Fei Xu - Representations of Categories

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Fei Xu

Representations and Cohomologyof Finite Categories (DRAFT)

Category Algebras & Simplicial Methods

June 23, 2011





Preface

These are expanded lecture notes that I used for a short course of the sametitle at Universitat Autonoma de Barcelona in 2010-2011 academic year. Thesubtitle reveals the main tools we are using. It means that we shall study

representations and cohomology of category algebras via simplicial modules.I assume the reader to have good background on homological algebra fromclassical books such as “A Course in Homological Algebra” by Hilton andStammbach. The main theme of these notes is to answer the question: what we may do with an abstract finite category? Here we regard a finite category asa generalization of an abstract finite group and of an abstract finite partiallyordered set. Starting from a finite group and a base ring, there are welldeveloped group representation theory and group cohomology theory. Paralleltheories have been introduced to finite partially ordered sets as well. Thisbook presents a theory that extending all of the above.

Category cohomology theory is a place where simplicial methods, homo-topy theory and representation theory naturally meet. The objectives of ourtheory may be summarized in the following diagram.

C classifying space

nerve

category algebra and modules

functor category

BC

singular cohomology

N C geo.

realiz.

simplicial cohomology

bar resolution

of the trivial RC−module R RC−mod

cohomology of modules

(R−mod)C

functor cohomology

H∗(C ; N ) ∼= lim←−∗

CN ∼= Ext∗RC(R, N )

Ext∗RC(M, N )

In the diagram, C is a finite category, N C is its nerve, a combinatorialconstruction, and BC is the geometric realization of N C (or C ) which is a CW-

v



vi Preface

complex. Also R is a commutative ring with identity and M , N ∈ (R-mod)C

are two covariant functors. (A functor in (R-mod)C should be considered as adiagram of R-modules.) In this picture, RC is the so-called category algebra .It was shown by B. Mitchell [56] that there exists an equivalence between

(R-mod)C

and RC -mod (the category of left RC -modules). Thus a functorfrom C to R-mod is always an RC -module. Notably when C is a group, thetheory becomes the usual group cohomology theory. When C is a transportercategory (see Chapter 6) defined over a group G, we recover the equivariantcohomology theory.

We can similarly study category homology by replacing the last two rowsin the picture by H∗(C ; N ) ∼= lim−→

∗

CN ∼= TorRC

∗ (R, N ) and TorRC∗ (M , N ) (M

is a right RC -module). In these notes we shall focus on cohomology theorybecause homology theory can be developed parallel to it.

Much of the general theory is indeed established for all small categories .To make this book as useful as we can, we shall state results in their generalforms, for small, not just finite, categories whenever it is the case. It is perhapsa good point to explain why small categories are of particular interests. In

fact this is a set-theoretic issue. In the above diagram, if we consider a smallcategory C then we can still construct a simplicial set N C , a topologicalspace BC , an algebra RC and a functor category (R-mod)C . In all theseconstructions, explicitly defined in the main text, the class of morphisms inC , Mor C , has to be a set, which is precisely the smallness condition on C . Itlies in the applications of category cohomology and representations that wehave in mind that C is frequently finite (Mor C is a finite set). The chief reasonis that we understand quite well the representations of RC , which becomesan associative algebra with identity and is of finite R-rank.

There exists another functor (co)homology theory [25], used in Steenrodalgebras and cohomology of finite group schemes. In that (co)homology the-ory, one mainly investigates representations and cohomology of the concrete(essentially small but infinite) category of finite-dimensional vector spacesover a field. The relationship between these two theories is comparable tothat between the cohomology theories of abstract finite groups and of gen-eral linear groups. They start off the same foundation but are of differentflavors and use quite different methods, even though they supply importantideas and results to each another. Thus although there are something in com-mon between [25] and this book, the main bulks of these two are differentand to some extent complementary to each other.

The ingredients shown in the previous diagram have been studied sincethe early stage of homological algebra in one way or another. In this bookwe collect many existing materials to form a source for self-learning as wellas a reference for researchers. The major inputs from the author are firstlyto provide a systematic introduction to the uses of simplicial methods, and

secondly to develop tools for comparing cohomology of two small categoriesconnected by a functor. Simplicial methods help us to construct some (com-plexes of) modules combinatorially while the tools we have mentioned tell us





viii Preface

G : D → T op is a functor, we can construct two functors C → T op. Thesetwo functors are called the left and right Kan extensions of G along u. It isintuitive if we record topologists’ notations

C ⊗D G and HomD(C ,G).

These two functors C ⊗D −, HomD(C , −) are the left and right adjoints of Resu. In Hollender-Vogt [37], one can see that these two functors enjoy manyproperties that an algebraist would expect for similar functors, induced by aring homomorphism R1 → R2, between R1-mod and R2-mod. In fact in [37],a covariant functor in T opC is called a left C -module while a contravariantfunctor is called a right C -module. I would like to point out here that since afunctor u : D → C does not lead to an algebra homomorphism from RD to RC ,one has to be very careful when constructing RC ⊗RD− and HomRD(RC , −).This is the reason why I refrain from using topologists’ notations of theKan extensions. Plus the intuitive notations do not seems to be convenientin computations as they are in other places such as group (co)homology.

However the Kan extensions are truly generalizations of well known functors.When i : H → G is the inclusion functor at the end of last paragraph, the leftand right Kan extensions along i are isomorphic to the induction RG ⊗RH −and co-induction HomRH (RG, −) (which happen to be isomorphic to eachother).

Let us now recall the history of (co)homology theory of small categoriesand related works. Homology theory of small categories, H∗(C ; N ), can befound in Gabriel-Zisman [28] (1967), as a special case of homology of sim-plicial sets with coefficients, in the appendices. There they also showed

H∗(C ; N ) ∼= lim−→∗

CN , where lim−→

∗

C are the derived functors of direct limit func-

tor lim−→C. By contrast, although the special case of partially ordered sets was

studied by J. E. Roos [66] (1961), cohomology of small categories began withBaues-Wirsching’s theory [3] (1985). They actually studied H∗(C ; N ), where

N is not a functor from C . Instead, it is a functor from another category F (C ),the category of factorizations in C . In their situation cohomology theory of small category can also be introduced by lim←−

∗

C, derived functors of inverse

limit functor lim←−C. Simplicial methods are used to define various chain or

cochain complexes whose (co)homology is the (co)homology of C .In these theories the coefficients of (co)homology are provide by functors.

Thus analyzing structures of functors should be of great importance. A sig-nificant achievement in this direction was obtained by Luck [51] and tomDieck [?] (1987), where they classified simple and projective functors underreasonable assumptions on the small category. An influential approach tofunctor categories interestingly began with P. Gabriel’s thesis [27] (1962).Given a small category, he made it into an additive category by linearizingall morphism sets. Then he showed that the category of addictive functorsfrom such an additive category to a module category is equivalent to anothermodule category, of modules over an algebra that now we may call the ad-



Preface ix

ditive category algebra of a small category. Later on B. Mitchell [56] (1972)introduced a different construction and defined an associative ring over asmall category. Consequently he established connections between (R-mod)C

and RC -mod. This is the result we pictured in the diagram.

At this point, one perhaps does not see a connection between Mitchell’swork and category (co)homology. These two seemingly unrelated subjectswere pieced up together by P. J. Webb (around 2000) who named these ringsappeared in Mitchell’s paper, as “category algebras”. Representation theoryof categories generalizes both representation theory of groups and of quivers,and thus is of great interest. Representations of categories are important tocategory (co)homology in the same way as group representations to group(co)homology. From category algebra point of view, we can put all necessaryingredients of category (co)homology under one framework, as shown in thediagram. Furthermore from the intrinsic structure of a category algebra, onecan see the similarities with and differences from a groups algebra. Hence itexplains why some classical results in group (co)homology may be generalizedto category (co)homology, while others may not.

At the early stage of category (co)homology, it seemed like a purely theo-retic construction with few calculations. The thrust of recent development wasbrought in by representation and homotopy theorist working on locally deter-mined structures. Their work completely reshaped the whole (co)homologytheory and provided many interesting concrete categories to work with. Weshall comment on it in Chapter 6. In the last decade many interesting re-sults on both abstract and concrete small categories have been obtained.For instance, a pivotal discovery is that the category of factorizations, F (C ),first used by Quillen to show homotopy equivalence between BC and BC op,and then by Baues-Wirsching to introduce their cohomology theory, pos-sesses the property that all (co)homology theories we consider here are indeed(co)homology of F (C ) with appropriate coefficients. However when one triesto teach oneself about category (co)homology, one finds materials scattered

in the literature and different writers have different background and styles.As an example the existing introduction to this subject by Webb [80] empha-sizes its module-theoretic aspects. Lack of a comprehensive treatment makesthe theory daunting for whoever wants to learn, and even for a researcherwho uses category (co)homology theory it may cause inconvenience. Thusa book, introducing basic ideas of category (co)homology theory, presentingstandard techniques and addressing the interactions between representationand homotopy theories, seems necessary. Such a book should provide a clearview of basic ideas, key methods, known results and unsolved conjectures,being a handy introduction that can be used to foster further investigationsand to search for future applications.

Finally we turn to the structure of this book. The first two chapters con-sists of preliminaries needed in category (co)homology. It means one can findthem in various classical books. The reason why I collect them here are,firstly it is convenient for the reader, secondly I try to provide some concrete



x Preface

examples to illustrate many abstract constructions which are in the center of this book. The first chapter recalls some basic definitions from category the-ory. The main focuses are limits of functors and their generalizations, thatis, the Kan extensions. The purpose of the second chapter is to equip the

reader with necessary knowledge about simplicial methods. We begin with areview of chain complexes. It is followed by an introduction to simplicial setsand the nerve of a small category where many combinatorially constructedchain complexes appear. Simplicial (co)homology is defined and several ex-amples are given. To tell how to compare (co)homology of small categories, wehave to inform the reader how to compare small categories and their nervesas well as classifying spaces. Thus some important categorical constructionsare provided, which are needed throughout this book, for example to stateQuillen’s Theorem A. For future references, and for the interested reader, weend Chapter 2 with bisimplicial sets and several key results. Although onlythe statements will be used in future, we nonetheless present their proofs.

The third chapter introduces category algebras and their representations.Examples are served at the beginning to motivate the reader. We shall classify

projective and injective modules under mild assumptions. Moreover we givean intrinsic characterization of category algebras so that they are comparablewith cocommutative bialgebras. It explains why category algebras possessinteresting homological properties.

The fourth chapter studies (ordinary) (co)homology of category algebras.We begin with the most economical way by using derived functors to de-fine category (co)homology. Then we recall Baues-Wirsching’s constructionon the way to introduce the bar resolution. The bar resolution is simpli-cially constructed and it leads to various important modules by applyingKan extensions on it. We will see they are the most powerful tools for us. Inthis chapter we also define the extension of a category by a group and theGrothendieck spectral sequences.

The fifth chapter discusses Hochschild (co)homology of category algebras.

The key result here is a theorem to interpret Hochschild (co)homology of category algebras by their ordinary (co)homology, and vice versa. A theoremwe prove here shows that all the previously mentioned (co)homology theoriesare just (co)homology of F (C ) with coefficients. Some examples are given todemonstrate explicit calculations.

The sixth chapter talks about connections between category and groupcohomology. It contains mostly unpublished results. We bring up the notionof a local category of a finite group. Local categories are the motivating casesfor research in category representations and cohomology. We put it in theend because we do need techniques developed earlier. This chapter containsmany concrete categories and we shall see clearly how one can compute usingvarious abstract machineries introduced before. A key concept in this chapteris a transporter category. We show how closely related are the transportercategories to the groups on which they are defined. The most important resultis perhaps the finite generation of cohomology of modules of finite transporter



Preface xi

categories. Also we will construct transfer maps for ordinary and Hochschildcohomology of transporter categories. This chapter should help the reader tocarry on further readings in advanced research papers.

There may be many errors or even mistakes in this unfinished manuscript,

all of which are my responsibility.

Universitat Autonoma de Barcelona, Fei Xu June 2011 [email protected]





Contents

1 Functors and their Kan extensions . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Functors and limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1.1 Basic category theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1.2 Adjoint functors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.1.3 Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.2 Restriction and Kan extensions. . . . . . . . . . . . . . . . . . . . . . . . . . . 151.2.1 Restriction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151.2.2 Overcategories and undercategories . . . . . . . . . . . . . . . . . 161.2.3 Kan extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2 Simplicial methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232.1 Complexes and homology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.1.1 Chain complexes, homology and chain homotopy . . . . . 232.1.2 Double complexes and operations on chain complexes . 26

2.2 Nerves, classifying spaces and cohomology . . . . . . . . . . . . . . . . . 292.2.1 Simplicial sets and nerves of small categories . . . . . . . . . 29

2.2.2 Classifying spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 422.2.3 Cup product and cohomology ring . . . . . . . . . . . . . . . . . . 44

2.3 Quillen’s work on classifying spaces . . . . . . . . . . . . . . . . . . . . . . . 482.3.1 Quillen’s Theorem A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 492.3.2 Constructions over categories and relevant functors . . . 50

2.4 Further categorical and simplicial constructions . . . . . . . . . . . . 532.4.1 Grothendieck constructions . . . . . . . . . . . . . . . . . . . . . . . . 542.4.2 Bisimplicial sets and homotopy colimits . . . . . . . . . . . . . 572.4.3 Proofs of Quillen’s Theorem A and Thomason’s

theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

3 Category algebras and their representations . . . . . . . . . . . . . . 653.1 Basic concepts and examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

3.1.1 Category algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 653.1.2 Representations of categories and Mitchell’s Theorem . 66

xiii



xiv Contents

3.1.3 Three examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 683.2 A closed symmetric monoidal category . . . . . . . . . . . . . . . . . . . . 70

3.2.1 Tensor structure and an intrinsic characterization of category algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

3.2.2 The internal hom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 753.3 Functors between module categories . . . . . . . . . . . . . . . . . . . . . . 76

3.3.1 Restriction on algebras and modules . . . . . . . . . . . . . . . . 763.3.2 Kan extensions of modules . . . . . . . . . . . . . . . . . . . . . . . . 783.3.3 Dual modules and Kan extensions . . . . . . . . . . . . . . . . . . 82

3.4 EI categories, projectives and simples . . . . . . . . . . . . . . . . . . . . . 833.4.1 EI condition and its implications . . . . . . . . . . . . . . . . . . . 843.4.2 Some representation theory . . . . . . . . . . . . . . . . . . . . . . . . 853.4.3 Classifications of projectives and simples . . . . . . . . . . . . 893.4.4 Projective covers, injective hulls and their restrictions . 93

4 Cohomology of categories and modules . . . . . . . . . . . . . . . . . . . 954.1 General theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

4.1.1 Cohomology of modules . . . . . . . . . . . . . . . . . . . . . . . . . . . 954.1.2 Cohomology of a small category with coefficients in a

functor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1004.1.3 Extensions of categories and low dimension cohomology107

4.2 Classical methods for computation . . . . . . . . . . . . . . . . . . . . . . . . 1124.2.1 Minimal resolutions and reduction . . . . . . . . . . . . . . . . . . 1124.2.2 Examples using classifying spaces. . . . . . . . . . . . . . . . . . . 114

4.3 Computation via adjoint functors . . . . . . . . . . . . . . . . . . . . . . . . . 1164.3.1 Adjoint restrictions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1164.3.2 Kan extensions of resolutions . . . . . . . . . . . . . . . . . . . . . . 117

4.4 Grothendieck spectral sequences . . . . . . . . . . . . . . . . . . . . . . . . . . 1224.4.1 Grothendieck spectral sequences for a functor . . . . . . . . 1224.4.2 Spectral sequences of category extensions . . . . . . . . . . . . 126

5 Hochschild cohomology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1315.1 Hochschild homology and cohomology . . . . . . . . . . . . . . . . . . . . . 131

5.1.1 Definition and general properties . . . . . . . . . . . . . . . . . . . 1315.1.2 Ring homomorphisms from the Hochschild

cohomology ring. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1355.2 Hochschild (co)homology of category algebras . . . . . . . . . . . . . . 137

5.2.1 Basic ideas and examples . . . . . . . . . . . . . . . . . . . . . . . . . . 1375.2.2 Hochschild (co)homology as ordinary (co)homology . . . 1385.2.3 EI categories. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

5.3 Examples of the Hochschild cohomology rings of categories . . 1485.3.1 The category E 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1485.3.2 The category E 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150

5.3.3 The category E 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1525.3.4 The category E 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153



Contents xv

6 Connections with group representations and cohomology . 1556.1 Local categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

6.1.1 G-categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1566.1.2 Homology representations of kG . . . . . . . . . . . . . . . . . . . . 157

6.1.3 Transporter categories as Grothendieck constructions . 1586.1.4 Local categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160

6.2 Properties of local categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1616.2.1 Two diagrams of categories . . . . . . . . . . . . . . . . . . . . . . . . 1616.2.2 Frobenius Reciprocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162

6.3 The functor π: group representations via transporter categories1646.3.1 Homology representations via transporter categories . . 1646.3.2 On finite generation of cohomology . . . . . . . . . . . . . . . . . 1696.3.3 Transfer for ordinary cohomology . . . . . . . . . . . . . . . . . . . 173

6.4 The functor ρ: invariants and coinvariants . . . . . . . . . . . . . . . . . 1796.4.1 Orbit categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1806.4.2 Brauer categories, fusion and linking systems . . . . . . . . 1826.4.3 Puig categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184

6.4.4 Orbit categories of fusion systems . . . . . . . . . . . . . . . . . . 1846.5 Hochschild cohomology. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185

6.5.1 Finite generation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1856.5.2 Transfer for Hochschild cohomology. . . . . . . . . . . . . . . . . 188

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199







xviii List of symbols

G ∝ P transporter category over a G-poset P (a Grothendieck construc-tion)

R a commutative ring with identityR-Mod the category of all R-modules

R-mod the category of finitely generated R-modulesk a field, usually algebraically closedV ectk the category of all k-vector spacesV ectk the category of finite-dimensional k-vector spacesRC the R-category algebra of C R the trivial RC -module, a constant functor

Hn(C , R), Hn(C , R) the nth simplicial (co)homology of C with coefficientsin R

Hn(C ; M ), Hn(C ; M ) the nth (co)homology of C with coefficients in a func-tor/module M

Hn(BC , R), Hn(BC , R) the nth singular (co)homology of BC with coeffi-cients in R

H∗(C , R) the simplicial cohomology ring

H∗(BC , R) the singular cohomology ringExt∗RC(R, R) the ordinary cohomology ring of RC HH∗(RC ) = Ext∗RCe(RC , RC ) the Hochschild cohomology ring of RC





2 1 Functors and their Kan extensions

If α ∈ HomC(x, y) is a morphism in C , very often we will picture it asα : x → y. A morphism α ∈ Mor C is a monomorphism if αβ = αγ fortwo morphisms β, γ ∈ Mor C , then β = γ . A morphism α ∈ Mor C is anepimorphism if βα = γα for two morphisms β, γ ∈ Mor C , then β = γ . A

morphism α ∈ HomC(x, y) is an isomorphism if there exists a morphism β ∈HomC(y, x) such that αβ = 1y and βα = 1x. An isomorphism is always bothmonomorphic and epimorphic. An isomorphism is also called an invertible morphism.

We say two objects x and y are isomorphic , written as x ∼= y, if thereexists an isomorphism α ∈ HomC(x, y). Let x be an object in a category C .Then the isomorphism class of x consists of all y ∈ Ob C that are isomorphicto x. The class of objects isomorphic to x is denoted by [x] ⊂ Ob C .

A category is called a groupoid if every morphism is an isomorphism.An object x in C is initial if to any other object y, there exists a unique

morphism x → y. An object x in C is terminal if to any other object y, thereexists a unique morphism y → x. An object is a zero object if it is both initialand terminal. All initial (or terminal or zero) objects are isomorphic.

Suppose C has a zero object 0, then for any two objects x, y we call thecomposite x → 0 → y the zero morphism in HomC(x, y), written as 0x,y. Notethat the composite of a zero morphism with any other morphism is a zeromorphism. Let α ∈ HomC(x, y) be a morphism. A kernel of α is a morphismβ ∈ HomC(w, x) satisfying αβ = 0w,y such that if β ∈ HomC(w, x) satisfiesαβ = 0w,x then there exists a unique morphism µ : w → w such thatβ = βµ. A cokernel of α is a morphism γ ∈ HomC(y, z) satisfying γ α = 0x,z

such that if γ ∈ HomC(x, z) satisfies γ α = 0x,z then there exists a uniquemorphism ν : z → z such that γ = νγ .

Let I ⊂ Ob C be a subset. Then the coproduct of objects in I , writtenas

I xi, is an object X ∈ Ob C such that each xi ∈ I is equipped with amorphism αi : xi → X and if Y is another object equipped with β i : xi → Y then there exists a unique morphism γ : X → Y such that β i = γαi. The

product of objects in I , written as I xi, is an object X ∈ Ob C such thateach xi ∈ I is equipped with a morphism αi : X → xi and if Y is anotherobject equipped with β i : Y → xi then there exists a unique morphismγ : Y → X such that αi = β iγ .

Definition 1.1.2. A category C is called preadditive if every HomC(x, y) isan abelian group and compositions are bilinear. A preadditive category isadditive if furthermore all finite coproducts and all products exists in C .

An additive category C is said to be preabelian if

1. it has a zero object;2. every morphism has a kernel and a cokernel.

A preabelian category is abelian if every monomorphism is a kernel of some

morphism and every epimorphism is a cokernel of some morphism.There are abundance of categories in mathematics, many of them are very

natural while one can also cook up all kinds of abstract categories.



1.1 Functors and limits 3

Example 1.1.3. 1. The trivial category • has exactly one object • and onemorphism 1•;

2. A group G gives rise to two categories: the first one has only one object• and its morphism set is G; the second, denoted by E G, has Ob E G = G

and Mor E G = g1,g2∈ObE GHomE G(g1, g2) with HomE G(g1, g2) = g2g−11 .

The former will usually be written just as G while the latter is called theCayley graph of G;

3. A partially ordered set (poset in short) P is naturally a category, stillnamed P , if we let Ob P be the set of elements in the poset and Mor P =x,y∈ObCHomC(x, y) in which the set HomP (x, y) = x ≤ y if the twoobjects are comparable, or empty otherwise. Posets can be characterizedas categories such that there exists at most one morphism between anytwo objects.

4. The category of sets and set maps is denoted by Set , ;5. Let R be a commutative ring with identity and A an associative R-algebra.

Then A-Mod and A-mod are the categories of all left A-modules andfinitely generated left A-modules;

6. The category Z-Mod is often written as Ab, the category of abelian groups;7. The category of topological spaces and continuous maps is written as T op.

In these notes, for an R-algebra A, if we do not specify, any A-module willbe a left A-module.

Definition 1.1.4. A category C is a small category if Ob C is a set, and is a finite category if Mor C is a finite set.

A finite category is necessarily small by Definition 1.1.1 (4). In Example1.1.3, the first three are small while the rest are abelian and not small.

Definition 1.1.5. Let C be a category. Then a subcategory D ⊂ C consistsof a subclass of objects Ob D ⊂ Ob C and a subclass of morphisms Mor D ⊂

Mor C , satisfying the axioms of a category with composition laws inheritedfrom C .

A subcategory D ⊂ C if full , if for any pair of objects x, y ∈ Ob D, wealways have HomD(x, y) = HomC(x, y).

Definition 1.1.6. A covariant functor F from a category D to another cat-egory C assigns to each x ∈ Ob D an object F (x) ∈ Ob C and to eachα ∈ HomD(x, y) a morphism F (α) ∈ HomC(F (x),F (y)) satisfying the con-ditions that

1. F (1x) = 1F(x) for every x ∈ Ob D;2. if α, β ∈ Mor D and β α exists, then F (β α) = F (β ) F (α).

A contravariant functor F from a category D to another category C assigns

to each x ∈ Ob D an object F (x) ∈ Ob C and to each α ∈ HomD(x, y) amorphism F (α) ∈ HomC(F (x),F (y)) satisfying the conditions that




1. F (1x) = 1F(x) for every x ∈ Ob D;2. if α, β ∈ Mor C and β α exists, then F (β α) = F (α) F (β ).

Example 1.1.7. Let C be a category and x ∈ Ob C an object. Then

1. there exists a covariant functor HomC(x, −) : C → Set such that for anyy ∈ Ob C , HomC(x, −)(y) := HomC(x, y) and for any α ∈ HomC(y, y) thecomposition HomC(x, −)(α) = α − : HomC(x, y) → HomC(x, y); and

2. there exists a contravariant functor HomC(−, x) : C → Set such that forany y ∈ Ob C , HomC(−, x)(y) := HomC(y, x) and for any α ∈ HomC(y, y)the composition HomC(−, x)(α) = − α : HomC(y, x) → HomC(y, x).

An object x ∈ Ob C is projective if HomC(x, −) : C → Set preservesepimorphisms. An object x ∈ Ob C is injective if HomC(−, x) : C → Set pre-serves monomorphisms. If C is an abelian category, we say that C has enough projective objects if for any y ∈ Ob C there exists a projective object x alongwith an epimorphism x → y and that C has enough injective objects if forany y ∈ Ob C there exists an injective object x along with a monomorphism

y → x.

Definition 1.1.8. Suppose C is a category. Its opposite category , named C op,share the same objects with C . Each α ∈ HomC(x, y) defines a unique mor-phism αop ∈ HomCop(y, x). If αop ∈ HomCop(y, x) and β op ∈ HomCop(z, y),their composite is αop β op := (β α)op ∈ HomCop(z, x).

Passing from a category to its opposite has a dualizing effect on many cat-egorical concepts, constructions and properties. For instance, it interchangesinitial objects with terminal objects, monomorphisms with epimorphisms,projective objects with injective objects, products with coproducts etc. More-over one can readily verify that a covariant functor F : C → D naturallydetermines a contravariant functor, the dual functor , F ∧ : C op → D , and vice

versa. Thus what we learn about covariant functors can be directly translatedto contravariant functors. From now on, if not specified, all functors will becovariant in these notes.

Definition 1.1.9. A functor F : D → C is full if F (D) ⊂ C is a full subcate-gory.

A functor F : D → C is faithful if for any α, β ∈ Mor D such that F (α) =F (β ) ∈ Mor C , then α = β .

If D and C are two preadditive categories, then a functor F : D → C is called additive if HomD(x, y) → HomC(F (x),F (y)) is a homomorphismbetween abelian groups.

Definition 1.1.10. Suppose F ,G : D → C are two functors. A natural transformation Φ : F → G assigns to each object x ∈ Ob D a morphismΦx : F (x) → G(x) so that we have a commutative diagram




F (x) Φx

F(α)

G(x)

G(α)

F (y)

Φy G(y)

for any given α ∈ HomD(x, y). If every Φx is an isomorphism in C , we callsuch Φ a natural equivalence and write F ∼= G in this situation.

Definition 1.1.11. Two categories C and D are equivalent , written as C D, if there exist functors F : C → D and G : D → C such that FG ∼= IdD andGF ∼= IdC.

Definition 1.1.12. Let C be a category. If we take exactly one object fromeach isomorphism class of objects in C , then we can form a full subcategoryconsisting of these chosen objects. This subcategory is called a skeleton of C .

Since skeletons of a category C are naturally equivalent to each other, we

can speak about the skeleton of C . We shall always denote by [C ] the skeletonof a category C . Then one can verify that [C ] C .

It is a fact that when D is small all the covariant functors from D to C forma category whose objects are these functors and morphisms are the naturaltransformations. We call it a functor category , written as C D. Sometimes wecall such a D an index category . Functor categories are of pivotal importancein these notes. We often consider functor categories whose index categoriesare finite (such as a finite group G) and whose target categories are large (suchas R-mod), in order to understand finite categories via their representations(see Chapter 3) in certain large categories.

The functors in Example 1.1.7 are very important to us. Here we presenta crucial property of those functors.

Definition 1.1.13. A covariant (respectively, contravariant) functor F : C →Set is called representable if it is naturally equivalent to HomC(x, −) (respec-tively, HomC(−, x)) for some x ∈ Ob C .

Lemma 1.1.14 (Yoneda Lemma). Let F : C → Set be a functor. Then we have

HomSetC(HomC(x, −),F ) ∼= F (x),

for any x ∈ Ob C .

Proof. Each natural transformation Φ in the left side is uniquely determinedby the morphism

Φx : HomC(x, x) → F (x)

which is uniquely determined by the image of Φx(1x) ∈ F (x). Hence we can

define a bijection Φ → Φx(1x) and then the statement follows.

In the end, we record a construction that we will use in the next section.






Ω (β ϕ) = G(β ) Ω (ϕ),

and when y = F (x) and ϕ = 1F(x), we get

Ω (β F (α)) = G(β ) α.

One should try to deduce similar formulas for future applications.2. Unit and counit : we define, for each x ∈ Ob C , Σ x = Ω (1F(x)) : x → GF (x)

and, for each y ∈ Ob D, Λy = Ω −1(1G(y)) : FG(y) → y . By naturality of Ω , we actually obtain two natural transformations

Σ : IdC → GF and Λ : FG → IdD.

For example, for any α : x → x, using various formulas we obtain fromthe naturality of Ω , we get

GF (α) Σ x = GF (α) Ω (1F(x))= Ω (F (α) 1F(x))

= Ω (1F(x) F (α))= Ω (1F(x)) α= Σ x α.

Thus we have a commutative diagram

x

α

Σ x GF (x)

GF(α)

x

Σ x GF (x).

We can deduce that the following

F FΣ −→FGF

ΛF−→F , G

Σ G−→GFG

GΛ−→G

compose to identities. For example, given some x ∈ Ob C , one can verifyΛF(x) F (Σ x) = 1F(x) by showing

Ω (ΛF(x) F (Σ x)) = Ω (ΛF(x)) Σ x = Σ x = Ω (1F(x)).

Moreover one can recover the adjunct by unit and counit: for any ϕ :F (x) → y and ψ : x → G(y)

Ω (ϕ) = G(ϕ) Σ x and Ω −1(ψ) = Λy F (ψ).

Theorem 1.1.18. Suppose F : C → D and G : D → C are two functors. If

there exist two natural transformations Σ : IdC → GF and Λ : FG → IdDsuch that the following two composites




F FΣ −→FGF

ΛF−→F , G

Σ G−→GFG

GΛ−→G

are identities, then F is a left adjoint of G. Moreover Ω : HomD(F (x), y) →HomC(x,G(y)) defined by Ω (ϕ) = G(ϕ) Σ x is the adjunct, with Σ, Λ the

unit and counit of the adjunction.

Proof. For x ∈ Ob C , y ∈ Ob D, φ ∈ HomD(F (x), y) and ψ ∈ HomC(x,G(y)),we define

Ω x,y(φ) = G(φ) Σ x and Ω x,y(ψ) = Λy F (ψ).

Then we can verify that they define two natural transformations of functorsC op × D → Sets. In fact for any α ∈ HomC(x, x), β ∈ HomD(y, y) andφ ∈ HomD(F (x), y) we have

G(β ) Ω (φ) α = G(β ) G(φ) Σ x α= G(β φ) GF (α) Σ x

= G(β φ F (α)) Σ x

= Ω (β φ F (α)),

which implies Ω is a natural transformation. The assertion for Ω is provedin a similar way.

Moreover Ω and Ω provide natural equivalences because we can show

( Ω Ω )(φ) = Ω (G(φ) Σ x)= Λy F (G(φ) Σ x)= Λy FG(φ) F (Σ x)= φ ΛF(x)F (Σ x)= φ,

and similarly (Ω Ω )(ψ) = ψ.

This theorem also implies that the adjoint of a functor is unique up tonatural equivalence.

Corollary 1.1.19. If G and G are right adjoints of F , then they are natu-rally equivalent. Similarly if F and F are two left adjoints of G, then they are naturally equivalent.

Proof. We only prove the first assertion. For any x ∈ Ob C and y ∈ Ob D wehave two isomorphisms

HomC(x,G(y)) Ω←−HomD(F (x), y)

Ω

−→HomC(x,G(y)).

Then (Ω Ω −1)(1G(y)

) : G(y) → G(y) and (Ω Ω −1)(1G(y)

) : G(y) → G(y)induce a natural equivalence between G and G.






It follows directly from the definition that, when a limit exists, it is uniqueup to isomorphism. If either the index category or the functor itself is struc-turally simple, we may explicitly compute the limits.

Example 1.1.21. 1. Let C = Z and T = Set . Then a functor M : Z → Set isrepresented by a chain of set maps

· · · → M (−1) → M (0) → M (1) → M (2) → · · · .

In particular if M 0 is given by the following diagram in which each mapis an inclusion

· · · ⊂ M 0(−1) ⊂ M 0(0) ⊂ M 0(1) ⊂ M 0(2) ⊂ · · · ,

then lim←−ZM 0 =

i∈Z M 0(i) and lim−→Z

M 0 =

i∈Z M 0(i).2. If C = Z and T = V ectk, the category of finite-dimensional k-vector spaces.

Suppose N is represented by the following

· · · → 0 N (1) N (2) N (3) · · · .

Then lim−→ZN does not exist in V ectk, the category of finite-dimensional

k-vector spaces, but lim←−ZN = 0.

3. Let T be a category and I a set considered as a discrete category. Thena functor M : I → T is simply an I -indexed set of objects in T . The thecoproduct of these objects is defined as

I xi = lim−→I

M and the product

of these objects is

I xi = lim←−I M .

4. Let C be the following poset

x

α

y β z

and T = V ectk, the category of all k-vector spaces. Each functor M ∈Ob T C is represented by

M (x)

M (α)

M (y)

M (β) M (z).

Then lim←−CM is called the pullback of the latter diagram and lim−→CM =

M (z).

Dually if D is the following poset




a f

g

b

c

and N ∈ Ob T D then lim−→DN is called the pushout of the diagram

N (a)

N (g)

N (f ) N (b)

N (c)

and lim←−DN = N (a).

The definition of limits can be rewritten by using a simple, yet very im-portant, construction.

Definition 1.1.22. There is a constant functor

K : T → T C

such that, for any t ∈ Ob T , K (t) is defined by K (t)(x) = t and K (α) = 1t forany x ∈ Ob C and α ∈ Mor C .

In the literature, the functor K : T → T C is often named the “diagonalfunctor”. Since the terminology is used for another purpose, see Definition2.2.34, we shall stick with our notion which seems to be more appropriate.

Now we present alternative characterizations of limits. The definition of an inverse limit can be rephrased as saying that there exists lim←−CM ∈ Ob C

with a natural transformation Γ : K (lim←−CM ) → M which is universal in the

sense that if there is another object t, along with a natural transformationΥ : K (t) → M , then there exists a unique morphism θt : t → lim←−CM hence a

natural transformation K (θt) : K (t) → K (lim←−CM ) such that Γ K (θt) = Υ .

K (t)

Υ

K(θt)

M

K (lim←−CM )

Γ

Especially we obtain a morphism

Ω : HomT C(K (t), M ) → HomT (t, lim←−CM ),




given by Ω (Υ ) = θt. It is an isomorphism because each t → lim←−CM extends

to a functor K (t) → M . Thus if every M ∈ Ob(T C) has a limit in T , thenlim←− : T C → T is the right adjoint of K and Ω becomes the correspondingadjunct. We characterize the direct limit lim−→C

in a similar way but we leave

it to the reader. When T is “large” enough with respect to C , we can introducethe limits in an economical way.

Proposition 1.1.23. If K has a right adjoint R : T C → T , then for each M ∈ Ob(T C), R(M ) is the inverse limit of M . If K has a left adjoint L :T C → T , then for each M ∈ Ob(T C), L(M ) is the direct limit of M .

For convenience, we introduce the following concepts.

Definition 1.1.24. A category T is called complete if for any small categoryC and any functor M ∈ T C the inverse limit lim←−C

M exists. A category T is

called cocomplete if for any small category C and any functor M ∈ T C thedirect limit lim−→C

M exists.

A category T is called finitely complete (respectively finitely cocomplete )

if for any finite category C and any functor M ∈ T C the inverse limit lim←−CM (respectively the direct limit lim−→C

M ) exists.

The categories Set and R-Mod are both complete and cocomplete.By Example 1.1.21, a coproduct is a direct limit and a product in an inverse

limit. In abelian categories, the existence of coproducts, respectively product,is equivalent to the cocompleteness, respectively completeness, condition ona category T .

Proposition 1.1.25. Let T be an abelian category. Then

1. it is complete (or finitely complete) if and only if all products (or all finite products) exist; and

2. it is cocomplete (or finitely cocomplete) if and only if all coproducts (or all

finite coproducts) exist.Proof. We only prove (1). Since a product is an inverse limit, we just have todemonstrate the opposite direction. Suppose C is a small (or finite) categoryand M : C → T is a functor. Let

C0(C ; M ) = f : Ob C →

x∈ObC

M (x) f (x) ∈ M (x)

andC1(C ; M ) = f : Mor C →

α∈Mor C

M (y) f (x

α→y) ∈ M (y).

We define δ : C0(C ; M ) → C1(C ; M ) by

δ (f )(x α→y) = f (y) − M (α)[f (x)].

Then we can verify that the kernel of δ is lim←−CM .




For example the category of finitely generated R-modules, R-mod, is bothfinitely complete and finitely cocomplete. Completeness and cocompletenesspass to functor categories. We record this and another result that shall beuseful in the next chapter.

Proposition 1.1.26. Let C be a small category. If T is complete (resp. co-complete), then T C is complete (resp. cocomplete).

Proof. Let D be another small category and M : D → T C a functor. Since(T C)D ∼= T C×D, M can be identified with a bifunctor from M (−, −) : C×D →T .

We may define a functor N : C → T by N (x) = lim←−DM (x, −) for every

x ∈ Ob C . Since [lim←−DM (−, −)](x) ∼= lim←−D[ M (x, −)] for all x ∈ Ob D, we can

show N is the limit lim←−DM . It means T C is complete. The cocompleteness of

T C can be proved in the same way.

The above result tells us that Set C and (R-Mod)C are both complete

and cocomplete if C is small. Furthermore (R-mod)C

is finitely complete andcocomplete if C is finite.

We record two results on preserving limits.

Proposition 1.1.27. Let T be both complete and cocomplete and C a small category. Given a functor M ∈ Ob T C and T ∈ Ob T , we have

HomT (T, lim←−CM ) ∼= lim←−CHomT (T, M (−))

and HomT (lim−→C

M, T ) ∼= lim←−CHomT (M (−), T )

Proof. To prove the first isomorphism, we take the diagram of M, which de-termines lim

←−CM . Applying the covariant functor HomT (T, −) to the diagram

it results in the diagram of the functor HomT (T, M (−)). From here one maycomplete the proof based by using the universal property of an inverse limit.The second isomorphism may be proved in the same way.

Proposition 1.1.28. Let F : T → T be the left adjoint of G : T → T .

1. Suppose M : C → T has a direct limit. Then F (lim−→CM ) ∼= lim−→C

F M .

2. Suppose N : D → T has an inverse limit. Then G(lim←−DN ) ∼= lim←−DG N .

Proof. We only prove (1). It is obvious that F (lim−→CM ) fits into the defining

diagram of lim−→CF M . We need to show F (lim−→C

M ) is universal. Suppose

t ∈ Ob T is another object satisfying the colimit defining diagram. It followsfrom the adjunction

HomT (F (lim−→CM ), t) ∼= HomT (lim−→C

M, G(t)),




along with the unit Id → F G that G(t) satisfies the defining diagram of lim−→C

M . From the universal property of lim−→CM , there exists a unique mor-

phism lim−→CM → G(t) which gives rise to a morphism F (lim−→C

M ) → t. The

universal property of F (lim−→CM ) follows from it.

In the end, we give a result that is used in the next section.

Theorem 1.1.29. Let C be a small category. Then any functor M : C → Set

is canonically a colimit of a diagram of representable functors.

Proof. We shall establish this result by firstly constructing a small categoryD from M : C → Set and secondly showing that M induces a functor M :D → Set C satisfying lim−→D

M ∼= M .

The objects in the category D are pairs (x, a) such that x is an object of C and a is an element in the set M (x). A morphism f : (x, a) → (y, b) is amorphism f ∈ HomC(x, y) satisfying M (f )(a) = b.

We define a functor M : D → Set C by M (x, a) = HomC(x, −). From

the Yoneda Lemma, there exists for each x ∈ Ob C an isomorphism of setsHomSetC (HomC(x, −), M ) ∼= M (x). It implies that each mx ∈ M (x) deter-mines uniquely a functor f (x,mx) : M (x, mx) → M making the enlarged

diagram of M commutative

HomC(x, −) = M (x, mx)

f (x,mx)

M (y, my) = HomC(y, −)f −

f (y,my)

M

Note that the commutativity of the diagram forces M (f )(mx) = my.In order to prove that M is the direct limit we need to show it is universal.

Suppose L is another functor fitting into the above diagram, and for each(x, mx) ∈ Ob D it comes with a functor g(x,mx) : HomC(x, −) → L. Againby the Yoneda Lemma, g(x,mx) determines uniquely an element lx ∈ L(x).From the commutativity of the diagram, we must have L(f )(lx) = ly. Nowwe define a functor h : M → L by

hz(mz) = lz

if z ∈ Ob C . We may readily verify that hzM (f ) = L(f )hz for any f : z → z .It implies that h is well defined. Since by definition hf (x,mx) = g(x,mx) for allobjects (x,mx) ∈ Ob D, M is universal.

The construction of D from a functor to Set in the proof is the predecessorof the Grothendieck constructions introduced in Section 2.4.



1.2 Restriction and Kan extensions 15

1.2 Restriction and Kan extensions

In last section we mentioned that, for a functor category T C , if the targetcategory are complete and cocomplete, then we may define the limits using

adjoints of the constant functor K : T → T C. When we examine closely, werealize that K itself is induced by another functor pt : C → •. This observationgenerates new ideas for constructing some extremely powerful functors, calledKan extensions.

1.2.1 Restriction

Definition 1.2.1. Suppose u : D → C is a functor between two small cate-gories. For any target category T , u induces a functor Resu : T C → T D viaprecomposition with u, called the restriction along u.

Example 1.2.2. The canonical functor pt : C → • induces a restriction T ∼=T • → T C, which is identical to K of Definition 1.1.22.

In last section, we know explicitly the left and right adjoints of K . As ageneralization, we shall describe the adjoints of an arbitrary restriction. Butbefore doing that, we provide several simple properties of the restriction.

Lemma 1.2.3. Suppose u : D → C is a functor. Then for any M ∈ Ob(T C),there are canonical morphisms lim←−CM → lim←−DResuM and lim−→D

ResuM →lim−→C

M .

Proof. The map between inverse limits follows from the universal property

lim←−CM

αy

αx

Θ

lim←−DResuM

φx

φy

ResuM (a) = M (u(a))

ResuM (φ)=M (u(φ)) ResuM (b) = M (u(b)) ,

in which φ : a → b is a morphism in D. The map between direct limits canbe established similarly.

Proposition 1.2.4. Suppose u : D → C has a left adjoint v : C → D. Then for any T , Resu has a right adjoint Resv. In particular, if u and v are category

equivalences, Resu and Resv are equivalences.If T is complete and cocomplete, then for any M ∈ Ob(T C), lim−→CM ∼=

lim−→DResuM , and for any N ∈ Ob(T )D, lim←−D

N ∼= lim←−CResvN .




Proof. We shall prove that for M ∈ Ob(T C) and N ∈ Ob(T D) there existsan isomorphism

HomT D(ResuM, N ) ∼= HomT C(M, ResvN ).

Let Σ : IdC → uv and Λ : vu → IdD be the unit and counit forthe adjunction between u and v. We define two natural transformationsΣ : IdT C → ResvResu and Λ : ResuResv → IdT D as follows. GivenM ∈ Ob(T C), x ∈ Ob C , N ∈ Ob(T D) and a ∈ Ob D,

(Σ M )x := M (Σ x) : M (x) → ResvResuM (x) = M (uv(x)),

and(ΛN )a := N (Λa) : ResuResvN (a) = N (vu(a)) → N (a).

One can verify that Σ and Λ provide the unit and counit of an adjunction.As for the second statement, we have for all t ∈ Ob T

HomT (lim−→DResuM, t) ∼= HomT

D

(ResuM, K (t))∼= HomT C(M, ResvK (t))∼= HomT C(M, K (t))∼= HomT (lim−→C

M, t).

andHomT (t, lim←−CResvN ) ∼= HomT C (K (t), ResvN )

∼= HomT D (ResuK (t), N )∼= HomT D (K (t), N )∼= HomT (t, lim←−DN ).

Thus lim−→DResuM ∼= lim−→C

M and lim←−CResvN ∼= lim←−DN .

In many places we will have to consider the adjoint functors of some re-

striction Resu. When u has an adjoint, we get an adjoint of Resu which isalso a restriction, by Proposition 1.2.4. The truth is that even if u does notadmit an adjoint, we can still construct the adjoints of Resu, and this is themain result of the upcoming two sections.

1.2.2 Overcategories and undercategories

In order to introduce the adjoints of a restriction, we have to provide some im-portant categorical constructions. These categorical constructions are of greatimportance in both homological algebra and homotopy theory of classifyingspaces. We shall be familiar with them as they appear almost everywhere

throughout these notes.




Definition 1.2.5. Let u : D → C be a functor between (small) categoriesand x ∈ Ob C . The category over x, u/x, consists of objects (a, α)

a ∈Ob D, α ∈ HomC(u(a), x). For any two objects (a, α), (b, β ), a morphismfrom (a, α) to (b, β ) is a morphism µ ∈ HomD(a, b) making the following

diagram commutativeu(a)

α

u(µ)

x,

u(b)

β

The category under x, written as x\u, is defined in a dual fashion. It consistsof objects (α, a)

a ∈ Ob D, α ∈ HomC(x, u(a)). For any two objects(α, a), (β, b), a morphism from (α, a) to (β, b) is a morphism µ ∈ HomD(a, b)making the following diagram commutative

u(a)

u(µ)

x

α

β u(b)

The categories defined above are customarily called overcategories andundercategories , associated with u : D → C . We will see later on that IdC/xand x\IdC, for any x ∈ Ob C , are already very interesting.

Remark 1.2.6. 1. From definition, an object in the overcategory u/x, (a, α),can be pictured as u(a)

α→x, and consequently a morphism µ : (a, α) →

(b, β ) can be equivalently interpreted as a sequence u(a)u(µ)→ u(b)

β→x. This

kind of rewritings will be useful for us when dealing with chains of mor-phisms in u/x and we shall come back to this point in later chapters.Similar reinterpretation can be made for objects and morphisms in under-categories too but we leave it to the reader.

2. There is a canonical functor Px : u/x → D (resp. Px : x\u → D givenon objects as projection to the first (resp. the second) component and onmorphisms as the identity. For simplicity, we shall denote such functors

just as P.3. If γ : x → y is a morphism in C , then it naturally induces a functor

γ ∗ : u

/x → u

/y and a functor γ

∗

: y\u

→ x\u

.Example 1.2.7. 1. Let pt : C → • be the canonical functor. Then pt/• ∼=

•\ pt ∼= C .




2. Let G be a group and H a subgroup. Then the inclusion functor iH :H → G gives exactly one overcategory iH /•. By direct calculation, theobjects are (•, g)|g ∈ G, and biject with the elements of G. There is amorphism from one object (•, g1) to another (•, g2) if there exists a h ∈ H

such that g1 = g2h. Obviously h = g−12 g1. Thus there is at most one

morphism from an object to another. Since h is invertible, there existsa morphism between two objects if and only if their are isomorphic iniH /•. In other words, two objects (•, g1) to another (•, g2) are isomorphicif and only if g1H = g2H . Because the category iH /• consists of [G : H ]many groupoids, each of which is equivalent to the trivial category, iH /• isequivalent to the discrete set G/H of left cosets (regarded as a category).Similarly the undercategory •\iH has objects (g, •)|g ∈ G. There isa morphism from (g1, •) to (g2, •) if and only if there exists a (unique)h ∈ H such that hg1 = g2 or equivalently h = g2g−1

1 . The undercategoryis equivalent to H \G, the set of right cosets. We have an isomorphismiH /• → •\iH given by

(•, g) → (g−1, •) a n d (•, g1)g−1

2 g1→ (•, g2) → (g−11 , •)g

−1

2 g1→ (g−12 , •).

When G = H , iG/• = IdG/• ∼= •\IdG = •\iG is the Cayley graph of G.Based on our observations, by Proposition 1.2.4, if M ∈ Ob(V ectH

k )(commonly called a k-representation of H ), lim−→iH/•

M ∼= lim−→G/H M ∼=

⊕g∈G/H gH ⊗k M . One can see that G permutes these direct summandsand the limit lim−→iH/•

M is isomorphic to kG ⊗kH M .

3. Suppose u : D → C is a functor between two posets. Then for any x ∈Ob C , u/x is isomorphic to the subposet of D consisting of objects a ∈Ob D

HomC(u(a), x) = ∅, while x\u is isomorphic to the subposet of Dconsisting of objects b ∈ Ob D

HomC(x, u(b)) = ∅.

The following observation follows directly from definitions and will be use-ful to us. For any functor u : D → C one can define a (covariant) opposite functor uop : Dop → C op such that uop(x) = x and uop(αop) = u(α)op. Beaware that it is different from the dual functor given before Definition 1.1.9.

Lemma 1.2.8. Suppose u : D → C is a functor. Consider its opposite functor uop : Dop → C op. Then for any x ∈ Ob C = Ob C op we have (u/x)op ∼= x\uop

and (x\u)op ∼= uop/x.

1.2.3 Kan extensions

In this section, we assume T to be a complete and cocomplete abelian cate-gory. The reader should bear Example 1.2.7 (1) in mind in order to see thatKan extensions generalize direct and inverse limits.




Theorem 1.2.9. Let u : D → C be a functor between small categories. Then the restriction Resu : T C → T D admits a left adjoint LK u, called the left Kan extension along u, as well as a right adjoint RK u, called the right Kan extension along u.

Proof. We only sketch the constructions of the Kan extensions and leavedetails to be filled by the reader.

Given M ∈ Ob(T D) we define its left and right Kan extensions along u as

LK uM = lim−→u/−ResPM and RK uM = lim←−−\uResPM,

where P is the functor in Remark 1.2.6 (2). Here we only prove the statementfor the left Kan extension because the proof for the right Kan extensionfollows the same pattern.

Step 1, we show LK uM is a functor from C to T . If γ ∈ HomC(x, y), thenwe have a functor γ ∗ : u/x → u/y. Since Resγ ∗ResPM = ResPM as functorsover u/x, by Lemma 1.2.3, it determines a canonical morphism

lim−→γ ∗ : LK uM (x) = lim−→u/xResPM → LK uM (y) = lim−→u/y

ResPM.

Hence LK uM = lim−→u/−ResPM ∈ Ob(T C).

Step 2, we state that LK u is a functor from T D to T C. For any natu-ral transformation Ψ : M → M between two objects of T D. We can usethe universal property to build a canonical natural transformation LK uΨ :LK τ M → LK uM .

