here the topological K-theory of acofibrant spectrum A is defined by homotopy classes of morphisms into the K-theory spectrum, i.e.,

KUm(A) = [A,Sm ∧KU ] .

We define a spectrum v−1M(p) by ‘inverting the self-map v’, i.e., as the mapping telescope (or the diagonal)of the sequence of symmetric spectra

M(p) v1−→ S−2p+2 ∧M(p) v1−→ S−4p+4 ∧M(p) v1−→ · · · .

Every spectrum in this sequence is essentially a suspension spectrum, shifted down finitely many dimensions,but the telescope is no longer a shift of any suspension spectrum. The Adams map v1 : M(p) −→ S−2p+2 ∧M(p) induces the trivial map in spectrum homology simply because the homology of source and target areconcentrated in different dimensions. Since the homology of a telescope is the colimit of the homologies[prove], the integral spectrum homology of the mapping telescope v−1

1 M(p) is trivial. On the other hand,the spectrum v−1

1 M(p) has non-trivial topological K-theory. Since a mapping telescope is a homotopycolimit in the triangulated stable homotopy category, we obtain the short exact Milnor sequence (4.8) ofChapter II for the KU -theory of the mapping telescope

0 −→ lim1KUm−1(Xi) −→ KUm(tel(Xi)) −→ lim KUm(Xi) −→ 0 .

[use K-homology to avoid lim1 ?] In the case of the mapping telescope v−1M(p), the Adams map v inducesan isomorphism in K-theory; so the limit is isomorphic to its first term KUmM(p) and the derived limit istrivial. In particular all the odd dimensional groups

SHC(v−11 M(p),S2i+1 ∧KU) = KU2i+1(v−1

1 M(p)) ∼= KU2i+1(M(p)) ∼= Z/p

are non-trivial. To sum up, the mapping telescope v−1M(p) has trivial spectrum homology, but it is notstably contractible. We conclude that v−1

1 M(p) must have non-trivial homotopy groups in arbitrarily lowdimensions. The homotopy groups of this mapping telescope were determined by Miller [57, Cor. 4.12],who showed that

πn(v−1M(p)

)=

Z/p for n ≡ −1, 0 modulo 2p− 2,0 else.

This calculation requires much more sophisticated tools than we have available.

We have a morphism

HA ∧L DX = HA ∧L F (X, S) −→ F (X,HA)

which is adjoint to Id∧εX,S : HA∧L F (X, S)∧LX −→ HA. Taking k-th homotopy group and applying theisomorphism πkF (X,HA) ∼= SHC(X, Sk ∧HA) we obtain a natural map

Hk(DX,A) = πk(HA ∧DX) −→ πkF (X,HA) ∼= Hk(X,A) .

For finite spectra X the morphism HA∧LDX −→ F (X,HA) is a stable equivalence, hence the morphism[...] is an isomorphism.

Construction 5.10. If R is a symmetric ring spectrum and A a ring, then the homology groupsH∗(R,A) naturally form a graded ring. If the multiplications of both R and A are commutative, thenthe product on H∗(R,A) is commutative. The product on H∗(R,A) arises as follows from a pairing of

6. CONNECTIVE COVERS AND POSTNIKOV SECTIONS 219

symmetric spectra. We consider symmetric spectra X and Y and abelian groups A and B; we have apairing

(5.11) HA ∧X ∧ HB ∧ Y −→ H(A⊗B) ∧ (X ∧ Y ) .

[...] This paring is associative, commutative and unital in a sense which by now is hopefully obvious. If wetake homotopy groups [and use...] we obtain a bilinear pairing

Hk(X,A)⊗Hl(Y,B) = πk(HA∧X)⊗πl(HB ∧Y ) −→ πk+l(H(A⊗B)∧ (X ∧Y ) = Hk+l(X ∧Y,A⊗B) .

So if R is a symmetric ring spectrum and A a ring, then HA ∧R becomes a symmetric ring spectrumwith multiplication map

HA ∧R ∧ HA ∧R Id∧τR,HA∧Id−−−−−−−−−→ HA ∧HA ∧ R ∧R µ∧µ−−−→ HA ∧R ∧R .

[compare ...above] So the homology of R with coefficients in A becomes a graded ring under the product

Hk(R,A)⊗Hl(R,A) −→ Hk+l(R ∧R,A⊗A)Hk+l(µR,µA)−−−−−−−−−→ Hk+l(R,A)

where µR : R ∧ R −→ R and µA : A ⊗ A −→ A are the products of R and A respectively. If R and Ahave commutative multiplications, then HA ∧ R is a commutative symmetric ring spectrum and then thehomology product is commutative in the graded sense.

[product factors over product over K]

Proposition 5.12 (Kunneth theorem). Let A be a commutative ring and let X and Y be symmetricspectra. Suppose that for each k ∈ Z the homology Hk(X,A) is projective as a A-module. Then for everyn ∈ Z the map

(5.13)⊕k+l=n

Hk(X,A) ⊗A Hl(Y,A) −→ Hn(X ∧L Y,A)

is an isomorphism.

Proof. Fix a symmetric spectrum X all of whose A-homology groups are projective as A-modulesand let Y be the class of spectra for which the map (5.13) is an isomorphism. Then Y contains the spherespectrum and is closed under sums and extensions. Since the sphere generates the stable homotopy category(Proposition 4.2), the class Y contains all symmetric spectra.

[Kunneth for cohomology.... Kunneth spectral sequence...in context of generalized CT? pairing ho andcohomology]

[Bocksteins of short exact sequences and some other cohomology operations...]

6. Connective covers and Postnikov sections

We denote by SHC≥n the full subcategory of the stable homotopy category with objects the (n − 1)-connected spectra. We recall from Proposition 4.16 that the (n−1)-connected spectra coincide with 〈Sn〉+,the smallest class of symmetric spectra which contains the n-dimensional sphere spectrum and is closedunder sums (possibly infinite) and ‘extensions to the right’.

The next theorem uses homotopy colimits, which were defined in Definition 4.3 for general triangulatedcategories, and we gave a more explicit construction in the stable homotopy category in Proposition 4.4.

Theorem 6.1. Let n be any integer. The inclusion of the full subcategory of (n− 1)-connected spectrainto SHC has a right adjoint 〈n〉 : SHC −→ SHC≥n. The adjunction counit qn : X〈n〉 −→ X is calledthe (n − 1)-connected cover of the symmetric spectrum X. There is a unique natural transformation in :X〈n〉 −→ X〈n − 1〉 satisfying qn−1 in = qn. Moreover, the morphisms qn : X〈n〉 −→ X express everysymmetric spectrum X as the homotopy colimit of the sequence of morphisms in as n goes to −∞.

Before we prove the theorem we spelling out explicitely the main properties of this cover:• the spectrum X〈n〉 is (n − 1)-connected and for every (n − 1)-connected symmetric spectrum A

the map [A, qn] : [A,X〈n〉] −→ [A,X] is an isomorphism.


• by taking A = Sk for k ≥ n and using that Sk represents the homotopy group functor πk we seethat qn induces isomorphisms of homotopy groups in dimensions n and above.• The (n− 1)-connected cover qn : X〈n〉 −→ X is a natural transformation,

The above conditions which refer to homotopy groups also characterizes the (n−1)-connected cover. Indeed,if f : A −→ X is any morphism of symmetric spectra which induces isomorphisms of homotopy groups indimensions n and above and such that A is (n−1)-connected, then by the above there is a unique morphismf : A −→ X〈n〉 in the stable homotopy category satisfying qnf = f , and f is necessarily a π∗-isomorphism,thus an isomorphism in SHC.

Proof of Theorem 6.1. We let C = Skk≥n be the set of sphere spectra of dimensions at leastk. For a given symmetric spectrum X we apply Proposition 4.10 to the representable functor [−, X]. Weobtain a spectrumX〈n〉 belonging to 〈C〉+ and a morphism qn : X〈n〉 −→ X which induces isomorphisms on[Sk,−] for all k ≥ n, i.e., isomorphisms on homotopy groups in dimensions n and above. Since 〈C〉+ = 〈Sn〉+equals the class of (n− 1)-connected spectra, X〈n〉 is (n− 1)-connected.

Now we claim that for every (n− 1)-connected spectrum A the map [A, qn] : [A,X〈n〉] −→ [A,X] is anisomorphism. We let Z be any symmetric spectrum such that the homotopy groups πkZ are trivial for allk ≥ n. We consider the class X of symmetric spectra A with the property that the groups SHC(ΣkA,Z)are trivial for all k ≥ n. The class X is closed under sums and contains the n-dimensional sphere spectrumSn. For a distinguished triangle

A −→ B −→ C −→ ΣA

such that A and B belong to X we apply [Σk(−), Z] and get an exact sequence

[Σk+1A,Z] −→ [ΣkC,Z] −→ [ΣkB,Z] −→ [ΣkA,Z] .

For k ≥ n the first and third group are trivial, hence so is the second, This shows that X is also closedunder extensions to the right, so altogether we have 〈Sn〉+ ⊂ X . By Proposition 4.16 we then know that[ΣkA,Z] = 0 for every connective spectrum A and k ≥ n.

We choose a distinguished triangle

X〈n〉 qn−−−→ X −→ Z −→ Σ(X〈n〉) .

Since πk(p) is bijective for k ≥ n and πn−1(X〈n〉) = 0, the long exact sequence of true homotopy groupsshows that πkZ = 0 for all k ≥ n. By the above we thus have [ΣA,Z] = 0 = [A,Z] for every (n − 1)-connected spectrum A. The exact sequence

0 = [ΣA,Z] −→ [A,X〈n〉] [A,qn]−−−−→ [A,X] −→ [A,Z] = 0

then shows that [A, p] is an isomorphism.So we now know that the restriction of the functor [−, X] to the category SHC≥0 of connective spectra

is representable, namely by the pair (X〈n〉, qn) constructed above. It is then a formal consequence thatthese choices of connective covers qn : X〈n〉 −→ X can be made into a right adjoint to the inclusions, ina unique way such that qn becomes the adjunction counit. We only show how to define 〈n〉 on morphismsand omit the remaining verifications. If f : X −→ Y is a morphism in SHC, then [−, f ] : [−, X] −→ [−, Y ]is a natural transformation of functors, which we can restrict to the full subcategory SHC≥n. On thissubcategory, the two functors are represented by X〈n〉 respectively Y 〈n〉, so by the Yoneda lemma there isa unique morphism Qnf : X〈n〉 −→ Y 〈n〉 which represents the natural transformation.

Since X〈n + 1〉 is n-connected, thus also (n − 1)-connected, the morphism qn+1 : X〈n + 1〉 −→ Xis adjoint to a morphism in+1 : X〈n + 1〉 −→ X〈n〉 which is in then a natural transformation satisfyingqn in+1 = qn+1.

It remains to show that every object X of the stable homotopy category is the homotopy colimit of itsn-connective covers as n goes to −∞. We let (X−∞, ϕn : X〈n〉 −→ X−∞n≤0) be a homotopy colimit, in

the sense of Definition 4.3, of the sequence X〈0〉 i0−→ X〈−1〉 i−1−−→ · · · . Since the morphism qn : X〈n〉 −→ X

6. CONNECTIVE COVERS AND POSTNIKOV SECTIONS 221

are compatible with the sequence, there is a morphism q : X−∞ −→ X, not necessarily uniquely determined,satisfying qϕn = qn for all n ≤ 0. For every integer k we have a commutative triangle

colimn−→−∞ πkX〈n〉

πk(qn)n ((QQQQQQQQQQQQQπk(ϕn)n // πk(X−∞)

πk(q)yyttttttttt

πkX

Since the sphere spectrum is compact, the horizontal map is an isomorphism by Lemma 4.6 (ii). Theleft diagonal map is also an isomorphism since πk(qn) is bijective for n ≤ k. Thus the map πk(q) is anisomorphism for all k ∈ Z, which proves that q : X−∞ −→ X is an isomorphism in the stable homotopycategory.

Remark 6.2. In Proposition I.8.17 we already constructed connective covers with a very differenttechnique, using Γ-spaces and an adjoint functor pair ((−)(S),Λ) which relates them to symmetric spectra.The main point was that for every injective [flat fibrant suffices] Ω-spectrum X the adjunction counit(ΛX)(S) −→ X is a connective cover.

Remark 6.3. In Section III.5 we will return to the topic of connected covers from a different angle,and in a much more general context. There we consider a connective operad O of symmetric spectra andconstruct connective (i.e., (-1)-connected) covers for algebras over the operad. When O is the initial operad,then its algebras are just symmetric spectra with no extra structure. In that case the general theory givesa refinement of the functor X 7→ X〈0〉 as constructed here, namely a connective cover functor on the levelof symmetric spectra which descends to 〈0〉 on the level of the stable homotopy category. For more detailswe refer to Example III.5.10.

Instead of ‘killing’ all the homotopy groups of a spectrum below a certain dimension, we can also ‘kill’the homotopy groups above a certain point. This leads to the following notion of ‘Postnikov section’ whichis somewhat dual to that of a connective cover.

Theorem 6.4. Let n be any integer. The inclusion of the full subcategory of (n + 1)-coconnectedspectra into SHC has a left adjoint Pn : SHC −→ SHC≤n. The adjunction unit pn : X −→ PnX iscalled the n-th Postnikov section of the symmetric spectrum X. There is a unique natural transformationjn : PnX −→ Pn−1X satisfying jn pn = pn−1.

There is a unique morphism δ : PnX −→ Σ(X〈n+ 1〉) such that the diagram

(6.5) X〈n+ 1〉 qn+1−−−→ Xpn−−−→ PnX

δ−−→ Σ(X〈n+ 1〉)

is a distinguished triangle in the stable homotopy category.Moreover, the morphisms pn : X −→ PnX express every symmetric spectrum X as the homotopy limit

of the tower of morphisms jn as n goes to ∞.

We take the time to spell out the properties of Postnikov sections ‘dual’ to certain properties of con-nective covers:

• the spectrum PnX is (n+ 1)-coconnected and for every (n+ 1)-coconnected symmetric spectrumY the map [pn, Y ] : [PnX,Y ] −→ [X,Y ] is an isomorphism.• by the long exact homotopy group sequence of the triangle (6.5) the Postnikov section pn induces

isomorphisms of homotopy groups in dimensions n and below.• The n-th Postnikov section pn : X −→ PnX is a natural transformation,

The above conditions which refer to homotopy groups also characterizes the n-th Postnikov section. Indeed,if f : X −→ Y is any morphism of symmetric spectra which induces isomorphisms of homotopy groups indimensions n and below and such that Y is (n+1)-connected, then by the above there is a unique morphismf : PnX −→ Y in the stable homotopy category satisfying fpn = f , and f is necessarily a π∗-isomorphism,thus an isomorphism in SHC.


Proof of Theorem 6.4. We define the spectrum PnX and the Postnikov section pn : X −→ PnXby choosing a distinguished triangle

X〈n+ 1〉 qn+1−−−→ Xpn−−−→ PnX

δ−−→ Σ(X〈n+ 1〉) .For every (n+1)-coconnected spectrum Y the groups [X〈n+1〉, Y ] and [Σ(X〈n+1〉), Y ] vanish by is trivialin SHC by Proposition 4.20 (or rather its (n+ 1)-fold shifted version). The exact sequence

[Σ(X〈n+ 1〉), Y ][δ,Y ]−−−→ [PnX,Y ]

[pn,Y ]−−−−→ [X,Y ][qn+1,Y ]−−−−−→ [X〈n+ 1〉, Y ]

then shows that the map [pn, Y ] : [PnX,Y ] −→ [X,Y ] is an isomorphism. The remaining arguments toextend Pn to a functor which is left adjoint to the inclusions and such that pn becomes the adjunction unitare formal, and ‘dual’ to the corresponding arguments for the connective covers in the proof of Theorem 6.1.The same goes for the construction of the natural transformation jn : Pn −→ Pn−1.

The triangle (6.5) came with the construction of the Postnikov section. Lemma 4.23, applied to thecomposable morphisms qn+1 : X〈n+ 1〉 −→ X and pn : X −→ PnX, shows that the connective morphismis uniquely determined.

For the last claim we let (X∞, ϕn : X∞ −→ PnXn≤0) be a homotopy limit, in the sense of [...] of the

tower · · · j2−→ P1Xj1−→ P0X. Since the morphism pn : X −→ PnX are compatible with the sequence, there

is a morphism p : X −→ X∞, not necessarily uniquely determined, satisfying ϕnp = pn for all n ≤ 0. Forevery integer k, the system of homotopy groups πkPnX eventually stabilizes to πkX, so in the short exactsequence [ref]

0 −→ lim1n πk(ΣPnX) −→ πk(holimn PnX) −→ limn πk(PnX) −→ 0 .

the lim1-term is trivial and the inverse limit is isomorphic to πkX, via the Postnikov sections pn : X −→PnX. Thus the morphism p : X −→ X∞ induces isomorphisms of all homotopy groups, so it is anisomorphism in the stable homotopy category.

Remark 6.6. For n ≥ m we have

(X〈m〉)〈n〉 ∼= X〈n〉 ∼= (X〈n〉)〈m〉 and Pn(PmX) ∼= PmX ∼= Pm(PnX)

via instances of the maps pk respectively qk. Moreover

Pn(X〈m〉) ∼= (PnX)〈m〉and this spectrum has its homotopy groups concentrated in dimensions m through n.

[Exercise: [HA,HB]n = 0 for n < 0. Thus X is coconnective if and only if it belongs to 〈HA〉− [check]][show — or exercise —

X〈n+ k〉 ∼= Sn ∧ ((S−n ∧X)〈k〉) and Pn+kX ∼= Sn ∧ Pk(S−n ∧X).

]By Proposition 4.20 (or rather its n-fold suspension) there is a unique morphism i : Sn ∧H(πnX) −→

PnX such that the composite

πnX = π0H(πnX) Sn∧−−−−−→ πn(Sn ∧H(πnX)) πni−−→ πn(PnX)

coincides with the isomorphism induced by the Postnikov section pn : X −→ PnX. Lemma 4.23 provides aunique morphism kn : Pn−1X −→ Σn+1H(πnX) such that the triangle

Sn ∧H(πnX) i−−→ PnXjn−−−→ Pn−1X

kn−−→ Sn+2 ∧H(πnX)

is distinguished. The morphism kn is an element of the cohomology group Hn+1(Pn−1X,πnX) and iscalled the n-th k-invariant of the spectrum X. [X is ‘determined’ by the sequence of homotopy groupsand k-invariants] We will see below [...] that for rational spectra X (i.e., all homotopy groups of X areuniquely divisible) all k-invariant are trivial since X decomposes as a product of suspended Eilenberg-MacLane spectra.

7. MOORE SPECTRA 223

[exercise: the k-invariant can also be defined as the composite

Pn−1X ∼= Pn−1(PnX) δ−→ Σ((PnX)〈n+ 1〉) ∼= Σn+2H(πnX)

where δ is the connecting morphism of the distinguished triangle (6.5) for the spectrum PnX and n replacedby n− 1.]

Remark 6.7. For spectra with additional structure one can typically refine the k-invariants and liftthem from the the spectrum cohomology group Hn+1(Pn−1X,πnX) to a ‘refined’ cohomology group whichdepends on the type of structure under consideration. We indicate a first example of this phenomenon inthe case of homotopy ring spectra. We intend to return to this later, treating in particular the cases ofsymmetric ring spectra (where the appropriate home for the k-invariant are topological derivation groups,closely related to topological Hochschild cohomology) and commutative ring spectra (where the appropriatetheory is topological Andre-Quillen cohomology).

Suppose R is a homotopy ring spectrum and M a homotopy R-bimodule. By a derivation of R withcoefficients in M we mean a morphism d : R −→M in the stable homotopy category which satisfies

dµ = d ∧ Id + Id∧da morphisms R ∧L R −→M in SHC.

If R is connective and n ≥ 0, one can show that the spectrum PnR inherits a unique structure ofhomotopy ring spectrum such that the Postnikov section pn : R −→ PnR is a morphism of homotopy ringspectra. Moreover, the k-invariant kn+1 : PnR −→ Σn+2(H(πn+1R)) is a derivation of PnR with coefficientsin the bimodule Σn+2(H(πn+1R)).

Example 6.8. The first k-invariant of the sphere spectrum S is a morphism k1(S) ∈ H2(HZ,Z/2)(where we identified P0S with HZ and the first stable stem πs

1∼= π1S with the group Z/2). This k-invariant

is non-zero [justify] and is in fact the pullback of the Steenrod operation Sq2 ∈ H2(HZ/2,Z/2) along theprojection morphism HZ −→ HZ/2.

The second (and first non-trivial) k-invariant of the connective complex K-theory spectrum ku is amorphism k2(ku) ∈ H2(HZ,Z) (where we identified P0S with HZ and π2ku with Z). This k-invariant isnon-zero [justify] and is in fact equal to β k1(S) where β : HZ/2 −→ Σ(HZ) is the Bockstein operator(see Proposition 4.24) associated to the extension Z ·2−→ Z −→ Z/2. The equation k2(ku) = β k1(S) is away of rephrasing the Toda bracket relation u ∈ 〈1, η, 2〉 (modulo 2u) in π2(ku) (compare I.6.15).

For an odd prime p the first non-trivial homotopy groups of the localized sphere spectrum S(p) inpositive dimension is a copy of Z/p generated by the class α1 in dimension 2p− 3. So the first non-trivialk-invariant of S(p) is a morphism k2p−3(S(p)) ∈ H2p−2(HZ(p),Z/p). As in the case p = 2, this k-invariant isnon-zero [justify] and is in fact the pullback of the Steenrod operation P 1 ∈ H2p−2(HZ/p,Z/p) along theprojection morphism HZ(p) −→ HZ/p.

The k-invariants k1(S), k2p−3(S(p)) and k2(ku) arise from symmetric ring spectra, so they are in factderivations by Remark 6.7. The only derivations (up to units) in the mod-p Steenrod algebra A∗ arethe Milnor elements Qn ∈ H2pn−1(HFp,Fp). We have Q0 = β, the mod-p Bockstein, and the otherclasses are inductively defined as commutators Qn+1 = [Qn, P p

n

]. These derivations are all realized ask-invariants of the suitable symmetric ring spectra, namely the connective Morava K-theory spectra k(n)(see Example I.6.32), i.e., we have Qn = k2pn−2(k(n)). [check this]

[Atiyah-Hirzebruch spectral sequences for homology and cohomology]

7. Moore spectra

Definition 7.1. A Moore spectrum is a connective symmetric spectrum X for which the homologygroups Hk(X,Z) are trivial for k 6= 0.

Given an abelian group A, a Moore spectrum for A is a Moore spectrumX together with an isomorphismbetween H0(X,Z) and A. When no confusion can arise we suppressed the isomorphism and abuse notationby writing H0(X,Z) = A (instead of using the isomorphism which is part of the data of a Moore spectrum).


We often write SA for a Moore spectrum for A; however, we emphasize that SA is not functorial in theabelian groups. In general, the construction of Moore spectra is not even functorial in the stable homotopycategory, although it is when we stay away from 2-torsion. We discuss the functoriality (or rather the lackthereof) of Moore spectra in Theorem 7.11 below.

Remark 7.2. The spectrum homology group Hk(X,Z) was defined as πk(HZ ∧ X), so if X is aMoore spectrum then the spectrum HZ ∧ X has trivial homotopy groups in all nonzero dimensions. ByTheorem 4.21 (ii) we deduce that HZ ∧X is then stably equivalent to the Eilenberg-Mac Lane spectrumHπ0X. We conclude that X is a Moore spectrum if and only if it is connective and HZ∧X and the stablyequivalent linearization Z[X] are stably equivalent to Hπ0X.

If the homotopy groups of X are bounded below, then by the Hurewicz theorem (Proposition 5.7) thefirst non-trivial homotopy groups of X is isomorphic to the first non-trivial integral homology group. Sothe condition that a Moore spectrum is connective can be weakened to the requirement that the homotopygroups are bounded below. However, some condition of this kind is necessary if we want Moore spectra tobe determined by the 0th homology groups. In fact, there exist spectra X with non-trivial homotopy groupsin arbitrary low dimensions such that the homology group Hk(X,Z) is trivial for all integers k. An exampleis mod-n topological K-theory for any n ≥ 2, which is the symmetric spectrum SZ/n ∧KU where SZ/n isa flat mod-n Moore spectrum and KU is the symmetric ring spectrum representing complex topologicalK-theory from Example I.1.18. Compare Example 5.8.

Example 7.3. The sphere spectrum S is a Moore spectrum for the group Z of integers. The spectrum‘sphere spectrum with m inverted’ S[1/m] of Example 3.26 is a Moore spectrum for the group Z[1/m] ofintegers with m inverted. The symmetric spectrum underlying the ring spectrum S[1/2, i] of Example 3.28is a Moore spectrum for the abelian group underlying the Gaussian integers with 2 inverted.

Example 7.4. If A is any uniquely divisible abelian groups (i.e., a vector space over the rationalnumbers), then the Eilenberg-Mac Lane spectrum HA is also a Moore spectrum for the group A. Indeed,HA is connective, so H0(HA,Z) ∼= π0HA = A by the Hurewicz Theorem (Proposition 5.7 (i)). The groupHk(HA,Z) is isomorphic to Hk(HZ, A) (induced by the symmetry isomorphism τ : HA∧HZ ∼= HZ∧HA)which by Proposition 5.7 (iii) is isomorphic to A⊗ πkHZ and thus trivial for k 6= 0.

Example 7.5. Suppose K is a Moore space for the abelian group A of dimension n, i.e., the reducedintegral homology of K is concentrated in dimension n where we have Hn(K,Z) ∼= A. Then the spectrumΩn(Σ∞K) is a Moore spectrum for the group A. Indeed, suspension spectra are connective, and loopingshifts homotopy groups, so Ωn(Σ∞K) is connective. A chain of isomorphisms

Hk(Ωn(Σ∞K),Z) ∼= Hk+n(Sn ∧ Ωn(Σ∞K),Z) ∼= Hk+n(Σ∞K,Z) ∼= Hk+n(K,Z)

is given by the suspension isomorphism for homology, the fact that the adjunction counit Sn ∧ Ωn(Σ∞K)is a stable equivalence [ref] and the isomorphism given by Proposition 5.6 (i). So the spectrum homologyof Ωn(Σ∞K) is indeed concentrated in dimension 0, where it is isomorphic to A.

Example 7.6. Let A be a subring of the ring Q of rational numbers. Then we can write down acommutative symmetric ring spectrum which is a Moore spectrum for A. First we introduce a more generalconstruction based on a pair (K, f) consisting of a base space (or simplicial set) K and a based mapf : S1 −→ K. From this data we define a commutative symmetric ring spectrum S(K, f) with levels

S(K, f)n = K∧n

with Σn permuting the smash factors. The multiplication map µn,m : K∧n ∧ K∧m −→ K∧(n+m) is thecanonical isomorphism. The unit map ι0 : S0 −→ K∧0 is the identity and the unit map ι1 : S1 −→ Kis the given map f . We have already seen special cases of this construction: S(S1, Id) = S is the spherespectrum, and if ϕm : S1 −→ S1 is a map of degree m, then S(S1, ϕm) = S[1/m] is the ‘sphere spectrumwith m inverted’ as defined in Example I.3.26. The functor S from the category of based space under S1

to commutative symmetric ring spectra is left adjoint to the ‘evaluation’ or forgetful functor which sends acommutative symmetric ring spectrum R to the pair (R1, ι1).


Back in the situation of a subring A of Q we choose a Moore space M for A, i.e., a CW-complex(resp. simplicial set) whose reduced integral homology is concentrated in dimension 1, where it is isomorphicto A. We also choose a based map ι : S1 −→M which sends the fundamental homology class of the circleto the class in H1(M ; Z) ∼= A which corresponds to the unit element of the ring A. We claim that then thecommutative symmetric ring spectrum S(M, ι) is a Moore spectrum for the ring A.

For n ≥ 1 the structure map

S(M, ι)n ∧ S1 = M∧n ∧ S1 σn−−−→ M∧n+1 = S(M, ι)n+1

is a homology isomorphism by the Kunneth theorem for space level singular homology and since A⊗ A isisomorphic to A. Since source and target of σn are simply connected CW-complexes, the structure map is aweak equivalence. As a consequence, the shifted spectrum sh(S(M, ι)) is level equivalent to the suspensionspectrum of M . So S(M, ι) is stably equivalent to Ω(Σ∞M), and thus a Moore spectrum by Example 7.5.This also shows that the symmetric spectrum S(M, ι) is semistable, which is not generally the case forS(X, f).

Construction 7.7. Now we give a construction of a Moore spectrum for a given group A in terms ofthe triangulated structure of the stable homotopy category. This construction also allows us to calculatethe homotopy groups of Moore spectra and the groups of maps out of Moore spectra.

We choose a free presentation of A, i.e., a short exact sequence

0 −→ Z[I] d−−→ Z[J ] e−−→ A −→ 0

where I and J are indexing sets. We have

π0(⊕I

S) ∼=⊕I

π0S ∼= Z[I]

and similarly for the sum of sphere spectra indexed by the set J . Since S represents the functor π0 wecan realize the map d : Z[I] −→ Z[J ] by a morphism d :

⊕I S −→

⊕J S, i.e., π0(d) equals d under the

isomorphisms. Now we choose a distinguished triangle

(7.8)⊕I

S d−−→⊕J

S −−→ SA −−→⊕I

ΣS .

We claim that SA is a Moore spectrum for the group A.We first describe the homotopy groups of SA in terms A and the stable homotopy groups of spheres.

The long exact homotopy sequence of this triangle contains the exact sequence

πn(⊕I

S) πnd−−→ πn(⊕I

S) −−→ πnSA −−→ πn−1(⊕I

S) −−→ πn−1(⊕J

S) .

Using that homotopy groups preserve sums we can rewrite this as an exact sequence

Z[I]⊗ πnS d⊗πnS−−−−→ Z[J ]⊗ πnS −→ πnSA −→ Z[I]⊗ πn−1S d⊗πn−1S−−−−−−→ Z[J ]⊗ πn−1S .

Since we started from a free resolution of A, kernel respectively cokernel of the morphism d⊗ πnS are thegroups A⊗πnS respectively Tor(A, πnS). So the long exact homotopy sequence decomposes into short exactsequences

(7.9) 0 −→ A⊗ πsn −−→ πnSA −−→ Tor(A, πs

n−1) −→ 0 .

This shows in particular that SA is connective and gives an isomorphism between π0SA and A. The longexact homology sequence of the triangle (7.8) and the vanishing of the homology of S in positive dimensionsshow that Hk(SA,Z) is trivial for k ≥ 1 (for k = 1 this also uses that H0(d,Z) is injective). So SA is indeeda Moore spectrum for the group A.

With a similar argument we can calculate the morphisms from SA to any other object X of the stablehomotopy category. We apply [−, X] to the distinguished triangle (7.8) and use the isomorphisms

[⊕I

S, X] ∼=∏

Iπ0X ∼= Hom(Z[I], π0X)


and similarly for J instead of I and for morphisms of degree 1. We obtain an exact sequence

Hom(Z[I], π0X)Hom(d,π0X)←−−−−−−−− Hom(Z[J ], π0X)←− [SA,X]←− Hom(Z[I], π1X)

Hom(d,π1X)←−−−−−−−− Hom(Z[J ], π1X) .

Since kernel respectively cokernel of the morphism Hom(d, πnX) are the groups Hom(A, π0X) respectivelyExt(A, π0X), the long exact homotopy sequence decomposes into short exact sequences

(7.10) 0 −→ Ext(A, π1X) −−→ [SA,X] π0−→ Hom(A, π0X) −→ 0 .

Theorem 7.11. Let X and Y be two Moore spectra. Then every homomorphism f : H0(Y,Z) −→H0(X,Z) can be realized by a morphism f : Y −→ X in the stable homotopy category. If f is an isomor-phism, then so is f . In particular, Moore spectra for a given group are unique up to isomorphism in thestable homotopy category.

Moreover, when restricted to Moore spectra for 2-divisible groups, the functor π0 becomes an equivalenceof categories. In particular, Moore spectra for 2-divisible groups can be chosen functorially in the stablehomotopy category.

Proof. In a first step we let Y = SA be a Moore spectrum of the special kind constructed in (7.7).Then the realizability of f is simply the surjectivity of the exact sequence (7.10), using that the zerothhomotopy and homology groups of a Moore spectrum are naturally isomorphic.

If f is an isomorphism and f : Y −→ X realizes f , then it induces an isomorphism on H0(−,Z), andthus on all integral homology groups (since X and Y are Moore spectra). Since X and Y are connective,f is an isomorphism in the stable homotopy category by the Whitehead theorem (Proposition 5.7).

The first step applied to Y = S(π0X) shows in particular that every Moore spectrum X is isomorphicin the stable homotopy category to S(π0X). We may thus assume without loss of generality that X = SAand Y = SB are both of the form constructed in (7.7).

The stable 0-stem πs0 is free abelian, hence flat, and πs

1∼= Z/2 generated by the Hopf map η. So

for n = 1 the short exact sequence (7.9) reduces to an isomorphism B ⊗ Z/2 ∼= π1(SB). So the exactsequence (7.10) becomes a short exact sequence

0 −→ Ext(A,B ⊗ Z/2) −−→ [SA,SB] π0−→ Hom(A,B) −→ 0 .

So if B is 2-divisible, then π0 : [SA,SB] −→ Hom(A,B) is bijective, i.e., π0 is fully faithful on this class ofMoore spectra.

There are certain similarities between Eilenberg-Mac Lane and Moore spectra; for example, both aredefined by the property that a certain homology theory (homotopy respectively integral homology) isconcentrated in dimension zero, and both exist for every abelian group and are unique up to isomorphismin the stable homotopy category. We want to emphasize however, that the functoriality properties of thesetwo kinds of spectra are very different.

Eilenberg-Mac Lane spectra have models which are functorial in the group on the point set level, i.e., asfunctors to the category of symmetric spectra. We have given such a construction in Example 1.14, and thatmodel also takes rings to symmetric ring spectra. One can also deduce this from the fact the Eilenberg-MacLane functor H with values in symmetric spectra is a symmetric monoidal functor with respect to smashproduct of spectra and tensor product of abelian groups (compare Example I.5.20).

For Moore spectra the situation is different. While they exist for every abelian group and we havegiven several different constructions, there are no pointset level models which are functorial in the groupA. Even worse, we will see in Example 7.13 below that in general, Moore spectra are not even functorial inthe homotopy category. Theorem 7.11 say that after inverting the prime 2, Moore spectra can be chosenfunctorially in the stable homotopy category, but even then, the construction is not compatible with tensorproduct respectively smash product. After inverting 2 and 3, Moore spectra can be made into a symmetricmonoidal functor to the stable homotopy category. This implies that away from 6 the Moore spectrumassociated to any ring can be made into a homotopy ring spectrum inducing the given multiplication on π0.However, Moore spectra of rings can in general not be realized as a symmetric ring spectrum. Specifically,for no n ≥ 2 can the mod-n Moore spectrum be realized as symmetric ring spectra. (prove this later) If


we invert all primes, i.e., for uniquely divisible abelian groups, all these problems go away since rationally,Moore and Eilenberg-Mac Lane spectra coincide, see Example 7.4.

Let us simplify the notation a little by writing S/p for the mod-p Moore spectrum S(Z/pZ). We calculatethe mod-p cohomology of S/p. Let us denote by ι : S −→ HZ/p the unit morphisms of the ring spectrumstructure, which we can view as a cohomology class in the group H0(S,Fp). Both rows in the diagram

S·p //

ι

Sj //

ι

S/p

e0

δ // ΣS

Σι

HZ/p ·p

// HZ/p2 // HZ/pβ

// ΣHZ/p

are distinguished triangles in the stable homotopy category, and the left square commutes. Here β isthe Bockstein morphism associated to the short exact sequence [...], compare [...]. So we can choose amorphism e0 : S/p −→ HZ/p which makes the entire diagram commute. The long exact sequence inmod-p cohomology associated to the upper distinguished triangle shows that the mod-p cohomology ofS/p is concentrated in dimensions 0 and 1, where they are 1-dimensional generated by e0 ∈ H0(S/p,Z/p)respectively e1 = β(e0) ∈ H1(S/p,Z/p).

Example 7.12. The mod-p Moore spectra for a prime p behave quite differently when p = 2 or p isodd. Let us first discuss the case of odd primes p. Since πs

1∼= Z/2 the exact sequence (7.9) shows that

π1(S/p) is trivial. So the map π0 : [S/p,S/p] −→ Hom(Z/p,Z/p) is an isomorphism. Thus p times theidentity of S/p is trivial in the stable homotopy category, and all groups of the form [X, S/p] or [S/p,X] forX in the stable homotopy category are Fp-vector spaces. In particular this holds for the homotopy groups,and so the exact sequence (7.9)

0 −→ Z/p⊗ πsn −→ πn(S/p) −→ pπs

n−1 −→ 0

splits. Here we write pA for x ∈ A | px = 0 ∼= Tor(Z/p,A).

Example 7.13. Now we discuss the mod-2 Moore spectrum S/2 and what is different compared tomod-p Moore spectra for odd primes p. Since the stable stems πs

1 and πs2 are both cyclic of order 2 with

generators η respectively η2, the exact sequence (7.9) specializes to a short exact sequence

(7.14) 0 −→ πs2 −→ π2(S/2) −→ πs

1 −→ 0 .

Proposition 7.15. The short exact sequence (7.14) does not split, and so the group π2(S/2) is cyclicof order four, generated by any of the two preimages of η.

Proof. We let η : S2 −→ S/2 be a morphism in the stable homotopy category whose composite withthe connective morphism δ : S/2 −→ S1 is η and suppose that 2η = 0. Then we could choose an extensionη : S2 ∧S/2 −→ S/2 of η to the mod-2 Moore spectrum. Let C(2, η, 2) be a mapping cone of this extension,i.e., a spectrum which is part of a distinguished triangle

S2 ∧ S/2 η−−→ S/2 j−−→ C(2, η, 2) δ−→ S3 ∧ S/2 .

Since the mod-2 cohomology of S/2 respectively its double suspension S2∧S/2 are concentrated in dimensions0 and 1 respectively 2 and 3, the morphism η must induce the zero map on mod-2 cohomology. The longexact cohomology sequence of the defining triangle for C(2, η, 2) thus show that H∗(C(2, η, 2),F2) is one-dimensional in dimensions 0, 1, 3 and 4 and trivial in all other dimensions. Since j : S/2 −→ C(2, η, 2)(respectively δ : C(2, η, 2) −→ Σ2S/2) are surjective (respectively injective) in mod-2 cohomology, we deducethat the Bockstein operation β = Sq1 is an isomorphism from dimension 0 to 1 and from dimension 3 to4. [draw picture] Since η is detected by the Steenrod operation Sq2, the composite operation Sq1Sq2Sq1 :H0(C(2, η, 2); F2) −→ H4(C(2, η, 2); F2) would be a non-trivial isomorphism. By the Adem relations wehave Sq1Sq2Sq1 = Sq2Sq2 which factors through the trivial group H2(C(2, η, 2); F2). We have obtained acontradiction, and so the class 2η must be nonzero, i.e., η ∈ π2(S/2) generates a cyclic group of order 4.


Now we calculate the ring of self maps of S/2 in the stable homotopy category. Since the stable 0-stemπs

0 is infinite cyclic and thus torsion free, the exact sequence (7.9) shows that π1(S/2) ∼= Z/2, generated bythe image of η. The sequence (7.10) thus specializes to a short exact sequence

(7.16) 0 −→ Ext(Z/2,Z/2) −→ [S/2,S/2] π0−→ Hom(Z/2,Z/2) −→ 0 .

The sequence does not split, because otherwise we would have 2 · IdS/2 = 0 and πn(S/2) would be anF2-vector space for all n, contradicting the calculation π2(S/2) ∼= Z/4 of the previous proposition. Thusthe endomorphism ring [SZ/2, SZ/2] is isomorphic to Z/4. The exact sequence (7.16) show that 2 IdSZ/2equals the image of the generator of Ext(Z/2,Z/2) which proves the relation

(7.17) 2 · IdS/2 = jηδ

in the group [S/2,S/2], where j : S −→ S/2 and δ : S/2 −→ S1 and the two morphisms from the definingtriangle for the mod-2 Moore spectrum.

In contrast to the case of odd primes, the exact sequence

0 −→ Z/2⊗ πsn

j·−−→ πn(S/2) δ·−→ 2πsn−1 −→ 0

does not generally split (as we already saw for n = 2). The relation (7.17) implies that η-multiplicationcompletely determines the class of this extension: if x ∈ πn(S/2) is a preimage of a 2-torsion elementx ∈ πs

n−1, then 2x is the image of ηx ∈ πsn (because 2x = jηδx = jηx). More generally, for any spectrum

X the sequence

0 −→ Z/2⊗ πnXj·−−→ πn(X; Z/2) δ·−→ 2πn−1X −→ 0

is short exact but need not split. The extension is determined by the action of η ∈ πs1 on the homotopy

groups of X in the same way as for X = S above.

8. Finite spectra

8.1. The Spanier-Whitehead category. In Proposition 4.1 we saw that the symmetric sphere S isa compact object in the stable homotopy category as well as a weak generator. We will now identity the fullsubcategory of all compact objects in the stable homotopy category with a more ‘concrete’ category definedfrom the homotopy category of finite CW-complexes by ‘inverting the suspension functor’. This categoryis known as the Spanier-Whitehead category, and historically it predates the stable homotopy category.

The Freudenthal suspension theorem asserts that for every n-connected pointed space Y and everypointed CW-complex X whose dimension is less than 2n (numbers right ?), the suspension map

Σ : [X,Y ] −→ [ΣX,ΣY ]

is bijective. So when defining the stable homotopy classes of maps

X,Y = colimn [ΣnX,ΣnY ] ,

the colimit system actually stabilizes whenever X is a finite-dimensional CW-complex. A natural idea isthus to define a category in which these stable values live, and this is the so-called Spanier-Whiteheadcategory.

Definition 8.1. The Spanier-Whitehead category SW has a objects the pairs (K,n) where K is apointed space which admits the structure of a finite CW-complex and n ∈ Z is an integer. Morphisms inSW are defined by

SW((K,n), (L,m)) = colimk [K ∧ Sn+k, L ∧ Sm+k] .

The colimit is taken over the suspension maps and ranges over large enough values of k for which both n+kand m+ k are non-negative. Composition is defined by composition of representatives, suitably suspendedso that composition is possible.

8. FINITE SPECTRA 229

By Freudenthal’s suspension theorem, the colimit is attained at a finite stage. We often identify a finiteCW-complex X with the object (X, 0). With this convention the morphism set

SW(Sn,K) = colimk [Sn+k,K ∧ Sk]∗ = colimk πn+k(K ∧ Sk)

agrees with the n-th homotopy group of the suspension spectrum Σ∞K, i.e., the n-th stable homotopygroup of K.

Tautologically, the identity map of K ∧ Sn+m represents an isomorphism between (K ∧ Sn,m) and(K,n +m) in SW, so suspension becomes invertible in SW. In fact, SW is in a certain precise sense theuniversal example of this...

