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THE MOTHER OF ALL ISOMORPHISM CONJECTURES

VIA DG CATEGORIES AND DERIVATORS

PAUL BALMER AND GONCALO TABUADA

Abstract. We describe a fundamental additive functor Efund on the orbitcategory of a group. We prove that any isomorphism conjecture valid for Efund

also holds for all additive functors, like K-theory, (topological) Hochschild orcyclic homology, etc. Finally, we reduce this universal isomorphism conjectureto K-theoretic ones, at the price of introducing some coefficients.

Contents

1. Introduction 12. Preliminaries 63. Assembly and isomorphism conjectures 84. Additive invariants of dg categories 105. The additive motivator 126. Universal additive assembly property 16Appendix A. Derivators, Stabilization and Localization 19References 22

1. Introduction

The Farrell-Jones so-called isomorphism conjectures in K- and L-theory and theBaum-Connes conjecture are important driving forces in contemporary mathemat-ical research. They are well known to imply many other conjectures, such as somedue to Bass, Borel, Kaplansky, Novikov, and more. See a survey in [17].

Given a discrete group G, the Farrell-Jones conjectures predict the value ofalgebraicK- and L-theory of the group ring R[G] in terms of the values on virtuallycyclic subgroups of G (see 3.1.3). Here R is some base ring (with involution), whichwe fix for the whole article and which can be taken to be Z or C, for instance.Similarly, the Baum-Connes conjecture predicts the value of topological K-theoryof the reduced C∗-algebra C∗

red(G) in terms of the values on finite subgroups.In their insightful article [6], Davis and Luck proposed the following unified

setting for all such isomorphism conjectures : Let F be a family of subgroups of G

Date: December 11, 2018.1991 Mathematics Subject Classification. 18E30, 19D55.Key words and phrases. Farrell-Jones conjectures, assembly map, algebraic K-theory, dg cat-

egories, derivators.The first author’s research is supported by NSF grant 0654397. The second author’s research

is supported by CMA-FCT/UNL.

1

http://arxiv.org/abs/0810.2099v3

2 PAUL BALMER AND GONCALO TABUADA

and let E : Or(G) → Spt be a functor from the orbit category of G (Def. 3.1.1) tospectra. The (E,F , G)-assembly map is the natural map of spectra

(1.0.1) hocolimOr(G,F)

E −→ hocolimOr(G)

E = E(G) ,

where Or(G,F) is the orbit category restricted on F in the obvious sense (Def. 3.1.4).

1.0.2. Definition. We say that the functor E has the F-assembly property (on G)when the (E,F , G)-assembly map given in (1.0.1) is a stable weak homotopy equiv-alence, i.e. when it induces an isomorphism on stable homotopy groups. Note thatfor random E, F and G, this assembly property has essentially no chance to hold.When we speak of the (E,F , G)-isomorphism conjecture, we refer to the expressedhope that this property holds for a particular choice of E, F and G.

Davis and Luck proved in [6] that the Farrell-Jones conjecture inK-theory for thegroup G is equivalent to the (Kalg(R[−]), VC,G)-isomorphism conjecture, whereVC is the family of virtually cyclic subgroups of G; and similarly for L-theory,mutatis mutandis. They also proved that the Baum-Connes conjecture is equivalentto the (Ktop(C∗

red(−)), Fin,G)-isomorphism conjecture, where Fin is the family offinite subgroups. (First of all, one must properly define those functors on orbitcategories and this can sometimes be non-trivial.)

In addition to the above conjectures, the literature contains many variations onthe theme, replacing the K-theory functors by other functors E defined on the orbitcategory. See for instance the Farrell-Jones conjectures for topological Hochschildhomology (THH) or for topological cyclic homology (THC), as stated by Luck [16,6.5].

It is worth observing that Definition 1.0.2 does not rely very heavily on thechoice of spectra Spt as the target category for the functor E. Indeed, in order todefine the assembly map (1.0.1), it is enough that E : Or(G) → M takes values ina category M where we can speak of homotopy colimits. In particular, M couldbe any cofibrantly generated model category, as proposed in [1]. In a somewhatmore elementary genre, we can also consider usual Hochschild homology (HH) andcyclic homology (HC), now taking values in suitable categories of complexes; seemore in 4.4 and 4.5 below.

This flexibility generates a profusion of potential isomorphism conjectures for allsorts of functors on the orbit category :

(1.0.3)

Spt

M1

Or(G)

Kalg

33hhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhh

HH 11ccccccccccccccccccccccccccccccccc HC //

THH--[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[

THC

++VVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVV M2

M3

. . . M4

Of course, some restraint must be exerted here. A random functor E : Or(G) → Mhas little chances of featuring a nice F -assembly property, that is, for F smallenough, like VC or Fin. We are therefore led to focus on a special class of functors,in the spirit of the above ones, which we shall call additive functors. We precisely

THE MOTHER OF ALL ISOMORPHISM CONJECTURES 3

define what we mean by additive functor in Definition 6.1.1 but be it enough forthis introduction to know that, except maybe for topological K-theory (see Re-mark 6.1.3), all the above functors are examples of additive functors, and so aremost of their obvious variants : connective K-theory, negative or periodic cyclichomology, etc.

Let us make clear that we do not seek here a new class of groups for which some ofthese conjectures would hold. We are rather interested in the general organizationand the deeper properties behind this somewhat exuberant herd of conjectures.

Using the above simple but crucial idea of letting the target categoryM float, wecan now make sense of the following result, which is an application of the techniquesdeveloped by the second author in his thesis [23].

Theorem. Let G be a group. There exists a fundamental additive functor

Efund : Or(G) −→ Mfund

through which all additive functors factor.

The precise statement (Theorem 6.1.6) requires the use of derivators but, at thelevel of sophistication of this introduction, this theorem essentially means that forany additive functor E : Or(G) → M there exists a functor E : Mfund → M suchthat E ≃ E Efund. In other words, we can comb the skein (1.0.3) from the left andisolate a fundamental additive functor

(1.0.4)

Spt

M1

Or(G)Efund

// Mfund

Kalg77ooooooooooooooooo

HH 33gggggggggggggggg HC //

THH++WWWWWWWWWWWWWWWW

THC ''OOOOOOOOOOOOOOOOOM2

M3

M4

Moreover, a key point in this construction is that the remaining functors E willcommute with homotopy colimits. This means that they will preserve any assemblyproperty that Efund might enjoy (see Cor. 6.1.8) :

Corollary. Let G be a group and let F be a family of subgroups. If the fundamentaladditive functor Efund has the F-assembly property, then so do all additive functors.

In good logic, we are led to consider the mother of all isomorphism conjectures :

1.0.5.Mamma Conjecture. The (Efund,VC,G)-isomorphism conjecture, namely :The fundamental additive invariant Efund : Or(G) → Mfund has the VC-assemblyproperty on G, where VC is the family of virtually cyclic subgroups of G.

The above corollary says that Conjecture 1.0.5 implies all additive isomorphismconjectures on the market. After this moment of exaltation, let us make clearthat the choice of virtually cyclic subgroups in this conjecture is merely borrowedfrom Farrell-Jones and that another family F might be preferable. There willunquestionably be some family F = F(G) for which Efund has the F -assemblyproperty (e.g. F = all subgroups). The main result is that once this is achieved


for some family F , then all additive functors will automatically inherit the sameF -assembly property.