Step 3, we construct an adjunct

Ω : HomT C(LK uM, N ) → HomT D(M, ResuN ).

For each M ∈ Ob(T D) we define the counit

Σ M : M → ResuLK uM

by the defining map of a limit

(Σ M )a : M (a) = ResPM [(a, 1u(a))] → lim−→u/u(a)ResPM,

for any a ∈ Ob D. For each N ∈ Ob(T C) we put the counit

ΛN : LK uResuN → N

such that, for every x ∈ Ob C , lim−→u/xResPResuN → N (x) comes from the

universal property of direct limit. We need to prove that the following are

identities

ResuResuΣ −→ ResuLK uResu

ΛResu−→ Resu and LK uΣLK u−→ LK uResuLK u

LK uΛ−→ LK u.




For the first we compute for any N ∈ Ob(T C) and a ∈ Ob D the followingcomposite

(ResuN )(a)[Resu(Σ N )]a

−→ (ResuLK uResuN )(a)(ΛResuN )a

−→ (ResuN )(a).

This composite is

(ResuN )(a) = (ResPResuN )[(a, 1a)] → lim−→u/u(a)ResPResuN → (ResuN )(a),

which is really an identity by the universal property of lim−→u/u(a)ResPResuN .

For the second we compute for any M ∈ Ob(T D) and x ∈ Ob C that

(LK uM )(x)[LK uΛM ]x

−→ (LK uResuLK uM )(x)(ΛLKuM )x

−→ (LK uM )(x)

gives the identity. But it is rewritten as

lim−→u/xResPM → lim−→u/x

(ResPResuLK uM ) → lim−→u/xResPM,

which in turn equals lim−→u/xResP applying to

M → ResuLK uM → M.

However the preceding morphisms compose to the identity because the fol-lowing composite

M (a) → lim−→u/u(a)ResPM → M (a)

is an identity for every a ∈ Ob D, due to the fact that (a, 1u(x)) is a terminalobject and that ResPM (a, 1u(a)) = M (a).

Remark 1.2.10. 1. One may choose T to be Ab, or R-Mod etc, for practicalapplications. When the index categories are finite, we can even use R-mod,the category of finitely generated R-modules.

2. When u : D → C is a full embedding and M ∈ Ob(AbD), thenLK uM ∈ Ob(AbC) restricted on u(D) is identified with M , which meansLK uM (u(d)) = M (d) for any d ∈ Ob D. This is why we call such functorsdiscovered by D. M. Kan, “the Kan extensions”.

Example 1.2.11. In Example 1.2.7 (2) where the over- and under-categoriesassociated with i : H → G are computed, we can continue to verify that Resiis the usual restriction ↓G

H , LK i is equivalent to the induction ↑GH = kG ⊗kH −

and RK i is equivalent to the coinduction ⇑GH = HomkH (kG, −). If G is finite,

the two Kan extensions are well known to be equivalent.

Corollary 1.2.12. Let u : D → C and v : E → D be two functors be-tween small categories. Suppose T is a complete and cocomplete category,




M ∈ Ob(T E ) and N ∈ Ob(T C). Then ResvResu = Resuv and consequently LK uLK v ∼= LK uv, RK uRK v ∼= RK uv.

Proof. The equality between restrictions follows directly from definition.

Then we haveHomT E (M, ResvResuN ) ∼= HomT D(LK vM, ResuN )

∼= HomT C(LK uLK vM, N ).

Hence LK uLK v ∼= LK uv. The isomorphism between right Kan extensionscan be proved similarly.





Chapter 2

Simplicial methods

Abstract We begin with chain complexes and their homology, the fundamen-tal concepts of homological algebra. Then we review simplicial constructionsin algebra and topology as they provide concrete examples and motivating

ideas. Out main interest lies in the nerve of a small category. This particularsimplicial set allows us to define simplicial homology and cohomology of acategory with coefficients in a commutative ring. The nerve of a small cate-gory has a geometric realization, called the classifying space. Hence we canalso consider the singular homology and cohomology of a classifying space.We shall show these two theories agree. We will introduce various impor-tant categorical constructions. Meanwhile we develop techniques for com-paring categoires, their nerves and classifying spaces. A major theorem isQuillen’s Theorem A. For future references and better understanding of sim-plicial methods, we also include a description of bisimplicial sets and severalrelevant results.

2.1 Complexes and homology

Here we recall basics about chain and cochain complexes as well as operationson them.

2.1.1 Chain complexes, homology and chain homotopy

Suppose A is an associative ring with identity and Z is the totally orderedset of integers. From Example 1.1.3 (3) we may deem Z as a category.

Definition 2.1.1. A chain complex of A-modules is an object C ∈ Ob(A-Mod)Z

op

such that C(n → n + 2) = 0 for all n ∈ Z. In other words it

23





2.1 Complexes and homology 25

Clearly a chain map between chain complexes φ : D → C induces mapsbetween homology groups φ∗ : Hn(D) → Hn(C) and a cochain map betweencochain complexes φ : D → C induces maps between cohomology groupsφ∗ : Hn(D) → Hn(C).

For convenience, we shall call both chain maps and cochain maps justchain maps. Given a chain complex Cn, ∂ nn∈Z of objects in an abeliancategory A, we can obtain a cochain complex Cn, ∂ nn∈Z, simply by askingCn = C−n and ∂ n = ∂ −n. Thus in some sense, chain and cochain complexesare the same. In practice, we will regard a complex as a chain complex or acochain complex depending on where it comes from.

Definition 2.1.4. If φ, ψ : D → C are two chain maps, we say φ and ψ arechain homotopic , written as φ ψ, if there are maps hn : Dn → Cn+1 suchthat Φn − Ψ n = hn−1∂ Dn + ∂ Cn+1hn

· · · Dn+1

∂ Dn+1

φn+1−ψn+1

Dn

hn

∂ Dn

φn−ψn

Dn−1

hn−1

φn−1−ψn−1

· · ·

· · · Cn+1

∂ Cn+1

Cn∂ Cn

Cn−1 · · ·

We say two complexes D and C are chain homotopy equivalent , writtenas D C if there are chain maps φ : D → C and ψ : C → D such thatφ ψ IdC and ψ φ IdD. We say a complex C is contractible , if it is chainhomotopy equivalent to the zero complex. We say a complex C is acyclic if

Hn(C) = 0 for all n. Acyclicity is strictly weaker than contractibility.

Proposition 2.1.5. If φ, ψ : D → C are chain homotopic, and if F : A-Mod → B-Mod is an additive functor, then

1. φ∗ = ψ∗ : Hn(D) → Hn(C). Thus a chain homotopy equivalence D Cinduces isomorphisms Hn(D) ∼= Hn(C).

2. F (φ) F (ψ) : F (D) → F (C), and F (φ)∗ = F (ψ)∗ : Hn(F (D)) → Hn(F (C)).

Let C be a bounded below chain complex of A-modules. A projective res-olution of C is a (bounded below) chain complex P of projective A-modules,along with a chain map P → C which induces isomorphisms on homologygroups. Any two projective resolutions of C are chain homotopy equivalent.Similarly we can consider an injective resolution I of a bounded above com-plex of A-modules D.





2.1 Complexes and homology 27

Hence the differential of the total complex Tot⊕(D⊗A C) is given by

∂ (x ⊗ y) = ∂ D p x ⊗ y + (−1) px ⊗ ∂ Cq y,

for x ∈ D p

and y ∈ Cq

.Suppose both E and C are chain complexes of left A-modules. Then we

introduce another double complex HomA(E,C) by (there are different con-ventions in the literature)

HomA(E,C)n =

q+ p=n

HomA(E− p,Cq),

pictured as

HomA(E− p+1,Cq)

∂ vp−1,q

HomA(E− p,Cq)

∂ vp,q

∂ hp,q

HomA(E− p+1,Cq−1) HomA(E− p,Cq−1)∂ hp,q−1

with ∂ h p,q(f ) = f ∂ E− p+1 and ∂ v p,q(f ) = (−1) p+q∂ Cq f .

The differential of the total complex Tot

(HomA(E,C)) is ∂ n = ∂ h p,q +∂ v p,q : HomA(E,C)n → HomA(E,C)n−1 given by

∂ nf = f ∂ E− p+1 + (−1)n∂ Cq f

for any f ∈ HomA(E− p,Cq).

Theorem 2.1.8. Suppose R is a commutative ring. Let C, D and E be com-plexes of R-modules. We have an isomorphism of double complexes

HomR(C⊗R D,E) ∼= HomR(C, HomR(D,E))

The following theorem is well known and its proof can be found in manyplaces. Here we record it for future references.

Theorem 2.1.9 (Kunneth formula). Suppose A is a ring. Let D be a chain complex of right A-modules and C,E chain complexes of left A-modules. Then

1. if Dn and ∂ (Dn) are flat for all n, there is a short exact sequence

0 →

p+q=n H p(D) ⊗A Hq(C) → Hn(D⊗A C)

→ p+q=n−1 TorA1 (H p(D), Hq(C)) → 0

2. if En and ∂ (En) are projective for all n, there is a short exact sequence



28 2 Simplicial methods

0 →

q− p=n+1 Ext1A(H p(E), Hq(C)) → Hn(HomA(E,C))

→

q− p=n HomA(H p(E), Hq(C)) → 0.

Corollary 2.1.10 (Universal Coefficient Theorem). Let D be a chain complex of right A-modules over a ring A, E a chain complex of left A-modules and M a left A-module considered as a complex concentrated in degree zero.Then

1. if Dn and ∂ (Dn) are flat for all n, there is a short exact sequence

0 → H p(D) ⊗A M → Hn(D⊗A M ) → TorA1 (Hn−1(D), M ) → 0,

2. if En and ∂ (En) are projective for all n, there is a short exact sequence

0 → Ext1A(Hn−1(E), M ) → H−n(HomA(E, M )) → HomA(Hn(E), M ) → 0.

In Corollary 2.1.10 (2), the middle term is often written as cohomology

Hn

(HomA(E, M )).The dual complex of C, denoted by C∧, is defined to be HomA(C, A), wherethe second A is regarded as a complex concentrated in degree zero. This isa complex of right A-modules. Note that if P is a projective A-module thenHomA(P, A) is a projective right module. We can define an evaluation map

ev : C∧ ⊗A C → A.

It is non-zero only at degree zero. More explicitly it is given by f ⊗ x → f (x)for any base elements f ⊗ x ∈ (C∧ ⊗C)0. Here x ∈ Cn and f ∈ C∧−n.

Proposition 2.1.11. Suppose k is a field. Let D and C be two complexes of finite-dimensional k-vector spaces. If for every n, Homk(D,C)n is finite-dimensional, then we have an isomorphism of complexes

D∧ ⊗k C ∼= Homk(D,C).

Proof. Note that if M, N are two stalk complexes of k-vector spaces concen-trated in degrees m and n respectively, then we have M ∧⊗kN ∼= Homk(M, N )as stalk complexes concentrated in degree n−m. Consequently we have an iso-morphism for every integer n, (D∧⊗kC)n

∼= Homk(D,C)n since Homk(D,C)n

is finite-dimensional and then Homk(D,C)n = ⊕q+ p=nHomk(D− p,Cq). Hencein order to finish the proof, we only need to verify that the two differentialsare identified under the vector space isomorphisms.

In the next section, we will see many combinatorially constructed com-plexes and double complexes.



2.2 Nerves, classifying spaces and cohomology 29

2.2 Nerves, classifying spaces and cohomology

By Dold-Kan Correspondence, the category of non-negatively graded com-plexes of A-modules is equivalent to the category of simplicial A-modules.

It implies also that the category of first quadrant double complexes of A-modules is equivalent to the category of bisimplicial A-modules. Since in thesenotes, we are mainly interested in various non-negatively graded complexessuch as projective resolutions of modules, and moreover these complexes comefrom corresponding simplicial sets, it is useful to introduce simplicial sets andmodules.

2.2.1 Simplicial sets and nerves of small categories

We recall the fundamental idea in algebraic topology of singular (co)homologytheory. From here we shall see how algebraic and combinatorial methods enterthe study of spaces. Then we will apply the same methods, called simplicialmethods, to investigate small categories (instead of spaces). Our first defi-nition of category (co)homology is given soon after the basic definitions arerecorded.

Let us start with the topological simplicies. For every integer n ≥ 0, astandard n-simplex is defined as a subspace of the (n + 1)-dimensional realvector space Rn+1

n = (x0, x1, · · · , xn) ∈ Rn+1 xi ≥ 0 and

ni=0

xi = 1.

An i- face of n for 0 ≤ i ≤ n is a subspace such that there are exactly

i + 1 chosen entries that are not constantly zero. Each i-face is isomorphicto the standard i-simplex. For example in the following picture, 2 has ex-actly one 2-face (e.g. 2), three 1-faces (isomorphic to 1) and three 0-faces(isomorphic to 0 which is a point).




X 2

X 3

X 1

(0,1,0)

(0,0,1)

(1,0,0)

2 ⊂ R3

Fix an integer n, there are many natural maps among all the faces of n. Since each face is isomorphic to a standard simplex, these maps can be

described as maps among all standard simplicies. The most distinguished arethe face maps di : n−1 → n (inserting a zero) and the degeneracy maps si : n+1 → n (adding up two adjacent entries), given explicitly by

di(x0, x1, · · · , xn−1) = (x0, · · · , xi, 0, xi+1, · · · , xn−1), 0 ≤ i ≤ n;si(x0, x1, · · · , xn+1) = (x0, · · · , xi + xi+1, · · · , xn+1), 0 ≤ i ≤ n.

Given a topological space X , in order to define and compute its singu-lar homology H∗(X,Z) we first form the sets of continuous maps S (X )n =HomT op(n, X ) and then produce free abelian groups ZS (X )n on top of them. Each face map di : n−1 → n induces a map di : S (X )n → S (X )n−1,and moreover ∂ n =

ni=0(−1)idi : ZS (X )n → ZS (X )n−1 satisfies ∂ n+1∂ n =

0. In this way we obtain a non-negatively graded chain complex

ZS (X )∗; ∂ ∗∗≥0,

in which every differential ∂ n is determined by an alternating sum of facemaps. The homology of this complex is defined to be the singular homologyof X , written as H∗(X,Z). The degeneracy maps are important to us as well,and we shall discuss their roles shortly.

In order to allow further applications of such a fundamental construction,we propose some alternative descriptions of the standard simplices basedon the following two observations. Firstly, to specify a standard complex, itsuffices to provide its vertices because n is the convex hull of the set of its vertices. Moreover since, in Rn+1, one can give the lexicographic order toits elements by asking (x0, · · · , xn) < (y0, · · · , yn) if there exists an integer

0 ≤ k ≤ n such that xi = yi for i < k and xk < yk, particularly there is anorder on the set of vertices of n. This totally ordered set of vertices uniquelydetermines n. For example in R3 we have (0, 0, 1) < (0, 1, 0) < (1, 0, 0), and




we can certainly identify this totally ordered set with 2. Thus giving n

is equivalent to giving the totally ordered set (0, · · · , 0, 1) < (0, · · · , 1, 0) <· · · < (1, 0, · · · , 0) of points in Rn+1. Moreover there exists a natural one-to-one correspondence between the set of all totally ordered subsets of vertices

and the set of faces of n

.Secondly, because the face and degeneracy maps on n are completely

determined by their values on the vertices, we can translate the face anddegeneracy maps accordingly. For example, we can write out values of d0 :2 → 3 and s0 : 2 → 1 on the vertices

d0(0, 0, 1) = (0, 0, 0, 1), d0(0, 1, 0) = (0, 0, 1, 0), d0(1, 0, 0) = (0, 1, 0, 0);s0(0, 0, 1) = (0, 1), s0(0, 1, 0) = (1, 0), s0(1, 0, 0) = (1, 0).

In general it is easy to see that di and si always send vertices to vertices.Furthermore they (weakly) preserve the order, in the sense that if a ≤ b thendi(a) ≤ di(b) and si(a) ≤ si(b). We shall illustrate it by an example. Let usexamine 1 = (0, 1) < (1, 0) and 2 = (0, 0, 1) < (0, 1, 0) < (1, 0, 0).

The face maps 1

→ 2

, adapted to our new combinatorial expression, aregiven by embeddings

d0 : (0, 1) < (1, 0) → (0, 0, 1) < (0, 1, 0)d1 : (0, 1) < (1, 0) → (0, 0, 1) < (1, 0, 0)d2 : (0, 1) < (1, 0) → (0, 1, 0) < (1, 0, 0).

The degeneracy maps 2 → 1 are the same as projecting the vertex (0, 1, 0)to one of the other two, upon identifying 1 with the line segment (0, 0, 1) −(1, 0, 0)

s0 : (0, 0, 1) < (0, 1, 0) < (1, 0, 0) → (0, 1) < (1, 0) = (1, 0) (0, 1) < (1, 0)s1 : (0, 0, 1) < (0, 1, 0) < (1, 0, 0) → (0, 1) = (0, 1) < (1, 0) (0, 1) < (1, 0).

In summary, the face and degeneracy maps are indeed weakly monotonicmaps among those totally ordered sets corresponding to the standard sim-plices. As a matter of fact, all weakly monotonic functions among those totallyordered sets, coming from standard simplices, are composites of these faceand degeneracy maps.

For various good reasons we continue to work on the combinatorial charac-terizations of standard simplices. Previously we have identified n with theposet (0, · · · , 0, 1) < (0, · · · , 1, 0) < · · · < (1, 0, · · · , 0), and have rewrittenthe face and degeneracy maps. Now we abstract the totally ordered set as0 < 1 < · · · < n. Consequently the face and degeneracy maps can be reformu-lated and will be denoted by di and si, respectively. This reformulation allowsus to forget the geometric definition of n, and thus all relevant constructionscan be made in an entirely combinatorial fashion. For future applications wewrite out the ith face map di (corresponding to di) for 0 ≤ i ≤ n




0 < 1 < · · · < n − 1 → 0 < 1 < · · · < n

di( j) =

j, if j < i j + 1 , if j ≥ i

and the ith degeneracy map si (corresponding to si) for 0 ≤ i ≤ n

0 < 1 < · · · < n + 1 → 0 < 1 < · · · < n

si( j) =

j, if j ≤ i j − 1 , if j > i

One can verify that these maps satisfy the relations (the cosimplicial identi-ties )

djdi = didj−1 i < j sjdi = di sj−1 i < j sjdi = 1 i = j or j + 1 sjdi = di−1 sj i > j + 1 sj si = si sj+1 i ≤ j .

For each n ∈ 0 ∪ N, we define an ordered set (which happens to be afinite category) n = 0 < 1 < 2 < · · · < n (or rather 0 → 1 → 2 → · · · → n).We denote by the category consisting of all such n, in which morphismsare weakly monotonic functions (equivalently, functors) among these orderedsets. Recall the motivating example at the very beginning of this section.Given a space X , in order to make use of , we associated to each n, thecombinatorial model of n, a set HomT op(n, X ), as well as an abeliangroup ZHomT op(n, X ). If we put all pieces together, we realize that weactually constructed a contravariant functor HomT op(−, X ) : → Set , aswell as a ZHomT op(−, X ) : → Ab. Since these functors give us the singularhomology of X , contravariant functors from to other categories may leadto interesting constructions too.

Definition 2.2.1. A simplicial object in a category T is a contravariant func-tor X : → T . Two simplicial objects X, Y in T are said to be isomorphic ,written as X ∼= Y , if as functors they are naturally equivalent.

A simplicial set is a simplicial object in Set .

Equivalently, since monotonic maps are compositions of di and si, a sim-plicial object X in T consists of a set of objects X n := X ([n]) ∈ Ob T (anelement of X n is called an n-simplex ) and morphisms among these simplicieswhich are composites of two special kinds of morphisms: di := X (di) : X n →X n−1, 0 ≤ i ≤ n, and si := X ( si) : X n → X n+1, 0 ≤ i ≤ n, satisfying (thesimplicial identities )




didj = dj−1di i < jdisj = sj−1di i < jdisj = 1 i = j or j + 1disj = sj di−1 i > j + 1

sisj = sj+1si i ≤ j .

In order to define a simplicial object X , it suffices to specify X nn≥0 ⊂ Ob T ,together with some maps di and si satisfying the above relations.

We shall be interested in the cases where T = Set or R-Mod for somecommutative ring R with identity. By Proposition 1.1.27, the two categoriesSimpSet and Simp(R-Mod) are both complete and cocomplete.

Example 2.2.2. 1. Let C be a small category. We define the nerve of C to bea simplicial set N C = HomCat(−, C ) such that N C n = C n (indeed Ob(C n)since we only need the underlying set). Alternatively N C n can be identifiedwith the set of n-chains of morphisms in C if n > 0 and Ob C if n = 0.When n > 0, the ith face map di : N C n+1 → N C n is

di(x0 → · · · → xi−1αi→xi

αi+1→ xi+1 → · · · → xn+1)

= x0 → · · · → xi−1αi→ xi

αi+1→ xi+1 → · · · → xn+1,

where ∧ means removing an object. For instance when n = 1 the threeface maps N C 2 → N C 1 are d0(x0

α1→x1α2→x2) = x1

α2→x2, d1(x0α1→x1

α2→x2) =

x0α2α1→ x2 and d0(x0

α1→x1α2→x2) = x0

α1→x1. The ith degeneracy map si :N C n → N C n+1 is given by

si(x0 → · · · → xi−1 → xi → xi+1 → · · · → xn)

= x0 → · · · → xi−1 → xi1xi→xi → xi+1 → · · · → xn.

One can verify that these maps satisfy the simplicial identities.

2. We consider a special situation of the first example. It shall fill the gapwhich may have occurred during transition from the geometric definition of standard n-simplicies to the abstract category-theoretic reformulation. Wefix a non-negative integer m. The totally ordered set m is indeed a category0 → 1 → · · · → m. Thus we can consider its nerve N m. By the precedingexample we have N mn = Ob(mn). In other words the combinatorial m-simplex m corresponds to the functor Hom(−, m) : → Set . Thiscorrespondence actually extends to a (covariant) functor, given by ι(m) =Hom(−, m),

ι : → SimpSet = Set .

By Yondeda Lemma we have

HomSimpSet(Hom(−, m), Hom(−, n)) ∼= Hom(m, n).

It means ι is fully faithful.




In the next section we shall talk about the geometric realization of simpli-cial sets. Then we will see that the simplicial set Hom(−, m) preciselygives rise to m.

3. Let A be an associative ring and A-Mod the category of all A-modules. A

simplicial object in A-Mod is called a simplicial A-module . From Set toA-Mod, there exists a natural covariant functor, given by constructing freeA-modules. Suppose X : → Set is a simplicial set. Then we naturallyobtain a simplicial A-module

A[X ] : → Set → A-Mod.

Given a simplicial A-module Y , one can construct a (non-negativelygraded) chain complex of A-modules by defining Cn(Y ) = Y n and ∂ n =n

i=0(−1)idi : Cn(Y ) → Cn−1(Y ). If Y comes from a simplicial set X ,that is, Y = A[X ], then we write the chain complex as C∗(X, A) insteadof C∗(Y ) or C∗(A[X ]) for various good reasons.

The following characteristic statement is enlightening. Suppose X is asimplicial set. Then we define the corresponding simplex category to be ι/X ,where ι : → SimpSet is the functor in Example 2.2.2 (2).

Theorem 2.2.3. Let X be a simplicial set. Then we have an isomorphism

X ∼= lim−→ι/XP,

where P : ι/X → SimpSet is given by P(n, Φ) = Hom(−, n).

Proof. From the contravariant version of Theorem 1.1.28 we know for thefunctor X : op → Set we can define a category D and a functor X : D →Set

op

= SimpSet such that

X ∼= lim−→DX.

We show the category D is exactly ι/X and subsequently X can be identifiedwith P.

The objects in the overcategory ι/X are of the form (n, Φ), where Φ :ι(n) = H o m(−, n) → X is a simplicial map. By the Yoneda LemmaHomSimpSet(Hom(−, n), X ) ∼= X (n) = X n. Thus Φ can be regarded asan element of the set X (n). This provides a bijection between Ob(ι/X ) andOb D. From here one can continue to finish the proof. We leave the detailsfor the reader.

Definition 2.2.4. Let C be a small category and N C its nerve. Suppose Ris a commutative ring with identity. Then the simplicial R-module R[N C ]

gives rise to a complex of R-modules, written as C∗(C , R). The n-th homol-ogy of C∗(C , R), denoted by Hn(C , R), is called the n-th homology of C with coefficients in R. The n-th cohomology of the cochain complex




C∗(C , R) := C∗(C , R)∧ = HomR(C∗(C , R), R),

denoted by Hn(C , R), is called the n-th cohomology of C with coefficients in R.

By direct calculation, one can see that H0(C , R) ∼= H0(C , R) is a free R-module with rank equal to the number of connected components of C . Forany chain complex C∗(C , R), define as above, we can insert the base ringR at degree -1 and obtain the so-called reduced chain complex C∗(C , R) =C∗(C , R) → R → 0. Then the homology of the reduced chain complex iscalled the reduced homology of C , written as H∗(C , R). We see H−1(C , R) = 0and H0(C , R) is a free R-module with rank equal to rkRH0(C , R) − 1. A smallcategory C is called R-acyclic if H∗(C , R) vanishes.

Now we address the issue of degeneracy maps. Suppose X is a simplicialobject in an abelian category T . Then we have a complex of objects in T ,denoted by X ∗. It has a subcomplex X ∗ such that X n =

i si(X n−1) if n > 0

and X 0 = 0. Then we continue to define a quotient complex X †∗ = X ∗/X ∗.

We can also define the normalized complex of X , N∗(X ), by

Nn(X ) =n−1i=0

Ker(di)

and ∂ n = (−1)ndn. This is a subcomplex of X ∗.

Theorem 2.2.5. 1. We have X ∗ = N∗(X ) ⊕ X ∗ and N∗(X ) ∼= X †∗. Further-

more the quotient map X ∗ X †∗ induces a chain homotopy equivalence and X ∗ is contractible.

2. (Dold-Kan Correspondence) Suppose T is an abelian category and Ch≥0(T ) is the category of non-negatively graded chain complexes in T .Then the normalized chain complex functor functor

N : SimpT → Ch≥0(T ),

is a category equivalence.

Proof. For the first part, see [53, Chapter VIII, Section?] and [84, Section8.3], and for the second part, the reader is referred to [84, Section 8.4].

Because of the above theorem, we also call X †∗ the normalized complex of X ∗.

When T = A-Mod, and X is a simplicial A-module, then X n, for anyn > 0, consists of degenerate elements of the form si(x) for some x ∈ X n−1.When X = R[N C ], a simplicial R-module defined over a small category C and

a commutative ring R, the degenerate elements in R[N C ]n are all the linearcombinations of the degenerate elements in N C n which are exactly thosen-chains containing an identity morphism. In practice, when we compute




(co-)homology of a category, we only need to use the normalized complex

C†∗(C , R).

Example 2.2.6. 1. Let C be the following category with two objects, two iden-tity and non-identity morphisms

xα

β

y

To compute H∗(C ,Z), we only have to write down the normalized chain

complex 0 → C†1(C ,Z) → C†0(C ,Z) → 0

0 → Zα, β → Zx, y → 0.

(By comparison, C∗(C ,Z) is infinite.) The non-trivial differential is givenby α → y − x and β → y − x. Then the only non-trivial homology groupsare H0(C ,Z) ∼= Z and H1(C ,Z) ∼= Z. The calculation of cohomology groups

is left to the reader.2. The second category D is slightly different from the first. One can easily

verify that the normalized complex C†∗(D,Z) is infinite, of dimension twoat each degree. By direct computation H0(D,Z) ∼= Z is the only non-trivialhomology group.

xα

yα−1

However there is an easy way to see it, if one notices that D • andknows that a category equivalence induces isomorphism on homology (seeProposition 2.2.19).

Definition 2.2.7. Let D and C be two small categories. Then the join of

D with C , denoted by D ∗ C , is a category whose objects are Ob D ∪ Ob C ,and whose morphisms are Mor D ∪ Mor C , plus exactly one extra morphismγ a,x ∈ HomD∗C(a, x) introduced for every pair of objects a ∈ Ob D andx ∈ Ob C . The composition laws in D ∗ C are determined by the compositionlaws in C and D, plus the equalities αγ a,x = γ a,y, γ a,xβ = γ b,x for anyα ∈ HomC(x, y) and β ∈ HomD(b, a).

By definition D ∗ C and C ∗ D are two different categories. One can easilyconstruct N (D ∗ C ) from N D and N C .

Proposition 2.2.8. Suppose both D and C are connected small categories. If R = k is a field, then

Hn(D ∗ C , k) ∼= i+j=n−1

Hi(D, k) ⊗ Hj (C , k)

if n ≥ 0. Particularly H∗(D ∗ C , k) ∼= H∗(C ∗ D, k).




Proof. For n ≥ 0, Cn(D ∗ C , k) = k[N (D ∗ C )]n has a basis consisting of thefollowing elements

x0 → · · · → xiγxi,yi+1

−→ yi+1 → · · · → yn

where x1, · · · , xi ∈ Ob D and yi+1, · · · , yn ∈ Ob C , for 0 ≤ i ≤ n. Here γ xi,yi+1

is the unique morphism in HomD∗C(xi, yi+1). Its differential is

i−1j=0(−1)j x0 → · · · → xj → · · · → xi

γxi,yi+1−→ yi+1 → · · · → yn

+ (−1)ix0 → · · · → xi−1γxi−1,yi+1

−→ yi+1 → · · · → yn

+ (−1)i+1x0 → · · · → xiγxi,yi+2

−→ yi+2 → · · · → yn

+n

l=i+2(−1)lx0 → · · · → xiγxi,yi+1

−→ yi+1 → · · · → yl → · · · → yn.

Suppose C∗(D ∗ C , k), C∗(D, k) and C∗(C , k) are the reduced chain complexeswith k inserted in degree -1. Then we can define a degree -1 chain map

C∗(D ∗ C , k) → C∗(D, k) ⊗ C∗(C , k)

such that, at degree −1 it is identity and at degree n ≥ 0, Cn(D ∗ C , k) →[C∗(D, k) ⊗ C∗(C , k)]n−1 is given by

x0 → · · · → xiγxi,yi+1

−→ yi+1 → · · · → yn → [x0 → · · · → xi]⊗[yi+1 → · · · → yn],

along withx0 → · · · → xn → [x0 → · · · → xn] ⊗ 1,

andy0 → · · · → yn → 1 ⊗ [x0 → · · · → xn].

Here 1 ∈ k = C−1(D, k) = C−1(C , k). This chain map is an isomorphism andthus by Kunneth formula we get

Hn(D ∗ C , k) ∼=

i+j=n−1 Hi(C∗(D, k)) ⊗ Hj (C∗(C , k))∼=

i+j=n−1 Hi(D, k) ⊗ Hj (C , k).

Example 2.2.6 (2) tells us that it is important to know how to comparevarious categories or more generally simplicial sets.

Definition 2.2.9. A natural transformation Φ : X → Y between simplicialobjects in T is called a simplicial map. The simplicial objects in T form acategory SimpT .

By definition a simplicial map Φ consists of a sequence of morphisms in

T , Φn : X n → Y n, commuting with the relevant face and degeneracy maps.




Lemma 2.2.10. Let R be a commutative ring with identity. A simplicial mapΦ : X → Y between simplicial sets induces a chain map CΦ : C∗(X, R) →C∗(Y, R).

Proof. Because every Φn : X n → Y n commutes with di, it commutes with thedifferentials of these chain complexes which are alternating sums of di’s.

Lemma 2.2.11. Suppose D and C are two small categories and Φ : N D →N C is a simplicial map. Then Φ determines a functor u : D → C which in turn gives rise to Φ as N u.

Proof. The map Φ0 : N D0 → N C 0 gives an assignment Ob D → Ob C whileΦ1 : N D1 → N C 1 gives an assignment Mor D → Mor C . We show that thesedefine a functor u : D → C such that u(d) = Φ0(d) if d ∈ Ob D and u(f ) =Φ1(f ) if f ∈ Mor D.

First of all, Φ0 and Φ1 are compatible in the sense that if c f →d ∈ N D1

then u(f ) = Φ1(f ) is a morphism from u(c) = Φ0(c) to u(d) = Φ0(d). Thisfollows from the commutative diagram by choosing i = 0 or 1

N D1Φ1

di

N C 1

di

N D0Φ0

N C 0

Second of all, we have to demonstrate u(f g) = u(f )u(g) for any two com-posable morphisms in Mor D. This time we just use a similar commutativediagram involving d1, Φ1 and Φ2.

At last we can show u(1d) = 1u(d) by invoking a degeneracy map

N D1

Φ1 N C 1

N D0Φ0

s0

N C 0

s0

Since Φ is completely determined by Φ0 and Φ1, one can prove that N u isexactly Φ.

Example 2.2.12. Suppose u : D → C is a functor between small categories andR is a commutative ring with identity. Then it induces a simplicial map N u :N D → N C . Furthermore it induces a chain map Cu : C∗(D, R) → C∗(C , R).

Corollary 2.2.13. A functor u : D → C induces a map u∗ : H∗(D, R) →H∗(C , R) as well as u∗ : H∗(C , R) → H∗(D, R).

Proof. The map u∗ is induced by Cu while u∗ is induced by HomR(Cu, R).




We shall illustrate the previous comparison results by an example. Notethat if there are two functors v : E → D and u : D → C , we can easily verifythat (uv)∗ = u∗v∗ and (uv)∗ = v∗u∗.

Example 2.2.14. Let us consider the following category C with four non-identity morphismsy

x

α

β

γ

µ z

Its opposite category is pictured as

yop

αop

βop xop

zop

γ op

µop

In order not to make confusion in the opposite category we write xop for xetc. By direct calculation we can find that H0(C ,Z) ∼= H0(C op,Z) ∼= Z and

H1(C ,Z) ∼= H1(C op,Z) ∼= Z ⊕ Z. These are the only non-trivial homologygroups.

We can list all (covariant) functors from C → C op, since there are notmany. Suppose u : C → C op is a functor. We write u(C ) to be the image of C ,a subcategory of C op.

1. If u(x) = xop, then both u(y) and u(z) have to be xop. In this case u(C )is a trivial category and has the only non-trivial homology at degree zero,which is Z.

2. If u(x) = yop, then there are several possibilities. In any case we must haveu(y) = u(z). Firstly, we may have u(C ) = yop. Then Hi(u(C ),Z) ∼= Z if i = 0 or zero otherwise. Secondly we may have u(C ) equals

yopαop

βop

xop

oryop xop .




In the former situation we will have H0(u(C ),Z) ∼= H1(u(C ),Z) ∼= Z andzero otherwise. In the latter situation we have only non-trivial homology

H0(u(C ),Z) ∼= Z.3. If u(x) = zop, then it is similar to 2.

Since u can be decomposed into a sequence C u(C ) → C op, u∗ is also thecomposite of H∗(C ,Z) → H∗(u(C ),Z) → H∗(C op,Z). Under the circumstancewe understand u∗ completely. A crucial fact is that, our previous calcula-tions assert that there exists no such functor u : C → C op that induces theisomorphism of graded abelian groups H∗(C ,Z) ∼= H∗(C op,Z).

Simplicial maps are used to compare two simplicial sets (and resultingcomplexes). Now we introduce a way to compare two simplicial maps.

Definition 2.2.15. For any two simplicial sets X, Y , we can define the Carte-sian product X × Y by (X × Y )n = X n × Y n with face and degeneracy mapsdi = (dX

i , dY i ), si = (sX

i , sY i ).

Example 2.2.16. If D and C are two small categories, then N (D × C ) ∼= N D ×N C .

For brevity we denote by 0, · · · , 0, 1, · · · , 1 the element 0 = · · · = 0 <1 = · · · = 1 of the set N 1n with exactly i copies of 1 for 0 ≤ i ≤ n + 1.

Definition 2.2.17. Let Φ, Ψ : X → Y be two simplicial maps between sim-plicial sets. We say Φ is simplicially homotopic to Ψ if there exists a natu-ral transformation : X × N 1 → Y such that n|Xn×(0,··· ,0) = Φn and n|Xn×(1,··· ,1) = Ψ n for all n ≥ 0. We call a simplicial homotopy from Φ

to Ψ , written as : Φ→Ψ or simply Φ Ψ .

We say two simplicial sets X and Y are homotopic , written as X Y , if there exist natural transformations Φ : X → Y and Ψ : Y → X such thatΨ Φ IdX and ΦΨ IdY .

There are two canonical simplicial maps ι0, ι1 : N 0 → N 1 by sending0, · · · , 0 to 0, · · · , 0 or 1, · · · , 1, respectively. Based on this observa-tion, being a simplicial homotopy is equivalent to having the followingcommutative diagram

X ∼= X × N 0

Φ

IdX×ι0 X × N 1

X × N 0 ∼= X IdX×ι1

Ψ Y .

Since (X × N 1)n consists of n + 2 copies of X n in the form of X n ×

0, · · · , 1, · · · , 1, combinatorially, the above definition is equivalent to sayingthat there exist maps i = si n|Xn×0,··· ,1,··· ,1 : X n → Y n+1 for 0 ≤ i ≤ nsuch that




d0 0 = Φn

dn+1 n = Ψ ndi j = j−1di i < jdj+1 j+1 = dj+1 j

di j =

j di−1 i > j + 1si j = j+1si i ≤ j

si j = j si−1 i > j.

Remember in the definition of i, 0, · · · , 1, · · · , 1 denotes the element 0 =· · · = 0 < 1 = · · · = 1 in N 1 with i copies of 1.

Definition 2.2.18. If the induced simplicial map N pt : N C → N • of thecanonical functor pt : C → • gives rise to a homotopy N C N •, then we sayC is contractible .

Note that there always exist various functors • → C . If C is contractible,then any functor ι : • → C will induce N • N C .

Proposition 2.2.19. If Φ : u → u is a natural transformation between two functors u, u : D → C , then N u is homotopic to N u. Consequently Cu and Cu are chain homotopic.

In particular, if N D N C , then C∗(D, R) and C∗(C , R) are chain homo-topy equivalent. Hence H∗(D, R) ∼= H∗(C , R) and H∗(D, R) ∼= H∗(C , R).

Proof. We can define a functor Φ : D × 1 → C by Φ(a, 0) = u(a), Φ(a, 1) =u(a) and Φ(α, 10) = u(α), Φ(α, 11) = u(α) and Φ(1a, <) = Φa for any

a ∈ Ob D and α ∈ Mor D. Then Φ = N Φ : N D × N 1 → N C is the desiredhomotopy between N u and N u. The combinatorial description of Φ directlyleads to the fact that Cu and Cu are chain homotopic.

The special case follows directly from the previous statements.

If u : D → C is a category equivalence, then we certainly have N D N C .

Corollary 2.2.20. 1. If u : D → C admits a (left or right) adjoint v : C → D.Then N D and N C are of the same simplicial homotopy type.

2. If C has an initial or a terminal object, then C is contractible.

Proof. Part (2) is a direct consequence of Part (1) because the inclusion of an initial (respectively, terminal) object into C has a right (respectively, left)adjoint functor. In order to prove Part (1) we notice that the unit and co-unitof the adjunction provide homotopies N uN v IdN C and N vN u IdN D bythe preceding proposition.

We end this section with an important concept although we will not useit in these notes.

Definition 2.2.21. Let s be the subcategory of whose morphisms arethe (monotonic) injections. A semi-simplicial object in a category T is acontravariant functor from s to T .




Suppose i : s → is the inclusion functor and Resi : T → T s admitsadjoint functors LK i, RK i : T s → T when T possesses all limits. This isthe historical context for the Kan extensions.

2.2.2 Classifying spaces

It is often very useful to have some knowledge about classifying spaces whencomputing the (co-)homology of a small category. In this section we recallseveral important results as facts, in order to obtain a balanced view towardskey results in the preceding section.

Definition 2.2.22. The geometric realization |X |, of a simplicial set X , isthe quotient of

n≥0

X n × n

by the equivalence relation given by (dix, y) ∼ (x, diy) and (six, y) ∼ (x, siy).

Definition 2.2.23. The geometric realization of N C , the nerve of a smallcategory C , is called the classifying space of C , customarily denoted by B C .

In the literature the classifying space of a small category is often writtenas |C|. However in these notes we try to avoid this notation since when C isa finite group, it has been used to denote the order of the group.

In the following examples, the first two are not hard to verify while theother two requires further knowledge from algebraic topology so we point outplaces where the reader may find proofs.

Example 2.2.24. 1. The classifying space of n is the standard n-simplex n.

2. The classifying space of the following category is S 1

. Recall its integralhomology and compare with Example 2.2.6 (1).

xα

β

y

3. The classifying space of Z2 is RP ∞. One can see this by using Milnor’sconstruction E Z2. Then up to homotopy BZ2 E Z2/Z2.

4. Let D and C be two small categories. There is a natural way to definethe join of N D and N C . Then N (D ∗ C ) ∼= N D ∗ N C and consequentlyB(D ∗ C ) BD ∗ BC , see [26].

The following theorem asserts that the classifying spaces are “good”.

Theorem 2.2.25. The space |X | is a CW-complex having one n-cell for each non-degenerate n-simplex of X .




It explains the structures of classifying spaces in Example 2.2.24.

Proposition 2.2.26. The singular homology and cohomology of BC with coefficients in R, namely H∗(BC , R) and H∗(BC , R), are identified with

H∗(C , R) and H

∗

(C , R).Proof. The normalized complex C†∗(C , R) is a cellular complex for computingthe singular homology of H∗(BC , R).

The essential connection between simplicial sets and topological spaces isgiven by the following result.

Theorem 2.2.27 (Kan). Let X be a simplicial set and Y a topological space.Then we have an adjunction

HomT op(|X |, Y ) ∼= HomSimpSet(X,SY ).

Moreover the adjunct preserves homotopies.

Proof. Let f : |X | → Y be a continuous map. Then we define a simplicialmap Φf by [(Φf )n(xn)](an) = f (xn, an) for n ≥ 0, xn ∈ X n and an ∈ n.Conversely if Φ : X → SY is a simplicial map, we define f Φ : |X | → Y byf Φ(xn, an) = [Φn(xn)](an) for n ≥ 0, xn ∈ X n and an ∈ n. These twoassignments give the adjunction.

As an example, every CW-complex is homotopy equivalent to the classi-fying space of the poset of its singular simplices.

Directly from Theorem 2.2.3 we have an alternative characterization of the geometric realization of a simplicial set X as

|X | ∼= lim−→ι/X|P|,

where |P| sends each object (n, Φ) to n, because a left adjoint functorpreserves direct limits. Given this homeomorphism we can prove Theorem2.2.27 alternatively as follows.

HomT op(|X |, Y ) ∼= HomT op(lim−→ι/X|P|, Y )

∼= lim←−ι/XHomT op(|P|, Y )

∼= lim←−ι/XHomSimpSet(P, SY )

∼= HomSimpSet(lim−→ι/XP, SY )

∼= HomSimpSet(X,SY ).

The third isomorphism does not depend on Theorem 2.2.27 and it is truebecause

HomT op(n, Y ) = SY n ∼= HomSimpSet(Hom(−, n), SY ).




Theorem 2.2.28. Let X, Y be two simplicial sets. Then there exists a natural homeomorphism

|X × Y | ∼= |X | × |Y |,

if |X | × |Y | is a CW-complex.

The space |X | × |Y | is a CW-complex if both X and Y are countable (thatis, both ∪n≥0X n and ∪n≥0Y n are countable) or if either |X | or |Y | is locally

finite (that is, every point is an inner point of a finite subcomplex).We collect some useful statements from last section which are adapted

to classifying spaces. The upshot is that a homotopy between two simplicialmaps or sets does give rise to a homotopy between continuous maps or spaces,in the usual topological sense. A small category C is contractible if BC hasthe same homotopy type of a point. By Theorem 2.2.27 it is equivalent tosaying that N C N •.

Corollary 2.2.29. A functor u : D → C induces a continuous map Bu :BD → BC .

1. If u : D → C is another functor and : N D × N 1 → N C is a homotopy between N u and N u, then we obtain a homotopy B : BD × 1 → B C between Bu and Bu.

2. A natural transformation Φ between u, u : D → C induces a homotopy B Φ : BD × 1 → BC between Bu and Bu.

3. If C has an initial or a terminal object, then B C is contractible.

The topological map Bu : BD → BC induces the maps u∗ and u∗ as inCorollary 2.2.13.

Remark 2.2.30. We must emphasize that continuous maps between classifyingspaces are not always realized by functors. In other words, not all topologicalmaps BD → BC come from a simplicial map N D → N C by Lemma 2.2.11.

For instance, there is a natural homeomorphism between B C and B C op, butone cannot construct a functor between C and C op realizing this homeomor-phism, except in very special situations (e.g. C is a group). See Example2.2.14.

We shall come back to comparing classifying spaces via functors by intro-ducing Quillen’s work.

2.2.3 Cup product and cohomology ring

In this section, we introduce cross product (or external product), cup product(or internal product) and the resulting multiplicative structure on simplicialcohomology.




Theorem 2.2.31 (Eilenberg-Zilber). Let X, Y be two simplicial sets. There is a natural chain homotopy equivalence

ξ : C∗(X × Y, R) → C∗(X, R) ⊗ C∗(Y, R).

Proof. The chain map, called the Alexander-Whitney map, can be explicitlyconstructed as follows

ξ (a, b) =n

i=0

di+1 · · · dn(a) ⊗ di0(b),

where (a, b) ∈ Cn(X × Y, R) = R[X n] ⊗ R[Y n].The inverse of ξ (up to chain equivalence), called the Eilenber-Zilber map

η : C∗(X, R) ⊗C∗(Y, R) → C∗(X × Y, R),

is defined by

η(c ⊗ d) = ( p,q)-shuffles σ

(−1)|σ|(sσ(1) · · · sσ( p)c, sσ( p+1) · · · sσ(n)d)

for c ∈ C p(X, R) and d ∈ Cq(Y, R) with p + q = n. Here a ( p, q )-shuffle σ isa permutation of n letters such that

σ(1) < · · · < σ( p) and σ( p + 1) < · · · < σ(n).

Remark 2.2.32. The Alexander-Whitney and Eilenberg-Zilber maps lead tochain homotopy equivalences between normalized chain complexes, in lightof Theorem 2.2.5.

Let C be a small category and R a commutative ring with identity. ForC∗(−, R) = C∗(−, R)∧, we have a cross product induced by the Alexander-Whitney map

× : Ci(X, R) ⊗Cj (Y, R) → HomR(Ci(X, R) ⊗Cj (Y, R), R)→Ci+j (X × Y, R)

with the left map given by

(f × g)(a, b) = f (di+1 · · · dn(a))g(di0(b)),

if f ∈ Ci(X, R), g ∈ Cj (Y, R), a ∈ Cn(X, R) and b ∈ Cn(Y, R). One cancheck that ∂ ∗(f × g) = ∂ ∗(f ) × g + (−1)if × ∂ ∗(g). Thus it induces a crossproduct on cohomology

× : Hi(X, R) ⊗ Hj (Y, R) → Hi+j (X × Y, R)

When R is a field, the map × is an isomorphism by Kunneth formula.




Proposition 2.2.33. Let X, Y be two simplicial sets and τ : X ×Y → Y × X the twist map, defined by τ (a, b) = (b, a) for any a ∈ X n and b ∈ Y n. Then τ induces an isomorphism τ ∗ : H∗(X × Y, R) → H∗(Y × X, R).

Suppose f ∈ Hi(X, R) and g ∈ Hj (Y, R). Then τ ∗(f × g) = (−1)ij g × f .

Proof. Since τ is an isomorphism of simplicial sets, it certainly induces anisomorphism on (co-)chain complexes and (co-)homology. Now we calculatethe value of τ ∗.

Define τ ∗ = ξτ ∗η : C∗(Y, R) ⊗C∗(X, R) → C∗(X, R) ⊗C∗(Y, R), as shownin the following commutative diagram

C∗(Y × X, R) τ ∗ C∗(X × Y, R)

ξ

C∗(Y, R) ⊗C∗(X, R)

τ ∗

η

C∗(X, R) ⊗ C∗(Y, R).

Then by direct calculation, it follows τ

∗ is chain homotopy equivalent to thecanonical chain equivalence τ ∗ : C∗(Y, R) ⊗C∗(X, R) → C∗(X, R) ⊗C∗(Y, R)determined by b ⊗ a → (−1)ij a ⊗ b for any a ∈ X i and b ∈ Y j . Henceτ ∗ : H∗(X × Y, R) → H∗(Y × X, R) is given by the chain equivalence inducedby τ ∗

HomR(C∗(X, R) ⊗ C∗(Y, R), R) → HomR(C∗(Y, R) ⊗ C∗(X, R), R)

On the other hand, there is a canonical chain equivalence

C∗(X, R) ⊗C∗(Y, R) → C∗(Y, R) ⊗C∗(X, R)

defined in the same fashion as τ ∗ . Thus there is a commutative diagram of cochain complexes

C∗(X, R) ⊗C∗(Y, R)∼=

C∗(Y, R) ⊗ C∗(X, R)

HomR(C∗(X, R) ⊗ C∗(Y, R), R) ∼=

HomR(C∗(Y, R) ⊗ C∗(X, R), R.

Since the vertical maps give rise to the cross products by sending f ⊗ g tof × g for any f ∈ Ci(X, R) and g ∈ Cj (Y, R), while the horizontal maps mapf ⊗ g and f × g to (−1)ij g ⊗ f and (−1)ij g × f , respectively, our formula forτ ∗ follows from it.

Definition 2.2.34. There exists a diagonal functor ∆ : C → C × C such

that ∆(x) = (x, x) and ∆(α) = (α, α) for any x ∈ Ob C and α ∈ Mor C ,respectively.




The diagonal functor induces a natural chain map, the simplicial diagonal map, also denoted by

∆ : C∗(C , R) → C∗(C × C , R).

Composing with the Alexander-Whitney map

C∗(C × C , R) → C∗(C , R) ⊗ C∗(C , R),

we get the diagonal map,

∆ : C∗(C , R) → C∗(C , R) ⊗C∗(C , R).

More explicitly for any x0 → x1 → · · · → xn, a base element of Cn(C , R),

∆(x0 → · · · → xn) =n

i=0

(x0 → · · · → xi) ⊗ (xi → · · · → xn).

By dualizing the diagonal map, we obtain the cup product on C∗(C , R) =C∗(C , R)∧ as

∪ : Ci(C , R) ⊗ Cj (C , R) ×−→Ci+j (C × C , R)

∆−→Ci+j(C , R).

Suppose f ∈ Ci(C , R) and g ∈ Cj (C , R) with i + j = n. Then

(f ∪ g)(x0 → · · · → xn) =n

k=0 f (x0 → · · · → xk)g(xk → · · · → xn)= f (x0 → · · · → xi)g(xi → · · · → xn).