The Spanier-Whitehead category is naturally endowed with the structure of a triangulated category asfollows. The shift functor is simply given by reindexing, i.e., Σ(K,n) = (K, 1 + n) The Spanier-Whiteheadcategory is additive because in it every objects is isomorphic to a double suspension, so the morphism-setsin SW are all abelian groups.

The distinguished triangles arise from homotopy cofiber sequences: for every based map f : K −→ Lbetween finite CW-complexes and every integer n the diagram

(K,n)f−−→ (L, n) i−−→ (C(f), n)

p−−→ (ΣK,n) ∼= (K, 1 + n)

is a distinguished triangle (where C(f) is the mapping cone), and a general triangle is distinguished if andonly if it is isomorphic to one of these.

We will not verify the axioms of a triangulated category for SW, but this can be found in [ref...]Alternatively, one can appeal to Proposition ?? (i) first to see that distinguished triangle in SW go todistinguished triangle in SHC and then use the fact that Σ∞ is fully faithful to deduce the axioms of atriangulated category for SW from those for SHC.

The following theorem says that the stable homotopy category contains the Spanier-Whitehead categoryas a full subcategory. For every pointed space K we have a suspension spectrum Σ∞K as in Example 1.13.We apply the singular complex functor S levelwise (compare Section I.1) to obtain a symmetric spectrumof simplicial sets and then use the localization functor γ : Sp −→ SHC to get into the stable homotopycategory. We also have to compensate the formal dimension attached to an object in the Spanier-Whiteheadcategory by shifting in the triangulated structure of the stable homotopy category, so altogether we definea functor

Σ∞ : SW −→ SHCon objects by

Σ∞(K,n) = γ(S(Σ∞K)) ∧L Sn .To define this functor on morphisms we define an isomorphism in SHC between Σ∞(K ∧ S1, n) andΣΣ∞(K,n).

γ(S(Σ∞(K ∧ S1))) ∧L Sn ∼= (S1 ∧ γ(S(Σ∞K))) ∧L SnId∧Lα1,n−−−−−−→ S1 ∧

(γ(S(Σ∞K)) ∧L Sn

)[define the remaining iso] Using these isomorphisms we get natural maps

[K ∧ Sn+k, L ∧ Sm+k] −→ SHC(γ(S(Σ∞K)) ∧L Sn+k, γ(S(Σ∞L)) ∧L Sm+k)

−∧LS−k−−−−−→ SHC(γ(S(Σ∞K)) ∧L Sn, γ(S(Σ∞L)) ∧L Sm)

where we have implicitly used the derived associativity isomorphisms and the isomorphism αn+k,−k :Sn+k ∧L S−k ∼= Sn. These maps are compatible as k increases [associativity of α’s ?], so the induce awell defined map

Σ∞ : SW((K,n), (L,m)) −→ SHC(Σ∞(K,n),Σ∞(L,m)) ,

which we take as the effect of the functor Σ∞ on morphisms. The following theorem and the characterizationof compact objects in Theorem 8.4 says that the Spanier-Whitehead category ‘is’ (up to equivalence oftriangulated categories which preserves the smash products) the full category of compact objects in SHC.


Theorem 8.2. The functorΣ∞ : SW −→ SHC

is fully faithful and exact.

Proof. [specify the isomorphism Σ∞(K, 1 + n) ∼= S1 ∧ Σ∞(K,n); prove exactness].Essentially by construction of Σ∞ there are natural isomorphisms Σ∞(K,n+m) ∼= Σ∞(K,n) ∧L Sm].

For showing that the map on morphism sets

Σ∞ : SW((K,n), (L,m)) −→ SHC(Σ∞(K,n),Σ∞(L,m))

is bijective we can thus assume n = m = 0.[...] Every finite CW-complex is homotopy equivalent, thusisomorphic in SW, to the geometric realization of a finite simplicial set, so we can assume that K = |K ′|and L = |L′| for finite pointed simplicial set K ′ and L′. Then the spectrum Σ∞(|K ′|, 0) = S(Σ∞|K ′|) islevel equivalent to the suspension spectrum Σ∞K ′, and similarly for L′. So in the special case the claimboils down to the statement that the map

|K ′|, |L′| = colimk [|K ′| ∧ Sk, |L′| ∧ Sk] −→ SHC(Σ∞K ′,Σ∞L′)

[which map?] is an isomorphism. This, however, was already shown in Example 1.18.

The Spanier-Whitehead has a symmetric monoidal smash product which is defined by (K,n)∧(L,m) =(K ∧ L, n + m) on objects, and with unit object (S0, 0). Moreover, the embedding Σ∞ of the Spanier-Whitehead category into SHC is compatible with smash products, i.e., it can be made into a strong sym-metric monoidal functor. We leave the details as Exercise 11.5.

Remark 8.3. There is an important conceptual difference in the construction of the Spanier-Whiteheadand the stable homotopy category. The difference is in which order the processes of ‘inverting suspension’and ‘passage to homotopy classes’ are taken. [...] There is no known construction of the stable homotopycategory where the two processes are taken in the other order.

Now we can explain why one usually restricts to finite CW-complexes when defining the Spanier-Whitehead category. The definition of morphisms in SW make sense for arbitrary pointed spaces, but thenatural map

X,Y = colimk[ΣkX,ΣkY ] −→ SHC(Σ∞X,Σ∞Y )is not generally a bijection. For example, the identity map of QS0 is adjoint to a morphism Σ∞QS0 −→Σ∞S0 in the stable homotopy category which is not in the image of QS0, S0. An injective Ω-spectrum Xis isomorphic to Σ∞(K,n) for some finite pointed simplicial set K and integer n if and only if it is compactas an object of the triangulated category SHC.

We recall that an object X of a triangulated category with infinite sums is called compact (sometimescalled small or finite ) if for every family Yii∈I of objects the natural map⊕

i∈I[X,Yi] −→ [X,

⊕i∈I

Yi]

is an isomorphism. We saw in Proposition 4.1 that the sphere spectrum is compact as an object of thestable homotopy category. We will now characterize the compact objects of the stable homotopy category,which are often referred to as ‘finite spectra’.

Some of the characterizations below use terminology which we have not yet introduced. The derivedfunction spectrum with sphere spectrum in the second variable provides a contravariant duality functorD = F (−,S) : SHCop −→ SHC. For every spectrumX there is an evaluation morphism ev : DX∧LX −→ Swhich is adjoint to the identity of DX. If Y is another spectrum, there is a natural morphism Y ∧LDX −→F (X,Y ) which is adjoint to the morphism Id∧ev : Y ∧L DX ∧L X −→ Y ∧L S ∼= Y .

A symmetric spectrum X is bounded below if there is an integer k such that all true homotopy groupsbelow dimension k are trivial.

Theorem 8.4. For an object X of the stable homotopy category the following five conditions are equiv-alent. Such objects are called finite spectra.

8. FINITE SPECTRA 231

(i) X is isomorphic to (Σ∞K) ∧L Sn for a finite pointed simplicial set K and an integer n;(ii) X is strongly dualizable, i.e., for every object Y of SHC the morphism Y ∧LDX −→ F (X,Y ) is

an isomorphism;(iii) X is a compact object of the triangulated category SHC;(iv) X is bounded below and its integral homology H∗(X,Z) is totally finitely generated;(v) X belongs to the thick subcategory generated by the sphere spectrum.

Proof. (i)=⇒(ii) induction on number of non-degenerate simplices of K(ii)=⇒(iii) For every symmetric spectrum X and every family Yii∈I of spectra we have a commutative

diagram ⊕i∈I(Yi ∧L DX)

∼= //

(⊕

i∈IYi) ∧L DX

⊕i∈IF (X,Yi) // F (X,

⊕i∈IYi)

in SHC in which the upper horizontal map is an isomorphism since the derived smash product is a leftadjoint. If X is strongly dualizable, then the two vertical morphism are isomorphism, hence so is the lowerhorizontal map. If we take the 0th homotopy group of the lower morphism, exploit that π0 commutes withsums, and use π0F (X,Z) ∼= SHC(X,Z), we see that X is compact.

(iii)=⇒(iv) The canonical morphism⊕

n∈Z HZ[n] −→∏n∈Z HZ[n] is a π∗-isomorphism, thus an iso-

morphism in SHC. If X is compact, the composite map⊕n∈Z

[X,HZ[n]] −→ [X,⊕n∈Z

HZ[n]]∼=−→ [X,

∏n∈Z

HZ[n]]∼=−→

∏n∈Z

[X,HZ[n]]

is thus an isomorphism. This means that the group [X,HZ[n]] ∼= Hn(X,Z) is trivial for almost all integersn, i.e., the integral cohomology ofX is concentrated in finitely many dimensions. By the universal coefficienttheorems, the integral homology is then also concentrated in finitely many dimensions.

We show next that Hn(X,Z) is finitely generated for every integer n. We consider a family Aii∈I ofabelian groups and form the sum of the associated Eilenberg-Mac Lane spectra, which is stably equivalent(even isomorphic as a symmetric spectrum) to the Eilenberg-Mac Lane spectrum of the sum. Since X iscompact, the map⊕

i∈IHn(X,Ai) ∼=

⊕i∈I

[X,HAi]n −→ [X,⊕i∈I

HAi]n∼=−→ [X, H(

⊕i∈I

Ai)]n∼=−→ Hn(X,

⊕i∈I

Ai)

is an isomorphism. We have a commutative diagram⊕i∈I H

n(X,Ai) //

Hn(X,⊕

i∈I Ai)

⊕i∈I Hom(Hn(X,Z), Ai) // Hom(Hn(X,Z),

⊕i∈I Ai)

in which the lower map is the canonical one. The upper map is an isomorphism and the two horizontalmaps are surjective by the universal coefficient theorem. So the lower map is also surjective. Since thisholds for all families Aii∈I of abelian groups the homology group Hn(X,Z) must be finitely generated.

The last thing we have to verify is that X is bounded below. By Proposition ?? X is a homotopycolimit of its connective covers, i.e., there is a distinguished triangle

⊕n≥0

P−nX1−shift−−−−→

⊕n≥0

P−nX1−shift−−−−→ X −→ Σ

⊕n≥0

P−nX

Since X is compact, Lemma 4.6 (ii) shows that the group [X,X] a colimit of the sequence of groups[X,P−nX] which implies that the identity morphism of X in the stable homotopy category factors through


the spectrum P−nX for some n ≥ 0. So X is a retract of a bounded below spectrum, hence it is itselfbounded below.

(iv)=⇒(v) Build X from spheres. Use induction on width of homology.(v)=⇒(i) We exploit that the functor Σ∞ : SW −→ SHC is fully faithful, compare Theorem 8.2,

and deduce from this that the essential image of the functor Σ∞ is closed under extensions in the stablehomotopy category. This is special case of the following more general fact:

Suppose F : S −→ T is a fully faithful and exact functor between triangulated category. Then theessential image of F is closed under extensions in T . Indeed, suppose that

Af−→ B

g−→ Ch−→ ΣA

is a distinguished triangle in T with A and B in the essential image of F . So there exist objects A′ and B′

of S and isomorphisms α : FA′ −→ A and β : FB′ −→ B in T . Since F is full, there exists a morphismf ′ : A′ −→ B′ in S such that F (f ′) = β−1fα. We choose a distinguished triangle

A′f ′−→ B′

g′−→ C ′h′−→ ΣA′

in S. The image of this triangle is distinguished in T , so the axiom (T3) of a triangulated category providesa morphism γ : FC ′ −→ C which makes the diagram

FA′F (f ′) //

α

FB′F (g′) //

β

FC ′

γ

τF (h′) // Σ(FA′)

Σ(α)

A

f// B g

// Ch

// ΣA

commute, and γ is then an isomorphism since α and β are. Since F commutes with suspensions anddesuspensions (up to natural isomorphism), the essential image of F is closed under suspensions and desus-pensions. Since triangles can be rotated, this also shows that if A and C are in the essential image of F ,then so is B, and if B and C are in the essential image of F , then so is A.

We show that (Σ∞K) ∧L Sn is compact for every a finite simplicial set K and integer n. We do aninduction on the number of cells in a CW-structure for K. The class of compact objects is closed underextensions [explain]

We draw some consequences of the characterization of compact objects.

Proposition 8.5. The class of compact objects in the stable homotopy category is closed under derivedsmash product, derived function spectra and duality. The duality functor restricts to a contravariant self-equivalence of SHCc.

Proof. To show that the restriction of the duality to SHCc is an self-equivalence it remains to showthat every finite spectrum X is dualizable, i.e., the double duality morphism X −→ DDX,, adjoint tothe identity of DX, is an isomorphism. However, the class of spectra X for which this morphism is anisomorphism is closed under extensions and retract, and it contains the sphere spectrum S. Since the thinksubcategory generated by S coincides with the class of compact spectra, X −→ DDX is an isomorphismfor all compact X.

An object X in a symmetric monoidal category is called invertible of there exists another object Y suchthat X ∧ Y is isomorphic to the unit object. In the stable homotopy category we have Sn ∧L S−n ∼= S, soSn is invertible for all integers n. We can now show that SHC has no other invertible objects.

Proposition 8.6. Every invertible object in the stable homotopy category is isomorphic to a spherespectrum Sn for some integer n.

9. BOUSFIELD LOCALIZATION 233

Proof. Suppose X is invertible. Then X ∧L− is an autoequivalence of the stable homotopy category,and thus it preserves all categorical properties. Since the sphere spectrum S is compact, so is X ∼= X ∧L S.By Theorem 8.4 X is connective and has totally finitely generated homology. Since X ∧L Y ∼= S theKunneth and universal coefficient theorems imply that the integral homology of X is concentrated in asingle dimension n, where it is free abelian of rank one. Since X is also connective, X is isomorphic to Snin the stable homotopy category.

Remark 8.7. The Freyd generating hypothesis is a prominent open problem about the stable homotopycategory. The question is whether the sphere spectrum is a (strong) categorical generator of the homotopycategory of finite spectra. This means the following: given a morphism f : X −→ Y between finite spectra(i.e., compact objects of the stable homotopy category) such that the induced map π∗f on homotopy groupsis trivial, is f then necessarily the trivial morphism ?

This notion of generator which asks whether the sphere spectrum detects morphisms should be con-trasted with the fact that the sphere spectrum is a weak generator, i.e., detects morphisms in the stablehomotopy category (not necessarily finite), see Proposition 4.1.

The restriction to finite spectra is clearly necessary in the generating hypothesis: the morphism β :HFp −→ ΣHFp between Eilenberg-Mac Lane spectra which represents the Bockstein operation in mod-pcohomology is non-trivial, but induces the trivial map on stable homotopy groups for dimensional reasons.

Remark 8.8. Nilpotence theorem gives a criterion for when a (shifted) selfmap f : X ∧L Sn −→ X isnilpotent, i.e., some iterate of f becomes trivial. [expand...]

Remark 8.9 (Spanier-Whitehead duality). The contravariant duality functor D = F (−,S) preservescompact objects and restricts to a contravariant self-equivalence of SHCc. Since the Spanier-Whiteheadcategory is equivalent to the category SHCc, the duality should be describable only in terms of finite CW-complexes. This is in fact possible as follows. [...] This duality on SW is called Spanier-Whitehead duality;historically, it was one of the origins for the stable homotopy theory. [classical definition for finite CW-complexes; prove that SW-duality is the restriction of ‘S-duality’ D = F (−,S) to compact objects; S-dualfor manifolds via Thom space of normal bundle]

9. Bousfield localization

[define Bousfield localization at a spectrum and discuss general properties; define Bousfield class

9.1. Localization at a set of primes. The localization of an abelian group at a set of prime numbershas an analogue in stable homotopy theory, also called ‘localization’. With this tool one can often study aproblem ‘one prime at a time’.

To fix our notation and language, we quickly review the localization of abelian groups. For every set Sof prime numbers we define a subring ZS of the ring of natural numbers by

ZS =ab∈ Q | b has only prime factors in S

.

For example, we have

Z∅ = Z , Zall primes = Q and Zall\p = Z(p) =ab| p does not divide b

.

Every subring of Q is of the form ZS for a unique set of primes S: given a subring R ⊂ Q, then R = ZS forthe set S defined by S = p | 1

p ∈ R.For a subring R ⊂ Q, the functor of abelian groups A 7→ R⊗A is exact. Since the multiplication map

R⊗R −→ R is an isomorphism, the functor is idempotent with respect to the natural map

A −→ R⊗A , a 7−→ 1⊗ a .This construction is called R-localization. An abelian group A is called R-local if the following equivalentconditions hold

(1) A has the structure of an R-module (necessarily unique).


(2) The map A −→ R⊗A is bijective.(3) For every prime p with 1

p ∈ R, multiplication by p on A is bijective.

Example 9.1. • Every abelian group is Z-local.• A is Z(p)-local (‘p-local’) if and only if multiplication by q is bijective on A for all primes q 6= p.• A is Q-local (‘rational’) if and only if A is uniquely divisible, i.e., for all a ∈ A and n ∈ N+ there

is a unique b ∈ A such that a = n · b.• Any finitely generated abelian group is a finite direct sum copies of Z and Z/qm for various primesq and m ≥ 1. Such a sum is p-local if and only if only Z/pm’s occur. The p-localization functorturns every copy of Z into a copy of Z(p), it leaves all summands of the form Z/pm untouched,and it kills summands of the form Z/qm for primes q 6= p.

Theorem 9.2. Let R be a subring of Q and SR a Moore spectrum for R.(i) For every symmetric spectrum X and integer k the natural map

R⊗ πkX −→ πk(SR ∧L X)

induced by the homotopy group pairing is an isomorphism. Hence a morphism f : X −→ Y is an SR-equivalence if and only if the map R ⊗ πkf is an isomorphism for all integers k. The Hurewicz mapX −→ SR ∧X is an SR-localization.(ii) For a symmetric spectrum X the following are equivalent.

(i) X is SR-local (in the sense of Bousfield localization).(ii) All homotopy groups of X are R-local.(iii) For every morphism f : A −→ B of symmetric spectra such that

R⊗ π∗(f) : R⊗ π∗A −→ R⊗ π∗Bis an isomorphism, the induced map category [f,X] : [B,X] −→ [A,X] of morphisms in the stablehomotopy is a bijection.

(iv) For every symmetric spectrum C such that R⊗π∗C is trivial the morphism group [C,X] is trivial.(v) The endomorphism ring SHC(X,X) in the stable homotopy category is an R-algebra.

If the homotopy groups of X are bounded below, then conditions (i)-(v) above are also equivalent to thecondition that the integral spectrum homology groups H∗(X,Z) are R-local.

Remark 9.3. A morphism f : A −→ B such that R ⊗ π0f : R ⊗ π∗A −→ R ⊗ π∗B is an isomorphismis called an R-equivalence. A spectrum C such that R⊗ π∗C = 0 is called R-acyclic.

Example 9.4. • If A is an R-local abelian group, then the Eilenberg-Mac Lane spectrum HAis R-local.

• Let X be any object of the stable homotopy category, p a prime and denote by X/p the mappingcone of an endomorphism of X which represents p · IdX in SHC(X,X). Proposition ?? gives along exact sequence of homotopy groups

· · · −→ πmX×p−−−→ πmX −→ πm(X/p) δ−−→ πm−1X

×p−−−→ πm−1X −→ · · ·which breaks up into short exact sequences

0 −→ Z/p⊗ πmX −→ πm(X/p) δ−−→ pπm−1X −→ 0 .

The groups Z/p⊗ πmX and pπm−1X are Fp-vector spaces, so the homotopy group πm(X/p) iskilled by multiplication by p2; hence the spectrum X/p is p-local. As a special case of this, theMoore spectrum M(p) is p-local.

• The connectivity assumption in part (c) of the above theorem is important. Indeed, for themapping telescope of the Adams map v : M(p) −→M(p)[−2p+ 2] (p an odd prime) we have

H∗(v−1M(p); Z) = 0 ,

as in Example ??. So the integral spectrum homology of v−1M(p) is rational. But π∗(v−1M(p))is a nontrivial graded Fp-vector space, so it is not rational.


9.2. Rational stable homotopy theory is easy. We shall now see that the rational stable homotopycategory, i.e., the full subcategory of SHC consisting of rational spectra, is very easy to describe. As above,an abelian group is rational if it admits the structure of a Q-vector space (necessarily unique). Equivalently,A is rational if and only it is uniquely divisible, i.e., for all a ∈ A and n > 0 there is a unique b ∈ A such thata = n · b. We call a symmetric spectrum rational if all its homotopy groups are rational. By Theorem 9.2 Xis rational if and only if it is SQ-local in the sense of Bousfield localization where SQ is a Moore spectrumfor the group Q. We recall from Example 7.4 that the Eilenberg-Mac Lane spectrum HQ is a possiblechoice for the rational Moore spectrum SQ.

Theorem 9.6 below shows that a rational spectrum is completely determined by its homotopy groups:the homotopy group functor is an equivalence from rational stable homotopy category to graded vectorspaces over Q. In particular, two rational spectra are stably equivalent if and only if the homotopy groupsare abstractly isomorphic. The key ingredient for all this is Serre’s calculation of homotopy groups ofspheres modulo torsion (Theorem 1.9). The integral analogue of this is not at all true, i.e., the homotopygroup functor

π∗ : SHC −→ (graded abelian groups)is very far from being an equivalence for general spectra.

Let V∗ = Vnn∈Z be a Z-graded abelian group. We defined the generalized Eilenberg-Mac Lanespectrum associated to V∗ as the product of suspended Eilenberg-Mac Lane spectra for the groups Vn,

HV∗ =∏n∈Z

Σn(HVn) .

Then there is a natural isomorphism

(9.5) πk(HV∗) =∏n∈Z

πk(Σn(HVn)) ∼= Vk

so that the generalized Eilenberg-Mac Lane spectrumHV∗ realizes the graded abelian group V∗ on homotopy.[the product is isomorphic in SHC to the sum]

Now we can show the main result of this section:

Theorem 9.6. (i) Let V∗ be a Z-graded Q-vector space and A a symmetric spectrum. Then the map

π∗ : [A,HV∗] −→ Homgr.Ab(π∗A, V∗)(f : A −→ HV∗) 7−→ (π∗f : π∗A −→ π∗(HV∗) ∼= V∗)

is an isomorphism of abelian groups.(ii) Every rational spectrum is a generalized Eilenberg-Mac Lane spectrum. More precisely, if A is a

symmetric spectrum with rational homotopy groups, then there exists a unique homotopy class of morphismA −→ H(π∗A) which realizes the isomorphism π∗A ∼= π∗H(π∗A) of (9.5) on homotopy groups.(iii) The homotopy group functor

π∗ : SHCQ −→ (graded Q-vector spaces)

is an equivalence from the rational stable homotopy category to the category of graded Q-vector spaces. Aninverse functor is given by the generalized Eilenberg-Mac Lane spectra.

Proof. (i) We start with the special case A = S of the sphere spectrum. Since the sphere spectrumrepresents π0 (Proposition 1.15), the group [S,HV∗] is isomorphic to π0(HV∗). Thus [S,HV∗] is trivial forn 6= 0 and isomorphic to V0 for n = 0. The right hand side

Homgr.Ab(π∗S, V∗) =∏n∈Z

HomAb(πnS, Vn)

is isomorphic to HomAb(π0S, V0) since πnS is a torsion group for n 6= 0 (by Serre’s theorem I.1.9) whereasVn is rational. Since π0S is free abelian generated by the fundamental class, the group Homgr.Ab(π∗S, V∗)is altogether isomorphic to V0, via evaluation at the fundamental class of S. So the claim is true for thesphere spectrum.


We let X denote the class of symmetric spectra A with the property that the map π∗ : [A,HV∗] −→Homgr.Ab(π∗A, V∗) is an isomorphism. By the above the spectrum S belongs to X . Moreover, the class Xis closed under sums and extensions in distinguished triangles. The latter needs that every rational abeliangroup is injective, i.e., the functor Homgr.Ab(−, V∗) is exact. Since the spectrum S generates the stablehomotopy category (see Proposition 4.2), the class X contains all spectra, which proves (i).

Part (ii) follows by applying part (i) to V∗ = π∗A. (iii) We have already seen that every graded Q-vectorspace is isomorphic to an object in the image of the functor π∗ (namely the generalized Eilenberg-Mac Lanespectrum). So it remains to show that π∗ is fully faithful, i.e., that the map

π∗ : SHCQ(X,Y ) −→ Homgr-Q-vs(π∗X,π∗Y )

is an isomorphism for all rational spectra X and Y . By (ii) we can assume that Y is a generalizedEilenberg-Mac Lane spectrum, i.e., Y = HV∗ for a graded Q-vector space V∗. But then part (i) give thefully-faithfulness. So the functor π∗ is an equivalence of categories.

Part (ii) of the previous theorem is very particular for rational spectra. A general spectrum is not ageneralized Eilenberg-Mac Lane spectrum. For example, the sphere spectrum S, the mod-p Moore spectrumSZ/p or the real and complex K-theory spectra KO and KU are not stably equivalent to any generalizedEilenberg-Mac Lane spectrum.

Remark 9.7 (move this earlier ?). By Proposition 5.7 (iii) the stable Hurewicz homomorphism

πnX −→ Hn(X; Z)

is a rational isomorphism for every spectrum X and every integer n, i.e., it becomes an isomorphism aftertensoring both sides with the group Q of rational numbers.

Now suppose that X is a spectrum whose integral spectrum homology groups H∗(X; Z) are trivial. IfX is not connective, then this need not imply that X is stably trivial, but it does have consequences forthe homotopy groups of X. Since the stable Hurewicz morphism is a rational isomorphism, the rationalizedstable homotopy groups Q ⊗ π∗X are the trivial. This is equivalent to the property that every homotopyelement is torsion, i.e., annihilated by multiplication by some positive natural number.

In Example 5.8 we have seen this phenomenon happen; the mapping telescope v−1M(p) of the Adamsmap on the mod-p Moore spectrum has trivial spectrum homology, but it is not stably contractible. In thatexample, all homotopy groups are Fp-vector spaces, so in particular annihilated by multiplication by p.

9.3. Completion. We let SQ/Z denote the mapping cone of the unit map S −→ HQ, i.e., by a choiceof distinguished triangle

S ι−−→ HQ p−−→ SQ/Z δ−−→ S1 .

Since S and HQ are Moore spectra for the groups Z respectively Q and the unit map ι : S −→ HQ inducesthe monomorphism on homology, the mapping cone SQ/Z is a Moore spectrum for the group Q/Z. Wedefine the profinite completion functor

(−)∧ : SHC −→ SHC

as the derived function spectrumX∧ = F (SQ/Z,ΣX) .

A natural morphism X −→ X∧ in the stable homotopy category is obtained as the adjoint of the morphism

X ∧L SQ/Z Id∧δ−−−→ X ∧L S1 ∼= ΣX

where δ is the connecting morphism.For a fixed prime p we similarly define the p-completion of X as

X∧p = Hom(SZ/p∞,ΣX) .

where SZ/p∞ is a Moore spectrum for the group Z/p∞, which can either be defined as the colimit ofthe groups Z/pn under multiplication by p maps or, equivalently, as the Z[1/p]/Z or, equivalently, as the


p-torsion subgroup of Q/Z. The p-adic completion comes with a natural map X −→ X∧p defined similarly

as for the profinite completion.Recall that we denote by X −→ XQ the rationalization of a spectrum, compare [...]

Theorem 9.8. Let X be a symmetric spectrum.(i) The square

X //

X∧

XQ // (X∧)Q

is homotopy cartesian, where the vertical maps a rationalizations and the horizontal maps are (obtainedfrom) completions.(ii) The map

X∧ −→∏

p prime

X∧p

whose p-component is obtained from the morphism SZ/p∞ −→ SQ/Z by applying F (−,ΣX) is an isomor-phism in SHC.(iii)

X∧p ' holimn(X ∧ S/pn) .

(iv) The map X −→ X∧p is a Bousfield localization with respect to the mod-p Moore spectrum S/p.

(v) There is a short exact sequence of abelian groups

0 −→ Ext(Z/p∞, πkX) −→ πk(X∧p ) −→ Hom(Z/p∞, πk−1X) −→ 0 .

If all homotopy groups of X are finitely generated as abelian groups, then the completion map X −→ X∧p

induces an isomorphism(πkX)∧p = lim

n(X/pnX) −→ πk(X∧

p )

from the algebraic p-completion of the homotopy groups to the homotopy groups of the p-adic completion.

If we combine parts (i) and (ii) of the previous theorem we obtain a homotopy cartesian square

X //

∏p primeX

∧p

XQ //(∏

p prime X∧p

)Q

which is called the arithmetic square. This square encodes the way in which a spectrum can be assembledfrom rational information and profinite information at each prime.

Proof of Theorem 9.8. (i) The triangle

Σ−1(SQ/Z) −Σ−1δ−−−−−−→ S ι−−→ HQ p−−→ SQ/Z

is a rotation of the defining triangle for SQ/Z, and thus distinguished. The functor F (−, X) and rational-ization are exact, so we have a commutative diagram of distinguished triangles

F (HQ, X)F (ι,X) //

'

X = F (S, X)F (p,X) //

F (SQ/Z,ΣX) = X∧

// ΣF (HQ, X)

'

F (HQ, X)QF (ι,X)Q

// XQ = F (S, X)QF (p,ΣX)Q

// F (SQ/Z,ΣX)Q = (X∧)Q // ΣF (HQ, X)Q


The derived function spectrum F (HQ, X) is a homotopy module spectrum over HQ, so it is already rationaland so the left and right vertical morphisms are stable equivalences. Thus the square in question is homotopycartesian.

(ii) We start from the algebraic fact that the map⊕p prime

Z/p∞ −→ Q/Z

is an isomorphism. So the map ∨p prime

SZ/p∞ −→ SQ/Z

is a stable equivalence. Hence the induced morphism of derived function spectra

X∧ = Hom(SQ/Z,ΣX) −→ Hom(∨

p prime

SZ/p∞,ΣX) ∼=∏

p prime

X∧p

is a stable equivalence.(iii) The group Z/p∞ is the colimit of the groups Z/pn under multiplication by p maps, so the spectrum

S/p∞ is the homotopy colimit of the spectra S/pn under the multiplication by p maps. Applying functionspectrum F (−,ΣX) is exact, so it takes homotopy colimits to homotopy inverse limits, which providesa stable equivalence between X∧

p = Hom(SZ/p∞,ΣX) and holimn F (S/pn, X). Since the mod-pn Moorespectra are self-dual, this proves the claim.

(iv) We show first that the p-adic completion X∧p is S/p-local. For every S/p-acyclic spectrum A.

distinguished triangles

A ∧ S/p Id∧(·p)−−−−−→ A ∧ S/pn+1 Id∧(·p)−−−−−→ A ∧ S/pn Id∧(·p)−−−−−→ Σ(X ∧ S/p)

allow an induction which shows that A∧S/pn is trivial in the stable homotopy category for all n ≥ 0. SinceS/p∞ is a homotopy colimit of the spectra S/pn, the spectrum A ∧ S/p∞ is also trivial. Thus the group[A,X∧

p ] = [A,F (SZ/p∞,ΣX)] ∼= [A ∧ SZ/p∞,ΣX] is trivial, so X∧p is indeed S/p-local.

It remains to show that the completion map X −→ X∧p is is a mod-p equivalence. We can take the

distinguished triangleS(p)

ι−−→ HQ p−−→ SZ/p∞ δ−−→ S1(p)

apply the exact functor S/p ∧ F (−,ΣX) and obtain a distinguished triangle

S/p ∧ F (HQ, X) −→ S/p ∧X −→ S/p ∧X∧p −→ Σ(S/p ∧ F (HQ, X)) .

The derived function spectrum F (HQ, X) is rational, so multiplication by p on it is a self-equivalence, andthus the smash product S/p ∧ F (HQ, X) is trivial on SHC. So the map S/p ∧X −→ S/p ∧X∧

p is a stableequivalence, as claimed.

(v) In (7.10) we derived a short exact sequence for morphisms in SHC out of a Moore spectrum foran arbitrary abelian group. The short exact sequence under consideration now is just the special caseA = S/Z/p∞. For a finitely generated abelian group A the group Hom(Z/p∞, A) is trivial and the groupExt(Z/p∞, A) is isomorphic to the completion A∧p [prove; specify the map]. Plugging this into the shortexact sequence yields the last claim.

9.4. Localization with respect to connective spectra.

9.5. Localization with respect to topological K-theory. We cannot refrain from giving someidea of what Bousfield localization with respect to a non-connective spectrum can look like. So we recallsome result about localization with respect to topological K-theory. However, we will need some factswhose proofs are beyond the scope of this book, so this section is much less self-contained than the rest ofthe book.

Proposition 9.9. The complex, self-conjugate and real topological K-theory spectra KU , KT respec-tively KO have the same Bousfield class.

10. THE ADAMS SPECTRAL SEQUENCE 239

We call a spectrum X K-local if it is local with respect to KU (or equivalently KT or KO).In Example 5.8 we discussed the Adams maps, a certain graded selfmap v1 : ΣqS/p −→ S/p of the

mod-p Moore spectrum where q = 8 for p = 2 and q = 2p−2 for odd primes p. [for p = 2 this should betterbe called v4

1 ] These selfmaps induce an isomorphism in complex topological K-theory (and hence also inKO and KT -theory). In particular, the map v1 is periodic in the sense that all iterates vn1 are non-trivialin the stable homotopy category. We recall that the mod-p homotopy groups of a spectrum X are definedby

πk(X,Z/p) = πk(S/p ∧L X) .Smashing with the Adams map and taking homotopy group gives an operator on mod-p homotopy groups

v1 : πk(X,Z/p) ∼= πk+q(ΣqS/p ∧X) −→ πk(S/p ∧L X) = πk+q(X,Z/p)which is called v1-multiplication. The mod-p homotopy groups are called v1-periodic if multiplication bythe operator v1 is an isomorphism for all integers k.

Theorem 9.10. A spectrum X is K-local if and only if for every prime p the mod-p homotopy groupsπk(X,Z/p) are v1-periodic. The K-localization functor is smashing, i.e., the morphism X −→ L1S(p) ∧LXis a K-localization for every p-local spectrum. [integrally ?]

So to complete the picture of K-localization we should describe the localized sphere spectrum in a moreexplicit way.

10. The Adams spectral sequence

10.1. Diagonals. Homology with coefficients in a field K has more structure; there is a diagonal onthe homology groupH∗(HK,K) which makes it into a Hopf algebra and theK-homology of every symmetricspectrum is naturally a comodule over the underlying coalgebra. The interesting cases are when K = Fp isa finite prime field, but we set this up more generally. It turns out (see below) that the extra structure israther boring for fields of characteristic 0, and if K has positive characteristic p, then the structure for Kis completely determined by the structure for Fp.

Construction 10.1. Let K be a field. For every symmetric spectrum X the unit map ι : S −→ HKinduces a morphism ι ∧ Id : X ∼= S ∧X −→ HK ∧X. We apply the homology functor to this morphismand get morphisms of K-vector spaces

H∗(ι ∧ Id,K) : H∗(X,K) −→ H∗(HK ∧X,K) .

The Kunneth isomorphism of Proposition 5.12 gives an identification of the target with the graded tensorproduct

H∗(HK,K)⊗K H∗(X,K) .We denote the resulting map

H∗(X,K) −→ H∗(HK,K)⊗K H∗(X,K)

by ∆ and refer to it as the diagonal.

Proposition 10.2. Let K be a field. The diagonal ∆ : H∗(X,K) −→ H∗(HK,K)⊗H∗(HK∧X,K) iscounital, coassociative and natural in the symmetric spectrum X. In the special case X = HK the diagonalis a morphism of graded K-algebras. Thus the diagonal makes the graded ring H∗(HK,K) into a Hopfalgebra over K and the K-homology of every spectrum into a comodule over H∗(HK,K).

Example 10.3 (Dual Steenrod algebra). As an example we describe the spectrum homologyH∗(HFp,Fp) of the Eilenberg-Mac Lane spectrum HFp with coefficients in the field Fp for every prime p.As we shall explain below, the graded ring H∗(HFp,Fp) is dual, as a Hopf algebra, to the Steenrod algebraof stable mod-p cohomology operations. So this graded ring is usually called the dual Steenrod algebra.

By the above discussion, these group H∗(HFp,Fp) are the homotopy groups of the semistable commu-tative symmetric ring spectrum Fp[HFp], the Fp-linearization of the Eilenberg-Mac Lane spectrum, andwe also describe the graded commutative product. The assembly map HFp ∧HFp −→ Fp[HFp] [ref] is a


homomorphism of symmetric ring spectra, and as we shall see later [ref] it is a π∗-isomorphism. So we arealso calculating the graded ring of homotopy groups of the smash product ring spectrum HFp ∧HFp.

The first space (HFp)1 = Fp[S1] is an Eilenberg-Mac Lane space of type (Fp, 1), and it mod-p homologyprovides a set of multiplicative generators for the dual Steenrod algebra. Indeed by the definition of spectrumhomology as a colimit we have the canonical morphism

(10.4) Hk+1(Fp[S1],Fp) −→ Hk+1(HFp,Fp) .

The Eilenberg-Mac Lane Fp[S1] is a simplicial abelian group, and so its mod-p homology has the structureof a graded commutative and cocommutative graded Hopf algebra over the field Fp. The multiplicationis induced by the group addition Fp[S1] × Fp[S1] −→ Fp[S1] and the comultiplication is induced by thediagonal map Fp[S1] −→ Fp[S1] × Fp[S1]. To describe the explicit structure of this Hopf algebra we letx ∈ H1(Fp[S1],Fp) denote the fundamental class and y ∈ H2(Fp[S1],Fp) the Bockstein operation applied tothe fundamental class x. Then H∗(Fp[S1],Fp) is the tensor product of the exterior Hopf algebra generatedby x and the divided power algebra generated by y, and both x and y are primitive. The diagonal of thenth divided power γn(y) of y is given by

∆(γn(y)) =∑i+j=n

γi(y)⊗ γj(y) .

This map factors over the indecomposables [...].For p = 2 we denote by

ξi ∈ H2i−1(HF2,F2)

the image of the non-trivial element of H2i(HF2,F2) under the canonical map (10.4). For odd primes pand i ≥ 0 we denote by

ξi ∈ H2pi−2(HFp,Fp) respectively τi ∈ H2pi−1(HFp,Fp)

the image of the non-trivial element of H2i(HF2,F2) under the canonical map (10.4). We have ξ0 = 1, thefundamental class and unit of the multiplication. Then we have

Proposition 10.5. (i) The graded F2-algebra

H∗(HF2,F2) ∼= π∗(F2[HF2])

is the polynomial algebra over F2 generated by the classes ξi for i ≥ 1. The diagonal is determined by therelation

∆(ξi) =i∑

j=0

ξ2j

i−j ⊗ ξj .

(ii) Let p be an odd prime. Then the graded Fp-algebra

H∗(HFp,Fp) ∼= π∗(Fp[HFp])

is the tensor product of the exterior algebra generated by the classes τi for i ≥ 0 and the polynomial algebragenerated by the classes ξi for i ≥ 1. The diagonal is determined by the relations

∆(ξi) =i∑

j=0

ξpj

i−j ⊗ ξj and ∆(τi) = τi ⊗ 1 +i∑

j=0

ξpj

i−j ⊗ τj

Proof. Since the diagonal is a homomorphism of graded rings it is completely determined by its effecton the generators.

[On homology, the induced map

ϕ∗ : H∗(Σ∞BF2; F2) −→ H∗−1(HF2; F2)

is trivial in dimensions ∗ 6= 2n.]


The antipode is determined inductively by the formula

ξn =n−1∑i=0

ξ2i

n−i · ξi .

For example, we have

ξ1 = ξ1 , ξ2 = ξ2 + ξ31 and ξ3 = ξ3 + ξ41ξ2 + ξ1ξ22 .

The mod-2 homology of the integral Eilenberg-Mac Lane spectrum and the connective real K-theoryspectrum are given by the sub-Hopf-algebras

(HF2)∗HZ = F2[ξ21 , ξ2, ξ3, ξ4, . . . ] and (HF2)∗ko = F2[ξ41 , ξ22 , ξ3, ξ4, . . . ]

The Bockstein β : HF2 −→ ΣHF2 viewed as a coefficient homomorphism induces (check !!!)

(β)∗HF2 : A∗ −→ A∗−1 , f 7→ df

dξ1.

The primitive elements in the dual Steenrod algebra and the classes they represents in Ext1,∗ are givenby

a = [τ0] ∈ Ext1,1 and hi+1 = [ξpi

1 ] ∈ Ext1,piq .

Example 10.6. If K is a field of characteristic 0, then H∗(HK,K) is concentrated in dimension 0where it is isomorphic to K.

Example 10.7. In the dual Steenrod algebra we have τ2i = 0, so Toda brackets of exterior generators

are defined. For p = 3 we have〈τ0, τ0, τ0〉 = ξ1

in A4. [show ! Do we have ξi+1 ∈ 〈τi, τi, τi〉?] For ≥ 5, the Toda bracket 〈τi, τi, τi〉 consists only of 0for dimensional reasons [check]. However, there are non-trivial higher Toda brackets (which we have notdefined) such as ξ1 ∈ 〈τ0, τ0, . . . , τ0〉 (p-fold bracket).

Construction 10.8. Let p be a prime number and X a symmetric spectrum. For ease of notation weabbreviate the Eilenberg-Mac Mane spectrum HFp to H for the course of this construction. Since H = HFpis a symmetric ring spectrum, the functor H ∧− is a triple on the category of symmetric spectra. Like anytriple, it gives rise to an augmented cosimplicial object H•X for every symmetric spectrum X.

In more detail, we define the symmetric spectrum of n-cosimplices by

HnX = H ∧ . . . ∧H︸︷︷︸n+1

∧X .

For i = 0, . . . , n the coface morphism di : Hn−1X −→ HnX is given by inserting the unit morphismι : S −→ H into the ith slot (counting from i = 0). For i = 0, . . . , n the codegeneracy morphism si :Hn+1X −→ HnX is given by using multiplication µ : H ∧H −→ H on the ith and (i+ 1)st factor.

The cosimplicial symmetric spectrum gives rise to a spectral sequences

Es,t1 = πt−s(Hs+1X) =⇒ πt−s TotH•X .

which converges (conditionally ?) to the homotopy groups of the totalization of the cosimplicial spectrum[ref]. We will now identify the E2-term of this spectral sequence as a homological invariant of the mod-phomology of X and investigate the relationship between the abutment and the homotopy groups of X.