The second objective of the paper is to reduce the F -assembly property forEfund, whose importance should now be clear, to F -assembly for more down-to-earth functors. For instance, does the F -assembly for E = Kalg(R[−]) imply thatfor Efund? In other words, is the Farrell-Jones conjecture already the Mamma?This, we do not know (and doubt) but we actually produce a collection of K-theoryfunctors whose VC-assembly property would imply Conjecture 1.0.5.

To do this, we need to consider functors which are cooked up via algebraicK-theory and differential graded (dg) categories as follows. For a survey of dgcategories, we refer the reader to Keller’s recent ICM address [15] or to 2.3 below.

Let B be a small dg category. Consider the functor Kalgc (−;B) : Or(G) → Spt,

defined for all G/H ∈ Or(G) by

(1.0.6) Kalgc (G/H ;B) := K

algc

(rep

(B, R[G/H]

)).

Some explanations are needed here. First, G/H stands for the transport groupoid

of the G-set G/H (Def. 3.3.1). Then R[G/H ] is the associated R-linear category(Def. 3.3.3). For any small dg category A, we denote by rep(B,A) the dg categoryof representations up to homotopy of B in A (Def. 2.3.5 and Rem. 2.3.6). Finally,K

algc stands for the connective algebraic K-theory spectrum.When B = R is the dg category with one object ∗ and with R as dg algebra of

endomorphisms, the functor Kalgc (−;B) reduces to the usual connective algebraic

K-theory functor Kalgc . When B is a general dg category, the functor Kalg

c (−;B)can be thought of as a coefficients version of Kalg

c (see 6.1.4).Unfortunately, the functor Kalg

c (−;B) is not additive in general, mainly becauseB might be “too large”. Hence we need to restrict to dg categories B which arehomotopically finitely presented (Def. 2.1.1). Heuristically, this condition is a ho-motopical version of the classical notion of finite presentation. The above exampleB = R is homotopically finitely presented. Restricting to a smaller class of dgcategories B is anyway an improvement in the logic of the following result :

Theorem. Let G be a group and F be a family of subgroups. Then the followingare equivalent :

(1) The fundamental additive functor Efund has the F-assembly property on G.

(2) The additive functors Kalgc (−;B) have the F-assembly property on G for all

homotopically finitely presented dg categories B (Def. 2.1.1).


strictly finite dg cells B (Def. 2.3.1).

The strictly finite dg cells of (3) form a set of homotopically finitely presented dgcategories which are especially small. Roughly speaking, they are the dg categoryanalogues of finite CW-complexes in topology, namely they are built by attachingfinitely many basic cells, chosen among the dg analogues S(n− 1) → D(n), n ∈ Z,of the topological inclusion Sn−1 → Dn; see a precise definition in 2.3.

If F is the family VC of virtually cyclic subgroups of G, Conditions (2) and(3) can be thought of as coefficients versions of the Farrell-Jones conjecture, withK-theory replaced by connective K-theory.


We now sum up several consequences of our main theorem.

Corollary. Suppose that for every strictly finite dg cell B, the additive functorKalg

c (−;B) has the VC-assembly property on G. Then we have :

(1) The Farrell-Jones isomorphism conjecture in non-connective K-theory holds.(2) Since the mixed complex construction C(−) is an additive functor, the in-

duced map

hocolimOr(G,VC)

C(R[G/H])∼−→ C(R[G])

is an isomorphism in D(Λ-Mod). This implies that by applying the homol-

ogy functor H∗(−) and the composed one H∗(−L

⊗Λ R) to the above inducedmap, we obtain an isomorphism in Hochschild homology

H∗(hocolimOr(G,F)

C(R[G/H]))∼−→ HH∗(R[G])

and in cyclic homology

H∗(hocolimOr(G,F)

(C(R[G/H ])L

⊗Λ R))∼−→ HC∗(R[G]) .

(3) Since THH(−) and THC(−) are additive functors, the induced maps ontopological Hochschild and topological cyclic homology groups

πs∗(hocolim

Or(G,VC)THH(R[G/H]))

∼−→ πs

∗ THH(R[G]) ,

πs∗(hocolim

Or(G,VC)THC(R[G/H ]))

∼−→ πs

∗ THC(R[G])

are isomorphisms. That is, the Farrell-Jones conjectures for THH and THCstated by W. Luck [16, 6.5] hold.

Note that (1) is not merely the hypothesis applied to B = R, for the latter wouldjust give the version of the conjecture for connective K-theory.

We make an extensive use of dg categories, not only in the second part where wereduce the F -assembly property of the fundamental additive functor Efund to thatof the functors Kalg

c (−;B) but already in the first part, in the very construction ofthe fundamental additive functor Efund. The other language that we use is the oneof Grothendieck’s derivators, as recalled in Appendix A.

There are many reasons for using derivators. The most basic one is to get ridof choices in the various model categories M that appear in the above discussion.In short, we could say that derivators provide the right framework in which tominimize the homotopical technicalities. See more explanations in A.1.

The second reason for using derivators is that the fundamental additive functorEfund : Or(G) → Mfund is produced by means of localization and stabilization. Iflocalization is reasonably under control in model categories via Bousfield techniques,stabilization of model categories gets messy and is better expressed in the languageof derivators. This is recalled in Appendix A, where we prove as well that theoperations of localization and stabilization commute (Theorem A.4.1).


2. Preliminaries

2.1. Notations. We denote by CAT the 2-category of categories and by Cat the2-category of small categories. For two categories C and D, we denote by Fun(C,D)the category of functors from C to D with natural transformations as morphisms.

Throughout the article we work over a fixed commutative ring R. We denote byR-Mod the symmetric monoidal category of R-modules and by sSet (resp. sSet•)the category of simplicial sets (resp. pointed simplicial sets), see [8, Chap. I].

Let M be a model category [21]. We denote by Map(−,−) : Mop × M −→Ho(sSet) [11, 17.4.1] its homotopy function complex. Recall that ifM is a simplicialmodel category [8, § II.3], its homotopy function complex is given, for X,Y ∈ M,by the simplical set

Map(X,Y )n = M(Xc ⊗∆[n], Yf )

whereXc is a cofibrant resolution ofX and Yf is a fibrant resolution of Y . Moreover,if Ho(M) denotes the homotopy category of M, we have the following isomorphismπ0Map(X,Y ) ≃ Ho(M)(X,Y ).

2.1.1. Definition. An object X in M is homotopically finitely presented if for anydiagram Y : J → M in M (for any shape, i.e. small category, J), the induced map

hocolimj∈J

Map(X,Yj) −→ Map(X, hocolimj∈J

Yj)

is an isomorphism in Ho(sSet).

2.2. Homotopy function spectrum. Let Spt be the (model) category of spec-tra [8, §X.4]. If X is a spectrum, we denote by X [n], n ≥ 0 its nth suspension, i.e.the spectrum defined as X [n]m = Xn+m, m ≥ 0. If X and Y are two spectra, wedefine its homotopy function spectrum Map(X,Y ) by

Map(X,Y )n = Map(X,Y [n]) ,

where the bonding maps are the natural ones.Let I be a small category. By [12, Thm. 3.3], the category of pre-sheaves of spec-

tra Fun(Iop, Spt) carries a natural simplicial model structure, the projective modelstructure, with weak-equivalences and fibrations defined objectwise. If we denoteby Map(−,−) its homotopy function complex, the homotopy function spectrum oftwo pre-sheaves F and G is given by (as in the case of spectra)

Map(F,G)n = Map(F,G[n]) ,

where G[n] is the nth objectwise suspension of G.