From direct calculation we get ∂ ∗(f ∪ g) = ∂ ∗(f ) ∪ g + (−1)if ∪ ∂ ∗(g) forany f ∈ Ci(C , R) and g ∈ Cj (C , R). Consequently the cup product passes tocohomology

∪ : Hi(C , R) ⊗ H

j (C , R) ×−→H

∗(C × C , R) ∆−→H

i+j (C , R),

which makes the graded R-module H∗(C , R) =

i≥0 Hi(C , R) a graded ring.This is the ordinary cohomology ring of C with coefficients in R. We mayreadily verify the following equality.

Proposition 2.2.35. f ∪ g = (−1)ij g ∪ f for any f ∈ Hi(C , R) and g ∈Hj (C , R).

Proof. It follows from Proposition 2.2.33 because the cup product is inducedby a cross product.

It means that the ordinary cohomology ring of a small category is graded

commutative . Since the cup product is exactly the same as the one for thetopological cohomology ring H∗(BC , R) =

i≥0 Hi(BC , R). We actually have

a ring isomorphism H∗(C , R) ∼= H∗(BC , R).




Remark 2.2.36. When C is a (discrete) group, H∗(C , R) is the usual groupcohomology ring.

Corollary 2.2.37. Let u : D → C be a functor. Then u∗ : H∗(C , R) →

H

∗

(D, R) is a ring homomorphism.This ring homomorphism is usually called the restriction .

Proposition 2.2.38. Let D and C be small categories. Suppose k is a field.Then the cup product of any two positive degree elements in H∗(D ∗ C , k) is constantly zero.

Proof. In Proposition 2.2.8 we computed homology groups of the join of twosmall categories. Accordingly there are formulas for cochain complexes andcohomology as well, i.e. for each n > 0

Cn(D ∗ C , k) ∼= [C∗(D, k) ⊗ C∗(C , k)]n−1,

and Hn(D ∗ C , k) =

i+j=n−1

Hi(D, k) ⊗ H

j(C , k).

It implies that Hn(D ∗ C , k) is spanned over a set of cohomology classesrepresented by functions of the form ηα such that ηα(β ) = 1 if β = α andzero otherwise, and such that α ∈ N n(D ∗C ) consists of exactly one morphismγ x,y where x ∈ Ob D and y ∈ Ob C , as introduced in Definition 2.2.7. Bydirect computation, the cup product of any two such functions is constantlyzero. Hence we are done.

There is a topological interpretation to the above result. In fact the co-homology ring of a join of two spaces always have trivial cup product. SeeExample 2.2.24 (4). There is a topological interpretation. Suppose X and Y

are two spaces. Then their join X ∗ Y is a suspension Σ (X × Y /X ∨ Y ).For any suspended space, its cohomology ring has trivial cup product. Thestructure of X ∗ Y also explains Proposition 2.2.8. In Example 2.2.6 (1), thecup product of any two elements of positive degree in H∗(C , R) is zero.

2.3 Quillen’s work on classifying spaces

In order to assure a functor u : D → C inducing a simplicial homotopyequivalence N D N C , one must have a functor on the opposite direction.However one may very often obtain a (topological) homotopy equivalenceBu : B D → B C without the existence of such v : C → D that Bv is also a

homotopy equivalence. Such homotopies certainly are not simplicial and willbe investigated now.



2.3 Quillen’s work on classifying spaces 49

2.3.1 Quillen’s Theorem A

Quillen’s Theorem A (also called Quillen’s Fibre Theorem ) provides sufficientconditions on a functor between two small categories which guarantee the

functor inducing a homotopy equivalence between classifying spaces.

Theorem 2.3.1 (Quillen’s Theorem A). Let u : D → C be a functor between two small categories. If either all the overcategories or all the under-categories are contractible. Then Bu is a homotopy equivalence.

Proof. The proof is given in Section 2.4.3.

Under certain circumstances, one may replace the over- or undercategorieswith some simpler constructions.

Definition 2.3.2. Let u : D → C be a functor between small categories. Foreach x ∈ Ob C , we define the fibre of u over x to be the category u−1(x) whoseobjects are preimages of x and whose morphisms are preimages of 1x.

Apart from intuition, the fibres are usually not good for comparing thehomotopy types of two classifying spaces. For instance if G is a group andH is a subgroup, then the inclusion functor has fibre at • equal to the trivialcategory. But there is no reason to say that BH would be homotopy equiva-lent to BG, especially when we think about the extreme case where H is thetrivial subgroup of G.

Definition 2.3.3. Let u : D → C be a functor between small categories. Wecall D prefibred over C if the natural functor ix : u−1(x) → x\u has a rightadjoint hx for each x ∈ Ob C . We call D precofibred over C if the naturalfunctor ix : u−1(x) → u/x has a left adjoint hx for each x ∈ Ob C .

Corollary 2.3.4. Suppose u : D → C such that D is either prefibred or pre-cofibred over C . If the fibre u−1(x) over every x ∈ Ob C is contractible, then Bu is a homotopy equivalence.

Proof. Since there are adjoint functors between u−1(x) and x\u (or u/x) forevery x ∈ Ob C . The condition implies the contractibility of all the undercat-egories (or overcategories). Then we apply Quillen’s Theorem A.

We will see some prefibred and precofibred examples in the next section.Note that, for G a group and H a subgroup, i : H → G is a simple examplethat H is neither prefibred nor precofibred over G, unless H = G.

Suppose u : D → C makes D precofibred over C . Then we have

Homu−1(x)(hx(a, α), b) ∼= Homu/x((a, α), ix(b))

for any b ∈ Ob u−1(x) and (a, α) ∈ Ob u/x. Let us try to understand hx.The set Homu/x((a, α), b) consists of exactly those f ∈ HomD(a, b) such that




u(f ) = α : u(a) → x = u(b). The identity 1hx(a,α) corresponds to a morphism

in Homu/x((a, α), ixhx(a, α)), which is some morphism f ∈ HomD(a, hx(a, α))

such that u( f ) = α. Then each f ∈ HomD(a, b), satisfying u(f ) = α, mustuniquely factor through f . If f = g f for g ∈ HomD(hx(a, α), b) such that

u(g) = 1x, then g corresponds to f ∈ Homu/x((a, α), ix(b)) in the isomor-phism.

We have alternative descriptions of hx and hx.

Lemma 2.3.5. Suppose u : D → C is as above and γ : x → y is a morphism in C .

1. If D is prefibred over C , then it induces a functor

γ : u−1(y) → y\u → x\u → u−1(x).

Furthermore the right adjoint of u−1(x) → x\u is given by (α, a) → α(a).2. If D is precofibred over C , then it induces a functor

γ : u−1(x) → u/x → u/y → u−1(y).

Furthermore the left adjoint of u−1(y) → u/y is given by (b, β ) → β (b).

Proof. We shall only prove (2) because the proof for (1) is similar.Let (b, β ) ∈ Ob u/y. Then

β : b → (b, 1y) → (b, β ) → hy(b, β ).

Hence β (b) = hy(b, β ).

Definition 2.3.6. Let u : D → C be a functor between small categories. Wecall D fibred over C if it is prefibred over C and moreover (βα) = αβ forany composable α, β ∈ Mor C .

We call D cofibred over C if it is precofibred over C and moreover (βα) =β α for any composable α, β ∈ Mor C .

2.3.2 Constructions over categories and relevant functors

We describe several important constructions here, which also serve an exam-ples to illustrate Quillen’s Theorem A.

Definition 2.3.7. Suppose C is a small category. The (first) category of fac-torizations in C , denoted by F (C ), has Ob F (C ) = Mor C . An object admits amorphism to another object if and only if as morphisms in C , the first objectfactors through the second.



2.3 Quillen’s work on classifying spaces 51

Iterating the construction, for non-negative integer n, we can defineF n(C ) = F (F n−1(C )) as the n-th category of factorizations in C . Here F 0(C )is defined to be C .

We shall see that in order to understand C very often we have to utilizeF (C ).In order not to cause confusions, when a morphism α ∈ Mor C is considered

as an object in F (C ), we write it as [α]. In the above definition, a morphismin F (C ) is a pair (α, α) : [α] → [β ] and is customarily pictured by

y

α

xα

y x

α

β

Example 2.3.8. 1. When C is a group, F (C ) has all group elements as objects.The category F (C ) is a groupoid whose skeleton is isomorphic to the groupitself.

2. When C is a poset, then F (C ) is another poset. For 1, we have F (1) asfollows

[1 ← 0]

(11,10)

[0 ← 0]

(α,10)

(10,10)

[1 ← 1]

(11,α)

(11,11)

For convenience we denote the non-identity morphism 1 ← 0 by α.

One can see that there are two natural covariant functors t : F (C ) → C and s : F (C ) → C op, where t stands for target while s means source .

Proposition 2.3.9. Suppose D C . Then F (D) F (C ).

Proof. Let u : D → C and v : C → D be such that vu ∼= IdD and uv ∼= IdC. Weconstruct two functors between F (C ) and F (D) which induce an equivalence.

We define F (u) : F (D) → F (C ) by F (u)([α]) = [u(α)], and F (u)(α, α) =(u(α), u(α)). One can verify this gives a functor. Similarly there exists afunctor F (v) : F (C ) → F (D). Since F (v)F (u)([α]) = [vu(α)] ∼= [α], we haveF (v)F (u) ∼= IdF (D). Similarly we get F (u)F (v) ∼= IdF (C). Thus F (D) F (C ).

Proposition 2.3.10. Consider the canonical functors C t←F (C )

s→C op. Then




1. F (C ) is cofibred over both C and C op. It implies, for any x ∈ Ob C , the natural functor ix : t−1(x) → t/x has a left adjoint, and, for any y ∈Ob C op, the natural functor iy : t−1(y) → s/y has a left adjoint;

2. for any x ∈ Ob C , the category t−1(x) ∼= (IdC/x)op has an initial object;

3. for any y ∈ Ob C op

, the category s−1

(y) ∼= y\IdC has an initial object.Consequently both t and s induce homotopy equivalences.

Proof. The category F (C ) is precofibred over C because we can definehx([α], α) = [αα] for any x ∈ Ob C and ([α], α) ∈ Ob t/x such that

Homt−1(x)(hx([α], α), [β ]) ∼= Homt/x(([α], α), ix(β )).

We can compute t−1(x) of t : F (C ) → C for every x ∈ Ob C and find t−1(x) ∼=(IdC/x)op. It has an initial object [1x] and thus is contractible. This meansthat t/x is always contractible and thus t induces an equivalence by Quillen’sTheorem A. Moreover from the description of t−1(x) we can see that F (C ) iscofibred over C .

We can similarly prove that s : F (C ) → C op induces an equivalence as well,with s−1(y) ∼= y\IdC.

A hidden fact in the proof is that whenever there is a category C , we canproduce many over- and undercategories IdC/x and x\IdC, all of which arecontractible. This will be used for an important construction later on.

Remark 2.3.11. Here the homotopy equivalence B t : BF (C ) → BC usually isnot a simplicial homotopy equivalence as we do not expect to have a func-tor from C to F (C ) giving a homotopy equivalence BC → BF (C ). If such afunctor were to exist, then we would obtain a functor inducing the homo-topy equivalence BC → BC op, and hence inducing isomorphism on homology

H∗(C , R) → H∗(C op, R), which rarely happens. See Example 2.2.14.

Definition 2.3.12. The functor ∇ = (t, s) : F (C ) → C e = C × C op is called

the skew diagonal functor . The category C e

is called the enveloping category of C .

The above construction is a special case of a more general definition F (u)for any u : D → C , naturally equipped with a functor F (u) → D×C op. IndeedF (IdC) = F (C ), see Section 2.4.3.

Example 2.3.13. When C is a poset, both F (C ) and C e are posets and ∇ :F (C ) → C e is an embedding. For the poset C = 1, we have F (1) → 1e asfollows

[1 ← 0] (1, 0)

[0 ← 0]

[1 ← 1]

→ (0, 0)

(1, 1)

(0, 1)



2.4 Further categorical and simplicial constructions 53

Proposition 2.3.14. The category F (C ) is cofibred over C e. It implies that, for any (y, x) ∈ Ob C e, the functor i(y,x) : ∇−1(y, x) → ∇/(y, x) has a left adjoint. Thus N ∇−1(y, x) N ∇/(y, x).

Furthermore ∇−1(y, x) ∼= HomC(x, y).

Proof. We first prove that F (C ) is precofibred over C e. To this end, we need toshow that for every (y, x) ∈ Ob C e, the functor i(y,x) : ∇−1(y, x) → ∇/(y, x)has a left adjoint. It is easy to identify ∇−1(y, x) with the discrete setHomC(x, y). Then we define h(y,x)([α], (α, α) ) = [ααα] ∈ HomC(x, y)which leads to an isomorphism

Hom∇−1(y,x)([ααα], [β ]) ∼= Hom∇/(y,x)(([α], (α, α)), ([β ], (1y, 1x))).

Note that in any case either morphism set has at most one element.If (γ , γ ) : (y, x) → (y, x) is a morphism, then (γ , γ ) is given by

[α] → ([α], (1y, 1x)) → ([α], (γ , γ )) → [γ αγ ],

for any α ∈ HomC(x, y). Hence one can verify that F (C ) is cofibred over C e.

We note that some of the overcategories or undercategories or fibres canbe empty. As a reminder, the homology of ∅ is constantly zero.

Remark 2.3.15. In the previous two propositions F (C ) are cofibred over C , C op

as well as C e. It is not fibred over these categories. For instance in Example2.3.13, ix : ∇−1(1, 0) = [1 ← 0] → ∇/(1, 0) ∼= F (1) has only a left adjoint andno right adjoint because [1 ← 0] is a terminal object in F (1). In these notes,it is very important to compute limits over various over and undercategories.We often do it by establishing adjoint functors between these categories andsome simpler categories. Then by Proposition 1.2.4 we will know whether we

can simplify the computations of either direct or inverse limits (usually not both! ). We shall see many explicitly calculations from Chapter 3.

2.4 Further categorical and simplicial constructions

In this section we will introduce the definition of a Grothendieck construc-tion which is important when we deal with finite categories constructed fromgroups. Grothendieck constructions are a certain kind of homotopy colimits.By Thomason’s Homotopy Colimit Theorem, every homotopy colimit is aGrothendieck construction. In order to provide the interested reader a neces-sary understanding of these constructions, we will define bisimplicial sets, and

since bisimplicial sets are introduced, there is no reason not to give proofsof Quillen’s and Thomason’s theorems. Although this chapter is the mostsuitable place to present these materials, it is possible to postpone reading




them until they are needed. Indeed, only understanding the Grothendieckconstructions and knowing the statements are sufficient for our purposes inthese notes.

2.4.1 Grothendieck constructions

Let C be a small category and F : C → Cat a functor. Then we can constructa small category, called the Grothendieck construction GrCF , as follows.

The objects are pairs of the form (x, a) with x ∈ Ob C and a ∈ Ob F (x).Let (x, a) and (y, b) be two objects. A morphism (x, a) → (y, b) is a pair (α, f )such that α ∈ HomC(x, y) and f ∈ HomF(b)(F (α)(a), b). It can be pictured as

y A B

x

α

C

D

y A B

x

α

C

D

Note that there always exists a functor π : GrCF → C .

Example 2.4.1. Let u : D → C be a functor between small categories. Thenthe over- and undercategories can be realized as Grothendieck constructions.Given an object x ∈ Ob C , we can define two functors u = HomC(u(−), x) :D → Set and u = HomC(x, u(−)) : D → Set . One can verify that GrDu and

GrDu are isomorphic to u/x and x\u, respectively.

We record one more important Grothendieck construction, although it isnot needed in this chapter.

Definition 2.4.2. A small category C is an EI category if all endomorphismsare isomorphisms.

Typical examples of EI categories are groups and posets. For each EIcategory C , one can introduce a partial order on the set of isomorphismclasses of objects by [x] ≤ [y] if and only if HomC(x, y) = ∅. Let us denote by[C ] the poset of isomorphism classes of objects in C . Obviously there existsa natural functor p : C → [C ]. Since [C ] is a poset, we know the barycentricsubdivision of [C ], written as S ([C ]), is another poset whose objects are chains

of non-isomorphism objects [x]n = [x0] → [x1] → · · · → [xn] and there is amorphism [x]n → [y]m if and only if [y]m is a subchain of [x]n. We define afunctor




p : S ([C ]) → Cat

by asking p([x]n) to be a category whose objects are the yn = y0 → y1 →· · · → yn in N C n such that p(yi) = [xi] for all 0 ≤ i ≤ n, and whosemorphisms are (n+1)-tuples of morphisms in C making the following diagram

commutative

y0α1

γ 0

y1

γ 1

α2 · · · αn yn

γ n

y0 β1

y1 β2 · · ·

βn

yn

Since C is EI, all γ i are isomorphism. This means that p([x]n) is always agroupoid. One can readily verify that for any xn → ym in Mor(sd[C ]), there is awell defined functor p(xn) → p(ym) given by first removing objects in xn whichare not isomorphic to any object in ym, and then composing appropriatemorphisms in what is left of xn.

Definition 2.4.3 (Slominska). Let C be an EI category. Then its subdivi-sion , written as S (C ), is defined to be GrS ([C])p.

Alternatively S (C ) is a category whose objects are chains of non-isomorphismsxn = x0 → x1 → · · · → xn in C , and a morphism from xn to ym = y0 →y1 → · · · → ym is an (m + 1)-tuple of isomorphisms (γ 0, · · · , γ m), making thefollowing diagram commutative

xn0

α1

γ 0

xn1

γ 1

α2 · · · αn xnm

γ n

y0

β1

y1β2

· · ·βn

ym

in which every xni ∈ xini=0 and xni ∼= yi in C . From this descriptionthere exists a functor s : S (C ) → C by xn → x0. Thus there are maps s∗ : H∗(S (C ), R) → H∗(C , R) and s∗ : H∗(C , R) → H∗(S (C ), R).

Proposition 2.4.4 (Slominska). The map s induces a homotopy equiv-alence BS (C ) BC . Hence we have isomorphisms s∗ : H∗(S (C ), R) →H∗(C , R) and s∗ : H∗(C , R) → H∗(S (C ), R).

Proof. We shall prove all undercategories are contractible. In order to do so,we show S (C ) is prefibred over C . Then one can easily verify that for eachx ∈ Ob C , s−1(x) has x as a terminal object and hence is contractible.

To show S (C ) is prefibred over C we only have to find a right adjoint to theinclusion ix : s−1(x) → x\ s. Suppose (α, x0 → x1 → · · · → xn) is an object

in x\ s. Then we define a functor x\ s → s−1

(x) by

(x0α1→x1 → · · · → xn, α) → (x

α1α→ x1 → · · · → xn).




Moreover if (γ 0, γ 1, · · · , γ m) is a morphism from (α, x0 → x1 → · · · → xn) to(α, x0 → x1 → · · · → xm), then its image is defined as (1x, γ 1, · · · , γ n). Nowone can check this functor is right adjoint to ix. Hence we are done.

These isomorphisms will be generalized in Chapter 4.

There are two or three occasions in this book that we need to understandthe classifying spaces of some categories. Thus we provide two examples onsubdivisions of categories.

Example 2.4.5. 1. Let C be the following EI category

xG α y H .

Then its underlying poset [C ] is x → y, and S ([C ]) is the poset [x] ← [x →y] → [y]. One can easily find that p[x] and p[y] are

xG , y H

respectively. The groupoid p[x → y] is

x α G×H y ,

in which we denote by α the only object in p[x → y]. The two functorsp[x → y] → p[x] p[x → y] → p[y] are the obvious projections. Thus the

category S (C ) has objects of the forms ([x], x), ([y], y) and ([x → y], x α→y).

All the morphisms in S (C ) are of the following forms

• ([x → y] → x, g), ∀g ∈ Mor(p[x]) = AutC(x) = G;• ([x → y] → y, h), ∀h ∈ Mor(p[y]) = AutC(y) = H ;• (1[x→y], (g, h)), ∀(g, h) ∈ Mor(p[x → y]) = AutC(x) × AutC(y) =

G × H .

As one can see the subdivision S (C ) is more complicated than C itself so itseems not helpful in terms of understanding BC through BS (C ). However,in the next section, we shall see it does help if we combine with Thomason’sHomotopy Colimit Theorem.For future reference, we record two more finite EI categories, which sharethe same underlying poset with C . The reader may try to figure out theirsubdivisions. We shall give only partial descriptions of these constructions.

2. Let D be the following category

x

1x

h

α β

y 1y ,

where h2 = 1x and βh = α. Then we also have poset [D] = x → y , andS ([D]) = [x] ← [x → y] → [y]. We can find p[x] and p[y] are




x

1x

h

, y 1y

respectively. The third groupoid p[x → y] has two isomorphic objects x α→y

and x β→y. Its skeleton is the trivial category.

3. Let E be the following EI category

x

1x

g

h

gh

α β

y 1y ,

where g2 = h2 = 1x, gh = hg,αh = α,αg = β,βh = β,βg = β . Then weagain have poset [E ] = x → y, and S ([E ]) = [x] ← [x → y] → [y]. By directcalculations, p[x] and p[y] are

x

1x

g

h

gh

, y 1y

respectively. The groupoid p[x → y] has two isomorphic objects x α→y and

x β→y. However in this case, its skeleton is isomorphic to C 2 = 1x, h,

which is the subgroup of AutC(x) × AutC(y) that stabilizes the set α, β .

We will come back to these examples at the end of the next section.

2.4.2 Bisimplicial sets and homotopy colimits

Definition 2.4.6. A bisimplicial set is a contravariant functor × to Set .

Because

HomCat(op × op, SimpSet ) ∼= HomCat(op, HomCat(op, SimpSet )),

clearly we may regard a bisimplicial set as a simplicial object in the categoryof simplicial sets. Thus a bisimplicial set is indeed a simplicial simplicial set.

Alternatively, a bisimplicial set is a bigraded sequence of sets X p,q for p, q ≥ 0, together with horizontal face and degeneracy maps dh : X p,q →

X p−1,q and sh : X p,q → X p+1,q as well as vertical face and degeneracy mapsdv : X p,q → X p,q−1 and sv : X p,q → X p,q+1. These maps must satisfysimplicial identities in either direction, and in addition every horizontal map




must commutes with every vertical map. The total complex of a bisimplicial set , Tot(X ), is the simplicial set

Tot(X )n :=

n≥0

X n,∗ × n/ ∼,

in which the equivalence is given by (dhi x, y) ∼ (x, diy) and (sh

i x, y) ∼(x, siy).

We can also consider bisimplicial objects in an abelian category A-Modfor some associative ring. Then by Dold-Kan Correspondence , the category of bisimplicial A-modules is equivalent to the category of first quadrant doublecomplexes of A-modules.

Fix a commutative ring R. Each bisimplicial set gives rise to a first quad-rant double complex R[X ] p,q with differentials ∂ h p,q =

i(−1)idh

i and ∂ v p,q =

(−1) p

i(−1)idvi . Consequently the category of bisimplicial R-modules is

equivalent to the category of first quadrant double chain complexes of R-modules.

Definition 2.4.7. The diagonal , diagX , of a bisimplical set X is the simpli-cial set obtained via restriction along the diagonal functor → × .

Theorem 2.4.8 (Bousfield-Kan ?). Let X, Y be two bisimplicial sets.

1. There is a natural simplicial isomorphism Tot(X ) ∼= diagX .2. Given a bisimplicial map Φ p,q : X p,q → Y p,q, if for all q , X ∗,q → Y ∗,q is a

weak homotopy equivalence, then so is diagΦ : diagX → diagY .

Proof. For the first part, we define a map Ψ , induced by the maps X p,∗× p →diagX p given by

(x, d py) → dh p x ∈ X p,p,

for any (x, d py) ∈ X p,q × pn. One may readily check that Ψ is a simplicial

isomorphism.As for Part (2), we will not provide a proof here but it is not surprising

from Part (1). See [9], [Tornehave], [Reedy] and [8, 16].

Definition 2.4.9. Suppose G : C → SimpSet is a functor. Then the sim-plicial replacement of G is the bisimplicial set

∗G which in dimension n

consists of the coproduct

(

∗G)n :=

x0→···→xn∈N Cn

G(x0).

The horizontal face map di maps G(s(σ)) to G(s(diσ)) by the identity mapif i > 0 and by α0 if i = 0. The horizontal degeneracy map si maps thesimplicial set G(s(σ)) to G(s(siσ)) by the identity map.

Definition 2.4.10. Let G : C → SimpSet be a functor. The homotopy col-imit of G, hocolimCG, is diag(

∗G).




For example one can easily see that hocolimCN pt = N C .

Theorem 2.4.11 (Thomason’s Homotopy Colimit Theorem). Sup-pose C is a small category and F : C → Cat is a functor. Let GrCF be the Grothendieck construction on F . Then there is a natural weak equivalence

N GrCF hocolimCN F .

The proof is in the next section. It is clear from the proof of Thomason’stheorem, and Definitions 2.2.2 and 2.4.9 that

BGrCF hocolimCBF .

Remark 2.4.12. The homotopy colimit is a replacement in T op of the colimitwe introduced earlier. In the category T op, the colimit of a diagram of spacesdoes exit. However it does not behave very well and thus is not good to workwith in algebraic topology. One can find a well-known example in [DH].

There is also a dual notion of homotopy limit but we shall not touch it in

this book.

The following examples provide the only three homotopy colimits that thereader needs to know for our purposes. In fact, they are only used in Section4.2.2, for computing certain ordinary (co)homology.

Example 2.4.13. When C is the poset P : a ← b → c, for any F : P → Cat thehomotopy colimit hocolimP BF is called a homotopy pushout because it fitsinto the right lower corner of the following diagram

BF (b) BF(b→c)

BF(b→a)

BF (c)

BF (a) hocolimP BF

such that it is commutative up to homotopy of continuous maps, andhocolimCBF enjoys a certain universal property, analogous to that of a col-imit.

Example 2.4.14. In Example 2.4.5, the classifying spaces of the subdivisionsof all three categories can be realized as homotopy pushouts, according toExample 2.4.13. For instance, in Example 2.4.5 (3) we have BE B(C 2 ×C 2)/BC 2 because we have the following homotopy pushout diagram

Bp([x → y]) = BC 2inclusion

Bpt

B(C 2 × C 2) = B p([x])

Bp([y]) = B• B(C 2 × C 2)/BC 2 hocolimP Bp BS (E )




Similarly, in Example 2.4.5 (1), we get BC BG ∗ BH (a join of spaces),and in Example 2.4.5 (2) BD BC 2. At least in these small examples,subdivisions are useful to obtain the homotopy type of classifying spaces of categories. This is one of the the reasons why we introduced the Grothendieck

construction earlier.

2.4.3 Proofs of Quillen’s Theorem A and Thomason’stheorem

The proofs are taken directly from the original papers of Quillen [61] andThomason [77].

In order to prove Quillen’s Theorem A, we shall first generalize the con-struction of category of factorizations.

Definition 2.4.15. Suppose u : D → C is a functor between two small

categories. We define F (u) to be a category whose objects are (a,α,x)with a ∈ Ob D, x ∈ Ob C and α ∈ HomC(x, u(a)). A morphism from(a,α,x) to (b,β,y) is a pair of morphisms (f, γ ) with f ∈ HomD(a, b) andγ ∈ HomC (y, x) such that β = u(f )αγ as in the diagram

u(a)

u(f )

xα

u(b) y

γ

β

Note that there exists two natural functors Pt : F (u) → Dop and Ps :F (u) → C , as well as Pt ×Ps : F (u) → D × C op. For any category C , F (IdC)

is exactly the category of factorizations in C , F (C ), such that Ps = s andPt = t.Let us notice that any element (a0, α0, x0) → (a1, α1, x1) → · · · →

(an, αn, xn) ∈ N F (u)n can be pictured as a commutative diagram

x0

α0

x1

α1

· · · xn

αn

u(a0) u(a1) · · · u(an)

Proof (of Quillen’s Theorem A). We will only prove the case where all under-categories associated wtih u are contractible. The other case can be shownsimilarly.

Let F (u) be as above. We define a functor N [IdC/u(−)]∗ : D → SimpSet and consider its simplicial replacement X =

∗N [IdC/u(−)]∗ the bisimpli-

cial set with X p,q consists of pairs of sequences of morphisms in C and D,




respectively,

(x p → · · · → x0 → u(a0), a0 → a1 → · · · → aq).

The horizontal and vertical face and degeneracy maps are given in the obviousway. We have a simplicial map X ∗q → N Dq as well as a simplicial mapX p∗ → N C op

p . Since the diagonal of X is the nerve of F (u), these two functorsprovide two more simplicial maps

a0→···→aq

N (IdC/u(a0))opq →

a0→···→aq

N •q = N Dq

and xp→···→x0

N (x0\u) p →

xp→···→x0

N • p = N C op p .

The geometric realizations of these two maps are homotopy equivalences be-cause both IdD/u(a0) and x0\u are contractible (the latter by assumption).

Hence by Theorem 2.4.7 (2) we obtain two homotopy equivalences

BD BF (u)BPt BPs BC op.

By examining the following diagram

C op F (u)Ps Pt

u

D

u

C op F (C )

Ps=s

Pt=t C

where u is defined by u(a,α,x) = (u(a), α , x). It forces Bu and thus Bu to

be homotopy equivalences.

The proof of Thomason’s homotopy colimit theorem is in spirit simi-lar to that of Quillen’s Theorem A. We remind the reader that N F (u) diagX = hocolimDN [IdC/u(−)] in the above proof, and one verifies thatGrDIdC/u(−) ∼= F (u).

Proof (of Thomason’s theorem). We shall establish a natural map

η : hocolimCN F → N GrCF

and construct a new functor F : C → Cat so that we have natural equivalences

hocolimCN F hocolimCN F λ1

λ2

N GrCF

satisfying ηλ1 λ2.




Step 1: Similar to the proof of Quillen’s Theorem A, hocolimCN F is thediagonal of

∗N F such that for any integer n ≥ 0, (

∗N F )n consists of

n-simplicies

(x0α1→ · · ·

αn→xn, a0f 1→ · · ·

f n→an)

where x0 → · · · → xn ∈ N C n and a0 → · · · → an ∈ N F (x0)n. We define η by

ηn(x0α1→ · · ·

αn→xn, a0f 1→ · · ·

f n→an)

= (x0, a0)(α1,F(α1)(f 1))

−→ (x1,F (α1)(a1))(α2,F(α2α1)(f 2))

−→ · · ·

· · ·(αn,F(αn···α1)(f n))

−→ (xn,F (αn · · · α1)(an))

Step 2: Let us construct λ1. We define F : C → Cat to be the functor

F (x) = π/x

for π : GrCF → C . Recall that the objects in F (x) are of the form ((y, a), α)such that y ∈ Ob C , a ∈ Ob F (y) and α : y → x, and moreover any α : y → xinduces F (α) : F (y) → F (x). There exists a canonical functor

F (x) → F (x)

for any x ∈ Ob C , given by ((y, a), α) → F (α)(a). This functor has a rightadjoint F (x) → F (x) by a → ((x, a), 1x). Consequently N F (x) → N F (x) isa simplicial homotopy equivalence. Since all functors constructed previously,F (x) → F (x), assemble to a natural transformation

F → F

of functors C → Cat , we obtain a simplicial homotopy equivalence N F → N F which gives rise to a homotopy equivalence, by Theorem 2.4.7 (2),

λ1 : hocolimCN F →hocolimCN F .

Step 3: Now we show there is a simplicial homotopy equivalence

λ2 : hocolimCN F → GrCF .

Similar to Step 1, hocolimCN F is the diagonal of

∗N F such that for any

integer n ≥ 0, (∗N F )n consists of n-simplicies

(x0α1→ · · ·

αn→xn, c0(h1,l1)

→ · · ·(hn,ln)

→ cn)

where x0 → · · · → xn ∈ N C n and c0 → · · · → cn ∈ N F (x0)n = [N (π/x0)]n.Note that each ci is of the form ((yi, bi), β i) such that yi ∈ Ob C , bi ∈ Ob F (yi)




and β i : yi → x0. Moreover, hi+1 : yi → yi+1 satisfies β i+1hi+1 = β i, andli+1 : F (hi+1)(bi) → bi+1, for every i ≥ 0. Note that there are morphismsαi · · · α1β i : yi → xi. We define λ2 by

λ2(x0α1

→ · · ·αn

→xn, ((y0, b0), β 0)(h1,l1)

→ · · ·(hn,ln)

→ ((yn, bn), β n))= (y0, b0)

(h1,l1)→ · · ·

(hn,ln)→ (yn, bn).

Let us consider N GrCF as a bisimplicial set with ( p, q )-simplicies N GrCF p,q =N GrCF q. This means it is constant in horizontal direction. Clearly N GrCF ∗ =diagN GrCF ∗,∗. The simplicial map λ2 is the diagonalization of an obvious

bisimlicial map λ :∗GrCF → N GrCF ∗,∗.

Thus, by Theorem 2.4.7 (1), we only have to show for each q , λ∗,q :∗N F ∗,q → N GrCF ∗,q is a homotopy equivalence. But λ∗,q is the coprod-

uct over all q -simplicies (x0, c0) → · · · → (xq, cq) of N GrCF of the map p xq\IdC →

p •. Then by Theorem 2.4.7 (2) λ∗,q, and so λ2, are homo-

topy equivalences.

Step 4: Finally we establish a simplicial hopmotopy : hocolimCN F × N 1 → N GrCF

from ηλ1 to λ2. We define it by

(x0α1→ · · ·

αn→xn, ((y0, b0), β 0) → · · · → ((yn, bn), β n)) × (

(i+1) zeros 0, · · · , 0 , 1, · · · , 1)

= (y0, b0)(h1,l1)

→ · · ·(hi,li)

→ (yi, bi)(αi+1···α1βi,F(αi+1···α1βi+1)(li+1))

−→

(xi+1,F (αi+1 · · · α1β i+1)(bi+1))(αi+2,F(αi+2···α1βi+2)(li+2))

−→

(xi+2,F (αi+2 · · · α1β i+2)(bi+2)) → · · · → (xn,F (αn · · · α1β n)(bn)).

One may verify that all simplicial identities are met. Hence we are done.





Chapter 3

Category algebras and theirrepresentations

Abstract The concept of a category algebra is introduced here. The relationship between the module category of a category algebra and an appropriatefunctor category is demonstrated at the very beginning. We shall characterize

the intrinsic properties of category algebras. Especially we show how onemay compare a category algebra with a cocommutative bialgebra so suchalgebras enjoy similar homological properties. We are particularly interestedin finite EI category algebras because we may classify their indecomposableprojective and simple modules. When dealing with category algebras, theusual homological tools, such as induction and coinduction, are replaced byKan extensions. Examples are supplied to show how one may compute Kanextensions of various modules.

Throughout this chapter, the base ring R is always a commutative ringwith identity. A module will normally be a left module, unless otherwisespecified. If S is a set, then RS stands for the free R-module generated by S .

3.1 Basic concepts and examples

3.1.1 Category algebras

The category algebra is a natural way to linearize a category.

Definition 3.1.1. Let C be a small category and R a commutative ring. Thecategory algebra RC is a free R-module whose basis is the set of morphismsof C . We define a product on the basis elements of RC by

α ∗ β = α β, if α and β can be composed in C ;

0 , otherwise

65



66 3 Category algebras and their representations

and then extend this product linearly to all elements of RC . With this prod-uct, RC becomes an associative R-algebra.

If Ob C is finite, it is easy to see that

x∈ObC 1x is the identity of RC

where 1x is the identity of AutC(x). If a category C is finite, then Ob C isfinite and RC is of finite R-rank.Another way to linearize a category C over a ring R is to construct a new

category C R whose objects are the same as C while the morphisms betweenany two objects x, y ∈ C R are HomCR(x, y) := RHomC(x, y). This categoryC R is additive and one can obtain the category algebra RC by forgetting thecategory structure in an obvious way.

Example 3.1.2. Suppose C is the following finite category

x1x α y

1y

g

with g2 = 1y and α = g α. Then the category algebra RC is of rank 4 withidentity 1RC = 1x + 1y. It contains two group algebras R1x and R1y, g.

We say C is connected if C as a (directed) graph is connected. Everycategory C can be written as the disjoint union of connected componentsC = i∈J C i, where each C i is a connected full subcategory and J is an indexset. As a consequence the category algebra RC becomes a direct product of ideals RC i, i ∈ J . Thus in order to study the properties of RC it suffices tostudy the properties of each RC i. For simplicity and some technical reasonswe often make the connectedness assumption.

3.1.2 Representations of categories and Mitchell’sTheorem

We shall show that a fundamental property of the category algebra RC isthat it provides a mechanism for investigating R-representations of C , whichwe define now.

Definition 3.1.3. An (R-)representation of a category C is a (covariant)functor M : C → R-mod.

When the base ring is understood, we often abbreviate an R-representation

as a representation of C . All representations of C form a functor category (R-mod)C, which is an abelian category with enough projectives and injectives



3.1 Basic concepts and examples 67

so we can talk about subfunctors and quotient functors. The following funda-mental theorem of category representations tells us an alternative descriptionof the functor category.

Theorem 3.1.4 (Mitchell). For any small category C with finitely many objects, there exist functors ι : (R-mod )C → RC -mod and σ : RC -mod → (R-mod )C such that

1. σ ι ∼= Id(R-mod )C , and 2. ι is fully faithful, and moreover if Ob C is finite then ι σ ∼= IdRC-mod .

Thus if Ob C is finite, the R-representations of C can be identified with the unital RC -modules.

Proof. Assume F : C → R-mod is a representation of C . We construct afree R-module MF =

x∈ObC F (x). For any m ∈ F (x) and morphism α ∈

Mor C , we ask α · m = F (α)(m), if t(α) = x, or α · m = 0 if t(α) = x. Byextending this operation linearly we obtain an RC -module structure on MF .This construction can be easily verified to define a functor ι : (R-mod)C →

RC -mod.Conversely, if M is an RC -module, we may define a functor F M by F M (x) =

1x · M . Since, if α ∈ HomC(x, y) and m ∈ 1x · M , α · m = (1y α) · m =1y · (α · m) ∈ 1y · M , we see that F M : C → R-mod is well defined. It inducesa functor σ : RC -mod → (R-mod)C.

We may readily check these two functors satisfy 1 and 2.

Similar statements can be made between right RC -modules and contravari-ant functors from C to R-mod.

Remark 3.1.5. Throughout these notes we will be particularly interested inRC -modules lying in ι(R-mod)C. One reason is that many important mod-ules are indeed of this form. Another reason is that, on top of module-theoretic

methods, we may apply simplicial or homotopy-theoretic tools. Hence wewant to focus on categories with finitely many objects, although many re-sults make sense for arbitrary small categories. Moreover since there is a welldeveloped representation theory of finite-dimensional algebras, from now onwe will restrict our attention to the finite categories. Consequently for prac-tical reasons we will not distinguish between RC -mod and (R-mod)C, andnormally refer to an object in these categories as an RC -module.

Among all RC -modules, there is a distinguished one that we introducebelow. It plays a key role throughout these notes.

Definition 3.1.6. For any category C , the constant functor or trivial module R : C → R-mod, is defined by R(x) = R for all x ∈ Ob C and R(α) = IdR forall α ∈ Mor C .

We provide several examples of simplicially constructed modules whereboth the module and functor aspects are described.




Example 3.1.7. 1. The constant functor R corresponds to the module R Ob C ,on which RC acts via α · x = y, if α ∈ HomC(x, y), or zero otherwise.

2. The regular module RC corresponds to a functor such that

RC (x) = 1x · RC = RHomC(−, x).

3. The opposite algebra (RC )op is isomorphic to the algebra of the oppositecategory C op, and thus a set of base elements are Mor C op. Consider theenveloping category of C , namely C e = C × C op. Then we have

RC e ∼= (RC )e = (RC ) ⊗R (RC )op,

the enveloping algebra of RC . Consequently RC is an RC e-module via(α, β op) ·γ = αγβ , for any α, β,γ ∈ Mor C . As a functor RC : C e → R-mod,we have

RC (y, x) = RHomC(x, y)

if (x, y) ∈ Ob C e.

Occasionally we will need to discuss right RC -modules. A dual versionof Mitchell’s theorem says that the right RC -modules corresponds to thecontra-variant functors/representations of C . There exists a natural functor(−)∧ = HomR(−, R) : RC -mod → mod-RC . We give an explicit descriptionof the dual module M ∧ = HomR(M, R) (a right RC -module) of M ∈ RC -mod. In case R is a field, ∧ induces an anti-isomorphism between RC -modand mod-RC .

Lemma 3.1.8. Suppose M ∈mod-RC . Then its dual M ∧ ∈ RC -mod has values M ∧(x) = M (x)∧ = HomR(M (x), R), and each α ∈ Mor C acts via M (α)∧ = HomR(M (α)(−), R).

Proof. As a free R-module, M ∧ =

x∈Ob C M (x)∧. Each α ∈ HomC(x, y)

acts as M (α) : M (y) → M (x) and thus induces a map M (α)∧ : M (x)∧ →M (y)∧. One can readily check that M ∧(x) is exactly M (x)∧ so we know thestructure of M ∧ as a right RC -module.

The next statement follows from direct calculation.

Corollary 3.1.9. We have R∧ = R. Here the first R is a left module and the second is a right module.

3.1.3 Three examples

Let us consider three motivating examples.

Example 3.1.10. Let G be a (discrete) group. Then the group can be regardedas a category with only one object •, whose morphisms are the elements of



3.1 Basic concepts and examples 69

G. The group algebra RG is the same as the category algebra RG. A leftRG-module M can be regarded as the representation of G given by a certainfunctor Φ : G → R-mod, sending • to M and AutG(•) = G into AutR(M ).The trivial RG-module R is exactly the trivial module of RG.

Note that in this case G ∼= Gop

and thus RG ∼= RGop

and RGe

∼= RG ⊗RG ∼= R(G × G).

A quiver q = (Γ 0, Γ 1) is a directed graph having Γ 0 and Γ 1 as the setof vertices and the set of arrows, respectively. The path algebra of q can bethought as a category algebra as follows. Any directed graph G = (Γ 0, Γ 1)may be used to generate a category C G on the same set Γ 0 of objects, wherethe morphisms of this category are the strings of composable arrows of G. Itis called the free category generated by G.

Example 3.1.11. The path algebra of a quiver q is the category algebra RC qof the free category C q.

It is necessary to point out that, given a category, its category algebra

is usually different from its path algebra, if we consider the category as aquiver at the same time. Nevertheless, there is a relationship between thesetwo algebras, as we now explain.

Proposition 3.1.12. Let q be a category. We may regard q as a quiver and form the free category C q over q. There is a natural functor u : C q → q,which extends to a surjective homomorphism, still denoted by u : RC q → Rq,

from the path algebra of q to the category algebra of q, such that its kernel I is generated by

α1←α2← −

α1α2← , where α1 and α2 are arrows of q. This epimorphism induces a natural isomorphism of R-algebras RC q/I and Rq.

Proof. The functor u is defined as follows. For each x ∈ Ob C q, φ(x) = x. Foreach α ∈ Mor C q, u(α) = the composite of the maps in the string α. It can be

extended linearly to an epimorphism u : RC q → Rq, having the kernel I .

The last example of a category algebra is the incidence algebra of a locallyfinite poset (partially ordered set). A (closed) interval of P is a subposet[x, y] ⊂ P which consists of all objects z such that x ≤ z ≤ y for a givenpair of objects x, y in P . The incidence algebra I(P , R) is an R-algebra of all functions f : Int(P ) → R, where Int(P ) is the set of intervals of P . Themultiplication (also called convolution ) if defined by

(f g)([x, y]) =

x≤z≤y

f ([x, z])g([z, y]).

It has an identity δ such that

δ ([x, y]) =1, if x = y;

0 , otherwise.




On the other hand since whenever x ≤ y in a poset we can replace ≤ withan arrow x → y, a poset is always a category.

Example 3.1.13. Let P be a finite poset. Then the incidence algebra I(P , R)

of P is isomorphic to the category algebra RP .

3.2 A closed symmetric monoidal category

Suppose C is finite. We describe the intrinsic structure of RC which makeit comparable with a cocommutative bialgebra. Later on we shall see RC -mod possesses similar homological properties as the module category of acocommutative bialgebra.

3.2.1 Tensor structure and an intrinsiccharacterization of category algebras

Suppose G is a group. The category RG-mod is a symmetric monoidal cate-gory equipped with a tensor product − ⊗R − and a tensor identity R. Thereason why RG-mod enjoys such good properties is that RG is a cocommuta-tive Hopf algebra. For general properties of Hopf algebras, we refer the readerto [?]. For an arbitrary unital associative R-algebra there are maps

1. There is a map µ : A ⊗R A → A, called the multiplication.2. The following diagram is commutative (associativity)

A ⊗ A ⊗ A µ⊗Id

Id⊗µ

A ⊗ A

µ

A ⊗ A

µ A

3. There is a map ι : R → A, called the unit, and the following diagram iscommutative

R ⊗ A ι⊗Id

∼=

A ⊗ A

µ

A ⊗ RId⊗ι

∼=

A

If A = RG, it is also a counital coalgebra, which means, on top of theprevious maps, we have the following extra maps, reversing maps in 1-3.

4. There is a comultiplication , RG → RG ⊗R RG.



3.2 A closed symmetric monoidal category 71

6. The following diagram is commutative (coassociativity)

RG ∆

∆

RG ⊗ RG

∆⊗Id

RG ⊗ RG

Id⊗∆ RG ⊗ RG ⊗ RG,

5. The group algebra has an augmentation map : RG → R, also called thecounit.

RG∼=

∆

∼=

R ⊗ RG RG ⊗ RG

⊗Id

Id⊗ RG ⊗ R

Any R-module satisfying conditions 4-6 is called a counital coassociativecoalgebra. Any R-algebra satisfying 4-6 is named a bialgebra.

A group algebra is usually not commutative, but

7. there is a twist map τ : RG⊗R RG → RG⊗R RG, given by τ (a⊗b) = b⊗a,such that the following diagram is commutative

RG

∆

∆

RG ⊗ RG

τ RG ⊗ RG.

An R-coalgebra or bialgebra satisfying 7 is called cocommutative.

8. Moreover there is an antipode η : RG → RG in the sense that if δ (a) =

i

bi ⊗ ci then i

biη(ci) = i

η(bi)ci = (a)e.

If a bialgebra has an antipode, then it is called a Hopf algebra.

Remark 3.2.1. For a group algebra, η is explicitly given by g → g−1. Thecomultiplication is given by g → g ⊗ g.

A cocommutative bialgebra A has the significant property that A-modis a symmetric monoidal category with tensor identity R. More precisely itmeans that for any two A-modules M and N , M ⊗R N is still an A-module,M ⊗R N ∼= N ⊗R M , and M ⊗R R ∼= M . In other words, A-mod inheritsvarious constructions on R-mod. Note that the tensor identity R plays therole of both the unit and co-unit of A. If A is Hopf, then even HomR(M, N )becomes an A-module such that

HomA(L ⊗ M, N ) ∼= HomA(L, HomR(M, N )).




In the literature, HomR(M, N ) is called a function object or the internal hom .Furthermore when R = k is a field, A is cocommutative Hopf and M , N arefinite-dimensional, we have an isomorphism of A-modules

M

∧

⊗ N ∼= HomR(M, N ).

Since R ∈ A-mod, M ∧ is a left A-module under the circumstance, satisfying(M ∧)∧ ∼= M .

In what follows, we begin with a description about how R-mod gives rise toa symmetric monoidal category structure on RC -mod. Then we characterizethe structure of RC using some structure maps comparable to those of acocommutative bialgebra. In this way we demonstrate why a category algebraand a cocommutative bialgebra, as well as their module categories, are similaryet different. It is the intrinsic structure of a category algebra that makes ita natural and interesting subject of investigation. We note that R-mod itself is the module category of the R-category algebra of the trivial category •,based on Mitchell’s theorem.

Fix a finite category C . The so-called internal product on (R-mod)C

RC -mod, in which the tensor product is denoted by ⊗R, is defined by(M ⊗RN )(x) = M (x) ⊗R N (x) for any M, N ∈ RC -mod (R-mod)C andx ∈ Ob C . The module structure of M ⊗N can be viewed as given by theco-multiplication ∆ : RC → RC ⊗R RC , induced by the canonical diagonalfunctor ∆ : C → C × C whose action on each α ∈ Mor C is ∆(α) = α ⊗ α.One can easily verify that the trivial module R is the tensor identity withrespect to ⊗R. For the sake of simplicity, we shall write ⊗ for ⊗R, and ⊗for ⊗R, throughout this book. We note that in the literature (see for ex-ample [25]), the symbol ⊗ is often used instead of ⊗ for the internal tensorproduct of functors. The new notation ⊗ is introduced because we need todistinguish between M ⊗ N and M ⊗N . In fact, as R-modules, the inclusionM ⊗N ⊂ M ⊗ N , for any RC -modules M and N , is often strict.

As we mentioned above, the diagonal functor ∆ : C → C × C induces a co-multiplication on the category algebra RC . The co-multiplication ∆ : RC →RC ⊗ RC almost gives us a co-algebra structure on RC except that there isnot a suitable choice of a map RC → R which would serve as the co-unitarymap. Recall from Example 3.1.7 (1) that the trivial module R can be realizedby R Ob C . The next result states a natural augmentation map from RC toR Ob C .

Lemma 3.2.2. There exists a surjective linear map

RC R Ob C = R,

defined on base elements by

(α) = t(α),




where t(α) is the target of α.

Proof. The surjective map induces a RC -module structure on R Ob C as fol-lows. If x ∈ Ob C and β ∈ HomC(x, y), then βx = y when x = x, and βx = 0

otherwise. If we return to ∆, we realize that the image of it really lies in RC ⊗RC , a

subspace of RC ⊗ RC , and moreover ∆ : RC → RC ⊗RC becomes a RC -map,since RC ⊗RC , unlike RC ⊗RC , is a well-defined RC -module. The above obser-vations hint that, in order to get a sound “co-algebra” structure, one needs touse ⊗ other than ⊗ to define the structure maps. This motivates us to writedown the following maps which resemble almost all of the structure maps fora cocommutative bialgebra. Abusing terminology, we adopt the same namesfor the structure maps of a category algebra, such as co-multiplication andco-unit , as their counterparts for a bialgebra. In a coalgebra, the augmenta-tion map and co-unit are the same, so by analogy the map in the precedinglemma will occasionally be called the co-unit. We emphasize that the unit ,given by the natural inclusion map R ∼

= R

·1

RC

ι

→R

C , is different from the

co-unit.

Proposition 3.2.3. Let RC be the category algebra of a finite category C .Then we have the following R-linear maps: a co-multiplication ∆ : RC →RC ⊗ RC , defined by ∆(

α λαα) =

α λαα ⊗ α, a co-unit : RC → R

defined as above, and a twist map τ : RC ⊗ RC → RC ⊗ RC defined on base elements by τ (α ⊗ α) = α ⊗ α, such that the following diagrams are commutative: 1. co-associativity

RC ∆

∆

RC ⊗ RC

∆⊗Id

RC ⊗ RC Id⊗∆ RC ⊗ RC ⊗ RC ,

2. co-unitary property

RC ∼=

∆

∼=

R⊗RC RC ⊗RC

⊗Id

Id⊗ RC ⊗R

and 3. co-commutativity

RC

∆

∆

RC ⊗ RC

τ RC ⊗ RC .