Proposition 10.9. (i) The E2-term of the spectral sequence is naturally isomorphic to

Es,t2∼= ExtsA(Fp,H∗(X,Fp)[t]) ,

the Ext-groups of comodules over the dual Steenrod algebra, from Fp to the t-fold shift of the mod-p homologyof X.(ii) Under some assumptions on the symmetric spectrum X, the natural map X −→ Tot(H•X) is a π∗-isomorphism [or an isomorphism on p-adic completions of homotopy groups]


Proof. (i) By [...]the group πt−s(Hs+1X) is naturally isomorphic to A⊗· · ·⊗A⊗H∗(X,Fp) (s factorsof the dual Steenrod algebra). So the bigraded abelian group Es,t1 is isomorphic to the cobar complex of theA-comodule H∗(X,Fp), and the d1-differential corresponds to the cobar differential. For every comodule M ,the homology of the cobar complex C(A,M) calculates the Ext-groups ExtsA(Fp,M [t]), which proves (i).

Definition 10.10. Let p be prime and X a (connective, finite type ?) symmetric spectrum. ByProposition 10.9 the cosimplicial spectrum H•X gives rise to a spectral sequence (how convergent?)

Es,t2 = Exts,tA (Fp,H∗(X,Fp)) =⇒ (πt−sX)∧pwhich is called the Adams spectral sequence for the spectrum X. (describe the filtration on homotopygroups)

[If R is a semistable symmetric ring spectrum then the Adams spectral sequence is a spectral sequenceof algebras]

Ext0,0A (Fp,Fp) = Fpand Ext0,tA (Fp,Fp) = 0 for t 6= 0.

Ext1,∗A (Fp,Fp) = Prim(A∗) =

F2ξ2

i

1 | i ≥ 1 p = 2

Fpτ0, ξpi

1 | i ≥ 1 p > 2.

Example 10.11. We discuss the Adams spectral sequence for the sphere spectrum

Es,t2 = Exts,tA (Fp,Fp) =⇒ (πt−sS)∧pand verify the table given in Example 1.11 for the homotopy groups spheres up to dimension 8. We startwith the prime p = 2, where the following chart displays the relevant portion of the spectral sequence.

0 2 4 6 8 10 12 140

2

4

6

8

••h0

••••••••

• h1

•

h21

•

h31 = h2

0h2

•h0h2

•h2

•h22

•h3

•••

•

••c0•

•〈h1, h0, h30h3〉

•

•

•〈h2, h30, h0h3〉

•

•h2

3

••

d0

•••

•h4

••••

•###

••

///////

WW

**********

TT

**********

TT

We explain how to interpret this chart (and similar pictures later on) and how to derived conclusionsabout the stable homotopy groups of spheres from it.

It is customary to draw Adams spectral sequences in Adams indexing which means that the homologicaldegree, usually denoted s, is drawn vertically, and the difference t − s of the degrees is drawn vertically.The number t− s is referred to as the topological degree. So the slot with coordinated (p, q) represents thegroup Extp,p+q. This way of drawing an Adams spectral sequence makes it easy to visialize which groupscontribute to the nth stable stem, since these all lie on the vertical line of topological degree t− s = n.

By the above calculation of Ext-groups in homological degrees 0 and 1, E0,∗2 is one-dimensional with

basis 1 ∈ E0,02 and E1,t

2 is one-dimensional with generator hi = [ξ2i

1 ] for t = 2i and trivial if t is not a powerof 2. Moreover Es,s2 is one-dimensional, generated by hs0 for all s ≥ 0, which means that the vertical line


with t−s = 0 continues indefinitely upwards; we only labeled a few of these classes in the chart. The verticallines represent multiplication by h0 and the diagonal lines represent multiplication by h1. So for example,the generator in bidegree (3, 3) is h3

1 = h20h2 and the generator in bidegree (7, 4) is h3

1h3. There are twonon-trivial h2-multiplications in this range (namely on 1 and h2) which are not represented graphically [addthat]. We have labeled all generator in the lower left part of the chart, and in the rest the multiplicativeyields basis elements.

The stable classes of the Hopf maps 2ι, η, ν and σ are all trivial in mod-2 homology, so the have haveAdams filtration 1. These classes are detected by the Steenrod operations Sq1, Sq2, Sq4 respectively Sq8;dually this means that their images under the map

F 1πsn −→ Ext1,n+1

is non-zero. In other words, the classes h0, h1, h2 and h3 are permanent cycles in the spectral sequence anddetect 2ι, η, ν and σ respectively.

Some more information, without proof. The other classes on the 1-line support a non-trivialdifferential

d2(hn+1) = h0h2n

which is non-trivial for n ≥ 3. The differential is a biproduct of Adams non-existence theorem for maps ofHopf invariant one [explain], and it is often referred to as the Adams differential.

In the 2-line of the Adams spectral sequence we have the relation hihi+1 = 0 for all i ≥ 0. Moreover,the classes

hihj for i = j or j + 1 < i

form a basis of the 2-line Ext2,∗.The classes h2

i are the Kervaire invariant classes; the class h2i is a permanent cycle in the Adams

spectral sequence if and only if there exists a framed (2i − 2)-manifold with Kervaire invariant 1. Then

Θi ∈ πs2i+1−2

denotes any stable homotopy element of filtration 2 which is detected by h2i . We have

Θ1 = η2 , Θ2 = ν2 , Θ3 = σ2 , Θ4 = 〈2,Θ3, 2,Θ3〉 ;

the class Θ5 exists, but is more difficult to describe [3]. The status of Θi for i ≥ 6 is unknown, and is oftenreferred to as the Kervaire invariant problem.

Mahowald constructs classes ηj ∈ πs2j which are detected by h1hj , or by the Adem relation

Sq2Sq2j

= · · · .

[Do all other 2-line classes support differentials?]After dividing out by commutativity and the relations

hihi+1 = 0 , hih2i+2 = 0 , and h2

ihi+2 = h3i+1 ,

the triple products hihjhk for i, j, k ≥ 0 become linearly independent in the 3-line Ext3,∗. Bruner [15]shows that the classes h2h

2j are permanent cycles for j ≥ 4 (??). The 3-line has another family of classes

which are multiplicatively indecomposable. There is a class

ci = 〈hi+1, hi, h2i+2〉 ∈ Ext3,?

[cobar representative ? bracket in exercise...] for i ≥ 0 which together with triple products of hi’s generatethe entire 3-line. The classes c0 is a permanent cycle and detects ε, the unique non-trivial element offiltration 3 in the 8-stem. The class c1 (outside our chart) is also a permanent cycle but for i ≥ 2, the classci supports a non-trivial d2-differential. There is one more multiplicative generator on the chart which wehave not yet introduced, namely

d0 = 〈h0, h1, h2, c0〉 ∈ Ext4,18


which is again the first of an infinite family of classes in the 4-line. This class d0 is a permanent cycle anddetects a homotopy class of filtration 4 in the 13-stem usually called κ. Using 4-fold Toda brackets (whichwe have not defined), the class κ can be decomposed as

κ ∈ 〈2, η, ν, ε〉 ∩ 〈ν, ε, 2, η〉 ∩ 〈ν, 2ν, ν, 2ν〉 .Some more systematic information is available for the 4-line and above, but we’ll stop our survey of the2-primary Adams spectral sequence here.

[vanishing line; Adams periodicity]We have chosen to display this particular portion of the spectral sequence because two phenomena occur

for the first time in topological degree t − s = 15: in bidegree (15, 5) sits the first Ext whose dimensionis larger than 1 and the first non-zero differentials originate in topological degree t − s = 15. The first ofthe Adams differentials can also be deduced from the fact that 2σ2 = 0 (since the stable homotopy groupsare graded commutative), whereas the class h0h

23 which detects 2σ2 is non-zero in Ext3,17. This can only

happen if h0h23 is the image of some differential, but there is nothing except the class h4 which could hit

h0h23. So we must have d2(h4) = h0h

23.

At an odd prime p, the primitive elements in the dual Steenrod algebra and the classes they representsin Ext1,∗ are given by

a0 = [τ0] ∈ Ext1,1 and hi = [ξpi

1 ] ∈ Ext1,piq .

[are these the standard names?]. Besides certain products [which are these, besides power of a?], the 2-linecontains a non-trivial class bi ∈ Ext2,qp

i

for each i ≥ 1 which is represented in the cobar complex byp−1∑k=1

1p

(p

k

)[ξp

ik1 |ξ

pi(p−k)1 ] .

There is also a decomposition of bi using ‘long’ Massey products (which we have not defined) as

bi ∈ 〈hi, . . . , hi〉 (p-fold bracket).

[vanishing line] Here is a chart for p = 3 up to topological degree t− s = 21

0 2 4 6 8 10 12 14 16 18 200

2

4

6

••a0

•••••

•jjjjjjjjjj

h0

••jjjjjjjjjj •b0

•jjjjjjjjjj

•h1

••

• ••jjjjjjjjjj •

b20

••jjjjjjjjjj

•

///////

WW

///////

WW

///////

WW

As the picture for p = 3 indicates, the classes h0 = a, h1 and b1 are permanent cycles since thereare no possible target for differentials. They detect homotopy classes denoted 3ι ∈ πs0, α1 ∈ πs2p−3 andβ1 ∈ πs2p(p−1)−2. The class β1 has a decomposition as a (in general ‘long’) Toda bracket

β1 = 〈α1, . . . , α1〉 (p-fold bracket) .

The first systematic families of differentials are the Adams differential

d2(hi) = h0bi−1

for i ≥ 1 and the Toda differentiald2p−1(bi+1) = h1b

pi ,

which imply various other differentials by the derivation property.

EXERCISES 245

The vertical lines represent multiplication by a, the diagonal lines of slope 1/3 represent multiplicationby a0 and the dotted diagonal lines of slope 1/7 represent the Massey product operation 〈h0, h0,−〉. and thedashed diagonal lines of slope 1/4 represent the Massey product operation 〈h0, a0,−〉. All the generatorson the line of 1/4 starting at h0 are obtained from one another by the Massey product operation 〈h0, a0,−〉(which we have not indicated graphically).

Exercises

Exercise 11.1. Let us denote by γss : Spss −→ SHC the restriction of the localization functorγ : Sp −→ SHC of Theorem 1.6 to the full subcategory of semistable symmetric spectra of simplicialsets. Show that γss is a localization of the category of semistable symmetric spectra at the class of π∗-isomorphism. [same for (Ω-spectra, level equivalences) and (injective Ω-spectra, homotopy equivalences).

Exercise 11.2. This exercise generalizes Lemmas ?? and 2.5. Let K be a pointed simplicial set whosereduced integral homology is concentrated in one dimension, where it is free abelian of rank 1. The degreeof a based self map τ : K −→ K is the unique integer deg(τ) such that τ induces multiplication by deg(τ)on reduced integral homology.

Show that for every injective Ω-spectrum X the induced morphism τ∗ : XK −→ XK equals deg(τ) · Idin the group [XK , XK ]. Show that for every symmetric spectrum A of simplicial sets, the image in thegroup SHC(K ∧ A,K ∧ A) of the self map τ ∧ Id of K ∧ A under the localization functor γ : Sp −→ SHCis multiplication by the degree of τ .

Exercise 11.3. In Proposition 3.4 we constructed the symmetric monoidal derived smash product onthe stable homotopy category of flat symmetric spectra and proved some properties. In this exercise weshow that these properties uniquely determine the derived smash product and the coherence isomorphismson SHC[.

(i) Show that there is only one functor ∧L : SHC[ × SHC[ −→ SHC[ which satisfies the equality∧L (γ[ × γ[) = γ[ ∧ as functors SpΣ,[ × SpΣ,[ −→ SHC[.

(ii) Show that there is only one way to define unit, associativity and symmetric isomorphisms for ∧Lon SHC[ if we want the functor γ[ : SpΣ,[ −→ SHC[ to strong symmetric monoidal with respect to theidentity transformation.

Exercise 11.4. Let T be a triangulated category and fn : Xn −→ Xn+1 a sequence of composablemorphism for n ≥ 0. Let (X, ϕn) and (X ′, ϕ′n) be two homotopy colimits of the sequence (Xn, fn). Constructan isomorphism ψ : X −→ X ′ satisfying ψϕn = ϕ′n and commuting with connecting morphisms to thesuspension of

⊕n≥0Xn. To what is extent it the isomorphism ψ unique?

Exercise 11.5. . Show that the Spanier-Whitehead has a symmetric monoidal smash product whichis defined by (X,n)∧ (Y,m) = (X ∧Y, n+m) on objects and with unit object (S0, 0). Make the embeddingΣ∞ of the Spanier-Whitehead category into SHC is compatible with smash products, i.e., it can be madeinto a strong symmetric monoidal functor.

Exercise 11.6. Let f : X −→ Y and g : Y −→ Z be morphisms in a triangulated category T suchthat the composite gf : X −→ Z zero and the group [ΣX,Z] is trivial. Show that there is at most onemorphism h : Z −→ ΣX such that (f, g, h) is a distinguished triangle.

Exercise 11.7. For spaces, (co)homology of spectra is defined from a certain functorially defined‘singular chain complex’; in this exercise we show that also spectrum (co-)homology can be calculatedfrom a chain complex which is assembled from the singular chain complexes of the individual spaces in thespectrum.

The definition of spectrum homology is very analogous to the definition of singular homology fortopological spaces. We recall that the definition of singular homology can be broken up as a composite ofseveral functors:

T S−→ sSZ[−]−−−→ sAb C−→ (chain complexes) Hk−−→ Ab .


The first functor associates to a spaces its singular complex, a simplicial set. By taking free abelian groupsin every simplicial dimension, the second step produces a simplicial abelian group. The third functor takesthe alternative sum of face morphisms to turn a simplicial abelian group into a chain complex. The singularhomology, finally, is the homology of this ‘singular chain complex’.

In the context of symmetric spectra we now define similar functors

(11.8) SpTS−→ SpsS

Z[−]−−−→ SpsAbC−→ (chain complexes) Hk−−→ Ab .

There is one qualitative difference to the previous case in that the singular chain complex of a space isconcentrated in non-negative dimensions, whereas the singular chain complex of a symmetric spectrum isin general not bounded below. Thus the homology groups of symmetric spectra are integer graded.

The first functor in the sequence (11.8) is an old friend, the singular complex applied dimensionswise toa symmetric spectrum of topological spaces. It yields a symmetric spectrum of simplicial sets as explainedin Section 1. The second functor takes reduced free abelian groups in every spectrum level and everysimplicial dimension. It naturally lands in a category of ‘symmetric spectra of abelian groups’ which wehave not yet introduced.

Definition 11.9. Let A be a ring. A symmetric spectrum of A-modules consists of the following data:• a sequence of simplicial left A-modules pointed spaces Mn for n ≥ 0• an A-linear simplicial left action of the symmetric group Σn on Mn for each n ≥ 0• morphisms of simplicial A-modules σn : Mn ⊗ Z[S1] −→Mn+1 for n ≥ 0.

This data is subject to the following condition: for all n,m ≥ 0, the composite(11.10)

Mn ⊗ Z[Sm]σn⊗ Id // Mn+1 ⊗ Z[Sm−1]

σn+1⊗Id // · · ·σn+m−2⊗Id// Mn+m−1 ⊗ Z[S1]

σn+m−1 // Mn+m

is Σn × Σm-equivariant. Here the symmetric group Σm acts by permuting the coordinates of Sm, andΣn × Σm acts on the target by restriction of the Σn+m-action.

A morphism f : M −→ N of symmetric spectra of A-modules consists of Σn-equivariant homomor-phisms fn : Mn −→ Nn for n ≥ 0, which are compatible with the structure maps in the sense thatfn+1 σn = σn (fn ∧ IdS1) for all n ≥ 0. The category of symmetric spectra is denoted by SpΣ

Amod.If A = Z is the ring of integers we also speak of symmetric spectra of abelian groups and write SpΣ

Ab forSpΣ

Zmod.

Symmetric spectra of abelian groups are symmetric spectra with extra structure. First, there is aforgetful functor

SpAb −→ Spfrom symmetric spectra of abelian group to symmetric spectra of simplicial sets which in every level forgetsthe abelian groups structure and chooses the zero element as the basepoint. The structure map of theunderlying symmetric spectrum is obtained as the composite

σn : Mn ∧ S1 −→ Mn ⊗ Z[S1] −→ Mn+1 .

[the adjoint σn : Mn −→ ΩMn+1 is a homomorphism]

Construction 11.11. Let A be a ring and M a symmetric spectrum of A-modules. We define thechain complex CM of A-modules by

CM = colimn N(Mn)[−n] .

In more detail, N(Mn) is the normalized chain complex (compare Section A.??) of the simplicial A-moduleMn, a non-negatively graded chain complex of A-modules. ThenN(Mn)[−n] is the shifted complex, which isnow concentrated in dimensions −n and above. The colimit is formed over the sequence of chain morphisms

N(Mn)[−n] ∼= N(Mn)[−n− 1]⊗N Z[S1] ∇−→ N(Mn ⊗ Z[S1])[−n− 1]N(σn)[−n−1]−−−−−−−−−→ N(Mn+1)[−n− 1]

EXERCISES 247

where ∇ is the (normalized) shuffle map, compare A.??.

Since the shuffle map is symmetric monoidal the complex CX is in fact the colimit, over inclusions,of a complex valued I-functor. So CX is naturally a chain complex of tame M-modules, and hence itshomology groups are tame M-modules.

The singular chain complex CX is the chain complex C(Z[X]) of the linearization of X. [is this acomplex of free abelian groups ? or do we take the mapping telescope ? that is a complex of free abeliangroups, and in fact cofibrant] If X is a symmetric spectrum of topological spaces, then its singular chaincomplex as the chain complex of the spectrum of simplicial sets S(X).

Since tensor product with an abelian group A commutes with normalized chain complex and colimits,the chain complex A⊗ CX is naturally isomorphic to the chain complex of the A-linearization A[X].

The chain complex CX is covariantly functorial in X. Hence the homology groups Hk(X,A) arecovariant functors in the symmetric spectrum X and the coefficient group A. Similarly, the cohomologygroups Hk(X,A) are contravariant functors in X and covariant functors in A.

Let X be a symmetric spectrum and pX : X −→ γX a stable equivalence whose target is an Ω-spectrum. Construct a chain of quasi-isomorphisms between the chain complexes Z ⊗LM CX and C(γX).Let f : X −→ Y be a morphism between connective symmetric spectra. Deduce a spectral sequenceconverging to H∗(γX,A) of a similar form as the one for homotopy groups. Show that f is a stableequivalence if and only if the morphism Z ⊗LM Cf : Z ⊗LM CX −→ Z ⊗LM CY is a quasi-isomorphism ofchain complexes.

Exercise 11.12. We recall that for a pointed space X the Hurewicz homomorphism h : πnX −→Hn(X,Z) sends the homotopy class of a based map f : Sn −→ X to the homology class f∗(ιn) whereιn ∈ Hn(Sn,Z) is the fundamental class, a chosen generator of the free abelian group Hn(Sn,Z) [chose theright one to make it coincide on the nose with inclusion of generators]. Show that the ‘unstable’ Hurewiczhomomorphisms for spaces converge to the ‘stable’ Hurewicz homomorphism for symmetric spectra in thefollowing sense.

(i) For every symmetric spectrum of topological spaces X and every integer k and n ≥ 0, the followingsquare commutes

πk+nXnι //

h

πk+n+1Xn+1

h

Hk+n(Xn,Z) // Hk+n+1(Xn+1,Z)

where the lower horizontal map is the one from the colimit system [...].(ii) The composite map

πkXcolimn h−−−−−→ colimn Hk+n(Xn,Z) −→∼= Hk(X,Z) i−→∼= πkZ[X]

agrees with the effect of the morphisms X −→ Z[X], by inclusion of generators, on homotopy groups. Herethe second map is the isomorphism of [...] and the third map is the isomorphism of [...].

Exercise 11.13. [Define Massey products in the cobar complex] Use the cobar complex of the dualSteenrod algebra at the prime 2 to deduce the following multiplicative and Massey product relations amongthe classes hn and cn:


(i) hnhn+1 = 0(v) h3

n+1 = h2nhn+2

(vi) h4n+1 = 0

(vii) hnh2n+2 = 0

(ix) h2nh

2n+3 = 0

(x) h2n

0 hn = 0 for n ≥ 1(xi) h2n

0 h2n+2 = 0

(ix) hncn = hn+2cn = hn+3cn = 0(iii) h2

n+1 = 〈hn, hn+1, hn〉(iv) hnhn+2 = 〈hn+1, hn, hn+1〉(viii) h3

n−1hn+2 = 〈hn+1, h3n−1, hn+1〉

cn = 〈hn+1, hn, h2n+2〉 = 〈hn, h2

n+2, hn+1〉(xi) 〈hn+1, hn+2, cn〉 3 0

(In (x), we supposedly already have h2n−10 hn = 0.) Hint: one can save some work by systematically

exploiting the juggling formulas for Massey products.Deduce that the products 2η, ην and νσ are trivial in the 2-primary homotopy groups of spheres and

that the Toda bracket relations

η2 ∈ 〈2, η, 2〉 , 2ν ∈ 〈η, 2, η〉 , ν2 ∈ 〈η, ν, η〉 and 8σ ∈ 〈ν, 8, ν〉hold. What can you conclude about the brackets 〈ν, η, ν〉 and 〈ν, σ, ν〉 ? Can you express ν3 in terms ofη2σ and ηε ?

Exercise 11.14. One can show that Ext3,?? is generated by the class h0h2n and that the class h0h

3n

is non-trivial in Ext4,???. Use this fact, the first Adams differential d2(h4) = h0h23 and the relations of

Exercise 11.13 to deduce the Adams differential

d2(hn) = h0h2n−1

for n ≥ 4 in the 2-primary Adams spectral sequence for the sphere spectrum.

Exercise 11.15. Show that the mod-3 Moore spectrum SZ/3 is not a homotopy ring spectrum becausethe multiplication morphism SZ/3 ∧L SZ/3 −→ SZ/3 is not associative in the stable homotopy category.This can be done in the following two steps:

(i) Show that the unique unit preserving morphism κ : SZ/3 −→ HZ/3 in SHC is compatible with themultiplications, i.e., the square

SZ/3 ∧L SZ/3 κ∧Lκ //

µ

HZ/3 ∧L HZ/3

µ

SZ/3

κ// HZ/3

commutes in the stable homotopy category.(ii) Use the Toda bracket relation 〈τ0, τ0, τ0〉 = ξ1 in H4(HZ/3,F3) to derive a contradiction from

the assumption that the multiplication of SZ/3 is homotopy associative.

History and credits

The stable homotopy category as we know it today is usually attributed to Boardman, who introducedit in his thesis [4] including the triangulated structure and the symmetric monoidal (derived !) smashproduct. Boardman’s stable homotopy category is obtained from a category of CW-spectra by passing tohomotopy classes of morphisms. Boardman’s construction was widely circulated as mimeographed notes [5],but he never published these. Accounts of Boardman’s construction appear in [78], [81], and [2, Part III].Strictly speaking the ‘correct’ stable homotopy category had earlier been introduced by Kan [39] based onhis notion of semisimplicial spectra. Kan and Whitehead [40] defined a smash product in the homotopy


category of semisimplicial spectra and proved that it is homotopy commutative, but neither they, noranyone else, ever addressed the associativity of that smash product. Before Kan and Boardman there hadbeen various precursors of the stable homotopy category, and I recommend May’s survey article [55] for adetailed discussion and an extensive list of references to these.

I am not aware of a complete published account that Boardman’s category is really equivalent to thestable homotopy category as defined in Definition 1.1 using injective Ω-spectra. However, here is a shortguide through the literature which outlines a comparison. In a first step, Boardman’s stable homotopycategory can be compared to Kan’s homotopy category of semisimplicial spectra, which is done in ChapterIV of Boardman’s unpublished notes [5]. An alternative source is Tierney’s article [78] where he promotesthe geometric realization functor to a functor from Boardman’s category of CW-spectra to Kan’s categoryof semisimplicial spectra. Tierney remarks that the singular complex functor from spaces to simplicial setdoes not lift to a pointset level functor in the other directions, but Section 3 of [78] then ends with thewords “(. . . ) it is more or less clear – combining various results of Boardman and Kan – that the singularfunctor exists at the level of homotopy and provides an inverse to the stable geometric realization, i.e. thetwo homotopy theories are equivalent. The equivalence of homotopy theories has also been announced byBoardman.” I am not aware that the details have been carried out in the published literature.

Kan’s semisimplicial spectra predate model categories, but Brown [14, Thm. 5] showed later that theπ∗-isomorphisms used by Kan are part of a model structure on semisimplicial spectra. In the paper [12]Bousfield and Friedlander introduce a model structure on a category of ‘sequential spectra’ which are justlike symmetric spectra, but without the symmetric group actions. In Section 2.5 of [12], Bousfield andFriedlander describe a chain of Quillen equivalences between semisimplicial and sequential spectra, whichthen in particular have equivalent homotopy categories. Hovey, Shipley and Smith show in [33, Thm. 4.2.5]that the forgetful functor is the right adjoint of a Quillen equivalence from symmetric spectra (with thestable absolute projective model structure in the sense of Chapter III) to the Bousfield-Friedlander stablemodel structure of sequential spectra. Since the weak equivalences used for symmetric spectra are thestable equivalences in the sense of Definition 4.8 we can conclude that altogether that Boardman’s stablehomotopy category is equivalent to the localization of the category of symmetric spectra at the class ofstable equivalences, which coincides with the stable homotopy category in our sense by Theorem 1.6.

A word of warning: the comparison which I just summarized passes through the intermediate homotopycategory of sequential spectra for which no intrinsic way to define a derived smash product has been studied.As a consequence, it is not clear to me if the combined equivalence takes Boardman’s derived smash productto the derived smash product as discussed in Section 3. However, I would be surprised if the compositeequivalence were not strongly symmetric monoidal.

CHAPTER III

Model structures

Symmetric spectra support many useful model structures and we will now develop several of these.We will mainly be interested in two kinds, namely level model structures (with weak equivalences the levelequivalences) and stable model structures (with weak equivalences the stable equivalences). The level modelstructures are really an intermediate steps towards the more interesting stable model structures. We willdevelop the theory for symmetric spectra of simplicial sets first, and later say how to adapt things tosymmetric spectra of topological spaces.

We have already seen pieces of some of the model structures at work. Our definition of the stablehomotopy category in Section 1 of Chapter II is implicitly relying on the absolute injective stable modelstructure in which every object is cofibrant (as long as we use simplicial sets, not topological spaces) and thefibrant objects are the injective Ω-spectra. However, this model structure does not interact well with thesmash product, so when we constructed the derived smash product in Section 3 of Chapter II we implicitlyworked in the flat model structures. So it should already be clear that it can be useful to play differentmodel structures off against each other.

Besides the injective and flat model structures there is another useful kind of cofibration/fibration pairwhich we will discuss, giving the projective model structures. Moreover, we will later need ‘positive’ versionsof the model structures which discard all homotopical information contained in level 0 of a symmetricspectrum.

So each of the model structures which we discuss has four kinds of ‘attributes’:• a kind of space (simplicial set or topological space)• a kind of cofibration/fibration pair (injective, flat or projective)• a type of equivalence (level or stable)• which levels are used (absolute or positive)

Since all of these attributes can be combined, this already makes 2×3×2×2 = 24 different model structureson the two kinds of symmetric spectra. More variations are possible: one can also take π∗-isomorphisms asweak equivalences, or even isomorphisms in some homology theory (giving model structures which realizeBousfield localizations), or one could study ‘more positive’ model structures which disregard even morethan the level 0 information. And this is certainly not the end of the story. . .

1. Level model structures

1.1. Types of cofibrations. The latching space LnA of a symmetric spectrum A was defined inDefinition I.5.22 as the nth level of the symmetric spectrum A∧ S, where S is the subspectrum of the spherespectrum with S0 = ∗ and Sn = Sn for positive n. We also gave a more explicit presentation of LnA asa quotient of Σ+

n ∧Σn−1 An−1 ∧ S1. The latching space has a based Σn-action and comes with a naturalequivariant map νn : LnA −→ An.

For a morphism f : A −→ B of symmetric spectra and n ≥ 0 we have a commutative square ofΣn-simplicial sets

LnALnf //

νn

LnB

νn

An

fn

// Bn

251

252 III. MODEL STRUCTURES

We thus get a natural morphism of Σn-simplicial sets

νn(f) : An ∪LnA LnB −→ Bn .

Definition 1.1. A morphism f : A −→ B of symmetric spectra of simplicial sets is• a projective cofibration if for every n ≥ 0 the morphism νn(f) is injective and the symmetric group

Σn acts freely on the complement of its image;• a flat cofibration if for every n ≥ 0 the morphism νn(f) is injective;• a level cofibration if it is a categorical monomorphism, i.e., if for every n ≥ 0 the morphismfn : An −→ Bn is injective.

By the criterion for flatness given in Proposition II.5.35 a symmetric spectrum A is flat in the originalsense (i.e., A∧− preserves monomorphisms) if and only if the unique morphism ∗ −→ A is a flat cofibration.We call a symmetric spectrum A projective if the unique morphism ∗ −→ A is a projective cofibration or,equivalently, if for every n ≥ 0 the morphism νn : LnA −→ An is injective and the symmetric group Σnacts freely on the complement of its image. Every symmetric spectrum of simplicial sets is level cofibrant.

Now we define the analogues of the three kinds of cofibrations for symmetric spectra of pointed topo-logical spaces. We refer to the standard model structure on the category of pointed compactly generatedweak Hausdorff spaces as described for example in [32, Thm. 2.4.25]. In this model structure, the weakequivalences are the weak homotopy equivalences and fibrations are the Serre fibrations. The cofibrationsare the retracts of ‘generalized CW-complexes’, i.e., cell complexes in which cells can be attached in anyorder and not necessarily to cells of lower dimensions. The term ‘Σn-cofibration’ in the next definition refersto the model structure on pointed Σn-spaces which is created by the forgetful functor to pointed spaces.These cofibrations are the retracts of ‘generalized free Σn-CW-complexes’, i.e., equivariant cell complexesin which only free Σn-cells are attached.

Definition 1.2. A morphism f : A −→ B of symmetric spectra of topological spaces is• a projective cofibration if for every n ≥ 0 the morphism νn(f) is a Σn-cofibration;• a flat cofibration if for every n ≥ 0 the morphism νn(f) is a cofibration of spaces;• a level cofibration if for every n ≥ 0 the morphism fn : An −→ Bn is a cofibration of spaces.

To see the analogy with the earlier definitions for symmetric spectra of simplicial sets one should remem-ber that in the standard model structure for pointed simplicial set the cofibrations are the monomorphisms.

By definition every projective cofibration is also a flat cofibration. Flat cofibrations are level cofibrationsby the following lemma.

Lemma 1.3. Let f : A −→ B be a morphism of symmetric spectra of simplicial sets or topologicalspaces. Then f is a flat cofibration if and only if for every level cofibration g : X −→ Y the pushout productmap

f ∧ g : B ∧X ∪A∧X A ∧ Y −→ B ∧ Yis a level cofibration. In particular, every flat cofibration is a level cofibration.

Proof. The inclusion S −→ S is a level cofibration and in level n the pushout product of f with thisinclusion is the morphism νn(f) : An ∪LnA LnB −→ Bn. So the pushout product condition for all levelcofibrations implies that f : A −→ B is a flat cofibration.

For the other direction we first consider the special case where f is of the form Gmi : GmK −→ GmLfor a Σm-equivariant map i : K −→ L of pointed simplicial sets or spaces which is a cofibration in theunderlying category. We use the isomorphism GmL ∧X ∼= L .m X (compare Proposition I.5.5) to rewritethe pushout product (Gmi) ∧ g as

L .m X ∪K.mX K .m Y −→ L .m Y .

In level m+n this morphism is given by first forming the pushout product of i∧gn : L∧Xn∪K∧XnK∧Yn −→L∧Yn as pointed simplicial set (or pointed spaces), and then inducing up from Σm×Σn to Σm+n. The map

1. LEVEL MODEL STRUCTURES 253

i ∧ gn is a cofibration by the pushout product property of simplicial sets respectively spaces, and inducingup preserves cofibrations. So in this special case the pushout product map is a level cofibration. [...]

If we apply this to the injective map ∗ −→ S the pushout product is isomorphic to the map f . So as aspecial csae we obtain that every flat cofibration is a level cofibration.

Lemma 1.4. Let f : A −→ B be a morphism of symmetric spectra of simplicial sets. Then f : A −→ Bis a projective cofibration if and only if it is a flat cofibration and the cokernel B/A is projective. [Is amorphism f : A −→ B is a flat cofibration if and only if it is an injective cofibration (i.e., monomorphism)and the cokernel B/A is flat?]

Proof. This is direct consequence of the definitions since a group acts freely on the complement ofthe image of an equivariant map A −→ B if and only if the induced action on the quotient B/A is freeaway from the basepoint.

Thus we have the following implications for the various kinds of cofibrations:

projective cofibration =⇒ flat cofibration =⇒ level cofibration

All these containments are strict, as the following examples show. The symmetric spectrum S is not flatsince its second latching object L2S is isomorphic to S1∨S1 and the map L2S −→ S2 = S2 is the fold map,which is not injective. Semifree symmetric spectra GmL are flat for all pointed Σm-simplicial sets L, butthey are projective only if Σm acts freely away from the basepoint (compare the following proposition).

Proposition 1.5. (i) Let K be a pointed Σm-simplicial set for some m ≥ 0. Then the nthlatching spaces of the semifree symmetric spectrum GmK is trivial for n ≤ m and for n > m themap νn : Ln(GmK) −→ (GmK)n is an isomorphism.

(ii) Given m ≥ 0 and an injective morphism f : K −→ L of Σm-simplicial sets, the induced mapGmf : GmK −→ GmL on semifree symmetric spectra is a flat cofibration, and it is a projectivecofibration if and only if Σm acts freely on the complement of the image of f . In particular, everysemifree symmetric spectrum GmL is flat and it is projective if and only if Σm acts freely on Laway from the basepoint.

(iii) If f : K −→ L is a monomorphism of simplicial sets, then for every m ≥ 0, the induced mapFmf : FmK −→ FmL is a projective cofibration. In particular, all free symmetric spectra areprojectively cofibrant.

Proof. (i) For n < m, the latching space Ln(GmK) consists only of the basepoint. For n ≥ m,substitution of the definitions and some rewriting gives

Lm+n(GmK) = (GmK ∧ S)m+n∼= (K .m S)m+n = Σ+

m+n ∧Σm×Σn K ∧ Sn

=

∗ for n = 0, and

Σ+m+n ∧Σm×Σn K ∧ Sn ∼= (GmK)m+n for n ≥ 1.

where we used the identification of GmK∧X with the twisted smash product K.mX (see Proposition I.5.5).(ii) We use part (i) to identify the terms in the commutative square of Σn-simplicial sets

Ln(GmK)Ln(Gmf) //

νn

Ln(GmL)

νn

(GmK)n

fn

// (GmL)n

For n < m all four terms are just points. For n = m the two upper objects are points and the lower verticalmap is injective. For n > m both vertical maps are isomorphisms. So the map

νn(Gmf) : (GmK)n ∪Ln(GmK) Ln(GmL) −→ (GmL)n


is an isomorphism for n 6= m and injective for n = m. So Gmf is always a flat cofibration. The only casein which νn(Gmf) is not an isomorphism is n = m, and then νn(Gmf) is isomorphic to f : K −→ L. SoGmf is a projective cofibration if and only if Σm acts freely away from the image of f .

The natural isomorphism between FmK and Gm(Σ+m ∧K) makes (iii) a special case of (ii).

Proposition 1.6. Let A be a flat symmetric spectrum and n ≥ 2. Then the symmetric power spectrum(A∧n)/Σn is again flat.[is the product of flat spectra flat ? how about AK and shA ?]

Theorem 1.7. Let f : X −→ Y be an injective morphism of Γ-spaces of simplicial sets. Then theassociated morphism f(S) : X(S) −→ Y (S) is a flat cofibration of symmetric spectra. In particular, forevery Γ-space of simplicial sets X, the associated symmetric spectrum X(S) is flat.

[how do the BF- and Q-cofibrations of Γ-spaces relate to the various cofibrations ?]

Proof.

Definition 1.8. A morphism f : K −→ L of Σn-simplicial sets is a Σn-fibration (respectively Σn-equivalence ) if the induced map on H-fixed points fH : KH −→ LH is a Kan fibration (respectively weakequivalence) of simplicial sets for all subgroups H of Σn.

Theorem 1.9. The category of symmetric spectra of simplicial sets admits the following three levelmodel structures in which the weak equivalences are those morphisms f : X −→ Y such that for all n ≥ 0the map fn : Xn −→ Yn is a weak equivalence of simplicial sets.

(i) In the projective level model structure the cofibrations are the projective cofibrations and a mor-phism f : X −→ Y is a projective level fibration if and only if for every n ≥ 0 the mapfn : Yn −→ Xn is a Kan fibration of simplicial sets.

(ii) In the flat level model structure the cofibrations are the flat cofibrations, and a morphism f :X −→ Y is a flat level fibration if and only if for every n ≥ 0 the map fn : Yn −→ Xn satisfiesthe following two equivalent conditions

– the map fn has the right lifting property for all injective morphisms of Σn-simplicial setswhich are weak equivalences on underlying simplicial set;

– map fn is a Σn-fibration and the commutative square

Xn//

fn

map(EΣ+n , Xn)

map(EΣ+n ,fn)

Yn // map(EΣ+

n , Yn)

is Σn-homotopy cartesian. Here map(EΣ+n , X) is the simplicial mapping space of all maps

from the contractible free Σn-simplicial set to X, with Σn-action by conjugation.A morphism f : X −→ Y is an acyclic fibration in the flat model structure if and only if for alln ≥ 0 the map fn : Xn −→ Yn is a Σn-equivariant acyclic fibration.

(iii) In the injective level model structure the cofibrations are the level cofibrations (i.e., monomor-phisms) and the injective fibrations are those morphisms which have the right lifting property withrespect to all morphisms which are simultaneously level equivalences and monomorphisms.

Moreover we have:• All three level model structures are proper, simplicial and cofibrantly generated.• The flat and projective level model structures are even finitely generated and monoidal with respect

to the smash product of symmetric spectra.

Proof. The category of symmetric spectra of simplicial sets has all set-indexed limits and colimits, thelevel equivalences satisfy the 2-out-of-3 property and in all three cases the classes of cofibrations, fibrationsand weak equivalences are closed under retracts. So it remains to prove the factorization and lifting axioms.


As usual we construct the factorizations using Quillen’s small object argument. We first defined therespective classes I lvproj , I

lvflat and I lvinj of generating cofibrations and J lvproj , J

lvflat and J lvinj of generating

acyclic cofibrations. As generating projective cofibrations we take

I lvproj =Fn∂∆[m]+ −→ Fn∆[m]+

n,m≥0

where Fn is the free symmetric spectrum generated by a pointed simplicial set in level n, see Example I.3.11.Since Fn is left adjoint to evaluation at level n, a morphism f : X −→ Y of symmetric spectra has the rightlifting property with respect to I lvproj if and only if for every n ≥ 0 the map fn : Yn −→ Xn has the RLPfor the boundary inclusions ∂∆[n] −→ ∆[n], i.e., if it is a Kan fibration and a weak equivalence. In otherwords, precisely the morphisms which are both level equivalences and projective level fibrations enjoy theright lifting property with respect to I lvproj .

As generating flat cofibrations we take

I lvflat =Gm (Σm/H × ∂∆[n])+ −→ Gm (Σm/H ×∆[n])+

n,m≥0, H≤Σn

where Gm is the semifree symmetric spectrum generated by a pointed Σm-simplicial set in level m, seeExample I.3.12. Since Gm is left adjoint to evaluation at level m with values in Σm-simplicial sets, amorphism f : X −→ Y of symmetric spectra has the right lifting property with respect to I lvflat if and onlyif for every m ≥ 0 and every subgroup H of Σn the map (fm)H : (Ym)H −→ (Xm)H on H-fixed points offm is a Kan-fibration and weak equivalence of simplicial sets. By Proposition A.4.5 this is equivalent tothe property that fn is simultaneously a Σn-fibration, a weak equivalence on underlying simplicial sets andthe square above is Σn-homotopy cartesian; in other words, precisely the flat level fibrations defined in (ii)above enjoy the right lifting property with respect to I lvflat.

Let f : X −→ Y be a flat level fibration. If H is a subgroup of Σn then the semifree symmetric spectrumGn(Σn/H)+ is flat. Since the flat model structure is simplicial, the induced map on mapping spaces

(Xn)H ∼= map(Gn(Σn/H)+, X) −→ map(Gn(Σn/H)+, Y ) ∼= (Yn)H

is a Kan fibration. Since this holds for all subgroup, the map fn : Xn −→ Yn is a Σn-fibration.Suppose that X is fibrant in the flat level model structure. Let L be any (unbased) Σm-simplicial set.

Then the projection EΣn×L −→ L is Σn-equivariant and a weak equivalence of underlying simplicial sets.So the induced map of semifree symmetric spectra

Gn(EΣn × L)+ −→ GnL+

is a level equivalence between flat symmetric spectra. Since the flat model structure is simplicial, theinduced map on mapping spaces

map(GnL+, X) −→ map(Gn(EΣn × L)+, X)

is a weak equivalence. By adjointness that map is isomorphic to

mapΣn(L+, Xn) −→ mapΣn((EΣn × L)+, Xn) ∼= mapΣn(L+,map(EΣ+n , Xn))

where map(EΣ+n , Xn) is the space (i.e., simplicial set) of all morphisms from EΣ+

n to X with conjugationaction by Σn. If we specialize to L = Σn/H for a subgroup H of Σn we see that the map

Xn −→ map(EΣ+n , Xn)

is a weak equivalence on H-fixed points, so it is an equivariant equivalence.To define the generating injective cofibrations we choose one representative for each isomorphism class

of pairs (B,A) consisting of a countable symmetric spectrum B and a symmetric subspectrum A.

We still have to show that the level model structures are simplicial and that the flat and projectivelevel model structures are monoidal with respect to the internal smash product of symmetric spectra. Sowe have to verify various forms of the pushout product property. We recall that the pushout product of


a morphism i : K −→ L of pointed simplicial sets or symmetric spectra and a morphism j : A −→ B ofsymmetric spectra is the morphism

i ∧ j : L ∧A ∪K∧A K ∧B −→ L ∧B .

The first proposition below is about smash products of simplicial sets with symmetric spectra, and it saysthat various model structures of symmetric spectra are simplicial model structures. The next propositionis about internal smash products of symmetric spectra, and it says that various flat and projective (but notinjective) model structures of symmetric spectra are monoidal model structures.

Proposition 1.10. Let i : K −→ L be a morphism of pointed simplicial sets and j : A −→ B amorphism of symmetric spectra.

(i) If i is injective and j a level cofibration, flat cofibration respectively projective cofibration, then thepushout product i ∧ j is also a level cofibration, flat cofibration respectively projective cofibration.

(ii) If i is an injective weak equivalence of simplicial sets, and j is a level cofibration (i.e, monomor-phism), then i ∧ j is also a level equivalence of symmetric spectra.

(iii) If i is injective and j a level cofibration (i.e, monomorphism) and a level equivalence, π∗-isomorphism respectively stable equivalence of symmetric spectra, then i ∧ j is also a level equiva-lence, π∗-isomorphism respectively stable equivalence.