2.2.1. Remark. Let S be a set of morphisms in Fun(Iop, Spt) and LS(Fun(Iop, Spt))

its left Bousfield localization [11, 4.1.1] with respect to the set S. Notice thatLS(Fun(I

op, Spt)) admits also a homotopy function spectrum given byMap(−, Q(G)),where Q(G) is a fibrant resolution of G in LS(Fun(I

op, Spt)).

2.3. Dg categories. For a survey article on dg categories see [15]. A dg category(over our base ring R) is an additive category enriched over cochain complexes ofR-modules (morphisms sets are such complexes) in such a way that compositionfulfills the Leibniz rule : d(f g) = (df) g + (−1)deg(f)f (dg).

We start by recalling from [23, § 1.3] some constructions concerning dg categories.If A is a dg category and x and y are two objects, we denote by A(x, y) the complexof morphisms from x to y. Let R be the dg category with one object ∗ and such


that R(∗, ∗) := R (in degree zero). For n ∈ Z, let Sn be the complex R[n] (withR concentrated in degree n) and let Dn be the mapping cone on the identity ofSn−1. We denote by S(n) the dg category with two objects 1 et 2 such thatS(n)(1, 1) = R , S(n)(2, 2) = R , S(n)(2, 1) = 0 , S(n)(1, 2) = Sn and compositiongiven by multiplication. We denote by D(n) the dg category with two objects 3and 4 such that D(n)(3, 3) = R, D(n)(4, 4) = R, D(n)(4, 3) = 0 , D(n)(3, 4) = Dn

and with composition given by multiplication. Finally, let ι(n) : S(n − 1) → D(n)be the dg functor that sends 1 to 3, 2 to 4 and Sn−1 to Dn by the identity on R indegree n− 1 :

S(n− 1)ι(n)

// D(n)

1

R

Sn−1

// 3

R

Dn

incl //

2

R

EE // 4

R

EE

where

Sn−1 incl // Dn

0 //

0

0 //

Rid

Rid //

R

(degree n−1)

0 // 0

We denote by I the set consisting of the dg functors ι(n)n∈Z and the dg functor∅ → R (where the empty dg category ∅ is the initial one).

2.3.1. Definition. A dg category A is a strictly finite dg cell [11, 10.5.8] if it isobtained from ∅ by a finite number of pushouts along the dg functors of the set I.

2.3.2. Notation. We denote by dgcatsf the full subcategory of the category dgcat ofsmall dg categories, whose objects are the strictly finite dg cells.

Let A be a small dg category. The opposite dg category Aop of A has the sameobjects as A and its complexes of morphisms are defined by Aop(x, y) = A(y, x).The homotopy category H0(A) of A has the same objects as A and its morphismsare defined by H0(A)(x, y) = H0(A(x, y)). Recall from [15, 3.1] that a right dgA-module is a dg functor M : Aop → Cdg(R), with values in the dg category Cdg(R)of complexes of R-modules. We note by C(A) (resp. by Cdg(A)) the category (resp.dg category) of dg A-modules and by D(A) the derived category of A, i.e. thelocalization of C(A) with respect to the class of quasi-isomorphisms. Finally, let

A −→ Cdg(A)A 7→ A(−, A)

be the Yoneda dg functor, which sends an object A to the dg A-module A(−, A)represented by A. See more in [15].

2.3.3. Notation. Let A be a dg category. We denote by perf(A) the full dg sub-category of Cdg(A), whose objects are the dg A-modules which become compactin D(A).

2.3.4. Remark. Recall from [23, Thm. 2.27] that the category dgcat is endowed witha cofibrantly generated Morita model structure, whose weak equivalences are theMorita dg functors, i.e. dg functors F : A → B which induce an equivalence on


the derived categories D(B)∼→ D(A). We sometimes denote by Hmo = Ho(dgcat),

the homotopy category hence obtained. By [23, Prop. 2.34], the fibrant objectsare the dg categories A such that H0(A) is equivalent to the thick triangulatedsubcategory of D(A) generated by the representable dg A-modules. Moreover, thecategory dgcat admits a tensor monoidal structure −⊗ −, which by [23, 2.40] canbe naturally derived into a bi-functor

−⊗L − : Hmoop × Hmo −→ Hmo .

Finally by [23, Proposition 3.10 and Example 3.9] a dg category is homotopicallyfinitely presented (Def. 2.1.1) if and only if it is Morita equivalent to a retract inHmo of a strictly finite dg cell (Def. 2.3.1).

2.3.5. Definition. (See [23, 3.72].) Let A and B be two dg categories. We denote

by rep(A,B) the full dg subcategory of dg AL

⊗Bop-modules, whose objects are thecofibrant dg A-B bimodules X such that X(?, A) is a compact object in D(B) forevery object A ∈ A.

2.3.6. Remark. Recall from [23, 2.40] that rep(−,−) induces the internal Hom-functor of the symmetric monoidal homotopy category (Hmo,−⊗L −, R).

Let A,B ∈ Hmo. Since B can be represented by a fibrant dg category, rep(A,B)is formed by the dg A-B bimodules X such that the tensor product

−⊗L

A X : D(A) −→ D(B)

takes the representable dg A-modules to objects which are isomorphic to repre-sentable dg B-modules. The bimodule X induces then a functor on the homotopycategories H0(A) → H0(B). Therefore we can think of rep(A,B) as the dg categoryof representations up to homotopy of A in B. Notice also that rep(R,B) is Moritaequivalent to perf(B) and so to B.

3. Assembly and isomorphism conjectures

In this Section, we recall Davis and Luck’s reformulation [6] of the Farrell-Jonesisomorphism conjecture in algebraic K-theory. Let G be a (fixed) discrete group.

3.1. The orbit category.

3.1.1. Definition. The orbit category Or(G) has as objects the homogeneous G-spaces G/H and G-maps as morphisms.

3.1.2. Definition. A family F of subgroups of G is a non-empty set of subgroups ofG closed under conjugation and finite intersection.

3.1.3. Examples. Fin is the family of finite subgroups of G. The cyclic subgroupsalso form a family. The family VC consists of virtually cyclic subgroups of G.Recall that H is virtually cyclic if it contains a cyclic subgroup of finite index.

3.1.4.Definition. The orbit category Or(G,F) restricted on F is the full subcategoryof Or(G) consisting of those objects G/H for which H belongs to F .


3.2. F-assembly property. Let F be a family of subgroups ofG and E : Or(G) →Spt be a functor from the orbit category to spectra. Recall the notion of assemblyproperty given in Definition 1.0.2 of the Introduction.

3.2.1. Remark. A typical approach in Davis and Luck’s philosophy is : Given G andE, find as small a family F as possible for which E has the F -assembly property. So,for the Farrell-Jones isomorphism conjecture, one expects F to reduce to virtuallycyclic subgroups, whereas for Baum-Connes one expects finite subgroups.