If we denote by µ the multiplication, then we have furthermore three commu-tative diagrams: 4. Multiplication and co-multiplication

RC ⊗ RC µ

∆⊗∆

RC ∆ RC ⊗ RC

RC ⊗ RC ⊗ RC ⊗ RC Id⊗τ ⊗Id

RC ⊗ RC ⊗ RC ⊗ RC ,

µ⊗µ

5. Unit and co-multiplication:

R ⊗ R

µ

ι⊗ι

RC

∆ RC ⊗RC ,

and 6. Unit and co-unit

R

ι

η

RC

R,

where the R-linear map η : R → R is defined by 1 →

x∈ObC x,

We note that the only missing diagram is the compatibility of multiplica-tion and co-unit. The reason is that one usually cannot give R a meaningfulalgebra structure so that : RC → R becomes an algebra homomorphism.The existence of the above tensor structure ⊗ is well known. Notably it hasbeen used to define the internal product in functor homology theory, see for

example [25].Remark 3.2.4. If we remove the finiteness condition on C , then there is noidentity in the algebra kC . Nevertheless many of the constructions are stillvalid, although in this case, we are forced to use the full subcategory V ectCkbecause kC -mod does not have a tensor structure.

Remark 3.2.5. If C = P happens to be a poset, then there is another co-multiplication that one can find in [72]. Let α ∈ Mor P , one has ∆ : k P →kP ⊗ kP such that ∆(α) =

β,γ |βγ =α β ⊗ γ . Nevertheless it is possible to

give an incidence algebra a Hopf algebra structure. The augmentation map : kP → k is given by (α) = 1 if α is an identity, or (α) = 0 otherwise. Theantipode kP → kP can also be explicitly constructed. Suppose α ∈ Mor P isa morphism in the poset. Then a factorization of α is a way of writing α as a

product α1α2 · · · αn for some non-identity αi ∈ Mor P . The length l(α) of αis the maximal n for such a factorization to exist. Then the antipode is givenby α →

β(−1)l(β)β , in which β runs over all possible factors of α.




Indeed one can do this for the category algebras of the so-called (finite)M¨ obius categories . A main feature of a Mobius category is that each objectonly has one automorphism.

It seems that RC can only be a (cocommutative) Hopf algebra if C is of the two extreme structures: either a group (all morphisms are invertible), ora Mobius category (identity morphisms are the only invertible morphisms) .

Remark 3.2.6. Let A be an R-algebra. Although A-mod is not monoidal ingeneral, the module category Ae-mod is always equipped with a tensor prod-uct ⊗A such that A is the tensor identity.

When A = RC is a category algebra, (RC )e ∼= RC e becomes the categoryalgebra of the category C e := C × C op, and hence there are two distinctmonoidal structures on RC e-mod. The monoidal structure given by ⊗RC ismore interesting to us, and we shall deal with it in Chapter 5 on Hochschildcohomology.

3.2.2 The internal hom

Usually one can not define an antipode for a category algebra. This causes aproblem when one attempts to define Homk(M, N ) as a sensible k C -module(generalizing the group case). We shall give a remedy below. Another rele-vant fact is that the product of two projective kC -modules is in general not projective, see Example 3.?. These make many homological properties, suchas the cohomology theory, of a finite-dimensional category algebra differentfrom those of a finite-dimensional cocommutative Hopf algebra.

Proposition 3.2.7 (Swenson). Suppose C is a finite category (or at least Ob C is finite). Let M, N ∈ RC -mod. Then we can define an internal hom Hom(M, N ) ∈ RC -mod.

Proof. We want to define Hom(M, N ) such that there is an isomorphism

HomRC(L⊗M, N ) ∼= HomRC(L, Hom(M, N )).

Let RC = RC · 1x. Assume the above isomorphism. Then the right hand sideis HomRC(RC · 1x, Hom(M, N )) ∼= Hom(M, N )(x), and the left hand side isHomRC(RC · 1x⊗M, N ). Hence we may define Hom(M, N ) by

Hom(M, N )(x) = HomRC(RC · 1x⊗M, N ).

When C = G is a group, it is straightforward from Swenson’s definitionthat Hom(M, N ) ∼= HomR(M, N ) as RG-modules.




Alternatively consider the diagonal functor ∆ : C → C × C . Because R(C ×C ) ∼= RC ⊗ RC , for L, M,N ∈ RC -mod, L ⊗ M is a R(C × C )-module, and

HomRC(L⊗M, N ) = HomRC(Res∆(L ⊗ M ), N )∼= HomR(C×C)(L ⊗ M,RK ∆N )∼= HomRC⊗RC(L ⊗ M,RK ∆N )∼= HomRC(L, HomRC(M,RK ∆N )).

Here in HomRC(M,RK ∆N ), RK ∆N ∈ RC ⊗ RC -mod is considered as a(R · 1RC) ⊗ RC -module and hence a RC -module. The RC ⊗ RC -, or ratherthe RC ⊗ (R · 1RC)-, module structure on RK ∆N provides a RC -modulestructure on HomRC(M,RK ∆N ). Then we may verify that Hom(M, N ) =HomRC(M,RK ∆N ).

With the internal hom, one might attempt to define a dual module of M by Hom(M, R) in order to generalize the group case. The following exampleexplain why this is not a good idea.

Example 3.2.8. Let k be a field of characteristic two and C the followingcategory

x1x α β

y 1y,g

with g 2 = 1y, gα = α and gβ = β .We consider two modules S x,k and S y,k such that S x,k(x) = k, S x,k(y) = 0,

S y,k(x) = 0 and S y,k(y) = k . Then Hom(S x,k, k) can be easily shown to beisomorphic to S x,k. However Hom(S y,k, k) ∼= S 2x,k⊕S y,k because kC 1x⊗S y,k

∼=S 2y,k and k C 1y ⊗S y,k

∼= kC 1y. Thus Hom(Hom(S y,k, k), k) ∼= S y,k.

3.3 Functors between module categories

We investigate functors for comparing two module categories over categoryalgebras.

3.3.1 Restriction on algebras and modules

We prove some basic properties, many of which follow directly from simplereasoning.

Definition 3.3.1. Suppose u : D → C is a (covariant) functor. We defineResu : RC -mod → RD-mod to be the restriction along u. Given a module

M ∈ RC -mod, we have ResuM = M u ∈ RD-mod.

Lemma 3.3.2. Let u : D → C be a functor. Then ResuR = R.



3.3 Functors between module categories 77

Since a functor u : D → C also extends linearly to a natural map of R-modules u : RD → RC , it is reasonable to ask if u is always an alge-braic homomorphism because if it is, the so-called change-of-base-ring or therepresentation-theoretic restriction, ↓RC

RD: RC -mod → RD-mod, should co-

incide with Resu. The answer to the question is no, and here is a simpleexample. Let D = 1 and C = •. There is a unique functor pt : D → C . Theinduced map ¯ pt : RD → RC is not an algebra homomorphism since the prod-uct of the two isomorphisms in D is zero while the product of their images isnot.

Proposition 3.3.3. A functor u : D → C extends linearly to an algebra homomorphism u : RD → RC if and only if u is injective on Ob D. When this happens, the induced functor followed by 1RD, 1RD ↓RC

RD: RC -mod → RD-mod, is exactly Resu.

Proof. We know u(βα) = u(β )u(α) for any pair of composable morphismsα, β in D. The injectivity of u implies two morphisms α, β ∈ Mor(D) arecomposable if and only if u(α), u(β )

∈ Mor(

C ) are composable.

If u is injective on Ob D, then we define a map u : RD → RC as the linearextension of functor u, i.e., u(

i riαi) =

i riu(αi) for any ri ∈ R, αi ∈

Mor(D). This u is indeed an algebra homomorphism because our previousobservation of u implies u((

j rj β j )(

i riαi)) = u(

j rj β j )u(

i riαi) is

always true.On the other hand if the linear extension u : RD → RC is an algebra

homomorphism then we must have u(0) = 0 and then u(1x)u(1y) = u(1x ·1y) = 0 unless x = y. This suggests that u is injective on Ob C .

When u : RD → RC is an algebra homomorphism, we show 1RD ↓RCRD=

Resu : RC -mod → RD-mod. Let M be an RC -module and γ ∈ Mor(D).Then it corresponds to some F M ∈ (R-mod)C. The RD-modules ResuM and 1RD · (M ↓RC

RD) are isomorphic because ResuM = MResu(FM ), γ · M =

u(γ )M for any γ ∈ Mor D and 1RD kills the elements of M ↓RCRD that are notsupported on any objects of D.

Proposition 3.3.4. Let D and C be equivalent small categories. Then (R-mod )C (R-mod )D, an equivalence which sends the constant functor to the constant functor. If both Ob C and Ob D are finite then RC and RD are Morita equivalent.

Proof. We show that the two functor categories (R-mod)C and (R-mod)D areequivalent. Then it implies the module categories RC -mod and RD-mod areequivalent, hence RC and RD are Morita equivalent. In fact if u : D → C andv : C → D are equivalences, we have ResuResv ∼= IdRD := IdRD-mod : (R-mod)D → (R-mod)D because of the following diagram




M (vu(x)) = (ResuResvM )(x)∼=

M (vu(α))=(ResuResvM )(α)

(IdRDM )(x) = M (x)

(IdRDM )(α)=M (α)

M (vu(y)) = (ResuResvM )(y) ∼= (IdRDM )(y) = M (y)

where M ∈ (R-mod)D, α : x → y ∈ Mor D and IdRD is the identity functor.Similarly we can show ResvResu ∼= IdRC : (R-mod)C → (R-mod)C. Clearlythe constant functor restricts to the constant functor always.

Remark 3.3.5. Recall from Chapter 2 that if D C then N D N C (andBD BC ). The above result is an algebraic consequence of a category equiv-alence. On one hand the existence of a Morita equivalence implies that thetwo (category) algebras are the same, in terms of homological properties. Onthe other hand there are non-equivalent finite categories whose algebras areMorita equivalent, see Example 3.1.16 (5).

By contrary a simplicial/topological equivalence does not provide a Moritaequivalence. For instance the posets 0 and 1 are simplicial homotopy equiv-alent, because there are natural adjoint functors between them. However if k is a field then k0 and k1 are not equivalent as algebras. The reason isthat k0 ∼= k has only one simple module k while k1 has two simple mod-ules because its identity can be written as a sum of orthogonal primitiveidempotents 10 + 11. Since their category algebras have different numbers of non-isomorphic simple modules, k0-mod and k1-mod cannot be equivalentas categories.

3.3.2 Kan extensions of modules

Since not every functor u : D → C has an adjoint, we have to look for otherfunctors for comparing RD-mod and RC -mod. Clearly the restriction Resupossesses two adjoint functors, the left and right Kan extensions LK u andRK u. Here will be our first attempt to study Kan extensions of modules.We note that, because Resu is exact, LK u preserves projectives while RK upreserves injectives.

In light of Proposition 3.2.3, if one is careful enough, the restriction canbe written using the usual module-theoretic notation ↓CD:=↓RC

RD, and thenin the literature one may see that its left adjoint and right adjoint are de-noted by ↑CD:= RC ⊗RD −, the induction, and ⇑CD:= HomRD(RC , −), thecoinduction. They are the Kan extensions in different forms. Since there isplenty of description of the induction and coinduction for algebra represen-

tations, in these notes we shall mainly discuss the functor-theoretic aspectof them through Kan extensions. A special situation of Kan extensions, the




group case, can be found in Example 1.2.11. One can see that using module-theoretic methods it is not easy to obtain most of the following results forgeneral category algebras.

Suppose M ∈ RD-mod and u : D → C is a functor. Then we obtain an

RC -module LK uM . As a functor, for each x ∈ Ob C , (LK uM )(x) is given by adirect limit, and if α : x → y is a morphism, it induces a map (LK uM )(x) →(LK uM )(y) by the universal property of lim−→. This specifies the RC -action onLK uM .

Suppose P is a poset and Q is a subposet. We call Q an ideal of P if for a pair of object x ∈ Ob Q and y ∈ Ob P such that x ≤ y in P then ybelongs to Q. Before we start the first calculation we mention that if D is afull subcategory of a small category C then any functor over D can be naivelyconsidered as a functor over C by asking its value to be zero on any objectsthat do not belong to D. In other words, any RD-module is naturally anRC -module under the assumption.

Proposition 3.3.6. Suppose P is a poset and Q is an ideal together with the

inclusion i : Q → P . Then for any M ∈ RQ-mod, the RP -module LK iM is exactly M , regarded as an RP -module.

Proof. One can find that (LK iM )(x) ∼= lim−→Q≤xM , where Q≤x is a subposet

of Q consisting of objects smaller or equal to x. By assumption this poset iseither empty or has a terminal object x. Consequently (LK iM )(x) is eitherzero if x ∈ Ob Q or M (x) otherwise. Hence we are done.

Alternatively we can compute RP ⊗RQ M . For any x ∈ Ob P , we have

1x · (RP ⊗RQ M ) =y≤x

RHomP (y, x) ⊗ M =y≤x

RHomP (y, x) ⊗ 1y · M.

This is not zero if and only if 1y · M = 0 which means y ∈ Ob Q. By as-

sumption, it forces x belongs to Q. Thus 1x · (RP ⊗RQ M ) = 0 if and only if x ∈ Ob Q. When this happens, 1x · (RP ⊗RQ M ) = 1x · M . Thus we obtainthe same result.

The concept of an idea can be defined for a certain class of categories,which are called EI-categories and will be introduced shortly. The aboveresult stays true in this generality. We will also give a dual concept of coideal later on.

Proposition 3.3.7. Suppose u : D → C satisfies the condition that for every x ∈ Ob C u/x is connected. Then LK uR = R.

Especially for t : F (C ) → C , we have LK tR ∼= R as an RC -module.

Proof. When u/x is connected for an x ∈ Ob C , we have lim

−→u/x

R = R. Then

by the universal property of direct limit we see any α : x → y must inducethe identity morphism on R. Thus LK uR = R.




For any M ∈ RF (C )-mod we can compute LK tM explicitly. If x ∈ Ob C ,then lim−→t/x

M ∼= lim−→t−1(x)M by Propositions 2.3.10 and 1.2.4. The category

t−1(x) has an initial object [1x] and thus in general we cannot simplify thedirect limit. We want to construct an example based on this description

showing that LK t(M ⊗N ) ∼= LK t(M )⊗LK t(N ) for M , N ∈ RF (C )-mod.

Example 3.3.8. Consider the following poset P and its factorization categoryF (P ) (a poset too)

1 [α] [β ]

2

α

3

β

, [12]

[11]

[13]

We define an RF (P )-module M by

R R

0

0

0

Since we know explicitly t−1(1), t−1(2) and t−1(3), we can compute directlythat LK tM (1) = R2, LK tM (2) = LK tM (3) = 0. Thus the RP -moduleLK tM ⊗LK tM is pictured by

R4

0

0

On the other hand, since M ⊗M = M , LK t(M ⊗M ) ∼= LK tM ⊗LK tM .

Proposition 3.3.9. Let C be a small category and ∇ : F (C ) → C e the skew diagonal functor. Then LK ∇R ∼= RC as an RC e-module.

Proof. We compute (LK ∇R)(y, x) = lim−→∇/(y,x)R for any (y, x) ∈ Ob C e. By

Proposition 2.3.13 the functor i(y,x) : ∇−1(y, x) → ∇/(y, x) has a left adjoint.Hence Proposition 1.2.4 implies that lim−→∇/(y,x)

R ∼= lim−→∇−1(y,x)R. However

∇−1(y, x) ∼= HomC(x, y) so we have lim−→∇−1(y,x)R = RC (y, x), where RC is

considered as a functor C e → R-mod. We are done.

For a finite group algebra RG and M , N ∈ RG-mod, we have a canonicalisomorphism HomRG(M, N ) ∼= HomRG(R, HomR(M, N )), where HomR(−, −)is the internal hom for RG-mod. The above proposition allows us to general-izes this isomorphism, but without using the previously defined internal homfor RC -mod.

Corollary 3.3.10. Let M, N ∈ RC -mod. Then HomR(M, N ) ∈ RC e-mod and




HomRC(M, N ) ∼= HomRF (C)(R, Res∇HomR(M, N ))∼= HomRC(R,RK tRes∇HomR(M, N )).

Proof. We can define an RC e-module structure on HomR(M, N ) by

[(α, β op) · f ](m) = αf (βm),

for any f ∈ HomR(M, N ) and m ∈ M . Then

HomRC(M, N ) ∼= HomRCe(RC , HomR(M, N ))∼= HomRCe(LK ∇R, HomR(M, N ))∼= HomRF (C)(R, Res∇HomR(M, N ))∼= HomRF (C)(RestR, Res∇HomR(M, N ))∼= HomRC(R,RK tRes∇HomR(M, N )).

Since F (G) is equivalent to G as categories, under Morita equivalencebetween RF (G) and RG, RK tRes∇HomR(M, N ) ∈ RG-mod is exactly the

usual RG-module HomR(M, N ). In Chapter 5, we shall give a higher versionof this corollary in terms of Ext, based on HomRC(M, N ) = Ext0

RC(M, N ) =Ext0

RCe(RC , HomR(M, N )). The reader should compare the above isomor-phism with HomRC(M, N ) ∼= HomRC(R, Hom(M, N )). We will explain lateron that the latter usually does not admit an higher version using Ext.

Proposition 3.3.11. Let p : C × C op → C be the projection to the first com-ponent. Then LK pRC = R as an RC -module.

Proof. For each x ∈ Ob C one can easily verify that p/x ∼= IdC/x × C op. SinceIdC/x has an terminal object (x, 1x), the inclusion (x, 1x) → IdC/x has aleft adjoint. Thus the natural functor C op ∼= • ×C op → IdC/x×C op ∼= p/x alsohas a left adjoint. By Proposition 1.2.4 (LK pRC )(x) = lim−→ p/x

RC ∼= lim−→CopRC .

As a functor from C op to R-mod (RC )(x) = RHomC(x, −) and thus we candefine a morphism (RC )(x) → R by i riαi → i ri so that R fits intothe defining diagram for direct limit. In fact if M fits into the limit definingdiagram

RHomC(y, −)

θy

RHomC(x, −)

θx

M ,

then θy(α) = θx(β ) for any β ∈ HomC(y, −) and α ∈ HomC(x, −). Thus(LK pRC )(x) = R and we are done.

The last proposition can be generalized to a functor isomorphism LK p ∼=− ⊗RC R, and we shall come back to it in Chapter 4.

Note that if we have two functors u : D → C and v : E → D thenResuv = ResvResu and as a consequence LK uv = LK vLK u. It is enlighteningif we summarize our three calculations in one diagram for t = p∇.




R

RF (C )-mod LK ∇

LK t

RC e-mod

LK p∼=−⊗RCR

RC

RC -mod .

R

Based on explicit formulas for computing LK ∇ and LK p, one should beable to find examples, similar to Example 3.3.8, that they do not commutewith corresponding tensor products.

Definition 3.3.12. Let C be a small category and x an object. Then wedefine a full subcategory C x which consists of one object x and all of itsendomorphisms. We also define C [x] to be the full subcategory of C containing

all objects that are isomorphic to x.

Proposition 3.3.13. Let C be a small category and x ∈ Ob C . Consider the inclusion ι : C x → C . Then LK ι[REndC(x)] ∼= RHomC(x, −).

Proof. For every y ∈ Ob C , the overcategory ι/y has objects (x, α) whereα runs over the set HomC(x, y). A morphism (x, α) → (x, β ) is an endomor-phism g ∈ EndC(x) such that α = βg. Hence we can fit RHomC(x, y) into thecommutative diagram

[REndC(x)](x, α)

α−

[REndC(x)](x, β )

β−

RHomC(x, y) ,

in which [REndC(x)](x, α) = [REndC(x)](x, β ) = REndC(x), and α −, β −are compositions. If M is another R-module that can replace RHomC(x, y)and make a new commutative diagram, then, by examining the images of EndC(x), one can easily establish a canonical map RHomC(x, y) → M suchthat the whole diagram is commutative. This means that RHomC(x, y) ∼=lim−→ι/y

[REndC(x)] and thus LK ι[REndC(x)] ∼= RHomC(x, −).

3.3.3 Dual modules and Kan extensions

In group representations, the two Kan extensions are isomorphic. Many im-portant properties of group representations rely on this fact. Unfortunately



3.4 EI categories, projectives and simples 83

it is not the case for category representations. To obtain a balanced under-standing of the homological properties of category algebras, we describe somecompatibility between the two Kan extensions. Here we assume the base ringis a field R = k.

Let M ∈ mod-kC be a right module. Its dual M ∧

= Homk(M, k) becomesa left kC -module. The functor (−)∧ = Homk(−, k) is an anti-isomorphismbetween the two module categories. Suppose G is a group and H is a sub-group. In group representation theory, there are two well-known isomor-phisms (M ↓G

H )∧ ∼= M ∧ ↓G

H , if M ∈ kG-mod, and (N ↑GH )∧ ∼= N ∧ ↑G

H , if N ∈ kH -mod. We shall extend these to category representations.

Lemma 3.3.14. Let τ : D → C be a functor between finite categories. Then we have Resτ M ∧ ∼= (Resτ M )∧, for any M ∈mod-kC .

Proof. The isomorphism follows from Lemma 3.1.8.

Lemma 3.3.15. With the same assumption as above, we have RK τ N ∧ ∼=(LK τ N )∧, for any N ∈ mod-kD.

Proof.RK τ N ∧ ∼= HomkC-mod(kC , RK τ N ∧)

∼= HomkD-mod(Resτ kC , N ∧)∼= Hommod-kD(N, (Resτ kC )∧)∼= Hommod-kD(N, Resτ kC ∧)∼= Hommod-kC(LK τ N, kC ∧)∼= HomkC-mod(kC , (LK τ N )∧)∼= (LK τ N )∧.

The last isomorphism can be immediately combined with computationsfrom the preceding section to get the right Kan extensions of certain modules.

Since this duality exchanges projective and injective modules, wheneverthe base ring is a field or a complete local ring if we can characterize theprojective modules then the structures of injective modules are understoodthrough duality.

3.4 EI categories, projectives and simples

In this section, we investigate the representation theory of EI-categories. Itwill help us in computing various (co)homology groups. For technical reasonswe always assume in this section the base ring R is a field or a completediscrete valuation ring, in order to have the unique decomposition property

for every RC -module.Some of the general theory of EI-categories is given by tom Dieck in [15]

(I.11), much of which was due to Luck, see also [51].




3.4.1 EI condition and its implications

Recall from Definition 2.4.2 that EI-category is a small category C in whichall endomorphisms are isomorphisms.

One of the important features of EI-categories is described as follows.Given an EI-category C , there is a preorder defined on Ob C , that is, y ≤ xif and only if HomC(y, x) = ∅. Let [y] be the isomorphism class of an objecty ∈ Ob C . This preorder induces a partial order on the set Is C of isomorphismclasses of Ob C (specified by [y] ≤ [x] if and only if HomC(y, x) = ∅), whichplays an important role in studying representations and cohomology of EI-categories. Because of the existence of an order for the isomorphism classes of objects in any EI-category, EI-categories are sometimes referred to as ordered categories by some authors, see [57], [42].

Definition 3.4.1. For any EI category C and any object x ∈ Ob C , we candefine a full subcategory D≤x ⊂ D consisting of all y ∈ Ob D such that[y] ≤ [x], or equivalently HomC(y, x) = ∅. Similarly we can define other full

subcategories of D : D<x, D≥x and D>x.

An object in an EI category C is called maximal if C >x = ∅, and is minimal if C <x = ∅.

Definition 3.4.2. Let C be an EI category. A subcategory D of C is calledan ideal if for any x ∈ Ob D, C ≥x ⊂ D.

A subcategory D of C is called an coideal if for any x ∈ Ob D, C ≤x ⊂ D.

By definition, ideals and coideals of an EI category are full subcategories.Naively speaking, an EI category consists of a bunch of groups and certain

non-isomorphisms connecting them. Note that usually a non-isomorphismdoes not induce a group homomorphism between the two automorphism

groups of the source and target of the non-isomorphism. Nonetheless, thepartial order on Is C allows us to filtrate an RC -module by modules of variousgroup algebras inside RC .

Suppose M ∈ RC -mod and x ∼= y ∈ Ob C . We easily deduce that 1xM ∼=1yM as R-modules. In the language of functors, we have M (x) ∼= M (y). If furthermore C is EI, then M (x) becomes an RAutC(x)-module.

Definition 3.4.3. Suppose C is a small category. An RC -module M is saidto be atomic if M as a functor satisfies the condition that M (y) = 0 if andonly if y ∼= x. Under the circumstance we say M is supported on C [x].

Let C be a finite EI-category and M ∈ RC -mod. Then for each maximalobject x ∈ Ob C , one can construct an atomic submodule M x ⊂ M suchthat M x(y) = M (y) if y ∼= x or M x(y) = 0 otherwise. Then we have a short

example0 → M x → M → M/M x → 0.




On the other hand, if x ∈ Ob C is minimal, one can introduce a submoduleM x ⊂ M such that M x(y) = M (y) if y ∼= x, and otherwise M x(y) = 0. Foreach minimal x we have a short exact sequence

0 → M x → M → M/M x → 0

such that M/M x is atomic. In either case, we can repeat the process for M x.Since C is finite, we will eventually obtain a filtration of M with every factoratomic. We can further refine this filtration until every factor is simple.

For each RC -module, even if C is not finite, the above analysis can visualizeits structure, which will help us in future investigations. For example, everysimple RC -module has to be atomic.

3.4.2 Some representation theory

Here we recall some standard facts in representation theory. Suppose A is afinite-dimensional algebra over a field k . An A-module M is indecomposable if M is not a direct sum of two non-trivial submodules. An element e ∈ A isan idempotent if e = 0 and e2 = e. For example 1 is always an idempotentand if e = 1 is an idempotent then so is 1 − e. Two idempotents e1 and e2

are orthogonal if e1e2 = e2e1 = 0. An idempotent e is primitive if one cannotwrite e = e1 + e2 such that both e1 and e2 are orthogonal idempotents.

Suppose e ∈ A is a non-identity idempotent. Then e and 1 − e are a pairof orthogonal idempotents with 1 = e + (1 − e). It results in a decomposition

A = Ae ⊕ A(1 − e).

The module Ae is projective and one can check that it is indecomposable

(not a direct sum of two non-trivial modules) if and only if e is primitive.Since A is finite-dimensional, 1 can be decomposed as a sum of finitelymany pairwise orthogonal primitive idempotents (called a primitive decom-position )

1 = e1 + e2 + · · · + en.

Consequently the regular module A becomes a direct sum

A = Ae1 ⊕ Ae2 ⊕ · · · ⊕ Aen.

and we obtain a list of indecomposable projective A-modules, Aei. Up toisomorphism this is a complete list of indecomposable projective A-modules.

In fact if 1 = e1 + e2 + · · · + em is another sum of primitive idempotentsin A, then m = n and there exists an invertible element a ∈ A such that

1 = a1a−1 = ae1a−1 +ae2a−1 +· · ·+aena−1 such that up to an reordering onehas ei = aeia−1 for all possible i. Following from this fact, the automorphism




of A, induced by a, maps each direct summand Aei of A = Ae1 ⊕ Ae2 ⊕ · · · ⊕Aen isomorphically to Aei. Thus the number of idempotents in a primitivedecomposition is equal to the number of indecomposable direct summand of A.

Example 3.4.4. Suppose C is a finite category and R is a commutative ringwith identity. Then 1RC is a sum of orthogonal idempotents

x∈ObC 1x.

Hence we have a decomposition of the regular module

RC =

x∈ObC

RC · 1x =

x∈ObC

RHomC(x, −),

and each summand RHomC(x, −) is a projective RC -module. However sincean idempotent 1x may be decomposable in RC , the above decompositioncan be refined, in a way depending on the choice of R. If this happens,the corresponding projective module RHomC(x, −) is a direct sum of otherprojective modules. See Example 3.4.7.

In order to describe the simple A-modules, we have to introduce a few moreterminologies. Let J (A) be the Jacobson radical of A. It is the intersectionof all left maximal ideals of A. In particular it is nilpotent in the sense thatthere exists an integer n > 0 such that J (A)n = 0. By contrast an elementa ∈ A is nilpotent if an = 0 for some positive integer n. If an ideal satisfiesthe condition that every element is nilpotent, then this ideal is contained inthe radical.

Example 3.4.5. If C is an EI-category, the any non-isomorphism, as an elementin the category algebra RC is nilpotent. Since they form an ideal of RC , itimplies that all non-isomorphisms are contained in J (RC ). A important factis that every element in J (A) is nilpotent, but the converse if not true (seeExample 3.4.7 (4)).

For each A-module, the radical of M , RadM , is the intersection of allmaximal submodules of M . For example the regular module has its radicalRadA = J (A) because left ideal of A is exactly the same as a left submoduleof A. In general RadM = J (A)M . It is easy to see that M/RadM has atrivial radical. Any A-module with trivial radical is called semi-simple . Asemi-simple module is called simple if it is indecomposable. Equivalently anA-module is simple if it does not contain any non-trivial submodule. Forexample A/RadA is semi-simple. Every simple A-module occurs as a directsummand in this semi-simple module up to isomorphism.

The quotient A/RadA is itself an algebra with identity 1, the image of 1 ∈ A. A pairwise-orthogonal primitive decomposition 1 = e1 + e2 + · · · +en gives rise to 1 ∈ A/RadA, 1 = e1 + e2 + · · · + en, which is again asum of pairwise-orthogonal primitive idempotents in A/RadA. This simple

observation actually establishes a one-to-one correspondence between the setsof isomorphism classes of indecomposable projective A-modules and of simpleA-modules.




Proposition 3.4.6. Every indecomposable projective A-module, up to iso-morphism, is of the form Ae, for some primitive idempotent e ∈ A. More-over, Ae/Rad(Ae) is a simple A-module and every simple A-module arises in this way.

Moreover the number of idempotents in a primitive decomposition of 1 ∈ A,the number of indecomposable summands of A and the number of indecom-posable summands of A/Rad(A) equal to each other.

There exists a large collection of good references on representation theoryof associative algebras. However for those who do not plan to go over thewhole theory, just bear in mind the basic constructions and important factsthat we record here. Then through upcoming examples one can see how theywork. We shall use them to classify projective and simple modules of certainfinite category algebras. It will be sufficient for us to develop (co)homologytheory of categories and modules.

Example 3.4.7. 1. Let G = g

g2 = 1• be the cyclic group of order 2,

regarded as a category with one object. If k = C is the field of complexnumbers, the identity 1• can be written as 1•+g

2 + 1•−g2 , a decomposition

into a sum of orthogonal primitive idempotents. The regular module is adirect sum of two one dimensional modules CG ∼= C(1• + g) ⊕ C(1• − g).Thus both C(1•+ g) and C(1•− g) are projective. They are simple as wellbecause they cannot have non-trivial submodules. It means CG is semi-simple with trivial radical. The module C(1•+ g) is the trivial module andC(1• − g) is called the sign representation.However when k is a field of characteristic 2, 1• is primitive. Hence kG isindecomposable. The regular module has exactly one non-trivial submod-ule k(1• + g), which has to be the radical Rad(kG). Then kG/Rad(kG) isone-dimensional and is simple. It is the only simple kG-module, the trivialmodule.

2. The poset 1 = 0 → 1 is a category with two objects 0 and 1. For anyfield k, the identity 1k1 = 10 + 11 in the category algebra k1. The twoidentity morphisms 10 and 11 are primitive orthogonal idempotents sok1 = k10, α ⊕ k11. The first indecomposable summand has exactly onenon-trivial submodule kα, the radical Rad(k10, α) of k10, α. Then itgives rise to a one-dimensional simple module S 0. As a functor, S 0(0) = kand S 0(1) = 0. The second summand is of dimension one so it is alreadysimple. If we denote it by S 1. As a functor S 1(0) = 0 and S 1(1) = k.

3. Now we examine the category C of Example 3.1.2 that is neither a groupnor a poset.

x1x α y

1y

g




with g2 = 1y and α = gα. We always have 1kC = 1x +1y so kC = k1x, α⊕k1y, g. Similar to 2, the first summand, named P x, is indecomposableand has radical kα. The quotient of P x by its radical is a one-dimensionalsimple module S x (analogues to S 0 as above). According to 1, the second

direct summand is decomposable if k = C. Whence we have CG ∼= C(1•+g) ⊕ C(1• − g). It means when CC has three indecomposable projectivemodules and the same number of simple modules.

The category algebra CC Indecomposable projective modules Simples module

P x,1 = C1x, α S x,1 = C1xP y,1 = C1y + g S y,1 = C1y + g

P y,−1 = C1y − g S y,−1 = C1y − g

When k is of characteristic 2, k1y, g is indecomposable. Whence kC onlyhas two indecomposable projective and simple modules.

The category algebra kC , chark = 2

Indecomposable projective modules Simple modulesP x,1 = k1x, α S x,1 = k1xP y,1 = k1y, g S y,1 = P y,1/k1y + g

4. A useful example to bear in mind is the following category (a groupoid)D that is equivalent to •

xα

y.

α−1

One can write RD = RD · 1x ⊕ RD · 1y = R1x, α ⊕ R1y, α−1. Observethat we have an isomorphism (−) α−1 : R1x, α → R1y, α−1. WhenR = k is a field, every non-zero element of a1x + bα ∈ kD · 1x generates

the whole module. Hence kD · 1x has no non-trivial submodule and thusis simple. Note that it corresponds to the functor S x,1 which takes valuesS x,1(x) = k1x and S x,1(y) = kα. Similarly kD · 1y is also a simple module,corresponding to S y,1 given by S y,1(x) = kα−1 and S y,1(y) = k1y, whichis isomorphic to S x,1. The algebra RD is semi-simple but it contains twonilpotent elements α and α−1.

5. Finally let us consider the following category E

xα

yβ

1y

e

such that αβ = e, βe = β , eα = α and e2 = e. Note that α and β arenot invertible in E . Hence E is not a groupoid and the two objects are




not isomorphic in E . However the category algebra RE is isomorphic toR1x, α , β , e × R1y ∼= RD × R•.This category is a simple example of inverse categories , see [51]. The cat-egory algebra of an inverse category is canonically isomorphic to a direct

product of groupoid algebras.Sometimes we may want to use injective modules, so we finish this section

with several remarks concerning the injectives. In general injective modulesbehave better than the projectives in the sense that for any ring A and any A-module M , there exists a minimal injective A-module I M such that M admitsan injection into I M . This module is called the injective hull of M . There aremodule categories with enough injective but not with enough projectives.However if A is a finite-dimensional algebra, then for any M ∈ A-mod thereexists a minimal projective module P M which admits a surjection onto M .It is called the projective cover of M . For instance, in the tables of Example3.4.7 (3), each row consists of a simple module as well as its projective cover.When we do representation theory we often prefer working with projectivemodules because simple modules come from their quotients.

As we pointed out earlier one has an anti-isomorphism (−)∧ = Homk(−, k)from A-mod to mod-A. Suppose P is a projective A-module, then P ∧ isan injective right A-module. The anti-isomorphism provides a bijection be-tween projective left (resp. right) A-modules and injective right (resp. left)A-modules. Thus knowing all projectives leads to getting all injectives. Animportant case is the group algebra of a finite group G, or more generallya finite-dimensional cocommutative Hopf algebra. They are self-injectives ,which means the regular module is an injective module. In this case, the setsof projective and injective modules coincide.

In the end we mention an important concept in algebra.

Definition 3.4.8. Let A and B be two rings. Then A is Morita equivalent to B if A-mod is equivalent to B -mod.

There are many interesting invariants under a Morita equivalence. For in-stance, a Morita equivalence preserves the number of isomorphism classes of simple modules. Hochschild (co)homology is invariant under Morita equiv-alence. Here we shall focus on category algebras only. In fact we will provethat a category equivalence induces a Morita equivalence between categoryalgebras. Thus in Example 3.4.7 (4) RC is Morita equivalent to R = R•.

3.4.3 Classifications of projectives and simples

Now we start describing the projective and simple modules of an EI category

algebra. This part of the work is due to Luck [51], as is described by tomDieck in [15]. The base ring R is assumed to be a field or a complete discretevaluation ring.




Let C be a small category and x ∈ Ob C an object. Suppose P x is a projective RC x-module (or in other words a projective REndC(x)-module). Thenits left Kan extension LK ιP x, along ι : C x → C , is a projective kC -module.Especially LK ι[REndC(x)] ∼= RHomC(x, −) by Proposition 3.2.11.

Let us assume furthermore C is a finite EI category. Then EndC(x) =AutC(x) for every x ∈ Ob C . From Example 3.4.4 we already learned thatRC decomposes into a direct sum ⊕x∈ObCRC · 1x. Now we try to analyze theindecomposable direct summands.

Suppose Is C is the full subcategory of C , consisting of all objects and allisomorphisms. Then R Is C is a subalgebra of RC .

Lemma 3.4.9. If 1RC = n

i=1 ei is a primitive decomposition of 1RC in R Is C , then it is also a primitive decomposition of 1RC in RC .

Proof. Given any primitive decomposition of 1RC in RC , the number of idem-potents in this decomposition is equal to the number of indecomposable directsummands of the regular module of RC , which is equal to the number of in-decomposable direct summands of RC /Rad(RC ) by Proposition 3.4.5. Let us

take the decomposition 1RC = ni=1 ei. We need to show it is primitive. Tothis end, we prove n equals the number of indecomposable direct summandsin RC /Rad(RC ).

Since all non-isomorphisms generate an ideal I of RC , which is contained inRad(RC ) and which induces an algebra isomorphism RC /I ∼= R Is C , from theisomorphism RC /Rad(RC ) ∼= (RC /I )/(Rad(RC )/I ), we know the two sideshave the same numbers of indecomposable direct summands. From definitionone can check that Rad(RC )/I is the radical of RC /I . Then by Proposition3.4.5, applied to both RC and RC /I ∼= R Is C , we see the primitive decompo-sitions of 1RC in both RC and R Is C must have the same number of idempo-tents. Hence we are done.

The category Is C is a disjoint union of groupoids, each of which comes from

an isomorphism class of some object. Recall that, for each object x ∈ Ob C , wedenote by [x] the set of objects isomorphic to x, and C [x] the full subcategoryconsisting of these objects.

Lemma 3.4.10. 1. If x ∼= y are two isomorphic objects, and f y ∈ HomC(x, y)is an isomorphism, then the assignment α → α · f y for each α ∈ RC · edefines an isomorphism of RC -modules RC · 1y → RC · 1x.

2. If 1x = n

i=1 ei is a primitive decomposition in RAutC(x), then 1y =ni=1 f yeif −1

y is a primitive decomposition in RAutC(y). Furthermore if we fix for each y ∼= x an isomorphism f y ∈ HomC(x, y), then

y∈ObC[x]

ni=1

f yeif −1y

is a primitive decomposition of the identity 1RC[x] in the groupoid algebra RC [x].




Proof. The isomorphism is straightforward to prove. Now if 1x = n

i=1 ei

is a primitive decomposition in RAutC(x), certainly 1y =n

i=1 f yeif −1y is a

decomposition in RAutC(y). It has to be primitive, because if it were not,then a primitive decomposition would be a sum of more than n idempotents

in RAutC(y). However f −1y (−)f y maps such a primitive decomposition of

1y to a decomposition of 1x, which contradicts with the assumption that1x =

ni=1 ei is a primitive decomposition.

The reader can compare the above statements with Example 3.4.7 (4).

Corollary 3.4.11. Let C be a finite EI category. One can write

1RC =

x∈Ob C

nxj=1

exj,

where nx is a positive integer for each x ∈ Ob C and 1x =nx

j=1 exj . As a con-sequence, RC = ⊕x∈ObC ⊕nx

j=1 RC · exj for some primitive pairwise orthogonal

idempotents exj ∈ RAutC(x), x ∈ Ob C .Any projective RC -module is isomorphic to a direct sum of indecomposable projective modules of the form RC · e, where e ∈ RAutC(x) is a primitive idempotent, for some x ∈ Ob C .

Given a primitive orthogonal decomposition 1RC =

i ei such that each ei

belongs to some group algebra RAutC(x), each summand of RC ∼=

eiRC ·ei

is indeed a left Kan extension

LK ι[RAutC(x)ei] = LK ι[RAutC(x)]ei ∼= RHomC(x, −)ei = RC ei

because LK ι commutes with direct sums. Here ι : C x → C is the inclusion.In each RC ei, its radical contains

z∼=x RHomx,zei which are linear combi-

nations of non-isomorphisms in HomC(x, −).

We continue to characterize the simple RC -modules. Directly from the EIcondition we have seen that a simple module S has to be atomic. It matcheswith our description of indecomposable projective modules. The quotient of RC ei, for a primitive idempotent ei ∈ RAutC(x), by its radical is an atomicmodule supported on C [x]. Moreover for each y ∼= x, S (y) must be a simpleRAutC(y)-module. In fact all simple RC -modules are exactly those simplemodules of R Is C , which are obtained from simple modules of automorphismgroup algebras RAutC(x).

Theorem 3.4.12 (Luck). Let C be a finite EI-category. The isomorphism classes of the simple RC -modules biject with the pairs ([x], V ), where x ∈ Ob C and V is a simple RAutC(x)-module, taken up to isomorphism.

Proof. First of all, we already know that all simple RC -modules are atomic.Thus simple RC -modules are exactly those simple RC [x]-modules, with x run-ning over Ob C .




Secondly for a fixed x, RC [x]-mod is equivalent to RAutC(y), for any y ∼= x,through restrictions induced by the equivalences C y → C [x] and C [x] → C y.Hence simple RC [x]-modules biject with simple RC y-modules, for any y ∼= x.Since C y is the group AutC(y), we have proved the assertion.

Because of the above theorem, it is natural to denote a simple RC -moduleby S x,V , if it comes from a simple RAutC(x)-module V , for some x ∈ Ob C .For consistency, we use P x,V for the projective cover of S x,V , whose structureis determined by its value at the object x. If RAutC(x) · e is the projectivecover of the simple RAutC(x)-module V , then RC · e is the projective coverof S x,V . The reader may revisit Example 3.4.7 to get better understandingof our results and notations in this section.

Example 3.4.13. Let k be a field of characteristic two and C the followingcategory

x1x α β

y 1y,g

with g2 = 1y, gα = α and gβ = β . Indeed the algebra kC has two(one-dimensional) simples S x,k, S y,k and their projective covers are P x,k =k1x, α , β , and P y,k = k1y, g, respectively. The product P x,k ⊗P x,k

∼=P x,k ⊕ S 2y,k is not projective because S y,k = P y,k.

Remark 3.4.14. Using the tensor product, one can introduce a “representa-tion ring” of RC , namely a(RC ), which consists of Z-linear combinations of symbols like [M ], representing an isomorphism class of a simple RC -moduleM . For any two elements [M ] and [N ], the multiplication is defined by[M ] · [N ] = [M ⊗N ]. However this product does not exist in K 0(RC ), which isspanned over the set of isomorphism classes of indecomposable projectives.

With the description of indecomposable projectives, we can show when

the trivial module R is projective.

Proposition 3.4.15. Let C be a finite EI-category. Then R is projective if and only if each connected component of C has a unique isomorphism class of minimal objects [x], with the properties that for all y in the same connected component as x, AutC(x) acts transitively on Hom(x, y), and |AutC(x)| is invertible in R.

Proof. Without loss of generality, we may assume C is connected.If R is projective then R ∼=

P y,W for certain indecomposable projective

modules P y,W . Since R is indecomposable and takes constant value at allobjects, we must have that R ∼= P x,V for some x ∈ Ob C , x is minimal,and all minimal objects are isomorphic. Moreover because P x,V (x) = R, the

projective cover of the simple kAutC(x)-module V , we get V = R and Rmust be projective as an RAutC(x)-module. Thus R is projective if and onlyif R ∼= P x,R, all minimal objects are isomorphic to x and |AutC(x)|−1 ∈ R.




Now, assume all minimal objects are isomorphic to x and |AutC(x)|−1 ∈ R.By Proposition 3.2.11 RHomC(x, −) ∼= LK ι[RAutC(x)]. Then R ∼= P x,R canbe explicitly constructed, using the idempotent e = 1

|AutC(x)|

g∈AutC(x) g, as

R ∼= P x,R ∼= LK ι[RAutC(x)e] = LK ι[RAutC(x)]e = RHomC(x, −)e

because LK ι commutes with direct sum. It is equivalent to saying that ateach y ∈ Ob C , R ∼= RHomC(x, y)e. This happens if and only if AutC(x) toact transitively on HomC(x, y).

3.4.4 Projective covers, injective hulls and their restrictions

Let C be a finite category and R = k a field. Then any finitely generatedkC -module M admits a minimal projective resolution

P ∗ → M → 0

in the sense that if P ∗ → M → 0 is another projective resolution of M ,then IdM induces a split injection of complexes from the minimal resolutionto the latter. Similarly we can define a minimal injective resolution of M ,0 → M → I ∗.

Proposition 3.4.16. Suppose C is an EI category.

1. If D ⊂ C is a coideal and P ∈ kC -mod is an indecomposable projective module, then ResιP ∈ kD-mod is either an indecomposable projective or zero.

2. If D ⊂ C is an ideal and I ∈ kC -mod is an indecomposable injective module,

then ResιI ∈ kD-mod is either an indecomposable injective or zero.

Proof. Let P = P x,V be an indecomposable projective kC -module. If x ∈Ob C , then ResiP x,V is brutally truncated from P x,V and is indecomposableprojective as an kD-module. On the other hand, if x ∈ Ob D, then ResiP x,V =0.

For the case of injective modules we recall (ResiP )∧ ∼= ResiP ∧ for anyright projective module, by Lemma 3.2.12. Note that Statement 1 stays truefor right projective modules if we replace the term “coideal” by “ideal”. Nowwe combine this with the duality between (indecomposable) right projectivesand left injectives.

For example if x is an object in C , then C ≤x is a coideal. Take any indecom-

posable projective module P x,V , then ResiP x,V ∼= P V , the projective coverof the simple kAutC(x)-module V . This module P V is an indecomposableprojective kC ≤x-module.




Definition 3.4.17. Let C be an EI category. Suppose M is a kC -module.Then we define C M ⊂ C to be the smallest ideal satisfying the condition thatif x ∈ Ob C M then M (x) = 0.

Suppose M is a kC -module. Then we define C M ⊂ C to be the smallest

coideal satisfying the condition that if x ∈ Ob C M

then M (x) = 0.Suppose (M, N ) is an ordered pair of kC -modules. We define a full sub-

category C N M to be C M ∩ C N .

Obviously if N ⊂ M then C N ⊂ C M and C N ⊂ C M .

Lemma 3.4.18. Let C be a finite EI category and M a kC -module. Then the projective cover P M of M satisfies the condition that C P M

⊂ C M , and the injective hull I M of M satisfies the condition that C I M ⊂ C M .

This result allows us to give a characterization of the minimal projectiveand injective resolutions of a module.

Corollary 3.4.19. Let C be a finite EI category and M ∈ kC -mod. Suppose

P ∗ and I ∗ are minimal projective and injective resolutions of M . Then for every n ≥ 0, C P n ⊂ C M and C I n ⊂ C M .

Proof. The kernel K 0 of P 0 M satisfies C K 0 ⊂ C P 0 ⊂ C M . We use thepreceding lemma repeatedly. Similar argument can be made on I ∗.

The last corollary will be useful when we compute cohomology of modules.



Chapter 4

Cohomology of categories and modules

Abstract We begin through investigation of (co)homology theories in thischapter. Extensions of modules over a category algebra is a concept of paramount importance here. We shall discuss various ways to examine Ext

groups, multiplicative structure and their relationship with previously de-fined simplicial and singular (co)homology. A particular important situationis when the first module is trivial. In this case, on top of the module theoretictools, simplicial methods are applicable. We shall provide a discussion of thebar resolution and its Kan extensions. Examples are used to illustrate variouscomputational methods. In the end, the Grothendieck spectral sequences areintroduced and we will study them in a couple of special cases.

4.1 General theory

4.1.1 Cohomology of modules

Since RC -mod is an abelian category with enough projectives and injec-tives, for any two modules M , N ∈ RC -mod we can consider the Ext groupsExti

RC(M, N ), with i ≥ 0. It is the i-th right derived functor of the left exactfunctor HomRC(M, −) (or HomRC(−, N )). In general for any M ∈ RC -modExt∗RC(M, M ) has a ring structure with product given by the Yoneda splice.Usually it is not graded commutative. However it is in the case of M = R.In this chapter we shall compare the Yoneda splice with the cup productintroduced earlier.

Similarly for a right RC -module M and a left RC -module N , we can studyTorRC

i (M , N ) as the i-th right derived functors of M ⊗RC− (or − ⊗RC N ).In these notes we shall focus on cohomology. However we shall remark on

homology whenever it is appropriate.

95



96 4 Cohomology of categories and modules

We recall some basics about extensions of modules and their relationshipwith cohomology classes.

Definition 4.1.1. Let M , N be A-modules. An n- fold extension , n ≥ 1, of M by N is an exact sequence of A-modules

0 → N → Ln−1 → · · · → L0 → M → 0.

Two n-fold extensions of M by N are equivalent if there is a commutativediagram

0 N Ln−1

· · · L0

M 0

0 N Ln−1 · · · L0 M 0

Then we can extend this by symmetry and transitivity to an equivalencerelation among n-fold extensions of M by N .

Proposition 4.1.2. There is an one-to-one correspondence between elements of Extn

A(M, N ) and equivalent classes of n-fold extensions of M by N .

Proof. Let P ∗ → M → 0 be a projective resolution. Then an extensiondetermines an element in Extn

RC(M, N ) by the following lifting

· · · P n+1

P n∂ n

f

P n−1

· · · P 0

M 0

0 N Ln−1 · · · L0

M 0

We see from here that two equivalent extensions give rise to the same elementin ExtnA(M, N ).

Conversely if two n-fold extensions determine the same elements in thegroup Extn

A(M, N ), then we can construct a commutative diagram (by en-larging P ∗ we may assume f is surjective)

0 N Ln−1 Ln−2

· · · L0 M 0

0 N P n−1/∂ n(Kerf )

P n−2

· · · P 0

M 0

0 N Ln−1 Ln−2

· · · L0 M 0

Definition 4.1.3. Let M ∈ A-mod. Then we can define the Yoneda splice on Ext∗A(M , M ) and Ext∗A(M, M ) so that for any η ∈ Exti

A(M , M ) and



4.1 General theory 97

η ∈ ExtjA(M, M ), η ∗ η ∈ Exti+j

A (M , M ) is given by

0 → M → N j−1 · · · → N 0 → N i−1 → · · · → N 0 → M → 0,

if η is represented by 0 → M → N i−1

· · · → N 0

→ M → 0 and η isrepresented by 0 → M → N j−1 · · · → N 0 → M → 0.

The Yoneda splice gives Ext∗A(M, M ) a ring structure which is not gradedcommutative in general. Moreover Ext∗A(M, M ) acts on Ext∗A(M, N ) andExt∗A(N, M ) for another N ∈ A-mod.