Thus the injective, flat and projective level model structures are simplicial model categories.

Proof. For every pointed simplicial set K and symmetric spectrum A the smash product K ∧ L isnaturally isomorphic to the smash product of the suspension spectrum Σ∞K with A. The suspensionspectrum functor takes injective maps of simplicial sets to projective cofibrations (see Proposition 1.5 (iii)for m = 0) and it takes weak equivalences to level equivalences. So this proposition is a special case ofProposition 1.11 below.

Proposition 1.11. Let i : K −→ L and j : A −→ B be morphisms of symmetric spectra.(i) If i is a level cofibration and j is a flat cofibration, then i ∧ j is a level cofibration.(ii) If both i and j are flat cofibrations, then so is i ∧ j.(iii) If both i and j are projective cofibrations, then so is i ∧ j.(iv) If i is a level cofibration, j a flat cofibration and one of i or j a level equivalence, π∗-isomorphism

respectively stable equivalence, then i ∧ j is also a level equivalence, π∗-isomorphism respectivelystable equivalence.

Thus the flat and projective level model structures are monoidal model categories with respect to the smashproduct of symmetric spectra.

Proof. Check on generators.

[State all adjoint forms of the simplicial and monoidal axiom]

Definition 1.12. A morphism f : K −→ L of Σn-spaces is a Σn-fibration (respectively Σn-equivalence)if the induced map on H-fixed points fH : KH −→ LH is a Serre fibration (respectively weak equivalence)for all subgroups H of Σn.

Theorem 1.13. The category of symmetric spectra of topological spaces admits the following two levelmodel structures in which the weak equivalences are those morphisms f : X −→ Y such that for all n ≥ 0the map fn : Xn −→ Yn is a weak equivalence of spaces.

(i) In the projective level model structure the cofibrations are the projective cofibrations and a mor-phism f : X −→ Y is a projective level fibration if and only if for every n ≥ 0 the mapfn : Yn −→ Xn is a Serre fibration.

(ii) In the flat level model structure the cofibrations are the flat cofibrations, and a morphism f :X −→ Y is a flat level fibration if and only if for every n ≥ 0 the map fn : Yn −→ Xn satisfiesthe following two equivalent conditions


– the map fn has the right lifting property for all cofibrations of pointed Σn-spaces which areweak equivalences on underlying spaces;

– map fn is a Σn-fibration and the commutative square

Xn//

fn

map(EΣ+n , Xn)

map(EΣ+n ,fn)

Yn // map(EΣ+

n , Yn)

is Σn-homotopy cartesian. Here map(EΣ+n , X) is the space of all maps from the contractible

free Σn-space to X, with Σn-action by conjugation.A morphism f : X −→ Y is an acyclic fibration in the flat model structure if and only if for alln ≥ 0 the map fn : Xn −→ Yn is a Σn-equivariant acyclic fibration.

Moreover, both level model structures are proper, topological and finitely generated, and monoidal withrespect to the smash product of symmetric spectra.

[Is there are an injective level model structure for symmetric spectra of spaces ?]Except for the injective level model structure, all the other level model structure can be produced by

one very general method. For this purpose we consider a category C which is complete, cocomplete andsimplicial , by which we mean enriched, tensored and cotensored over the category of pointed simplicialsets. In this situation, the category G-C of G-objects in C is also complete, cocomplete and simplicial forevery group G; indeed, limits, colimits, tensors and cotensors in G-C are created in the underlying categoryC. [note: C is pointed]

Definition 1.14. Let C be a simplicial category. A symmetric spectrum in C consists of the followingdata:

• an object Xn of C equipped with an action of the symmetric group Σn for n ≥ 0• C-morphisms σn : Xn ∧ S1 −→ Xn+1 for n ≥ 0.

This data is subject to the following condition: for all n,m ≥ 0, the composite

(1.15) Xn ∧ Smσn ∧ Id // Xn+1 ∧ Sm−1

σn+1∧Id // · · ·σn+m−2∧Id // Xn+m−1 ∧ S1

σn+m−1 // Xn+m

is Σn × Σm-equivariant.A morphism f : X −→ Y of symmetric spectra in C consists of Σn-equivariant morphisms fn : Xn −→

Yn for n ≥ 0, which satisfy fn+1 σn = σn (fn ∧ IdS1) for all n ≥ 0. We denote the category of symmetricspectra in C by SpΣ(C).

Of course, when C = T is the category of based compactly generated weak Hausdorff spaces or C = sSis the category of based simplicial sets, we recover the definitions of symmetric spectra of spaces respectivelyof simplicial sets of Section I.1. As in these special cases, we denote the composite map [...] by σm and werefer to the object Xn as the nth level of the symmetric spectrum X. If we want a ‘level’ model structure onSp(C) we need to start with compatible model structure on the categories of Σn-objects in C. We formalizewhat we mean by ‘compatible model structures’ in the following definition. [define action of Sp, latchingmap]

Definition 1.16. Let C be a pointed simplicial category. A global model structure for C consists ofa simplicial model structure on the category GC of G-objects in C for every finite group G such that forevery monomorphism f : H −→ G between finite groups the restriction functors f∗ : GC −→ HC preservesfibrations and acyclic fibrations.

In the situation of the previous definition, the restriction functor f∗ : GC −→ HC automatically hasa left adjoint f∗ : HC −→ GC given by f∗X = G+ ∧H X. So in the presence of a global model structureand a group monomorphism f : H −→ Gthe adjoint functor pair (f∗, f∗) forms a Quillen pair between themodel categories of G-objects and H-objects in C. We will give lots of examples of global model structuresin [...] below.


Definition 1.17. Let C be a pointed simplicial category equipped with a global model structure, andlet f : X −→ Y be a morphism of symmetric spectra in C. We call f

• a level equivalence if fn : Xn −→ Yn is weak equivalence in the model category ΣnC for all n ≥ 0,• a level fibration if fn : Xn −→ Yn is fibration in the model category ΣnC for all n ≥ 0,• a cofibration if the latching morphism νn(f) : Xn ∪LnX LnY −→ Yn is a cofibration in the model

category ΣnC for all n.

In the previous definition, we really only need the model categories GC when G = Σn is a symmetricgroup. [...]

Lemma 1.18. Let i : D −→ E be a flat cofibration of symmetric spectra of simplicial sets and f : A −→B a cofibration in Sp(C). Then the pushout product morphism

if : E ∧A ∪D∧A D ∧B −→ E ∧B

is a cofibration in Sp(C). If in addition i or f is a level equivalence, then so is if . [no ! OK for projectivecofibrations of if (mono of H-sset)Box (cof) = cof]

Proof. The filtration of a symmetric spectrum reduces to the case where i = Gm(j) for an injectivemap j : K −→ K ′ of based Σm-simplicial sets. The analogous filtration of a symmetric spectrum in Creduces to the case where f = Gk(g) for a cofibration g : X −→ X ′ in the model category ΣnC.

The morphismj ∧ g : K ′ ∧X ∪K∧X K ∧X ′ −→ K ′ ∧X

is a cofibration in the model category ΣnC by axiom (SM7a). So the morphism

Σ+m+k ∧Σm×Σk (j ∧ g) : Σ+

m+k ∧Σm×Σk (K ′ ∧X ∪K∧X K ∧X ′) −→ Σ+m+k ∧Σm×Σk (K ′ ∧X)

is a cofibration in We have GmK ∧GkX ∼= Gm+k(Σ+m+k ∧Σm×Σk K ∧X), so in this special case we need to

show thatj .m f : K ′ .m A ∪K.mA K ′ .m B −→ K ′ .m B

is a cofibration in Sp(C), and a level equivalence if in addition i or f is one.The latching objects of K .m A below level m are trivial, and we have a natural isomorphism

Lm+k(K .m A) ∼= Σ+m+k ∧Σm×Σk (K ∧ LkA)

for k ≥ 0.

Lemma 1.19. The following are equivalent for a morphism f : A −→ B of symmetric spectra in C.(i) The morphism f is a cofibration and a level equivalence.(ii) For every n ≥ 0 the latching morphism νn(f) : An ∪LnA LnB −→ Bn is an acyclic cofibration in

the model category of Σn-objects in C.• Generating (acyclic) cofibrations for ΣnC give rise to generating (acyclic) cofibrations for Sp(C)• cofibrantly generated ? combinatorial ? stable ?

Proof. We consider a cofibration f : A −→ B and assume that the morphisms fk : Ak −→ Bk areweak equivalences for 0 ≤ n ≤ n− 1. We show that then the morphism Ln(f) : LnA −→ LnB is an acycliccofibration in ΣnC. [...]

(i)=⇒(ii) By the above, the morphism Ln(f) : LnA −→ LnB is an acyclic cofibration of Σn-objects,hence so is its cobase change An −→ An∪LnALnB. Since the composite of this with the latching morphismνn(f) equals the weak equivalence fn, the latching morphism is a weak equivalence.

(ii)=⇒(i) We use induction on n to show that fn is a weak equivalence for all n. The equality f0 =ν0(f) : A0 −→ B0 let’s us start. In the inductive step the above guarantees that the morphism Ln(f) :LnA −→ LnB is an acyclic cofibration of Σn-objects, hence so is its cobase change An −→ An ∪LnA LnB.Since the latching morphism νn(f)is a weak equivalence so is the composite fn.


Proposition 1.20. Let C be a pointed simplicial category equipped with a global model structure. Thenthe level equivalences, level fibrations and cofibrations define a simplicial model structure on the categorySp(C) of symmetric spectra in C. Moreover, the following properties holds.

(i) If the model category ΣnC is left proper (respectively right proper, respectively proper) for all n ≥ 0,then the level model structure on Sp(C) is left proper (respectively right proper, respectively proper).

(ii) If the model category ΣnC has functorial factorizations for all n ≥ 0, then the level model structureon Sp(C) has functorial factorizations.

[remark and/or exercise: we even have ‘if and only if’ in the items]

Proof. Limits, colimits, tensors and cotensors are defined levelwise [...] The 2-out-of-3 property forlevel equivalences and the closure of the three distinguished classes under retracts are direct consequencesof the corresponding properties in the global model structure on C.

For the lifting properties we consider a cofibration i : A −→ B and a level acyclic fibration f : X −→ Yin Sp(C), as well as a commutative square

Aϕ //

i

X

f

B

ψ// Y

We construct the required lifting l : B −→ X by induction over the level. Suppose we have alreadyconstructed Σk-equivariant morphisms lk : Bk −→ Xk for k = 0, . . . , n − 1 and some n ≥ 0 which satisfylkik = ϕk and fklk = ψk. Assume moreover that this data constitutes a partial morphism in the sense thatlk+1 σn = σn (lk ∧ IdS1) holds for 0 ≤ k < n− 1. This data assembles into a Σn-equivariant morphisml : LnB −→ Xn [say more] such that l Ln(i) : LnA −→ Xn coincides with ϕn νn(i). In other words, wehave the lifting problem in the category ΣnC

An ∪LnA LnBϕn∪l //

νn(i)

Xn

fn

Bn

ψn

//

ln

66

Yn

Since i is a cofibration, the left vertical morphism νn(i) is a cofibration in the model structure on Σn-objects, while the right vertical morphism fn is an acyclic fibration of Σn-objects. There is a Σn-equivariantmorphism ln : Bn −→ Xn which solves the lifting problem. [...] The induction can continue.

For the other lifting property we consider a cofibration i : A −→ B which is also a level equivalence and alevel fibration f : X −→ Y in Sp(C). We claim that then the latching morphism νn(i) : An∪LnALnB −→ Bnis not only a cofibration in ΣnC (which holds by definition), but also a weak equivalence in the given modelstructure ΣnC. Granted this for a moment, the same inductive construction as in the previous paragraphprovides the required filler l : B −→ X.

[show that νn(i) is acyclic cofibration; does this need hypotheses on the global model structure ?]Now we tackle the factorizations of a morphism f : A −→ X of symmetric spectra in C as a cofibration

followed by a level acyclic fibration. We morphisms i : A −→ B and p : B −→ X levelwise by induction.Suppose we have already constructed Bk in ΣkC for k = 0, . . . , n−1 for some n ≥ 0 as well as Σk-equivariantmorphisms ik : Ak −→ Bk and acyclic fibrations pk : Bk −→ Xk which form partial morphism and satisfypkik = fk. As above the data assembles into a morphism of Σn-objects fn ∪ i : An ∪LnA LnB −→ Xn. Wechoose a factorization

An ∪LnA LnBϕ−→ Bn

pn−→ Xn

as a cofibration followed by an acyclic fibration in the model category of Σn-objects. This gives us the nextlevel of the symmetric spectrum B, the next level of the morphism p : B −→ X, and the next level of the


morphism i : A −→ B as the composite

Ancan.−−→ An ∪LnA LnB

ϕ−→ Bn .

In the end, by construction, the morphism i : A −→ B is a cofibration and p : B −→ X is a levelacyclic fibration. Moreover, if the factorizations in the global model structure are functorial, then so is thisfactorization.

The problem of factoring a morphism f : A −→ X as a cofibration which is also a level equivalencefollowed by a level fibration is similar, but need a small extra argument. In the same way as in the lastparagraph we can produce a factorization f = qj in Sp(C) where q : Z −→ X is a level fibration andj : A −→ Z has the property that all the latching morphisms νn(j) : An ∪LnA LnZ −→ Zn are acycliccofibrations in ΣnC. Moreover, functoriality of factorization is inherited from the global model structure onC. It remains to show that j : A −→ Z is levelwise a weak equivalence in ΣnC. [...]

The additional claim (i) about right properties is straightforward because weak equivalences, fibrationsand pullbacks in Sp(C) are all given levelwise. [left properness]

For every finite group G the category of based G-spaces (compactly generated and weak Hausdorff)and the category of based G-simplicial sets admit three model structures in which equivalence are theweak equivariant equivalences, i.e., those equivariant morphisms which are weak equivalences of underlyingspaces (respectively simplicial sets). We call these three model structure the weak, tight and mixed modelstructures. Propositions 1.20 can be applied to each of these and yields the projective, tight respective flatlevel model structure of Theorem 1.13 respectively of Theorem 1.9. We discuss each of the three cases inslightly more detail.

Example 1.21 (Projective level model structure). For every finite group G the category of basedG-spaces (compactly generated and weak Hausdorff) and the category of based G-simplicial sets admitthe following weak equivariant model structures. An equivariant morphism f : A −→ B of G-spaces (G-simplicial sets) is a weak equivalence (resp. fibration) if and only if it is a weak equivalence (resp. Serre/Kanfibrations) of spaces (simplicial sets) after forgetting the G-action. In topological context the morphism fis a cofibration if it is a retract of a free G-cell complex. In the simplicial context, f is a cofibration if andonly if it is injective and the action of G on B is dimensionwise free away from the image of f .

In both contexts, the cofibrations can equivalently be described as those retracts G-cell complexesG-CW-complex such that for every non-trivial subgroup K of G the induced map on K-fixed points fK :AK −→ BK is an isomorphism. [simplicial...] [More details in the appendix...] [First reference: Quillen [59]]Since fibrations and weak equivalences are defined on underlying spaces (simplicial sets), restriction alongevery group homomorphism f : H −→ G preserves fibrations and weak equivalences, so f∗ is a rightQuillen functor. So as G-varies, the weak equivariant model structures define a global model structure onthe category of based spaces respectively simplicial sets.

Proposition 1.20 provides an associated level model structure on symmetric spectra of spaces (resp.simplicial sets), which is exact the projective level model structure of Theorem 1.13. The weak equivariantmodel structures are proper and have functorial factorizations, so the projective level model structureare also proper and have functorial factorizations. This proves part (i) of Theorem 1.13 respectively ofTheorem 1.9.

Example 1.22 (Tight level model structure). For every finite group G the category of based G-spaces(compactly generated and weak Hausdorff) and the category of based G-simplicial sets admit the followingtight equivariant model structures. The weak equivalences are again the equivariant weak equivalences(equivariant morphisms which are weak equivalences on underlying spaces or simplicial sets). A morphismf : A −→ B of G-spaces (G-simplicial sets) is a fibration if and only if it is a strong equivariant fibration,i.e., for every subgroup K of G the induced map on K-fixed points fK : AK −→ BK is a Serre fibration(respectively Kan fibration). The cofibrations are those retracts G-cell complexes G-CW-complex such thatfor every non-trivial subgroup K of G the induced map on K-fixed points fK : AK −→ BK is a weakequivalence of spaces (resp. simplicial sets). [simplicial...] [More details in the appendix...] [First reference:Cole [17]] Since the weak equivalences are defined on underlying spaces (simplicial sets), restriction along


every group homomorphism f : H −→ G preserves weak equivalences. [Fibrations...], so f∗ is a rightQuillen functor. So as G-varies, the tight equivariant model structures define a global model structure onthe category of based spaces respectively simplicial sets.

Proposition 1.20 provides an associated level model structure on symmetric spectra of spaces (resp.simplicial sets), which is exact the tight level model structure of Theorem 1.13. The tight equivariant modelstructures are proper and have functorial factorizations, so the tight level model structure are also properand have functorial factorizations. This proves part (ii) of Theorem 1.13 respectively of Theorem 1.9.

Example 1.23 (Flat level model structure). [change projective to flat...] For every finite group Gthe category of based G-spaces (compactly generated and weak Hausdorff) and the category of based G-simplicial sets admit the following weak equivariant model structures. An equivariant morphism f : A −→ Bof G-spaces (G-simplicial sets) is a weak equivalence (resp. fibration) if and only if it is a weak equivalence(resp. Serre/Kan fibrations) of spaces (simplicial sets) after forgetting the G-action. In topological contextthe morphism f is a cofibration if it is a retract of a free G-cell complex. In the simplicial context, f is acofibration if and only if it is injective and the action of G on B is dimensionwise free away from the imageof f .

In both contexts, the cofibrations can equivalently be described as those retracts G-cell complexesG-CW-complex such that for every non-trivial subgroup K of G the induced map on K-fixed points fK :AK −→ BK is an isomorphism. [simplicial...] [More details in the appendix...] [First reference: Quillen [59]]Since fibrations and weak equivalences are defined on underlying spaces (simplicial sets), restriction alongevery group homomorphism f : H −→ G preserves fibrations and weak equivalences, so f∗ is a rightQuillen functor. So as G-varies, the weak equivariant model structures define a global model structure onthe category of based spaces respectively simplicial sets.

Proposition 1.20 provides an associated level model structure on symmetric spectra of spaces (resp.simplicial sets), which is exact the projective level model structure of Theorem 1.13. The weak equivariantmodel structures are proper and have functorial factorizations, so the projective level model structureare also proper and have functorial factorizations. This proves part (i) of Theorem 1.13 respectively ofTheorem 1.9.

[positive model structures]

Lemma 1.24. A morphism of symmetric spectra is an injective fibration if and only if it has the rightlifting property with respect to all injective level equivalences between countable symmetric spectra.

Proof. Suppose that f : X −→ Y has the right lifting property with respect to all injective levelequivalences with countable target (and hence source). Consider a lifting problem

A //

i

X

f

B // Y

in which i : A −→ B is an injective level equivalence, with no restriction on the cardinality of B.We denote by P the set of ‘partial lifts’: an element of P is a pair (U, h) consisting of a symmetric

subspectrum U of B which contains the image of A and such that the inclusion U −→ B (and hence themorphism A −→ U) is a level equivalence and a morphism h : U −→ X making the following diagramcommute

A //

i

X

f

U

h

77nnnnnnnnnnnnnnincl.

// B // Y

The set P can be partially ordered by declaring (U, h) ≤ (U ′, h′) if U is contained in U ′ and h′ extends h.Then every chain in P has an upper bound, namely the union of all the subspectra U with the commonextension of the morphisms h. [uses that transfinite composite of injective level equivalences is an injective


level equivalence] So by Zorn’s lemma, the set P has a maximal element (V, k). We show that V = B, so kprovides the required lifting showing that f is an injective fibration.

We argue by contradiction and suppose that V is strictly smaller than B. Then we can find a countablesubspectrum W of B which is not contained in V and such that the morphism V ∩W −→ W is a levelequivalence [justify; the argument is in Lemmas 5.1.6 and 5.1.7 of [33]]. Since W is countable, f has theright lifting property with respect to the inclusion V ∩W −→W . We have a pushout square

V ∩W //

V

W // V ∪W

so f also has the right lifting property with respect to the inclusion V −→ V ∪W . But that means thatthe morphism k : V −→ X can be extended to V ∪W , which contradicts the assumption that (V, k) is amaximal element in the set P extensions.

2. Stable model structures

We recall from Definition II.4.8 that a morphism f : A −→ B if symmetric spectra of simplicial sets isa stable equivalence if for every injective Ω-spectrum X the induced map

[f,X] : [B,X] −→ [A,X]

on homotopy classes of spectrum morphisms is a bijection.For every morphism f : X −→ Y the natural morphism λ∗X : X −→ Ω(shX) adjoint to λX : S1∧X −→

shX gives rise to a commutative square of symmetric spectra

(2.1) Xλ∗X //

f

Ω(shX)

Ω(sh f)

Y

λ∗Y

// Ω(shY )

Theorem 2.2. The category of symmetric spectra of simplicial sets admits the following three stablemodel structures in which the weak equivalences are the stable equivalences.

(i) In the projective stable model structure the cofibrations are the projective cofibrations and thefibrations are those projective level fibrations f : X −→ Y for which the commutative square (2.1)is levelwise homotopy cartesian.

(ii) In the flat stable model structure the cofibrations are the flat cofibrations, and the fibrations arethose flat level fibrations f : X −→ Y for which the commutative square (2.1) is levelwise homotopycartesian.

(iii) In the injective stable model structure the cofibrations are the level cofibrations (i.e., monomor-phisms) and the fibrations are those those injective fibrations f : X −→ Y for which the commu-tative square (2.1) is levelwise homotopy cartesian.

Moreover we have:• All three stable model structures are proper, simplicial and cofibrantly generated.• The flat and projective stable model structures are even finitely generated and monoidal with respect

to the smash product of symmetric spectra.• In all three cases a morphism is a stable acyclic fibration if and only if it is a level acyclic fibration.

Proof. We reduce the proof of the stable model structures to the level model structures by applyinga general localization theorem of Bousfield, see Theorem 1.9 of Appendix A. In Proposition II.?? weconstructed a functor Q : Sp −→ Sp with values in Ω-spectra and a natural stable equivalence α : A −→QA. We note that a morphism f : A −→ B of symmetric spectra is a stable equivalence if and only ifQf : QA −→ QB is a level equivalence. Indeed, since αA : A −→ QA and αB : B −→ QB are stable

3. OPERADS AND THEIR ALGEBRAS 263

equivalences, f is a stable equivalence if and only if Qf is. But Qf is a morphism between Ω-spectra, so itis a stable equivalence if and only if it is a level equivalence.

We now apply Bousfield’s Theorem A.1.9 to the injective, flat and projective level model structures. Allthree level model structures are proper by Theorem 1.9. Axiom (A1) holds since we have a commutativesquare

(2.3) Xα //

f

QX

Qf

Y α

// QY

If f is a level equivalences, then Qf is a stable equivalence between Ω-spectra, hence a level equivalence.Axiom (A2) holds: αQX is a stable equivalence between Ω-spectra, hence a level equivalence. Then QαX :QX −→ QQX is a level equivalence since Q takes all stable equivalences, in particular αX , to levelequivalences.

We prove (A3) in the projective level model structure. Since the projective fibrations include the flatand injective fibrations, it then also holds in the flat and injective level model structures. So we are givena pullback square

Vi //

f

X

g

W

j// Y

of symmetric spectra in which X and Y are Ω-spectra (possibly not levelwise Kan), f is levelwise a Kanfibration and j is a stable equivalence. We showed in part (iv) of Proposition II.4.22 that then i is alsoa stable equivalence. This proves (A3), and thus Bousfield’s theorem provides three model structureswith stable equivalences as weak equivalence and with cofibrations the projective, flat or level cofibrationsrespectively.

Bousfield’s theorem characterizes the fibrations as those level fibrations f : X −→ Y for which thecommutative square (2.3) is homotopy cartesian. So it remains to shows that for a morphism f : X −→Y which is levelwise a Kan fibration the square (2.1) is levelwise homotopy cartesian if and only if thesquare (2.3) is levelwise homotopy cartesian.

Corollary 2.4. The following categories are equivalent

• the stable homotopy category, i.e., the homotopy category of injective Ω-spectra of simplicial sets;• the homotopy category of those flat Ω-spectra of simplicial sets for which all Xn are Σn-fibrant

and the maps Xn −→ map(EΣn, Xn) are Σn-equivalences;• the homotopy category of projective Ω-spectra which are levelwise Kan complexes.

3. Operads and their algebras

Definition 3.1. An operad O of symmetric spectra consists of

• a collection O(n)n≥0 of symmetric spectra,• an action of the symmetric group Σn [on the right ?] on the spectrum O(n) for all n ≥ 0,• a unit morphism ι : S −→ O(1) and• composition morphisms

γ : O(n) ∧ O(i1) ∧ . . . ∧ O(in) −→ O(i)

for all n, i1, . . . , ın ≥ 0 where i = i1 + · · ·+ in.

Moreover, this data has to satisfy the following three (?) conditions:


(Associativity) The square

O(n) ∧ O(i1) . . .O(in) ∧ O(j11) . . .O(j1i1) ∧ . . . ∧ O(jn1 ) . . .O(jnin)γ∧Id //

shuffle

O(i) ∧ O(j11) . . .O(j1i1) ∧ . . . ∧ O(jn1 ) ∧ . . . ∧ O(jnin)

γ

O(n) ∧ O(i1) ∧ O(j11) . . .O(j1i1) ∧ . . . ∧ O(in) ∧ O(jn1 ) . . .O(jnin)

Id∧γ...∧γ

O(n) ∧ O(j1) ∧ . . . ∧ O(jn) γ// O(j)

commutes for all n, i1, . . . , in, [...] ≥ 0, where the indices run over all natural numbers and i = i1 + · · ·+ in,jk = jk1 + · · · jkik and j = j1 + · · · jn.

(Equivariance)(Unit) The two composite morphisms

O(n) ∼= S ∧ O(n) ι∧Id−−−→ O(1) ∧ O(n)γ−−→ O(n)

andO(n) ∼= O(n) ∧ S ∧ · · · ∧ S︸︷︷︸

n

Id∧ι∧...∧ι−−−−−−−→ O(n) ∧ O(1) ∧ · · · ∧ O(1)︸︷︷︸n

γ−−→ O(n)

are the identity for all n ≥ 0, where the first maps in both composites are unit isomorphisms.A morphism f : O −→ P of operads is a collection of Σn-equivariant morphisms of symmetric spectra

f(n) : O(n) −→ P(n) for all n ≥ 0 which preserve the unit morphisms in the sense that f(1) ιO = ιP andwhich commute with the structure morphisms in the sense that [...]

Definition 3.2. Given an operad O, an O-algebra is a symmetric spectrum A together with morphismsof symmetric spectra

αn : O(n) ∧A(n) −→ A

for n ≥ 0 which satisfy the following conditions.(Associativity) The square

O(n) ∧ O(i1) ∧ · · · ∧ O(in) ∧A(i1) ∧ · · · ∧A(in)

shuffle

γ∧Id // O(i) ∧A(i)

αi

O(n) ∧ O(i1) ∧A(i1) ∧ · · · ∧ O(in) ∧A(in)

Id∧αi1∧···αin

O(n) ∧A(n)γ

// A

commutes for all n, i1, . . . , in ≥ 0, where i = i1 + · · ·+ in.(Equivariance)(Unit) The composite

A ∼= S ∧A ι∧Id−−−→ O(1) ∧A γ−→ A

is the identity.A morphism f : A −→ B of O-algebras is a morphism of symmetric spectra which commutes with the

action morphisms in the sense that [...]

If we realize geometrically we obtain an operad |O| in the category of pointed compactly generatedspaces, which can similarly act on symmetric spectra of topological spaces.

3. OPERADS AND THEIR ALGEBRAS 265

Remark 3.3. In the special case n = 1 = i1 the associativity condition says that the morphismγ : O(1) ∧ O(1) −→ O(1) is an associative product. Moreover, the unit condition for n = 1 says thatι : S −→ O(1) is unital for γ : O(1) ∧ O(1) −→ O(1). In other words, for any operad O, the object O(1)is a monoid in the monoidal category C. In the case of operads of symmetric spectra this means that forany operad O, the spectrum O(1) is a symmetric ring spectrum with unit morphism ι : S −→ O(1) andmultiplication γ : O(1) ∧ O(1) −→ O(1). [O(n) is an O(1)-module for all n ≥ 0]

[O-algebras as monoids in (O(1)mod-,, I)? This is only a weak monoidal product...]The notion of an operad is not restricted to symmetric spectra; indeed, an operad can be defined in any

symmetric monoidal category (C,⊗, I). In the end we will mainly care for the case of symmetric spectra(with respect to the smash product) but we can use the more general context, among other things, toproduce examples of operads of symmetric spectra. [operads in spaces, simplicial sets, categories, sets, andchain complexes]

[Operads can be described as monoids with respect to the circle product of symmetric sequences]

Example 3.4 (Operads of spaces or simplicial sets). ... give rise to operads of symmetric spectra bytaking suspension spectra (based and unbased versions)

Important special cases: A∞-operads (e.g. Stasheff polytopes) and E∞-operads.

Example 3.5 (Operads of categories).

Example 3.6 (Operads of sets). [gives rise to an operad in any symmetric monoidal (C,⊗, I) as longas C has coproducts]

We can formalize the previous examples as follows. We have a diagram of symmetric monoidal categoriesand strong monoidal functors

(set,×, ∗) // (categories,×, ∗) N // (sS,×, ∗)Σ∞+ //

| |

(SpsS,∧,S)

| |

(T,×, ∗)Σ∞+

// (SpT,∧,S)

We can (and will) take an operad in any one category and push it forward by applying the respective strongmonoidal functor to all objects in the operad.

Example 3.7 (Associative operad). We let Ass denote the operad of sets with Ass(n) = Σn [operadstructure]; Because of the equivariance conidition, the action morphism αn : Ass(n)⊗A(n) = qΣnA

(n) −→ Ais completely determined by its restriction to the summand indexed by the identity element of Σn.

So the Ass-algebras in the category of sets ‘are’ then the associative (and unital) monoids. Moreprecisely, the forgetful functor Ass -alg −→ (monoids) which remembers only the unit morphism and themorphism

M ⊗M 1⊗Id⊗ Id−−−−−−→ Ass(2)⊗M ⊗M −→M

is an isomorphism of categories. [more details for C = Sp in the section on symmetric ring spectra]In the special case of symmetric spectra under smash product we deduce that the category of Ass-

algebras is isomorphic to the category of symmetric ring spectra. [Ass-algebras in the stable homotopycategory are homotopy ring spectra]

Example 3.8 (Commutative operad). We let Com denote the operad of sets with Com(n) = ∗, theone point set, for all n ≥ 0, and unique operad structure. This is the terminal operad of sets and theCom-algebras in the category of sets are then the commutative (and associative and unital) monoids. Thisphenomenon persists to any symmetric monoidal category (C,⊗, I) [explain in appendix;reference].

In the special case of symmetric spectra under smash product we deduce that the category of Com-algebras is isomorphic to the category of commutative symmetric ring spectra.


Example 3.9 (A∞ operads).

Example 3.10 (E∞ operads). Several different E∞ operads have been discussed in the literature. Hereare some specific examples, where we only recall the spaces O(n) and refer to the original sources for theremaining structure.

The Barratt-Eccles operad is the categorical operad with n-th category given by EΣn. The operadis mostly used in its simplicial or topological version, i.e., after taking nerves and possibly also geometricrealization.

The Dold operad [...] The surjection operad [...] The linear isometries operad L [...]

Example 3.11 (Injection operad). The injection operad M is the operad of sets defined by lettingM(n) be the set of injections from the set ω × n into ω, for n ≥ 0. Note that for n = 0 the source is theempty set, soM(0) has exactly one element, andM(1) is the injection monoidM. The symmetric groupspermute the second coordinates in ω × n. The operad structure is via disjoint union and composition, i.e.,M is a suboperad of the endomorphism operad of the set ω in the symmetric monoidal category of setsunder disjoint union. More precisely, the operad structure morphism

γ : M(n)×M(i1)× · · · ×M(in) −→ M(i1 + · · ·+ in)

sends (ϕ, f1, . . . , fn) to ϕ (f1 + · · ·+ fn).The injection operad is, in a sense, the discrete analog of the linear isometries operad discussed above.[operad map M−→ L]

The injection operad came up before because the injection monoid M = M(1) acts naturally on thenaive homotopy groups of a symmetric spectrum. The entire operad injection is relevant because the naivehomotopy groups of every symmetric ring spectrum are naturally an algebra over the injection operad,compare Exercise 8.32

Example 3.12 (Endomorphism operad). Given any symmetric monoidal category (C,⊗, I) and anobject X of C, the endomorphism operad End(X) is defined as follows.

In the case of symmetric spectra, a monoid with respect to the smash product is a symmetric ringspectrum, and the monoid End(X)(1) coincides with the endomorphism ring spectrum End(X) as definedin Example 3.21 in Chapter I.

[O-algebra structures on X are the same as operad morphisms O −→ End(X)]

Example 3.13 (Operads from monoids). As we explained in Remark 3.3, every operad O gives rise toa monoid O(1) by neglect of structure. We can also go the other direction: suppse that M is a monoidin the symmetric monoidal category C with unit morphism ι : I −→ M and multiplication morphismµ : M ⊗M −→M . We define a operad oM in C by

oM(n) =

M for n = 1,∅ else.

where ∅ denotes the initial object of C. All symmetric groups act trivially, and the unit morphism I −→uM(1) of the operad is the unit morphism ι : I −→ M of the monoid. The composition morphism γ isthe mutliplication morphism µ : M ⊗M −→ M for n = i1 = 1. In all other cases, the source of the unitmorphism for oM involves at least one factor which is the initial object ∅, hence the entire source object isan initial object, which only has one morphism out of it.

The associativity constraint specializes to the associtivity of µ in the case n = 1, and it is automaticallysatisfied in all other cases since then the source is an initial object.

The functor which associates the operad oM to a monoid M is left adjoint to the functor which takesan operad O to the monoid O(1).

The terminology is somewhat unfortunate in this particular case: algebras over the operad uM are ‘thesame as’ modules over the monoid M . More precisely, the forgetful functor oM -alg −→ M mod- whichforgets all action morphisms except oM(1)⊗A = M ⊗A −→ A is an isomorphism of categories.

4. MODEL STRUCTURES FOR ALGEBRAS OVER AN OPERADS 267

Since simplicial sets act on symmetric spectra in a way compatible with the smash products, we canconsider O-algebras in category of symmetric spectra of simplicial sets.

[Operads versus symmetric operads; every operad O gives rise to a symmetric operad Σ×O such thatthe O-algebras coincide with the symmetric algebras over Σ×O]

[Modules over an algebra over an operad; universal envelopping algebra]

4. Model structures for algebras over an operads

In this section we let O be an operad of symmetric spectra (under smash product), either in the contextof spaces or simplicial sets. We will lefi various stable model structure from the category of symmetricspectra to the categorie of O-algebras. This contains the cases of module spectra over a fixed symmetricring spectrum, symmetric ring spectra, and of commutative symmetric ring spectra in the following sense.These special cases are particlarly important, and we devote a seperate chapter (or section?) to each ofthem [ref...].

Theorem 4.1. Let O be an operad of symmetric spectra. The category of O-algebras admits the fol-lowing positive stable model structures in which the weak equivalences are those morphisms of O-algebraswhich are stable equivalences on underlying symmetric spectra.

(i) In the positive projective stable model structure the fibrations are those morphisms of O-algebraswhich are positive projective stable fibrations on underlying symmetric spectra.

(ii) In the positive flat stable model structure the fibrations are those morphisms of O-algebras whichare positive flat stable fibrations on underlying symmetric spectra.

If the object O(n) is projective (flat enough?) as a Σn-symmetric spectrum for every n ≥ 0, then thecategory of O-algebras in R-modules also admits the following absolute stable model structures in whichthe

(iii) In the absolute flat stable model structure the fibrations are those morphisms of O-algebras whichare absolute flat stable fibrations on underlying symmetric spectra.

(iv) In the absolute projective stable model structure the fibrations are those morphisms of O-algebraswhich are absolute projective stable fibrations on underlying symmetric spectra.

All model structures are cofibrantly generated, simplicial/topological and right proper.For every (positive resp. absolut, flat) cofibrant O-algebra A the unique morphism O(0) −→ A from the

initial O-algebra is a flat cofibration of underlying symmetric spectra. Thus every cofibrant O-algebra A isflat as a symmetric spectrum. [is this right?]

[free actions on operad of simplicial sets gives projective operad of symmetric spectra][explain restriction and extension along an operad morphism]

Theorem 4.2. Let f : O −→ P be a morphism of operads of symmetric spectra.(i) The functor pair

P -algf∗ // O -algf∗

oo

is a Quillen adjoint functor pair with respect to the positive projective and the positive flat stablemodel structures on both sides.

(ii) If for every n ≥ 0 the group Σn acts freely on O(n) and P(n), then (f∗, f∗) is a Quillen adjointfunctor pair with respect to the absolut projective and absolute flat stable model structures on bothsides.

(iii) If for every n ≥ 0 the map f(n) : O(n) −→ P(n) is a stable equivalence of symmetric spectra afterforgetting the Σn-action, then the adjoint functor pair (f∗, f∗) is a Quillen equivalences in all thecases whenever it is a Quillen functor pair.

The Quillen equivalence between commutative and E∞-ring spectra is a special case of Quillen equiv-alences associated to weak equivalences of suitable operads[...]


Remark 4.3. Before we prove the theorem, let us explain why the absolute stable model structures doesnot generally lift from symmetric spectra to O-algebras without the freeness assumption on the operad O.We illustrate this for the operad Com (where the Σn-action is certainly not free in general), whose algebrasare commutative symmetric ring spectra. So suppose that the forgetful functor from commutative symmetricring spectra creates a model structure relative to one of the absolute (injective, flat or projective) stablemodel structure on symmetric spectra. Then we could choose a fibrant replacement S −→ Sf of the spherespectrum in the respective stable model structure. The target is then Sf a commutative symmetric ringspectrum which is also an Ω-spectrum. Since Sf is stably equivalent, thus π∗-isomorphic, to the spherespectrum, its 0-th space Sf0 has the homotopy type of QS0 = colimn ΩnSn. However, the space in level 0of any commutative symmetric ring spectrum R is a simplicial (or topological) commutative monoid, viaµ0,0 : R0∧R0 −→ R0. [in pointed ssets...] If the monoid of components forms a group (which is the case forQS0), then such a commutative monoid is weakly equivalent to a product of Eilenberg-Mac Lane spaces.So altogether an absolute stable model structure on commutative symmetric ring spectra would imply thatspace QS0 is weakly equivalent to a product of Eilenberg-MacLane spaces, which is not the case.

From the operad O we define a functor On[−] : Sp −→ Σn-Sp for n ≥ 0 to the category of Σn-symmetricspectra by

On[X] =∨k≥0

O(k + n) ∧Σk X(k) .

In the definition, the symmetric group Σk acts on O(k + n) by restriction along the ‘inclusion’ − + 1n :Σk −→ Σk+n and on X(k) by permuting the smash factors. The symmetric group Σn acts on each wedgesummand by restriction of the action on O(k + n) along the monomorphism 1k +− : Σn −→ Σk+n. As weshall see, the functor O[−] = O0[−] has the structure of a triple on the category of symmetric spectra andit acts from the right on the functors On[−]. We define a unit transformation X −→ O[X] = O0[X] as thecomposite of

X ∼= S ∧X ι∧Id−−−→ O(1) ∧Xwith the wedge summand inclusion for k = 1.

We now define a natural transformation m : On[O0[X]] −→ On[X] of functors to Σn-symmetric spectra.First we generalize the operad composition to a morphism of symmetric spectra

O(k + n) ∧ O(i1) ∧ · · · ∧ O(ik)Id∧ι∧···∧ι−−−−−−−→ O(k + n) ∧ O(i1) ∧ · · · ∧ O(ik) ∧ O(1) ∧ · · · ∧ O(1)︸︷︷︸

n

γ−−−−→ O(i1 + · · ·+ ik + n) .

For fixed k ≥ 0 we get a morphism

O(k + n) ∧

∨m≥0

O(m) ∧Σm X(m)

(k)

∼=∨

(i1,...,ik)∈NkO(k + n) ∧

(O(i1)× · · · × O(ik) ∧Σi1×···×Σik

X(i1+···+ik))

−→∨

(i1,...,ik)∈NkO(i1 + · · ·+ ik + n) ∧Σi1+···+ık

X(i1+···+ik) −→∨m≥0

O(m+ n) ∧Σm X(m) .

(4.4)

The morphism (4.4) is Σk-equivariant with respect to the diagonal action on the source and the trivialaction on the target. The morphism (4.4) is also Σn-equivariant for these action. So altogether (4.4) passesto a natural Σn-equivariant map

O(k + n) ∧Σk

∨m≥0

O(m) ∧Σm X(m)

(k)

−→∨m≥0

O(m+ n) ∧Σm X(m) .

As k varies, these morphism add up to the natural transformation m : On[O[X]] −→ On[X] of functorswith values in Σn-symmetric spectra.


These maps m are associative and unital in the sense that the diagrams of functors and natural trans-formations

On[O[O[X]]]On[mX ] //

mO[X]

On[O[X]]

m

On[X]

Id &&MMMMMMMMMMOn[unit]// On[O[X]]

m

On[O[X]]

m// On[X] On[X]

commute for all n ≥ 0. These properties ultimately come from associativity and unitality of the operadstructure, and we omit the details.

For n = 0 we have the additional property that the composite

O[X]unitO[X]−−−−−→ O[O[X]] m−−→ O[X]

is the identity, and so this structure makes the functor O[−] = O0[−] into a triple (also called monad) onthe category of symmetric spectra. An O[−]-algebra is a symmetric spectrum A together with a morphismα : O[A] −→ A which is associative and unital in the sense that the following diagrams commute

O[O[A]]O[α] //

m

A

α

A

Id%%KKKKKKKKKKKK

unit // O[A]

α

O[A]

α// A A

Given such an O[−]-algebra (A,α) we can give the symmetric spectrum A the structure of an algebra overthe operad O by declaring the n-th part of the action map to be the composite

O(n) ∧Σn A(n) incl−−→ O[A] α−→ A .

This assignment is an isomorphism of categories between the algebras over the triple O[−] and algebrasover the operad O. So we will now use these two notions of algebras interchangeably, and just talk aboutO-algebras even when we think about O[−]-algebras.

One immediate consequence of the comparison between algebras over O[−] and O is the following. Forevery symmetric spectrum X the spectrum O[X] has the structure of an O[−]-algebra, thus of an O-algebra,such that X 7→ O[X] is left adjoint to the forgetful functor from O-algebras to symmetric spectra. By aslight abuse of notation we also denote this free functor by

O[−] : Sp −→ O -alg .