3.2.2. Remark. Conceptually, the F -assembly property for a functor E : Or(G) →Spt essentially means that it is induced from its restriction to Or(G,F), up tohomotopy, i.e. it belongs to the image of the functor on homotopy categories

LInd : Ho(Fun(Or(G,F), Spt)

)−→ Ho

(Fun(Or(G), Spt)

)

left adjoint to the obvious functor in the other direction, defined by restrictionfrom Or(G) to Or(G,F). This is explained in [1], where we say that the functor Esatisfies Or(G,F)-codescent if E belongs to the image of LInd up to isomorphismin Ho

(Fun(Or(G), Spt)

). This is equivalent to the F -assembly property for G and

all its subgroups.

3.3. Functorial K-theory construction.

3.3.1. Definition. Let S be a left G-set. The transport groupoid S associated to S,has S as set of objects and for s, t ∈ S, we define

S(s, t) := g ∈ G | gs = t .

The composition is given by group multiplication. This defines a functor

Or(G)?

−→ Grp ,

with values in the category of groupoids.

3.3.2. Remark. For every subgroup H of G, the groupoid G/H is connected. Henceit is equivalent to the full subcategory on any of its objects, for instance the canoni-cal one given by eH ∈ G/H . The automorphisms of that object is clearly H . So, ifwe think of the group H as a one-object category, say H , we have a equivalence of

groupoidsH∼→ G/H . In other words, the groupoid G/H is a natural several-object

replacement of the group H .

We now show how to pass from groupoids to R-categories, i.e. additive categoriesenriched over the symmetric monoidal category (R-Mod,−⊗R −, R).

3.3.3.Definition. Let C be a category. The associated R-category R[C] has as objectsthe formal finite direct sums of objects of C and as morphism the obvious matri-ces with entries in the free R-modules R[C(X,Y )] generated by the sets C(X,Y ).Composition in R[C] is induced from composition in C.

3.3.4. Remark. The construction C 7→ R[C] is functorial and so we obtain a well-defined functor

GrpR[−]−→ R-Cat ,

with values in the category of additive R-categories.


We can then consider the non-connective algebraic K-theory spectrum functorof additive categories (see Pedersen-Weibel [20] for details)

Kalg : R-Cat −→ Spt .

Finally we obtain, by concatenating, our functor Kalg : Or(G) → Spt

Or(G)? //

Kalg

))SSSSSSSSSSSSSSSSGrp

R[−]// R-Cat

Kalg

Spt .

As usual, one obtains the K-theory groups Kalg∗ by taking the stable homotopy

groups of the spectrum Kalg.

3.3.5. Remark. Remark 3.3.2 implies the following isomorphism

Kalg∗ (R[H ]) = πs

∗ Kalg(R[H ]) = πs

∗ Kalg(R[H ])

∼−→ πs

∗ Kalg(R[G/H ]),

which explains that the K-theory functor on Or(G) is indeed the expected one.

4. Additive invariants of dg categories

Here, the reader needs some familiarity with derivators. See Appendix A.

4.1. The additive motivator.

4.1.1. Definition. Recall from [23, 3.86] that an additive invariant of dg categoriesis a morphism of derivators

A : HO(dgcat) −→ T ,

from the derivator HO(dgcat) associated to the Morita homotopy theory of dgcategories to a triangulated one T, which commutes with filtered homotopy colimits(only), preserves the point and sends split short exact sequences to split exacttriangles.

In [23], we have constructed the universal such additive invariant

Uadddg : HO(dgcat) −→ Motadddg ,

with values in what deserves the name of additive motivator of dg categories Motadddg .

4.1.2. Theorem. [23, Thm. 3.85] Let D be a strong triangulated derivator. Themorphism Uadd

dg induces an equivalence of categories

Hom!(Motadddg ,D)(Uadd

dg )∗

−→ Homflt, p,Add(HO(dgcat),D) ,

where Homflt, p,Add(HO(dgcat),D) denotes the category of additive invariants of dgcategories (Def. 4.1.1).

We now give examples of additive invariants of dg categories and we postponeto Section 5 below the discussion of how the universal one Motadddg is constructed.


4.2. Connective algebraic K-theory. Let A be a dg category. Its Waldhausen’sconnective K-theory spectrum Kalg

c (A) is defined by applying Waldhausen’s con-struction [27] to the category of cofibrant and perfect A-modules. Recall that thecofibrations are the split monomorphisms and the weak-equivalences the quasi-isomorphisms. By Waldhausen’s additivity [27, Thm. 1.4], the assignment A 7→Kalg

c (A) yields a morphism of derivators

Kalgc : HO(dgcat) −→ HO(Spt) ,

which is an additive invariant of dg categories.

4.3. Non-connective algebraic K-theory. Let A be a small dg category. Itsnon-connective K-theory spectrum Kalg(A) is defined by applying Schlichting’sconstruction [22] to the Frobenius pair associated with the category of cofibrantperfect A-modules (to the empty dg category we associate 0). Recall that theconflations in the Frobenius category of cofibrant perfect A-modules are the shortexact sequences which split in the category of graded A-modules.

By [23, Thm. 3.49], the assignmentA 7→ Kalg(A) yields a morphism of derivators

Kalg : HO(dgcat) −→ HO(Spt) ,


4.4. Hochschild and cyclic homology. Let A be a small R-flat R-category. TheHochschild chain complex of A is the complex concentrated in homological degreesp ≥ 0 whose pth component is the sum of the

A(Xp, X0)⊗A(Xp−1, Xp)⊗A(Xp−2, Xp−1)⊗ · · · ⊗ A(X0, X1) ,

where X0, . . . , Xp range through the objects of A, endowed with the differential

d(fp ⊗ . . .⊗ f0) = fp−1 ⊗ · · · ⊗ f0fp +

p∑

i=1

(−1)ifp ⊗ · · · ⊗ fifi−1 ⊗ · · · ⊗ f0 .

Via the cyclic permutations tp(fp−1 ⊗ · · · ⊗ f0) = (−1)pf0 ⊗ fp−1 ⊗ · · · ⊗ f1, thiscomplex becomes a precyclic (see [14, § 1.3]) chain complex and thus gives rise to amixed complex C(A), i.e. a dg module over the dg algebra Λ = R[B]/(B2), where Bis of degree −1 and dB = 0. All variants of cyclic homology only depend on C(A)considered in the derived category D(Λ). For example, the cyclic homology of A is

the homology of the complex C(A)L

⊗Λ R. See details in [13].By [23, Thm. 3.47], the map A 7→ C(A) yields a morphism of derivators

C : HO(dgcat) −→ HO(Λ-Mod) ,


4.5. Topological Hochschild and cyclic homology. Let A be a small dg cat-egory. For the definition of the topological Hochschild homology spectrum of A,denoted THH(A), and for the definition of its topological cyclic homology, denotedTHC(A), the reader is invited to consult [25, § 10]. We obtain morphisms of deriva-tors

THH,THC : HO(dgcat) −→ HO(Spt) ,

which by [25, Prop. 10.7] are additive invariants of dg categories.


4.6. Coefficients Version. Let B be a dg category. The functor rep(B,−) ofDefinition 2.3.5 induces naturally a morphism of derivators

rep(B,−) : HO(dgcat) −→ HO(dgcat) .