Definition 4.1.4. We call Ext∗RC(R, R) =

i≥0 ExtiRC(R, R) the ordinary

cohomology ring of the category algebra A. The product in this ring is definedby the Yoneda splice of Ext classes.

We shall show later on that this ring is isomorphic in a natural way to

H∗(C , R) ∼= H∗(BC , R) so it deserves the name.Now based on the tensor structure on RC -mod, we provide a module theo-

retic description to the ring Ext∗

RC(R, R). In the meantime we pave the road

to studying the ring action of Ext∗RC(R, R) on various Ext groups. We com-ment that since (RC -mod, ⊗, R) is a monoidal category with an exact tensorproduct, it gives rise to a suspended monoidal category (D−(RC ), ⊗, R) andthen following a general statement [71] on the endomorphisms of the identityin a suspended monoidal category, EndD−(RC)(R) is a graded commutativering. It will be clear in this section that this endomorphism ring is isomorphicto what we call the ordinary cohomology ring Ext∗RC(R, R).

Let M, M , N , N ∈ RC -mod which are projective as R-modules. We willdefine the cup product to be

∪ : ExtiRC(M, N ) ⊗ Extj

RC(M , N ) → Exti+jRC (M ⊗M , N ⊗N ).

Since R is the identity with respect to ⊗, this will give us a ring structureon Ext∗RC(R, R), as well as an action of Ext∗RC(R, R) on Ext∗RC(M, N ) forarbitrary M, N ∈ RC -mod. One shall compare our construction with [?,Section 3.2] for cocommutative Hopf algebras.

Now we are ready to give a precise definition to the cup product. Supposeζ ∈ Extm

RC(M, N ) is represented by an exact sequence

0 → N → Lm−1 → P m−2 → · · · → P 0 → M → 0,

where P i’s are projective RC -modules, and ζ ∈ ExtnRC(M , N ) is represented

by an exact sequence

0 → N → Ln−1 → Qn−2 → · · · → Q0 → M → 0,

where Qj ’s are projective RC -modules. Since projective RC -modules are projective R-modules, all modules in these two exact sequences are projectiveR-modules. Furthermore an RC -module L being a projective R-module is




equivalent to L(x) being projective for all x ∈ Ob C . Then applying theKunnethe formula to the following two complexes of projective R-modules

0 → N (x) → Lm−1(x) → P m−2(x) → · · · → P 0(x)

and0 → N (x) → Ln−1(x) → Qn−2(x) → · · · → Q0(x),

for all x ∈ Ob C , we get exact sequences

0 → (N ⊗N )(x) → (Lm−1⊗N )(x)

(N ⊗Ln−1)(x) → · · ·

→ (P 0⊗Q0)(x) → (M ⊗M )(x) → 0,

with x running over Ob C . Thus we get an exact sequence of RC -modules

0 → N ⊗N → (Lm−1⊗N )

(N ⊗Ln−1) → · · · → P 0⊗Q0 → M ⊗M → 0,

which is defined to be the cup product of ζ and ζ

,

ζ ∪ ζ ∈ Extm+nRC (M ⊗M , N ⊗N ).

Lemma 4.1.5. Let ζ, ζ be as above. The cup product ζ ∪ ζ is the Yoneda splice of

ζ ⊗N ∈ ExtiRC(M ⊗N , N ⊗N )

with M ⊗ζ ∈ Extj

RC(M ⊗M , M ⊗N ).

There exists a natural map φM = −⊗M : Ext∗RC(R, R) → Ext∗RC(M, M )lies in the graded center, for any M ∈ RC -mod. Particularly, Ext∗RC(R, R) is graded commutative.

Proof. By using exact sequences representing ζ and ζ , one can easily estab-lish a map between (n + m)-fold extensions (ζ ⊗IdN ) ∗ (IdM ⊗ζ ) → ζ ∪ ζ .Let

D : 0 → N → P m−1 = Lm−1 → P m−2 → · · · → P 0

andC : 0 → N → Qn−1 = Ln−1 → Qn−2 → · · · → Q0

come from the given two extensions. Then we have a map of (m + n)-foldextensions

0 N ⊗N (P m−1⊗N )

(N ⊗Qn−1)

f m+n−1

· · · P 0⊗Q0

f 0

M ⊗M 0

0 N ⊗N

P m−1⊗N

· · · M ⊗Q0 M ⊗M

0

given by






If for two integers m, n, ζ ∈ ExtmRC(M, N ) and ζ ∈ Extn

RC(M , N ) are repre-sented by two cocycles f : P m → N and g : Qn → N , then the product ζ ∪ ζ

is represented by (f ⊗g) δ : Rm+n → N ⊗N . In Theorem 4.1.13 we shallshow how to establish the algebra isomorphism Ext∗RC(R, R) ∼= H∗(BC , R).

4.1.2 Cohomology of a small category with coefficientsin a functor

In this section we introduce a particular important case, the Baues-Wirschingcohomology theory of small categories, and go over some basic properties. Thecohomology theory of small categories has been discussed in various placesin literature, see Baues-Wirsching [3], Generalov [29] and Oliver [?]. One canalso find in Gabriel-Zisman [28] and Hilton-Stammbach [35] the homologytheory of small categories.

Definition 4.1.6. Let C be a small category and R a commutative ring. Wedefine Hn(C ; M ), the n-th cohomology of C with coefficients in a covariant

functor M : C → R-mod, as the homology of the following cochain complexCi(C ; M ), δ ii≥0, where

Ci(C ; M ) = f : N C i →

[x0→x1→···→xi]

M (xi) f ([x0 → · · · → xi]) ∈ M (xi)

for all i ≥ 0; and for f ∈ Ci(C ; M ),

δ n(f )([x0 → · · · → xiφ

→xi+1]) =

ij=0

(−1)j f ([x0 → · · · → xj → · · · → xi+1])

+(−1)i+1M (φ)(f ([x0 → · · · → xi]))

We define Hn(C ; M ), the n-th homology of C with coefficients in a covariant functor M : C → R-mod, as the homology of the following chain complexCi(C ; M ), δ ii≥0, where

Ci(C ; M ) =

[x0→x1→···→xi]

M (x0),

for all i ≥ 0; and for any m[x0→x1→···→xi] ∈ Ci(C ; M ) which is an element insome m ∈ M (x0) indexed by x0 → · · · → xj → · · · → xi,

δ i(m[x0

φ→x

1→···→x

i−1→x

i]) = [M (φ)(m)][x1→···→xi−1]




+i

j=0

(−1)j m[x0→···→xj→···→xi],

in which m [x0→···→xj→···→xi] means m ∈ M (x0) is considered as an element

of Ci−1(C ; M ) indexed by the (i − 1)-sequence x0 → · · · → xj → · · · → xi.

We now give different interpretations to the above homology and cohomol-ogy theory. They will give us better understanding of the preceding defini-tions and then lead us to a more general cohomology theory. Recall thatfor any functor u : D → C between two small categories, we can natu-rally define two functors u/− : C → Cat , the category of small categories,and B(u/−) : C → T op, the category of topological spaces. Thus for anyu : D → C , we may write C∗(u/x,R), x ∈ Ob C as the chain complex comingfrom the simplicial set associated to the small category u/x, which can beregarded as the cellular chain complex on the space B (u/x).

Definition 4.1.7. Let C be a small category. We define the bar resolution of R ∈ RC -mod as B C

∗ = C

∗(Id

C/−, R), a complex of RC -modules.

The bar resolution is a complex reconstructed from C∗(C , R) and its nameis justified as follows.

Proposition 4.1.8. For any small category C , B C∗ is a complex of projective RC -modules such that

1. B C0 ∼= RC ;2. B C∗ → R → 0 is an exact sequence of RC -modules;3. There is a degree 1 isomorphism of complexes of R-modules from

C∗(IdC/−, R) → R → 0

to R[N C ]∗ = C∗(C , R) → 0.4. HomRC(B C∗ , M ) ∼= x0→···→xi

M (xi).

Remark 4.1.9. We emphasize that there is no conflict between 2 and 3. Acomplex of RC -modules is exact if and only if it is pointwise exact.

Proof. For each i ≥ 0, Ci(IdC/−, R) : C → R-mod is a well defined func-tor and thus an RC -module. There is an RC -map ∂ i : Ci+1(IdC/−, R) →Ci(IdC/−, R) as follows. For any x ∈ Ob C ,

∂ i((x0, α0) → · · · → (xj , αj ) → · · · → (xi+1, αi+1))

=i+1

j=0(−1)j [(x0, α0) → · · · → (xj , αj ) → · · · → (xi+1, αi+1)],

where αj ∈ HomC(xj , x). Thus for each x ∈ Ob C , C∗(IdC/x,R) → R → 0 is

the augmented chain complex of IdC/x. When we assemble these augmentedchain complexes together, C∗(IdC/−, R) → R → 0 becomes a complex of RC -modules. Since every IdC/x has a terminal object (x, 1x) and thus is




contractible, the complex C∗(IdC/−, R) → R → 0 is acyclic because its eval-uation at any x, i.e. C∗(IdC/x,R) → R → 0 computes the reduced homology

H∗(IdC/x,R) and is acyclic. Moreover, we can show every Ci(IdC/−, R) isprojective. Indeed for each i > 0, Ci(IdC/x,R) has base elements of the

form (x0, α0) → · · · → (xi, αi). The following bijection provides a differentpresentation of these base elements

(x0, α0)β1→ · · ·

βi→(xi, αi) x0αiβi···β1−→ · · ·

αiβi−→xiαi→x.

This bijection in fact induces an isomorphism of complexes of R-modules (seeDefinitions 1.2.5 and 2.2.4)

Ci(IdC/−, R) ∼=−→R[N C ]i+1 = Ci+1(C , R), ∀i ≥ 0.

We get immediately C0(IdC/−, R) ∼= C1(C , R) ∼= RC . In general let us takethe new R-basis of Ci(IdC/−, R), x0 → · · · → xi → x, and regroup themby putting two such sequences into one subset of the basis if, after removing

the rightmost object, they become identical. Then each subset determines anRC -module

Rx0 → · · · → xiα

→− s(α) = xi,

isomorphic to the projective module RHomC(xi, −). It means as RC -modules

Ci(IdC/−, R) ∼=

x0→x1→···→xi

RHomC(xi, −).

Hence B C∗ → R → 0 is a projective resolution. In the end for any M ∈ RC -mod, there is an isomorphism by Yoneda lemma

HomRC(Ci(IdC/−, R), M ) ∼=

x0→···→xi

M (xi).

We remind the reader that R can be defined as R Ob C ∼= R[N C ]0 ∼= C0(C , R).

Thus we have a degree one isomorphism of chain complexes (of R-modules)from C∗(IdC/−, R) → R → 0 to R[N C ]∗ = C∗(C , R) → 0 if in the firstcomplex we insert R in degree -1.

Example 4.1.10. In Example 2.2.6 (1) we considered the following categorywith two objects, two identity and non-identity morphisms

xα

β

y

Its nerve gives rise to a non-normalized chain complex C∗(C , R) which is of

rank 2 in degree zero, rank 4 in degree 1 and rank 2n + 2 at degree n ≥ 2. Itis certainly not exact.




The bar resolution B C∗ can be written down by computing the two overcat-egories IdC/x and IdC/y. In fact IdC/x is a trivial category with one object(x, 1x), and IdC/y is a poset with three objects (x, α), (x, β ) and (y, 1y).

(y, 1y)

(x, α)

α

(x, β )

β

Hence the bar resolution is given by B C∗ (y) and B C∗ (x). The former has rank3 in degree zero, rank 5 in degree 1, and rank 2n + 3 in degree n ≥ 2;while the latter has rank 1 at every degree. Thus B C∗ has rank 4 in degreezero, rank 6 in degree 1 and rank 2n + 4 in degree n ≥ 2. One can see thatRank(B Ci ) =Rank(Ci+1(C , R)) for all i ≥ 0. This equality is actually inducedby the R-isomorphism we explained in the proof of Proposition 3.1.3. Notethat both B C∗ (y) and B C∗ (x) are exact because they come from the nerves of the two overcategories Id

C/y and Id

C/x which are contractible. Furthermore

we can explicitly verify that B C0 ∼= RC , B C1 ∼= P x,1 ⊕ P 3y,1, etc.If one examines the normalized complexes of C∗(C , R), C∗(IdC/y,R) and

C∗(IdC/x,R) then the above equality between ranks are not true any more.

Remark 4.1.11. Proposition 4.1.8 summarizes to a characterization of the barresolution in terms of the nerve, as well as an isomorphism

HomRC(B Ci , M ) ∼=

x0→···→xi

M (xi).

By contrast, when we consider R as a right RC -module. The bar resolutionof it is C∗(−\IdC, R). We shall also denote it by B C∗ when it does not causeany confusion. For each i ≥ 0,

Ci(−\IdC , R) ∼=

x0→x1→···→xi

RHomC(−, x0).

Since RHomC(−, x0) = 1x0 · RC , we can verify that

B Ci ⊗RC M ∼=

x0→x1→···→xi

M (x0).

When we examine the special case for M = R, we get isomorphisms of complexes

HomRC(B Ci , R) ∼=

x0→···→xi

R ∼= HomR(R[N C ]i, R).

and




B Ci ⊗RC R ∼=

x0→···→xi

R ∼= R[N C ]i.

Proposition 4.1.12. Let C be a small category and R a commutative ring.If M : C → R-mod is a functor, then

lim−→n

CM ∼= TorRC

n (R, M ) ∼= Hn(C ; M ),

and lim←−

n

CM ∼= Extn

RC(R, M ) ∼= Hn(C ; M ).

Proof. Let U be an arbitrary R-module. Then we have

HomR(U, HomRC(R, M )) ∼= HomRC(R ⊗R U, M ) ∼= HomRC(U , M ),

where U ∈ RC -mod is a constant functor. Hence by Proposition 1.1.23,lim←−C

M ∼= HomRC(R, M ).Similarly

HomR(R ⊗RC M, U ) ∼= HomRC(M, HomR(R, U )) ∼= HomRC(M, U ),

and consequently R ⊗RC M ∼= lim−→CM .

We will only prove the isomorphisms for cohomology because the state-ments for homology can be obtained similarly. Since the kernel of the differ-ential δ 0 : C0(C ; M ) → C1(C ; M ) can be identified with HomRC(R, M ), weget H0(C ; M ) ∼= HomRC(R, M ).

Since lim←−C is left exact and lim←−n

C is its n-th right derived functor, we

have lim←−n

CM ∼= Extn

RC(R, M ). Hence we only have to show ExtnRC(R, M ) ∼=

Hn(C ; M ). Because ExtnRC(R, M ) can be computed by using any projective

resolution of R in RC -mod, we can use the bar resolution to do it. But wehave an isomorphism for each i ≥ 0

HomRC(B Ci , M ) ∼= x0→···→xi

M (xi) ∼= Ci(C ; M ).

Suppose C = G is a group and M, N are two left RG-modules, since a leftRG-module can be naturally regarded as a right RG-module (and vice versa),it makes sense to consider both TorRG

∗ (M, N ) and TorRG∗ (N, M ). Indeed we

always have TorRG∗ (M, N ) ∼= TorRG

∗ (N, M ), see [12, Chapter 3]. It is not thecase in general for category homology.

We deduce now that the preceding homology and cohomology theories arethe same as the simplicial and singular homology and cohomology theorieswe introduced in Chapter 2.

Theorem 4.1.13. We have isomorphisms

TorRC∗ (R, R) ∼= H∗(C ; R) ∼= H∗(C , R) ∼= H∗(BC , R)




and Extn

RC(R, R) ∼= H∗(C ; R) ∼= H∗(C , R) ∼= H∗(BC , R).

The latter are algebra isomorphisms.

Proof. In order to prove the isomorphisms between graded R-modules, weonly need to show that H∗(C ; R) ∼= H∗(C , R) and H∗(C ; R) ∼= H∗(C , R). Butthese follow from Remark 4.1.11 which compare Definitions 2.2.4 with 4.1.6.

In order to prove the isomorphism between cohomology rings is an algebraisomorphism we just have to compare the cup products. Here we only needto explain the first algebra isomorphism because the second is clear.

Using the bar resolution B C∗ → R → 0, we can describe the cup product onExt∗RC(R, R). In fact, we can explicitly write out a diagonal approximation map (unique up to chain homotopy)

B C∗

D

R 0

B C∗ ⊗B C∗ R 0

as a natural transformation, given by

Dx(x0α1→x1 → · · ·

αn→xnα

→x) =n

i=0

(x0α1→ · · · → xi

α···αi+1→ x)⊗(xi

αi+1→ · · ·

αn→xnα

→x),

for any x ∈ Ob C and integer n. Since for each n ≥ 0, (B C∗ ⊗B C∗ )n =i+j=n B Ci ⊗B Cj and there is a natural map

HomRC(B Ci , R) ⊗ HomRC(B Cj , R) → HomRC(B Ci ⊗B Cj , R),

one can easily see, under the isomorphism HomRC(B C∗ , R) ∼= HomR(R[N C ]∗, R),

that the diagonal approximation map D induces a map

HomRC(B C∗ , R) ⊗ HomRC(B C∗ , R) → HomRC(B C∗ , R)

identical to the following

HomR(R[N C ]∗, R) ⊗ HomR(R[N C ]∗, R) → HomR(R[N C ]∗, R)

induced by the Alexander-Whitney map. Since these two maps are used tocalculate cup products in the two cohomology rings, our observations implythat Ext∗RC(R, R) ∼= H∗(BC , R) as algebras.

Recall that the bar resolution can also be constructed via the nerve of

overcategories associated with the identity functor IdC : C → C , as B C∗ ∼=C∗(IdC/−, R) := RN ∗(IdC/−). In this form, B Cn(x) ∼= Cn(IdC/x), for eachx ∈ Ob C and integer n ≥ 0, consists of the following chains as base elements




(x0, β 0)γ 1→(x1, β 1) → · · · → (xn−1, β n−1)

γ n→(xn, β n),

in which β i is a morphism in HomC(xi, x), and γ i ∈ HomC(xi−1, xi) suchthat β i−1 = β iγ i−1. The previously defined diagonal approximation map D

is given by

Dx((x0, β 0)γ 1→(x1, β 1) → · · · → (xn−1, β n−1)

γ n→(xn, β n))

=n

i=0[(x0, β 0)γ 1→ · · ·

γ i→(xi, β i)] ⊗ [(xi, β i)γ i+1→ · · ·

γ n→(xn, β n)]

This is exactly the Alexander-Whitney map for the chain complex comingfrom the nerve of IdC/x.

All homology and cohomology theories of small categories that we haveintroduced so far coincide whenever they are comparable. Hence we shall onlydeal with the most general form Ext∗RC(M, N ) and TorRC

∗ (M , N ) from nowon, where M, N ∈ RC -mod and M ∈ mod-RC .

Proposition 4.1.14. Suppose u : D → C is a functor. Then we have a re-striction Res

u :

RC -mod

→ RD-mod. Given two

RC -modules

M, N , Res

uinduces a natural map

resu : Ext∗RC(M, N ) → Ext∗RD(ResuM, ResuN ),

called the restriction.

Proof. Take a projective resolution P ∗ of M . Then ResuP ∗ → ResuM → 0 isan exact sequence of RD-modules. Hence for any projective resolution Q∗ →ResuM → 0, the usual lifting, a chain map, Q∗ → ResuP ∗ induces a cochainmap HomRC(P ∗, N ) → HomRD(Q∗, ResuN ) because we have a cochain mapHomRC(P ∗, N ) → HomRD(ResuP ∗, ResuN ). This is a well defined map sinceit does not depend on the choice of P ∗ and Q∗.

When M = N = R, by using bar resolutions, one can easily see that theabove restriction is the same as the one we defined for simplicial cohomologyin Chapter 2.

Remark 4.1.15. Suppose u : D → C is a functor. Let us consider

resu : Ext∗RC(R, N ) → Ext∗RD(R, ResuN ).

One way to obtain some information about the restriction is to examine itsaction in degree zero. In fact, the restriction is the same as Hom RC(R, N ) →

HomRC(LK uR, N )∼=→HomRD(R, ResuN ) induced by the counit LK uR → R.

When LK uR ∼= R, resu becomes an isomorphism in degree zero. GivingLK uR ∼= R is equivalent to saying that every overcategory of u is R-acyclic ,

that is, H∗(u/x,R) vanishes for every x ∈ Ob C . In other words, every over-category u/x has to be connected, see Proposition 3.2.7.




4.1.3 Extensions of categories and low dimension cohomology

We describe low dimension cohomology groups ExtnRC(M, N ) for n = 1, 2. The

results and their proofs are taken from Webb’s notes [80]. Let us recall thatthere is an augmentation map : RC → R by sending α to t(α), its target.The kernel of is a left ideal of RC , which we call the (left) augmentation ideal , and write as I C := Ker.

Lemma 4.1.16. The augmentation ideal I C is a free R-module with basis the elements α − 1t(α), where α runs over all the non-identity morphisms in C .

Definition 4.1.17. Let M ∈ RC -mod. We define a derivation d : C → M tobe a mapping from Mor C to M so that d(α) ∈ M (t(α)) and so that d(αβ ) =M (α)d(β ) + d(α). The set of derivations forms an R-module Der(C , M ).

Given any set of elements ux ∈ M (x)x ∈ Ob C we obtain a derivation

specified by d(α) = M (α)(us(α)) − ut(α). Any derivation obtained in thisway is called an inner derivation . The inner derivations form an R-moduleIDer(C , M ).

Proposition 4.1.18. 1. Der (C , M ) ∼= HomRC(I C , M ) as R-modules.2. H1(C ; M ) ∼= Der(C , M )/IDer(C , M ).

Proof. Given a homomorphism δ : I C → M , we can define a derivationd : C → M by d(α) = δ (α−1t(α)). It makes sense because M (α)d(β )+d(α) =δ [(α(β − 1t(β))] + d(α) = δ (αβ − α) + d(α) = δ (αβ − 1t(αβ))) = d(αβ )since t(αβ ) = t(α). Conversely given a derivation d we can define a RC -homomorphism δ : I C → M by δ (α − 1t(α)) = d(α), and we can verify that δ is an RC -map. Hence we have proved (1).

As for (2), consider the short exact sequence 0 → I C → RC → R → 0 andapply Ext∗RC(−, M ). We obtain an exact sequence

0 → HomRC(R, M ) → HomRC(RC , M ) → HomRC(I C , M ) → Ext1RC(R, M ) → 0.

The image of HomRC(RC , M ) ∼= M consists of f m|I C, where f m is theRC -homomorphism determined by some m ∈ M via the canonical mapm → f m such that f m(1RC) = m. Because f m(α − 1t(α)) = [M (α)](m) −[M (1t(α))](m) = [M (α)](1s(α) · m) − [M (1t(α))](1t(α) · m), f m|I C is an inner

derivation determined by the set of elements 1x · m ∈ M (x) x ∈ Ob C.

Moreover all inner derivations are of this form. We are done.

In order to characterize H2(C ; K), we have to introduce category extensionsin the sense of G. Hoff.

Definition 4.1.19 (Hoff). An extension E of a category C via a category Kis a sequence of functors

K ι−→E π−→C ,

which has the following properties:




1. Ob K = Ob E = Ob C , ι is injective and π is surjective on morphisms; and2. if π(α) = π(β ), for two morphisms α, β ∈ Mor(E ), if and only if there is a

unique g ∈ Mor(K) such that β = ι(g)α.

We may readily deduce some useful information from the above definition.

Proposition 4.1.20. 3. If αι(h) exists for α ∈ Mor(E ) and h ∈ Mor(K),then there exists a unique hα ∈ Mor(K) such that ι(hα)α = αι(h). More-over any α ∈ HomE (x, y) induces a group homomorphism K(x) → K(y);and

4. for any α ∈ HomC(x, y), K(y) acts regularly (i.e. freely and transitively)on π−1(α).

Proof. Suppose αι(h) exists for a given α ∈ Mor E . Then since π(αι(h)) =π(α), by Definition 4.1.19 (2), there exists a unique hα ∈ Mor K such thatαι(h) = ι(hα)α. If h1, h2 ∈ K(x), then we have αι(h1h2) = ι[(h1h2)α]α.On the other hand, we also get αι(h1h2) = αι(h1)ι(h2) = ι(h1)α[αι(h2)] =ι(hα

1 )ι(hα2 )α. Hence hα

1 hα2 = (h1h2)α and α induces a group homomorphism.

Now assume β ∈ HomE (x, y) and π(β ) = α. Let h ∈ K(y). If ι(h)β = β ,then we have π(ι(h)β ) = π(β ) = α. Definition 4.1.19 (2) forces h = 1y

because of the uniqueness property.

One can see that K is indeed a disjoint union of the groups π−1(1x) for all1x ∈ Mor(C ) (regarded as categories), and can be identified with a functorK : E → Groups. Usually from the context, one knows when we take K to bea category and when it is regarded as a functor.

Since in practice one often crosses a concept dual to the category extension,for future reference, we first state its definition.

Definition 4.1.21. An opposite extension E of C via K is a sequence of func-tors K

ι→E

π→C such that the following sequence is an extension of C op

Kop ιop−→E op πop

−→C op,

Sometimes we just say K → E → C is an opposite extension. Dually thereare characterizations as follows.

1op. Ob K = Ob E = Ob C , ι is injective and π is surjective on morphisms; and2op. if π(α) = π(β ), for two morphisms α, β ∈ Mor(E ), if and only if there is a

unique g ∈ Mor(K) such that β = αι(g).3op. If ι(h)α exists for α ∈ Mor(E ) and h ∈ Mor(K), then there exists a unique

h ∈ Mor(K) such that αι(h) = ι(h)α; and4op. for any α ∈ HomC(x, y), K(x) acts regularly on π−1(α).

One of the main difference between extensions and opposite extensions is

that, for an extension, K can be considered as a covariant functor from C to Groups while, for an opposite extension, K gives a contra-variant func-tor. When we study H1 and H2, extensions are used because we deal with left






quotient categories, e.g. the orbit category OP (G), many of which formextension sequences, e.g.

Ks → G ∝ P → OP (G),

where Ks(H ) = H for any H ∈ Ob P = Ob(G ∝ P ). We shall come backto this in Chapter 6.

Definition 4.1.23. Two extensions are equivalent if we have the followingcommutative diagram

K E

C

K E C

Definition 4.1.24. An extension is split if it admits a functor s : C → E such that π s = 1C.

Proposition 4.1.25. An extension is split if and only if it is equivalent to a Grothendieck construction.

Proof. Assume an extension K → E → C is split. Then K can be restrictedto a functor C → E → Groups. The Grothendieck construction GrCK has thesame objects of the form (x, •x) where x ∈ Ob C and •x is the only object of K(x). A morphism from (x, •x) to (y, •y) is a pair (α, h) with α ∈ HomC(x, y)and h ∈ K(y). Hence we can define a funtor GrCK → E by (x, •x) → x and(α, h) → hα.

On the other hand for an extension K → GrCK → C , we can define afunctor C → GrCK by x → (x, •x) and α → (α, 1y) if α ∈ HomC(x, y). Thusthis extension is split.

Proposition 4.1.26. The equivalence classes of extensions of C by K are in bijection with H2(C ; K) in such a way that the zero element corresponds tothe Grothendieck construction GrCK.

Proof. We will provide correspondence between equivalence classes of exten-sions and elements of H2(C ; K).

Let K → E → C be an extension. We can choose a section for E → C ,that is, for each morphism α : x → y in C a morphism q (α) : x → y in E with pq (α) = α. If α : x → y and β : y → z are two morphisms in C , thenq (βα) = ι[τ (β, α)]q (β )q (α) for a unique τ (β, α) ∈ K(z). Thus we obtain amapping τ : C × C → C . The associativity of morphisms implies the 2-cocyclecondition by direct calculation

τ (γβ,α) + τ (γ, β ) = τ (γ,βα) + K(γ )τ (β, α).

If f ∈ HomRC(RC , M ), then mappings of the form




f (β, α) := f (βα) − f (β ) − K(β )f (α)

automatically satisfies the 2-cocycle condition. If we examine HomRC(B C∗ , M ),used to define H∗(C ; M ) in Definition 3.1.1, we can verify that all these map-

pings τ and ˜f biject with the 2-cocycles and 2-coboudaries in the the cochaincomplex HomRC(B C∗ , M ). In fact we already learned that

HomRC(B C2 , M ) ∼=

xα→y

β→z

K(z)

andHomRC(B C1 , M ) ∼=

v γ→w

K(w).

With explicitly given differential we can prove the above bijection. In the end,we may verify that 2-cocycles are homologous if and only if the extensionswhich produce them are equivalent, and also a change in the choice of asection gives rise to a cohomologous 2-cocycle. Hence we are done.

To finish this section, we comment on the connection between Hi(C ; M )and Exti

RC(R, M ), for i = 1, 2. In fact from the short exact sequence 0 →I C → RC → R → 0, we get

ExtiRC(R, M ) ∼= Exti−1

RC (I C , M ).

Let 0 → M → E → I C → 0 be an RC -module extension representing anelement in Ext1

RC(I C , M ). We can construct an extension of categories

M → GrCE → GrCI C .

There is a splitting functor C → GrCI C , which is identity on objects and

sends a morphism α to (α − 1t(α), α). Now we form a pullback diagram

M E

C

M GrCE GrCI C

which gives rise to an extension M → E → C , an element in H2(C ; M ). Notethat if 0 → M → E → I C → 0 splits, then GrCE ∼= GrGrCI CM , in H1(C ; M ).

Conversely if M → E → C is an extension and IM is the kernel of thealgebra homomorphism RE → RC . Then we can define an extension of RC -modules

0 → M → I E /(IM · I E ) → I C → 0,

representing an element in Ext2RC(R, M ) ∼= Ext1

RC(I C , M ).




4.2 Classical methods for computation

4.2.1 Minimal resolutions and reduction

Recall in Section 3.3.3 we described the minimal projective and injectiveresolutions of a module. With Definition 3.3.13 we establish the followingisomorphism.

Proposition 4.2.1. Let C be an EI category, M , N two k C -modules and M

a right kC -module. Then

Ext∗kC(M, N ) ∼= Ext∗kCN M

(ResiM, ResiN )

and TorkC

∗ (M , N ) ∼= Tor∗kCM

N (ResiM , ResiN ).

Proof. Suppose P ∗ is a minimal projective resolution of M . Then each di-

rect summand P x,V of any P n must satisfy the condition that x ∈ Ob C M

by Corollary 3.3.15. Hence HomkC(M, N ) ∼= HomkCM (ResiM, ResiN ) andExt∗kC(M, N ) ∼= Ext∗kCM

(ResiM, ResiN ). Now we apply Corollary 3.3.15again to a minimal injective resolution of N when computing

Ext∗kCM (ResiM, ResiN )

and the isomorphism follows.To prove the isomorphism for Tor the method is the same. One just need

to take the right projective resolution of M .

This result is useful because one may replace the original category by afull subcategory before carrying on any computations. It is particular helpful

if both M and N are atomic. Since any module of an EI category algebrahas a filtration by atomic modules. Repeatedly using the above corollarymay significantly simplify things. In following examples we will shall how onecombines resolutions, module filtrations and reductions through Proposition4.2.1 to compute various Ext groups.

Example 4.2.2. 1. Suppose C = n is a poset. Then k ∼= kHomn(0, −) is aprojective module. Thus Ext∗kn(k, M ) ∼= M (0) for any M ∈ kn-mod.

2. Suppose C is the following category

xα

β

y.

We will only write down non-vanishing Ext groups in this example.The minimal projective resolution of k is 0 → P y,k → P x,k → k → 0.Thus for any M ∈ kC -mod Ext∗kC(k, M ) is given by the homology of 0 →






Since Ext∗kD(k, k) ∼= k, both Ext∗kD(S x,k, S y,k ) and Ext∗kD(S y,k, S y,k) arenot finitely generated modules.If we look at the opposite category Dop, again we have Ext∗kDop(k, k) ∼=Ext∗kD(k, k) ∼= k , and Ext∗kDop(k, S y,k) ∼= Ext∗kDop(S y,k , S y,k) is of infinite

dimensional. It means we do not have the finite generation of Ext∗kD(k, M )

over Ext∗kD(k, k) even if they can be calculated by using the same projec-tive resolution.

4.2.2 Examples using classifying spaces

Understanding the homotopy type of a classifying space certainly will be agreat help for compute cohomology rings. Here we list several cases wheredirect calculation is possible.

Example 4.2.3. In Example 4.2.2, part (1) is a contractible category n so we

know Ext∗

kn(k, k) ∼= k. Part (2) is a category whose classifying space is thecircle S 1. Part (3) is a contractible category because it has an initial object.

However, there are more things we can take from homotopy theory. If weknow all Ext∗kC(k, −), then we may readily write down Ext∗kCop(k, −).

Now we examine some more sophisticated examples. Although they do nottell much about cohomology with local coefficients, they do provide substan-tial information about the ordinary cohomology ring of a category.

Example 4.2.4. 1. Let C be the following category

xG α y H .

Then this is the join of G and H . Thus its classifying space is a join of spaces BC BG ∗ BH . Thus its cohomology is completely determined byG and H . Also the cup product is trivial.

2. Let D be the following category

x

1x

h

α β

y 1y ,

where h2 = 1x and β h = α. Then the inclusion i : AutD(x) → D inducesa homotopy equivalence between their classifying spaces. In particular, wehave a ring isomorphism Ext∗RD(R, R) = Ext∗RAutD(x)(R, R) for any ringR. When R = C, the cohomology ring is just C, while when R is a fieldof characteristic 2 the ordinary cohomology ring is a polynomial algebrawith indeterminant in degree one.



4.2 Classical methods for computation 115

3. Let E be the following category

x

1x

g

h

gh

α

β

y 1y

,

where g2 = h2 = 1x, gh = hg,αh = α,αg = β,βh = β,βg = β . Itsclassifying space is the homotopy pushout of

BC 2

BAutE (x) = B(C 2 × C 2)

BAutE (y) ∼= •

where the cyclic group of order 2 is 1x, h, the stabilizer of HomE (x, y) =α, β . Thus BE BAutE (x)/BC 2. In fact, by Definition 2.4.3 and Propo-

sition 2.4.4, we can consider the subdivision of E , which is a Grothendieckconstruction Grsd[E ]M . Then by Theorem 2.4.10, BGrsd[E ]M , and thusBE , is homotopy equivalent to hocolimsd[E ]BM . Since sd[E ] is a poset• ← • → •, this homotopy colimit is just the homotopy pushout of theabove diagram, if one follows Definition 2.4.3 to compute B M .We can use the exact sequence for computing relative cohomology (see forinstance [34])

H∗(BAutE (x), BC 2; R)

to find the positive degree cohomology of BE , while in degree zero we just have R because E is connected. In fact H∗(BE , k), for k a field of characteristic 2, is a subring of the polynomial ring k [X,Y], with degX =degY = 1 and with all linear combinations of Xii>0 removed. This ring

has no nilpotent elements and is not finitely generated. In fact, this is thesmallest example in commutative algebra that a subring of a Noetherianring is not finitely generated.However when R = C, C is projective (by direct calculation or by Propo-sition 3.3.11). That means H0(BE ,C) = C is the only non-vanishingcohomology. Note that C as a RE op-module is not projective, although

H∗(BE op,C) = C as well.





4.3 Computation via adjoint functors 117

Hom(y, x) = ∅, then every RHomC(y, x) is a free RAutC(x)-module. As anexample the standard resolution evaluated at x, P C(x), becomes a projectiveresolution for the RAutC(x)-module R.

4.3.2 Kan extensions of resolutions

For an arbitrary functor u : D → C we can still construct the left and rightadjoints of Resu, i.e. the Kan extensions. In the situation that u admitsa left (or right) adjoint v, then the left (or right) Kan extension enjoys aparticularly simple form Resv, which is exact. In general, the Kan extensionscannot have simplified forms, and consequently computing Kan extensionsof an arbitrary module is difficult and the new modules do not come withexplicit descriptions. However if a module is simplicially constructed, in thesense that it comes from the nerve of a category, then we do have somesatisfactory results on their Kan extensions. This is a fundamental step to

study cohomology rings and higher limits.

Proposition 4.3.2. Suppose u : D → C is a functor between small categories.Then

LK uB D∗ = LK uC∗(IdD/−, R) ∼= C∗(u/−, R),

a complex of projective RC -modules.Particularly for pt : D → •, we have

LK ptB D∗ = LK ptC∗(IdD/•, R) ∼= C∗( pt/•, R) ∼= C∗(D, R),

Proof. For the sake of convenience, we suppress the base ring R in our nota-tions. Suppose x ∈ Ob C . Fix an integer n ≥ 0. Then

LK uCn(IdD/−) ∼= lim−→u/xCn(IdD/−).

If (y, α) ∈ Ob u/x, that is, there is an α : u(y) → x, then by definition[Cn(IdD/−)](y, α) = Cn(IdD/y), and we can define a morphism

θ(y,α) : Cn(IdD/y) → Cn(u/x)

by [(y0, α0)β1→ · · ·

βn→(yn, αn)] → [(y0, αα0)β1→ · · ·

βn→(yn, ααn)]. One can easilyverify that we have a commutative diagram for any γ : (y, α) → (z, β )

Cn(IdD/y) = [Cn(IdD/−)](y, α) γ ∗

θ(y,α)

[Cn(IdD/−)](z, β ) = Cn(IdD/z)

θ(z,β)

Cn(u/x)




so that Cn(u/x) fits into the limit defining diagram. If L is another R-module that fits into the limit defining diagram, equipped with maps θL(y,α) :

[Cn(IdD/−)](y, α) → L for all (y, α) ∈ Ob(u/x). Then we can introduce amorphism Θ : Cn(u/x) → L such that

Θ[(y0, αα0)β1→ · · ·

βn→(yn, ααn)]

:= θL(yn,ααn)[(y0, u(β n · · · β 1))β1→(y0, u(β n · · · β 2))

β2→ · · ·βn→(yn, 1u(yn))].

This is a well defined morphism. In the end, since αn : (yn, ααn) → (y, α)in a morphism in u/x, it induces a functor Cn(IdD/yn) → Cn(IdD/y). Inparticular this functor gives

[(y0, u(β n · · · β 1))β1→(y0, u(β n · · · β 2))

β2→ · · ·βn→(yn, 1u(yn))]

→ [(y0, α0)β1→ · · ·

βn→(yn, αn)],

by composing with αn. By assumptions on L we get

ΘL(yn,ααn)[(y0, u(β n · · · β 1))β1→(y0, u(β n · · · β 2))

β2→ · · ·βn→(yn, 1u(yn))]

= ΘL(y,α)[(y0, α0)β1→ · · ·

βn→(yn, αn)].

This equality implies that we can insert Θ into the following diagram andmake it commutative

Cn(IdD/y) = [Cn(IdD/−)](y, α) γ ∗

θ(y,α)

θL(y,α)

[Cn(IdD/−)](z, β ) = Cn(IdD/z)

θ(z,β)

θL(z,β)

Cn(u/x)

Θ L

Consequently Cn(u/x) ∼= lim−→u/xCn(IdD/−) for all x ∈ Ob C . Since all these

isomorphisms assemble to an isomorphism C∗(u/−) ∼= LK uC∗(IdD/−), bynaturality, we are done.

Note the second statement generalizes the obvious fact

LK ptB D∗ = LK ptRD ∼= lim−→DRD ∼= R ⊗RD RD ∼= R = R Ob D = C0(D, R).

Remark 4.3.3. To understand the complex of projective RC -modulesC∗(u/−),we describe the structure of each Cn(u/−). As we observed in the proof,each base element (y0, α0) → · · · → (yn, αn) ∈ Cn(u/−) can be written asαn · [(y0, u(β n · · · β 1)) → · · · → (yn, 1u(yn))]. Thus similar to the special case

of bar resolution where u = IdC, we know




LK uB Dn ∼= Cn(u/−, R) ∼=

(y0,u(βn···β1))β1→···

βn→(yn,1u(yn))

RHomC(u(yn), −).

For future reference, the dual version for the right bar resolution is

LK uB Dn ∼= Cn(−\u, R) ∼=

(1u(y0),y0)β1→···

βn→(u(βn···β1),yn)

RHomC(−, u(y0)).

Recall that a category is R-acyclic if its reduced (simplicial) homologywith coefficients in R vanishes.

Corollary 4.3.4. Suppose u : D → C is a functor, M ∈ RC -mod and N ∈mod-RC . Then

1. if u/x is R-acyclic for every x ∈ Ob C ,

Ext∗RC(R, M ) ∼= Ext∗RD(R, ResuM ) and TorRC∗ (N, R) ∼= TorRD

∗ (ResuN, R)

2. if x\u is R-acyclic for every x ∈ Ob C ,

Ext∗RC(R, N ) ∼= Ext∗RD(R, ResuN ) and TorRC∗ (R, M ) ∼= TorRD

∗ (R, ResuM ).

Proof. We prove part (1). Under the assumption, LK uR ∼= R and LK uB D∗ ∼=C∗(u/−, R) → R → 0 becomes a projective resolution. Hence the isomor-phism follows from

HomRC(C∗(u/−, R), M ) ∼= HomRD(B D∗ , ResuM ).

For the second isomorphism, we also have

N ⊗RC C∗(u/−, R) ∼=

(y0,u(βn···β1))β1→···

βn→(yn,1u(yn))N ⊗RC RHomC(u(yn), −)

∼= (y0,u(βn···β1))β1→···βn→(yn,1u(yn))(ResuN )(yn)∼=

y0β1→···

βn→yn(ResuN )(yn)

∼= ResuN ⊗RD B Dn .

The above isomorphisms are Eckmann-Shapiro type results.

Example 4.3.5. Let D be the category from Example 4.2.4 (2)

x

1x

h

α β

y 1y ,

where h2 = 1x and β h = α. The inclusion i : AutD(x) → D has two overcat-egories i/x and i/y, both contractible. Thus we have




Ext∗RAutD(x)(R, M ) ∼= Ext∗RD(R, ResiM ).

This calculation can be generalized to all categories that possess a uniqueminimal object x whose automorphism group acts regularly on every non-

empty morphism set HomD(x, y).Recall from Section 2.3.1 that we introduced the subdivision S (C ) of an EI

category C and showed that their classifying spaces are homotopy equivalentand hence they have the same simplicial (co)homology.

Corollary 4.3.6. Let C be an EI category and S (C ) its subdivision. Suppose M ∈ RC -mod and M ∈ RC op-mod. Then

TorRC∗ (R, M ) ∼= TorRS (C)

∗ (R, RestM )

and Ext∗RCop(R, M ) ∼= TorRS (C)op

∗ (R, RessM )

where s : S (C ) → C is the natural functor.

Proof. In the proof of Proposition 2.3.?, we actually proved that x\ s is con-tractible for all x ∈ Ob C . This is equivalent to having sop/x contractible, byLemma 1.2.8. Hence we can apply Corollary 4.3.4.

Remark 4.3.7. Let u : D → C be a functor. We can use LK u to see how oneobtains the maps introduced in Chapter 2 u∗ : H∗(D, R) → H∗(C , R) andu∗ : H∗(C , R) → H∗(D, R). In fact we have the following diagram

D u

pt

C

pt

•

Thus by LK pt = LK ptLK u we get LK ptB D∗ = LK ptLK uB D∗ . Since LK uB D∗ ∼=C∗(u/−, R) is a complex of projective modules and meanwhile there is acanonical map LK uR → R between RC -modules, there exists a lifting, whichis a chain map unique up to chain homotopy (Comparison Theorem),

· · ·

LK uB D∗

LK uR

0

· · · B C∗ R 0

Conceptually we see the left Kan extension induces chain maps

R ⊗RD B D∗ ∼= R ⊗RC LK uB D∗ → R ⊗RC B C∗

because R ⊗RD B D∗ = LK ptB D∗ = LK ptLK uB D∗ ∼= R ⊗RC LK uB D∗ and




HomRC(B C∗ , R) → HomRC(LK uB D∗ , R) ∼= HomRD(B D∗ , R).

Hence these give exactly the same map we introduced in Chapter 2. Alter-natively by applying LK pt to the previous commutative diagram we obtain

a chain map

· · ·

LK ptLK uB D∗

LK ptLK uR

0

· · · LK ptB C∗ LK ptR 0

The positive degree part LK ptB D∗ = LK ptLK uB D∗ → LK ptB C∗ is exactly thechain map induced by u in Definition 2.2.4, i.e. C∗(D, R) → C∗(C , R), in lightof Proposition 4.3.2.

Remark 4.3.8. The following descriptions balance the understanding of leftand right Kan extensions. Especially we know the right Kan extension of a

certain injective resolution of k. Since (−)∧ establishes a one-to-one corre-spondence between right projective kC -modules and left injective kC -modules,we can extend the above lemmas to the bar resolution of k ∈mod-kD andobtain a complex of left injective kC -modules. Suppose B D∗ → k → 0 is thebar resolution of k ∈mod-kD, where B D∗ ∼= C∗(−\IdD). Then

0 → k → (B D∗ )∧ ∼= C∗(−\IdD)∧

is an injective resolution of the left kD-module k. Applying RK u we get acomplex of injective kC -modules, excluding RK uk, as follows

0 → RK uk → RK u(B D∗ )∧.

By Lemma 3.2.13 the complex RK u(B D∗ )∧ ∼

= (LK uB D∗ )∧ ∼

= C∗(−\u)∧

. Thuswe have a simplicially constructed injective resolution of k, and we havecontrol on the complex after applying the right Kan extension

0 → RK uk → (B D∗ )∧ ∼= C∗(−\u)∧.

Remark 4.3.9. At this stage, the reader shall be well prepared to go overChapter 5 where the main results in Section 5.2 are applications of Proposi-tion 4.3.2.




4.4 Grothendieck spectral sequences

Let u : D → C be a functor. We can construct Grothendieck spectral se-quences. They generalize the Lydon-Hochschil-Serre spectral sequences for

group extensions as well as the Leray-Serre spectral sequences for fibrations.

4.4.1 Grothendieck spectral sequences for a functor

Let u : D → C be a functor between two small categories. Consider thecomposites of following functors

RD-mod RK u RC -mod

lim←−C R-mod,

as well as

RD-mod LK u RC -mod

lim−→C R-mod.

The second isomorphism of the following is used in Remark 4.3.7.

Lemma 4.4.1. lim←−CRK u ∼= lim←−D and lim−→CLK u ∼= lim−→D

.

Proof. This is a special case of Corollary 1.2.12 for u : D → C and pt : C → •.

Since RK u preserves injectives, there exists a Grothendieck cohomologyspectral sequence for any functor u : D → C and M ∈ RD-mod, which comesfrom a double complex E ∗,∗

0 (M ) that we will describe shortly. The spectralsequence E ∗,∗

2 (M ) ⇒ E ∗,∗

∞ (M ) is

H∗(C ; H

∗(−\u; M )) ⇒ H∗(D; M ).

Remember that for any small category E , one has lim←−i

E ∼= Hi(E ; −) ∼=

ExtiRE (R, −). It means H∗(−\u; M ) is some sort of higher right Kan exten-

sion of M . Since LK u preserves projectives, we get a Grothendieck homologyspectral sequence as well. However we will only construct the cohomologyspectral sequence as the construction for the homology spectral sequence issimilar. For future reference, we record the Grothendieck homology spectralsequence for u : D → C and M ∈ RD-mod

H∗(C ; H∗(u/−; M )) ⇒ H∗(D; M ),

in which H∗(u/−; M ) should be considered as higher left Kan extensions of M .






lim←−CJ 0,0

lim←−CJ 1,0

lim←−CJ 2,0

· · ·

lim

←−CJ 0,1

lim

←−CJ 1,1

lim

←−CJ 2,1

· · ·

lim←−CJ 0,2

lim←−CJ 1,2

lim←−CJ 2,2

· · ·

......

... ,

and it gives rise to the Grothendieck spectral sequence recorded above. Weomit the details as the construction is standard and we are more interestedin finding a pairing. Suppose we also have a double complex E ∗,∗

0 (N ) foranother RD-module N

lim←−CJ 0,0

lim←−CJ 1,0

lim←−CJ 2,0

· · ·

lim←−CJ 0,1

lim←−CJ 1,1

lim←−CJ 2,1

· · ·

lim←−CJ 0,2

lim←−CJ 1,2

lim←−CJ 2,2

· · ·

......

... ,

and furthermore a double complex E ∗,∗0 (M ⊗N ) for the RD-module M ⊗N

lim←−CJ 0,0

lim←−CJ 1,0

lim←−CJ 2,0

· · ·

lim←−CJ 0,1

lim←−CJ 1,1

lim←−CJ 2,1

· · ·

lim←−CJ 0,2

lim←−CJ 1,2

lim←−CJ 2,2

· · ·

.

.....

.

.. .

We want to establish a natural map



4.4 Grothendieck spectral sequences 125

lim←−CJ i,j ⊗ lim←−CJ s,t → lim←−CJ i+s,j+t,

which is compatible with the differentials. In fact, since there is a uniquemap, given by the universal property of lim←−,

lim←−CJ i,j ⊗ lim←−CJ s,t → lim←−CJ i,j ⊗J s,t,

we only need to construct a map lim←−CJ i,j ⊗J s,t → lim←−C

J i+s,j+t. Our defi-nition is again based on the universal property of lim←−, along with the ten-sor product of complexes of functors in Section 3.4.1. We emphasize thatlim←−C

J i,j ⊗ lim←−CJ s,t → lim←−C

J i,j ⊗J s,t respects the differentials in E ∗,∗0 due to

its construction via the universal property. This is the case when we definelim←−CJ i,j ⊗J s,t → lim←−CJ i+s,j+t and thus we will not verify the map we areabout to construct does respect differentials.

From the two injective resolutions0 → M → I ∗ and 0 → N → I ∗, we canbuild a commutative diagram

0 M ⊗N I ∗⊗I ∗

0 M ⊗N I ∗ ,

in which the upper row is an exact sequence and the lower one is the injectiveresolution used to define E ∗,∗

0 (M ⊗N ). Applying RK u we obtain a chain mapRK u(I ∗⊗I ∗) → RK uI ∗ . Especially we have for any non-negative integers iand s a map RK u(I i⊗I s) → RK uI i+s. The universal property of lim←− provides

a morphism RK uI i⊗RK uI s → RK u(I i⊗I s). Thus we have a natural map

RK uI i⊗RK uI s → RK uI i+s.

Next we repeat the above tensor construction for the two injective res-olutions 0 → RK uI i → J i,∗ and 0 → RK uI s → J s,∗. It follows from ourdiscussions that there is a commutative diagram

0 RK uI i⊗RK uI s

J i,∗⊗J s,∗

0 RK uI i+s

J i+s,∗+∗.

In particular there exists J i,j ⊗J s,t → J i+s,j+t, and consequently the desired

map lim←−CJ i,j ⊗J s,t → lim←−CJ i+s,j+t. Hence we do obtain a pairing E ∗,∗0 (M ) ⊗

E ∗,∗

0 (N ) → E ∗,∗

0 (M ⊗N ).