Two examples of this have already come up in Example I.5.19: if O = Ass is the associative operad, thenAss[X] = TX is the tensor algebra generated by the symmetric spectrum X; for the commutative operadO = Com we get Com[X] = PX, the symmetric algebra generated by X.

We can also deduce from general principles that the category of O-algebras has limits and colimits.

Proposition 4.5. For every operad O of symmetric spectra, the category of O-algebras has [enriched?]limits and colimits. The forgetful functor from O-algebras to symmetric spectra commutes with limits andwith filtered colimits.

Proof. A limit ofO-algebras is given by the limit of the underlying symmetric spectra, with a canonicalO-algebra structure. Colimits of algebras over a triple are slightly more subtle, but they exist here becausethe underlying functor of the triple O preserves filtered colimits.

The key ingredient in the proof of Theorem 4.1 is a homotopical analysis of certain pushouts of O-algebras. For this we use a certain filtration of such pushout which we now define.


We consider a morphism f : X −→ Y of symmetric spectra and define a Σn-symmetric spectrum Qn(f)and an equivariant morphism Qn(f) −→ Y (n) for n ≥ 0. To define Qn(f) we first describe an n-dimensionalcube of symmetric spectra; by definition, such a cube is a functor

W : P(1, 2, . . . , n) −→ Spfrom the poset category of subsets of 1, 2, . . . , n and inclusions. If S ⊆ 1, 2, . . . , n is a subset, the vertexof the cube at S is defined as

W (S) = C1 ∧ C2 ∧ . . . ∧ Cnwith

Ci =X if i 6∈ SY if i ∈ S.

For S ⊆ T the morphism W (S) −→ W (T ) is a smash product of copies of the map f : X −→ Y at allcoordinates in T − S with identity maps of X or Y . For example for n = 2, the cube is a square and lookslike

X ∧Xf∧Id

Id∧f // X ∧ Yf∧Id

Y ∧X

Id∧f// Y ∧ Y.

We denote by Qn(f) the colimit of the punctured cube, i.e., the cube with the terminal vertex removed.[explain the Σn-action] It comes with a natural equivariant morphism Qn(f) −→ W (1, . . . , n) = Y (n) tothe terminal vertex of the cube.

Definition 4.6. Let G be a finite group. A morphism f : A −→ B of G-symmetric spectra is aflat G-cofibration (respectively projective G-cofibration) if it has the left lifting property with respect to allmorphisms of G-symmetric spectra which are flat trivial fibrations (respectively projective trivial fibrations)of underlying spectra.

G-symmetric spectra ‘are’ modules over the spherical group ring S[G]; under this isomorphism ofcategories, the flat (projective) G-cofibrations are the same as the cofibrations in the flat (projective)absolute stable model structure on S[G]-modules [see below].

Proposition 4.7. Let f : X −→ Y be a flat (respectively projective) cofibration of symmetric spectrasuch that f0 : X0 −→ Y0 is an isomorphism. Then for every n ≥ 0 the morphism of Σn-symmetric spectra

γn : Qn(f) −→ Y (n)

is a flat (respective projective) Σn-cofibration.

We note that hypothesis that f0 be an isomorphism is essential. Indeed, if K is a non-empty cofibrantspace (or simplicial set), then the suspension spectrum Σ∞+ K is projective. So the morphism from thetrivial spectrum to Σ∞+ K is a projective cofibration which violates the conclusion of Proposition 4.7: thesmash power (Σ∞+ K)(n) is isomorphic to Σ∞+ (Kn), but for any n ≥ 2 the cartesian product Kn has Σn-fixedpoints, so it is not a free Σn-space. Consequently, the spectrum (Σ∞+ K)(n) is not Σn-flat [ref] (although itsunderlying non-equivariant spectrum is projective, thus flat).

Proof of Proposition 4.7. Let us call a morphism f of symmetric spectra a power cofibration if forall n ≥ 0 the morphism γn : Qn(f) −→ Y (n) is a flat Σn-cofibration. We claim that:

(a) The class of power cofibrations is closed under pushouts. Indeed, for every pushout squares ofsymmetric spectra

Xf //

Y

Z g

// P


the square

Qn(f)γn(f) //

Y (n)

Qnn−1(g)

γn(g)// P (n)

is a pushout of Σn-spectra. So if f is a power fibration, then so is g, since flat Σn-cofibrations are stableunder pushouts.

(b) The class of power cofibrations is closed under countable composition. Indeed, if fi : Xi −→ Xi+1

are power cofibrations for all i ≥ 0 and f∞ : X0 −→ X∞ = colimXi is the canonical morphism, then(c) For every m ≥ 0 and every cofibration ϕ : K −→ L of Σm-simplicial sets, the map of semifree

symmetric spectra Gmϕ : GmK −→ GmL is a power cofibrations.In the general case of a flat cofibration f : X −→ Y we use the filtration FmX defined in Section II.??.

The latching map νm(f) : Xm ∪LmX LmYm −→ Ym is a Σm-cofibration of based spaces (or simplicial sets),so the morphism Gmνm(f) of semifree spectra is a power cofibration by (c). The pushout squares

Gm(Xm ∪LmX LmYm)Gmνm(f) //

GmYm

X ∪Fm−1X Fm−1Y

jm// X ∪FmX FmY

[check...] show that the upper horizontal morphism jm is a power cofibration by (a). The spectrumY is the colimit of the spectra X ∪FmX FmY along the morphisms jm, so finally the morphism X =X ∪F−1X F−1Y −→ Y is a power cofibration by (b).

We start with the special case X = GmL for some m ≥ 0 and a pointed Σm-simplicial set L. We have

(GmL)(n) ∼= Gmn(Σ+mn ∧Σm×···×Σm L ∧ . . . ∧ L) ;

in this description the permutation action of γ ∈ Σn in level mn+ k of (GmL)(n) is given by

γ∗ : Σ+mn+k ∧(Σm)n×Σk L

(n) ∧ Sk −→ Σ+mn+k ∧(Σm)n×Σk L

(n) ∧ Sk

τ ∧ a1 ∧ . . . an ∧ x 7−→ τ∆(γ) ∧ aγ−1(1) ∧ . . . aγ−1(n) ∧ x .

Here τ ∈ Σmn+k, ai ∈ L, x ∈ Sk and ∆ : Σn −→ Σmn is the diagonal embedding. The space((GmL)(n)

)mn+k

is a wedge of copies of L(n) ∧ Sk indexed by the cosets of the group (Σm)n × Σk inΣmn+k. The diagonal subgroup ∆(Σn) normalizes (Σm)n×Σk inside Σmn+k, and so Σn acts from the righton the set of the cosets Σmn+k/(Σm)n × Σk by

[g ((Σm)n × Σk)] · γ = g∆(γ) ((Σm)n × Σk) .

By the formula above, this is how the permutation action of Σn permutes the wedge summands in levelmn + k. For m ≥ 1 the diagonal embedding ∆ : Σn −→ Σmn+k is injective and its image intersects thesubgroup Σmn+k/(Σm)n×Σk only in the identity element. Thus the right action of Σn on Σmn+k/(Σm)n×Σkis free, and thus the Σn-action on (GmL)(n) is levelwise free away from the basepoint.

Construction 4.8. Given an O-algebra (A,α : O(A) −→ A) and a number n ≥ 0 we define a Σn-symmetric spectrum UnA as the coequalizer, in the category of Σn-symmetric spectra, of the two morphisms

On(O(A))mA //On(α)

// On(A) .

The role of the spectrum UnA is that the underlying symmetric spectrum of an O-algebra coproductAqO(X) is isomorphic to ∨

n≥0

UnA ∧Σn X∧n .


For example, we have U0A = A since the diagram

O(O(A))mA //O(α)

// O(A) α // A

is a (split) coequalizer, even as O-algebras. [So we should have Un(O(∗)) = O(n) to recover O(X) ∼=O(∗)qO(X)]

We consider an O-algebra A, a morphism f : X −→ Y of symmetric spectra and a morphism g :O(X) −→ A of O-algebras. We define a filtration of the O-algebra pushout AqO(X)O(Y ). We set A0 = Aand define a symmetric spectrum An inductively as the pushout

(4.9) UnA ∧Σn Qnn−1

Id∧γn //

UnA ∧Σn Y(n)

An−1

// An

The left vertical map is obtained by passage to coequalizers from the composite

On(A) ∧Σn Qnn−1 =

∨k≥0

O(k + n) ∧Σk×Σn A(k) ∧Qnn−1

∼=−→

∨k≥0

O(k + n) ∧Σk×Σn A(k)∧ : limS 6=1,...,nW (S)

use f : X −→ A−−−−−−−−−−→

Lemma 4.10. The colimit A∞ = colimnAn of the sequence of symmetric spectra has the structure ofan O-algebra which makes it a pushout in the category of O-algebras of the diagram

Ag←−− O(X)

O(f)−−−→ O(Y )

in such a way that the canonical morphism of symmetric spectra A −→ A∞ becomes the canonical morphismof O-algebras from A to the pushout.

Proof. There are several things to check:

(i) A∞ is naturally a O-algebra so that(ii) A −→ A∞ is a map of O-algebras and(iii) A∞ has the universal property of the pushout in the category of O-algebras.

Define the O-algebra structure O(A∞) −→ A∞ as the composite of A −→ A∞ with the unit of A. Themultiplication of A∞ is defined from compatible maps An∧Am −→ An+m by passage to the colimit. Thesemaps are defined by induction on n + m as follows. Note that An ∧ Am is the pushout in mod-R in thefollowing diagram.

Qn ∧ ((A ∧ L)m ∧A) ∪(Qn∧Qm) ((A ∧ L)n ∧A) ∧Qm

// (A ∧ L)n ∧A) ∧ ((A ∧ L)m ∧A)

(An−1 ∧Am) ∪(An−1∧Am−1) (An ∧Am−1) // An ∧Am

The lower left corner already has a map to An+m by induction, the upper right corner is mapped thereby multiplying the two adjacent factors of A followed by the map (A ∧ L)n+m ∧ A −→ An+m from thedefinition of An+m. We omit the tedious verification that this in fact gives a well defined multiplicationmap and that the associativity and unital diagrams commute. Hence, A∞ is a O-algebra. Multiplicationin A∞ was arranged so that A −→ A∞ is a O-algebra map.


For (iii), suppose we are given another O-algebra B, a O-algebra morphism A −→ B, and a mod-R-mapL −→ B such that the outer square in

K

// L

111

1111

1111

1111

A

((QQQQQQQQQQQQQQQQ // A∞

!!B

commutes. We have to show that there is a unique O-algebra map A∞ −→ B making the entire squarecommute. These conditions in fact force the behavior of the composite map W (S) −→ Pn −→ A∞ −→ B.Since A∞ is obtained by various colimit constructions from these basic building blocks, uniqueness follows.We again omit the tedious verification that the maps W (S) −→ B are compatible and assemble to aO-algebra map A∞ −→ B.

Proposition 4.11. Let O be an operad of symmetric spectra, A and O-algebra, f : X −→ Y be a flatcofibration of symmetric spectra and let O(X) −→ A be a morphism of O-algebras. Suppose in additionthat

• f is an isomorphism in level 0, [or• the symmetric groups Σn act freely on O(n) for all n ≥ 0].

Then then the pushout A −→ A∪O(X) O(Y ) is an injective cofibration [flat if source is flat?] of underlyingsymmetric spectra. If moreover f is a stable equivalence or π∗-isomorphism, then so is the morphismA −→ A ∪O(X) O(Y ).

Proof. Since f : X −→ Y is a flat acyclic cofibration, the morphism γn : Qn −→ Y (n) is also a flatacyclic cofibration by Proposition 4.7. We show by induction on n that the morphisms An−1 −→ An definedin (4.9) are injective stable equivalences.

Since γn is a flat acyclic cofibration, the morphism

Id∧γn : UnA ∧Qnn−1 −→ UnA ∧ Y (n)

is injective and a stable equivalence. In particular, the quotient spectrum UnA ∧ (Y/X)(n) is stably con-tractible. Now we pass to quotients by the Σn-action. We deduce that the morphism

Id∧Σnγn : UnA ∧Σn Qnn−1 −→ UnA ∧Σn Y

(n)

is again injective, and its cokernel is isomorphic to the spectrum

UnA ∧Σn (Y/X)(n) .

Under the assumption that f : X −→ Y is an isomorphism in level 0 the symmetric spectrum Y/X istrivial in level 0. Again by Proposition 4.7 the permutation action of Σn on the smash power (Y/X)(n) is thenfree away from the basepoint, hence so is the diagonal action on UnA ∧ (Y/X)(n). Since UnA ∧ (Y/X)(n)

is stably constractible (respectively has trivial homomotopy groups) and has a free action, the quotientspectrum UnA ∧Σn (Y/X)(n) is again stably contractible (respectively has trivial homomotopy groups) byProposition 4.12.

Now we can left the various stable model structures from symmetric spectra to O-algebras.

Proof of Theorem 4.1. We define cofibrations of O-algebras by the lifting property with respect toacylic fibrations of O-algebras.

The category ofO-algebras has limits and colimits by Proposition 4.5. The 2-out-of-3 property for stableequivalences of O-algebras and the closure under retracts for stable equivalences and stable fibrations ofO-algebras follow from that corresponding properties for symmetric spectra. Cofibrations are defined by alifting property, so the are closed under retracts.


The factorizations are produced by the small object argument. We define the necessary sets of gener-ating cofibrations and acyclic cofibrations as

IO = O(I) and JO = O(J) ,

where I and J are the generating cofibrations (respectively acyclic cofibrations) for the [...adjectives...]stable model structure of symmetric spectra, defined in [...].

The key non-trivial step is to verify that every IO-cell complex is also a stable equivalence. This is aspecial case of Proposition 4.11.

Is this needed:

Proposition 4.12. Let G be a finite group and X a G-symmetric spectrum such that the G-action isfree away from the basepoint.

• If X is stably contractible, then so is the quotient symmetric spectrum.• If X is k-connected for some integer k, then so is the quotient symmetric spectrum.

Proof. Since the G-action is free away from the basepoint the morphism EG+ ∧G X −→ S0 ∧G Xinduced by the unique map EG −→ ∗ is a level equivalence. So it suffices to show that the left hand sideis stably contractible (respectively k-connected) if X is. We show by induction over n that the symmetricspectrum EnG+ ∧G X is stably contractible (respectively k-connected), where EnG is the simplicial n-skeleton of EG. The induction start with n = −1 where we interpret E−1G as the empty simplicial set.The quotient EnG/En−1G is isomorphic as a G-simplicial set to G+ ∧∆[n]/∂∆[n]∧G(n), where the smashpowers are taken with the unit 1 ∈ G as basepoint. So we get a cofibre sequence of symmetric spectra

En−1G+ ∧G X −→ EnG+ ∧G X −→(G+ ∧∆[n]/∂∆[n] ∧G(n)

)∧G X ∼= ∆[n]/∂∆[n] ∧G(n) ∧X .

The last term is stably contractible (respectively (k + n)-connected) and the first morphism is injective.So the inclusion En−1G+ ∧G X −→ EnG+ ∧G X is a stable equivalence (respectively (k + n)-connected)and EnG+ ∧G X is stably contractible (respectively k-connected) for all n ≥ 0. Thus the filtered colimitEG+ ∧G X is also stably contractible (respectively k-connected).

5. Connective covers of structured spectra

Construction 5.1. The category of graded abelian groups form a symmetric monoidal category undergraded tensor product (with sign in the symmetry isomorphism).

Let O be an operad of symmetric spectra. Then we can define an operad πO in the category of gradedabelian groups (under graded tensor product) by

(πO)(n) = π∗(O(n)) ,

the graded abelian group of true homotopy groups of the symmetric spectrum O(n).For every operad O of symmetric spectra and every O-algebra A we now make the true homotopy

groups π∗A into a graded πO-algebra. For the associative respectively commutative operad this recovers thestructure of (commutative) graded ring on the homotopy groups of a semistable (commutative) symmetricring spectrum.

Example 5.2. Suppose E is an operad of sets, and let O = Σ∞+ E the operad of symmetric spectraobtained by taking suspensions spectra. Then O(n) is a wedge of sphere spectrum S, indexed by theelements E(n), and so by [...] we have π0O(n) ∼= Z[E(n)]. In other words, the degree part of the operad πOis the free abelian group operad generated by E .

Example 5.3. Algebras over the associative operad ‘are’ symmetric ring spectra, and the observationabove [degree zeo part] reduces to the fact, already observed in Proposition I.6.5, that the homotopy groupsof a symmetric ring spectrum naturally form a graded ring.

For the commutative operad, we similar re-obtain that the homotopy groups of a sommutative sym-metric ring spectrum naturally form a graded commutative ring.

5. CONNECTIVE COVERS OF STRUCTURED SPECTRA 275

Example 5.4. We identify the Σn-symmetric spectra UnA in the case of the initial, the commutativeand the associative operad. The key to this identification is the isomorphism

AqO(L) ∼=∨n≥0

UnA ∧Σn L(n)

which is natural in the O-algebra A and the symmetric spectrum L.Commutative operad: algebras over the commutative operad are commutative symmetric ring spectra

and the coproduct is given by the smash product of the underlying symmetric spectra. So for O = Com, asymmetric spectrum L and an O-algebra A we have

AqO(L) = A ∧ PL =∨n≥0

A ∧ (L(n)/Σn) .

Thus we have UnA = A with trivial Σn-action. In particular we have that UnA is connective whenver A is.Associative operad: algebras over the associative operad are symmetric ring spectra and the coproduct

of a symmetric ring spectrum A with the tensor algebra TL of a symmetric spectrum L is given

AqO(L) =∨n≥0

A ∧ (L ∧A)(n) .

SinceA ∧ (L ∧A)(n) = (Σ+

n ∧A(n+1)) ∧Σn L(n)

we have UnA = Σ+n ∧A(n+1). In particular we have that UnA is connective whenver A is and the underlying

symmetric spectrum of A is flat, which for example is the case when A is cofibrant as a symmetric ringspectrum.

Initial operad: algebras over the initial operad are symmetric spectra and the coproduct is given by thewedge. So we have

AqO(L) = A∨L .

This means that

Un(A) =

A for n = 0,S for n = 1, and∗ for n ≥ 2.

In particular UnA is connective whenver A is.

Proposition 5.5. Let O be an operad of symmetric spectra such that O(n) is connective for all n ≥ 0.Then for every connective cofibrant O-algebra A the Σm-symmetric spectrum UmA is also connective.

Recall that a morphism of symmetric spectra is n-connected, for some integer n, if it induces iso-morphisms of homotopy groups below dimension n and epimorphism on πn. If the morphism is a levelcofibration, then the long exact sequence of homotopy groups shows that the morphism is n-connected ifand only if its cokernel is n-connected.

Lemma 5.6. Let n ≥ 0. Then for every (n− 1)-connected flat symmetric spectrum X with X0 = ∗ andevery cofibrant connective O-algebra A the O-algebra morphism A −→ AqO(X) O(CX) is n-connected.

Proof. In a first step we show that if X is (n−1)-connected, then for every O-algebra A the summandinclusion A −→ A q O(X) is (n − 1)-connected. The coproduct is a special case of a pushout along theinitial object, so as a special case of Lemma 4.10 the coproduct is isomorphic to∨

m≥0

UmA ∧Σm X(m) .

The morphism A −→ A q O(K) corresponds to the summand inclusion for m = 0, so it suffices to showthat the remaining summands are (n−1)-connected. Since A is connective and cofibrant the Σm-symmetricspectrum UmA is also connective. [we only know this part for the associative or the commutative operad...]Form ≥ 1 the smash powerX(m) is again (n−1)-connected, flat and has a free Σm-action by Proposition 4.7.


So the spectrum UmA∧X(m) is also (n−1)-connected and has a free Σm-action, and thus by Proposition 4.12the orbit spectrum UmA ∧Σm X(m) is (n− 1)-connected.

The O-algebra AqO(X) O(CX) is the realization of the simplicial O-algebra

[k] 7−→ AqO(X)q · · · q O(X)︸︷︷︸k

= AqO(X∨ . . . ∨X︸︷︷︸k

) .

The object of 0-simplices is exactly A. Since the realization of simplicial O-algebras in performed onunderlying symmetric spectra, we argue with the underlying simplicial object of symmetric spectra todeduce that the vertex map A −→ AqO(X) O(CX) is n-connected.

We consider the simplicial spectrum obtained by dimensionwise collapsing the vertex object A is n-connected. By the first part the spectrumAqO(X∨ . . . ∨X︸︷︷︸

k

)

/A(quotient as symmetric spectra) is (n − 1)-connected for all k ≥ 1, and it is trivial for k = 0. So thegeometric realization is this simplicial quotient spectrum is k-connected, hence so is the vertex morphismA −→ AqO(X) O(CX).

Proposition 5.7. Let O be an operad of symmetric spectra such that O(n) is connective for all n ≥ 0and let ϕ : A −→ B be a morphism of O-algebras. For every n ≥ 0 there exists a functorial factorization

Ag−→ A〈n〉

h−→ B

of ϕ in the category of O-algebras such that:• g is a positive projective cofibration of O-algebras and h is a positive projective level fibration;• if A is connective [and cofibrant?], the morphism g induces isomorphisms of homotopy groups

below dimension n and an epimorphism on πn and the morphism h induces a monomorphism onπn and an isomorphism of homotopy groups above dimension n.

Proof. The idea is to ‘kill homotopy groups’ in the world of O-algebras. In other words, we cone offall morphisms from O-algebra spheres O(FmSk+m) for k ≥ n to A and iterate the process. Since we wantfunctoriality, we cannot choose generators of homotopy groups, but we should rather cone off all morphismfrom O(FmSk+m) to A. The ‘small object argument’ is exactly the process to achieve this.

We define a set of morphisms of symmetric spectra as K〈n〉 = J lv,+proj ∪ C〈n〉. Here

J lv,+proj =FmΛi[k]+ −→ Fm∆[k]+

m≥1,k≥0,0≤i≤k

of generating acyclic cofibrations for the positive projective level model structure and

C〈n〉 =FmS

k+m −→ FmCSk+m

m≥1,k≥n

where Sk+m −→ CSk+m is the cone inclusion. A morphism f : X −→ Y of symmetric spectra has theright lifting property with respect to the set J lv,+proj if and only if it is a projective level fibration in positivelevels, i.e., if and only if the morphisms fm : Xm −→ Ym are Kan fibrations for m ≥ 1.

If in addition f has the RLP for the set C〈n〉, then so has its fibre F over the basepoint. This meansthat for positive m the simplicial set Fm is Kan and has the right lifting property for the cone inclusionsSk+m −→ CSk+m for all k ≥ n. Thus the homotopy groups of Fm vanish in dimensions ≥ n + m,and so we have πkF = 0 for k ≥ n. The long exact sequence of homotopy groups shows that the mapπnf : πnX −→ πnY is injective and f induces isomorphisms of homotopy groups above dimension n.

We now apply the small object argument, in the category of O-algebras, to the morphism ϕ : A −→ Bwith respect to the set OK〈n〉. It produces a functorial factorization

Ag−→ A〈n〉

h−→ B

5. CONNECTIVE COVERS OF STRUCTURED SPECTRA 277

of ϕ in the category of O-algebras such that g is a OK〈n〉-cell complex and h has the right lifting propertywith respect to the set OK〈n〉. Since O is left adjoint to the forgetful functor, this means that the underlyingmorphism of symmetric spectra of h has the RLP with respect to the set K〈n〉, so by the above, h is apositive projective level fibration, induces a monomorphism on πn and isomorphisms of homotopy groupsabove dimensions n.

Since every morphism in the set K〈n〉 is a positive projective cofibration of symmetric spectra, everymorphism in the set OK〈n〉 is a positive projective cofibration of O-algebras, and so every OK〈n〉-cellcomplex, in particular the morphsm g, is a positive projective cofibration of O-algebras,

It remains to identify the effect of the morphism g : A −→ A〈n〉 on homotopy groups. Let f lv : X −→ Ybe a positive projective cofibration which is also a level equivalence. Then A −→ A qO(X) O(Y ) is a π∗-isomorphism by Proposition 4.11.

If X is an (n− 1)-connected projective symmetric spectrum with X0 = ∗ and fc : X −→ CX the coneinclusions, the by Lemma 5.6 the morphism A −→ AqO(X) O(CX) is n-connected.

Every OK〈n〉-cell complex is a countable compositition of pushouts along morphisms O(f lv∨fc) wheref lv and fc are positive projective cofibrations, f lv is a level equivalence and fc is a cone inclusion ofan (n − 1)-connected spectrum. Such pushout can be done in two seperate steps, so by the above, themorphism from A to the pushout is n-connected. So every OK〈n〉-cell complex is n-connected. This appliesin particular to the morphism g : A −→ A〈n〉 which thus induces a bijection of homotopy groups belowdimension n and an epimorphism in dimension n.

Theorem 5.8. Let O be an operad of symmetric spectra such that O(n) is connective for all n ≥ 0.

(i) There exists a functorτ≥0 : O -alg −→ O -alg

and a natural morphism of O-algebras τ≥0A −→ A with the following property:• the O-algebra τ≥0A is connective• the morphism τ≥0A −→ A induces an isomorphism on πk for all k ≥ 0.

We refer to τ≥0A as the connective cover of the O-algebra A.(ii) For every n ≥ 0 there exists a functor

Pn : O -alg −→ O -alg

and a natural morphism of O-algebras A −→ PnA such that for every connective and cofibrant O-algebraA the following properties hold:

• the homotopy groups πk(PnA) are trivial for k > n• for k ≤ n the morphism A −→ PnA induces an isomorphism on πk.

We refer to PnA as the n-th Postnikov section of the connective O-algebra A.

Proof. For part (i) we first produce a morphism ϕ : A+ −→ A whose source is a connective O-algebra and such that ϕ is surjective on π0. For example, we can choose a family of maps of pointedspaces Snj −→ Anj with nj ≥ 1 whose classes generate π0A as an abelian group and define A+ as the freeO-algebra

A+ = O

∨j

FnjSnj

.

The morphism A+ −→ A is adjoint to wedge of the adjoints FnjSnj −→ A of the maps Snj −→ Anj . We

can make A+ depend functorially on A by using all maps Sj −→ Aj for all j ≥ 1.Now we apply Proposition 5.7 with n = 0 to the morphism ϕ : A+ −→ A. Since the source is connective

and cofibrant we obtain a functorial O-algebra factorization

A+g−→ A〈0〉

h−→ A


of ϕ such that g induces isomorphisms of negative dimensional homotopy groups and h induces a monomor-phism on π0 and an isomorphism of homotopy groups in positive dimensions. Since A+ is connective, thefirst statement says that A〈0〉 is also connective. Since the composite

π0(A+)π0g // π0A

〈0〉 π0h // π0A

coincides with the surjective map π0ϕ, the map π0h : π0A〈0〉 −→ π0A is not only injective, but in fact

bijective. So altogether we conclude that the morphism of O-algebras h : A〈0〉 −→ A serves as a connectivecover.

Part (ii) is the special case of Proposition 5.7 for the unique morphism A −→ ∗ to the terminal O-algebra. In the factorization

Ag−→ A〈n+1〉 h−→ ∗

the morphism g induces an isomorphism of homotopy groups below dimension n + 1. The morphism h isinjective on homotopy group of dimension n+ 1 and above. The terminal O-algebra is a trivial symmetricspectrum, so in this case we deduce that πkA〈n+1〉 = 0 for k ≥ n + 1. So the morphism of O-algebrasg : A −→ A〈n+1〉 serves as the n-th Postnikov section.

Let O be an operad of symmetric spectra such that O(n) is connective for all n ≥ 0. Then the collectionof zeroth homotopy groups π0O is an operad of abelina groups (under tensor product). For every π0O-algebra A the collection of Eilenberg-Mac Lane spectrum HA (see Example I.1.14) is naturally an algebraover the operad O [...] [We could also deduce this from an operad morphism O −→ H(π0O)]

Proposition 5.9 (Uniqueness of Eilenberg-Mac Lane algebras). Let O be an operad of symmetricspectra such that O(n) is connective for all n ≥ 0. If A is O-algebra with true homotopy groups concentratedin dimension 0, then A is stably equivalent to H(π0A) as an O-algebra.

Remark 5.10. It seems worth spelling out the results of this section in the case of the initial operadoS with objects

oS(n) =

S for n = 1,∗ else.

Then oS-algebra ‘are’ S-modules, which in turn ‘are’ symmetric spectra. More precisely, the forgetful functoroS -alg −→ Sp is an isomorphism of categories. Every object in the operad S is connective, so Theorem 5.8provides a functor τ≥0 : Sp −→ Sp and a natural morphism of symmetric spectra τ≥0A −→ A such thatτ≥0A is connective and the morphism τ≥0A −→ A induces an isomorphism on πk for all k ≥ 0.

Since the homotopy groups of a τ≥0A depend functorially on the homotopy groups of A, the functor τ≥0

and the natural transformation descend to the level of homotopy categories. [ref to universal property] Sothis section gives a somewhat different way to construct connective covers in the stable homotopy category,which we first discussed in Theorem II.6.1.

In much the same way, the uniqueness result for Eilenberg-Mac Lane algebras is Proposition 5.9 spe-cializes, in the case of the initial operad oS to the uniqueness result for Eilenberg-Mac Lane spectra in thestable homotopy category as stated in Theorem II.4.21.

6. Bousfield localization of structured spectra

What is the right homology theory to localize with respect to ? For R-algebras, use R-modules.

Exercises

Exercise 4.1. Let C be a pointed simplicial category equipped with a global model structure in the senseof Definition 1.16. Show that the level model structure on the category Sp(C) established in Proposition 1.20is model-enriched over the projective level model structure on symmetric spectra of simplicial sets (i.e.,satisfies the pushout product axiom). Find conditions on the global model structure which guarantee thatthe level model structure on Sp(C) is model enriched with respect to the mixed or flat level model structureon symmetric spectra.


Exercise 4.2. Give instructions on how to prove the projective and flat level and stable model struc-tures for orthogonal spectra. Show that the two adjoint functor pairs

SpT

P((

?

66SpOUoo

are Quillen equivalences with respect to the projective stable model structure on SpO; for the pair (P, U) weuse the projective stable model structure, for (U, ?) the flat stable model structure on symmetric spectra.Need that the forgetful functor takes projective cofibrations in SpO to flat cofibrations in Sp. Key steps:(1) A projective symmetric spectrum, X orthogonal Ω-spectrum, then in commutative triangle

A

'""EE

EEEE

EEϕ // UX

UPAϕ

;;wwwwwwwww

the adjunction unit is a stable equivalence [ref], so ϕ is a stable equivalence of symmetric spectra if andonly if its adjoint ϕ is a stable equivalence of orthogonal spectra.

(2) B projective orthogonal spectrum, Y flat symmetric Ω-spectrum, then in commutative triangle

UB

U(ψ) ##GGGGGGGGψ // Y

U?Y

'

<<zzzzzzzz

the adjunction counit is a stable equivalence [ref], so ψ is a stable equivalence of orthogonal spectra if andonly if its adjoint ψ is a stable equivalence of symmetric spectra.

Same for unitary spectra.

History and credits

The projective and injective level and stable model structures for symmetric spectra are constructed inthe original paper [33] of Hovey, Shipley and Smith. The flat model structures show up in the literatureunder the name of S-model structure. (the ‘S’ refers to the sphere spectrum). The cofibrant objects in thismodel structure (which we call ‘flat’ and Hovey, Shipley and Smith call ‘S-cofibrant’) and parts of the modelstructures show up in [33] and in [66], but the first verification of the full model axioms appears in Shipley’spaper [73]. I prefer the term ‘flat’ model structure because the cofibrant objects are very analogous to flatmodules in algebra and because we can then also use the term ‘flat model structure’ for algebras over anoperad and modules over a symmetric ring spectrum.

The stable model structures for algebras over operads were obtained by different people in successivelymore general situations. The first special case were the stable model structure for modules over a symmetricring spectrum and for algebra spectra of a commutative symmetric ring spectrum, which are examplesof the general theory of Schwede and Shipley [68]. The positive stable model structure for commutativesymmetric ring spectra was first established by Mandell, May, Schwede and Shipley in [51] for the projectivecofibration/fibration pair and by Shipley [73] for the flat cofibration/fibration pair (called the S-modelstructure there). [known to Smith, and actually a motivitation for why symmetric spectra are ’right’]

A proof of the stable model structures for algebras over an operad of simplicial sets first appeared in thepaper [25] by Elmendorf and Mandell. They actually work more generally with algebras over multicategories,and operads are multicategories with a single object. [but only the flat positive model structure ?] In thefull generality of Section 4, i.e., for algebras over an operad of symmetric spectra, the stable model structure


were first established by Harper [30]; Harper also shows that for any such operad O, the category of leftO-modules admits various stable model structure.

CHAPTER IV

Module spectra

1. Model structures for modules

With the symmetric monoidal smash product and a compatible model structure in place, we are readyto explore ring and module spectra. In this section we construct model structures on the category ofmodules over a symmetric ring spectrum. We restrict our attention to stable model structures and showthat the forgetful functor to symmetric spectra ‘creates’ various such model structure. The forgetful functoralso creates various level model structures, but we have no use for that and so will not discuss level modelstructures for R-modules.

The various stable model structures are also ‘stable’ in the technical sense that the suspension functoron the homotopy category is an equivalence of categories. As consequence of this is that stable homotopycategory of modules over a ring spectrum is a triangulated category. The free module of rank one is a smallgenerator.

We originally defined a symmetric ring spectrum in Definition I.1.3 in the ‘explicit’ form, i.e., as a familyRnn≥0 of pointed simplicial sets with a pointed Σn-action on Rn and Σp ×Σq-equivariant multiplicationmaps µp,q : Rp ∧Rq −→ Rp+q and two unit maps subject to an associativity, unit and centrality condition.Using the internal smash product of symmetric spectra we saw in Theorem I.5.17 that a symmetric ringspectrum can equivalently be defined as a symmetric spectrum R together with morphisms µ : R∧R −→ Rand ι : S −→ R, called the multiplication and unit map, which satisfy certain associativity and unitconditions. In this ‘implicit’ picture a morphism of symmetric ring spectra is a morphism f : R −→ S ofsymmetric spectra commuting with the multiplication and unit maps, i.e., such that f µ = µ (f ∧ f) andf ι = ι.

Given a symmetric ring spectrum R, we have three different (but isomorphic) ways to think of R-modules, namely explicitely, implicitely and operadic. Indeed, a right R-module was originally definedexplicitly, but it can also be given in an implicit form as a symmetric spectrum M together with an actionmap M ∧ R −→ M satisfying associativity and unit conditions. A morphism of right R-modules is amorphism of symmetric spectra commuting with the action of R. We denote the category of right R-modules by mod-R. There is yet another way to view right modules over a ring, namely as the algebrasover an operad oR associated to the ring spectrum R. We define the spectra in this operad by

oR(n) =

R for n = 1, and∗ else.

Since most of the terms in the operad are the trivial spectrum, all symmetric groups have to act triviallyand the only composition map which can be non-trivial arises for n = i1 = 1, where it is the multiplicationmorphism of the ring spectrum R. Again since oR(n) is trivial for n 6= 1, most instances of the symmetryand associativity constraints have trivial source, and are thus automatically satisfied. The only non-trivialcondition is associativitiy for n = 1, which holds by associativity of the multiplication of R.[unit]

If M is an algebra over the operad oR, then the structure morphisms oR(n) ∧ M (n) −→ M arenecessarily trivial for n 6= 1, so the only genuine information encoded in the algebra structure is themorphism oR(n) ∧M = R ∧M −→ M . The associativity respectivyly unitality conditions for the actionsin particular say that this action is associative and unital. In other words, M is a left module over thering spectrum oR. Conversely, any module over the ring spectrum R can be made into an oR-algebra bydefining oR(n) ∧M −→M to be the action morphism. So we conclude:

281

282 IV. MODULE SPECTRA

Proposition 1.1. The functoroR -alg −→ mod-R

is an isomorphism of categories. [right vs left]

The unit S of the smash product is a ring spectrum in a unique way, and S-modules are the sameas symmetric spectra. The smash product of two ring spectra is naturally a ring spectrum. For a ringspectrum R the opposite ring spectrum Rop is defined by composing the multiplication with the twist mapR ∧R −→ R ∧R (so in terms of the bilinear maps µp,q : Rp ∧Rq −→ Rp+q, a block permutation appears).The definitions of left modules and bimodules is hopefully clear; left R-modules and R-T -bimodule can alsobe defined as right modules over the opposite ring spectrum Rop, respectively right modules over the ringspectrum Rop ∧ T .

A formal consequence of having a closed symmetric monoidal smash product of symmetric spectra isthat the category of R-modules inherits a smash product and function objects. The smash product M ∧RNof a right R-module M and a left R-module N can be defined as the coequalizer, in the category of

symmetric spectra, of the two maps

M ∧R ∧N //// M ∧Ngiven by the action of R on M and N respectively. Alternatively, one can characterize M ∧R N as theuniversal example of a symmetric spectrum which receives a bilinear map from M and N which is R-balanced, i.e., all the diagrams

(1.2)

Mp ∧Rq ∧Nr

αp,q∧Id

Id∧αq,r // Mp ∧Nq+rιp,q+r

Mp+q ∧Nr ιp+q,r

// (M ∧N)p+q+r

commute. If M happens to be a T -R-bimodule and N an R-S-bimodule, then M ∧R N is naturally aT -S-bimodule. If R is a commutative ring spectrum, the notions of left and right module coincide andagree with the notion of a symmetric bimodule. In this case ∧R is an internal symmetric monoidal smashproduct for R-modules. There are also symmetric function spectra HomR(M,N) defined as the equalizerof two maps

Hom(M,N) −→ Hom(R ∧M,N) .The first map is induced by the action of R on M , the second map is the composition of R ∧ − :Hom(M,N) −→ Hom(R ∧M,R ∧ N) followed by the map induced by the action of R on N . The in-ternal function spectra and function modules enjoy the ‘usual’ adjointness properties with respect to thevarious smash products. [spell out]

Proposition 1.3. Given a morphism f : R −→ S of symmetric ring spectra, the functor f∗ :mod-S −→ mod-R of restriction of scalar has a left and a right adjoint, and hence commutes with limitsand colimits.

In particular, for every symmetric ring spectrum R the forgetful functor to symmetric spectra has a leftand a right adjoint, and hence commutes with limits and colimits.

Theorem 1.4. Let R be a symmetric ring spectrum of topological spaces or simplicial sets. The categoryof right R-modules admits the following four stable model structures in which the weak equivalences arethose morphisms of R-modules which are stable equivalences on underlying symmetric spectra.

(i) In the absolute projective stable model structure the fibrations are those morphisms of R-moduleswhich are absolute projective stable fibrations on underlying symmetric spectra.

(ii) In the positive projective stable model structure the fibrations are those morphisms of R-moduleswhich are positive projective stable fibrations on underlying symmetric spectra.

(iii) In the absolute flat stable model structure the fibrations are those morphisms of R-modules whichare absolute flat stable fibrations on underlying symmetric spectra.

1. MODEL STRUCTURES FOR MODULES 283

(iv) In the positive flat stable model structure the fibrations are those morphisms of R-modules whichare positive flat stable fibrations on underlying symmetric spectra.

Moreover we have:• All four stable model structures are proper, simplicial and cofibrantly generated.• If R is commutative then all four stable model structures are monoidal with respect to the smash

product over R.If underlying symmetric spectrum of R is flat, then the category of right R-modules admits the following

two injective stable model structures in which the weak equivalences are those morphisms of R-modules whichare stable equivalences on underlying symmetric spectra.

(v) In the absolute injective stable model structure the fibrations are those morphisms of R-moduleswhich are absolute injective stable fibrations on underlying symmetric spectra.

(vi) In the positive injective stable model structure the fibrations are those morphisms of R-moduleswhich are positive injective stable fibrations on underlying symmetric spectra.

Moreover, both injective stable model structures are proper, simplicial and cofibrantly generated.In all six model structures, a cofibration of R-modules is a monomorphism of underlying symmetric

spectra.

Proof. In the language of Definition 1.4 of Appendix A we claim that in all of the six cases the forgetfulfunctor from R-modules to symmetric spectra creates a model structure on R-modules. In Theorem A.1.5we can find sufficient conditions for this, which we will now verify.

The category of R-modules is complete, cocomplete and simplicial; in fact all limits, colimits, tensorsand cotensors with simplicial sets are created on underlying symmetric spectra. In particular the forgetfulfunctor preserves filtered colimits. The forgetful functor has a left adjoint free functor, given by smashingwith R. [Smallness]

It remains to check the condition which in practice is often the most difficult one, namely that every(J ∧R)-cell complex is a weak equivalence. We claim that in all six cases the free functor X 7→ X ∧R takesstable acyclic cofibrations of symmetric spectra of the respective kind to stable equivalences of R-moduleswhich are monomorphisms. In the first four cases (where we have no assumption on R) this uses that everygenerating acyclic cofibration i : A −→ B is in particular a flat cofibration, so i ∧ Id : A ∧ R −→ B ∧ Ris injective and a stable equivalence by parts (i) and (iv) of Proposition 1.11. In the ‘injective’ cases (v)and (vi) the argument is slightly different; then the assumption that R is flat assures that for every injectivestable equivalence i : A −→ B the morphism i ∧ Id : A ∧ R −→ B ∧ R is again injective (by the definitionof flatness) and a stable equivalence (by Proposition II.5.38).

So in all the six cases, the free functor −∧R takes the generating stable acyclic cofibrations to injectivestable equivalences of R-modules. Since colimits of R-modules are created on underlying symmetric spectra,the class of injective stable equivalences is closed under wedges, cobase change and transfinite composition.So every (J ∧ R)-cell complex is a stable equivalence. So we have verified the hypothesis of Theorem 1.5,which thus shows that the forgetful functor creates the six model structure. It also shows that the modelstructures are simplicial and right proper.

[left proper] [monoidal if R commutative] [preservation of cofibrations]

[Is there an ‘strongly injective’ stable model structure in which cofibrations are the monomorphisms ofR-modules ? make exercise?]

Proposition 1.5. A morphism f : M −→ N of right R-modules is a flat cofibration if and only if forevery morphism g : V −→W of left R-modules the pushout product map

f ∧R g : M ∧RW ∪M∧RV N ∧RW −→ N ∧RW

is an injective morphism of symmetric spectra.

There are also characterizations of flat and projective cofibrations in terms of ‘R-module latchingobjects’, see Exercise 2.6.


As we just proved, cofibrations of R-modules are always monomorphisms of underlying symmetricspectra, but sometimes more is true. As the special case S = S of Theorem 1.6 (iii) below we will see thatif R is flat as a symmetric spectrum, then every flat cofibration of R-modules is also a flat cofibration onunderlying symmetric spectra. Similarly, if R is projective as a symmetric spectrum, then every projectivecofibration of R-modules is also a projective cofibration on underlying symmetric spectra.