4.6.1. Lemma. Suppose that the dg category B is homotopically finitely presented(Def. 2.1.1). Then the morphism of derivators rep(B,−) commutes with filteredhomotopy colimits, preserves the point and sends split short exact sequences to splitshort exact sequences.

Proof. Clearly rep(B,−) preserves the point [23, § 3.6]. Since B is homotopicallyfinitely presented, an argument analogous to the one of [26, Lemma2.10] showsthat rep(B,−) commutes with filtered homotopy colimits. Finally, if

0 // A′ // Aoo // A′′

oo // 0

is a split short exact sequence of dg categories [23, 3.68], so is

0 // rep(B,A′) // rep(B,A)oo // rep(B,A′′)

oo // 0 .

4.6.2. Notation. By Lemma 4.6.1, if B is a homotopically finitely presented dg cate-gory and A : HO(dgcat) → T is an additive invariant of dg categories (Def. 4.1.1), wecan construct a new additive invariant A(−;B) : HO(dgcat) → T by pre-composingwith rep(B,−)

A(A;B) := A(rep(B,A)) , A ∈ dgcat .

4.6.3. Remark. For B = R, the dg category rep(R,A) is Morita equivalent to Aand so A(−;R) reduces to A. If B is a general homotopically finitely presented dgcategory, A(−;B) can be thought of as a coefficients version of A.

5. The additive motivator

The purpose of this Section is to recall the additive motivator Motadddg of dgcategories [23] and to provide a new Quillen model for it.

5.0.4. Notation. Let M be a pointed proper simplicial model category. We denoteby Spt(M) the stable model category of (usual) spectra of objects of M.

5.1. Old model. In [23, § 3.5], we have constructed the small category dgcatf offinite I-cells as the smallest full subcategory of dgcat whose set of objects con-tains strictly finite dg cells (Def. 2.3.1) and which is stable under the co-simplicialand fibrant resolution functors of [23, 3.11]. Then, we considered the model cate-gory Fun(dgcatopf , sSet•) of pre-sheaves of pointed simplicial sets with the projectivemodel structure and took its left Bousfield localization

LgEs

un, p,Σ

Fun(dgcatopf , sSet•) ,

with respect to sets of morphisms Esun, p and Σ (see [23, § 3.14] for details). Heuris-

tically, inverting Σ is responsible for inverting Morita equivalences, inverting p is

responsible for pointing and finally, inverting Esun is responsible for mapping split

short exact sequences of dg categories to split triangles in the homotopy category.See more in Proposition 5.3.1 below and in its proof.


5.1.1. Remark. In Fun(dgcatopf , sSet•), sequential homotopy colimits commute withfinite products and homotopy pullbacks and so by Remark A.1.3, the associatedderivator is regular (Def. A.1.2). Since the domains and codomains of the sets

of morphisms Esun, p and Σ (see [23, § 3.14]) are homotopically finitely presented

(Def. 2.1.1), Remark A.3.2 implies that the associated derivator to the Bousfieldlocalization LgEs

un, p,Σ

Fun(dgcatopf , sSet•) is also regular.

5.1.2. Definition. We defined (see [23, 3.82]) the additive motivator Motadddg asthe triangulated derivator associated (as in A.1) to the stable model categorySpt

(LgEs

un, p,Σ

Fun(dgcatopf , sSet•)):

(5.1.3) Motadddg := HO(Spt

(LgEs

un, p,Σ

Fun(dgcatopf , sSet•)))

.

5.2. New model. Recall from Subsection 2.2 that since dgcatf is a small category,the category Fun(dgcatopf , Spt) carries naturally a simplicial model structure.

5.2.1. Remark. We have a natural (Quillen) identification

Spt(Fun(dgcatopf , sSet•)) ≃ Fun(dgcatopf , Spt) .

5.2.2. Notation. Consider the Yoneda functor

dgcatfh

−→ Fun(dgcatopf , Spt)B 7→ Σ∞dgcatf(−,B) ,

where dgcatf(−,B) is a constant pointed simplicial set and Σ∞(−) denotes theinfinite suspension spectrum functor. If F is a fibrant object in Fun(dgcatopf , Spt),we have the following weak equivalences:

Map(h(B), F ) ≃ F (B)0 Map(h(B), F ) ≃ F (B) .

We have also a homotopical Yoneda functor

dgcath

−→ Fun(dgcatopf , Spt)A 7→ Σ∞Map(−,A) ,

where Map(−,−) denotes the homotopy function complex (see Section 2) of theMorita model structure on dgcat. By construction, homotopy limits and homotopycolimits in Fun(dgcatopf , Spt) are calculated objectwise. This implies that the shiftmodels in Spt (see [12, § 1]), X [1] and X [−1], for the suspension and loop spacefunctors in Ho(Spt) induce objectwise shift models in Fun(dgcatopf , Spt) for the sus-pension and loop space functors in the triangulated category Ho(Fun(dgcatopf , Spt)).

Observe that Remarks 5.1.1 and 5.2.1, the Commutativity Theorem A.4.1 and[23, Theorems 3.8 and 3.31] imply the following proposition.

5.2.3. Proposition. The additive motivator Motadddg (5.1.3) admits a new Quillenmodel given by

LΩ(gEs

un),Ω(p),Ω(Σ)

Fun(dgcatopf , Spt) ,

where Ω(Esun), Ω(p) and Ω(Σ) are obtained by stabilizing the sets Es

un, p and Σ inFun(dgcatopf , Spt) under the objectwise loop space functor.

5.2.4. Notation. For the sake of simplicity, we will denote byN the set of morphisms

Ω(Esun),Ω(p),Ω(Σ) in Fun(dgcatopf , Spt) and consequently by LNFun(dgcatopf , Spt)

our new Quillen model.


Let us now sum up (compare with [23, 3.83]) our new construction of the uni-versal additive invariant Uadd

dg of dg categories

HO(dgcatf) //

HO(h)

HO(dgcat)

Rh

uu

Uadddg

qq

HO(LΩ(Σ)Fun(dgcat

opf , Spt)

)

Motadddg

,

where HO(dgcatf) is the prederivator associated with the full subcategory dgcatfof dgcat (see A.1.1 and [23, § 3.5] for details) and Uadd

dg is the composition of thefunctor Rh induced by Yoneda and the localization morphism

HO(LΩ(Σ)Fun(dgcat

opf , Spt)

)−→ HO

(LNFun(dgcatopf , Spt)

)= Motadddg .

5.3. Fibrant objects.

5.3.1. Proposition. An object F ∈ LNFun(dgcatopf , Spt) is fibrant if and only if thefollowing four conditions are verified:

(1) F (B) ∈ Spt is stably fibrant, for all B ∈ dgcatf .

(2) For every Morita equivalence B∼→ B′ in dgcatf , the induced morphism

F (B′)∼→ F (B) is a stable weak equivalence in Spt.

(3) F (∅) ∈ Spt is contractible.(4) Every split short exact sequence

0 // B′ // Boo // B′′

oo // 0

in dgcatf (see [23, 3.68]), induces a homotopy fiber sequence

F (B′′) → F (B) → F (B′)

in Ho(Spt).