4.4.2 Spectral sequences of category extensions

We introduced category extensions and opposite extensions in Section 4.1.3.Here we want to show that the Grothendieck spectral sequences for a functor

have simpler forms when the functor fits into an extension sequence.

Lemma 4.4.2. Let K → E → C be an extension. Then there exists a natural functor ι : K(y) → π/y such that every undercategory associated with it is contractible. Hence

1. H∗(π/−; M ) ∼= H∗(K(−); M (−)) in RC -mod for any M ∈ RE -mod; and 2. H∗(−\πop; N ) ∼= H∗(Kop(−); N (−)) in RC op-mod for any N ∈ RE op-mod.

Proof. The category π/y has objects of the form (x, α), where x ∈ Ob E =Ob C and α ∈ HomC(x, y). From the definition of π/y, one can see that themaximal objects are (y, g), g ∈ AutC(y), which are isomorphic to each otherand have automorphism groups isomorphic to K(y). Let us take the fullsubcategory [(y, 1y)] of π/y consisting of all maximal objects. Its skeletonis isomorphic to the group K(y). Using Quillen’s Theorem A, we show theundercategories associated with ι : [(y, 1y)] → π/y are contractible.

Fix an object (x, α) ∈ Ob(π/y). The undercategory (x, α)\(π/y) has objects of the form (β, (y, g)), where β : (x, α) → (y, g) is an morphism in π/ysatisfying gπ(β ) = α. Since π(β ) = g−1α, by the definition of a categoryextension, β = g−1αh for a unique h ∈ K(x). From here we can deduce that(β, (y, g)) ∼= (β , (y, g)) for any (y, g) and β : (x, α) → (y, g), and that(β, (y, g)) ∈ (x, α)\(π/y) has a trivial automorphism group. These imply(x, α)\(π/y) is equivalent to a point, and hence is contractible.

The first isomorphism follows from Corollary 4.3.4 (2)

TorR(π/y)∗ (R, M ) ∼= TorRK(y)

∗ (R, ResπM ),

and the naturality in y .The proof of the second is the same. By Lemma 1.2.8 the overcategories

associated with ιop : K(y)op → (π/y)op are contractible. Then we applyCorollary 4.3.4 (1).

We can obtain similar statements for opposite extensions. Keep in mindthat since K is a group, K ∼= Kop.

Lemma 4.4.3. Let K → E → C be an opposite extension. Then there exists a natural functor ι : K(y) → y\π such that every undercategory associated with it is contractible. Hence

1. H∗(πop/−; M ) ∼= H∗(Kop(−); M (−)) in RC op-mod for any M ∈ RE op-

mod; and 2. H∗(−\π; N ) ∼= H∗(K(−); N (−)) in RC -mod for any N ∈ RE -mod.




Proof. Since Kop → E op → C op is an extension, we know by Lemma 4.4.2that there is a functor ι : Kop(y) → πop/y such that all undercategoriesare contractible. From Lemma 1.2.8, this is equivalent to saying that allovercategories associated to ιop : K(y) → y\π are contractible.

For brevity we write H∗(K; M ) etc, instead of H∗(K(−); M (−)) etc, forthe functors in above lemmas.

Proposition 4.4.4. Given a functor M ∈ RE -mod, there are two spectral sequences associated with an extension K → E → C as follows:

1. a homology spectral sequence

E 2ij = Hi(C ; Hj (K; M )) ⇒ Hi+j (E ; M );

and 2. a cohomology spectral sequence

E ij

2 =

H

i(C

op;H

j (K

op; M op)) ⇒ H

i+j (E

op; M op).

Dually we have results for opposite extensions. Recall that RE op-mod =mod-RE . If M ∈ RE -mod, it gives M op ∈ RE op-mod.

Proposition 4.4.5. Given a functor M ∈ RE -mod, there are two spectral sequences associated with an opposite extension K → E → C as follows:

1. a homology spectral sequence

E 2ij = Hi(C op; Hj (Kop; M op)) ⇒ Hi+j (E op; M op);

and 2. a cohomology spectral sequence

E ij2 = Hi(C ; Hj (K; M )) ⇒ Hi+j(E ; M ).

Spectral sequences naturally give rise to some long exact sequences, andwe record them below.

Remark 4.4.6. If K → E → C is an extension and M ∈ RE -mod, one canobtain two five term exact sequences

H2(E ; M ) → H2(C ; M ) → H0(C ; H1(K; M )) → H1(E ; M ) → H1(C ; M ) → 0,

and

0 → H1(C op; M op) → H

1(E op; M op) → H0(C op; H

1(Kop; M op)) → H2(C op; M op) → H

2(E op; M op).

When K → E → C is a group extension then these two exact sequences arethe usual five term sequence in group homology and cohomology.






Definition 4.4.8. Let K → E → C be an extension and D ⊂ C a subcategory.The subextension of D in E via K|D, named E D, is a subcategory of E whoseobject set is the same as D and whose morphism set consists of morphismsin E which are preimages of morphisms in D .

If D is a full subcategory of C then E D is a full subcategory of E . Givenan extension K → E → C , AutK(x) → AutE (x) → AutC(x) is a subextensionfor any x ∈ Ob C .

Proposition 4.4.9. Let K → E → C a sequence of functors and D a full subcategory of C with the inclusion ιD : D → C . Then

1. if E is an extension of C , K|D → E D → D is the subextension and ιE D :E D → E is the inclusion, then for any y ∈ Ob C = Ob E , the undercategory y\ιD is isomorphic to a subcategory of the undercategory y\ιE D , which is equivalent to y \ιE D ;

2. if E is an opposite extension of C , K|D → E D → D is the opposite subex-tension and ιE D : E D → E is the inclusion, then for any y ∈ Ob C = Ob E ,

the overcategory ιD/y is isomorphic to a subcategory of the overcategory ιE D/y, which is equivalent to ιE D/y.

Proof. We will only prove (2). In ιE D/y, any two objects (x, α) and (x, β )

are isomorphic if and only if π(α) = π(β ). Let ιE D/y ⊂ ιE D/y be the fullsubcategory consisting of one object from each isomorphism class of objectsdescribed above. Then ιE D/y and ιE D/y are equivalent. We prove the formeris isomorphic to ιD/y.

There is a natural bijection between objects sets of these two categories(x, α) → (x, π(α)) (π is surjective on morphisms). We show there is a bijectionbetween the morphism sets and the bijections extend to a functor which gives

an isomorphism between two categories. Any (x, α) γ →(z, β ) in Mor(ιE D/y)

gives rise to a morphism (x, π(α))π(γ )

→ (z, π(β )) in ι

D/y. On the other hand,

a morphism (x, π(α))π(γ )→ (z, π(β )) in ιD/y implies π(β )π(γ ) = π(α), which

means there exists a unique g ∈ K(x) such that βγ = αg. Thus we have

a uniquely defined morphism (x, α)g−1

→ (x,αg) γ →(z, β ) = (x, α)

γg−1

→ (z, β ) inMor(ιE D/y). Note that a different γ such that π(γ ) = π(γ ) gives the same

morphism (x, α)γg−1

→ (z, β ), so the map from Mor(ιD/y) to Mor(ιE D/y) is well-defined. It’s straightforward to check these two assignments on morphismsare mutually inverse to each other.

In order to show the bijections on objects and morphisms defining anisomorphism between categories, we need to verify they preserve compo-sition and identity. We will just prove the former and leave the proof of preserving identity to the reader. Suppose (x, α) → (z, β ) → (w, γ ) is a

composite of two morphisms in ιE D/y. Then our map naturally sends it toa composite of morphisms (x, π(α)) → (z, π(β )) → (w, π(γ )). Conversely,




if (x, π(α))π(u)→ (z, π(β ))

π(v)→ (w, π(γ )) = (x, π(α))

π(vu)→ (w, π(γ )) is the compos-

ite of two morphisms in ιD/y, then we need to show the two morphisms

(x, α)ug−1

→ (z, β )vh−1

→ (w, γ ) = (x, α)vh−1ug−1

→ (w, γ ) and (x, α)vut−1

→ (w, γ ) areequal, where g,h ,t are isomorphisms, described in the preceding paragraph.Since π(vh−1ug−1) = π(vut−1), there is a unique isomorphism s satisfy-ing vh−1ug−1 = vut−1s. But then we have α = γvut−1 = γvh−1ug−1 =γvut−1s, and this forces s = 1 because Definition 4.1.21 (4op). Hence we getvh−1ug−1 = vut−1.

The following corollary is a useful outcome of the proposition.

Corollary 4.4.10. Let K → E → C be a sequence of functors and D ⊂ C a full subcategory with the inclusion ιD : D → C . Then

1. if E is an extension of C , then y\ιD is contractible (or R-acyclic or con-nected) if and only if y\ιE D is;

2. if E is an opposite extension of C , then ιD/y is contractible (or R-acyclic or connected) if and only if ιE D/y is.

Example 4.4.11. Let K → E → C be an extension with a unique maximalobject x such that AutC(x) acts freely and transitively on HomC(y, x) for anyy ∈ Ob C . Then it’s easy to check that ι : AutC(x) → C induces a homotopyequivalence since all undercategories associated to it are contractible. Hencewe know AutE (x) → E is a homotopy equivalence as well.

Since any category can be regarded as a trivial extension of itself, thefollowing result is some sort of generalization of Corollary 4.3.4.

Corollary 4.4.12. Suppose there is an extension K → E → C . If ιD : D →C is an inclusion such that y\ιD is contractible for every y ∈ Ob C , then H ∗(E ; M ) ∼= H∗(E D; M ) for any M ∈ RE op-mod, and H∗(E ; N ) ∼= H∗(E D; N )

for any N ∈ RE -mod. Here E D is the subextension corresponding to D.

Suppose there is an opposite extension K → E → C . If ιD : D → C is an inclusion such that ιD/y is contractible for every y ∈ Ob C , then H ∗(E ; M ) ∼=H∗(E D; M ) for any M ∈ RE -mod, and H∗(E ; N ) ∼= H∗(E D; N ) for any N ∈RE op-mod. Here E D is the opposite subextension corresponding to D.

Proof. We prove the statements for cohomology. Since y\ιD is contractible forevery y ∈ Ob C , y\ιE D is contractible for every y ∈ Ob E as well by Corollary4.4.10. Then we apply Corollary 4.3.4.

When we have an opposite extension, using Corollary 4.4.10 again we getthat ιE D/y is contractible for every y ∈ Ob E . Hence Corollary 4.3.4 can beused to obtain the isomorphism.

As an example when K → E → C is an extension (resp. an oppositeextension) and C has a unique maximal (resp. minimal) object x and AutC(x)

acts regularly on HomC(y, x) (or HomC(x, y)) for any y ∈ Ob C , we haveH∗(C ; M ) ∼= H∗(AutC(x), M (x)) hence H∗(E ; M ) ∼= H∗(AutE (x), M (x)) forany contra-variant (resp. covariant) functor M .



Chapter 5

Hochschild cohomology

Abstract We consider Hochschild cohomology of category algebras here.The upshot is that we can translate Hochschild cohomology into functor co-homology, which provide a context for comparing Hochschild and ordinary

cohoomology of a small category. A very general theorem on the relationshipof these two cohomology theories will be stated and proved, combining mod-ule theoretic and simplicial methods. Some examples are computed to helpthe reader to understand.

5.1 Hochschild homology and cohomology

Let A be an associative R-algebra with identity. Then we consider A as anAe-module. We shall assume A is a finitely presented flat R-module. Thisimplies that A is a projective R-module.

5.1.1 Definition and general properties

Definition 5.1.1. The acyclic Hochschild complex of A is B An n≥−1 suchthat

B An = A ⊗R · · · ⊗R A (n+2)−copies

and such that the differential ∂ n : A⊗n → A⊗(n−1) is given by

∂ n(a0 ⊗ a1 ⊗ · · · ⊗ an) →n

i=0

(−1)ia0 ⊗ · · · ⊗ aiai+1 ⊗ · · · ⊗ an.

131



132 5 Hochschild cohomology

By direct calculations, one can see ∂ 2 = 0 and so we do have a complex of Ae-modules. Moreover since we can establish a chain contraction sn : B An−1 →B An by sn(a0 ⊗ · · · ⊗ an) = a0 ⊗ · · · ⊗ an ⊗ 1, this complex is exact.

Suppose M ∈ Ae-mod. Then we can regard it as a right Ae-module by

m · (a, a

) := a

·m · a. Thus we shall not distinguish left and right Ae

-modulesin this chapter.

Definition 5.1.2. For any M ∈ Ae-mod, we define

1. the Hochschild homology HH∗(A, M ) to be the homology of

· · · → M ⊗Ae B An → · · · → M ⊗Ae B A1 → M ⊗Ae B A0 → 0;

and2. the Hochschild cohomology HH∗(A, M ) to be the homology of

0 → HomAe(B A0 , M ) → HomAe(B A1 , M ) → · · · → HomAe(B An , M ) → · · · .

Remark 5.1.3. Since for any n ≥ 0 there is an isomorphism of Ae-modules

B An → Ae ⊗R B An ,

in which B A0 = R and B An for n ≥ 1 is

A ⊗R · · · ⊗R A n−copies

,

and furthermore because B An is a projective R-module B An is a projectiveAe-module, we actually obtain a complex of Ae-modules with A in degree −1

· · · → B An → · ·· → B A1 → B A0 → A → 0.

We call B A∗ the bar resolution of the Ae-module A. Note that

M ⊗Ae B An ∼= M ⊗R B An and HomAe(B An , M ) ∼= HomR(B An , M ).

Because of our assumption on A we can also define Hochschild homologyand cohomology as TorAe

n (M, A) and ExtnAe(A, M ), respectively. When M =

A, we usually write HHn(A) and HHn(A) for the Hochschild homology andcohomology.

When we regard the bar resolution B A∗ → A → 0 as a complex of right A-modules, it splits. Hence for any left A-module M , B An ⊗A M → A ⊗A M → 0is a projective resolution of the left A-module M . Based on this observation,we prove the following well-known results [14].

Proposition 5.1.4. Let M, N ∈ A-mod and M ∈ mod-A. Suppose R is a field. Then HomR(M, N ) and N ⊗R M are Ae-modules such that



5.1 Hochschild homology and cohomology 133

Ext∗A(M, N ) ∼= H∗(A, HomR(M, N )) ∼= Ext∗Ae(A, HomR(M, N )).

and TorA

∗ (M , N ) ∼= H∗(A, N ⊗R M ) ∼= TorAe

∗ (N ⊗R M , A).

Proof. We only prove the isomorphism for Ext. The Ae-module structure onHomR(M, N ) is given by (a1, a2)f (m) = a1f (a2m) for any (a1, a2) ∈ Ae andm ∈ M . Let B An → A be the previously introduced projective resolution of the Ae-module A. Then

HomAe(B An , HomR(M, N )) = HomR(B An , HomR(M, N ))∼= HomR(B An ⊗R M, N )∼= HomA(B An ⊗A M, N ).

Since R is a field, B An ⊗A M is projective and thus B A∗ ⊗A M becomesa projective resolution of M . Consequently the last term above computesExt∗A(M, N ).

Let B A∗ → A → 0 be the bar resolution of the Ae-module A. There is achain map D : B A∗ → B A∗ ⊗A B A∗ given by

D(a0 ⊗ a1 ⊗ · · · ⊗ an+1) =n

i=0

(a0 ⊗ · · · ai ⊗ 1) ⊗ (1 ⊗ ai+1 ⊗ · · · ⊗ an+1).

Let ζ ∈ Hn(A) and ζ ∈ Hm(A). Then the cup product ζ ∪ ζ is given by

B A∗D→B A∗ ⊗A B A∗

ζ ⊗ζ

−→A ⊗A A = A.

More explicitly ζ ∪ ζ : B An+m → A is given by

(ζ ∪ζ )(a0

⊗a1

⊗ · · · ⊗an+m+1

) = ζ (a0

⊗ · · · an

⊗1)ζ (1⊗an+1

⊗ · · · ⊗an+m+1

).

Based on previous construction, HH∗(A) forms a ring, called the Hochschild cohomology ring of A. The multiplicative structure was known to Yonedawhen R is a field.

By [67], one can use an arbitrary projective resolution P ∗ → A → 0 toconstruct the cup product, and there exists a unique chain map D : P ∗ →P ∗ ⊗A P ∗ such that (∂ 0 ⊗ ∂ 0)D = ∂ 0. In fact one just have to show thatP ∗ ⊗A P ∗ → A → 0 is also a projective resolution.

Theorem 5.1.5 (Gerstenhaber). With the above cup product the Hochschild cohomology ring

HH∗(A) =i≥0

HHi(A)

is graded commutative. Moreover the cup product coincides with the Yondea splice.




Proof. Let 0 → A → Ln−1 → · · · → L0 → A → 0 and 0 → A → M m−1 →· · · → M 0 → A → 0 be two n-fold and m-fold extensions. They uniquelydetermine two cohomology classes by liftings

· · · B An

ζ =ζ n

B An−1

ζ n−1

· · · B A0

ζ 0

A 0

0 A = Ln Ln−1

· · · L0 A 0

and

· · · B Am

ξ=ξm

B Am−1

ξm−1

· · · B A0

ξ0

A 0

0 A = M m M m−1 · · · M 0 A 0.

The Yoneda splice ζ ∗ ξ is given by any lifting of ξ = ξ m

· · · B An+m

θn

B An+m−1

θn−1

· · · B Am

θ0 ξ

B Am−1

· · · A 0

0 A Ln−1 · · · L0

M m−1 · · · A 0

A

0

0 .

To finish our proof, we will define two explicit liftings. On one hand, thecomposition of the following chain maps

θ∗ : B A∗ D−→B A∗ ⊗A B A∗ Id⊗ξ−→B A∗ ⊗A A[m] → B A∗ [m].

provides a lifting of ξ by

θi : B Ai+mD

−→(B A∗ ⊗A B A∗ )i+mId⊗ξ−→B Ai ⊗A A[m] → B Ai [m]

ξi→Li,

for each 0 ≤ i ≤ n. Here [m] denotes a shift of the chain complex. From itsdefinition we can check that ζ ∗ ξ = ζ (Id ⊗ θn) D = ζ ∪ ξ .

On the other hand we can define ξ i : B An+i → B Ai by

ξ i (a0 ⊗ · · · ⊗ an+i+1) = (−1)niξ (a0 ⊗ · · · ⊗ an ⊗ 1) ⊗ an+1 ⊗ · · · ⊗ an+i+1.

They give rise to another lifting of ξ . Then we have



5.1 Hochschild homology and cohomology 135

ζ ∗ ξ (a0 ⊗ · · · ⊗ an+m+1) = ζ θm(a0 ⊗ · · · ⊗ an+m+1)= (−1)nmζ (ξ (a0 ⊗ · · · ⊗ an ⊗ 1) ⊗ an+1 ⊗ · · · ⊗ an+m+1)= (−1)nmξ (a0 ⊗ · · · ⊗ an ⊗ 1)ζ (an+1 ⊗ · · · ⊗ an+m+1)= (−1)nm(ξ ∪ ζ )(a0 ⊗ · · · ⊗ an+m+1).

The proof is taken from [70].

Theorem 5.1.6 (Dennis,?). If A and B are Morita equivalent, then HH∗(A) ∼=HH∗(B) and there is a ring isomorphism HH∗(A) ∼= HH∗(B).

Proof. The proof to the general situation is not difficult, but one needs toknow the functors realizing the given equivalence between two module cate-gories. See [?]?

In the situation of category algebras, if D C are two categories, then theHochschild (co)homology of RD and RC are isomorphic. These isomorphismscan be seen after we express Hochschild (co)homology as certain ordinary

(co)homology.Hochschild homology and cohomology are important invariants of rings.

However they are very difficult to compute. The main result in this chapteris that when A = RC is a category algebra, we can express HH∗(RC ) byordinary cohomology, and this allows effective calculation.

5.1.2 Ring homomorphisms from the Hochschild cohomology ring

In general, if A and B are two associative k-algebras and M is a A ⊗k Bop-module, or equivalently a A-B-bimodule, we can define a ring homomorphisminduced by the tensor product − ⊗A M

φM : Ext∗Ae(Λ, Λ) → Ext∗A⊗RBop(M, M ).

Let 0 → A → Ln−1 → P n−2 → · · · → P 0 → A → 0 represent an element inExtn

Ae(A, A). We may assume P i are projective Ae-modules. Then consideredas a complex of right A-modules, it is split exact. Thus tensoring with M givesan exact sequence of A-B-modules

0 → M → Ln−1 ⊗A M → P n−2 ⊗ −AM → · · · → P 0 ⊗A M → M → 0.

This induces a ring homomorphism φM : Ext∗Ae(A, A) → Ext∗A⊗RBop(M, M ).If N is another A⊗RBop-module, we see Ext∗A⊗RBop(M, N ) has an Ext∗Ae(A, A)-

module structure via the ring homomorphisms φM and φN together with theYoneda splice. We quote the following theorem of Snashall and Solberg [ ?],which generalizes Gerstenhaber’s theorem.




Theorem 5.1.7. Let A and B be two associative R-algebras. Let η be an element in Extn

Ae(A, A) and θ an element in ExtmA⊗RBop(M, N ) for two A-

B-bimodules M and N . Then φN (η)θ = (−1)mnθφM (η).

Proof. Let us fix an element η for n ≥ 10 → A → Ln−1 → P n−2 → · · · → P 0 → A → 0

with P i projective.First we consider the case for θ ∈ HomA⊗Bop(M, N ) (m = 0). When n = 0,

Ext0Ae(A, A) = Z (A). It means each element in Ext0

Ae(A, A) is a multiplica-tion by some a ∈ Z (A). Such a map induces an element in HomA⊗Bop(M, M )by m → am. One can easily verify that (η ⊗A N )θ = θ(η ⊗A M ).

When n = 1, we can construct the following commutative diagram

0 A ⊗A M

A⊗θ

L0 ⊗A M

L0⊗θ

A ⊗A M 0

0 A ⊗A N X

A ⊗A M

A⊗θ

0

0 A ⊗A N L0 ⊗A N A ⊗A N 0

where the middle row is (η ⊗A N )θ = θ(η ⊗A M ).Next let θ ∈ Ext1

A⊗RBop(M, N ) represented by 0 → N → Y → M → 0.Since all syzygies of A are projective as right A-modules, we have the followingcommutative diagram

0 Ω i(A) ⊗A N

P i−1 ⊗A N

Ω i−1(A) ⊗A N

0

0 Ω i(A) ⊗A Y

P i−1 ⊗A Y

Ω i−1(A) ⊗A Y

0

0 Ω i(A) ⊗A M P i−1 ⊗A M Ω i−1(A) ⊗A M 0

for all i with Ω 0(A) = A. Denote the upper row by σi, the rightmost columnby θi, the leftmost column by θi+1 and the lower row by σi. Then by the3 × 3-splice of [52, Lemma 3.3], we get σiθi = −θi+1σi for all i ≥ 1. Since

η ⊗A N = (0 → A ⊗A N → L0 ⊗A N → Ω n−1(A) ⊗A N → 0)σn−1 · · · σ1

andη ⊗A M = (0 → A ⊗A M → L0 ⊗A M → Ω n−1(A) ⊗A M → 0)σn−1 · · · σ1



5.2 Hochschild (co)homology of category algebras 137

we obtain an equality (η ⊗A N )θ = (−1)nθ(η ⊗A M ) by combining all above.When m > 1, since every θ ∈ Extm

Ae(M, N ) is the Yoneda splice of mextensions, it follows directly that (η ⊗A N )θ = (−1)mnθ(η ⊗A M ). We aredone.

5.2 Hochschild (co)homology of category algebras

5.2.1 Basic ideas and examples

Let C be a small category. Recall from Section 3.1.1 that we call C e = C × C op

the enveloping category of a small category C . We also showed in Exam-ple 3.1.5 that there is a natural isomorphism kC e ∼= (kC )e. As a functor,kC (x, y) = kHomC(y, x) if HomC(y, x) = ∅ and kC (x, y) = 0 otherwise. Here(x, y) ∈ Ob C e. This result is just a simple observation. It implies the envelop-ing algebra of a category algebra of

C is the category algebra of its enveloping

category, so later on we will just use the terminology kC e when dealing withHochschild cohomology. This identification enables us to apply functor coho-mology theory to the investigation of the Hochschild cohomology theory of category algebras.

A key ingredient is F (C ), the category of factorizations in C . Recall thatthe category F (C ) has the morphisms in C as its objects. In order to avoidconfusion, we write an object in F (C ) as [α], whenever α ∈ Mor C . A mor-phism from [α] ∈ Ob F (C ) to [α] ∈ Ob F (C ) is given by a pair of u, v ∈ Mor C ,making the following diagram commutative

x

u

yα

vop

x y.

α

In other words, there is an morphism from [α] to [α] if and only if α = uαvfor some u, v ∈ Mor C , or equivalently α is a factor of α in Mor C . Thecategory F (C ) admits two natural covariant functors to C and C op

C F (C )t s C op ,

where t and s send an object [α] to its target and source, respectively. Usinghis Theorem A and its corollary, Quillen showed these two functors inducehomotopy equivalences of the classifying spaces. We will be interested in the

functor ∇ = (t, s) : F (C ) → C e = C × C op,




sending an [α] ∈ Ob F (C ) to (x, y) ∈ Ob C e if α ∈ HomC(y, x) and a morphism(u, vop) ∈ Mor F (C ) to (u, vop) ∈ Mor(C e).

The importance of the functor ∇ : F (C ) → C e lies in the fact that itstarget category gives rise to the Hochschild cohomology ring of C , while its

source category determines the ordinary cohomology ring of C F (C ). In thesituation of (finite) posets and groups, the functor is well-understood and inthe group case it has been implicitly used to establish the homomorphismfrom the Hochschild cohomology ring to the ordinary cohomology ring.

Example 5.2.1. 1. When C is a poset, ∇ : F (C ) → C e sends F (C ) isomorphi-cally onto a full category C e∆ ⊂ C e, where

Ob C e∆ = (x, y) ∈ Ob C e HomC(y, x) = ∅

(the full subcategory C e∆ is well-defined whenever C is EI). One can easilysee that RC as a functor only takes non-zero values at objects in Ob C e∆.Furthermore as a RC e∆-module, RC ∼= R is the trivial module. SinceC e

∆

∼= F (C ) is a co-ideal in the poset C e, we obtain Ext∗

RCe(RC , RC ) ∼=

Ext∗RCe∆(RC , RC ) ∼= Ext∗RF (C)(k, k) ∼= Ext∗kC(k, k), where the last isomor-

phism comes from the fact that B F (C ) BC . This isomorphism betweenthe two cohomology rings was first established in [30];

2. When C is a group, the category F (C ) is a groupoid and is equivalent to asubcategory of the one object category C e with morphism set

(g, g−1op) g ∈ Mor C ⊂ Mor C e.

Based on this description, one can prove the existence of the surjectivehomomorphism from the Hochschild cohomology ring to the ordinary co-homology ring of a group, which is basically the same as the classicalapproach. See for example [?].

5.2.2 Hochschild (co)homology as ordinary (co)homology

The two examples of last section hint that we may express Hochschild(co)homology in terms of ordinary (co)homology. Moreover they show usthe way to establish such an expression. Let us examine the following com-mutative diagram of small categories

F (C )

t

∇ C e = C × C op

p

C ,




where p is the projection onto the first component. Recall from Section 3.2.2that the preceding diagram leads to another

R

RF (C )-mod LK ∇

LK t

RC e-mod

LK p∼=−⊗RCR

RC

RC -mod .

R

In the rest of this section, we will establish and describe the following maps,induced by the three left Kan extensions LK t, LK p and LK ∇ respectively,

t∗ : Ext∗RF (C)(R, R) → Ext∗RC(R, R),

p∗ : Ext∗RCe(RC , RC ) → Ext∗RC(R, R)∇∗ : Ext∗

RF (C)(R, R) → Ext∗

RCe(RC , RC ).

Theorem 5.2.2. Let C be a small category. For any functor M ∈ RC e-mod,we have

Ext∗RCe(RC , M ) ∼= Ext∗RF (C)(R, Res∇M ).

The maps t∗ is an ring isomorphism, p∗ ∼= φC is a split surjective ring ho-momorphism and ∇∗ is a split injective ring homomorphism.

The theorem is proved in a series of lemmas.

Lemma 5.2.3. The following complex of RC e-module

LK ∇B F (C)∗ → LK ∇R → 0

is a projective resolution of the RC e-module RC .

Proof. Let B F (C)∗ → R → 0 be the bar resolution. By Proposition 4.3.2,

LK ∇B F (C)∗

∼= C∗(∇/−, R). In Proposition 2.3.13, due to Quillen, we assertedthat the category F (C ) is a cofibred category over C e. More explicitly for any(x, y) ∈ Ob C e the overcategory ∇/(x, y) is homotopy equivalent to the fibre∇−1(x, y), which is the discrete category HomC(y, x). In other words, for any(x, y) ∈ Ob C e,

C∗(∇/(x, y), R) → RHomC(y, x) → 0

is exact. ThusC∗(∇/−, R) → RC → 0

is a projective resolution of the RC e-module RC . Furthermore we have

LK ∇R ∼= RC by direct calculation

(LK ∇R)(x, y) = lim−→∇/(x,y)R ∼= lim−→HomC(y,x)

R ∼= RHomC(y, x).




For any functor M ∈ RC e-mod, the isomorphism

Ext∗RCe(RC , M ) ∼= Ext∗RF (C)(R, Res∇M )

is a direct consequence of Lemma 5.2.3.Before we study the ring homomorphisms, we give a interesting result

on Res∇RC . For an arbitrary u : D → C , since LK u is the left adjoint of Resu, there are natural transformations Id → ResuLK u and LK uResu →Id. We pay attention to the case of ∇ : F (C ) → C e. There exists anRF (C )-homomorphism R → Res∇LK ∇(R) = Res∇(RC ) as well as a RC e-homomorphism RC = LK ∇Res∇(R) → R. The latter gives rise to a RF (C )-homomorphism

Res∇(RC ) = Res∇LK ∇Res∇(R) → R = Res∇R.

In case that C is a poset, one has R = Res∇(RC ). When C is a group, F (C )

is a groupoid, equivalent to the automorphism group of [1C ] ∈ Ob F (C ),that is, (g, g−1op

) g ∈ Mor C. If we name the full subcategory of F (C ),

consisting of one object [1C], by ∆C and the inclusion (an equivalence) byi : ∆C → F (C ). Then Res∇i(RC ) = Res∇(RC )([1C]) is a R ∆C -module withthe action (g, g−1op

) · a = gag−1, a ∈ Res∇i(RC ). Thus Res∇i(RC ) =

Rcg,where cg is the conjugacy class of g ∈ Mor C . In particular R = Rc1C isa direct summand of Res∇i(RC ) and it implies R

Res∇(RC ) as RF (C )-modules because i is an equivalence of categories.

Lemma 5.2.4. Let C be a small category. Then R Res∇(RC ) as RF (C )-

modules.

Proof. One needs to keep in mind that the restriction of a module usually hasa larger R-rank than the module itself since ∇ is not injective on objects. Wedefine a RF (C )-homomorphism (a natural transformation) ι : R → Res∇(RC )by the assignments ι[α](1R) = α ∈ Res∇(RC )([α]) for each [α] ∈ Ob F (C ).If [β ] is another object in Ob F (C ) and (u, vop) ∈ HomF (C)([α], [β ]) is anarbitrary morphism, then by the definition of an F (C )-morphism, (u, vop)·α =uαv = β . Hence ι maps R isomorphically onto a submodule of Res∇(RC ). Onthe other hand, we may define a RF (C )-homomorphism : Res∇(RC ) → Rsuch that, for any [α] ∈ Ob F (C ), [α] : Res∇(RC )([α]) → R([α]) = R sendseach base element in Res∇(RC )([α]) = RHomC(y, x) to 1R. One can readilycheck the composite of these two maps is the identity

R ι→Res∇(RC )

→R,

and this means R Res∇(RC ) or Res∇(RC ) = R ⊕ N C for some RF (C )-module N C.

The module N C as a functor can be described by




N C([α]) = Rβ − γ β, γ ∈ HomC(y, x),

if [α] ∈ Ob F (C ) and α ∈ HomC(y, x). It will be useful to our computationsince it determines the “difference” between the Hochschild and ordinary

cohomology rings of a category. The next lemma finishes off our proof of themain theorem.Using adjunction between LK ∇ and Res∇, along with Lemmas 5.2.3 and

5.2.4, we get a commutative diagram

HomRF (C)(B F (C)∗ , R)

LK ∇ HomRCe(LK ∇B F (C)∗ , LK ∇R)

HomRF (C)(B F (C)∗ , Res∇RC )

splitting

∼= HomRCe(LK ∇B F (C)

∗ , RC )

The top row gives rise to ∇∗. From this diagram one can see it can alsoobtained as

Ext∗RF (C)(R, R) → Ext∗RF (C)(R, Res∇RC ) ∼= Ext∗RCe(RC , RC ).

Now we turn to establish the ring homomorphisms.

Lemma 5.2.5. The map t∗ : Ext∗RF (C)(R, R) → Ext∗RC(R, R) is a ring iso-morphism.

Proof. As we explained in Remark 4.3.7, t∗ is induced by LK t, and is thesame as the map between simplicial/singular cohomology, induced by thefunctor t in Chapter 2. Since F (C ) C , it is an isomorphism.

Lemma 5.2.6. The map p∗ : Ext∗RCe(RC , RC ) → Ext∗RC(R, R) is equal toφC, induced by − ⊗RC R.

Proof. Suppose M ∈ RC e-mod. We show LK pM ∼= M ⊗RCR as RC -modules.Let x ∈ Ob C . Then

1x · LK pM = (LK pM )(x) = lim−→ p/xM.

Because p/x ∼= (IdC/x) × C op (x, 1x) × C op, we have

lim−→ p/xM ∼= lim−→IdC/x×Cop

M ∼= lim−→(x,1x)×Cop

M ∼= lim−→Cop

M (x, −)∼= lim−→Cop

1x · M ∼= R ⊗RCop (1x · M )∼= 1x · M ⊗RC R.

In particular it implies LK p(RC e) ∼= RC e ⊗RC R ∼= RC ⊗R R. Thus if P ∗ →RC → 0 is the projective resolution in Section 5.1.1, it splits when regarded




as a complex of right RC -modules. Consequently LK p ∼= − ⊗RC R maps anyprojective resolution of RC to an exact sequence of left RC -modules

LK p P ∗ → LK p(RC ) ∼= R → 0,

which is a projective resolution. Hence LK p induces a chain map

HomRCe( P ∗, RC ) → HomRC(LK p P ∗, R),

and the ring homomorphism

p∗ = φC : Ext∗RCe(RC , RC ) → Ext∗RC(R, R).

Lemma 5.2.7. Let C be a small category. Then ∇∗ is a ring homomorphism and there is another ring homomorphism ∗

Ext

∗

RCe(RC , RC )

Ext

∗

RF (C)(R, R),

such that ∗∇∗ ∼= 1 and p∗ ∼= t∗∗. It means ∇∗ and ∇∗(t∗)−1 are injective ring homomorphisms while p∗ and ∗ = (t∗)−1 p∗ are surjective ring homo-morphisms.

Proof. We prove the map ∇∗ is a ring homomorphism. Then from p∗∇∗ = t∗

we get [(t∗)−1 p∗]∇∗ = 1Ext∗RF (C)

(R,R) and thus we can define ∗ = (t∗)−1 p∗

which is a surjective ring homomorphism.

Take the bar resolution B F (C)∗ → R → 0. We have seen that LK ∇B F (C)

∗ →LK ∇R ∼= RC → 0 is a projective resolution of the RC e-module RC . Let f, gbe two cocycles representing two cohomology classes. Then we construct thefollowing diagram

B F (C)∗

DF (C) B F (C)

∗ ⊗B F (C)∗

⇓LK ∇

f ⊗g R⊗R∼= R

LK ∇B F (C)∗

LK ∇DF (C)

LK ∇(B F (C)∗ ⊗B F (C)

∗ ) LK ∇(f ⊗g) LK ∇(R⊗R)

∼= RC

LK ∇B F (C)∗

DCe LK ∇B F (C)

∗ ⊗RC LK ∇B F (C)∗

LK ∇(f )⊗LK ∇(g)

Θ∇

LK ∇(R) ⊗RC LK ∇(R)

∼=Θ0

∼= RC .

The first row represents f ∪ g, the cup product of f and g. The left Kanextension LK ∇ maps it to the second row of RC e-modules which representsthe image of the cup product, and we want to show it gives rise to the cupproduct of LK ∇(f ) and LK ∇(g) as Hochschild cohomology classes. Since

we have LK ∇(DF (C)

) : LK ∇B F (C)

∗ → LK ∇(B F (C)

∗ ⊗B F (C)

∗ ), and LK ∇(B F (C)

∗ )and LK ∇(B

F (C)∗ ) ⊗RC LK ∇(B

F (C)∗ ) are chain homotopy equivalent as both of

them are projective resolutions of RC , we can construct chain maps DCe and




Θ∇, unique up to chain homotopy, such that the above diagram is commuta-tive. Because the Hochschild diagonal approximation map always exists andis unique up to chain homotopy, independent of the choice of a projectiveresolution of RC [67], DCe will serve as the Hochschild diagonal approxi-

mation map. Then since the lower two rows form a commutative diagram,we know they represent the same cohomology class, i.e. the cup productLK ∇(f ) ∪ LK ∇(g), in Ext∗RCe(RC , RC ).

The surjective ring homomorphism

∗ = (t∗)−1 p∗ : Ext∗RCe(RC , RC ) → Ext∗RF (C)(R, R)

is the composite of

Ext∗RCe(RC , RC )∼=→Ext∗RF (C)(R, Res∇RC )Ext∗RF (C)(R, R).

Remark 5.2.8. Slightly modifying the previous argument, we can also demon-strate the action of Ext∗RCe(RC , RC ) on Ext∗RCe(RC , M ) alternatively via ⊗on RF (C )-mod. For any M ∈ RC e-mod, one gets

Ext∗RCe(RC , M ) ∼= Ext∗RF (C)(R, Res∇M ).

It was also shown that the RF (C )-module Res∇RC natural splits as R ⊕ N Cfor some N C ∈ RF (C )-mod. This provides a surjective homomorphism ρ :Resτ RC ⊗Res∇M → Res∇M , and hence a map

ρ∗ : Ext∗RF (C)(R, Res∇RC ⊗Res∇M ) → Ext∗RF (C)(R, Res∇M ).

The latter fits into the following commutative diagram

Ext∗RF (C)(R,Res∇RC)⊗ Ext∗RF (C)(R,Res∇M )

∪

Ext∗RCe(RC, RC)⊗ Ext∗RCe(RC,M )

∪

Ext∗RF (C)(R,Res∇RC⊗Res∇M )

ρ∗

Ext∗RF (C)

(R,Res∇M ) Ext∗RCe(RC,M ),

which reduces to [67, Proposition 3.1] when C = G is a group. The top row isthe so-called cup product with respect to the pairing ρ. Since Ext∗RF (C)(R, R)is a direct summand of Ext∗RF (C)(R, Res∇RC ), it also exhibits the action of Ext∗RC(R, R) on Ext∗RCe(RC , M ), via its identification with Ext∗RF (C)(R, R).

Note that when C is an abelian group, we obtain an isomorphism [36, ?]

Ext∗RCe(RC , RC ) ∼= Ext∗RF (C)(R, Res∇(RC )) ∼= RC ⊗R Ext∗RC(R, R).

Finally we comment on Hochschild homology. It is a direct consequenceof Lemma 5.2.3. Its proof is entirely analogues to that of Corollary 4.3.4 (1).




We do not have a counterpart for Corollary 4.3.4 (2) because we do not knowthe structure of (y, x)\∇ (? for posets ?).

Theorem 5.2.9. Suppose N ∈ RC e-mod. Then we have

TorRCe∗ (N, RC ) ∼= TorRF (C)

∗ (Res∇N, R).

Particularly

TorRCe

∗ (RC , RC ) ∼= TorRF (C)∗ (Res∇RC , R) ∼= TorRF (C)

∗ (R, R)⊕TorRF (C)∗ ( N C, R).

Thus TorRC∗ (R, R) is isomorphic to a direct summand of TorRCe

∗ (RC , RC ).

Remark 5.2.10. Suppose u : D → C is an equivalence. By Proposition 2.3.9we naturally obtain another equivalence F (u) : F (D) → F (C ). Because wehave a commutative diagram

F (D) F (u)

∇

F (C )

∇

De

ue C e

we can deduce that the Hochschild (co)homology of RD and RC are isomor-phic. For instance, given any M ∈ RC e-mod, we have

TorRCe

∗ (RC , RC ) ∼= TorRF (C)∗ (Res∇RC , R)

∼= TorRF (D)∗ (ResF (u)Res∇RC , R)

∼= TorRF (D)∗ (Res∇ResueRC , R)

∼= TorRDe

∗ (ResueRC , RD).

Finally using Hochschild (co)homology, we can establish the following use-ful isomorphisms between ordinary (co)homology. Part (1) extends Proposi-tion 3.3.10.

Theorem 5.2.11. Suppose k is a field and C is a small category. Let M, N ∈kC -mod and M ∈ mod-kC . Then we have

1. Ext∗kC(M, N ) ∼= Ext∗kF (C)(k, Res∇Homk(M, N )); and

2. TorkC∗ (M , N ) ∼= TorkF (C)

∗ (Res∇(N ⊗k M ), k).

Proof. By Proposition 5.1.4, Ext∗kC(M, N ) ∼= Ext∗kCe(kC , Homk(M, N )). FromTheorem 5.2.2, Ext∗kCe(kC , Homk(M, N )) ∼= Ext∗kF (C)(k, Res∇Homk(M, N )).

Similarly by Proposition 5.1.4 and Theorem 5.2.9, we get the isomorphismfor homology. Note that it is natural to give Res

∇(N ⊗ M ) a right RF (C )-

module structure.




The above isomorphisms generalize the well known isomorphisms in groupcohomology Ext∗kG(M, N ) ∼= Ext∗kG(k, Homk(M, N )) and TorkG

∗ (M, N ) ∼=TorkG

∗ (N ⊗k M, k).Proposition 3.3.10 does not generalize to cohomology. Let B C∗ → k → 0 be

the bar resolution. Then the lefting B F (C)∗ → RestB C∗ induces

HomkC(B C∗ , RK tRes∇Homk(M, N )) ∼= HomkF (C)(RestB C∗ , Res∇Homk(M, N ))

→ HomkF (C)(B F (C)∗ , Res∇HomR(M, N )),

and thus

Ext∗kC(k,RK tRes∇Homk(M, N )) → Ext∗kF (C)(k, Res∇HomR(M, N ))∼= Ext∗kC(M, N ).

These maps have no reason to be isomorphisms in general. Similarly we havea map

TorkC∗ (M , N ) ∼= TorkF (C)

∗ (Res∇(N ⊗k M ), k)

→ Tor

kC

∗ (LK tRes∇(N ⊗k M

), k),by Corollary 4.3.4 (1) and Lemma 4.4.1.

5.2.3 EI categories

In this section, we assume our categories are EI. The purpose is to compareHochschild (co)homology of C with that of an automorphism group of anobject.

Suppose A is the full subcategory of C which consists of all ob jects and allisomorphisms in C . The category A is a disjoint union of finitely many finitegroups. Its category algebra RA = x∈ObC RAutC(x) is an RC e-module, a

direct sum of atomic modules supported on minimal objects in C RC. Thereis a surjective RC e-morphism π : RC → RA, with kernel written as ker π.Considered as a functor ker π ⊂ RC takes non-zero values only at (x, y) forwhich there exists a C -morphism from y to x and x ∼= y. Since there is aninclusion functor i : A → C , we have maps between ordinary (co)homologyi∗ : H∗(C , R) → H∗(A, R) and i∗ : H∗(A, R) → H∗(C , R). Here we show thereare maps between their Hochschild (co)homology as well.

The short exact sequence of RC e-modules

0 → ker π → RC π→RA → 0

induces a long exact sequence

· · · → ExtnRCe(RC , ker π) → ExtnRCe(RC , RC ) π→ExtnRCe(RC , RA) η→ · · ·




By Proposition 4.2.1, one can see Ext∗RCe(RC , RA) is naturally isomorphic to

Ext∗RAe(RA, RA),

which is isomorphic to the direct product of the Hochschild cohomology ringsof the automorphism groups of objects in C :

[x]⊂Ob C

Ext∗RAutC(x)e(RAutC(x), RAutC(x)),

where [x] stands for the isomorphism class of x ∈ Ob C . The following mapwill still be written as π

π : Ext∗RCe(RC , RC ) → Ext∗RAe(RA, RA).

We show π can be identified with the algebra homomorphism induced by−⊗RCRA, where as a left RC -module RA is a direct sum of atomic modules.

φA : Ext∗RCe(RC , RC ) → Ext∗RCe(RA, RA) ∼= Ext∗RAe(RA, RA).

Hence we do not need to distinguish the maps φA and π.

Lemma 5.2.12. The following diagram is commutative

Ext∗RCe(RC , RC ) π

φA

Ext∗RCe(RC , RA)

∼=

Ext∗RCe(RA, RA) ∼=

Ext∗RAe(RA, RA).

Proof. This can be seen on the cochain level. Suppose R∗ → RC → 0 is theminimal projective resolution of the RC e-module RC . Then Ext∗

RCe(RC , RC ) is

the cohomology of the cochain complex HomRCe(R∗, RC ). The tensor product− ⊗RC RA induces a map

HomRCe(R∗, RC ) → HomRC−RA(R∗ ⊗RC RA, RA).

Since B RC∗ ⊗RC RA is a projective resolution of the RC -RA-module RA,R∗ ⊗RC RA is also a projective resolution of RA. Moreover because R∗ isminimal, it is supported on C RC. It implies R∗ ⊗RC RA is also supported onC RC. But the RC e-module RA is supported on minimal objects of C RC, wehave

HomRC−RA(R∗ ⊗RC RA, RA) ∼= HomRAe(Resi(R∗ ⊗RC RA), RA)∼= HomRAe(ResiR∗, RA),

which gives rise to φA. Here i : Ae → C e is the inclusion, induced by anotherinclusion, also written as i : A → C . On the other hand π is exactly given by




the same chain map

HomRCe(R∗, RC ) → HomRCe(R∗, RA) ∼= HomRAe(ResiR∗, RA).

Thus we are done.

We have the following commutative diagram, involving four cohomologyrings.

Theorem 5.2.13. Let C be an EI-category. Then we have the following com-mutative diagram

Ext∗RCe(RC , RC ) φA=π

φC

Ext∗RAe(RA, RA)

φA

Ext∗RC(R, R)

Resi

Ext∗RA(R, R).

Proof. We prove it on the cochain level. Let R∗ → RC → 0 be the minimalprojective resolution of the RC e-module RC . Then we have the followingcommutative diagram

HomRCe(R∗, RC ) −⊗RCRA

−⊗RCR

HomRC−RA(R∗ ⊗RC RA, RA)

−⊗RAR

HomRC(R∗ ⊗RC R, R)

HomRC(R∗ ⊗RC R, R)

HomRC(R

∗, R)Resi

HomRA(ResiR∗, R)

HomRA(R

∗ , R),

in which R∗ → R → 0 and R

∗ → R → 0 are the projective resolutions of thetrivial RC - and RA-modules satisfying the following commutative diagramsof RC -modules and RA-modules, respectively,

R∗

R 0 R∗

R 0

R∗ ⊗RC R R 0 and ResiR∗ R 0.

In the main diagram, upper left cochain complex computes Ext∗RCe(RC , RC ),upper right corner computes Ext∗RAe(RA, RA) by Lemma 5.2.12, lower left




corner computes Ext∗RC(R, R) and lower right corner computes Ext∗RA(R, R).Hence our statement follows.

We note that in the theorem the category A may be replaced by any

full subcategory of it. Especially, we have a commutative diagram for eachAutC(x) ⊂ A

Ext∗RCe(RC , RC )φRAutC(x)

φC

Ext∗RAutC(x)e(RAutC(x), RAutC(x))

φAutC(x)

Ext∗RC(R, R)

Resi

Ext∗RAutC(x)(R, R).

5.3 Examples of the Hochschild cohomology rings of

categories

In this section we calculate the Hochschild cohomology rings for four finiteEI-categories, with base field k of characteristic 2.

5.3.1 The category E 0

In [87] we presented an example, by Aurelien Djament, Laurent Piriou andthe author, of the mod-2 ordinary cohomology ring of the following categoryE 0

x

1x

g

h

gh

α β

y 1y ,

where g 2 = h2 = 1x, gh = hg,αh = βg = α, and αg = βh = β . The ordinarycohomology ring Ext∗kE 0(k, k) is a subring of the polynomial ring H∗(Z 2 ×Z 2, k) ∼= k[u, v], removing all un, n ≥ 1, and their scalar multiples. It has nonilpotents and is not finitely generated. By Theorem 2.3.4, it implies that theHochschild cohomology ring Ext∗kE e0 (kE 0, kE 0) is not finitely generated either.We compute its Hochschild cohomology ring using Proposition 2.3.5.

The category of factorizations in E 0, F (E 0), has the following shape



5.3 Examples of the Hochschild cohomology rings of categories 149

[α] [β ]

[1x]

[1y]

[h]

[gh]

[g]

,

in which [1x] ∼= [h] ∼= [g] ∼= [gh] and [α] ∼= [β ]. For the purpose of computation,we use the skeleton F (E 0) of F (E 0) (which is equivalent to F (E 0) hence thetwo category algebras and their module categories are Morita equivalent)

[α]

(1y,1opx )

[1x]

(α,1opx ),(α,hop),(β,gop),(β,(gh)op)

(1x,1opx ),(h,hop),(g,gop),(gh,(gh)op)

[1y].

(1y,αop)

(1y,1opy )

In the above category, next to each arrow is the set of homomorphisms inF (E 0) from one object to another. The module N E 0 ∈ kF (E 0)-mod (seeProposition 2.3.5) takes the following values

N C([1x]) = k1x + h, g + gh, 1x + g , N C([h]) = k1x + h, g + gh, 1x + g,N C([g]) = k1x + h, g + gh, 1x + g , N C([gh]) = k1x + h, g + gh, 1x + g,N C([α]) = kα + β , N C([β ]) = kα + β ,N C([1y]) = 0.