For a morphism f : S −→ R of symmetric ring spectra, there is are two adjoint functor pairs relating themodules over S and R. The functors are analogous to restriction and extension respectively coextensionof scalars. Every R-module becomes an S-module if we let S act through the homomorphism f ; moreprecisely, given an R-module M we define an S-module f∗M as the same underlying symmetric spectrumas M and with S-action given by the composite

(f∗M) ∧ S = M ∧ S Id∧f−−−→ M ∧R α−−→ M .

We call the resulting functor f∗ : mod-R −→ mod-S restriction of scalars along f and note that it has both aleft and right adjoint. We call the left adjoint extension of scalars and denote it by f∗ : mod-S −→ mod-R.The left adjoint takes an S-module N to the R-module f∗N = N ∧S R, where S is a left R-module via f ,and with right R-action through the right multiplication action of R on itself. We call the right adjoint off∗ the coextension of scalars and denote it by f! : mod-S −→ mod-R. The right adjoint takes an S-moduleN to the R-module f!N = Hommod-S(R,N), where S is a right R-module via f , and with right R-actionthrough the left multiplication action of R on itself.

Theorem 1.6. Let f : S −→ R be a homomorphism of symmetric ring spectra.(i) The functor pair

mod-Sf∗ // mod-Rf∗

oo

is a Quillen adjoint functor pair with respect to the absolute projective, the positive projective, theabsolute flat and the positive flat stable model structures on both sides.

(ii) If S and R are flat as symmetric spectra then (f∗, f∗) is a Quillen adjoint functor pair with respectto the absolute injective and the positive injective stable model structures on both sides.

(iii) Suppose that the morphism f : S −→ R makes R into a flat (respectively projective) right S-module. Then the functor pair

mod-Rf∗ // mod-Sf!

oo

is a Quillen adjoint functor pair with respect to the absolute and positive flat stable (respectivelyabsolute and positive projective stable) model structures on both sides. In particular, the restrictionof scalars f∗ then takes flat (respectively projective) cofibrations of R-modules to flat (respectivelyprojective) cofibrations of S-modules.

(iv) If the homomorphism f : S −→ R is a stable equivalence, then the adjoint functor pairs (f∗, f∗)and (f∗, f!) are a Quillen equivalences in all the cases when they are Quillen adjoint functors.

Proof. (i) In each case, the weak (i.e., stable) equivalences and the various kinds of fibrations aredefined on underlying symmetric spectra, hence the restriction functor preserves fibrations and acyclicfibrations. By adjointness, the extension functor preserves cofibrations and trivial cofibrations.

(iv) If f : S −→ R is a stable equivalence, then for every flat right S-module N the morphism

N ∼= N ∧S S −→ N ∧S R = f∗N

is a stable equivalence. Thus if Y is a fibrant left R-module, an S-module map N −→ Y is a weakequivalence if and only if the adjoint R-module map f∗N −→ Y is a weak equivalence.

Theorem 1.7. Let R be a connective symmetric ring spectrum and n any integer. There are functors

τ≥n : mod-R −→ mod-R and Pn : mod-R −→ mod-R

1. MODEL STRUCTURES FOR MODULES 285

and natural morphisms of R-modules τ≥nM −→M and M −→ PnM with the following property:• the R-module τ≥nM is (n− 1)-connected• the morphism τ≥nM −→M induces an isomorphism on πk for all k ≥ n.• the homotopy groups πk(PnM) are trivial for k > n• for k ≤ n the morphism M −→ PnM induces an isomorphism on πk.

] We refer to τ≥nA as the (n−1)-connected cover and to PnM as the n-th Postnikov section of the R-moduleM .

Proof. A special case of Theorem III.5.8, provides connective covers of algebras over an operad []Careful: as opposed to general operads, we get connected covers and Postnikov sections for all integers

Remark 1.8. The hypothesis in Theorem 1.7 that the ring spectrum R be connective is essential.To see this we consider a ‘periodic’ ring spectrum R, i.e., one which has a unit u 6= 0 in πkR for somek 6= 0 (and hence cannot be connective). Examples of such ring spectra are the complex or real K-theoryspectra KU and KO (see Example I.1.18) or the Morava K-theory spectra K(n) at a prime p and leveln (see Example I.6.32) (although we will only see much later that the homotopy ring spectrum K(n) asconstructed in Example I.6.32 can be rigidified to a symmetric ring spectrum).

Suppose there was a connective cover ϕ : τ≥nR −→ R for the free module of rank one. Replacingu by its inverse, if necessary, we can suppose that the dimension k of the unit u is positive. Taking asufficiently higher power of u, if necessary, we can moreover suppose that the dimension k of the unit usatisfies −k < n ≤ k. Then there exists a preimage v ∈ πk(τ≥nR) satisfying ϕ∗(v) = u. The class vu−2 liesin π−k(τ≥nR) and is thus zero. Since the map induced by ϕ on homotopy groups is π∗R-linear, we wouldget the contradiction

0 = ϕ∗(vu−2) = ϕ∗(v)u−2 = u−1 6= 0 .There cannot be a Postnikov section R −→ PnR of the free module of rank 1, either. Indeed, if there

was one, then the morphism from its homotopy fibre to R would be an (n − 1)-connected cover, and wejust showed that this does not exist.

Example 1.9 (Modules over Eilenberg-Mac Lane spectra). For every ring A we have an associatedEilenberg-Mac Lane ring spectrum, see Example I.1.14. This symmetric spectrum arises from a Γ-spaceby evaluation on spheres, so it is flat as a symmetric spectrum (Proposition II.8.16). Hence all six modelstructure of Theorem 1.4 are defined on the category of HA-modules, and they are Quillen equivalent toeach other.

The homotopy category of HA-modules can be described purely algebraically in terms of A-modules.More precisely, the stable model structures of HA-modules are Quillen equivalent to the category of chaincomplexes of A-modules in any of the model structures which have the quasi-isomorphisms as weak equiv-alences. In particular, we get an equivalence of triangulated categories

Ho(mod-HA) ∼= D(A)

to the unbounded derived category of the ring A.

1.1. The derived category of a ring spectrum.

Theorem 1.10. Let R be a symmetric ring spectrum. Then the derived category D(R) is a triangulatedcategory. The image in D(R) of the free R-module of rank one is a compact weak generator of D(R). If Ris commutative, then the derived smash product over R makes D(R) into a tensor triangulated category,.

The first part of Theorem 1.10 is really a special case of a Theorem A.?? about stable model categories.

Remark 1.11. Theorem 1.10 says that modules over a symmetric ring spectrum form a stable modelcategory with single compact generator. The converse is also true, at least up to Quillen equivalenceand under some technical hypothesis. More precisely, let C be a stable model category which is proper,cofibrantly generated and simplicial. Theorem 3.1.1 of [69] says that if the triangulated homotopy categoryhas a single compact generator, then C is Quillen equivalent to the stable model category of modules over a


symmetric ring spectrum. A ring spectrum which does the job can be obtained as a suitable endomorphismring spectrum (in a similar sense as in Example I.3.21) of a cofibrant-fibrant weak generator.

Corollary 1.12 (of the appendix). Let R be a connective symmetric ring spectrum. Then R-moduleshave Postnikov section, and this gives a t-structure on D(R) whose heart is equivalent, via the functor π0

to the abelian category of modules over the ring π0R.

2. Toda brackets in triangulated categories

Construction 2.1. Let T be a triangulated category and α : Y −→ Z, β : X −→ Y and γ : W −→ Xthree composable morphisms which satisfy αβ = 0 = βγ. We define a Toda bracket 〈α, β, γ〉, subset of themorphism group T (ΣW,Z), as follows.

We start by choosing a distinguished triangle

(2.2) Xβ−−→ Y

i−−→ C(β)p−−→ ΣX .

Since αβ = 0 there exists a morphism α : C(β) −→ Z such that αi = α; since βγ = 0 there exists amorphism γ : ΣW −→ C(β) such that pγ = Σγ, compare the commutative diagram.

ΣW

γ

Σγ

""FFFF

FFFF

F

Yi //

α!!DD

DDDD

DDD C(β)

p //

α

ΣX

Z

The bracket 〈α, β, γ〉 then consists of all morphisms of the form αγ : ΣW −→ Z for varying α and γ. Notethat any two choices of triangles (2.2) are isomorphic, and so the bracket 〈α, β, γ〉 does not depend on thischoice.

Proposition 2.3. Let α : Y −→ Z, β : X −→ Y and γ : W −→ X be morphisms in a triangulatedcategory T satisfying αβ = 0 = βγ. Then the Toda bracket 〈α, β, γ〉 ⊂ T (ΣW,X) is a coset of the subgroup(α T (ΣW,Y )) + (T (ΣX,Z) Σγ). If δ : V −→W is another morphism such γδ = 0, then the relation

α〈β, γ, δ〉 = 〈α, β, γ〉(Σδ)

holds as subsets of T (ΣV,Z).

Proof. Let γ′ : ΣW −→ C(β) be another morphism satisfying pγ′ = Σγ. Then p(γ′−γ) = 0, so thereis a morphism u : ΣW −→ Y such that iu = (γ′ − γ). We have

αγ − αγ′ = α(γ′ − γ) = αiu = αu .

So different lifts for γ change the bracket representative by an element in αT (ΣW,Y ). The analogousargument shows that extensions of α change the bracket representative by an element in T (ΣX,Z)(Σδ).So the bracket 〈α, β, γ〉 lies in a single coset of the subgroup (α T (ΣW,Y )) + (T (ΣX,Z) Σγ).

Conversely, let α : C(β) −→ Z and γ : ΣW −→ C(β) satisfy αi = α and pγ = γ, so that αγ ∈ 〈α, β, γ〉.Given arbitrary morphisms v : ΣX −→ Z and u : ΣW −→ Y , then α + vp is another extension of α andγ + iu is another lift of γ. Hence

(α+ vp)(γ + iu) = αγ + αiu+ vpγ = αγ + αu+ v(Σγ)

is also an element of the bracket 〈α, β, γ〉. So 〈α, β, γ〉 indeed consists of the entire coset.

2. TODA BRACKETS IN TRIANGULATED CATEGORIES 287

We now have two notions of Toda bracket: the construction above works in triangulated categories andis based on composition of morphisms. In Construction I.6.15 we defined Toda brackets for homotopy classesof symmetric ring spectra, and this was based on smash product of morphisms. When the ring spectrumis the sphere spectrum S, its homotopy groups can be identified with morphisms in the triangulated stablehomotopy category, and we shall now see that the two notions of Toda brackets coincide. In more detail,evaluation at the fundamental class is an isomorphism of abelian groups

SHC(Sk,S) −→ πkS , α 7−→ (πkα)(ιk)

(compare Proposition 1.15) and this map is multiplicative in the sense that it takes composition in thestable homotopy category to product in π∗S = πs

∗.

Proposition 2.4. The isomorphism of graded rings

ev : SHC(Sk,S) −→ πkSgiven by evaluation at the fundamental classes is compatible with Toda brackets. More precisely, let α :Sk −→ S, β : Sl −→ S and γ : Sm −→ S be morphisms between sphere spectra in a stable homotopy categorywhich satisfy αβ = 0 = βγ. Then the relation

ev(〈α,Σk(β),Σk+l(γ)〉) = 〈ev(α), ev(β), ev(γ)〉holds as subsets of the group πk+l+j+1S. [strictly speaking we should throw in some coherence isomorphisms]

Proof. We represent the morphisms in SHC as ‘flat fractions’ of morphisms of symmetric spectra,i.e., we write α = γ(g) γ(f)−1 where Z is a flat symmetric spectrum, f : Z −→ Sk a stable equivalenceand g : Z −→ S a morphism of symmetric spectra [ref]. Then

ev(α) = (πkg)((πkf)−1(ιk)

),

and the right hand side was also denoted by 〈Z, f, g〉 in [...] We choose similar representations of β and γas flat fractions, and then have ev(β) = 〈Z ′, f ′, g′〉 and ev(γ) = 〈Z ′′, f ′′, g′′〉.

Exercise 2.5 (Modules as continuous functors). There is way to interpret modules over a symmetricring spectrum R as continuous functors on a based topological (or simplicial) category ΣR which generalizesthe isomorphism between the categories of symmetric spectra and continuous functors Σ −→ T.

Given a symmetric ring spectrum of topological spaces R we define a based topological category ΣR asfollows. The objects of ΣR are the natural numbers 0, 1, 2, . . . and the based space of morphisms from nto m is given by ΣR(n,m) = Σ+

m ∧1×Σm−n Rm−n, which is to be interpreted as a one-point space if m < n.Composition is defined by : ΣR(m, k) ∧ΣR(n,m) −→ ΣR(n, k) is defined by

[τ ∧ z] [γ ∧ y] = [τ(γ × 1) ∧ µn,m(y ∧ z)]where τ ∈ Σk, γ ∈ Σm, z ∈ Rk−m and y ∈ Rm−n. The identity in Σ(n, n) = Σ+

n ∧R0 is the identity of Σnsmashed with the identity element of R0.

Show that ΣR is a category and construct an isomorphism between the category of R-modules and thecategory of based continuous functors from ΣR to the category T of based compactly generated spaces.[Analog for simplicial sets]

Exercise 2.6. Let R by a symmetric ring spectrum. We define an R-bimodule R by

Rn =

∗ for n = 0Rn for n ≥ 1.

We define the n-latching object LRnM of a right R-module M by LRnM = (M ∧R R)n. [use previous exercisefor latching objects] The latching object has a left action of the symmetric group Σn and a right action ofthe pointed monoid R0. The inclusion R −→ R is a morphism of R-bimodules and thus induces a morphismof Σn-R0 simplicial bisets

νn : LRnM = (M ∧R R)n −→ (M ∧R R)n ∼= Mn .

Show:


(i) A morphism f : M −→ N is a flat cofibration of R-modules if and only if the maps νn(f) :LRnN ∪LRnM Mn −→ Nn are cofibrations of right R0-simplicial sets.

(ii) A morphism f : M −→ N is a projective cofibration of R-modules if and only if the mapsνn(f) : LRnN ∪LRnM Mn −→ Nn are cofibrations of Σn-R0-simplicial bisets.

(Hint: define a suitable R-module analog of the filtration FmA of a symmetric spectrum A so that theproof of Proposition II.5.35 can be adapted.)

Exercise 2.7. [adapt to new defn of homotopy category; check if works] Let R be a symmetric ringspectrum of simplicial sets. An R-module M is strongly injective is it has the extension property for allhomomorphisms of R-modules which are levelwise injective and a weak equivalence of underlying simplicialsets. We define the derived category D(R) of the ring spectrum R as the homotopy category of thosestrongly injective R-modules whose underlying symmetric spectra are Ω-spectra.

(i) Suppose that R is flat as a symmetric spectrum. Show that then the underlying symmetric spectrumof a strongly injective R-modules is injective. Give an example showing that the converse is not true.

(ii) Show that the derived category D(R) has the structure of a triangulated category with shift anddistinguished triangles defined after forgetting the R-action.

(iii) Show that D(R) is the target of a universal functor from R-modules which takes stable equivalencesto isomorphisms.

(iv) Let f : R −→ S be a homomorphism of symmetric ring spectra which makes R a flat rightS-module. Show that restriction of scalars from S-modules to R-modules passes to an exact functor oftriangulated categories f∗ : D(S) −→ D(R).

(v) Suppose that the underlying symmetric spectrum of R is semistable. Show that then R, consideredas a module over itself, has a strongly injective Ω-spectrum replacement γR as an R-module. Show thatthe map

[γR, γR]D(R)k

∼= πk(γR) ∼= πkR

is an isomorphism of graded rings, where the first map is evaluation at the unit 1 ∈ π0(γR) ∼= πk(γR)[k].Show that the map

[γR,M ]D(R)k

∼= πkM

is an isomorphism of graded modules over π∗R for every strongly injective Ω-R-module M . Show that γRis a compact weak generator of the triangulated category D(R).

We shall see later that for R = HA the Eilenberg-Mac Lane ring spectrum associated to a ring A (compareExample I.1.14) the derived category D(HA) is triangle equivalent to the unbounded derived category ofthe ring A. In fact, the equivalence of triangulated categories will come out as a corollary of a Quillenequivalence of model categories.

History and credits

Shipley [73] calls the flat model structure for modules over a symmetric ring spectrum R the ‘R-modelstructure’.

APPENDIX A

Miscellaneous tools

In more detail, a triple is a pointed functor T : set∗ −→ set∗ together with natural transformationsm : T T −→ T and η : Id −→ T such that the following diagrams commute

T T T Id m //

mId

T Tm

TId η //

=

&&MMMMMMMMMMMM T Tm

TηIdoo

=

xxqqqqqqqqqqqq

T T m// T T

An algebra over a triple T is a pointed set A together with a pointed map α : TA −→ A such that α ηAis the identity and the following associativity diagram commutes:

T (TA)T (α) //

mA

TA

α

TA α

// A

For every pointed set K, the pointed set TK is a T -algebra with structure map mK : T (TK) −→ TK; theT -algebra (TK,mK) will be denoted fTK and it is the free T -algebra generated by K in the sense that thefree functor fT : set∗ −→ T -alg is left adjoint to the forgetful functor fT from T -algebras to pointed sets.Note that we have T = fT fT and that the structure maps of T -algebras provides a natural transformationα : fT fT −→ Id.

1. Model category theory

The main references for model categories are Quillen’s original book [59], the modern introduction byDwyer and Spalinski [22] and Hovey’s monograph [32].

Definition 1.1. Let i : A −→ B and g : X −→ Y be morphisms in some category C. We say that ihas the left lifting property for g (or g has the right lifting property for i, or the pair (i, g) has the liftingproperty) if for every commutative square (solid arrows only)

A

i

// X

g

B

>>~~

~~

// Y

there exists a lifting, i.e., a morphism B −→ X (dotted arrow) which makes both resulting trianglescommute.

1.1. Cofibrantly generated model categories and a lifting theorem. In this section we reviewcofibrantly generated model categories and a general method for creating model category structures. If amodel category is cofibrantly generated, its model category structure is completely determined by a set ofcofibrations and a set of acyclic cofibrations. The transfinite version of Quillen’s small object argument

289

290 A. MISCELLANEOUS TOOLS

allows functorial factorization of maps as cofibrations followed by acyclic fibrations and as acyclic cofibra-tions followed by fibrations. Most of the model categories in the literature are cofibrantly generated, e.g.topological spaces and simplicial sets, as are all model structures involving symmetric spectra which wediscuss in this book.

The only complicated part of the definition of a cofibrantly generated model category is formulatingthe definition of relative smallness. For this we need to consider the following set theoretic concepts. Thereader might keep in mind the example of a compact topological space which is ℵ0-small relative to closedembeddings.

Ordinals and cardinals. An ordinal γ is an ordered isomorphism class of well ordered sets; it can beidentified with the well ordered set of all preceding ordinals. For an ordinal γ, the same symbol will denotethe associated poset category. The latter has an initial object ∅, the empty ordinal. An ordinal κ is acardinal if its cardinality is larger than that of any preceding ordinal. A cardinal κ is called regular if forevery set of sets Xjj∈J indexed by a set J of cardinality less than κ such that the cardinality of each Xj

is less than that of κ, then the cardinality of the union⋃J Xj is also less than that of κ. The successor

cardinal (the smallest cardinal of larger cardinality) of every cardinal is regular.Transfinite composition. Let C be a cocomplete category and γ a well ordered set which we identify

with its poset category. A functor V : γ −→ C is called a γ-sequence if for every limit ordinal β < γ thenatural map colimV |β −→ V (β) is an isomorphism. The map V (∅) −→ colimγV is called the transfinitecomposition of the maps of V . A subcategory C1 ⊂ C is said to be closed under transfinite compositionif for every ordinal γ and every γ-sequence V : γ −→ C with the map V (α) −→ V (α + 1) in C1 for everyordinal α < γ, the induced map V (∅) −→ colimγ V is also in C1. Examples of such subcategories are thecofibrations or the acyclic cofibrations in a closed model category.

Relatively small objects. Consider a cocomplete category C and a subcategory C1 ⊂ C closed undertransfinite composition. If κ is a regular cardinal, an object C ∈ C is called κ-small relative to C1 if forevery regular cardinal λ ≥ κ and every functor V :λ −→ C1 which is a λ-sequence in C, the map

colimλ HomC(C, V ) −→ HomC(C, colimλ V )

is an isomorphism. An object C ∈ C is called small relative to C1 if there exists a regular cardinal κ suchthat C is κ-small relative to C1.

I-injectives, I-cofibrations and I-cell complexes. Given a cocomplete category C and a class I of maps,we denote

• by I-inj the class of maps which have the right lifting property with respect to the maps in I.Maps in I-inj are referred to as I-injectives.

• by I-cof the class of maps which have the left lifting property with respect to the maps in I-inj.Maps in I-cof are referred to as I-cofibrations.

• by I-cell ⊂ I-cof the class of the (possibly transfinite) compositions of pushouts (cobase changes)of maps in I. Maps in I-cell are referred to as I-cell complexes.

In [59, p. II 3.4] Quillen formulates his small object argument, which immediately became a standardtool in model category theory. In our context we will need a transfinite version of the small object argument,so we work with the ‘cofibrantly generated model category’, which we now recall. Note that here I has tobe a set, not just a class of maps. The obvious analogue of Quillen’s small object argument would seem torequire that coproducts are included in the I-cell complexes. In fact, any coproduct of an I-cell complex isalready an I-cell complex, see [32, 2.1.6].

Lemma 1.2. Let C be a cocomplete category and I a set of maps in C whose domains are small relativeto I-cell. Then

• there is a functorial factorization of any map f in C as f = qi with q ∈ I-inj and i ∈ I-cell andthus

• every I-cofibration is a retract of an I-cell complex.

Definition 1.3. A model category C is called cofibrantly generated if it is complete and cocompleteand there exists a set of cofibrations I and a set of acyclic cofibrations J such that

1. MODEL CATEGORY THEORY 291

• the fibrations are precisely the J-injectives;• the acyclic fibrations are precisely the I-injectives;• the domain of each map in I (resp. in J) is small relative to I-cell (resp. J-cell).

Moreover, here the (acyclic) cofibrations are the I (J)-cofibrations.

For a specific choice of I and J as in the definition of a cofibrantly generated model category, themaps in I (resp. J) will be referred to as generating cofibrations (resp. generating acyclic cofibrations).In cofibrantly generated model categories, a map may be functorially factored as an acyclic cofibrationfollowed by a fibration and as a cofibration followed by an acyclic fibration.

Definition 1.4. Let C be a model category

R : D −→ C

a functor. We say that R creates a model structure on the category D if the following definitions make Dinto a model category: a morphism f in D is a

• weak equivalence if the morphism R(f) is a weak equivalence in C,• fibration if the morphism R(f) is a fibration in C,• cofibration if it has the left lifting property with respect to all morphisms in D which are both

fibrations and weak equivalences.

Theorem 1.5. Let C be a model category, D a category which is complete and cocomplete and let

R : D −→ C : L

be a pair of adjoint functors such that R commutes with filtered colimits. Let I (J) be a set of generatingcofibrations (resp. acyclic cofibrations) for the cofibrantly generated model category C. Let LI (resp. LJ) bethe image of these sets under the left adjoint L. Assume that the domains of LI (LJ) are small relative toLI-cell (LJ-cell). Finally, suppose every LJ-cell complex is a weak equivalence. Then R : D −→ C creates amodel structure on D which is cofibrantly generated with LI (LJ) a generating set of (acyclic) cofibrations.

If the model category C is right proper, then so is the model structure on D.If C and D are simplicially enriched, the adjunction (L,R) is simplicial, and the model structure of C

is simplicial, then the model structure on D is again simplicial.If C and D are topologically enriched, the adjunction (L,R) is continuous, and the model structure of

C is topological, then the model structure on D is again topological.

Proof. Model category axiom MC1 (limits and colimits) holds by hypothesis. Model category axiomsMC2 (saturation) and MC3 (closure properties under retracts) are clear. One half of MC4 (lifting properties)holds by the definition of cofibrations in D.

The proof of the remaining axioms uses the transfinite small object argument (Lemma 1.2), whichapplies because of the hypothesis about the smallness of the domains. We begin with the factorizationaxiom, MC5. Every map in LI and LJ is a cofibration in D by adjointness. Hence every LI-cofibration orLJ-cofibration is a cofibration in D. By adjointness and the fact that I is a generating set of cofibrationsfor C, a map is LI-injective precisely when the map becomes an acyclic fibration in C after application ofR, i.e., an acyclic fibration in D. Hence the small object argument applied to the set LI gives a (functorial)factorization of any map in D as a cofibration followed by an acyclic fibration.

The other half of the factorization axiom, MC5, needs the hypothesis. Applying the small objectargument to the set of maps LJ gives a functorial factorization of a map in D as an LJ-cell complexfollowed by a LJ-injective. Since J is a generating set for the acyclic cofibrations in C, the LJ-injectives areprecisely the fibrations among the D-morphisms, once more by adjointness. We assume that every LJ-cellcomplex is a weak equivalence. We noted above that every LJ-cofibration is a cofibration in D. So we seethat the factorization above is an acyclic cofibration followed by a fibration.

It remains to prove the other half of MC4, i.e., that any acyclic cofibration A −→ B in D has the leftlifting property with respect to fibrations. In other words, we need to show that the acyclic cofibrations are


contained in the LJ-cofibrations. The small object argument provides a factorization

A −→W −→ B

with A −→ W a LJ-cofibration and W −→ B a fibration. In addition, W −→ B is a weak equivalencesince A −→ B is. Since A −→ B is a cofibration, a lifting in

A //

W

∼

B

>>

B

exists. Thus A −→ B is a retract of a LJ-cofibration, hence it is a LJ-cofibration.

In cofibrantly generated model categories fibrations can be detected by checking the right lifting propertyagainst a set of maps, the generating acyclic cofibrations, and similarly for acyclic fibrations. This is incontrast to general model categories where the lifting property has to be checked against the whole class ofacyclic cofibrations. Similarly, in cofibrantly generated model categories, the pushout product axiom andthe monoid axiom only have to be checked for a set of generating (acyclic) cofibrations:

Lemma 1.6. Let C be a cofibrantly generated model category endowed with a closed symmetric monoidalstructure. If the pushout product axiom holds for a set of generating cofibrations and a set of generatingacyclic cofibrations, then it holds in general.

Proof. For the first statement consider a map i :A −→ B in C. Denote by G(i) the class of mapsj :K −→ L such that the pushout product

A ∧ L ∪A∧K B ∧K −→ B ∧ Lis a cofibration. This pushout product has the left lifting property with respect to a map f :X −→ Y ifand only if j has the left lifting property with respect to the map

p : [B,X] −→ [B, Y ]×[A,Y ] [A,X].

Hence, a map is in G(i) if and only if it has the left lifting property with respect to the map p for allf :X −→ Y which are acyclic fibrations in C.

G(i) is thus closed under cobase change, transfinite composition and retracts. If i : A −→ B isa generating cofibration, G(i) contains all generating cofibrations by assumption; because of the closureproperties it thus contains all cofibrations, see Lemma 1.2. Reversing the roles of i and an arbitrarycofibration j : K −→ L we thus know that G(j) contains all generating cofibrations. Again by the closureproperties, G(j) contains all cofibrations, which proves the pushout product axiom for two cofibrations.The proof of the pushout product being an acyclic cofibration when one of the constituents is, follows inthe same manner.

We now spell out the small object argument for symmetric spectra.

Theorem 1.7 (Small object argument). Let I be a set of morphisms of symmetric spectra based onsimplicial sets. Then there exists a functorial factorization of morphisms as I-cell complexes followed byI-injective morphisms.

Proof. In the first step we construct a functor F from the category of morphisms of symmetric spectrato symmetric spectra as follows. Given a morphism f : X −→ Y and a morphism i : Si −→ Ti in the set Iwe let Di denote the set of all pairs (a : Si −→ X, b : Ti −→ Y ) of morphisms satisfying fa = bi, i.e., whichmake the square

Sia //

i

X

f

Ti

b// Y

1. MODEL CATEGORY THEORY 293

commute. We define F (f) as the pushout in the diagram∨i∈I∨DiSi

∨a //

∨i

X

j

∨i∈I∨DiTi // F (f)

The morphisms b : Ti −→ Y and f : X −→ Y glue to a morphism p : F (f) −→ Y such that pj = f .The factorization we are looking for is now obtained by iterating this construction infinitely often, possiblytransfinitely many times.

We define functors Fn : Ar(Sp) −→ SpecΣ and natural transformations Xjn−→ Fn(f)

pn−→ Y for everyordinal n by transfinite induction. We start with F 0(f) = X, j0 = Id and p0 = f . For successor ordinal weset Fn+1(f) = F (pn : Fn(f) −→ Y ) with the morphisms jn+1 = j jn respectively pn+1 = p(pn). For limitordinals λ we set Fλ(f) = colimµ<λ F

µ(f) with morphisms induced by the jµ and pµ. By construction, allmorphisms jn : X −→ Fn(f) are I-cell complexes.

We claim that there exists a limit ordinal κ, depending on the set I, such that for every morphism fthe map pκ : Fκ(f) −→ Y is I-injective. Then f = pκjκ is the required factorization.

We prove the claim under the simplifying hypothesis that for each morphism i ∈ I the source Si isfinitely presented as a symmetric spectrum, i.e., for every sequence Z0 −→ Z1 −→ Z2 −→ . . . the naturalmap

colimn Sp(Si, Zn) −→ Sp(Si, colimn Zn)

is bijective. In that case, the first infinite ordinal ω will do the job. Indeed, Fω(f) is the colimit over thesequence

X = F 0(f)j1−→ F 1(f)

j2−→ F 2(f) · · · .Given a morphism i ∈ I and a lifting problem

(1.8) Sia //

i

Fω(f)

pω

Ti

b// Y

there exists a factorization a = can for some n ≥ 0 and some morphism an : Si −→ Fn(f) since Si is finitelypresented (where c : Fn(f) −→ Fω(f) is the canonical morphism to the colimit). The commutative square

Sian //

i

Fn(f)

pωc=pn

Ti

b// Y

is an element in the set Di which is used to define Fn+1(f) = F (pn). Thus the canonical morphismC : Ti −→ Fn+1(f) makes the diagram

Sian //

i

Fn+1(f)

pn+1

Ti

b//

C

;;wwwwwwwwwY

commute. Then the composite of C with the canonical morphism Fn+1(f) −→ Fω(f) solves the liftingproblem (1.8).


1.2. Bousfield’s localization theorem.

Theorem 1.9 (Bousfield). Let C be a proper model category with a functor Q : C −→ C and a naturaltransformation α : 1 −→ Q such that the following three axioms hold:

(A1) if f : X −→ Y is a weak equivalence, then so is Qf : QX −→ QY ;(A2) for each object X of C, the maps αQX , QαX : QX −→ QQX are weak equivalences;(A3) for a pullback square

Vk //

g

X

f

W

h// Y

in C, if f is a fibration between fibrant objects such that α : X −→ QX, α : Y −→ QY and Qh : QW −→ QYare weak equivalences, then Qk : QV −→ QX is a weak equivalence.

Then the following notions define a proper model structure on C: a morphisms f : X −→ Y is aQ-cofibration if and only if it is a cofibration, a Q-equivalence if and only if Qf : QX −→ QY is a weakequivalence, and Q-fibration if and only if f is a fibration and the commutative square

Xα //

f

QX

Qf

Y α

// QY


The reference is [10, Thm. 9.3]. [note: if Q preserves pullbacks and fibrations, then (A3) is automatic]

1.3. Some useful lemmas.

Lemma 1.10. We consider a commutative diagram

A

α

Bg //

β

foo C

γ

A′ B′

g′//

f ′oo C ′

in a model category C in which the morphisms α : A −→ A′ and g′ ∪ γ : B′ ∪B C −→ C ′ are cofibrationsrespectively acyclic cofibrations. then the induced morphism on pushouts α ∪ γ : A ∪B C −→ A′ ∪B′ C ′ is acofibration respectively acyclic cofibration.

Proof. The map α ∪ γ factors as the composite

A ∪B Cα∪C−−−→ A′ ∪B C

A′∪γ−−−→ A′ ∪B′ C ′ .

The first map α ∪ C is a cofibration (resp. acyclic cofibration) since α is. The second map A′ ∪ γ is acofibration (resp. acyclic cofibration) by the assumption of g′ ∪ γ and since

B′ ∪B C

f∪C

g′∪γ // C ′

A′ ∪B C

A′∪γ// A′ ∪B′ C ′

is a pushout.

2. COMPACTLY GENERATED SPACES 295

2. Compactly generated spaces

In this section we review some properties of compactly generated spaces. The most comprehensivereference for this category is Appendix A of Lewis’ thesis [42] which, although widely circulated, is un-published. So during this section we depart from our standing assumption that every space of compactlygenerated.

Let us fix some terminology. A topological space is compact if it is quasi-compact (i.e., every open coverhas a finite subcover) and satisfies the Hausdorff separation property (i.e., every pair of distinct points canbe separated by disjoint open subsets).

Definition 2.1. Let X be a topological space.

• X is weak Hausdorff if for every compact spaces K and every continuous map f : K −→ X theimage f(K) is closed in X.• A subset U of X is compactly closed if for every compact space K and every continuous mapf : K −→ X, the inverse image f−1(U) is closed in K.

• X is a Kelley space if every compactly open subset is open.• X is a compactly generated space if it is a Kelley space and weak Hausdorff.

We denote by Spc the category of topological spaces and continuous maps, by K its full subcategoryof Kelley spaces and by T the full subcategory of Spc and K of compactly generated spaces. We collectsome immediate observations. If K is compact and f : K −→ X continuous, then the image f(K) is alwaysquasi-compact; if X is Hausdorff, then any quasi-compact subset such as f(K) is closed. In other words,Hausdorff spaces are also weak Hausdorff spaces. Any point of any space is the continuous image of acompact space. So in weak Hausdorff spaces, all points and thus all finite subsets are closed. If X is weakHausdorff, K compact and f : K −→ X continuous, then the image f(K) is compact [42, Lemma 1.1].

Every closed subset is also compactly closed. One can similarly define compactly open subsets of X bydemanding that for every compact space K and every continuous map f : K −→ X, the inverse image isopen in K. A subset is then compactly open if and only if its complement is compactly closed. Thus Kelleyspaces can equivalently be defined by the property that all compactly open subsets are open.

All compact spaces are compactly generated. [also locally compact ? see Lewis] If X is weak Hausdorff(respectively a Kelley space, respectively compactly generated) and K is compact, then the X×K with theproduct topology is also weak Hausdorff (respectively a Kelley space, respectively compactly generated).[check; ref or proof]

[ref. to Lewis’ thesis]If X is any topological space we let kX be the space which has the same underlying set as X, but such

that the open subsets of kX are the compactly open subsets of X. This indeed defines a topology whichmakes kX into a Kelley space and such that the identity Id : kX −→ X is continuous. Part (i) of the nextproposition is a fancy way of saying that any continuous map Y −→ X whose source Y is a Kelley space isalso continuous when viewed as a map to kX. If X is weak Hausdorff, then so is kX.

If X is any space we let wX denote the maximal weak Hausdorff quotient of X [defined; does thispreserve Kelley spaces ?].

Proposition 2.2. (i) The functor k : Spc −→ K is left inverse and right adjoint to the inclusion.Unit and counit of the adjunction are the identity maps.

(ii) The functor w : K −→ T is right inverse and left adjoint to the inclusion.

The construction of the left adjoint w : K −→ T to the inclusion is not particularly instructive, as it isobtained by Freyd’s adjoint functor theorem. [give the proof]

There is a useful criterion, due to Lewis [42, App. A, Prop. 3.1], for when a Kelley space X is weakHausdorff: if and only if the diagonal subset in X ×X is closed in the K-topology (i.e., compactly closedin the usual product topology).

It follows formally from part (i) of this proposition that the category K of Kelley spaces has small limitsand colimits. Colimits can be calculated in the ambient category of all topological spaces; equivalently, any


colimits of Kelley spaces is again a Kelley space. To construct limits, we can first take a limit in the ambientcategory of all topological spaces; this ‘ambient limit’ need not be a Kelley space, but applying the functork : Spc −→ K yields a limit in K. Since k does not change the underlying set, the categories K and Spcshare the property that the forgetful functor to sets preserves all limits and colimits. More loosely speaking,the underlying set of a limit or colimit in K is what one first thinks of.

An important example where a limit in K resp. T can differ from the limit in Spc is the product oftwo CW-complexes X and Y . All CW-complexes are compactly generated [ref?], and the product X × Ywith the usual product topology is a Hausdorff space which comes with a natural filtration (X × Y )(n) =∪p+q=nX(p)× Y(q), where X(p) is the p-skeleton of the CW structure on X. If X or Y is locally finite, thenthe product topology is compactly generated, and then the above filtration makesX×Y into a CW-complex.In general, however, X × Y may not be a Kelley space, and hence cannot have a CW structure. But theproduct in the category K, i.e., the space k(X ×Y ), is always compactly generated and a CW-complex viathe above filtration.

It follows formally from the above and part (ii) of the proposition that the category T of compactlygenerated spaces has small limits and colimits. Limits can be calculated in the category K of Kelley spacesas explained in the previous paragraph. To construct colimits, we can first take a colimit in the category Kof Kelley spaces (or equivalently in Spc); while a Kelley space, this colimit need not be weak Hausdorff, butapplying the functor w : K −→ T yields a colimit in T. The ‘maximal weak Hausdorff quotient’ functorw : K −→ T is not particularly explicit and may change the underlying set; so one has to be especiallycareful with general colimits in T: unlike for Spc of K, the forgetful functor from T to sets need notpreserve colimits. More loosely speaking, the underlying set of colimit in T may be smaller than one firstthinks.

It will be convenient to know some particular instances of diagrams in T where it makes no differenceif we calculate the colimit in the category T or in K respectively Spc.

We call a continuous map f : X −→ Y between topological spaces a closed embedding if f is injective,the image f(X) is closed in Y and f is a homeomorphism onto its image. The basechange, in Spc or K,of a closed embedding is again a closed embedding. (Note that there is an ambiguity with the meaningof ‘embedding’ in general, due to the fact that a general subset of a Kelley space, endowed with thesubspace topology, need not be a Kelley space, and so one may or may not want to apply ‘Kelleyfication’k : Spc −→ K to the subspace topology. However, closed subsets of Kelley spaces are again Kelley spaceswith the usual subspace topology [this is OK for subsets which are the intersection of an open and a closedsubset], so there is not such ambiguity with the notion of ‘closed embedding’.)

A partially ordered set is a set P equipped with a binary relation ‘≤’ which is reflexive (i.e., x ≤ xfor all x ∈ P ), antisymmetric (i.e., x ≥ y and y ≤ x imply x = y) and transitive (i.e., x ≥ y and y ≤ zimply x ≤ z). The partially ordered set P is filtered if for every pair of elements x, y ∈ P there exists anelement z ∈ P such that x ≤ z and y ≤ z. [non-empty?] We will routinely interpret a partially ordered setP as a category without change in notation. In the associated category, the objects are the elements of Pand there is a unique morphism from x to y if x ≤ y, and no morphism from x to y otherwise. Via thisinterpretation we can consider functors defined on partially ordered sets.

Proposition 2.3. (i) Given a pushout in the category K of Kelley spaces

Xf //

Z

Y g

// Y ∪f Z

such that f is a closed embedding, then g is also a closed embedding. If moreover, X,Y and Z are compactlygenerated, the so is Y ∪f Z, and hence the diagram is a pushout in T.(ii) Let P be a filtered partially ordered set and F : P −→ T a functor from the associated poset category. LetF∞ be the colimit of F in the category K of Kelley spaces (or, equivalently, in Spc) and κi : F (i) −→ F∞the canonical map. If for every i ≤ j in P the map F (i) −→ F (j) is injective, then the maps κi are also

2. COMPACTLY GENERATED SPACES 297

injective and the colimit F∞ is weak Hausdorff, thus a colimit of F in the category T. If moreover, all mapsF (i) −→ F (j) are closed embedding, then so are the maps κi : F (i) −→ F∞.(iii) Let λ be a regular (?) cardinal and X : λ −→ T a λ-sequence of injective maps. Then the colimitcolimλX in the category Spc is again compactly generated and thus a colimit of X in T.

Proof. (i) is [42, Prop. 7.5], (ii) is [42, Prop. 9.3] and (iii) is in Hovey [32, ].

An example of Lewis [42, ] shows that a pushout in T along a non-closed embedding need not even beinjective.

The following proposition says that compact spaces are ‘small with respect to closed embeddings’.Note that if all spaces in the λ-sequence are compactly generated, then by Proposition 2.3 (ii) it makes nodifference whether the colimit is calculated in the category Spc, K or T.

Proposition 2.4. [32, Prop. 2.4.2] Let λ be an ordinal and X : λ −→ T a λ-sequence of closedembeddings [def? what happens at limit cardinals ?]. Then for every compact space K the natural map

colimi>λ C(K,Xi) −→ C(K, colimλX))

is bijective.

It is not in general true that C(K,−) commutes with filtered colimits over closed embeddings. Anexample from Lewis’ theses (which he credits to Myles Tierney) is the unit interval [0, 1]. A subset of [0, 1]is closed if and only if its intersection with every countable closed subset J of [0, 1] is closed in J , whichimplies that [0, 1] is the filtered colimit of its countable closed subsets, ordered by inclusion. Since [0, 1] isuncountable, the identity map of [0, 1] does not factor through any of the spaces in the filtered system.

[is there a criterion in terms of the poset P when for every compact space K the natural map

colimi∈P C(K,F (i)) −→ C(K, colimP F ))

is bijective. ? [or even a homeomorphism ?]]

Proposition 2.5. Let Xii∈I be a family of based compactly generated spaces. Then the wedge (one-point union)

∨i∈I Xi is compactly generated, thus the coproduct of the family in T. Moreover, for every

compact space K and every continuous map f : K −→∨i∈I Xi there is a finite subset J of I such that f

factors through the sub-wedge∨i∈J Xi.

Proof.

Corollary 2.6. Let λ be an ordinal and X : λ −→ T a λ-sequence of compactly generated spaces.If all maps in the sequence are closed embeddings [def? what happens at limit cardinals ?], then for everycompact point x ∈ X∅ and every n ≥ 0 the natural map

colimi>λ πn(Xi, x) −→ πn(colimλX,x)

is bijective. If in addition all maps in the sequence are weak equivalences, then so is the transfinite compositeX∅ −→ colimλX.

[check out also Lemma 9.3 of [74]]There is a suitable version of the compact open topology which gives mapping spaces in the category

T. For spaces X and Y , we let C(X,Y ) denote the set of continuous maps from X to Y . A subbasis for atopology is given by all sets S(U, f : K −→ X) where U is an open subset of Y , K a compact space and fa continuous map; the set S(U, f) consists of all those continuous ϕ : X −→ Y such that ϕ(f(k)) ⊂ U . IfX and Y are compactly generated, then the space C(X,Y ) is weak Hausdorff, but not necessarily a Kelleyspace. So the mapping space map(X,Y ) is defined as kC(X,Y ), which is then a compactly generated space.[ref to Lewis]

Theorem 2.7. The category of compactly generated spaces is cartesian closed, i.e., the natural map

map(X × Y, Z) ∼= map(X,map(Y, Z))

is a homeomorphism for all compactly generated spaces X,Y and Z, where the product on the left hand sideis taken in the category T.