Proof. Condition (1) corresponds to the fact that F is fibrant in Fun(dgcatopf , Spt)since we use the projective model. By the objectwise definition of the shift models inFun(dgcatopf , Spt) for the suspension and loop space functors in Ho(Fun(dgcatopf , Spt)),the construction of the localized model structure yields:

• An object F is Ω(Σ)-local if and only if for every Morita equivalence B∼→ B′

in dgcatf , the morphism F (B′) → F (B) is a levelwise weak equivalencein sSet•. Since F (B′) and F (B) are stably fibrant this is equivalent toCondition (2).

• An object F is Ω(p)-local if and only if F (∅)n is contractible for every n ≥ 0.Since F (∅) is stably fibrant this is equivalent to Condition (3).

We now discuss Condition (4). The construction of the set Ω(Esun) (see [23, 3.74]

and Proposition 5.2.3) and the fact that the functor

Map(?, F ) : Ho(Fun(dgcatopf , Spt))op −→ Ho(sSet•)

transforms homotopy cofiber sequences into homotopy fiber sequences, implies that:


• An object F is Ω(Esun)-local if and only if every split short exact sequence

0 // B′ // Boo // B′′

oo // 0

in dgcatf induces a homotopy fiber sequence

F (B′′)n → F (B)n → F (B′)n

in Ho(sSet•) for every n ≥ 0 (see [23, 3.76]). Once again since F (B′), F (B)and F (B′′) are stably fibrant, this is equivalent to Condition (4).

We then conclude by general Bousfield localization non-sense, see [11, Prop. 3.4.1].

5.4. Homotopy colimits. Using the above description of the fibrant objects, wenow prove our key technical result.

5.4.1. Proposition. For every dg category C ∈ dgcatf , the functor

Map(h(C),−) : Ho(LNFun(dgcatopf , Spt)) −→ Ho(Spt)

commutes with homotopy colimits.

Proof. We start by observing that by construction, the result holds in the sta-ble model category Fun(dgcatopf , Spt). By Remark 2.2.1 it is enough to provethe following: If (Fj)j∈J is a diagram of fibrant objects in the localized categoryLNFun(dgcatopf , Spt), its homotopy colimit hocolim

j∈JFj satisfies Conditions (2), (3)

and (4) of Proposition 5.3.1. Since homotopy colimits in Fun(dgcatopf , Spt) are cal-culated objectwise, Conditions (2) and (3) are clearly verified. In what concernsCondition (4), notice that Ho(Spt) is a triangulated category and so the homotopyfiber sequences

Fj(B′′) −→ Fj(B) −→ Fj(B

′), j ∈ J

are also homotopy cofiber sequences. This implies, as in the case of Condition (2)and (3), that

hocolimj∈J

Fj(B′′) −→ hocolim

j∈JFj(B) −→ hocolim

j∈JFj(B

′)

is a homotopy cofiber sequence and so a homotopy fiber sequence. In conclusionhocolim

j∈JFj satisfies Condition (4) and the proof is finished.

5.5. Homotopic generators. Let I be a small category and M a left Bousfieldlocalization of Fun(Iop, Spt), see Subsection 2.2.

5.5.1. Definition. We say that a set of objects Gjj∈J satisfies condition (HG) (forHomotopic Generators) in M if the following condition holds:

(HG) A morphism f : F → F ′ is a weak equivalence in M if (and only if) forevery object Gj the induced map of spectra

f∗ : Map(Gj , F ) −→ Map(Gj , F′)

is a stable weak equivalence.

If f is a weak equivalence in M, the induced maps of spectra are stable weakequivalences, so the “only if” part is redundant.

Let Gjj∈J be a set of objects in M and S a set of morphisms in M.

5.5.2. Lemma. If the set Gjj∈J satisfies condition (HG) in M then it satisfiescondition (HG) in LS(M).


Proof. By Remark 2.2.1, the left Bousfield localization LS(M) admits also a homo-topy function spectrum given by Map(−, Q(−)) where Q is a (functorial) cofibrantreplacement in LS(M). Let f : F → F ′ be a morphism in M which induces astable equivalences under Map(Gj , Q(−)) for all j ∈ J . Consider the followingcommutative square

F

f

∼ // Q(F )

Q(f)

F ′

∼// Q(F ′) .

Since by hypothesis, the set Gjj∈J satisfies condition (HG) in M, Q(f) is a weakequivalence in M and so a weak equivalence in LS(M). By the two out of threeproperty, we conclude that f is a weak equivalence in LS(M).

Now, recall from Definition 2.3.1 the full subcategory dgcatsf of dgcatf , whoseobjects are the strictly finite dg cells.

5.5.3. Proposition. The set of objects h(B) | B ∈ dgcatsf satisfies condition (HG)in LNFun(dgcatopf , Spt).

Proof. Notice that the set h(B) | B ∈ dgcatf satisfies condition (HG) in the modelcategory Fun(dgcatopf , Spt) by the very definition of weak equivalences. From Defi-nition [23, 3.11], every object in dgcatf is Morita equivalent to an object in dgcatsf .This implies by Lemma 5.5.2 and Proposition 5.3.1, that the set h(B) | B ∈ dgcatsfsatisfies condition (HG) in LΩ(Σ)Fun(dgcat

opf , Spt). Once again by Lemma 5.5.2 we

can localize further with respect to N and the proof is finished.

6. Universal additive assembly property

The purpose of this Section is to describe the fundamental additive functor Efund

of the Introduction and to prove our main result (Theorem 6.2.1 below). Let usstart by connecting Davis and Luck’s construction (Section 3) to dg categories andtheir additive invariants.

6.1. Additive functors. Notice that an R-category can be naturally consideredas a dg category (with complexes of morphisms concentrated in degree 0). For a(discrete) group G, we thus obtain a composed functor

Or(G)?

−→ GrpR[−]−→ R-Cat −→ dgcat .

6.1.1. Definition. Let M be a stable model category (Rem.A.1.5) and E : Or(G) →M a functor. We say that E is additive if it factors through a functor dgcat →M whose associated morphism of derivators HO(dgcat) → HO(M) is an additiveinvariant of dg categories in the sense of Definition 4.1.1.

6.1.2. Remark. This factorization should not be confused with the one we want toestablish in Theorem 6.1.6 (that is, via the fundamental additive functor Efund),otherwise that result would sound like a tautology. We rather restrict attention tofunctors on the orbit category that only depend on the associated dg category. Thisis a mild restriction since all functors classically considered have been extended todg categories, as explained in Section 4.


6.1.3. Remark. At the time of writing, we do not know whether Ktop(C∗red(−)) can

be seen as an additive functor or not, simply because we do not know how to extendit to dg categories. This problem seems interesting, especially in view of [6, § 2].

6.1.4. Examples. Recall from Section 4 several examples of functors A : dgcat → M(e.g. algebraicK-theory, Hochschild and cyclic homology, their topological variants)whose associated morphism of derivators A : HO(dgcat) → HO(M) is an additiveinvariant of dg categories. By pre-composing them with the functor

Or(G)?

−→ GrpR[−]−→ R-Cat −→ dgcat ,

we obtain several examples of additive functors Or(G) → M in the sense of Defi-nition 6.1.1. Moreover, if B is a homotopically finitely presented dg category B, wehave additive functors A(−;B) (see 4.6) defined as follows

A(G/H ;B) := A(rep(B, R[G/H])) , G/H ∈ Or(G) .