Thus N E 0 = S [1x],k(1x+h) ⊕ S [1x],k(g+gh) ⊕ k 1x+g, where S [1x],k(1x+h) andS [1x],k(g+gh) are simple kF (E 0)-modules such that S [1x],k(1x+h)([1x]) = k(1x+h) and S [1x],k(g+gh)([1x]) = k(g + gh), and k1x+g is a kF (E 0)-module such

that k1x+g([1x]) = k(1x + g), k1x+g([α]) = k(α + β ) and k1x+g([1y]) = 0.Note that S [1x],k(1x+h)([1x]) = k(1x + h), S [1x],k(g+gh)([1x]) = k(g + gh) andk1x+g([1x]) = k(1x + g) are all isomorphic to the trivial kAutF (E 0)([1x])-module, and have the same trivial ring structure in the sense that the prod-uct of any two elements is zero. Hence we have (along with the result quoted

in Section 2.4, paragraph two)

Ext∗kF (E 0)(k, S [1x],k(1x+h)) ∼= k(1x + h) ⊗k Ext∗kAutF (E0)([1x])(k, k)






x

1x

h

α y 1y ,

where h2 = 1x and αh = α. The contractibility implies the ordinary coho-mology ring is simply the base field k. In this case F (E 1) is the followingcategory

[α]

(1x,1opy )

(h,1opy )

[1x]

(1x,1opx )

(h,hop)

(α,AutE1(x)op)

(h,1opx )

[1y]

(1y,αop)

(1y,1opy )

[h]

(1x,hop)

(h,1opx )

(α,AutE1(x)op)

(1x,hop)

We calculate its Hochschild cohomology ring. By proposition 2.3.5, weonly need to compute Ext∗kF (E 1)(k, N E 1), where N E 1 has the following valueat objects of F (E 1)

N E 1([1x]) = k1x + h , N E 1([h]) = k1x + h,N E 1([1y]) = 0 , N E 1([α]) = 0.

One can easily see that N E 1 = S [1x],k(1x+h) is a simple module of dimensionone with a specified value k(1x + h) at [1x]. Since [1x] ∼= [h] ∈ Ob F (E 1) are

minimal objects, using quoted result in Section 2.4 paragraph two, we get

Ext∗kF (E 1)(k, N E 1) ∼= Ext∗kAutF (E1)([1x])(k, k(1x+h)) ∼= k(1x+h)⊗kExt∗kZ 2(k, k),

which is isomorphic to k (1x + h) ⊗k k[u]. Here k [u] is a polynomial algebrawith an indeterminant u at degree one and k(1x + h) is at degree zero. Thus

Ext∗kE e1 (kE 1, kE 1) ∼= Ext∗kE 1(k, k)⊕Ext∗kF (E 1)(k, N E 1) ∼= k ⊕k(1x +h)⊗k k[u].

The kernel of φE 1 consists of all nilpotents in the Hochschild cohomology ring.





The following category has its classifying space homotopy equivalent to the join, BZ 2 ∗ BZ 2 = Σ (BZ 2 ∧ BZ 2) = Σ [B(Z 2 × Z 2)/(BZ 2 ∨ BZ 2)], of the

classifying spaces of the two automorphism groups:

x

1x

h

α y

1y

g

,

where h2 = 1x, αh = α = gα and g2 = 1y. As direct consequences, itsordinary cohomology groups are equal to k, 0, 0 at degrees zero, one and two,and kn−2 at each degree n ≥ 3, and furthermore the cup product in this ringis trivial [87]. We compute its Hochschild cohomology ring. The categoryF (E 2) is as follows

[α]

(AutE2(x),AutE2(y)op)

[1x]

(α,AutE2(x)op)

(1x,1opx )

(h,hop)

[g]

(AutE2(y),αop)

(1y,gop)

(g,1opy )

[h]

(α,AutE2 (x)op)

(1x,hop)

(h,1opx )

[1y]

(AutE2 (y),αop)

(1y,1opy )

(g,gop)

By Proposition 2.3.5, we need to compute Ext∗kE 2(k, N E 2). In this case we

have N E 2([1x]) = k1x + h , N E 2([h]) = k1x + h,N E 2([1y]) = k1y + g , N E 2([g]) = k1y + g,N E 2([α]) = 0.

It means N E 2 = S [1x],k(1x+h) ⊕ S [1y],k(1y+g) and thus by Proposition 2.2.5

Ext∗kE 2(k, N E 2) ∼= Ext∗kAutF (E2)([1x])(k, k(1x + h)) ⊕ Ext∗kAutF (E2)([1y ])(k, k(1y + g))∼= k(1x + h) ⊗k Ext∗kZ 2(k, k) ⊕ k(1y + g) ⊗k Ext∗kZ 2(k, k).

Hence

Ext∗kE e2 (kE 2, kE 2) ∼= Ext∗kE 2(k, k) ⊕ k(1x + h) ⊗k k[u] ⊕ k(1y + g) ⊗k k[v],

where k [u] and k[v] are two polynomial algebras with indeterminants in de-gree one. Both the Hochschild and ordinary cohomology rings modulo nilpo-tents are isomorphic to the base field k .





The following category has a classifying space homotopy equivalent to thatof AutE 3(x) ∼= Z 2 (by Quillen’s Theorem A [?], or see [86])

x

1x

h

α β

y 1y ,

where h2 = 1x and αh = β . We compute its Hochschild cohomology ring.The category F (E 3) is as follows (not all morphisms are presented since onlyits skeleton is needed)

[α](1y,hop)

(1y,1opx )

[β ]

(1y,1opx )

[1x]

(α,1opx ),(β,h

op

)

(1x,hop)

(h,1opx )

[1y]

(1y,βop

)

(1y,1opy )

[h]

(1x,hop)

(h,1opx )

.

The module N E 3 takes the following values

N E 2([1x]) = k1x + h , N E 2([h]) = k1x + h,N E 2([1y]) = 0 , N E 2([α]) = kα + β ,

N E 2([α]) = kα + β .

Thus N E 3 fits into the following short exact sequence of k F (E 3)-modules

0 → N E 3 → k → S [1y ],k → 0.

Just like in our first example, using the long exact sequence coming from it, weknow Ext0

kE 2(k, N E 3) = 0 and Ext∗>0kE 2 (k, N E 3) ∼= k(1x +h)⊗k Ext∗>0

kF (E 3)(k, k) ∼=

k(1x + h) ⊗k Ext∗>0kE 3 (k, k). Hence

Ext∗kE e3 (kE 3, kE 3) ∼= Ext∗kE 3(k, k) ⊕ k(1x + h) ⊗k Ext∗>0kE 3 (k, k).

The kernel of φE 3 contains all nilpotents in the Hochschild cohomology ring.





Chapter 6

Connections with group representationsand cohomology

Abstract We study various local categories of finite groups. The purpose isto compare representations and cohomology of groups and of these categories.The transporter categories play a key role to bridge up these two concepts. In

fact a group and all its subgroups are some sort of transporter categories. Wepay attention to the transporter categories and try to demonstrate their sim-ilarities with and differences from groups, in terms of homological properties.We shall apply tools developed in the preceding two chapters to investigatetransporter categories, a special kind of finite EI categories which partiallymotivate modern research on category representations and cohomology. Inthis chapter all categories are finite and all modules are finitely generated.

6.1 Local categories

Let G be a finite group and p a positive prime that divides |G|, the order of G. We study various collections of subgroups of G and resulting categories.Finite categories naturally appear in both group representation and homo-topy theory. For a finite group G and a positive prime p that divides theorder of G, various posets of p-subgroups have been investigated intensivelyto establish connections between representations of G and those of its localsubgroups [61, 7, 76, 80, 81, 69]. Over the past two decades, mathematiciansrealized that categories build upon those previously mentioned posets shouldplay an important role in modular representation theory [60]. Indeed, to anysuch poset P , it naturally comes with a G-action. There is a Grothendieckconstruction on P which results in a category G ∝ P , called a transportercategory, containing P as a subcategory. A transporter category admits someinteresting quotient categories, namely Brauer categories, Puig categories and

orbit categories [75]. For many good reasons, we shall call any quotient cat-egory of a transporter category a local category, as it reveals some p-localinformation about G. Representations and cohomology of these local cate-

155



156 6 Connections with group representations and cohomology

gories are currently in the center of group representation theory. In homotopytheory of classifying spaces, the geometric realization of the nerve of a trans-porter category is a Borel construction, E G ×G BP . It was shown by severalauthors, especially Dwyer [17], that transporter categories play a key role

in homology decompositions of classifying spaces. Recently, Broto, Levi andOliver [10, 11] developed a theory of p-local finite groups, which is motivatedby the use of local categories associated with BG (or its p-completion). Intheir terminology, a p-local finite group is a triple (S, F , L) such that S isa finite p-group, F is a finite category called a fusion system on S , and Lis an extension of a full subcategory of F . The ( p-completion of) classifyingspace of L behaves like the classifying space of a finite group. In this theory,cohomology of small categories is an essential ingredients. See the new book[2] for fusion systems in group theory, homotopy theory and representationtheory.

6.1.1 G-categories

Definition 6.1.1. Consider G as a category with one object •. A G-category is a functor F : G → Cat .

In other words, a G-category is a category C , equipped with a group ho-momorphism G = AutG(•) → AutCat(C ). The simplest example is a pointwith trivial action by G. Recall that a set is regarded as a poset with trivialrelations, and a poset is regarded as a category.

Example 6.1.2. Suppose H ⊂ G is a subgroup. Consider the discrete set of left cosets G/H = gH

g ∈ G. Then G acts on it by left multiplication,permuting these cosets.

Example 6.1.3. Suppose H ⊂ G is a subgroup. Then the discrete set of conjugacy class GH = gH

g ∈ G is also a G-category with G acting byconjugations.

The above two examples are the examples of G-sets.

Definition 6.1.4. A collection of subgroups of G is a set of subgroups of G,closed under conjugations by elements in G.

Example 6.1.5. A collection of subgroups of G is naturally a poset with in-clusions as relations. Hence any collection of subgroups is a G-poset.

There are various interesting collections of subgroups of G.

Example 6.1.6. 1. The collection of all p-subgroups of G, denoted by S e p .



6.1 Local categories 157

2. The collection of all non-identity p-subgroups of G, denoted by S p. Belowis a concrete example of the poset of all non-trivial 2-subgroups of Σ 4,S 2(Σ 4):

· · ·

· · ·

D8 D8 D8

C 2 × C 2

C 4

V

C 2

C 2

C 2

C 2

C 2

Here D8 is the dihedral group of order 8, V is a Kleine four group andthose C ? are cyclic groups with order specified in the subscripts. Due tothe size, we do not record the full poset. In deed we only write down the

subgroups of one of the three D8. However, in order to obtain the fullposet, one just needs to copy the same thing under the leftmost D8 andput it for each omitted part, where the dots appear. Note that V is asubgroup of all three D8.

3. The collection of all non-identity elementary p-subgroups of G, denotedby A p.

4. The collection of all p-radical subgroups of G, denoted by B p.

Moreover, there are various refinements of some of the above posets.

Example 6.1.7. 1. Let b be a p-block of k G. The the b-Brauer pairs S b forma G-poset. When b is the principal block, it is isomorphic to S e p .

2. Let A be an interior G-algebra. Then the pointed subgroups of G form a

G-poset.

6.1.2 Homology representations of kG

Suppose C is a small G-category. Its nerve N ∗C is a simplicial set, fromwhich one can construct a chain complex C∗(C , k) such that, for each i > 0,Ci(C , k) is a k-vector space with a basis the set of i-chains of morphisms inC , while C0(C , k) = k Ob C . Then one can see that G acts on each Ci(C , k)and Ci(C , k) by permutating its base elements. Consequently we obtains G-modules Hi(C , k) and Hi(C , k).

Definition 6.1.8. Let C be a small category. Assume the normalized complexC†∗(C , k) is finite. Then the Euler characteristic χ(C ) = χ(C , k) is defined to

be

i≥0(−1)i dimk C†i (C , k).




Note that C†∗(C , k) being finite is equivalent to BC being a finite CW

complex. Since

i≥0(−1)i dimk C†i (C , k) =

i≥0(−1)i dimk Hi(C , k), χ(C , k)

is invariant under homotopy equivalence.

Example 6.1.9. 1. In Example 6.1.2, C∗(G/H ) is a stalk complex C∗(G/H ) =C0(G/H ) ∼= k ↑GH with χ(G/H ) = [G : H ].

2. In Example 6.1.3, C∗(GH ) is a stalk complex C∗(GH ) = C0(GH ) ∼=k ↑G

N G(H ) with χ(GH ) = [G : N G(H )].

3. In Example 6.1.4 (1), S e p is not a stalk complex. However since S e p has aninitial object and thus is contractible, χ(S e p ) = 1.

4. The inclusions A p ⊂ S p and B p ⊂ S p induce homotopy equivalences, byusing Quillen’s Theorem A. Hence χ(S p, k) = χ(A p, k) = χ(B p, k).

6.1.3 Transporter categories as Grothendieck constructions

Various G-posets of subgroups of G demonstrate certain local structural in-formation of G. However it is not complete. For example given H ∈ G andthe discrete set GH , any two non-identical objects are conjugate in G whilethis relationship is not seen in GH . Thus we want a category that presentsall intrinsic connections among any chosen collection of subgroups of G.

Definition 6.1.10. Let G be a group and P a G-poset. We define the trans-porter category on P to be a Grothendieck construction G ∝ P := GrGP .More precisely, G ∝ P has the same objects as P , that is, Ob(G ∝ P ) = Ob P .For x, y ∈ Ob(G ∝ P ), a morphism from x to y is a pair (g,gx ≤ y) for someg ∈ G.

In the literature the transporter categories are mostly considered as auxil-iary constructions before passing to various quotient categories of them. Herewe want to stress on the perhaps unique property, among various categoriesconstructed from a group, that transporter categories admit natural functorsto the group itself. It singles out this particular type of categories and is thestarting point of this chapter. Here in order to emphasize the similarities andconnections between transporter categories and subgroups, we follow a defi-nition which is well known to some algebraic topologists. The symbol G ∝ P is not standard and is used because this particular Grothendieck constructionresembles a semidirect product, yet is different.

This neat but seemingly abstract definition can be easily seen to give theusual transporter categories. For example, when P is the poset of non-trivial

p-subgroups, we get G ∝ P = Tr p(G), the p-transporter category of G. The

advantage of taking our approach is shown by the upcoming examples, whereeach subgroup of G is identified as a transporter category, up to a categoryequivalence.



6.1 Local categories 159

From Definition 6.1.10 one can easily see that there is a natural embeddingιP : P → G ∝ P via (x ≤ y) → (e, x ≤ y). On the other hand, the transportercategory admits a natural functor πP : G ∝ P → G, given by x → • and(g,gx ≤ y) → g. Thus we always have a sequence of functors

P ιP→G ∝ P

πP→G

such that πP ιP (P ) is the trivial subgroup or subcategory of G. For conve-nience, in the rest of this chapter we often neglect the subscript P and writeι = ιP , π = πP .

Example 6.1.11. 1. If G acts trivially on P , then G ∝ P = G × P .2. Let G be a finite group and H a subgroup. We consider the set of left cosets

G/H which can be regarded as a G-poset: G acts via left multiplication.The transporter category G ∝ (G/H ) is a connected groupoid whose skele-ton is isomorphic to H . In this way one can recover all subgroups of G,up to category equivalences. Consequently we have k(G ∝ G/H ) kH as

well as a homotopy equivalence B (G ∝ G/H ) BH .3. From Example 6.1.6 (2) we build a concrete transporter category Σ 4 ∝S 2(Σ 4) :

· · ·

· · ·

D8

8

8

D88

8 D8

8

C 2 × C 2

8

8

C 4 8

8

V 24

24

24

24

C 2

4

4

8

C 2

4

4

8

C 2

8

8

8

24

8 C 2

8

24

8

8 C 2

8

24

8

Here the numbers are the numbers of morphisms from one object to an-other. Note that again this is part of the whole category. However, itcontains a skeleton so we know what is missing.

Remark 6.1.12. One can check directly that if HomG∝P (x, y) = ∅ then bothAutG∝P (x) and AutG∝P (y) act freely on HomG∝P (x, y).

Another notable structural fact is that G ∝ P is a category with subob- jects, which means each morphism can be uniquely factorized as an isomor-phism followed by a morphism in P (regarded as a subcategory of G ∝ P ). In

fact we have (g,gx ≤ y) = (e,gx ≤ y) (g,gx = gx) as shown in the diagram




y

x

(g,gx≤y)

(g,gx=gx) gx.

(e,gx≤y)

Definition 6.1.13. We call B (G ∝ P ) the Borel construction on P .

For any G-space X , one can define a Borel construction E G ×G X . In ourdefinition, we actually have B(G ∝ P ) EG×G BP . It explains the concept.In particular B (G ∝ (G/H )) B H . Forming the transporter category overa G-poset eliminates the G-action, as an algebraic analogy of introducing aBorel construction over a G-space.

6.1.4 Local categories

Our concept of a transporter category is quite general because each subgroupH of G can be recovered as a transporter category G ∝ (G/H ), for the G-poset G/H , up to a category equivalence. Thus it makes sense if we deemtransporter categories as generalized subgroups for a fixed finite group.

Definition 6.1.14. Any quotient category C of G ∝ P is called a local cate-gory of G. When P consists of p-subgroups of G, for a prime p

|G|, we alsocall such a quotient category a p-local category.

A local category is connected with the group by the following diagram

G ∝ P π

ρ

G C

Transporter categories were implicitly considered by Mark Ronan and SteveSmith [65] in the 1980s for constructing group modules, and later on playeda key role in Bill Dwyer’s work [17] on homology decomposition of classifyingspaces. Dwyer used this diagram to establish connections among various ho-motopy colimits (e.g. classifying spaces), while Ronan and Smith constructedkG-modules via representations of G ∝ P (using the language of G-presheaveson P ).

Example 6.1.15. 1. The p-transporter category Tr p(G) = G ∝ S p has all non-identity subgroups as its objects. For any p-subgroups P, Q, the morphismset is often written as HomG(P, Q) = gP ⊂ Q

g ∈ G. In particular

AutG(P ) = N G(P ), the normalizer.2. The p- fusion system F p(G) is a quotient category of G ∝ S p, given by

HomF p(G)(P, Q) = HomG(P, Q)/C G(P ).






to [17] for Dwyer’s beautiful results on homology decompositions of classifyingspaces of groups.

In practice one often finds that the functor G ∝ P → C is part of anextension (or an opposite extension) sequence of categories. (Among examples

are various orbit categories, Brauer categories and Puig categories.) It meansthat for such a quotient category C there exists a category K which is a disjointunion of subgroups of AutG∝P (x), x running over Ob P = Ob(G ∝ P ), suchthat we can add K into the picture

K

G ∝ P

π

ρ

G C

and moreover K → G ∝ P

C satisfies some natural conditions provided inSection 4.1.?. It will helps us to understand relationship between k(G ∝ P )-mod and kC -mod.

For any M ∈ kG-mod, sometimes we denote by κM the restriction ResπM and call it a constant value module .

6.2.2 Frobenius Reciprocity

Applying classical tools in homological algebra, particularly the Kan exten-sions, we obtain a Frobenius Reciprocity between kG-mod and kC -mod, wherekC is the (k-)category algebra of C . It implies, to some extent, comparing kG-

mod with kC -mod provides an extended context for local representation the-ory. This observation illustrates a possible way to understand group represen-tations and cohomology via those of suitable categories. From the precedingdiagram we obtain a diagram of module categories

k(G ∝ P )-modLK π ,RK π

LK ρ,RK ρ

kG-mod

Resπ

kC -mod.Resρ

The adjunctions between the restrictions and Kan extensions have the fol-lowing consequences.

Proposition 6.2.1 (Frobenius Reciprocity). Suppose P is a G-poset and C is a quotient category of G ∝ P as in the preceding diagrams. Let M, N ∈kG-mod and m,n ∈ kC -mod. Then



6.2 Properties of local categories 163

1. HomkG(M,RK πResρn) ∼= HomkC(LK ρResπM, n);2. HomkG(LK πResρm, N ) ∼= HomkC(m, RK ρResπN ).

By direct calculations, these particular Kan extensions in Proposition 6.2.1

are simplified:• LK π ∼= lim−→P

, RK π ∼= lim←−P (used by Ronan-Smith. See Corollary 6.3.? for

a proof);• LK ρ ∼=↑kC

k(G∝P ) (the induction), RK ρ ∼=⇑kCk(G∝P ) (the co-induction), since

ρ induces an algebra homomorphism k(G ∝ P ) → kC when C is a quotientcategory.

Remark 6.2.2. For any n ∈ kC -mod, we shall write the k(G ∝ P )-moduleResρn as n because they share the same underlying vector space. Recall thatκM = ResπM . Then the Frobenius Reciprocity can be rewritten as

(i’) HomkG(M, lim←−P n) ∼= HomkC(κM ↑kC

k(G∝P ),n);

(ii’) HomkG(lim−→P m, N ) ∼= HomkC(m, κN ⇑kC

k(G∝P )).

When P = G/H for some subgroup H , we have natural isomorphismslim←−G/H

∼= lim−→G/H ∼=↑G

H . Then the above isomorphisms certainly become the

usual adjunctions between ↑GH and ↓G

H (the usual Frobenius Reciprocity) withC = G ∝ (G/H ) and ρ = Id, in light of the Morita equivalence between kC and kH .

Remark 6.2.3. Our Frobenius reciprocity is different from a similar result of Ronan-Smith, see [6, 7.2.4], where they (implicitly) had a diagram of the sameshape. However their C = G ∝ Q, not necessarily a quotient of G ∝ P , isanother transporter category and ρ is induced by a G-map P → Q. This pro-hibits us from considering various quotients of transporter categories. More-over since a G-map P → Q usually does not induce an algebra homomorphism

from k (G ∝ P ) to k(G ∝ Q), their Kan extensions cannot be interpreted asinduction and coinduction.

The functors ↑kCk(G∝P ) and ⇑kC

k(G∝P ) admit interesting interpretations when

G ∝ P → C is part of an extension (or an opposite extension) sequence of categories. Under the circumstance ↑kC

k(G∝P ) and ⇑kCk(G∝P ) on certain k(G ∝

P )-modules can be very well understood. We shall discuss it in Section 5.




6.3 The functor π: group representations viatransporter categories

6.3.1 Homology representations via transporter categories

Suppose P is a G-poset and π is the natural functor from the transportercategory G ∝ P to G, regarded as a category with one object •. The naivefibre of the functor π, i.e. π−1(•), is exactly P . On the other hand, theovercategory π/• provides the (topological) fibre of the map Bπ in the sensethat the sequence of categories

π/• → G ∝ P → G

corresponds to a fibration after passing to classifying spaces

B (π/•) → B(G ∝ P ) → BG.

It leads to an action of G = π1(BG) on B (π/•), which is realized by a G-action on the category π/•, described shortly after Proposition 6.3.1. We shallsee that there exists a functor π/• → P as well as an inclusion P → π/•,inducing a category equivalence. Interestingly the former is a G-functor, whilethe latter is not.

The objects of π/• are of the form (x, h), in which x ∈ Ob(G ∝ P ) = Ob P and h ∈ G. A morphism from (x, h) to (x, h) is a morphism (g,gx ≤ x) ∈Mor(G ∝ P ) such that hg = h or equivalently g = h−1h. It implies thateach object (x, h) ∼= (x, h) if and only if x = gx and h = hg−1 for someg ∈ G. Indeed the objects isomorphic to (x, h) are (gx,hg−1)

g ∈ G.Particularly (x, h) ∼= (hx,e) for the identity e ∈ G.

There is also an undercategory •\π. One can put it in place of the over-category and all our observations stay true. The objects of •\π are of theform (h, x), in which x ∈ Ob(G ∝ P ) = Ob P and h ∈ G. A morphismfrom (h, x) to (h, x) is a morphism (g,gx ≤ x) ∈ Mor(G ∝ P ) such thatgh = h. It implies that each object (h, x) ∼= (h, x) if and only if x = gxand h = g−1h for some g ∈ G. Indeed the objects isomorphic to (h, x) are(g−1h,gx)

g ∈ G. Particularly (h, x) ∼= (e,hx) for the identity e ∈ G.When P = •, G ∝ • ∼= G and the functor π can be identified with

IdG. Consequently π/• ∼= IdG/•. The following result generalizes this specialsituation and gives a characterization of the overcategory and undercategorycoming from G ∝ P → G.

Proposition 6.3.1. The category π/• is isomorphic to P ×(IdG/•), and •\πis isomorphic to P × (•\IdG). Consequently π /• ∼= •\π.

Proof. We establish an isomorphism φ : P × (IdG/•) → π/• as follows.The objects of P × (IdG/•) are (x, (•, g))

g ∈ G, x ∈ Ob P. We define



6.3 The functor π : group representations via transporter categories 165

φ((x, (•, g))) = (g−1x, g) ∈ Ob(π/•). For a morphism (x1 ≤ x2, g−12 g1) :

(x1, (•, g1)) → (x2, (•, g2)) we put

φ((x1 ≤ x2, g−12 g1)) := (g−1

2 g1, (g−12 g1)(g−1

1 x1) ≤ g−12 x2),

a morphism from (g−11 x1, g1) t o (g−1

2 x2, g2). We can write down its in-verse ψ : π/• → P × (IdG/•), given by ψ((y, h)) := (hy, (•, h)). For anymorphism in π/•, (h−1

2 h1, h−12 h1y1 ≤ y2) : (y1, h1) → (y2, h2), we de-

fine ψ((h−12 h1, h−1

2 h1y1 ≤ y2)) := (h1y1 ≤ h2y2, h−12 h1) : (h1y1, (•, h1)) →

(h2y2, (•, h2)).The isomorphism for •\π can be similarly obtained.In the end, from Example 1.2.7 (1), we have an isomorphism Id G/• ∼=

•\IdG and hence the isomorphism between the over category and undercat-egory.

As we have shown in Example 1.2.7 (2) that both •\IdG and IdG/• areequivalent to •. For instance, there is the canonical functor pt : IdG/• → •as well as a functor • → IdG/•, given by • → (•, e), where e is the identityof G. The reader can quickly verify that they provide a category equivalence.These two categories are actually the Cayley graph.

Corollary 6.3.2. There exists a natural embedding P → π/• (or P → •\π)making P a skeleton of π/• (or •\π). Consequently LK πm ∼= lim−→P

m and

RK πm ∼= lim←−P m for any m ∈ k(G ∝ P )-mod.

Proof. By the preceding proposition, π/• ∼= P × IdG/•. Since we know ex-plicitly the equivalence-inducing functors between IdG/• and •, we can easilytranslate them for π/•.

The natural functor P → π/•, given by x → (x, e) and x ≤ y → (e, x ≤ y),is an embedding, sending P to a skeleton of π/•. It is straightforward to checkthat there is a natural surjective functor π/• → P , induced by (x, h) → hx.For any morphism in π/•, (h−1

2 h1, h−12 h1y1 ≤ y2) : (y1, h1) → (y2, h2), we

define ψ((h−12 h1, h−1

2 h1y1 ≤ y2)) := h1y1 ≤ h2y2.These two functors provide an equivalence between π/• and P . Same

equivalence can be established between P and •\π.The existence of equivalences between these categories forces lim−→π/•

m ∼=

lim−→P m and lim←−•\π

m ∼= lim←−P m.

Next we shall show that both π/• and •\π are G-categories. Moreover westudy whether or not the above functors, inducing equivalences with P , arecompatible with G-action. Before we do so, we need to specify what we meanby this compatibility.

Definition 6.3.3. Let D, C be two G-categories and u : D → C a functor. Wesay u is a G-functor if for any g ∈ G and x ∈ Ob D we have u(gx) = gu(x),and for any α ∈ Mor D, gu(α) = u(gα).




We first describe the G-actions on IdG/• and on •\IdG. They are given onobjects by u · (•, g) = (•, ug) and u · (g, •) = (gu−1, •), respectively. Onmorphisms,

u · [(•, g1)g−12 g1−→ (•, g2)] = (•, ug1)

g−12 g1−→ (•, ug2)

while

u · [(g1, •)g2g1

−1

−→ (g2, •)] = (g1u−1, •)g2g1

−1

−→ (g2u−1, •).

Immediately we see that the surjection IdG/• → • is a G-functor, and bycontrast the embedding • → IdG/• is not a G-functor, because g• = • →(•, e) by the embedding while g(•, e) = (•, g), for any g ∈ G. Hence we donot expect the embedding P → π/• to be a G-functor.

Based on the G-actions on IdG/• and •\IdG and Proposition 6.3.1, we candefine G-actions on π/• and •\π. For any object (x, h) ∈ Ob(π/•) and u ∈ G,we have u · (x, h) = (x,uh), and for any morphism (g,gx ≤ x) : (x, h) →(x, h) we have u · (g,gx ≤ x) = (g,gx ≤ x) : (x,uh) → (x, uh). Notethat g = h−1h = (uh)−1(uh). Similarly For any object (x, h) ∈ Ob(•\π)

and u ∈ G, we have u · (x, h) = (x,hu−1

), and for any morphism (g,gx ≤x) : (x, h) → (x, h) we have u · (g,gx ≤ x) = (g,gx ≤ x) : (x,hu−1) →(x, hu−1), with g = hh−1 = (hu−1)(hu−1)−1. From our constructions of G-actions on π/• and •\π, we reach the following statements.

Corollary 6.3.4. 1. The isomorphisms π/• ∼= P × IdG/• and •\π ∼= P ×•\IdG are G-isomorphisms.

2. The natural functors π /• P and •\π P are G-functors.

As we mentioned earlier, a G-category C gives rise to a complex of kG-modules C∗(C , k). Furthermore if u : D → C is a G-functor between two G-categories, then it induces a G-simplicial map u∗ : N D∗ → N C ∗, and hencea chain map between complexes of kG-modules u : C∗(D, k) → C∗(C , k).

Example 6.3.5. If H ⊂ K are subgroups of G, then there is a G-functorG/H → G/K . Hence we have a chain map C∗(G/H,k) → C∗(G/K,k). Sinceboth complexes concentrate in degree zero, this chain map consists of onlyone kG-map: k(G/H ) = C0(G/H,k) → k(G/K ) = C0(G/K,k).

Corollary 6.3.6. The G-functors π/• → P and •\π → P induce chain maps between complexes of kG-modules. The complex LK πB G∝P ∗

∼= C∗(π/•, k) is a projective resolution of the finite complex of k G-modules C∗(P ).

Proof. Since LK π preserves projectives, LK πB G∝P ∗ ∼= C∗(π/•, k) is a com-

plex of projective kG-modules. The existing G-functor π/• → P gives rise to achain map of complexes of kG-modules LK πB G∝P ∗

∼= C∗(π/•, k) → C∗(P , k).However since that G-functor is a category equivalence, it induces an isomor-phism between the homology of complexes.

Remark 6.3.7. Any left kG-module is naturally a right kG-module. If we con-sider C∗(P , k) as a complex of right kG-modules through (−) · g := g−1 · (−),




then a projective resolution can be obtained as C∗(•\π, k) ∼= RK πB G∗ inwhich we take the right bar resolution of k ∈mod-k(G ∝ P ). If we regardthe complex of right modules C∗(•\π, k) as a complex of left modules viag · (−) := (−) · g−1, then it is isomorphic to C∗(π/•, k). In fact, the isomor-

phism comes from the isomorphism of categories π/• ∼= •\π.It should be useful to understand the complexes C∗(π/•, k) and C∗(•\π, k).

Example 6.3.8. By Proposition 6.3.1, if P = G/G = •, then πG/G = IdG andπG/G/• = IdG/• is the Cayley graph, giving rise to the total space EG whosecomplex is the bar resolution B G∗ . More generally for P = G/H , we have anisomorphism of complexes of kG-modules C∗(π/•) ∼= B G∗ ⊗ k(G/H ), whichprovide a projective resolution of k(G/H ) = C0(G/H ) = C∗(G/H ).

For any small category C , there is an augmentation map : C∗(C , k) → k,which is zero on positive degrees and which maps every base element inC0(C , k) = k Ob C to 1 ∈ k.

Corollary 6.3.9. 1. The G-isomorphisms π /• ∼= P × IdG/• and •\π ∼= P ×

•\IdG induce isomorphisms between complexes of kG-modules.2. We have equivalences of complexes of kG-modules C∗(π/•, k) C∗(P , k)⊗C∗(IdG/•, k) and C∗(•\π, k) C∗(P , k) ⊗ C∗(•\IdG, k). Furthermore C∗(π/•) C∗(P ) ⊗ B G∗ and C∗(•\π) C∗(P ) ⊗ B G∗ .

3. The chain map C∗(P × IdG/•, k) → C∗(IdG/•, k), induced by P → •,corresponds to the chain map C∗(P , k) ⊗ C∗(IdG/•, k) → C∗(IdG/•, k),induced by the augmentation map C∗(P , k) → k.

Proof. The first statement is a consequence of the preceding corollary.Since π/• ∼= P × IdG/•, we have isomorphism C∗(π/•, k) ∼= C∗(P ×

IdG/•, k). By Theorem 2.2.31, there is a natural chain homotopy equivalenceC∗(P × IdG/•, k) → C∗(P , k) ⊗C∗(IdG/•, k), called the Alexander-Whitneymap. From its definition, one can see it is a chain map of k G-modules.

To see the third statement, we draw a commutative diagram

C∗(P × IdG/•, k)

C∗(P , k) ⊗C∗(IdG/•, k)

⊗1

C∗(IdG/•, k) k ⊗C∗(IdG/•, k)

Suppose (x0 → · · · → xn) ⊗ (g0 → · · · → gn) in a base element of Cn(P ×IdG/•, k). Then the Alexander-Whitney map sends it to

ni=0

(x0 → · · · → xi) ⊗ (gi → · · · → gn),

whose image under the right vertical map equals ⊗ 1(x0 ⊗ (g0 → · · · →gn)) = g0 → · · · → gn. This is exactly the image of (x0 → · · · → xn) ⊗ (g0 →· · · → gn) under the left vertical map, which is a projection map.




The last corollary of Proposition 6.3.1 is actually a consequence of Corol-lary 6.3.2. Since it is closely related to the second main result in this section,we put it here.

Corollary 6.3.10. Suppose M ∈ kG-mod and n ∈ k(G ∝ P )-mod. Then LK π(κM ⊗n) ∼= M ⊗ LK πn and RK π(κM ⊗n) ∼= M ⊗ RK πn as k G-modules.In particular LK π(κM ) ∼= M ⊗ LK πk and RK π(κM ) ∼= M ⊗ RK πk, where LK πk ∼= H0(BP , k) ∼= H0(BP , k) ∼= RK πk of dimension equal to the number of connected components of P .

Proof. For the left Kan extension we have

LK π(κM ⊗n) ∼= lim−→P (κM ⊗n) ∼= M ⊗ lim−→P

n ∼= M ⊗ LK πn.

The second isomorphism is true because κM as a kP -module admits trivialaction. The statement for the right Kan extension is similar.

The above corollary implies that LK πκM ∼= RK πκM , suggesting the exis-

tence of a transfer map, which we will construct later on.The next result is a direct generalization of the fact that the two obvious

kG-module structures on P ⊗ M are isomorphic, for P, M ∈ kG-mod with P projective. It reveals another connection between representations of groupsand of transporter categories.

Theorem 6.3.11. Let P ∈ k(G ∝ P )-mod be a projective module and κM = ResπM for some M ∈ kG-mod. Then P⊗κM is a projective k(G ∝ P )-module. Consequently B G∝P ∗

⊗κM → k⊗κM = κM → 0 is a projective reso-lution.

Proof. We will prove P⊗κM ∼= P ⊗ M , with k (G ∝ P ) acting on the latter

via left multiplication. To this end, we assume P = kHomG∝P (x, −). The

proof is entirely analogues to the case when P = •, i.e. when G ∝ • = G.We define a k-linear map ϕ : kHomG∝P (x, −)⊗M → kHomG∝P (x, −)⊗κM

as follows. On base elements ϕ((g,gx ≤ y) ⊗m) = (g,gx ≤ y)⊗(g,gx ≤ y)m,where the latter m is considered as an element in κM (x). For any (h,hy ≤ z),we readily verify

(h,hy ≤ z)ϕ[(g,gx ≤ y) ⊗ m] = ϕ[(h,hy ≤ z)((g,gx ≤ y) ⊗ m)].

Thus ϕ is a homomorphism of k(G ∝ P )-modules.We remind the reader that, following definition, given any pair of x, y ∈

Ob(G ∝ P ) with HomG∝P (x, y) non-empty, both AutG∝P (x) and AutG∝P (y)act freely on HomG∝P (x, y). This implies that kHomG∝P (x, y) is a freekAutG∝P (x)- or kAutG∝P (y)-module. Consequently ϕ restricts on each y

to the classical k AutG∝P (y)-isomorphism (see [5, 3.1.5])kHomG∝P (x, y) ⊗ M → (kHomG∝P (x, −)⊗κM )(y) = kHomG∝P (x, y) ⊗ M.




Furthermore because as vector spaces

kHomG∝P (x, −)⊗κM =

y∈Ob(G∝P )

kHomG∝P (x, y)⊗M = kHomG∝P (x, −)⊗M,

the linear map ϕ is actually one-to-one and hence an isomorphism of k(G ∝P )-modules.

For an arbitrary category algebra, the tensor product of a projective mod-ule with a non-trivial module usually does not stay projective, as one canfind a counter-example in [?, 2.5]. The key point here is that the structure of G ∝ P allows us to apply some results on group algebras.

Assume C∗ is a complex of kG-modules. Then we naturally obtain a com-plex of k(G ∝ P )-modules via restriction, written as κC∗ .

Lemma 6.3.12. The functor π : G ∝ P → G induces a natural chain mapΠ from B G∝P ∗ → k → 0 to κBG∗ → k → 0. More generally for any M ∈ kG-mod, it naturally induces a chain map between exact sequences of k (G ∝ P )-

modules

Π M : B G∝P ∗ ⊗κM → κM → 0 → κBG∗ ⊗κM = κBG∗ ⊗M → κM → 0.

Proof. The reason is that π induces a natural map between the nerves of these categories, while the bar resolutions are constructed from the chaincomplexes from the nerves. More explicitly the complexes B G∝P ∗ → k → 0and κBG∗ → k → 0 evaluated at any x ∈ Ob(G ∝ P ) are the augmented chain

complexes C∗(IdG∝P /x) → k → 0 and B G∗ → k → 0, respectively. The chainmap Π is induced by π [(g,ga ≤ b)] = g.

The construction of Π M for a fixed M ∈ kG-mod is similar.

6.3.2 On finite generation of cohomology

Subsections 6.4.2 and 6.4.3 contains two applications of results from Subsec-tion 6.4.1 to transporter category cohomology and its connection with groupcohomology. To study the functor ρ : G ∝ P → C , we will start a new Section6.5.

Let G be a finite group and P a finite G-poset. Then there exists a sequenceof functors

P ι→G ∝ P

π→G,

where ι is the natural embedding, and whose topological realization is afibration






modules over Ext∗k(G∝•)(k, k) = Ext∗kG(k, k). We point out that the groupcohomology ring H∗(G, k) ∼= Ext∗kG(k, k) acts on

H∗(G ∝ P ; N ) ∼= Ext∗k(G∝P )(k, N )

via the algebra homomorphism induced by π (or B π),

Ext∗kG(k, k) ∼= H∗(G, k) → H

∗(|G ∝ P|, k) ∼= H∗(G ∝ P ; k) ∼= Ext∗k(G∝P )(k, k).

Since P has the property that k is of finite projective dimension, weknow Hj (•\π; −) ∼= Extj

k(•\π)(k, −) vanishes for large j. Furthermore, the

well-known theorem of Evens and Venkov says that, for each j, the mod-ule Ext∗kG(k, Extj

k(•\π)(k, N )) is a finitely generated over Ext∗kG(k, k). Since

E ∞ is a subquotient of E 2 of a cohomology spectral sequence, we have thefollowing statement.

Lemma 6.3.13. For any N ∈ kTrP (G)-mod, Ext∗k(G∝P )(k, N ) is a finitely generated Ext∗kG(k, k)- and Ext∗k(G∝P )(k, k)-module.

Theorem 6.3.11 helps us to extend this finite generation property.

Proposition 6.3.14. For any M ∈ kG-mod and n ∈ k(G ∝ P )-mod,Ext∗k(G∝P )(κM , n) is finitely generated over Ext∗k(G∝P )(k, k).

Proof. One can easily deduce from Theorem 6.4.11, together with the internalhom in Section 3.4.?, an Eckmann-Sharpiro type isomorphism

Ext∗k(G∝P )(κM , n) ∼= Ext∗k(G∝P )(k, Hom(κM , n)).

Then we apply the finite generation result that we just quoted.

The ultimate goal is to establish the finite generation of Ext∗k(G∝P )(m, n),

for any m, n ∈ k(G ∝ P )-mod. However the existence of an internal hom is notenough because, unless m is induced from a k G-module, Ext∗k(G∝P )(m, n) ∼=Ext∗k(G∝P )(k, Hom(m,n)). We shall establish the finite generation theorem,Theorem 6.5.3, in the very last section because we have to rely on theHochschild cohomology of transporter categories.

Example 6.3.15. Let G = C 2 = g g2 = e and chark = 2. Then the

G-poset S e2 is e → C 2, and the transporter category C := C 2 ∝ S e2 is

ee e g

C 2.

e

g

There are two simple modules S e = S e,k and S C 2 = S C 2,k, both of di-mension 1. Furthermore there is a short exact sequence 0 → S C 2 → P e →




S e → 0. If we apply Ext∗kC(−, S C 2) to it, then we get ExtikC(S e, S C 2) ∼=

Exti−1kC (S e, S e) for i > 0 and Ext0

kC(S e, S C 2) = 0. The cohomology ringis finitely generated because

Ext∗

kC(k, k) ∼= Ext∗

kC(S e, S e) ∼= Ext∗

kC 2(k, k) ∼= k[x]

it is a polynomial ring with one indeterminant x of degree 1.We can see that Exti

kC(S e, S C 2) is of dimension 1 for i > 0 andExt∗kC(S e, S C 2) is finitely generated over Ext∗kC(k, k). By contrast,

Ext∗kC(k, Hom(S e, S C 2)) = 0

since Hom(S e, S C 2) = 0 by direct calculation.

Remark 6.3.16. Using Dan Swenson’s definition [72] of the internal hom oneverifies that

κHomk(M,N ) ∼= Hom(κM , κN ).

Consequently,

Ext∗k(G∝P )(k, κHomk(M,N )) ∼= Ext∗k(G∝P )(k, Hom(κM , κN )) ∼= Ext∗k(G∝P )(κM , κN ).

Before moving to the next section, we record a connection between coho-mology of transporter categories and equivariant cohomology, which is per-haps known to the experts.

Proposition 6.3.17. The left Kan extension induces an isomorphism

λM : Ext∗k(G∝P )(k, κM ) ∼= H∗G(BP , M ),

where H∗G(BP , M ) is the equivariant cohomology group for some M ∈ kG-mod.

Proof. Take the bar resolution B G∝P ∗ → k → 0 and consider the complexHomk(G∝P )(B G∝P ∗ , κM ). The left Kan extension induces a chain map

Homk(G∝P )(B G∝P ∗ , κM ) ∼= HomkG(LK πB G∝P ∗ , M ) ∼= HomkG(B G∗ ⊗C∗(P ), M ).

But the rightmost term is

HomkG(B G∗ ⊗ C∗(P ), M ) ∼= HomkG(B G∗ , Homk(C∗(P ), M )),

which gives rise to H∗G(BP , M ).

In light of the above proposition, we may introduce the Tate cohomologyof transporter categories as Tate equivariant cohomology. With Remark 3 in

mind, one can further define negative degree Ext groups Ext∗k(G∝P )(κM , κN ).




6.3.3 Transfer for ordinary cohomology

Let u : D → C be a functor between small categories. There is always a re-striction resu : H∗(C ; k) → H∗(D; k). However usually one cannot construct a

map in the opposite direction, unless the two Kan extensions are connectedby a natural transformation. In this section, based on our knowledge aboutrepresentations of k(G ∝ P ), we establish various transfer maps, includingthe Becker-Gottlieb transfer map with respect to πP : G ∝ P → G. Herewe provide an algebraic alternative to the construction of Becker-Gottlieb[4], and the core idea is taken from Dwyer-Wilkerson [21, 9.13]). Essentiallyour construction incorporates [21, 9.13] in an entirely representation-theoreticsetting. The upshot is that our construction is analogues to the classical situ-ation, see for instance [5, 3.6.17]. Keep in mind that Resπ and LK π generalize↓G

H and ↑GH , respectively, used in group cohomology.

In Corollary 6.3.9 (3), we described a chain map

LK πB G∝P ∗ ∼= C∗(π/•, k) ∼= C∗(P × IdG/•, k) −→ C∗(IdG/•, k) ∼= B G∗ ,

induced by P → •. This chain map gives rise to a cochain map

HomkG(B G∗ , k) → HomkG(LK πB G∝P ∗ , k) ∼= Homk(G∝P )(B G∝P ∗ , k)

and hence the restriction

resπ : Ext∗kG(k, k) → Ext∗k(G∝P )(k, k).

We want to establish a map, called transfer , on the opposite direction

Ext∗k(G∝P )(k, k) → Ext∗kG(k, k).

Example 6.3.18. Before we construct the transfer, we examine the special case

for group cohomology. Suppose H ⊂ G is a subgroup. We want to define amap trG

H : H∗(H, k) → H∗(G, k).For the functor πG/H : G ∝ (G/H ) → G. We get

LK πG/HB G∗ = C∗(πG/H /•, k) ∼= C∗((G/H ) × IdG/•, k)

as complexes of kG-modules. The rightmost admits a map to C∗(IdG/•, k),induced by the G-functor pt : G/H → •. If we invokeC∗((G/H )×IdG/•, k) k(G/H )⊗B G∗ , then this chain map is identified with the chain map k(G/H )⊗B G∗ → C∗(IdG/•, k), induced by the augmentation

: k(G/H ) = k ↑GH → k.

By applying HomkG(−, k), this chain map induces the restriction

resπG/H : H

∗(G, k) → H∗(G ∝ (G/H ), k) ∼= H

∗(H, k).




To produce another map in the opposite direction, we notice that althoughthere is no G-functor • → G/H (unless G = H ), we can always build a kG-map after linearization

k = k• → k(G/H ) = k ↑G

H .

It induces a chain mapB G∗ → k ↑G

H ⊗B G∗ .

When we apply HomkG(−, k) to the following and take cohomology

B G∗ → k ↑GH ⊗B G∗

⊗1−→B G∗ ,

sincek ↑G

H ⊗B G∗ ∼= (k ⊗ B G∗ ↓GH ) ↑G

H = B G∗ ↓GH ↑

GH ,

we get HomkG(B G∗ ↓GH ↑

GH , k) ∼= HomkH (B G∗ ↓G

H , k) and consequently two in-duced maps

H∗(G, k)trGH

←−H∗(H, k)resGH

←−H∗(G, k).

Since the composite k → k(G/H ) = k ↑GH → k equals a scalar multiplication

by [G : H ], we know trGH resG

H = [G : H ] = |G/H |. Here the restriction resGH is

identified with resπG/H upon the isomorphism H∗(G ∝ (G/H ), k) ∼= H∗(H, k).

Since for arbitrary G ∝ P , we have constructed the restriction

resπ : H∗(G, k) → H

∗(G ∝ P , k).

Now we would like to have a generalized transfer map in the opposite direc-tion. As in the group case, it should be induced by some B G∗ → LK πB G∝P ∗ ,or rather

C∗(IdG/•, k) → C∗(π/•, k) ∼= C∗(P × IdG/•, k) C∗(P , k) ⊗ C∗(IdG/•, k).

Indeed, it should be reduced to establishing a map

k = k• = C∗(•, k) → C∗(P , k).

Note that, although there are many functors from • → P , all of them havethe problem that they are not G-functors. Thus they will not directly producechain maps between the above two complexes of kG-modules. Now we do haveto work on the chain level. If such a chain map exists, then we immediatelyobtain a chain map

C∗(•, k) ⊗C∗(IdG/•, k) → C∗(P , k) ⊗C∗(IdG/•, k) C∗(P × IdG/•, k).

As the first attempt, we record the following observation.




Lemma 6.3.19. For any n ≥ 0 and any x0 → · · · → xn ∈ N P n, one can construct a chain map ci : C∗(•, k) → C∗(P , k), for each 0 ≤ i ≤ n.

Proof. Since C∗(•, k) = k, we define the chain maps as

C0(•, k) → C0(P , k), 1 → I

gxi,

Here I is the G-orbit of xi.

Obviously various linear combinations of the above maps are still kG-maps.

Example 6.3.20. Suppose x ∈ N P 0. Then C0(•, k) maps isomorphicallyto a 1-dimensional submodule of C0(P , k), that is, kO+(x) for O+(x) =

y∈O(x) y where O(x) is the G-orbit of x in G ∝ P . Hence we get

C∗(•, k) kO+(x)

C∗(•, k)

C∗(•, k) kO(x)

C∗(•, k)

C∗(•, k) C∗(P , k) C∗(•, k)

Here kO(x) is isomorphic to k(G/StabG(x)) ∼= k ↑GStabG(x). Tensoring with

C∗(IdG/•, k) we get

C∗(IdG/•, k)∼= kO+(x) ⊗C∗(IdG/•, k)

C∗(IdG/•, k)

C∗(IdG/•, k) kO(x) ⊗C∗(IdG/•, k)

C∗(IdG/•, k)

C∗(IdG/•, k) C∗(P , k) ⊗C∗(IdG/•, k)

C∗(IdG/•, k)

In the central column, the top one is isomorphic to C∗(IdG/•, k), and themiddle is isomorphic to C∗(πO(x)/•, k). Thus by applying HomkG(−, k) andtaking cohomology we a commutative diagram




H∗(G, k) H∗(G, k)∼= H∗(G, k)∼=

H∗(G, k) H∗(G ∝ O(x), k)trO(x)

tr

H∗(G, k)resO(x)

H∗(G, k) H∗(G ∝ P , k)

res

H∗(G, k)resπ

The transfer map is the usual one for group cohomology, see Example 6.3.17.The restrictions are induced by the functors G ∝ O(x) → G ∝ P → G. Theabove diagram reduces to a sequence of maps

H∗(G, k) → H

∗(G ∝ P , k) → H∗(G ∝ O(x), k) → H

∗(G, k),

or equally

H∗(G, k) → H

∗(G ∝ P , k) → H∗(StabG(x), k) → H

∗(G, k).

We can actually take any G-set sitting inside P . The natural candidates areall the sets N P n, n ≥ 0. For instance if α = x0 < · · · < xn ∈ N P n and O(α)denotes the G-orbit of α, then we will get a sequence of maps

H∗(G, k) → H∗(G ∝ P , k) → H∗(G ∝ O(α), k) → H∗(G, k),

or equally

H∗(G, k) → H

∗(G ∝ P , k) → H∗(StabG(α), k) → H

∗(G, k).

Note that StabG(α) =

i StabG(xi).

The preceding example implies that using naively constructed chain maps

C∗(•, k) → C∗(P , k) it is unlikely to obtain a novel H∗(G ∝ P , k) → H∗(G, k)which does not depend on the transfer in group cohomology H∗(StabG(α), k) →H∗(G, k). Thus we must try something more complicated. In the proof of thenext theorem, we construct the Becker-Gottlieb transfer, also seen in Dwyer-Wilkerson.

It seems like many important theorems in group cohomology share thesame nature and live in the same context. Using the double complex HomkG(C†∗(P )⊗B G∗ , k) to obtain Webb’s Theorem, Dwyer’s sharp decompositions results???