[ref to Lewis ?]A continuous map f : X −→ Y of topological spaces is a weak equivalence if f induces a bijection on

path components and for every point x ∈ X and n ≥ 1 the map πn(f) : πn(X,x) −→ πn(Y, f(x)) is anisomorphism. An equivalent condition is that for every CW-complex A (possibly empty) the induced map[A, f ] : [A,X] −→ [A, Y ] on homotopy classes of continuous maps is bijective.

The map f is a cofibration if and only if it is a retract of a relative cell complex [spell out]. The mapf is a fibration if it is a Serre fibration, i.e., has the right lifting property with respect to the inclusionsDn −→ Dn × [0, 1], x 7→ (x, 0) for all n ≥ 0.

Theorem 2.8. [32, Thm. 2.4.25] The cofibrations, weak equivalences and (Serre) fibrations make thecategory T of compactly generated spaces into a proper model category. The model structure is monoidalwith respect to the cartesian product. [generators]

Although this will not be relevant for us we feel obliged to mention that the weak equivalences, cofi-brations and (Serre) fibrations also define model structures on the categories Spc of all topological spaces(this is originally due to Quillen [59, II.3, Thm. 1], proofs can also be found in [22, Prop. 8.3] or [32,Thm. 2.4.19]) and K of Kelley spaces [32, Thm. 2.4.23]. Since weak equivalences are defined by takinghomotopy classes of continuous maps out of compact spaces the identity kX −→ X is a weak equivalencefor every topological space X. So weak equivalences don’t see any difference between the categories Spcand K. A more structured way to say this is that the left of the two adjoint functor pairs

Spck

// Kincl.oo w // T

incl.oo

is a Quillen equivalence. The second pair is also a Quillen equivalence because the right adjoint preservesand detects weak equivalences and fibrations and because for every cofibrant object A of K the adjunctionunit A −→ wA is an isomorphism, thus a weak equivalence. However, the category T is the only one amongthese three model categories which is monoidal, and that is the reason why we work in T. [no: K is alsoclosed symmetric monoidal...] [Strom model structures a la Cole]

[based versions]We consider a sequence

X0f0−−→ X1

f1−−→ X2f2−−→ · · ·

of based continuous maps between based topological spaces. We want to define the reduced mappingtelescope teliXi of the sequence. We first define the ‘partial telescopes’ Fj and based homotopy equivalencesαj : Xj −→ tel[0,j]Xi by induction on j ≥ 0. We start with tel[0,0]Xi = X0 and α0 = Id. The next partialtelescope tel[0,j+1]Xi is defined as the pushout of the diagram

Fjαj←−−− Xj

x7→x∧0−−−−−→ Xj ∧ [0, 1]+ ∪fj Xj+1 .

The map αj+1 is the composite of the homotopy equivalence Xj+1 −→ Xj ∧ [0, 1]+ ∪fj Xj+1 and thecanonical map from the mapping cylinder Xj ∧ [0, 1]+ ∪fj Xj+1 to the pushout Fj+1. The compositeαj+1 fj : Xj −→ Fj+1 is homotopic, in a basepoint preserving fashion, to the composite of αj and thecanonical map Fj −→ Fj+1. We can now define the reduced mapping telescope as the colimit of thesequence

F0 −→ F1 −→ F2 −→ · · · .The composite maps α′j : Xj −→ Fj −→ teliXi then have the property that α′j+1 fj : Xj −→ Fj+1 isbased homotopic to α′j . Thus for every n ≥ 0 the induced maps on homotopy groups (or sets, for n = 0)satisfy

πn(α′j+1) πn(fj) = πn(α′j)

and so they assemble into a map

colimi πn(Xi, xi) −→ πn(teliXi, x∞) .

3. SIMPLICIAL SETS 299

Proposition 2.9. For every sequence of based compactly generated spaces Xi and based continuousmaps fi : Xi −→ Xi+1 the reduced mapping telescope teliXi is again compactly generated and the naturalmap

colimi πn(Xi, xi) −→ πn(teliXi, x∞)is bijective for every n.

Proof. We claim that if all the spaces Xi are compactly generated, then the maps Fj −→ Fj+1 areall closed embeddings and all the partial telescopes Fj are compactly generated. Proposition 2.3 (ii) thenshows that the telescope is compactly generated and Corollary 2.6 shows that the canonical map

colimj πn(Fj , x∞) −→ πn(teliXi, x∞)

is bijective for every n. Since αj : Xj −→ Fj is a homotopy equivalence we can replace the group (or set)πn(tel[0,j]Xi, x∞) by the isomorphic group πn(Xj , xj) to obtain the isomorphism we want.

It remains to prove the claim. The map from Xj to the reduced mapping cone Xj ∧ [0, 1]+ ∪fj Xj+1

which sends x to the class of x ∧ 0 is a closed embedding [...] So its basechange Fj −→ Fj+1 is a closedembedding and we can use Proposition 2.3 (i) to argue inductively that Fj+1 is compactly generated.

3. Simplicial sets

The books by May [53], Lamotke [41] and Goerss-Jardine [28] or Chapter 3 of [32] can serve as generalreferences for simplicial sets and their homotopy theory. [search all these for references...]

We denote by ∆ the simplex category whose objects are the partially ordered sets [n] = 0, . . . , n(with the usual order) and whose morphisms are weakly monotone maps. A simplicial set is a contravariantfunctor X from ∆ to the category of sets. We often denote the value X([n]) of X at the object [n] by Xn,and refer to its elements as the n-simplices of X. [introduce di and si]

[morphisms] We denote the category of based simplicial sets by sSSome important examples of simplicial sets are the representable simplicial sets ∆[n] = ∆(−, [n]) called

the standard n-simplex and its boundary ∂∆[n] defined by

(∂∆[n])m = α : [m] −→ [n] | α is not surjective .Moreover, for every 0 ≤ k ≤ n there is a simplicial subset Λk[n] of ∂∆[n] called its k-th horn and given by

(Λk[n])m = α : [m] −→ [n] | k 6∈ α([m]) .Every small category C give rise to a simplicial set NC, called the nerve of C. To define the nerve we

introduce the category [[n]] associated to the ordered set [n]; so the object set of [[n]] is 0, . . . , n andthere is a unique morphism from i to j if and only if i ≤ j. Every weakly monotone map α : [k] −→ [n]is the object function of a unique functor [α] : [[k]] −→ [[n]]. In total this gives a fully faithful functor[−] : ∆ −→ Cat from the simplex category ∆ to the category Cat of small categories. We can then definethe nerve of C by NC = Cat([−], C). In more detail, the k-simplices of the nerve NC are given by

(NC)k = Cat([[k]], C) ,the set of functors from [[k]] to C. The structure map α∗ : (NC)n −→ (NC)k associated to a weaklymonotone map α : [k] −→ [n] is given by precomposition with the functor [α]. We note that a functorϕ : [[k]] −→ C is determined by a string of k composable morphisms

i0a1−→ i1

a2−→ · · · ak−→ ik

in C. In particular, (NC)0 is the set of objects of C and (NC)1 is the set of morphisms of C.[BM for a monoid M ]A simplicial set can be thought of as a combinatorially defined CW-complex. This is made precise by

the functor of geometric realization. For a simplicial set X the geometric realization |X| is the topologicalspace

|X| =∫

[n]∈∆

Xn ×∆[n] .


This need some explanation. We denote by

∆[n] = (x0, . . . , xn) ∈ Rn+1 | xi ≥ 0,n∑i=0

xi = 1

the topological n-simplex. As n varies, we get a covariant functor ∆[−] : ∆ −→ Spc by sending a weaklymonotone map α : [k] −→ [n] to the affine linear map α∗ : ∆[k] −→ ∆[n] given by

α∗(x0, . . . , xk) = (y0, . . . , yn)

with yj =∑α(i)=j xi. [write as coequalizer, quotient space]

For every simplicial set X, the geometric realization |X| comes with a natural filtration whose p-th termis the image of the space qpn=0Xn×∆[n] under the quotient map. This filtration gives |X| the structure ofa CW-complex; in particular, the geometric realization of a simplicial set is a compactly generated space.Thus we can, and will, view geometric realization as a functor | − | : sS −→ T to the category of compactlygenerated spaces.

A homeomorphism

(3.1) ∆[n]∼=−→ |∆[n]|

from the topological n-simplex to the geometric realization of the standard n-simplex is given by the com-posite of the summand inclusion ∆[n] −→ qm ∆[n]m×∆[m] indexed by Id[n] ∈ ∆[n]n and the quotient mapto |∆[n]|. Under this homeomorphism, the realization |∂∆[n]| of the boundary of the simplex correspondsto the geometric boundary of the topological n-simplex, i.e., the subspace of tuples (x0, . . . , xn) ∈ ∆[n]such that xi = 0 for at least one coordinate Xi. Moreover, the realization |Λk[n]| of the k-th horn corre-sponds to the union of all except the k-th faces of the topological n-simplex, i.e., the subspace of tuples(x0, . . . , xn) ∈ ∆[n] such that there exists an i 6= k with xi = 0.

The geometric realization |NC| of the nerve is often called the classifying space of the small category C.

Theorem 3.2. The geometric realization functor | − | : sS −→ T has the following properties.(i) For all simplicial sets K and L the natural map

|K × L| −→ |K| × |L|

is a homeomorphism where the target is the product in the category T of compactly generated spaces.

(i’) For all based simplicial sets K and L the natural map

|K ∧ L| −→ |K| ∧ |L|

is a homeomorphism where the target is the smash product in the category T of compactly generated basedspaces.

(ii) If A and X are simplicial sets and X is fibrant, then the natural map

|map(A,X)| −→ map(|A|, |X|)

is a weak equivalence.

(ii’) If A and X are based simplicial sets and X is fibrant, then the natural map

|map(A,X)| −→ map(|A|, |X|)

[this time based; sort out the notation...] is a weak equivalence.

(iii) The geometric realization of a Kan fibration between simplicial sets is a Serre fibration of spaces.

3. SIMPLICIAL SETS 301

References for the proofs: (i): [32, Lemma 3.2.4], plus the fact that limits in the category T ofcompactly generated spaces can be formed in the category K of Kelley spaces. [also: Ch. 3 of Gabriel-Zisman [27] ?] (iii): the original reference is [60], but see also [32, Cor. 3.6.2] [is this in Gabriel-Zisman [27]?].

As a special case of part (ii’) above we have that for a fibrant based simplicial set X the natural map

|ΩX| −→ Ω|X|is a weak equivalence.

The geometric realization functor has an adjoint, the singular complex functor S : Spc −→ sS. For atopological space X the n-simplices of the simplicial set S(X) are given by

S(X)n : Spc(∆[n], X) ,

the set of continuous maps from the topological n-simplex to X. For weakly monotone map α : [k] −→ [n]the induced map α∗ : S(X)n −→ S(X)n is precomposition with the affine linear map α∗ : ∆[k] −→ ∆[n].The adjunction bijection

sS(A,S(X)) ∼= T(|A|, X)is quite tautological. A morphism of simplicial sets f : A −→ S(X) gives, for every n-simplex a ∈ An, acontinuous map f(a) : ∆[n] −→ X. As the n-simplices vary, these maps give a continuous map

An ×∆[n] −→ X , (a, y) 7→ f(a)(y) .

These maps are compatible as n varies, so they assemble into a continuous map f : |A| −→ X.[Since the simplex ∆[n] is a compact space, the identity Id : kX −→ X induces an isomorphism after

taking singular complexes. Same for X −→ wX ?][The singular complex functor commuted with filtered colimits over closed embeddings.][compare combinatorial homotopy groups with homotopy groups of realization whenever X is Kan]

[realization of simplicial object is diagonal]A morphism f : X −→ Y of simplicial sets is a weak equivalence if the geometric realization |f | is a

weak equivalence of topological spaces. [equivalent characterizations: |f | is homotopy equivalence; Ex∞ ishomotopy equivalence of ssets; [f,X] is bijection for all Kan simplicial sets X]

The map f is a cofibration if it is dimensionwise injective, or, equivalently, a categorical monomorphism.The map f is a fibration if it is a Kan fibration, i.e., has the right lifting property with respect to theinclusions Λk[n] −→ ∆[n] of all horns into simplices. [equivalently: RLP for all injective weak equivalences.]

For every topological space X (in Spc, K or T?) the adjunction counit |S(X)| −→ X is a weakequivalence and for every simplicial set Y , the adjunction unit Y −→ S(|Y |) is a weak equivalence. [refs...the second is a formal consequence of the first by the def’n of weak equivalences in sS]

The following model structure on the category of simplicial sets is due to Quillen [59, II.3, Thm. 3];proofs can also be found in [32, Thm. 3.6.5] and [28, Thm I.11.3]. The Quillen equivalence can be obtainedby combining Theorems 2.4.25 and 3.6.7 of [32]; however the equivalence of the homotopy categories ofCW-complexes and simplicial sets has been know since the 1950’s [ref’s: Gabriel-Zisman [27] ? May [53]?according to Goerss-Jardine also Kan’s [38]]

Theorem 3.3. The cofibrations, weak equivalences and Kan fibrations make the category sS of simplicialsets into a proper model category. The model structure is monoidal with respect to the cartesian product.The adjoint functors of geometric realization and singular complex are a Quillen equivalence

sS|−| // T .S

oo

[generators]

[based version; realization of simplicial spaces and bisimplicial sets]The augmented simplicial category ∆+ is the category with objects the finite ordered sets [n] =

0, . . . , n for n ≥ −1, where [−1] = ∅ is the empty set, and all weakly monotone maps as morphisms.


Thus ∆+ contains the simplicial category ∆ as a full subcategory and has one additional object [−1] whichis initial and receives no morphisms from [n] for n ≥ 0. An augmented simplicial object is a contravariantfunctor from the augmented simplicial category ∆+ to C. An augmented simplicial object X determines,and is determined by

• the simplicial object obtained by restriction of X to ∆,• the object X−1 = X([−1]) and• the morphism d0 : X0 −→ X−1, called the augmentation, induced by the unique morphism

[−1] −→ [0] in ∆+ which has to coequalize the two morphism d0, d1 : X1 −→ X0.The augmentation d0 : X0 −→ X−1 of an augmented simplicial space X gives rise to a continuous map

|uX| −→ X−1 .

Proposition 3.4. Let X be an augmented simplicial space which admits extra degeneracies. Then themap |res(X)| −→ X−1 induced by the augmentation is a based homotopy equivalence.

Proof. We let cX−1 denote the constant simplicial space with values X−1. We can define a morphismof simplicial spaces s−1 : cX−1 −→ uX in simplicial dimension k by the composite of s−1 : X−1 −→ X0 withthe degeneracy morphism s : X0 −→ Xk. We can define a morphism of simplicial spaces d : uX −→ cX−1

in simplicial dimension k by the unique face morphism Xn −→ X−1. These morphisms induce basedcontinuous maps on geometric realizations

|s−1| : |cX−1| −→ |uX| respectively |d| : |uX| −→ |cX−1| .

Since the composite ds−1 : cX−1 −→ cX−1 is the identity, so is the composite of the realizations of dand s−1. A homotopy of the other composite |s−1| |d| : |uX| −→ |uX| is given by the realization of themorphism of simplicial spaces

H : uX ×∆[1] −→ uX

defined in simplicial dimension k by [...]

Proposition 3.5. Let X be an augmented simplicial abelian group which admits extra degeneracies.Then the chain complex

· · · −→ Xn+1∂−→ Xn

∂−→ · · ·X1∂−→ X0

∂−→ X−1 −→ 0

is exact.

Proof.

4. Equivariant homotopy theory

In this section we review some facts about equivariant homotopy theory over finite groups, with emphasison model structures. We let G be a finite group, the most important case for us will be when G = Σn is asymmetric group.

We denote by TG respectively sSG the categories of G-objects in T or sS. So in the first case anobject is a based compactly generated spaces equipped with a continuous, based left G-action. In thesecond case an object is a based simplicial X set equipped with an associative and unital action morphismG × X −→ X which fixes the basepoint; the category sSG can equivalently be described as the categoryof simplicial objects of left G-sets equipped with G-invariant basepoints. Morphisms in TG respectivelysSG are those morphisms in T or sS which commute with the G-action. [limits and colimits in underlyingcategories T respectively sS; symmetric monoidal closed structure...]

The following terminology is useful for naming the various model structures. A morphism f : X −→ Yof based G-spaces is a

• G-cell complex [...]• G-cofibration if it is a retract of a G-cell complex;• free G-cofibration if it is a retract of a free G-cell complex;

4. EQUIVARIANT HOMOTOPY THEORY 303

• weak G-equivalence (respectively weak G-fibration) if the underlying map of spaces is a weakequivalence (respectively fibration) after forgetting the group action;

• strong G-equivalence (respectively strong G-fibration) if for every subgroup K of G the map ofK-fixed points fK : XK −→ Y K is a weak equivalence (respectively fibration) of spaces.

We note that relative G-CW complexes are G-cell complexes, and thus G-cofibrations, but in a G-cellcomplex the cells need not be attached in the order of dimension.

A morphism f : X −→ Y of based G-simplicial sets is a

• G-cofibration if it is a monomorphism, i.e., injective in every simplicial dimension;• free G-cofibration if it is a monomorphism a nd in every simplicial dimension n the action of G onYn is free away from the image f(Xn);

• weak G-equivalence (respectively weak G-fibration) if the underlying map of simplicial sets is aweak equivalence (respectively fibration) after forgetting the group action;• strong G-equivalence (respectively strong G-fibration) if for every subgroup K of G the map ofK-fixed points fK : XK −→ Y K is a weak equivalence (respectively fibration) of simplicial sets.

Theorem 4.1 (Weak equivariant model structure). For every finite group G the weak G-equivalences,weak G-fibrations and free G-cofibrations make the categories TG of based G-spaces and sSG of based G-simplicial sets into proper model categories. [generators, monoidal]

The weak equivariant model structure is a special case of Quillen’s model structure [59, Ch. II.4,Thm. 4] on the category of simplicial objects in a category with sufficiently many projectives. [other refs ?case of spaces ?]

Theorem 4.2 (Strong equivariant model structure). For every finite group G the strong G-equivalences,strong G-fibrations and G-cofibrations the G-cofibrations, strong G-fibrations and strong G-equivalencesmake the categories TG of based G-spaces and sSG of based G-simplicial sets into proper model categories.[generators, monoidal]

[can do this more generally for a family of subgroups...][for TG: Strom model structures a la Cole]There is another pair of equivariant model structure which is relevant for the study of symmetric

spectra, namely the mixed model structure. [better name ?] The word ‘mixed’ refers to the fact that thismodel structure has the same weak equivalences as the weak equivariant model structure, but the samecofibrations as the strong equivalent model structure. There has to be a new class of fibrations then, whichwe now define. The homotopy fixed points of a G-space (or G-simplicial set) X is the space (simplicial set)XhG = map(EG,X)G of G-equivariant maps from the free contractible G-space EG to X. The uniquemap EG −→ ∗ is equivariant and induces a natural map XG = map(∗, X)G −→ map(EG,X)G = XhG

from the fixed points to the homotopy fixed points. A morphism f : X −→ Y of bases G-spaces or basedG-simplicial sets is a mixed G-fibration [better name? ‘mixed‘ is stronger than ‘strong’...] if it is a strongG-fibration and for every subgroup K of G the square of spaces (simplicial sets)

(4.3) XK //

fK

XhK

fhK

Y K // Y hK


Theorem 4.4 (Mixed equivariant model structure). For every finite group G the weak G-equivalences,mixed G-fibrations and G-cofibrations make the categories TG of based G-spaces and sSG of based G-simplicial sets into proper model categories. [generators, monoidal]


In the case of G-simplicial sets, the mixed model structure was established by Shipley in [73, Prop. 1.3].However, Shipley defines the mixed fibrations as the morphism with the right lifting property for G-cofibration which are also weak G-equivalences, so we have to prove that this class coincides with ourdefinition of mixed G-fibration.

Proposition 4.5. Let G be a finite group and f : X −→ Y a morphism of based G-spaces or basedG-simplicial sets. Then the following are equivalent:

(i) f is is a mixed G-fibration,(ii) f is a strong G-fibration and the square

(4.6) X //

f

map(EG,X)

map(EG,f)

Y // map(EG, Y )

is homotopy cartesian in the strong G-equivariant model structure,(iii) f has the right lifting property for all G-cofibrations which are weak G-equivalences.

fix the proof.... (i)=⇒(ii) Mixed G-fibrations are strong G-fibrations by definition, so it remainsto show that the square 4.6 is strongly G-homotopy cartesian. Since f is a strong G-fibration and EG isG-cofibrant, the map map(EG, f) is a strong G-fibration by the [adjoint of] the pushout product property.So we may show that the map X −→ Y ×map(EG,Y ) map(EG,X) is a strong G-equivalence.

We fix a G-equivariant based map f : X −→ Y . Let us first assume that f has the right lifting propertyfor all G-cofibrations which are weak G-equivalences. Then in particular f has the right lifting propertyfor all G-cofibrations which are strong G-equivalences, so f is a strong G-fibrations, and it remains to showthat the square (4.3) is homotopy cartesian for every subgroup H of G.

We claim that for every G-cofibration K −→ L which is also a weak G-equivalence the induced map

(4.7) map(L,X) −→ map(K,X)×map(K,Y ) map(L, Y )

is a strong G-fibration and strong G-equivalence. To prove this, we note that for all subgroups H of G andall boundary inclusions, the G-equivariant pushout product morphism

G/H × (L× ∂∆[n] ∪K×∂∆i[n] K ×∆[n]) −→ G/H × L×∆[n]

is a G-cofibration and weak G-equivalence. Since f has the RLP for such maps, by adjointness the H-fixed points of the map (4.7) have the right lifting property for all boundary inclusions, so they are acyclicfibrations of spaces (simplicial sets). Since this holds for all subgroups H of G, the map (4.7) is a strongG-acyclic fibration.

[...fix this...] Now we show that the square of property (ii) is G-homotopy cartesian. Since f is G-fibration, so is the morphism map(EG, f), and hence it suffices to show that the morphism

X −→ Y ×map(EG,Y ) map(EG,X)

is a G-weak equivalence. The inclusion EG −→ C(EG) of EG into its cone is G-equivariant and aninjective weak equivalence of underlying simplicial sets (but not a G-weak equivalence !). So by the previousparagraph the induced morphism

map(C(EG), X) −→ map(EG,X)×map(EG,Y ) map(C(EG), Y )

is a G-acyclic fibration. In the commutative square

X = map(∗, X) //

map(EG,X)×map(EG,Y ) map(∗, Y )

map(C(EG), X) // map(EG,X)×map(EG,Y ) map(C(EG), Y )

5. ENRICHED CATEGORY THEORY 305

the vertical maps are induced by the unique morphism C(EG) −→ ∗ which is a G-equivariant homotopyequivalence, so induces a homotopy equivalence on mapping spaces. So the top horizontal map is a G-weakequivalence since the other three maps are.

For the other direction we assume that f is a mixed G-fibration. Let i : K −→ L be a G-cofibrationwhich is also a weak G-equivalence; we have to show that the pair (i, f) has the lifting property. Then i is acofibration in the strong equivariant model structure. Since that model structure is monoidal, the inducedmap

map(L,X) −→ map(K,X)×map(K,Y ) map(L, Y )

is a strong G-fibration. We show that it is also a strong G-equivalence, [thus a G-acyclic fibration]. Since thesquare is G-homotopy cartesian, we can replace the G-fibration f : X −→ Y be the G-fibration map(EG, f)and show that the G-fibration.

map(L,map(EG,X)) −→ map(K,map(EG,X))×map(K,map(EG,Y )) map(L,map(EG, Y ))

is a G-weak equivalence. This map is isomorphic to

map(L× EG,X) −→ map(K × EG,X)×map(K×EG,Y ) map(L× EG, Y ) .

What we have gained now is that the morphism i × Id : K × EG −→ L × EG is a G-equivariant weakequivalence between free G-simplicial sets, thus a G-weak equivalence. So the latter morphism is a G-acyclicfibration by the adjoint of the pushout product property.

By taking G-fixed points we then get an acyclic fibration of simplicial sets

mapG(L,X) −→ mapG(K,X)×mapG(K,Y ) mapG(L, Y )

which is in particular surjective on vertices. This exactly means that f : X −→ Y has the right liftingproperty with respect to i : K −→ L.

The fact that the strong and mixed equivariant model structures share the same class of cofibrationsimplies that they also share the same acyclic fibrations. In other words, for a morphism f : X −→ Y in TG

or sSG the following are equivalence:

• f is a strong G-equivalence and strong G-fibration;• f is a weak G-equivalence and strong G-fibration and for every subgroup K of G the square (4.3)

of spaces (simplicial sets) is homotopy cartesian;• f is a weak G-equivalence and mixed G-fibration.

5. Enriched category theory

As we explain in Section 8.1, symmetric spectra can be viewed as ‘enriched functors’ defined on a certaincategory Σ. Several properties of symmetric spectra which we use are special cases of enriched categorytheory. In this section we summarize the relevant material.

Enriched category theory can be set up relative to a symmetric monoidal base category. We will mostlyneed the categories T of based compactly generated spaces and sS of based simplicial set, under smashproduct. For definiteness we develop the theory for the base category T and indicate and the end of thissection which changes have to be made in the case of simplicial sets.

We let J be a small T-category, or based topological category. In more detail, J consists of

• a set obJ of objects,• for every pair i, j of objects a compactly generated based space J(i, j)• and for every triple of objects i, j, k a based continuous composition map

: J(j, k) ∧ J(i, j) −→ J(i, k) .


Moreover, composition should be associative in the sense that for every quadrupel of objects i, j, k, l thesquare

J(k, l) ∧ J(j, k) ∧ J(i, j) Id∧ //

∧Id

J(k, l) ∧ J(i, k)

J(j, l) ∧ J(i, j) // J(i, l)

commutes and composition should have two sided units.A continuous functor X : J −→ T consists of• an space Xj for every object j of J and• based continuous maps X : J(i, j) −→ map(Xi, Xj)

which are unital and associative. We denote the category of continuous functors from J to T by TJ.If X is a continuous functor, the effect on morphisms can also be given in adjoint form as continuous

maps J(i, j)∧Xi −→ Xj . The associativity condition is then equivalent to the requirement that the squares

J(j, k) ∧ J(i, j) ∧XiId∧ //

∧Id

J(j, k) ∧Xj

J(i, k) ∧Xi // Xk

commute.Ordinary category theory can be subsumed under this branch of enriched category theory as follows.

Every category C in the ordinary sense gives rise to a T-category C+ by adding disjoint basepoints tothe morphisms sets and endowing them with the discrete topology and extending composition by [...]Continuous functors from C+ to T are in bijective correspondence [isomorphism of categories] with functorsC −→ T in the classical sense.

Example 5.1. Here are some examples of enriched categories J which are relevant for this book. Ourmain example is the category Σ introduced in Section I.8.1 which parameterizes symmetric spectra ofspaces. More generally we have an enriched category ΣR for every symmetric ring spectrum R such thatenriched functors ΣR −→ T are R-modules. Similarly, there is an enriched category O which parameterizesorthogonal symmetric spectra, see Exercise I.8.10 Other examples are the injection category I or the simplexcategory ∆.

[Limits, colimits, ends, coends, realization objectwise]For every object j of J there is a representable continuous functor Fj defined by Fj = J(j,−). It comes

with a special element Idj ∈ Fj(j).

Proposition 5.2 (Enriched Yoneda lemma). For every continuous functor X : J −→ T and everyobject j ∈ J the assignment

mapTJ(Fj , X) −→ Xj , τ 7→ τj(Idj)is a homeomorphism.

Corollary 5.3. For every every object j ∈ J the functor T −→ TJ which sends a based space K tothe continuous functor K ∧ Fj is left adjoint to evaluation at j.

For every continuous functor X and every object i of J there is map κi : Fi∧Xi −→ X which is adjoint[...]. Thus the value of κi at an object j is given by the map J(i, j) ∧ Fi −→ Xj which is adjoint to thestructure map X : J(i, j) −→ T(Xi, Xj). As i varies the squares

commute, which means that the maps κi assemble into a natural map F• ∧JX• −→ X from the coend.

Corollary 5.4. For every continuous functor X the natural map

κ : F• ∧J X• = coequalizer( ∨

j,k Fk ∧ J(j, k) ∧XjF∧Id //Id∧X

//∨i Fi ∧Xi

)−→ X


is an isomorphism of continuous functors.This really means that for every object j ∈ J the map

J(−, j) ∧J X −→ Xj

is a homeomorphism. Also, for every G : Jop −→ T and every object j ∈ J the map

G ∧J Fj −→ Gj

is a homeomorphism [follows from previous by replacing J by Jop]

Let α : J −→ J′ be an enriched functor between T-categories. Precomposition with α gives a ‘restriction’functor α∗ : TJ′ −→ TJ.

We call α fully faithful if the map of spaces α : J(i, j) −→ J′(α(i), α(j)) is a homeomorphism for allobjects i, j of J. Since there are bijective continuous maps which are not homeomorphisms ‘fully faithful’as an enriched functor is a stronger condition than demanding that the underlying ordinary functor is fullyfaithful.

Proposition 5.5. For every enriched functor α : J −→ J′ the restriction functor α∗ : TJ′ −→ TJ hasleft adjoint α∗ called left Kan extension along α and a right adjoint α! called right Kan extension along α.

If α is fully faithful then the unit X −→ α∗(α∗X) of the adjunction (α∗, α∗) and the counit α∗(α!X) −→X of the adjunction (α∗, α!) are isomorphisms for every enriched functor X : J −→ T.

Proof. We construct the left Kan extension α∗X of an enriched functor X : J −→ T and leave therest to references. We set

α∗X = FJ′

• ∧J X = coequalizer( ∨

j,k FJ′

k ∧ J(j, k) ∧XjF∧Id //Id∧X

//∨i F

J′

i ∧Xi

).

Now for some homotopy theory. Given enriched functors α : J −→ J′ and X : J −→ T where J andJ′ are small, we construct left and right homotopy Kan extensions of X along α. For this purpose we firstdefine a simplicial object B•(α,J, X) in the category of enriched functors from J′ to T, called the two-sidedbar construction. The functor of k-simplices is given by

Bk(α,J, X) =∨

(i0,...,ik)∈(obJ)k+1

FJ′

ik∧ J(ik−1, ik) ∧ · · · ∧ J(i0, i1) ∧Xi0 .

The face maps are given by the coaction of J on FJ′

ikthrough α, composition in J respectively the action on

X [spell out]. The degeneracy maps are given by the identity morphisms in J. We note that the coequalizerof the two face map d0.d1 : B1(α,J, X) −→ B0(α,J, X) is precisely the coend FJ′

• ∧J X•, which is the(ordinary) left Kan extension of X along α.

We define the homotopy Kan extension αh∗(X) as the realization of this simplicial functor, i.e.,

αh∗(X) = |B•(α,J, X)| .[define augmented simplicial object with B−1(α,J, X) = X] We thus obtain a natural morphism αh∗(X) −→α∗(X).

Now consider the special case where J = J+ and J′ = (J ′)+ arise from an ordinary categories J by givingthen the discrete topology and adding disjoint basepoints to the morphism sets, and where α : J −→ J′

arises similarly from an ordinary functor a : J −→ J ′. Then the homotopy Kan extensions αh∗(X) can bepresented in a slightly different form. In fact the J′-functor Bk(α,J, X) is then isomorphic to∨

(a1,...,ak)∈(NJ)k

FJ′

ik∧Xi0 ;

this wedge is indexed over all k-simplices of the nerve of the category J , i.e., strings of k composablemorphisms

i0a1−→ i1

a2−→ · · · ak−→ ik


in J . The face maps are given by the coaction of J on FJ′

ikthrough α, composition in J respectively the

action on X [spell out]. The degeneracy maps are given by the identity morphisms in J.We discuss two important special cases of homotopy Kan extensions.Homotopy colimits. We can take the target category J′ as the trivial category ∗, i.e., with only one

object and one identity morphism. If α : J −→ ∗ [should the target have morphisms S0?] is the uniquefunctor (which is automatically enriched), then the (ordinary) Kan extension α∗X of an enriched functorX : J −→ T is the enriched colimit colimJX [explain]. If we now use the ‘homotopy’ version of the Kanextension we obtain the homotopy colimit αh∗(X). Since the functor α is uniquely determined it is customaryto use a different notation and write hocolimJX for αh∗(X).

In the case where J = J+ arises from an ordinary category J by giving it the discrete topology andadding disjoint basepoints to the morphism sets, the enriched colimit colimJX is just the ordinary colimitcolimJ X of X over J . The homotopy colimit hocolimJX is then also written hocolimJ X. [nerve andclassifying space as hocolims]

Homotopy coends. Let G : Jop −→ T and X : J −→ T be continuous functors. We can definea simplicial object B•(X,J, α) in the category of enriched functors from J′ to T, called the two-sided barconstruction. The functor of k-simplices is given by

Bk(G,J, X) =∨

(i0,...,ik)∈(obJ)k+1

Gik ∧ J(ik−1, ik) ∧ · · · ∧ J(i0, i1) ∧Xi0 .

The face maps are given by the coaction of J on Gik through α, composition in J respectively the action onX [spell out]. The degeneracy maps are given by the identity morphisms in J. We note that the coequalizerof the two face map d0.d1 : B1(G,J, X) −→ B0(G,J, X) is precisely the coend G ∧J X.

We define the homotopy coend G ∧hJ X as the realization of this simplicial functor, i.e.,

G ∧hJ X = |B•(G,J, X)| .

[define augmented simplicial object with B−1(G,J, X) = G ∧J X] We thus obtain a natural morphismG ∧hJ X −→ G ∧hJ X from the homotopy coend to the coend.

[under cofibrancy conditions, weak equivalences in G or X induce weak equivalences of homotopycoends]

Proposition 5.6. (0) Homotopy coend commutes with colimits and smash product with a based spacein both variables.

(?) Natural isomorphism G ∧hJ X ∼= Xop ∧hJop Gop

(i) Let g : G −→ G′ be a natural weak equivalence of enriched Jop-functors and let ϕ : X −→ X ′ be anatural weak equivalence of enriched J-functors. Suppose in addition that G, G′, X and X ′ are objectwisecofibrant as based spaces [+cofibrancy in J]. Then the induced map g ∧hJ ϕ : G ∧hJ X −→ G′ ∧hJ X ′ ofhomotopy coends is a weak equivalence.

(ii) Let X be an enriched J-functor and j an object of J. Then the augmentation

J(−, j) ∧hJ X −→ Xj

is a homotopy equivalence.

(iii) Let G be an enriched Jop-functor and j an object of J. Then the augmentation

G ∧hJ J(j,−) −→ Gj

is a homotopy equivalence.

Proof. (?) Use the ‘orientation reversal’ automorphism of the simplicial category ∆.


(ii) The space of k-simplices of the simplicial space B•(J(−, j),J, X) is given by

Bk(J(−, j),J, X) =∨

(i0,...,ik)∈(obJ)k+1

J(ik, j) ∧ J(ik−1, ik) ∧ · · · ∧ J(i0, i1) ∧Xi0 .

Since j is fixed we can define ‘extra degeneracy maps’

s−1 : Bk(J(−, j),J, X) −→ Bk+1(J(−, j),J, X)

by including into the summand where ik = j via the identity morphism of the object j. For k ≥ −1, whereB−1(J(−, j),J, X) = Xj . The extra degeneracies satisfy the relations [...] which means that the realize toa homotopy equivalence between J(−, j) ∧hJ X = |B•(J(−, j),J, X)| and Xj .

(iii) If we replace J by the opposite category Jop, this becomes an instance of (ii).

Bar resolutions. We have a bifunctor J(−,−)opJ ∧J −→ T; if we fix a particular value in one if the twovariables, we obtain co- respectively contravariant representable functors. For fixed j, the homotopy coendJ(−, j) ∧hJ X yields a based space, and as j varies we obtain another covariant functor X] : J −→ T givenby (X])j = J(−, j) ∧hJ X. The augmentations provide a natural transformation of J-functors X] −→ Xwhich is objectwise a homotopy equivalence by Proposition 5.6.

[Can also view this as a special case of the homotopy Kan extension, X] = Idh∗(X), by taking α asthe identity functor the category J. Then restriction along α does not do anything, and so the ordinaryleft and right Kan extension functors along α = IdJ do not do anything either.] Homotopy Kan extension,however, do have an effect. We refer to X] as the bar resolution of X. The advantage of the resolution X]

over the original functor is that X] tends to be ‘free’ (or ‘projective’, or ‘cofibrant’), see for example [...]

Bibliography

[1] J. Adamek, J. Rosicky, Locally presentable and accessible categories, London Math. Soc. Lecture Note Series 189,Cambridge University Press, Cambridge, 1994. xiv+316 pp.

[2] J. F. Adams, Stable homotopy and generalised homology. Chicago Lectures in Mathematics. University of Chicago Press,Chicago, Ill.-London, 1974. x+373 pp.

[3] Barratt, M. G.; Jones, J. D. S.; Mahowald, M. E. Relations amongst Toda brackets and the Kervaire invariant indimension 62. J. London Math. Soc. (2) 30 (1984), no. 3, 533–550.

[4] J. M. Boardman, On Stable Homotopy Theory and Some Applications. PhD thesis, University of Cambridge (1964)[5] J. M. Boardman, Stable homotopy theory, Various versions of mimeographed notes. University of Warwick 1966 and

Johns Hopkins University, 1969–70.[6] J. M. Boardman, Conditionally convergent spectral sequences. Homotopy invariant algebraic structures (Baltimore, MD,

1998), 49–84, Contemp. Math., 239, Amer. Math. Soc., Providence, RI, 1999.[7] M. Bokstedt, Topological Hochschild homology, Preprint (1985), Bielefeld.[8] M. Bokstedt, W. C. Hsiang, I. Madsen, The cyclotomic trace and algebraic K-theory of spaces. Invent. Math. 111 (1993),

no. 3, 465–539.[9] A. K. Bousfield, The localization of spaces with respect to homology, Topology 14 (1975), 133–150.

[10] A. K. Bousfield, On the telescopic homotopy theory of spaces. Trans. Amer. Math. Soc. 353 (2001), no. 6, 2391–2426.[11] A. K. Bousfield, D. M. Kan, Homotopy limits, completions and localizations, Lecture Notes in Mathematics, Vol. 304.

Springer-Verlag, 1972. v+348 pp.[12] A. K. Bousfield, E. M. Friedlander, Homotopy theory of Γ-spaces, spectra, and bisimplicial sets, Geometric applications

of homotopy theory (Proc. Conf., Evanston, Ill., 1977), II Lecture Notes in Math., vol. 658, Springer, Berlin, 1978,pp. 80–130.

[13] E. H. Brown, F. P. Peterson, A spectrum whose Zp cohomology is the algebra of reduced pth powers. Topology 5 (1966),149–154.

[14] K. S. Brown, Abstract homotopy theory and generalized sheaf cohomology. Trans. Amer. Math. Soc. 186 (1974), 419–458.[15] Bruner, Robert R. An infinite family in π∗S0 derived from Mahowald’s ηj family. Proc. Amer. Math. Soc. 82 (1981),

no. 4, 637–639.[16] R. Bruner, J. P. May, J. McClure, M. Steinberger, H∞ ring spectra and their applications. Lecture Notes in Mathematics,

1176. Springer-Verlag, Berlin, 1986. viii+388 pp.[17] M. Cole, Mixing model structures. Topology Appl.153 (2006), no. 7, 1016–1032.[18] B. Day, On closed categories of functors. 1970 Reports of the Midwest Category Seminar, IV pp. 1–38 Lecture Notes in

Mathematics, Vol. 137 Springer, Berlin[19] A. Dold, Homology of symmetric products and other functors of complexes, Ann. Math. 69 (1958), 54-80.[20] B. Dundas, O. Rondigs, P. A. Østvær, Enriched functors and stable homotopy theory. Doc. Math. 8 (2003), 409–488.[21] B. Dundas, O. Rondigs, P. A. Østvær, Motivic functors. Doc. Math. 8 (2003), 489–525.[22] W. G. Dwyer, J. Spalinski, Homotopy theories and model categories, Handbook of algebraic topology (Amsterdam),

North-Holland, Amsterdam, 1995, pp. 73–126.[23] S. Eilenberg and S. Mac Lane, On the groups H(Π, n), I, Ann. of Math. (2) 58, (1953), 55–106.[24] A. D. Elmendorf, I. Kriz, M. A. Mandell, J. P. May, Rings, modules, and algebras in stable homotopy theory. With

an appendix by M. Cole, Mathematical Surveys and Monographs, 47, American Mathematical Society, Providence, RI,1997, xii+249 pp.

[25] A. D. Elmendorf, M. A. Mandell, Rings, modules, and algebras in infinite loop space theory, Adv. Math. 205 (2006),163-228.

[26] T. Geisser, L. Hesselholt, Topological cyclic homology of schemes, Algebraic K-theory (Seattle, WA, 1997), 41–87, Proc.Sympos. Pure Math., 67, Amer. Math. Soc., Providence, RI, 1999.

[27] P. Gabriel, M. Zisman, Calculus of fractions and homotopy theory. Ergebnisse der Mathematik und ihrer Grenzgebiete,Band 35 Springer-Verlag 1967 x+168 pp.

[28] P. Goerss, J. F. Jardine, Simplicial homotopy theory, Progress in Mathematics, 174. Birkhauser Verlag, Basel, 1999.xvi+510 pp.

[29] T. Gunnarsson, Algebraic K-theory of spaces as K-theory of monads, Preprint, Aarhus University, 1982.[30] J. E. Harper, Homotopy theory of modules over operads in symmetric spectra, arXiv:0801.0193

311

312 BIBLIOGRAPHY

[31] L. Hesselholt, I. Madsen, On the K-theory of finite algebras over Witt vectors of perfect fields. Topology 36 (1997),29–101.

[32] M. Hovey, Model categories, Mathematical Surveys and Monographs, vol. 63, American Mathematical Society, Provi-dence, RI, 1999, xii+209 pp.