If B = R, the additive functor A(−;B) reduces to the following composition

Or(G)?

−→ GrpR[−]−→ R-Cat −→ dgcat

A−→ M .

If B is a general homotopically finitely presented dg category, A(−;B) can bethought of as a coefficients version of the above additive functor A(−;R) (see Re-mark 4.6.3).

6.1.5. Definition. The fundamental additive functor Efund is given by the followingcomposition

Or(G)?


Uadddg

−→ LNFun(dgcatopf , Spt) =: Mfund ,

where Mfund := LNFun(dgcatopf , Spt) is our new Quillen model (see 5.2.4) of the

additive motivator Motadddg .

6.1.6. Theorem. Let G be a group and E : Or(G) → M an additive functor

(Def. 6.1.1). Then there exists a morphism of derivators E : Motadddg → HO(M),which commutes with homotopy colimits, such that the diagram

Or(G)Efund //

E((RRRRRRRRRRRRRR

Ho(Mfund)

E(e)

Ho(M)

commutes up to isomorphism.

Proof. By Definition 6.1.1, E factors through a functor dgcat → M whose associ-ated morphism of derivators HO(dgcat) → HO(M) is an additive invariant of dgcategories. By Theorem 4.1.2, this additive invariant descends to a morphism ofderivators E : Motadddg → HO(M), which commutes with homotopy colimits, makingthe above diagram commute up to isomorphism.

Generalizing Definition 1.0.2 to any target category, we have the obvious :

6.1.7. Definition. Let F be a family of subgroups of G and E : Or(G) → M anadditive functor (Def. 6.1.1). The (E,F , G)-assembly map is the natural map

hocolimOr(G,F)

E −→ hocolimOr(G)

E = E(G)


in the homotopy category Ho(M). We say that E has the F-assembly property(on G) when the (E,F , G)-assembly map is an isomorphism in Ho(M).

Notice that Theorem 6.1.6 has the following corollary.

6.1.8. Corollary. Let G be a group and let F be a family of subgroups. If thefundamental additive functor Efund has the F-assembly property, then so do alladditive functors.

Proof. Apply E to the (Efund,F , G)-assembly map and use that E commutes witharbitrary homotopy colimits.

6.2. Main theorem.

6.2.1. Theorem. Let G be a group and F be a family of subgroups. Then thefollowing are equivalent :

(1) The fundamental additive functor Efund has the F-assembly property on G.


homotopically finitely presented dg categories B (Def. 2.1.1).


strictly finite dg cells B (Def. 2.3.1).

Proof. By Corollary 6.1.8, Condition (1) implies Condition (2). Since every strictlyfinite dg cell dg category is homotopically finitely presented (Rem. 2.3.4), Condi-tion (2) implies Condition (3). We now show that Condition (3) implies Condi-tion (1).

Recall from Definition 6.1.5 the construction of the fundamental additive functor

Efund : Or(G)?


Uadddg

−→ LNFun(dgcatopf , Spt) .

Assuming Condition (3), we need to show that the induced map

hocolimOr(G,F)

Uadddg (R[G/H ]) −→ Uadd

dg (R[G/G])

is an isomorphism in Ho(LNFun(dgcatopf , Spt)) = Motadddg (e). Since the category

Motadddg (e) is triangulated, it is equivalent to prove that the suspension map

hocolimOr(G,F)

Uadddg (R[G/H ])[1] ≃ (hocolim

Or(G,F)Uadddg (R[G/H]))[1] −→ Uadd

dg (R[G/G])[1]

is an isomorphism. By Proposition 5.5.3, the set of objects h(B) | B ∈ dgcatsfsatisfies Condition (HG) in LNFun(dgcatopf , Spt) and so it is enough to prove that,for every B ∈ dgcatsf , the induced map of spectra

Map(h(B), hocolim

Or(G,F)Uadddg (R[G/H ])[1]

)−→ Map

(h(B),Uadd

dg (R[G/G])[1])

is a stable weak equivalence. By Proposition 5.4.1, the functor Map(h(B),−) com-mutes with homotopy colimits and so we have

Map(h(B), hocolim

Or(G,F)Uadddg (R[G/H])[1]

)≃ hocolim

Or(G,F)Map

(h(B),Uadd

dg (R[G/H])[1]).

Moreover, the co-representability theorem [24, Thm. 16.10] for connective algebraic

K-theory in Motadddg provides stable weak equivalences

Map(h(B),Uadddg (R[G/H])[1]) ≃ K

algc (rep(B, R[G/H])) ,


for every B ∈ dgcatsf and H ∈ Or(G,F). In conclusion, we are reduced to showthat for every strictly finite dg cell B, the map

hocolimOr(G,F)

Kalgc (rep(B, R[G/H])) −→ K

algc (rep(B, R[G/G]))

is a stable weak equivalence. This is precisely our hypothesis, namely that theadditive functors Kalg

c (−;B) have the F -assembly property on G.

Appendix A. Derivators, Stabilization and Localization

We quickly recall basic facts on derivators and then prove that the operationsof stabilization (see [23, § 3.8]) and left Bousfield localization (see [23, § 3.4]) ofderivators commute.

A.1. Derivators. The original reference for derivators is Grothendieck’s manu-script [9]. See also Maltsiniotis [19] or a short account in Cisinski–Neeman [5, § 1].Derivators originate in the problem of higher homotopies in derived categories. Fora non-zero triangulated category D and for X a small category, it essentially neverhappens that the diagram category Fun(X,D) remains triangulated (it already failsfor the category of arrows in D, that is, for X = [1] = (• → •)).

Now, very often, our triangulated category D appears as the homotopy (or de-rived) category D = Ho(M) of some model M. In this case, we can consider thecategory Fun(X,M) of diagrams in M, whose homotopy category Ho(Fun(X,M))is often triangulated and provides a reasonable approximation for Fun(X,D). Moreimportant maybe, one can let X move. This nebula of categories Ho(Fun(X,M)),indexed by small categories X , and the various functors and natural transforma-tions between them is what Grothendieck formalized into the concept of derivator.A derivator D consists of a strict contravariant 2-functor from the 2-category ofsmall categories to the 2-category of all categories (a. k. a. a prederivator)

D : Catop −→ CAT,

subject to certain conditions. We shall not list them here for it would be too longbut we refer to [5, § 1]. The essential example to keep in mind is the derivatorD = HO(M) associated to a (cofibrantly generated) Quillen model category Mand defined for every small category X by

(A.1.1) HO(M) (X) = Ho(Fun(Xop,M)

).

See more in Cisinski [2].We denote by e the 1-point category with one object and one (identity) mor-

phism. Heuristically, the category D(e) is the basic “derived” category under con-sideration in the derivator D. For instance, if D = HO(M) then D(e) = Ho(M).

A.1.2. Definitions. We now recall three slightly technical properties of derivatorsand refer the non-specialist reader to the quoted literature for further details onthe terminology. See also Remark A.1.3 in the case of D = HO(M).

(1) A derivator D is strong if for every finite free category X and every smallcategory Y , the natural functor D(X × Y ) −→ Fun(Xop,D(Y )) is full andessentially surjective. See details in [10].

(2) A derivator D is regular if in D, sequential homotopy colimits commute withfinite products and homotopy pullbacks. See details in [10].