When P = S p, by Brown’s theorem, χ(P ) = 1 in k . By Quillen’s(?) The-orem resπ : H∗(G, k) → H∗(G ∝ P , k) is an isomorphism. What about A p

etc?

Theorem 6.3.21. Suppose Resπ : kG-mod → k(G ∝ P )-mod is the restric-

tion along π and write κM = ResπM for any M ∈ k G-mod. Then we have the following two maps, restriction and transfer,




Ext∗kG(M, N )resP−→Ext∗k(G∝P )(κM , κN )

trP−→Ext∗kG(M, N ),

which compose to χ(P ; k) ·1, multiplication by the Euler characteristic of (the order complex of) P .

Proof. We shall construct these two maps. Then in the sequel we can deducethe statement on their composite. The restriction is a generalized version of the one shown at the beginning of this section.

Suppose B G∗ and B G∝P ∗ are the bar resolutions of k ∈ kG-mod and k ∈k(G ∝ P )-mod, respectively. Then B G∗ ⊗ M is a projective resolution of a fixedM ∈ kG-mod. For another kG-module N , the cochain complex HomkG(B G∗ ⊗M, N ) computes Ext∗kG(M, N ). The exact functor Resπ sends this cochaincomplex to Homk(G∝P )(κBG∗ ⊗M , κN ). Applying Lemma 6.3.12 we obtain thecomposite of two chain maps, unique up to chain homotopy,

HomkG(B G∗ ⊗M, N )Resπ→ Homk(G∝P )(κBG∗ ⊗M , κN )

Π ∗M → Homk(G∝P )(B G∝P ∗ ⊗κM , κN ).

These seemingly abstract maps can be written down explicitly, but we willleave it to the interested reader. The composite Π ∗M Resπ induces a map oncohomology, which we call the restriction ,

resP : Ext∗kG(M, N ) → Ext∗k(G∝P )(κM , κN ).

It is helpful to have a different characterization of the restriction. In orderto do so we use a series of obvious isomorphisms to rewrite the previouslymentioned complex Homk(G∝P )(B G∝P ∗

⊗κM , κN ). Firstly by adjunction, it isisomorphic to

HomkG(LK π(B G∝P ∗ ⊗κM ), N ).

Here (Π ∗M Resπ)α is mapped to (ΛN LK π)(Π ∗M Resπα), where ΛN :LK πκN → N , determined by the counit of adjunction, is isomorphic to ⊗

IdN . The natural map : H0(P ) → k is induced by the augmentation map : C0(P ) → k (or rather P → •), by Corollary 6.3.10. Secondly from thesame corollary our cochain complex is canonically isomorphic to

HomkG(LK π(B G∝P ∗ ) ⊗ M, N ).

Thirdly by Corollary 6.3.9 LK πB G∝P ∗ B G∗ ⊗C∗(P ), the above complex is

HomkG(LK π(B G∝P ∗ ) ⊗ M, N ) HomkG(B G∗ ⊗C∗(P ) ⊗ M, N )∼= HomkG(B G∗ ⊗ M ⊗ C∗(P ), N ).

The observations imply that resP : Ext∗kG(M, N ) → Ext∗k(G∝P )(κM , κN ) isthe same as the map induced by the following chain map

HomkG(B G∗ ⊗ M, N ) → HomkG(B G∗ ⊗ M ⊗ C∗(P ), N ),




which is given by the augmentation : C∗(P ) → k. Now we are ready to buildthe transfer map. From the cochain complex HomkG(B G∗ ⊗ M ⊗ C∗(P ), N )we continue to establish a map which leads back to group cohomology

Θ

∗

: HomkG(B G

∗ ⊗ M ⊗ C∗(P ), N ) → HomkG(B G

∗ ⊗ M, N ).

The map Θ∗ is induce by the following chain map Θ : k → C∗(P ) of Dwyer-Wilkerson [21, 9.13], as the composite of

ka →a·Id

→ Homk(C∗(P ),C∗(P ))∼=→C∗(P )∧⊗C∗(P )

Id⊗∆→ C∗(P )∧⊗C∗(P )⊗C∗(P )

ev⊗Id→ C∗(P ).

Here C∗(P )∧ = Homk(C∗(P ), k), non-positively graded, is the k-dual of C∗(P ). The composite of

Homk(G∝P )(B G∝P ∗ ⊗κM , κN )

∼=→HomkG(LK π(B G∝P ∗ ⊗κM ), N )

Θ∗

→HomkG(B G∗ ⊗M, N )

defines a map

trP : Ext∗k(G∝P )(κM , κN ) → Ext∗kG(M, N ).

which is called the transfer .In fact we have the following commutative diagram of cochain complexes

HomkG(B G∗ ⊗ M, N )

=

Homk(G∝P )(B G∝P ∗ ⊗κM , κN )

HomkG(B G∗ ⊗ M, N )

=

HomkG(B G∗ ⊗ M, N ) HomkG(B G∗ ⊗ M ⊗ C∗(P ), N ) HomkG(B G∗ ⊗ M, N ).

Upon passing to cohomology, both rows give rise to

Ext∗

kG

(M, N )resP→ Ext∗

k(G∝P )

(κM , κN )trP→Ext∗

kG

(M, N ).

In the end we prove that the composite resP trP = χ(P ) · 1. Since the

normalization C∗(P , k) → C†∗(P , k) is a G-chain homotopy equivalence, we

can replace C∗(P , k) by C†∗(P , k) in our calculation. The normalized chain

complex is finite so we can find an integer d such that C†d(P , k) = 0 butC†n(P , k) = 0 for all n > d. The following proof is due to Dwyer-Wilkerson[21, 9.13] too, which shows by direct calculation that

k → C†∗(P ) → k

is a scalar multiplication by χ(P ). Write the natural basis of C†n(P ) as cindn

i=1

for dn = dimk C†n(P ) (see Section 6.2.2). The step-by-step images of 1 ∈ kunder Θ are



6.4 The functor ρ: invariants and coinvariants 179

1 → IdC†∗(P )

→d

n=0(−1)ndn

i=1(cin)∧ ⊗ ci

n

→d

n=0(−1)ndn

i=0(cin)∧ ⊗ [ci

n ⊗ t(cin)]

= dn=0(−1)n

dni=1[(ci

n)∧ ⊗ cin] ⊗ t(ci

n)

→ dn=0(−1)ndn

i=1 t(cin)

→d

n=0(−1)ndn

= χ(P ).

Here t(cin) ∈ C†0(P ) denotes the last object, i.e. the target, of the n-chain of

morphisms cin ∈ C†n(P ).

Remark 6.3.22. In fact, by Remark 6.3.15, the above restriction and transfercoincide with

Ext∗kG(k, Homk(M, N ))resP→ Ext∗k(G∝P )(k, κHomk(M,N ))

trP−→Ext∗kG(k, Homk(M, N )).

When M = N = k, our construction is exactly the Becker-Gottlieb transfer

([4],[21]), because Ext∗k(G∝P )(k, κM ) ∼= H∗G(BP , M ).We emphasize that if either M ∈ kG-mod is not acted trivially by kG or if

G sends a connected component of P to a different one, then the constantlyvalued κM ∈ k(G ∝ P )-mod is not truly constant since H0(G ∝ P ; κM ) ∼=lim←−G∝P

κM ∼= lim←−G

lim←−P κM ∼= (M ⊗ H0(P ))G.

Corollary 6.3.23. If χ(P ; k) is invertible in k, then resP is an injective homomorphism.

According to Dwyer, a collection of subgroups of G is a set of subgroupsthat is closed under conjugation. If P is a collection then it is naturally aG-poset in which the relations are inclusions and G-acts by conjugation. Acollection is called ample if for a fixed prime p and a field k of characteristic

p the restriction resP : Ext∗k(G∝P )(k, k) → Ext

∗kG(k, k) is an isomorphism.

There is an extensive discussion on such posets in [17] or [18]. When P isample, we get trP = χ(P )res−1

P .

6.4 The functor ρ: invariants and coinvariants

We have exploited the functor π : G ∝ P → G. Now we turn to study theother functor ρ : G ∝ P → C , where C is a quotient category of G ∝ P .In general situation it seems hard to make group theoretic interpretation of ↑kC

k(G∝P ) and ⇑kCk(G∝P ). However we can do so when we have certain quotient

categories, which are part of some category extension sequences in the sense

of Hoff.An extension E of a category C via a category K is a sequence of functors




K ι

−→E ρ−→C ,

which satisfies certain properties. A sequence K → E C is called an opposite extension if Kop → E op

C op is an extension.

The advantage of considering π : G ∝ P → C which is part of an extension(or opposite extension) is that it enables us to provide a good characterizationof the left (or right) Kan extension. Indeed it is the case for many familiarcategory constructions in representation theory and homotopy theory.

Remark 6.4.1. We emphasize that for any quotient category C of G ∝ P , P is naturally a subcategory of C . Thus for any kC -module, it makes sense toconsider its limits. Indeed if m ∈ kC -mod then both lim←−P

m ∼= lim←−P Resρm andlim−→P

m ∼= lim−→P Resρm are k G-modules.

Lemma 6.4.2. Let K → E → C a sequence of three EI-categories and m ∈kE -mod.

1. Suppose K → E → C is an extension. Then LK ρm ∼= mK, where mK as a functor over C is given by mK(x) = m(x)K(x) ( K(x)-coinvariants of the

kAutE (x)-module m(x)), for any x ∈ Ob C = Ob E = Ob K.2. Suppose K → E → C is an opposite extension. Then RK ρm ∼= mK, where mK as a functor over C is given by mK(x) = m(x)K(x) ( K(x)-invariants of the kAutE (x)-module m(x)), for any x ∈ Ob C = Ob E = Ob K.

In the above lemma, there is another way to express the Kan extensions.Under the same assumptions, they are mK = H0(K;m) and mK = H0(K;m)respectively.

In what follows, we shall apply the above statements to various local cat-egories of G, in combination with the Frobenius reciprocity

(i’) HomkG(M, lim←−P n) ∼= HomkC(LK ρκM , n);

(ii’) HomkG(lim−→P m, N ) ∼= HomkC(m, LK ρκN ).

6.4.1 Orbit categories

Suppose P is a collection of subgroups of G on which G acts by conjugation.Then it forms a G-poset and we can define an orbit category OP as thequotient category of G ∝ P by asking

HomOP (P, Q) = Q\N G(P, Q).

Then we have an extension sequence

S → G ∝ P OP ,

where S is the disjoint union of all objects in P , regarded as a subcategoryof G ∝ P .




When m = κM for some M ∈ kG-mod, (LK ρκM )(P ) = M P for anyP ∈ Ob P . We denote such a kOP -module by M S := (κM )S = LK ρκM . Sincegiving a morphism (g, gP ≤ Q) is the same as giving a group homomorphismP → Q, the conjugation induced by g, there is a natural way to construct

a map M P → M Q, identical to the natural map H0(P ; M ) = k ⊗kP M →k ⊗kQ M = H0(Q; M ). Hence we know how kOP acts on M S .

Proposition 6.4.3. Let M ∈ kG-mod and n ∈ kOP -mod. Then

HomkG(M, lim←−P n) ∼= HomkOP (M S ,n),

where M S is as above.

Corollary 6.4.4. lim←−P n ∼= HomkOP (kGS ,n) and HomkG(k, lim←−P

n) ∼= HomkOP (k, n).

As an example we let H be a subgroup of G and P the subgroups that areconjugate to H . The size of the discrete poset P is G/N G(H ). Note that bothG ∝ P and OP are connected groupoids, the former equivalent to N G(H )

and the latter N G(H )/H . Thus the above isomorphism can be interpreted as

HomkG(M, HomkN G(H )(kG,N )) ∼= HomkN G(H )(M, N )∼= Homk(N G(H )/H )(k(N G(H )/H ) ⊗kN G(H ) M, N )∼= Homk(N G(H )/H )(M H , N ),

where M ∈ kG-mod and N ∈ k(N G(H )/H )-mod.By looking at the special case we have just mentioned, the following state-

ment implies that Proposition 5.2 may be useful in a greater generality.

Lemma 6.4.5. Suppose M is an indecomposable kG-module. Let P be a p-subgroup and N a projective simple k(N G(P )/P )-module. Then HomkN G(P )/P (M P , N ) =0 if and only if there exists a surjective map f : M → N ↑G

N G(P ). In this case

N ↑GN G(P ) is indecomposable and f (M ) ∼= k ↑

GP .

Proof. If HomkN G(P )/P (M P , N ) = 0, then N has to be a direct summand of M P because N is projective simple. By adjunction

HomkG(M, N ↑GN G(P )) ∼= HomkG(M, HomkN G(P )(kG,N )) ∼= Homk(N G(P )/P )(M P , N )

we know there is a non-trivial map f : M → N ↑GN G(P ). If we restrict this map

back to N G(P ), the right side is a semisimple module and thus the image of M ↓N G(P ) contains at least a copy of g ⊗ N for some g ∈ G. It forces the mapf : M N ↑G

N G(P ) to be surjective which implies that N ↑GN G(P )= f (M )

is indecomposable. Furthermore since as a kN G(P )-module N has P as a

vertex, we get f (M ) (N ↓P ↑N G(P )) ↑G. But N ↓N G(P )P is a direct sum of

trivial modules. Simultaneously we obtain f (M ) ∼= k ↑GP .The converse is straightforward by the adjunction.




6.4.2 Brauer categories, fusion and linking systems

Suppose b is a p-block of the group algebra kG and P b is the set of b-Brauerpairs. Then for any G-subposet P ⊂ P b we can introduce the Brauer category

B P as the quotient category of G ∝ P such that

HomBP (P, Q) = HomG∝P (P, Q)/C G(P ).

This gives us an opposite extension, which means that the following sequence

C G → (G ∝ P )op → B opP

is an extension sequence, given that C G is the disjoint union of all C G(P ),P ∈ Ob P . Dual to the extension situation we examined before, now we areable to describe the right Kan extension of modules.

If m = κM for some M ∈ k G-mod, we denote by M C G the kF P -moduleRK ρκM . Since a morphism (g, gP ≤ Q) provides a group homomorphism

P → Q and thus induces an injection cg−1 : C G(Q) → C G(P ), we obtain aninjection M C G(P ) → M C G(Q). This leads to the k B P -action on M C G .

Proposition 6.4.6. Let m ∈ kB P -mod and N ∈ kG-mod. Then

HomkG(lim−→P m, N ) ∼= HomkBP (m, N C G),

where N C G is as above.

As an example we assume b is the principal block b0 and P is the conjugacyclass of a fixed p-subgroup H . Then the discrete poset P has |G/N G(H )|objects. Both G ∝ P and B P are connected groupoids, the former equivalentto N G(H ) and the latter N G(H )/C G(H ). Thus the above isomorphism canbe interpreted as

HomkG(kG ⊗kN G(H ) M, N ) ∼= HomkN G(H )(M, N )∼= Homk(N G(H )/C G(H ))(M, HomkN G(H )(k(N G(H )/C G(H )), N ))∼= Homk(N G(H )/C G(H ))(M, N C G(H )),

where N ∈ kG-mod and M ∈ k(N G(H )/C G(H ))-mod.

Corollary 6.4.7. We have HomkG(lim−→P m, k) ∼= HomkBP (m, k) for any m ∈

kB P -mod.

Suppose b is a nilpotent block. Then, for every b-Brauer pair (H, e),N G(H, e)/C G(H ) is a p-group.

Let B b = B P b . If we fix a maximal object (S, eS ) and take all objects

(Q, eQ) with Q ⊂ S , then the full subcategory of B b, consisting of all theseobjects, is a fusion system, usually written as F b or F S . Note that the in-clusion F b ⊂ B b is an equivalence. There is a general theory of p-local finite




groups introduced by Broto, Levi and Oliver. A p-local finite group consistsof three categories (S, F , L), where S is a p-group, F is an (abstract) fusionsystem–a finite category whose objects are subgroups of S . Let us take thefull subcategory F c ⊂ F consisting of F -centric subgroups of S . (If F = F b,

the F -centric subgroups correspond to the so-called self-centralizing b-Brauerpairs in modular representation theory.) A centric linking system Lc, if it ex-ists, situates in the middle of a sequence

Z → Lc F c

which is an opposite extension. Here Z is the disjoint union of the centers of all F -centric subgroups.

Proposition 6.4.8. Let n ∈ kLc-mod and m ∈ kF c-mod. Then

HomkLc(Resρm, n) ∼= HomkF c(m,nZ ),

where nZ is defined by nZ (P ) = n(P )Z (P ).

Let B cb be the full subcategory of B b for a block b, consisting of self-centralizing b-Brauer pairs. Since F cb naturally identifies with a full sub-category of B cb which induces an equivalence, we similarly can consider anopposite extension

Z → Lcb B cb.

If Lcb exists, then so is Lc

b, and vice versa. Moreover between the corresponding

extensions there exists a natural embedding Lcb → Lc

b inducing an category

equivalence. By taking the larger category Lcb (but essentially the same as

Lcb), we can write down

C G/Z → G ∝ P cb Lcb,

another opposite extension. Here C G/Z is the disjoint union of C G(P )/Z (P )in which P runs over all F -centric subgroups. For the sake of convenience,we introduce a notation C G = C G/Z so that C G(P ) = C G(P )/Z (P ) ∼=P C G(P )/P for each P .

When m = κM for some M ∈ kG-mod, we write the k Lcb-module RK ρκM

as M C G . Given a morphism (g, gP ≤ Q) it induces an injection C G(Q) →C G(P ) thus a morphism M C G(P ) → M C G(Q). Hence we get the k Lc

b-action

on M C G

Proposition 6.4.9. Let m ∈ k(G ∝ P cb )-mod and N ∈ kG-mod. Then

HomkG(lim−→P m, N ) ∼= Homk Lcb

(m, N C G),

where N C G is defined as above.In particular if m = Resρm

for some m ∈ kB cb-mod then




HomkG(lim−→P m, N ) ∼= HomkBcb

(m, N C G),

a special situation of Proposition 6.5.3.

Proof. The first part is a direct consequence of Proposition 5.4. As for the

special case We need to notice that lim−→P Resρm ∼= lim−→P m as kG-modules.Then

HomkG(lim−→P Resρm

, N ) ∼= Homk Lcb(Resρm

, N C G)∼= HomkBcb

(m, (N C G)Z )∼= HomkBcb

(m, N C G).

6.4.3 Puig categories

If we take P A to be the poset of pointed subgroups on an interior G-algebra

A, then analogues to the Brauer category for any G-subposet P ⊂ P A we canintroduce the Puig category LP as a quotient category of G ∝ P such that

HomLP (P γ , Qδ) = HomG∝P (P γ , Qδ)/C G(P ).

Then some results in last section can be obtained accordingly.If m = κM for some M ∈ kG-mod, we denote by M C G the kLP -module

RK ρκM . Since a morphism (g, gP ≤ Q) provides a group homomorphismP → Q and thus induces an injection cg−1 : C G(Q) → C G(P ), we obtain an

injection M C G(P ) → M C G(Q). This leads to the k LP -action on M C G .

Proposition 6.4.10. Let m ∈ kLP -mod and N ∈ kG-mod. Then

HomkG(lim

−→P m, N ) ∼= HomkLP (m, N C G),

where N C G is as above.

6.4.4 Orbit categories of fusion systems

This method also works for the orbit category of a fusion system. Suppose F is an abstract fusion system. The one may define the orbit category OF in asimilar fashion as above by

HomOF (P, Q) = Q\HomF (P, Q).

Again we obtain an extension sequence

S → F OF ,



6.5 Hochschild cohomology 185

where S is the disjoint union of objects in F .

Proposition 6.4.11. If m ∈ kF -mod and n ∈ kOF -mod, then

HomkF (m, Resρn) ∼= HomkOF (mS ,n).

Moreover for F = F cb , HomkG(lim−→P m, k) ∼= HomkOF c

b(mS , k).

Proof. When F = F cb , we have isomorphisms

HomkG(lim−→P m, k) ∼= HomkF (m, k)

∼= HomkOF cb

(mS , k).

6.5 Hochschild cohomology

In this section we continue to demonstrate the similarities between trans-porter categories and groups. The first main assertion is the the Hochschildcohomology ring of a finite transporter category is finitely generated. Fromhere we will establish the finite generation of cohomology. The second is thatwe can construct a transfer map between Hochschild cohomology. Both re-sults are established by passing between Hochschild cohomology and ordinarycohomology of F (G ∝ P ) and G ∝ P . Thus computing various over and un-dercategories for functors from F (G ∝ P ) to appropriate categories are themajor auxiliary statements.

6.5.1 Finite generation

In Section 4.2.2 we have seen that the ordinary cohomology ring of a fi-nite category can be infinitely generated even after quotient out nilpotentelements. Based on the Theorem 5.2.2, the Hochschild cohomology ring of such a finite category algebra is not finitely generated either. So the questionreduces to finding out whether or not Ext∗kCe(kC , kC ) modulo nilpotents isfinitely generated over Ext∗kC(k, k) if the latter is Noetherian. On the firstattempt to solve this question, one may want to check if the Evens-VenkovTheorem on the finite generation of group cohomology could be generalizedto category cohomology. This is not true since Example 4.2.2 (3) implies thatwe can not expect a finite generation property of Ext∗kC(M, N ) over a finitelygenerated Ext∗kC(k, k). Thus we have to look at particular families of finite

categories for the finite generation property. In what follows, we show finitetransporter categories constructed over a finite group are very close to whatwe expect.




To examine the finite generation of Hochschild cohomology ring, we usethe isomorphism

Ext∗k(G∝P )e(k(G ∝ P ), k(G ∝ P )) ∼= Ext∗kF (G∝P )(k, Res∇(k(G ∝ P ))),

because it allows us to use the Grothendieck spectral sequence. Let us intro-duce a functor π = π t

F (G ∝ P ) t

π

G ∝ P

π

G .

When we investigated the finite generation of H∗(G ∝ P , k) we used theGrothendieck spectral sequence. Based on the fact that •\π P , we obtainedthe finite generation. Naturally we want to look at the Grothendieck spectralsequence for

π. In order to understand the spectral sequence, we have to

know the undercategory •\π. Since t induces a homotopy equivalence, theundercategory •\π should be similar to •\π. If we take P = • in the abovediagram, then π is a canonical isomorphism and •\π •. By contrast π canbe identified with tG : F (G) → G and we also have •\π = •\tG •.

Lemma 6.5.1. Let tG : F (G) → G be the target functor. Then we have twoisomorphic categories

1. •\π ∼= F (P ) × •\tG.2. π/• ∼= F (P ) × tG/•.3. •\tG

∼= tG/• are equivalent to •.

Proof. We only prove Parts 1 and 3. First we prove (1). The objects in •\

π are

(•, [(g,gx ≤ y )]). A morphism from (h, [(g,gx ≤ y )]) to (h, [(g, gx ≤ y )])

is a morphism

((l1, l1y ≤ y ), (l2, l2x ≤ x)) : [(g,gx ≤ y)] → [(g, gx ≤ y )]

such that l1h = h and g = l1gl2. Thus from one object in •\π to an-other there is at most one morphism. It implies this finite undercategory isequivalent to a finite poset. Furthermore we notice that (h, [(g,gx ≤ y)]) isisomorphic to (e, [(h−1g, (h−1g)x ≤ h−1y)]). But [(h−1g, (h−1g)x ≤ h−1y)] isisomorphic to [(e, e(h−1gx) ≤ h−1y)] in F (G ∝ P ). Now we define a functor•\π → F (P ) × •\tG by

(h, [(g,gx ≤ y)]) → [h−1gx ≤ h−1y] × (h, [g]).

Its inverse is given by

[x ≤ y] × (h, [g]) → (h, [(g, g(g−1hx) ≤ hy)]).




The isomorphism in (3) is easy to write down as (h, [g]) → ([g], h−1).Furthermore, any object ([g], h) ∈ Ob(tG/•) is isomorphic to ([e], e) whichhas only one endomorphism.

Now we can state the consequence of Lemma 6.5.1 (1). It is similar to themain result in Section 6.3.2.

Proposition 6.5.2. Let G be a finite group and P a finite G-poset. Then for any M ∈ kF (G ∝ P )-mod, Ext∗kF (G∝P )(k, M ) becomes a finitely gen-erated Ext∗kG(k, k)- and Ext∗kF (G∝P )(k, k)-module. Especially the Hochschild cohomology ring Ext∗k(G∝P )e(k(G ∝ P ), k(G ∝ P )) is finitely generated.

Proof. We apply the Grothendieck spectral sequence to π. Since •\π is equiv-alent to a finite poset F (P ), Hj (•\π, M ) vanishes for all j larger than a chosenpositive integer. Consequently the E 2 page of the spectral sequence only hasfinitely many non-zero rows in the first quadrant, and thus we have the finitegeneration of Ext∗kF (G∝P )(k, M ).

The preceding result enables us to prove a finite generation theorem. Itwill be the foundation for developing a support variety theory over the ringExt∗k(G∝P )(k, k).

Theorem 6.5.3. Suppose M, N are two k(G ∝ P )-modules. Then the module Ext∗k(G∝P )(M, N ) is finitely generated over Ext∗k(G∝P )(k, k).

Proof. By Theorem 5.2.11,

Ext∗k(G∝P )(M, N ) ∼= Ext∗kF (G∝P )(k, Res∇Homk(M, N )).

Hence by Proposition 6.5.2, Ext∗kF (G∝P )(k, Res∇Homk(M, N )) is finitely gen-erated over Ext∗kF (G∝P )(k, k) ∼= Ext∗k(G∝P )(k, k). We are done.

If P = •, we get the usual assertion that Ext∗kGe(kG,kG) is finitely gen-erated over Ext∗kG(k, k). If P = S b as in Example 6.1.?, we have Ext∗kG(k, k)acting on Ext∗k(G∝S b)e(k(G ∝ S b), k(G ∝ S b)) via Ext∗k(G∝S b)(k, k). Espe-cially when b = b0, S b ∼= S p and Ext∗k(G∝S p)(k, k) ∼= Ext∗kG(k, k).

Corollary 6.5.4. If k is a field with positive characteristic p |G|, and b is

a p-block, then, for any full subcategory Tr ⊂ Trb(G) whose objects are closed under G-conjugation, Ext∗kTre(kTr, kTr) is a finitely generated algebra.

Given the principal block b0 of a group algebra kG, we have a fusion sys-tem F b0 = F p over a fixed Sylow p-subgroup S . As we mentioned earlier,there exists a centric linking system Lb0 = Lc

p which is determined by thefull subcategory Trc

p(G)≤S of the transporter category Tr p(G), consisting of

all p-centric subgroups contained in S [10]. In fact Lc p is a quotient categoryof Trc

p(G)≤S by some p-groups. In other words, if one looks at the canoni-cal functor π : Trc

p(G)≤S → Lc p, each undercategory has the property such




that it has a minimal object whose automorphism group is p and, if oneregards this p-automorphism group as a subcategory, the left Kan extensionalong the inclusion is exact. Furthermore the left Kan extension of the triv-ial group module is the trivial module of the undercategory. This implies,

by an Eckmann-Shapiro type result, the cohomology of each undercategorycan be reduced to the cohomology of the automorphism group of the above-specified minimal object in it. Consequently, the mod- p cohomology of eachundercategory of π with arbitary coefficients vanishes in positive degrees. Tosummarize, since Trc

p(G)≤S is equivalent to Trc p(G) and the Grothendieck

spectral sequence for π collapses, we have an isomorphism

Ext∗kTrcp(G)(k, V ) ∼= Ext∗kTrcp(G)≤S(k, V ) ∼= Ext∗kLcp(k,RK πV ),

where RK πV = H0(?\π; V ) ∼= lim←−?\πV is the right Kan extension along

π of V . It is similar to [10, Lemma 1.3 (iii)] in which right modules areconsidered and thus the left Kan extension is applied. Especially, the functorsG ← Trc

p(G) ← Trc p(G)≤S → Lc

p induce isomorphisms of mod- p ordinary

cohomology rings.

Proposition 6.5.5. Let Lc p be the centric linking system associated to the

principal block of a finite group algebra kG. Then Ext∗kLcp(k,RK πV ) is finitely

generated as an Ext∗kLcp(k, k)-module.

However this is still far from understanding the finite generation of theHochschild cohomology ring Ext∗k(Lcp)e(kLc

p, kLc p) ∼= Ext∗kF (Lcp)(k, Resτ (kLc

p)).

6.5.2 Transfer for Hochschild cohomology

We end this chapter with a transfer map between Hochschild cohomology.Consider the following commutative diagram of functors

(G ∝ P )e πe Ge

F (P )

t

F (G ∝ P )

t

F (π)

∇

F (G)

t

∇

P G ∝ P π G

Recall that in Section 6.3.3, based on the bottom row, we were able to es-

tablish a transfer map between ordinary cohomology. Since the three targetfunctors all induce homotopy equivalences, the middle row may as well givea transfer map Ext∗kF (G∝P )(−, −) → Ext∗kF (G)(−, −), for suitable modules.






[h] = F (π)([(h,hx ≤ y)])

(h1,h2)

(v1,v2)=F (π)((v1,v1y≤y),(v2,v2x≤x))

[g]

[h] = F (π)([(h, hx ≤ y)])

(h1,h2)

It implies various identities: g = h1hh2 = h1hh2, h1 = h1v1 and h2 = v2h2.We define a functor F (π)/[g] → F (P ) × IdF (G)/[g] such that on objects

([(h,hx ≤ y)], (h1, h2)) → ([h−12 x ≤ g−1h1y], ([h], (h1, h2))),

and on morphisms ((v1, v1y ≤ y), (v2, v2x ≤ x)) → ((e, e), (v1, v2)), be-

cause v1y ≤ y implies g−1h1y ≤ g−1h1y while v2x ≤ x implies h2−1x ≤

h−12 x. The inverse of this functor is defined by ([x ≤ y], ([h], (h1, h2))) →

([(h, h(h2x) ≤ h

−1

1 gy)], (h1, h2)) on objects, and on morphisms is defined by((e, e), (v1, v2)) → ((v1, v1h−11 gy ≤ h1

−1gy), (v2, v2h2x ≤ h2x)). Thus weobtain an isomorphism

F (π)/[g] ∼= F (P )[g] × IdF (G)/[g],

where F (P )[g] denotes a copy of F (P ) indexed by the object [g]. If (l1, l2) :[g] → [g] is a morphism in F (G). Then it induces a functor F (π)/[g] →F (π)/[g] given by

([(h,hx ≤ y)], (h1, h2)) → ([(h,hx ≤ y)], (l1h1, h2l2))

and

((v1, v1y ≤ y), (v2, v2x ≤ x)) → ((v1, v1y ≤ y ), (v2, v2x ≤ x))

Using the isomorphisms F (π)/[g] ∼= F (P )[g] × IdF (G)/[g] and F (π)/[g] ∼=F (P )[g] × IdF (G)/[g], one can see it induces an isomorphism F (P )[g] →F (P )[g].

Finally since G is isomorphic to the automorphism group of [e] in thegroupoid F (G), we have an equivalence F (G) G. Thus C∗(F (P )[e] ×IdF (G)/[e]) C∗(F (P )) ⊗ C∗(IdF (G)/[e]) is a complex of projective kG-

modules. Furthermore because C∗(IdF (G)/[e]) = B F (G)∗ ([e]), it has to be a

projective resolution of the trivial kG-module k . Hence we get the chain ho-motopy equivalence as stated.

This lemma allows us to describe the transfer map in Theorem 5.2.2 in

terms of factorization categories:

trP : Ext∗kF (G∝P )(k, k) → Ext∗kF (G)(k, k).




Indeed we have a commutative diagram

F (G ∝ P )F (π)

t

F (G)

t G ∝ P π

G

Thus we get a commutative diagram of cochain complexes

HomkF (G∝P )(B F (G∝P )∗ , k)

∼=

∼=

HomkF (G)(LK F (π)B F (G∝P )∗ , k)

∼=

Homk(G∝P )(LK tB F (G∝P )∗ , k) ∼=

HomkG(LK πB F (G∝P )∗ , k)

Homk(G∝P )(B G∝P ∗ , k) HomkG(C∗(P ) ⊗ B G∗ , k)

Homk(G∝P )(B G∝P ∗ , k) HomkG(B G∗ , k)

HomkG(LK tB F (G)∗ , k)

∼=

HomkF (G)(B F (G), k)

The isomorphisms are given by adjunctions and the chain homotopy equiv-alences are induced by changing projective resolutions. From the upper leftcorner to the lower right corner is the transfer that we want to describe. Itfactors through the lowest square, including the chain map which gives riseto the transfer constructed in Theorem 6.3.21. Note that F (G) is a groupoidand G is a skeleton of F (G). Comparing with [45, Section 4], F (G) plays therole of ∆G if one identifies Ge with G × G.

Theorem 6.5.8. There exists a map

htrP : Ext∗k(G∝P )e(k(G ∝ P ), k(G ∝ P )) → Ext∗kGe(kG,kG).

Proof. Using Lemma 6.5.6 we have cochain maps




HomkF (G∝P )(B G∝P ∗ , ResF (π)Res∇kG) ∼= HomkF (G)(LK F (π)B G∝P ∗ , Res∇kG)∼= HomkF (G)(C∗(F (π)/−), Res∇kG)∼= HomkG(C∗(F (π)/[e]), kG) HomkG(C∗(F (P )) ⊗ B G∗ , kG)

→ HomkG(B G

∗ , kG)

Here the module kG in HomkG(−, kG) is acted by kG via conjugations. Thelast map is induced by k → C∗(F (P )), a chain map constructed in the sameway as the one in the proof of Theorem 6.3.21 k → C∗(P ). Indeed sincet : F (P ) → P is a G-functor, it induces an chain homotopy equivalenceC∗(F (P )) C∗(P ).

By Lemma 6.5.5 we also have a chain map

HomkF (G∝P )(B G∝P ∗ , Res∇kF (G ∝ P )) → HomkF (G∝P )(B G∝P ∗ , ResF (π)Res∇kG)

induced by the inclusion Res∇kF (G ∝ P ) → ResF (π)Res∇kG. Hence alto-gether we obtain a chain map

HomkF (G∝P )(B G∝P ∗ , Res∇kF (G ∝ P )) → HomkG(B G∗ , kG).

Passing to cohomology we get a map between Hochschild cohomology byTheorem 5.2.2

Ext∗kF (G∝P )(k, Res∇kF (G ∝ P )) htrP

∼=

Ext∗kG(k,kG)

∼=

Ext∗k(G∝P )e(kF (G ∝ P ), kF (G ∝ P ))

htrP

Ext∗kGe(kG,kG)

The above map deserves to be called a transfer since k Res∇kF (G ∝ P ),k kG and k

Res∇kF (G). In the construction of htrP if we replace thesecond module in all Hom−(−, −) and Ext∗−(−, −) by k then it is exactly thetransfer created in Theorem 6.3.21 and reinterpreted before Theorem 6.5.7.In other words, we have a commutative diagram




Ext∗k(G∝P )(k, k) trP

∼=

Ext∗kG(k, k)

∼=

Ext

∗

kF (G∝P )(k, k)

trP

injection

Ext

∗

kF (G)(k, k)injection

Ext∗kF (G∝P )(k, Res∇kF (G ∝ P ))

∼=

Ext∗kF (G)(k, Res∇kF (G))

∼=

Ext∗k(G∝P )e(k(G ∝ P ), k(G ∝ P ))

htrP

Ext∗kGe(kG,kG)






References 195

References

1. J. Alperin, Local Representaion Theory , Cambridge Studies in Adv. Math. 11,Cambridge University Press (1986).

2. M. Aschbacher, R. Kessar, B. Oliver, Fusion Systems in Algebra and Topology ,in print 2011.3. H.-J. Baues, G. Wirsching, Cohomology of small categories, J. Pure Appl. Algebra

38 (1985) 187-211.4. J. C. Becker, D. H. Gottlieb, Transfer maps for fibrations and duality , Compositio

Math. 33 (1976), 107-133.5. D. Benson, Representations and Cohomology I , Cambridge Studies in Adv. Math.

30, Cambridge University Press (1998).6. D. Benson, Representations and Cohomology II , Cambridge Studies in Adv.

Math. 31, Cambridge University Press (1998).7. S. Bouc, Homologie de certains ensembles ordonnes, Comptes Rendues Acad.

Sci. Paris 299, Serie I (1984), 49-52.8. A. Bousfield, E. Friedlander,9. A. Bousfield, D. Kan, Homotopy Limites, Completions and Localizations, Lecture

Notes in Math 304 (1972) Springer.10. C. Broto, R. Levi, B. Oliver, Homotopy equivalences of p-completed classifying

spaces of finite groups, Invent. Math.11. C. Broto, R. Levi, B. Oliver, The homotopy theory of fusion systems, J. Amer.

Math. Soc. 16 (2003) 779-856.12. K. Brown, Cohomology of Groups, GTM87, Springer 1982.13. L. Budach, B. Graw, C. Meinel, S. Waack, Algebraic and Topological Properties

of Finite Partially Ordered Sets, Teubner-Texte zur Mathematik, Band 109,Teubner Publishing House, 1987.

14. H. Cartan, S. Eilenberg, Homological Algebra , Princeton University Press 1956.15. T. tom Dieck, Transformation groups, de Gruyter Studies in Math. 8, Walter de

Gruyter, 1987.16. A. Dold, D. Puppe, Homologie nicht-additiver funktoren , Ann. Inst. Fourier

(Grenoble) 11 (1961) 201-312.17. W. G. Dwyer, Homology decompositions for classifying spaces of finite groups,

Topology 36 (1997), 783-804.18. W. G. Dwyer, H.-W. Henn, Homotopy Theoretic Methods in Group Cohomology

(Birkhauser 2001).

19. W. G. Dwyer, D. Kan, A classification theorem for diagrams of simplicial sets,Topology, 23 (1984) 139-155.

20. W. G. Dwyer, D. Kan,21. W. G. Dwyer, C. W. Wilkerson, Homotopy fixed-point methods for Lie groups

and finite loop spaces, Ann. of Math. (2) 139 (1994), 395-442.22. S. Eilenberg, Homological dimension and syzygies, Ann. of Math., 64 (1956)

328-336.23. P. Etingof, V. Ostrik, Finite tensor categories, Mosc. Math. J. 4 (2004) 627-654,

782-783.24. L. Evens, The cohomology ring of a finite group, Trans. Amer. Math. Soc., 101,

(1961) 224-239.25. E. Friedlander, V. Franjou, T. Pirashivili, L. Schwartz, Functor Homology The-

ory , Panoramas et Syntheses 16, Soc. Math. France 2003.26. R. Fritsch, M. Golasinski, Topological, simplicial and categorical joins, Arch.

Math. (Basel) 82 (2004) 468-480.27. P. Gabriel, Des Categories Abelinnes, Bull. Soc. Math. France 90 (1962) 323-448.28. P. Gabriel, M. Zisman, Calculus of Fractions and Homotopy Theory , Ergebnisse

der Mathematik und ihrer Grenzgebiete, New Series, 35 Springer-Verlag NewYork (1967).




29. A. I. Generalov, Relative homological algebra, cohomology of categories, posets

and coalgebras, Handbook of Algebra, Vol 1, M. Hazewinkel ed., North-Holland,1996.

30. M. Gerstenhaber, S. D. Schack, Simplicial cohomology is Hochschild cohomology ,J. Pure and App. Algebra, 30 (1983) 143-156.

31. J. Grodal, Higher limits via subgroup complexes, Ann. of Math. 155 (2002) 405-457.

32. K. Gruenberg, Cohomological topics in group theory , Lecture Notes in Math.143, Springer-Verlag, 1970.

33. D. Happel, Hochschild cohomology of finite-dimensional algebras, Lecture Notesin Math. 1404 (1989) 108-126.

34. A. Hatcher, Algebraic Topology , Cambridge University Press 2002.35. P. Hilton, U. Stammbach, A Course in Homological Algebra (Second edition),

GTM 4 Springer (1997).36. G. Hoff, Cohomologies et extensions de categories, Math. Scand. 74 (1994) 191-

207.37. J. Hollender, R. M. Vogt, Modules of topological spaces, applications to homotopy

limits and E ∞ structures, Arch. Math. 59, 115-129 (1992).38. M. Hovey, J. Palmieri, N. Strickland, Axiomatic Stable Homotopy Theory , Amer.

Math. Soc. Memoirs 610 (1997).

39. K. Igusa, D. Zacharia, On the cohomology of incidence algebras of partially or-dered sets, Comm. Algebra, 18 (1990) 873-887.40. S. Jackowski, J. McClure, Homotopy decomposition of classifying spaces via ele-

mentary abelian subgroups, Topology 31 (1992), 113-132.41. S. Jackowski, J. McClure, B. Oliver, Homotopy decomposition of self-maps of

BG via G-actions, I and II , Ann. of Math. 135 (1992) 183-226; 227-270.42. S. Jackowski, J. Slominska, G-functors, G-posets and homotopy decompositions

of G-spaces, Fundamenta Mathematicae 169 (2001) 249-287.43. R. Knorr, G. Robinson, Some remarks on a conjecture of Alperin , J. London

Math. Soc. (2) 39 (1989), 48-60.44. N. Kuhn, The generic representation theory of finite fields: a survey of basic

structure , Infinite Length Modules, Bielefeld 1998.45. M. Linckelmann, Transfer in Hochschild cohomology of blocks of finite groups,

Alg. Rep. Theo. 2 (1999) 107-135.46. M. Linckelmann, Varieties in block theory , J. Algebra 215 (1999) 460-480.47. M. Linckelmann, Fusion category algebras, J. Algebra 277 (2004) 222-235.48. M. Linckelmann, Alperin’s conjecture in terms of equivariant Bredon cohomology ,

Math. Z. 250 (2005), 495-513.49. M. Linckelmann, Hochschild and block cohomology varieties are isomorphic ,

Preprint 2009.50. M. Linckelmann, On inverse categories and transfer in cohomology , Preprint

2010.51. W. Luck, Transformation groups and algebraic K-Theory , Lecture Notes in Math.

1408, Springer-Verlag, 1989.52. S. Mac Lane, Categories for the Working Mathematician (Second edition), GTM

5 (Springer 1998).53. S. Mac Lane, Homology , Classics in Mathematics, Springer-Verlag, 1995.54. J. P. May, Simplicial Objects in Algebraic Topology , Chicago University Press

1967.55. J. McCleary, A User’s Guide to Spectral Sequences Cambridge Studies in Adv.

Math. 58, Cambridge University Press 2000.

56. B. Mitchell, Rings with several objects, Adv. Math. 8 (1972) 1-161.57. B. Oliver, Higher limits via Steinberg representations, Comm. Algebra 22 (1994)

1381-1393.



References 197

58. B. Oliver, J. Ventura, Extensions of linking systems with p-group kernel , Math.Ann. 338 (2007) 983-1043.

59. R. S. Pierce, Associative Algebras, GTM 88, Springer-Verlag New York 1982.60. L. Puig, Structure locale dans les groupes finis, Bull. Soc. Math. France Suppl.

Mem. 47 (1976).

61. D. Quillen, Higher algebraic K -theory I , in: Lecture Notes in Math. 341(Springer-Verlag, 1973), pp. 85-147.

62. D. Quillen, Homotopy properties of the poset of nontrivial p-subgroups of a group,Adv. Math. 28 (1978), 101-128.

63. Reedy,64. V. Reiner, P. J. Webb, Combinatorics of the bar resolution in group cohomology ,

J. Pure Appl. Algebra 190 (2004), no. 1-3, 291-327.65. M. A. Ronan, S. D. Smith, Sheaves on buildings and modular representations of

Chevalley groups, J. Algebra 96 (1985), 319-346.66. J.-E. Roos, Sur les foncteurs derives de lim

←−, Applications, Compte Rendue Acad.

Sci. Paris 252 (1961) 3702-3704.67. S. Siegel, S. Witherspoon, The Hochschild cohomology ring of a groups algebra ,

Proc. London Math. Soc. 79 (1999) 131-157.68. J. Slominska, Decompositions of categories over posets and cohomology of cate-

gories, Manuscripta Math. 104 (2001) 21-38.

69. S. D. Smith, Subgroup Complexes, Book draft.70. O. Solberg, Support varieties for modules and complexes, Comtep. Math. 406,Amer. Math. Soc. (2006) 239-270.

71. M. Suarez-Alvarez, The Hilton-Eckmann argument for the anti-commutativity of

cup product , Proc. Amer. Math. Soc. 132 (2004) 2241-2246.72. M. Sweedler, Hopf Algebras, Math. Lecture Notes, No. 44, Addison-Wesley Pub

Co 1969.73. D. Swenson, The Steinberg Complex of an Arbitrary Finite Group in Arbitrary

Positive Characteristic , PhD thesis, University of Minnesota 2009.74. P. Symonds, The Bredon cohomology of subgroup complexes, J. Pure Appl. Al-

gebra 199 (2005), no. 1-3, 261-298.75. J. Thevenaz, G-algebras and Modular Representation Theory , Oxford University

Press 1995.76. J. Thevenaz, P. J. Webb, Homotopy equivalence of posets with a group action ,

J. Combin. Theory Ser. A 56 (1991), 173-181.77. R. W. Thomason, Homotopy colimits in the category of small categories, Math.

Proc. Cambridge Philoso. Soc. 85 (1979), 91-109.78. Tornehave,79. B. B. Venkov, Cohomology algebras for some classifying spaces, Dokl. Akad.

Nauk SSSR vol. 127 (1959) 943-944.80. P. J. Webb, Supgroup complexes, Arcata Conf. on Representations of Finite

Groups, Proc. Symp. Pure Math. 47 (Part I), Amer. Math. Soc. 1987, 349-365.81. P. J. Webb, A local method in group cohomology , Comment. Math. Helvetivi 62

(1987), 135-167.82. P. J. Webb, An introduction to the representations and cohomology of categories,

in: Group Representation Theory , (EPFL Press 2007), pp. 149-173.83. P. J. Webb, Standard stratifications of EI categories and Alperin’s weight con-

jecture , J. Algebra 320 (2008) 4073-4091.84. C. Weible, An Introduction to Homological Algebra , Cambridge Studies in Adv.

Math. 38, Cambridge University Press 1994.85. V. Welker, G. Ziegler, R. Zivaljevic, Homotopy colimits-comparison lemmas for

combinatorial applications, J. Reine Angew. Math. 509 (1999) 117-149.86. F. Xu, Representations of small categories and their applications, J. Algebra 317

(2007) 153-183.




87. F. Xu, Hochschild and ordinary cohomology rings of small categories, Adv. Math.219 (2008) 1872-1893.

88. F. Xu, Tensor structure on kC-mod and cohomology , to appear in Skye 2009Conference Proceedings.

89. F. Xu, On local categories of finite groups, Preprint 2010.

90. F. Xu, Becker-Gottlieb transfer for Hochschild cohomology , Preprint 2011.91. F. Xu, Support varieties for transporter category algebras, Preprint 2011.



Index

G-category, 154R-acyclic category, 35

abelian category, 2acyclic, 25, 104, 117additive, 2additive functor, 4adjunct, 6Alexander-Whitney map, 45, 104antipode, 72atomic module, 82augmentation ideal, 105augmentation map, 70, 72, 165

bar resolution, 99, 130bicomplex, 26bifunctor, 6bisimplicial set, 57

Borel construction, 158boundary, 24bounded, 24bounded above, 24bounded below, 24bounded double complex, 26Brauer category, 159

category, 1category algebra, 63category of factorizations, 50Cayley graph, 3chain complex, 23chain complex from a simplicial module,

34chain homotopy, 25chain homotopy equivalence, 25chain isomorphism, 24chain map, 24

change of base, 75classifying space, 42co-associativity, 71

coboundary, 24cochain complex, 24cochain map, 24cocommutativity, 71cocomplete, 12cocycle, 24cofibred category, 49cohomology of a small category, 35cohomology spectral sequence, 120cohomology with coefficients in a

functor, 98coideal of a poset, 77coideal of an EI category, 82coinduction, 20collection of subgroups, 107complete, 12comultiplication, 68, 71connected category, 64constant functor, 11, 65constant value module, 160contractible, 25, 41, 44contravariant functor, 3coproduct, 2, 10cosimplicial identities, 32counit, 7, 71counitary property, 71countable simplicial set, 44covariant functor, 3cross product, 45cup product, 46cycle, 24

degeneracy map, 30, 32degenerate element, 35

199



200 Index

degree, 24derivation, 105diagonal, 57diagonal approximation map, 103, 104diagonal functor, 46

diagonal map, 46diagram of a functor, 9differential, 24direct limit, 9Dold-Kan Correspondence, 35, 57double complex, 26dual complex, 28dual functor, 4dual module, 66

Eckmann-Shapiro type lemma, 117EI category, 54, 82EI-category, 77Eilenberg-Zilber Theorem, 45enough injectives, 4

enough projectives, 4enveloping category, 51enveloping category algebra, 66epimorphism, 2equivalent categories, 5equivalent extensions of categories, 108equivariant cohomology, 170Euler characteristic, 155evaluation map, 28extension of a category, 105extension of module, 94

face, 29face map, 30, 31faithful functor, 4

fibre of a functor, 48fibred category, 49finite category, 3finitely cocomplete, 12finitely complete, 12first quadrant double complex, 26five term exact sequence, 125free category, 67Frobenius Reciprocity, 160full functor, 4full subcategory, 3function object, 70functor category, 5fusion system, 158

geometric realization, 42graded commutative, 47Grothendieck construction, 53group algebra, 67

group cohomology ring, 47groupoid, 2

higher left Kan extensions, 120higher right Kan extensions, 120

Hochschild cohomology ring, 131Hochschild complex, 129homology of a chain complex, 24homology of a cochain complex, 24homology of a small category, 34homology spectral sequence, 120homology with coefficients in a functor,

98homotopy, 44homotopy colimit, 58homotopy limit, 58homotopy of simplicial sets, 40homotopy pushout, 58

ideal of a poset, 77

ideal of an EI category, 82idempotent, 83identity morphism, 1incidence algebra, 67indecomposable module, 83index category, 5induction, 20initial object, 2injective hull, 87injective object, 4internal hom, 70inverse category, 87inverse limit, 9invertible morphism, 2isomorphic objects, 2isomorphism, 2isomorphism class of an object, 2isomorphism of simplicial objects, 32

join, 36

Kan extension, 18Kan extension of modules, 76kernel of a morphism, 2

left adjoint functor, 6locally finite, 44

Mobius category, 73maximal ideal, 84maximal object, 82minimal injective resolution, 91minimal object, 82minimal projective resolution, 91



Index 201

Mitchell’s theorem, 65monomorphism, 2Morita equivalence, 75, 87morphism, 1

natural equivalence, 5natural transformation, 4nerve of a small category, 33nilpotent element, 84nilpotent ideal, 84normalized chain complex, 35

object, 1opposite category, 4opposite category algebra, 66opposite extension of a category, 106

i f 18

right adjoint functor, 6

self-injective algebra, 87semi-simple module, 84semi-simplicial object, 41

simple module, 84simplex, 32simplex category, 34simplicial diagonal map, 46simplicial homotopy of simplicial maps,

40simplicial identities, 32simplicial map, 37simplicial module, 34simplicial object, 32simplicial replacement, 58

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