[33] M. Hovey, B. Shipley, J. Smith, Symmetric spectra, J. Amer. Math. Soc. 13 (2000), 149–208.[34] P. Hu, S-modules in the category of schemes, Mem. Amer. Math. Soc. 161 (2003), no. 767, viii+125 pp.[35] J. F. Jardine, Motivic symmetric spectra. Doc. Math. 5 (2000), 445–553.[36] M. Joachim, A symmetric ring spectrum representing KO-theory. Topology 40 (2001), no. 2, 299–308.[37] M. Joachim, S. Stolz, An enrichment of KK-theory over the category of symmetric spectra. Preprint [2005].[38] D. M. Kan, Adjoint functors. Trans. Amer. Math. Soc. 87 (1958), 294–329.[39] D. M. Kan, Semisimplicial spectra. Illinois J. Math. 7 (1963), 463–478.[40] D. M. Kan, G. W. Whitehead, The reduced join of two spectra. Topology 3 (1965) suppl. 2, 239–261.[41] K. Lamotke, Semisimpliziale algebraische Topologie. Die Grundlehren der mathematischen Wissenschaften, Band 147,

Springer-Verlag, Berlin-New York 1968 viii+285 pp.[42] L. G. Lewis, Jr., The stable category and generalized Thom spectra. Ph.D. thesis, University of Chicago, 1978[43] L. G. Lewis, Jr., Is there a convenient category of spectra ? J. Pure Appl. Algebra 73 (1991), 233-246.[44] L. G. Lewis, Jr., J. P. May, M. Steinberger, Equivariant stable homotopy theory, Lecture Notes in Mathematics, 1213,

Springer-Verlag, 1986.[45] M. Lydakis, Smash products and Γ-spaces, Math. Proc. Cambridge Philos. Soc. 126 (1991), 311–328.[46] M. Lydakis, Simplicial functors and stable homotopy theory, Preprint (1998). http://hopf.math.purdue.edu/[47] S. Mac Lane, Homology, Grundlehren der math. Wissensch. 114, Academic Press, Inc., Springer-Verlag, 1963 x+422 pp.[48] I. Madsen, Algebraic K-theory and traces. Current developments in mathematics, 1995 (Cambridge, MA), 191–321,

Internat. Press, Cambridge, MA, 1994.[49] M. A. Mandell, Equivariant symmetric spectra, Homotopy theory: relations with algebraic geometry, group cohomology,

and algebraic K-theory, 399–452, Contemp. Math., 346, Amer. Math. Soc., Providence, RI, 2004.[50] M. A. Mandell, J. P. May, Equivariant orthogonal spectra and S-modules, Mem. Amer. Math. Soc. 159 (2002), no. 755,

x+108 pp.[51] M. A. Mandell, J. P. May, S. Schwede, B. Shipley, Model categories of diagram spectra, Proc. London Math. Soc. 82

(2001), 441-512.[52] H. R. Margolis, Spectra and the Steenrod algebra. Modules over the Steenrod algebra and the stable homotopy category.

North-Holland Mathematical Library, 29. North-Holland Publishing Co., Amsterdam, 1983. xix+489 pp.[53] J. P. May, Simplicial objects in algebraic topology, Chicago Lectures in Mathematics, Chicago, 1967, viii+161pp.[54] J. P. May, E∞ ring spaces and E∞ ring spectra. With contributions by F. Quinn, N. Ray, and J. Tornehave. Lecture

Notes in Mathematics, Vol. 577. Springer-Verlag, Berlin-New York, 1977. 268 pp.[55] J. P. May, Stable algebraic topology, 1945–1966. History of topology, 665–723, North-Holland, Amsterdam, 1999.[56] J. P. May, The additivity of traces in triangulated categories. Adv. Math. 163 (2001), no. 1, 34–73.[57] H. R. Miller, On relations between Adams spectral sequences, with an application to the stable homotopy of a Moore

space. J. Pure Appl. Algebra 20 (1981), 287–312.

[58] F. Morel, V. Voevodsky, A1-homotopy theory of schemes. Inst. Hautes Etudes Sci. Publ. Math. 90, 45–143 (2001).[59] D. G. Quillen, Homotopical algebra. Lecture Notes in Mathematics, 43, Springer-Verlag, 1967.[60] D. G. Quillen, The geometric realization of a Kan fibration is a Serre fibration. Proc. Amer. Math. Soc. 19 (1968),

1499–1500.[61] D. G. Quillen, Elementary proofs of some results of cobordism theory using Steenrod operations Advances in Math. 7

1971 29–56 (1971).[62] D. C. Ravenel, Complex cobordism and stable homotopy groups of spheres. Pure and Applied Mathematics, 121. Academic

Press, Inc., Orlando, FL, 1986. xx+413 pp.[63] A. Robinson, The extraordinary derived category. Math. Z. 196 (1987), no. 2, 231–238.[64] Y. B. Rudyak, On Thom spectra, orientability, and cobordism. Springer Monographs in Mathematics. Springer-Verlag,

Berlin, 1998. xii+587 pp.[65] R. Schwanzl, R. Vogt, F. Waldhausen: Adjoining roots of unity to E∞ ring spectra in good cases — a remark. Homotopy

invariant algebraic structures (Baltimore, MD, 1998), 245–249, Contemp. Math., 239, Amer. Math. Soc., Providence,RI, 1999.

[66] S. Schwede, S-modules and symmetric spectra, Math. Ann. 319 (2001), 517-532.[67] S. Schwede, On the homotopy groups of symmetric spectra. Preprint (2006).

http://www.math.uni-bonn.de/people/schwede

[68] S. Schwede, B. Shipley, Algebras and modules in monoidal model categories, Proc. London Math. Soc. 80 (2000), 491-511[69] S. Schwede, B. Shipley, Stable model categories are categories of modules. Topology 42 (2003), no. 1, 103–153.[70] G. Segal, Categories and cohomology theories, Topology 13 (1974), 293–312.[71] B. Shipley, Symmetric spectra and topological Hochschild homology, K-Theory 19 (2) (2000), 155-183.[72] B. Shipley, Monoidal uniqueness of stable homotopy theory, Advances in Mathematics 160 (2001), 217-240.

BIBLIOGRAPHY 313

[73] B. Shipley, A convenient model category for commutative ring spectra. Homotopy theory: relations with algebraicgeometry, group cohomology, and algebraic K-theory, 473–483, Contemp. Math., 346, Amer. Math. Soc., Providence,RI, 2004.

[74] N. E. Steenrod, A convenient category of topological spaces. Michigan Math. J. 14 (1967), 133-152.[75] R. E. Stong, Notes on cobordims theory. Mathematical notes Princeton University Press, Princeton, N.J.; University of

Tokyo Press, Tokyo 1968 v+354+lvi pp.[76] N. P. Strickland, Realising formal groups. Algebr. Geom. Topol. 3 (2003), 187–205.[77] R. M. Switzer, Algebraic topology—homotopy and homology. Die Grundlehren der mathematischen Wissenschaften,

Band 212. Springer-Verlag, New York-Heidelberg, 1975. xii+526 pp.[78] M. Tierney, Categorical constructions in stable homotopy theory. A seminar given at the ETH, Zrich, in 1967. Lecture

Notes in Mathematics, No. 87 Springer-Verlag, Berlin-New York 1969 iii+65 pp.[79] J.-L. Verdier, Des categories derivees des categories abeliennes. With a preface by Luc Illusie. Edited and with a note

by Georges Maltsiniotis. Astrisque 239 (1996), xii+253 pp. (1997).[80] V. Voevodsky, A1-homotopy theory. Doc. Math. ICM I (1998), 417-442.[81] R. Vogt, Boardman’s stable homotopy category. Lecture Notes Series, No. 21 Matematisk Institut, Aarhus Universitet,

Aarhus 1970 i+246 pp.[82] F. Waldhausen, Algebraic K-theory of spaces. Algebraic and geometric topology (New Brunswick, N.J., 1983), 318–419,

Lecture Notes in Math., 1126, Springer, Berlin, 1985.[83] A. Weiner, Symmetric spectra and Morava K-theories. Diplomarbeit, Universitat Bielefeld, 2005.

Index

A[X], linearization 214

A∨, Pontryagin dual 206

B(n), 116

BO, 153

BP , Brown-Peterson spectrum 116

BP 〈n〉, 116

BSO, 153

BSU , 153

BSp, 153

BSpin, 153

BU , 153

CM , 246

D, Spanier-Whitehead dual 230

E(n), Johnson-Wilson spectrum 116

F (Y, Z), derived function spectrum 195

F kX, 89

FmK, free symmetric spectrum 34

GmL, semifree symmetric spectrum 35

HA, Eilenberg-Mac Lane spectrum 15

Hk(X, A), cohomology with A-coefficients 213

Hk(X, A), homology with A-coefficients 213

K(n), periodic Morava K-theory 116

KO, real topological K-theory 19, 112, 163

KU , complex topological K-theory 19, 163, 218

L .m X, twisted smash product 36

LmX, latching space 90

M(p), mod-p Moore spectrum 218

MO, unoriented Thom spectrum 16

MSO, oriented Thom spectrum 17

MSU , special unitary Thom spectrum 17

MSp, symplectic Thom spectrum 17

MSpin, spin Thom spectrum 17

MU , unitary Thom spectrum 17

MUP , periodic unitary Thom spectrum 140

M ∧R N , smash product of R-modules 282

Mk(R), matrix ring spectrum 41

NC, nerve of a category 299

P (n), 116

PX, symmetric algebra 88

PnX, 221

Qi, 223

R-local, 233

RX, = Ω(sh X) 51, 126

R[1/x], 42, 112

R[M ], monoid ring spectrum 40

R∞X, = teln Ωn(shn X) 51, 126

Rop, opposite ring spectrum 107

S(K, f), 224

S1 ∧ −, suspension 65

Sn, n-sphere 5

TX, tensor algebra 88

W (1), shift of an M-module 123

X〈n〉, 219

X], standard resolution 132

∆[n], standard simplex 299

∂∆[n], boundary of a simplex 299

∆[n], topological simplex 300

HomR(M, N), function spectrum for R-modules 282

Λk[n], horn of a simplex 299

L, linear isometries operad 266

M, injection monoid 116

Pn, 120

SW, Spanier-Whitehead category 228

S(X), singular complex 30

Σ, suspension in the stable homotopy category 182

Σ∞, suspension spectrum 15

Θi, Kervaire invariant class 243

πk, naive homotopy group 11

∆, simplex category 299

∆+, augmented simplex category 301

Γ, category of finite based sets 45, category of finitebased sets 143

I, category of finite sets and injections 148

K, category of Kelley spaces 295

O, orthogonal index category 159

Σ, symmetric index category 136

ΣR, 287

Σ≤k, 89

Spc, category of topological spaces 295

T, category of compactly generated topological spaces5, 295

U, unitary index category 159

ξX,Y , 83

β, mod-p Bockstein operation 213

sS, 5, category of based simplicial sets 299

End(X), endomorphism operad 266

χn,m, shuffle permutation 10

diag H, 149

diagi Xi, 29

η, Hopf map in 1-stem 12, 243

ηj , Mahowald family of stable homotopy classes 243

γ, 172

315

316 INDEX

ι, 11, 20

ιmn , true fundamental class 70

ιmn , naive fundamental class 70

ιn, 200

κ, 244

λA, natural morphism S1 ∧A −→ sh A 33

λA, natural morphism .A −→ ΩA 53

λA, natural morphism .(S1 ∧A) −→ A 53

λA, natural morphism A −→ Ω(sh A) 53

λ(m)A , natural morphism Sm ∧A −→ shm A 39

D, detection functor 154

SA, Moore spectrum 224

S, sphere spectrum 12

S, truncated sphere spectrum 92

S[k], truncated sphere spectrum 162

S[1/m], 42, 224

S[M ], spherical monoid ring 40

Sn, n-dimensional sphere spectrum 200

map(X, Y ), mapping space 38

ν, Hopf map in 3-stem 15, 243

πsk, k-th stable stem 12

ρx, 109

SHC, stable homotopy category 171

SHC[, stable homotopy category of flat spectra 192

σ, Hopf map in 7-stem 15, 243

Sp, category of symmetric spectra 9, 30

SpT, category of symmetric spectra of spaces 9

SpsS, category of symmetric spectra of simplicial sets30

SpO, orthogonal spectrum 137

SpU, unitary spectrum 140

SpN, sequential spectrum 20

SpΣ(C), 257

Sp[, category of flat symmetric spectra 193

Sp≤k, 89

τi, exterior generator of dual Steenrod algebra 240

τ≤mX, 163

〈x, y, z〉, Toda bracket 107

teli Xi, mapping telescope 28

., shift adjoint 34

ε, class in 8-stem 15, 243

ξi, polynomial generator of dual Steenrod algebra 240

X, Y , fraction category 72

X, Y [, flat fraction category 98

c, tautological map πkA −→ πkA 62

ci, class in Ext3,∗A (F2, F2) 243

d, cycle operator 118

f∗, restriction of scalars 284

f!, coextension of scalars 284

f∗, extension of scalars 284

hi, class in Ext1,2i

A (F2, F2) 242

k(n), connective Morava K-theory 116

|Y |, geometric realization 30

A∞ operad, 266

Adams differential, 243

Adams map, 217, 239

Adams spectral sequence, 242

adjunction bijection

for (K ∧ −. map(K,−)), 6

for πk(ΩmZ), 51

arithmetic square, 237

assembly map, 141, 146, 214

associative operad, 265

augmentation, 302

bar resolution, 309

Barratt-Eccles operad, 266

bimorphism, 77

Bockstein operation

mod-p, 213

of an extension, 211

Bott periodicity theorem, 19

boundary

of a simplex, 299

bounded below, 230

Bousfield localization, 233

at a set of primes, 234

Brown-Comenetz dual, 206

Brown-Peterson spectrum, 13, 116

canonical filtration, 92, 162

category with cofibrations and weak equivalences, 44

cell complex, 290

chain complex

of a symmetric spectrum, 246

classifying space

of a category, 300

closed embedding, 296

cobordism spectrum, see Thom spectrum

coextension of scalars, 284

cofibrantly generated, 290

cofibration

flat, 252

level, 93, 252

of simplicial sets, 301

of spaces, 298

projective, 252

cohomology


cohomotopy group, 181

colimit

derived, 76

in T, 296

of symmetric spectra, 32

commutative operad, 265

compact, 202, 204, 230

compactly generated, 295

compactly open, 295

completion

p-adic, 236

profinite, 236

connected cover, 219

of a module spectrum, 285

connecting homomorphism, 24, 25

for true homotopy groups, 68

connective, 208

connective cover

of a spectrum, 278

of an O-algebra, 277

continuous functor, 141

INDEX 317

cycle operator, 118

derivation, 223

detection functor, 139, 154

Shipley’s, 154

diagonal, 29

of an I-spectrum, 149

distinguished triangle

elementary, 183

in the stable homotopy category, 183

Dold operad, 266

dual

Brown-Comenetz, see Brown-Comenetz dual

Pontryagin, see Pontryagin dual

Spanier-Whitehead, see Spanier-Whitehead duality

duality

of the stable homotopy category, 230

dualizable, 232

E∞ operad, 266

Eilenberg-Mac Lane spectrum, 15, 58, 142, 144, 180, 209,224, 234, 278, 285

generalized, 235

of an I-functor, 164

of an M-module, 120

endomorphism operad, 266

endomorphism ring spectrum, 40

evaluation morphism, 195

exact functor, 197

extension of scalars, 284

fibration

flat, 254

injective, 254


of spaces, 298

projective, 254

stable flat, 262

stable injective, 262

stable projective, 262

finite spectrum, 230

flat symmetric spectrum, 93, 129, 130

fraction category, 72

flat, 98

free orthogonal spectrum, 138

free symmetric spectrum, 34, 59, 71, 122

function ring spectrum, 41

function spectrum, 39

derived, 195

functor

cohomological, 203

enriched, 136

exact, 197

fundamental class

of FmSn

naive, 70

true, 70

of an Eilenberg-Mac Lane space, 181

Γ-space, 143

Eilenberg-Mac Lane, 144

special, 144

very special, 144, 147

generating hypothesis, 233

generator

of a triangulated category, 205

weak, 202, 205

geometric realization

of a simplicial set, 299

of a symmetric spectrum of spaces, 30

h-cofibration, 26

Hausdorff space

weak, see weak Hausdorff space

heart

of a t-structure, 210

homology


homotopy

of spectrum morphisms, 48

homotopy colimit, 308

in a triangulated category, 203


of a sequence, 203


homotopy equivalence


homotopy fiber, 22, 68

homotopy fixed points, 303

homotopy group

mod-p, 239

naive, see naive homotopy group

of a space, 6

of spheres, 12

relative to a class, 160

true, see true homotopy group, see true homotopygroup

homotopy Kan extension, 307

homotopy limit


homotopy orbits, 76

homotopy ring spectrum, 199, 201

Hopf map, 12, 15, 243

horn

of a simplex, 299

Hurewicz homomorphism, 216, 236

Hurewicz theorem

for spectra, 216

I-functor, 163

I-space, 148

indeterminacy

of a Toda bracket, 108

injection monoid, 116, 118, 266

injection operad, 106, 165, 266

invertible, 200, 232

Johnson-Wilson spectrum, 116

juggling formula

for Toda brackets, 108

k-invariant, 222

K-local, 239

K-theory

318 INDEX

algebraic, 44

Morava, 116

topological, 19, 163, 217, 236

Kan extension, 307

homotopy, see homotopy Kan extension

Kan fibration, 301

Kelley space, 295

Kervaire invariant, 243

kill

homotopy class, 110

regular sequence, 110

latching map, 252

latching space, 90, 92, 94, 251

of twisted smash product, 91

left lifting property, 289

level

of a sequential spectrum, 20


level cofibration, 93

level model structure

flat, 261

projective, 260

tight, 260

limit

derived, 76

in T, 296

of symmetric ring spectra, 32


linear isometries operad, 266

linearization

of a space, 15

of a spectrum, 214

localization

Bousfield, see Bousfield localization

of a spectrum, 234

of an abelian group, 233

smashing, 239

localizing subcategory, 202

long exact sequence

of naive homotopy groups, 25

of true homotopy groups, 69

loop space, 6

loop spectrum, 20, 33, 122

M-module, 116

tame, 116

map, 38

mapping cone, 22, 24, 68, 183

mapping space, 38

mapping telescope, 28, 218

reduced, 298

matrix ring spectrum, 41, 107

Milnor element, 223

Milnor sequence, 204

model category

cofibrantly generated, 290

model structure

creating, 291

for G-spaces/G-simplicial sets

mixed, 303

strong, 303

weak, 303

for R-modules, 282

for simplicial sets, 301

for spaces, 298

for symmetric spectra

level, 254, 256

stable, 262

global, 257

module

over a symmetric ring spectrum, 10, 281

module spectrum, 281

monoid ring

spherical, see spherical monoid ring

monoid ring spectrum, 40, 107

Moore spectrum, 223, 234, 236

mod-p, 199, 218, 227

mod-2, 227

Morava K-theory

connective, 116

periodic, 116

morphism

of module spectra, 10

of sequential spectra, 20


of symmetric spectra, 9, 246, 257

of triangles, 181

multiplication

in homotopy of symmetric ring spectrum, 103

in naive homotopy of symmetric ring spectrum, 166

in stable stems, 13

n-connected, 207

naive homotopy group, 11, 20

of a loop spectrum, 122

of a product, 28, 123

of a shift, 33

of a suspended spectrum, 21, 122

of a symmetric ring spectrum, 166

of a wedge, 27

nerve

of a category, 299

Ω-spectrum, 16

positive, 20, 144

operad, 263

A∞, see A∞ operad

E∞, see E∞ operad

associative, see associative operad

Barratt-Eccles, see Barratt-Eccles operad

commutative, see commutative operad

Dold, see Dold operad

injection, see injection operad

linear isometries, see linear isometries operad

of categories, 265

of sets, 265


of spaces, 265


surjection, see surjection operad

orthogonal ring spectrum, 137

INDEX 319

orthogonal spectrum, 137

coordinate free, 159

free, 138

partially ordered set

filtered, 296

periodic self-map, 217

Pontryagin dual, 206

Postnikov section

of a module spectrum, 285

of a spectrum, 221

of an O-algebra, 277

product

on naive homotopy groups, 103

on true homotopy groups, 100

pushout product, 255

realization

of a simplicial spectrum, 71

regular ideal, 111

regular sequence, 110

restriction of scalars, 284

ring spectrum

orthogonal, see orthogonal ring spectrum

symmetric, see symmetric ring spectrum

up to homotopy, see homotopy ring spectrum

roots of unity, 43

S·-construction, 44

S-duality, see Spanier-Whitehead duality

S-module, 143

semifree symmetric spectrum, 35, 59, 71, 93, 122

semistable symmetric spectrum, 60, 126, 130

sequential spectrum, 20

Serre fibration, 298

shift, 33, 123, 183

of free spectrum, 38, 85

of semifree spectrum, 38, 85

of twisted smash product, 83

shift adjoint, 34

shuffle, 37

simplex, 299

standard, 299

topological, 300

simplex category, 299

simplicial functor, 142

simplicial object

augmented, 302

simplicial set, 299

simplicial spectrum, 71

singular chain complex


singular complex

of a space, 301

of a symmetric spectrum of simplicial sets, 30

skeleton


skeleton filtration, 89

small object argument, 290

smash product

derived, 191

of R-modules, 282

of a space and symmetric spectrum, 32

of an I-space and symmetric spectrum, 149



twisted, 36, 121

with semifree symmetric spectrum, 81

Spanier-Whitehead category, 228

Spanier-Whitehead duality, 233

special Γ-space, 144

spectral sequence

for homotopy colimit, 76

for homotopy limit, 76

for true homotopy groups, 133

naive-to-true, 133

spectrum

finite, 230

orthogonal, see orthogonal spectrum

rational, 235

sequential, see sequential spectrum

symmetric, see symmetric spectrum

unitary, see unitary spectrum

spectrum cohomology, 213

spectrum homology, 213, 234

sphere, 5

sphere spectrum, 12, 236

truncated, 92, 94, 162

spherical monoid ring, 40, 142

stabilization map, 11, 20

stable equivalence, 49, 262

stable homotopy category, 171

rational, 235

universal property, 172, 245

stable homotopy group

of a space, 15

of spheres, 12

stable stem, 12

standard resolution, 132

Steenrod algebra

dual, 239

surjection operad, 266

suspension, 21, 122


suspension homomorphism

for naive homotopy groups, 21

suspension spectrum, 15

suspensions, 33

symmetric algebra, 269


symmetric ring spectrum, 9

commutative, 10

implicit, 87

opposite, 107

symmetric sequence, 161

symmetric spectrum, 9

co-free, 46

co-semifree, 46

coordinate free, 158

flat, 93, 129, 130

free, 34, 59, 71, 122

in C, 257

injective, 46

320 INDEX

of A-modules, 246of simplicial sets, 30projective, 252semifree, 35, 59, 71, 93, 122semistable, 60, 126, 130truncated, 89

t-structure, 210tame, 116tensor algebra, 269

of a symmetric spectrum, 88Thom spectrum

oriented, 17special unitary, 17spin, 17symplectic, 17unitary, 17unoriented, 16

Toda bracket, 107, 111, 163in KO, 108in stable stems, 108in topological K-theory, 109in triangulated categoies, 286

topological Andre-Quillen cohomology, 223topological Hochschild cohomology, 223transfinite composition, 290triangle, 181triangulated category, 181

compactly generated, 206true homotopy group, 61, 160

of a free spectrum, 71of a loop spectrum, 66of a product, 68of a semifree spectrum, 71of a shift, 70of a shift adjoint, 67of a suspended spectrum, 66of a symmetric ring spectrum, 103of a wedge, 68

truncationof a symmetric spectrum, 89, 163

unit maps, 9, 11unitary spectrum, 140unitary Thom spectrum, 13universal property

of smash product, 77

v1-multiplication, 239v1-periodic, 239

weak equivalenceof simplicial sets, 301of spaces, 298

weak Hausdorff space, 295Whitehead theorem

for spectra, 216

Yoneda lemmaenriched, 306

Solutions to selected exercises

Chapter I

Exercise 8.1:

Exercise 8.2: If Σn+2 acts trivially on Xn+2, then the double structure map σ2 : Xn∧S2 −→ Xn+2 factorsover the quotient map Xn ∧ S2 −→ Xn ∧ (S2/Σ2) and so the iterated stabilization map ι2 : πk+nXn −→πk+n+2Xn+2 factors through πk+n+2(Xn ∧ (S2/Σ2)). The orbit space S2/Σ2 is contractible, so the doublestabilitation map is trivial. If this happens infinitely often, then the colimit πkX is trivial.

If infinitely many of the alternating groups act trivially on the levels of X, then X is semistable andevery naive homotopy group πkX is uniquely divisible (i.e., a Q-vector space).

Exercise 8.3:

Exercise 8.4: This should go as follows: the isomorphism

πk(|A[∆[n]/∂∆[n]]|, 0) ∼= Hk(∆[n], ∂∆[n];A)

between the homotopy groups of KAn and the relative homology groups of the pair (∆[n], ∂∆[n]) showsthat the geometric realization of KAn is an Eilenberg-Mac Lane space of type (A,n). However, the adjointstructure map σn : KAn −→ Ω(KAn+1) is multiplication by n+1 on πn. In particular, the naive homotopygroup πkKA is trivial for k 6= 0, and KA is an Ω-spectrum if an only if multiplication by n is invertible onA for all n, i.e., if and only if A is uniquely divisible.

The naive homotopy group π0KA is isomorphic to the colimit of the sequence

A·1−→ A

·2−→ A·3−→ A

·4−→ A · · ·and thus isomorphic to the rationalisation Q⊗A.

Exercise 8.5:

Exercise 8.7:

Exercise 8.8:

Exercise 8.9: The associativity condition in particular shows that this is in fact an associative action. Wedefine the structure map σn : Xn ∧ S1 −→ Xn+1 as the map

ι∗ : Xn ∧ S1 ∼= Xn ∧ Sn+1 −→ Xn+1

where ι : n −→ n + 1 is the inclusion, and the homeomorphism Sn+1 ∼= S1 arises from the linearisomorphism Rn+1 ∼= R1 respecting the preferred bases.

The associativity condition implies that the iterated structure map σm : Xn ∧ Sm −→ Xn+m equalsthe composite

Xn ∧ Sm ∼= Xn ∧ Sn+1,...,n+m ιm∗−−→ Xn+m

where ιm : n −→ n + m is the inclusion. The equivariance property is seen as follows: for γ ∈ Σn we haveιm γ = (γ × 1) ιm, and associativity for this injection n −→ n + m amounts to Σn × 1-equivariance.For τ ∈ Σm we have ιm = (1 × τ) ιm, and associativity for this injection n −→ n + m amounts to1× Σm-equivariance.

Let X be a symmetric spectrum. For each finite set A, choose a bijection κA : A −→ n where n = |A| isthe cardinality of A, insisting that κn is the identity. Define XA = Xn. For each injective map α : A −→ Bof finite sets define a structure map α∗ : XA ∧ SB−α(A) −→ XB as the composite

Xn ∧ SB−α(A) Id∧(γκB)∗−−−−−−−→ Xn ∧ Sm−nσm−n−−−−→ Xm

γ−1∗−−→ Xm

322 INDEX

where n = |A| and m = |B| and γ ∈ Σm is any permutation such that γκBα = ιm−nκA. The firstisomorphism comes from the bijection B − α(A) ∼= m − n = n + 1, . . . ,m given by the restriction ofγκB : B −→m.

Exercise 8.10:

Exercise 8.11:

Exercise 8.12:

Exercise 8.13 We fix a level n. By Proposition 4.13 the set πl(R∞A)n is in natural bijectionwith πl−nA, and similarly for the spectrum B. Since f : A −→ B is a π∗-isomorphism, the map(R∞f)n : (R∞A)n −→ (R∞B)n induces bijections of all homotopy groups with respect to thedistinguished basepoint of (R∞A)n. We claim that (R∞A)n has a natural loop space structure,which implies that (R∞f)n induces bijections of all homotopy groups with respect to anybasepoint of (R∞A)n, and is thus a weak equivalence.

To provide deloopings for the levels of R∞A we let R∞A denote the mapping telescope ofthe sequence of symmetric spectra

shAsh(λA)−−−−−−→ shRA

shR(λA)−−−−−−−→ shR2AshR2(λA)−−−−−−−−→ · · · .

Taking loop spaces commutes with mapping telescope up to weak equivalence [ref], so thenatural map

teln(Rn+1A

)= teln (Ω(shRnA)) −→ Ω(teln shRnA) = Ω(R∞A)

is a level equivalence. The source of this morphism is the mapping telescope of the se-quence (4.12) with initial term omitted, so it maps to R∞A by a level equivalence. Altogetherthis provides a chain of two natural level equivalences between R∞A and the symmetric spec-trum Ω(R∞A).

Exercise 8.14:

Exercise 8.15:

Exercise 8.16:

Exercise 8.17:

Exercise 8.18:

Exercise 8.19:

Exercise 8.20: Note that two consecutive truncated sphere spectra are related by the equationS1 ∧ S[k] = sh(S[1+k]). We construct natural maps

?(k,m) : (FmX)m+k −→ (X ∧ S[k])m+k

for m, k ≥ 0 by induction on m. For m = 0 we have F 0X = Σ∞X0 which gives

(F 0A)k = (Σ∞A)k = A0 ∧ Ski0,k−−→ (A ∧ S[k])k

INDEX 323

where i0,k is a component of the universal bimorphism for the pair (A,S[k]). The pushoutsquare (5.28) of symmetric spectra in level k +m gives a pushout square of Σk+m-spaces

Σ+k+m ∧Σk×Σm Sk ∧ LmA //

Σ+k+m ∧Σk×Σm Sk ∧Am

(Fm−1A)k+m // (FmA)k+m

.

So we can define ?(k,m) by the universal property of the pushout from compatible maps(Fm−1A)k+m and Σ+

k+m ∧Σk×Σm Sk ∧Am.Now we show that for every symmetric spectrum X and numbers m, k ≥ 0 the natural

map (FmX)m+k −→ (X ∧ S[k])m+k is an isomorphism. In particular, m-th latching space LmAis naturally isomorphic to the m-th level of the symmetric spectrum A ∧ S. under this iso-morphism, the latching map νm : LmA −→ Am corresponds to the m-th level of the morphismId∧i : A ∧ S −→ A ∧ S rA−−→∼= A for i : S −→ S the inclusion.

The commutative square of symmetric spectra

Sk .k S //

Sk .k S = GkSk

S[k+1] // S[k]

is a pushout, where .k is the twisted smash product, see Example 3.14. Indeed, in levelsn 6= k both horizontal maps are isomorphisms and in level n = k both vertical maps areisomorphisms. So after smashing with A we obtain a pushout square

Sk .k (S ∧A) //

Sk .k A

S[k+1] ∧A // S[k] ∧A

Evaluating at level k +m we obtain a pushout square of Σk+n-spaces (or simplicial sets)

Σ+k+m ∧Σk×Σm Sk ∧ LmA //

Σ+k+m ∧Σk×Σm Sk ∧Am

(S[k+1] ∧A)k+m // (S[k] ∧A)k+m

.

By induction, the lower left corner is isomorphic to (Fm−1A)m+k.

Exercise 8.21: In levels n 6= m both horizontal maps in the square are isomorphisms, and inlevel m both vertical maps are isomorphisms. So the square is levelwise a pullback, hence apullback in the category of symmetric spectra.

Exercise 8.23: (i) Since π∗KU is 2-periodic and trivial in odd dimensions, any graded π∗KU-module M∗ is isomorphic to (M0⊗π∗KU)⊕(M1⊗π∗KU)[1]. Since homotopy groups take wedgesto direct sums and suspension to shift, it suffices to realize modules of the form A ⊗ π∗KUfor abelian groups A. But this π∗KU-module is realized by SA∧KU where SA is a flat Moorespectrum for the group A.

(ii) We abbreviate the graded ring π∗KO to KO∗. Since the Bott class β ∈ π8KO is a unit,the graded KO∗-modules KO∗/x ·KO∗ and KO∗/xβ ·KO∗ are isomorphic for all x, so we onlyhave to investigate the cyclic modules KO∗/x ·KO∗ for classes x in dimensions 0 through 7.

324 INDEX

For any odd integer n the KO∗-module KO∗/nξ · KO∗ is isomorphic to Z/n ⊗ KO∗ bymultiplication by ξβ−1. So the remaining cases to consider are KO∗/x ·KO∗ for x ∈ η, η2 ∪n · 1, n · ξ | n even.

For each such class x we exhibit a Toda bracket 〈x, y, z〉 which does not contain zero, andthen Proposition 6.25 proves the non-realizability. The brackets

ξ ∈ 〈η, η2, 2〉 ∩ 〈η2, η, 2〉 , kξ ∈ 〈2k, η, η2〉 and kβ ∈ 〈2kξ, η, η2〉

do the job, where the classes on the left do not lie in the respective indeterminacy subgroups.(iii) The following table (and Bott periodicity) summarizes the homotopy groups of KO/2.

Here j : KOn −→ πn(KO/2) is induced by the mapping cone inclusion KO −→ KO/2 and thenotation x refers to an element in πn(KO/2) which maps to x under the connecting mapδ : πn(KO/2) −→ KOn−1.

n 0 1 2 3 4 5 6 7πn(KO/2) Z/2 Z/2 Z/4 Z/2 Z/2 0 0 0generator j(ι) j(η) η η2 η · η2

relations j(η2) = 2η j(ξ) = η · η2

The last line shows only the ‘hidden extensions’, i.e., multiplicative relations which do notfollow directly from the fact that j and δ are KO∗-linear.

The table can be derived as follows. We have a short exact sequence of KO∗-modules

0 −→ Z/2⊗KO∗j−−→ π∗(KO/2) δ−−→ KO∗−12 −→ 0

which shows that π∗(KO/2) is trivial in dimensions congruent to 5,6 and 7 modulo 8, cyclic oforder 2 in dimensions congruent to 0, 1, 3 and 4 modulo 8 and has four elements in dimensionscongruent to 2 modulo 8. Exercise 8.22 (ii) and the Toda bracket η2 ∈ 〈2, η, 2〉 in KO2 showthat the extension

0 −→ KO2j−−→ π2(KO/2) δ−−→ KO1 −→ 0

is non-split and so π2(KO/2) is cyclic of order 4.Most of the η-multiplications in π∗(KO/2) are determined by KO∗-linearity of the maps j

and δ in the short exact sequence above. However, there is one less obvious ‘hidden extension’:Exercise 8.22 (ii) and the Toda bracket 〈2, η2, η〉 show that ·η : π3(KO/2) −→ π4(KO/2) isan isomorphism. For dimensional reasons, the only possible nonzero ξ-multiplications inπ∗(KO/2) start in dimensions divisible by 4. We have ξ · j(1) = j(ξ) = η · η2. Since ξ2 = 4β isdivisible by 4, it follows that the map ·ξ : π4(KO/2) −→ π8(KO/2) is zero.

The homotopy groups of KO/ξ can similarly be determined from the short exact sequenceof KO∗-modules

0 −→ KO∗−2/ξ ·KO∗−6j−−→ π∗(S2 ∧KO/ξ) δ−−→ KO∗−7ξ −→ 0 .

This time there is no room for additive extensions and so the multiplicative structure of KO∗completely determines the additive structure of π∗(KO/ξ) as follows:

n 0 1 2 3 4 5 6 7πn(KO/ξ) Z/4 Z/2 Z/2 0 0 0 Z/2 Z/2

Again most η-multiplications follow from the multiplicative structure in KO∗, except for onehidden extension implied by the bracket 2β ∈ 〈ξ, η2, η〉 which shows that ·η : π7(KO/ξ) −→π8(KO/ξ) is injective. The bracket relation η2β ∈ 〈ξ, η, ξ〉 implies a nonzeo ‘hidden’ ξ-multiplication in π∗(KO/ξ) from dimension 6 to dimension 10. If we shift the table periodicallyby two dimensions we see that the homotopy groups of S2 ∧KO/ξ are isomorphic as gradedKO∗-modules to π∗(KO/2).

Now we construct a stable equivalence of KO-modules from S2 ∧ KO/ξ to KO/2. Werepresent the class η ∈ π2(KO/2) by a KO-morphism f : S2 ∧KO −→ KO/2. Since ξ · η = 0, the

INDEX 325

morphism f can be extended to a morphism f : S2∧KO/ξ −→ KO/2 from the double suspensionof the mapping cone. By construction the class η is in the image of π∗(f), so π∗f is surjectivein dimensions 2, 3 and 4 by KO∗-linearity. The additive structure and η-multiplications thenforce π∗f to be surjective, hence bijective, in all dimensions.

Exercise 8.24: (iii) We let Iω denote the category with objects the sets n for n ≥ 0 and the set ωand with all injective maps as morphisms. So Iω contains I as a full subcategory and containsone more object ω whose endomorphism monoid is M. We will now extend an I-functor F toa functor from the category Iω in such a way that the value of the extension at the object ωis the colimit of F , formed over the subcategory N of inclusions. It will thus be convenient,and suggestive, to denote the colimit of F , formed over the subcategory N of inclusions, byF (ω) and not introduce new notation for the extended functor. The M-action on the colimitof F is then the action of the endomorphisms of ω in Iω on F (ω).

So we set F (ω) = colimN F and first define β∗ : F (n) −→ F (ω) for every injection β : n −→ ωas follows. We set m = maxβ(n), denote by β|n : n −→ m the restriction of β and take β∗(x)to be the class in the colimit represented by the image of x under

(β|n)∗ : F (n) −→ F (m) .

It is straightforward to check that this is a functorial extension of F , i.e., for every morphismα : k −→ n in I we have (βα)∗(x) = β∗(α∗(x)).

Now we let f : ω −→ ω be an injective self-map of ω, and we want to define f∗ : F (ω) −→ F (ω).If [x] ∈ F (ω) is an element in the colimit represented by a class x ∈ F (n), then we set f∗[x] =[(f |n)∗(x)] where f |n : n −→ ω is the restriction of f and f∗ : F (n) −→ F (ω) was defined in theprevious paragraph. Again it is straightforward to check that this definition does not dependon the representative x of the class [x] in the colimit and that the extension is functorial, i.e.,we have (fα)∗(x) = f∗(α∗(x)) for injections α : n −→ ω as well as (fg)∗[x] = f∗(g∗[x]) when g isanother injective self-map of ω. As an example, if we also write ι : n −→ ω for the inclusion,then we have ι∗(x) = [x] for x ∈ F (n).

The definition just given is in fact the universal way to extend an I-functor F to a functoron the category Iω, i.e., we have just constructed a left Kan extension of F : I −→ Ab alongthe inclusion I −→ Iω. However, we do not need this fact, so we omit the proof.

A trivial but important observation straight from the definition is that the action of theinjection monoid M on the colimit of any I-functor F is tame in the sense of Definition 7.1:every element in the colimit F (ω) is represented by a class x ∈ F (n) for some n ≥ 0; then forevery element f ∈M which fixes the numbers 1, . . . , n, we have f · [x] = [x].

Exercise 8.25: To define the I-functor W (•) on morphisms α : n −→ m in the category I wechoose any extension α : ω −→ ω of α and define α∗ : W (n) −→ W (m) as the restriction ofα· : W −→ W . This really has image in W (m) by part (ii) of Lemma 7.7 and is independent ofthe extension by (i) of that lemma. The rest is immediate.

Exercise 8.26: Smashing with the identity of S1 from the left provides a map S1 ∧− : πsmK −→

πs1+m(S1 ∧K) which is a special case of the suspension isomorphism (compare Proposition ??)

for the suspension spectrum of K.For a symmetric spectrum we can then define an I-functor πs

kX by setting (πskX)(n) =

πsk+nXn on objects (with no restriction on k+n). We let the inclusion ι : n −→ n + 1 act as the

composite

πsk+nXn

S1∧−−−−−→ πs1+k+n(S

1 ∧Xn)(−1)k+ntwist−−−−−−−−−→ πs

k+n+1(Xn ∧ S1)πsk+n+1σn−−−−−−→ πs

k+n+1Xn+1 .

[check that 1 × Σm acts trivially on the image of ιm.] and defining the action of a morphismn −→m in the same way as for the I-functor πkX of unstable homotopy groups.

326 INDEX

The map from the initial term to the colimit of the sequence (??) provides a naturaltransformation πmK −→ πs

mK which is compatible with stabilization [check], so it defines amorphism of I-functors πkX −→ πs

kX for every symmetric spectrum X. The induced map oncolimits

colimN πkX∼=−−→ colimN πs

kX

is bijective and thus an isomorphism of M-modules.

Exercise 8.27:

Exercise 8.28:

Exercise 8.29:

Exercise 8.30: We show that the collection of functorsF 7→ TorZ[M]

p (Z, colimN F )p≥0

has all the properties that characterize the left derived functors of the colimit. Clearly

colimI F ∼= Z⊗M (colimN F ) ,

so the functors agree for p = 0. Taking colimit over the filtered category N is exact, so theTor functors of the colimit take short exact sequences of I-functors to long exact sequences ofabelian groups. The least obvious part is that the Tor groups TorZ[M]

p (Z, colimN F ) vanish for allprojective functors F and all p ≥ 1. The I-functors Z[I(n,−)] arising as the linearizations of therepresentable functors form a set of projective generators for the category of I-functors, so itsuffices to show the vanishing of higher Tor groups for these. But the colimit over N of theI-functor Z[I(n,−)] is precisely the M-module Pn by Exercise 8.24 (iv); so Proposition 7.37 (ii)provides the vanishing result and finishes the proof.

Exercise 8.31:

Exercise 8.32: (ii) It is clear that the map is well-defined and a homomorphism of M-Mn+m-bisets. To define a map in the other direction we choose two bijections sn ∈ M(n) andsm ∈M(m) and define a bijection sn + sm : N× 1, . . . , n+m −→ N× 2 by

(sn + sm)(i, j) =

sn(i, j) for j = 1, . . . , n,sm(i, j − n) for j = n+ 1, . . . , n+m.

Then we define b :M(n+m) −→M(2)×M(1)2 (M(n)×M(m)) by

b(κ) = (κ(sn + sm)−1)(sn, sm)

and get a(b(κ)) = κ and

b(a(ϕ(ψ, λ)) = (ϕ(ψ + λ)(sn + sm)−1)(sn, sm) = (ϕ(ψs−1n , λs−1

m ))(sn, sm)

= ϕ((ψs−1n )sn, (λs−1

m )sm) = ϕ(ψ, λ) .

(ii’) Compare [24, Ch. I, Lemma 5.5].(iii) Hint: the corresponding property for the linear isometries operad is [24, Ch. I,

Lemma 8.1].

Exercise 8.33: We define a morphism in the other direction by

(.mF )(ω) −→ Z[M]〈m〉 ⊗Σm×M F (ω) , [γ ⊗ x] 7−→ γ ⊗ [x]

where x ∈ F (n) and γ ∈ Σm+n. The main point is to check that the formulas for both maps areactually well-defined, i.e., they indeed factor over the tensor product over Σm×M respectivelyover Σm × Σn and are independent of the representative x respectively [γ ⊗ x] in the colimit

INDEX 327

F (ω) respectively (.mF )(ω). We omit these routine verifications, and after that it is clear thatboth maps are M-linear and inverse to each other.

Exercise 8.35:

Exercise 8.36:

Chapter II

Chapter III

Date post:	17-Jul-2020
Category:	Documents
Upload:	others
View:	2 times
Download:	0 times

Symmetric spectra - kuweb.math.ku.dk/~larsh/teaching/restricted/SymSpec.pdf · over structured ring...

Documents