(3) A derivator D is pointed if for any closed immersion i : Z → X in Cat

the cohomological direct image functor i∗ : D(Z) −→ D(X) has a rightadjoint, and if moreover and dually, for any open immersion j : U → Xthe homological direct image functor j! : D(U) −→ D(X) has a left adjoint.See details in [5, 1.13].

A.1.3. Remark. A strong derivator is the same thing as a small homotopy theoryin the sense of Heller [10]. By [4, Prop. 2.15], if M is a Quillen model category,its associated derivator HO(M) is strong. Moreover, if sequential homotopy col-imits commute with finite products and homotopy pullbacks in M, the associatedderivator HO(M) is regular. Finally if M is pointed, then so is HO(M). In short,the reader who wishes to restrict attention to derivators of the form HO(M) canas well consider the three properties of Definition A.1.2 as mild ones.

A.1.4. Definition. A derivator D is triangulated or stable if it is pointed and if everyglobal commutative square in D is cartesian exactly when it is cocartesian. Seedetails in [5, 1.15].

A.1.5. Remark. If a pointed model category M is stable, i.e. its suspension functorΣ : Ho(M) → Ho(M) is an equivalence, then its associated derivator HO(M) istriangulated.

A.1.6. Theorem (Maltsiniotis [18]). For any triangulated derivator D and smallcategory X the category D(X) has a canonical triangulated structure.

An explicit description of the triangulated structure is also given [5, 7.9].

A.1.7. Notation. Let D and D′ be derivators. We denote by Hom(D,D′) the categoryof all morphisms of derivators and by Hom!(D,D

′) the category of morphisms ofderivators which commute with homotopy colimits. See details in [3, § 3.25].

A.2. Stabilization. LetD be a regular pointed strong derivator (Def. A.1.2). In [10],Heller constructed the universal morphism to a triangulated strong derivator

Dstab−→ St(D) ,

which commutes with homotopy colimits. Heller proved the following universalproperty.

A.2.1. Theorem ([10]). Let T be a triangulated strong derivator. Then the mor-phism stab induces an equivalence of categories

Hom!(St(D),T)stab∗

−→ Hom!(D,T) .

A.3. Left Bousfield localization. Let D be a derivator and S a class of mor-phisms in the base category D(e).

A.3.1. Definition (Cisinski). The derivator D admits a left Bousfield localizationwith respect to S if there exists a morphism of derivators

γ : D → LSD ,

which commutes with homotopy colimits, sends the elements of S to isomorphismsin LSD(e) and satisfies the following universal property: For every derivator D′ themorphism γ induces an equivalence of categories

Hom!(LSD,D′)

γ∗

−→ Hom!,S(D,D′) ,


where Hom!,S(D,D′) denotes the category of morphisms of derivators which com-

mute with homotopy colimits and send the elements of S to isomorphisms in D′(e).

A.3.2. Remark. Let M be a left proper, cellular model category and LSM its leftBousfield localization [11, 4.1.1] with respect to a set of morphisms S. Then theinduced morphism of derivators HO(M) → HO(LSM) is a left Bousfield localiza-tion of derivators with respect to the image of S in Ho(M). See [23, Thm. 3.8].Moreover, if the domains and codomains of the set S are homotopically finitely pre-sented objects (Def. 2.1.1), the functor Ho(LSM) → Ho(M) (right adjoint to thelocalization functor) preserves filtered homotopy colimits. See [23, Lemma 3.23].Under these hypothesis, if HO(M) is regular (Def. A.1.3) so it is HO(LSM).

A.3.3. Remark. By [23, Lemma 3.7], the Bousfield localization LSD of a triangulatedderivator D remains triangulated as long as S is stable under the loop space functor.For more general S, to remain in the world of triangulated derivators, one has tolocalize with respect to the set Ω(S) generated by S and loops, as follows.

A.3.4. Proposition. Let D be a triangulated derivator and S a class of morphismsin D(e). Let us denote by Ω(S) the smallest class of morphisms in D(e) whichcontains S and is stable under the loop space functor Ω : D(e) → D(e). Then forany triangulated derivator T, we have an equality of categories

(A.3.5) Hom!,Ω(S)(D,T) = Hom!,S(D,T) .

Heuristically, LΩ(S)D is the triangulated localization of D with respect to S.

Proof. For F an element of Hom!(D,T), the functor F (e) : D(e) → T(e) commuteswith homotopy colimits, hence it commutes in particular with the suspension func-tor. Since both D and T are triangulated, suspension and loop space functors areinverse to each other. Hence F (e) also commutes with Ω. It is then obvious thatF (e) sends S to isomorphisms if and only if it does so with Ω(S).

A.4. Commutativity. Let D be derivator that we assume pointed, strong andregular (Def. A.1.2). Let S be a class of morphisms in D(e). Assume that D admitsa left Bousfield localization LSD with respect to S. We obtain then a derivator LSDwhich is still pointed and strong. If it is also regular (see Remark A.3.2), we canconsider its stabilization St(LSD) as in A.2.

On the other hand, we can first consider the triangulated derivator St(D). We

still denote by S the image of the class S under the morphism of derivators Dstab−→

St(D). Suppose that the left Bousfield localization LΩ(S)St(D) by Ω(S) exists.We now have two constructions

Dγ

ttiiiiiiiiiii stab

++WWWWWWWWWWWWWW

LSD

stab $$II

II

St(D)

γxxqqqqq

St(LSD) LΩ(S)St(D)

and we claim that they agree, namely :

A.4.1.Theorem. With the above notations and hypotheses, the derivators LΩ(S)St(D)and St(LSD) are canonically equivalent, under D.


Proof. Both derivators are triangulated (for LΩ(S)St(D), see Remark A.3.3) andstrong. So, it suffices to show that for any triangulated strong derivator T, we havethe following equivalences of categories :

Hom!,S(D,T)

Hom!(LSD,T)≃

γ∗22fffffffffffffff

Hom!,S(St(D),T)(A.3.5)

≃ stab∗iiSSSSSSS

Hom!,Ω(S)(St(D),T)

Hom!(St(LSD),T)

≃stab∗

hhPPPPPP

Hom!(LΩ(S)St(D),T) .

≃γ∗

44iiiiiii

The two equivalences on the left-hand side as well as the lower-right one all fol-low from Theorem A.2.1 or Definition A.3.1. Finally, the equivalence stab∗ :Hom!,S(St(D),T)

∼−→ Hom!,S(D,T) requires a comment. Indeed, we have by Theo-

rem A.2.1 an equivalence stab∗ : Hom!(St(D),T)∼−→ Hom!(D,T) and it is straight-

forward to check that it preserves the above subcategories.

Acknowledgments : We would like to thank Bernhard Keller and Bertrand Toenfor some useful discussions. We would like also to thank Denis-Charles Cisinski forkindly pointing out a missing hypothesis in Theorem A.4.1.

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Paul Balmer, Department of Mathematics, UCLA, Los Angeles, CA 90095-1555, USA

E-mail address: [email protected]

URL: http://www.math.ucla.edu/∼balmer

Goncalo Tabuada, Departamento de Matematica, FCT-UNL, Quinta da Torre, 2829-

516 Caparica, Portugal

E-mail address: [email protected]

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