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Book project on derivators, volume I

(under construction)

Moritz Groth

January 26, 2016

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Contents

Abstract 7

Preface 9

1 Introduction and overview 13

1.1 From classical derived functors to derived categories . . . . . . . 14

1.2 From derived categories to derivators . . . . . . . . . . . . . . . . 16

1.3 Derivators arising in algebra, geometry, and topology . . . . . . . 18

1.4 Derivators and the calculus of Kan extensions . . . . . . . . . . . 21

1.5 The basic calculus in pointed derivators . . . . . . . . . . . . . . 23

1.6 Stable derivators and canonical triangulations . . . . . . . . . . . 24

1.7 Outlook on outlook and appendices . . . . . . . . . . . . . . . . . 27

I Motivation and background 29

2 Abelian categories and classical derived functors 31

2.1 Review of abelian categories . . . . . . . . . . . . . . . . . . . . . 31

2.2 Classical derived functors . . . . . . . . . . . . . . . . . . . . . . 35

2.3 Group cohomology as a derived limit . . . . . . . . . . . . . . . . 39

3 Derived categories of abelian categories 45

3.1 Towards derived cokernels . . . . . . . . . . . . . . . . . . . . . . 45

3.2 Cones and cofibers . . . . . . . . . . . . . . . . . . . . . . . . . . 49

3.3 Derived categories and derived functors . . . . . . . . . . . . . . 53

3.4 Derived categories as 2-localizations . . . . . . . . . . . . . . . . 57

3.5 Cones as derived cokernels . . . . . . . . . . . . . . . . . . . . . . 60

4 Coherent versus incoherent diagrams 63

4.1 Underlying diagram functors . . . . . . . . . . . . . . . . . . . . 63

4.2 The case of category algebras . . . . . . . . . . . . . . . . . . . . 67

4.3 Some explicit examples . . . . . . . . . . . . . . . . . . . . . . . . 71

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5 Derived categories as triangulated categories 735.1 The homotopy category of an abelian category . . . . . . . . . . 745.2 Distinguished triangles in the homotopy category . . . . . . . . . 765.3 Triangulated categories . . . . . . . . . . . . . . . . . . . . . . . 795.4 Exact morphisms and classical triangulations . . . . . . . . . . . 825.5 Beyond triangulated categories . . . . . . . . . . . . . . . . . . . 85

6 Kan extensions 916.1 Motivation and definition . . . . . . . . . . . . . . . . . . . . . . 916.2 Final functors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 946.3 Pointwise Kan extensions . . . . . . . . . . . . . . . . . . . . . . 976.4 Basic properties and first examples . . . . . . . . . . . . . . . . . 1026.5 Cokernels and kernels via Kan extensions . . . . . . . . . . . . . 106

II The basic calculus in derivators 111

7 Basics on derivators 1137.1 Prederivators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1137.2 Derivators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1197.3 Examples of derivators . . . . . . . . . . . . . . . . . . . . . . . . 1247.4 Limits versus homotopy limits . . . . . . . . . . . . . . . . . . . . 1307.5 The case of topological surfaces . . . . . . . . . . . . . . . . . . . 1337.6 Commuting limits . . . . . . . . . . . . . . . . . . . . . . . . . . 137

8 Homotopy exact squares 1398.1 The calculus of mates . . . . . . . . . . . . . . . . . . . . . . . . 1398.2 Homotopy exact squares and Kan extensions . . . . . . . . . . . 144

9 Basics on pointed derivators 1539.1 Definition and first properties . . . . . . . . . . . . . . . . . . . . 1539.2 Suspensions, loops, cofibers, and fibers . . . . . . . . . . . . . . . 1569.3 Cartesian and cocartesian squares . . . . . . . . . . . . . . . . . . 1629.4 First applications of cocartesian squares . . . . . . . . . . . . . . 1689.5 Fiber and cofiber sequences . . . . . . . . . . . . . . . . . . . . . 1719.6 Exceptional inverse image functors . . . . . . . . . . . . . . . . . 174

10 Parametrized Kan extensions 18110.1 Parametrized Kan extensions in categories . . . . . . . . . . . . . 18110.2 Exponentials for derivators . . . . . . . . . . . . . . . . . . . . . 18410.3 Parametrized Kan extensions in derivators . . . . . . . . . . . . . 188

11 The yoga of colimiting cocones 19311.1 Colimiting cocones in derivators . . . . . . . . . . . . . . . . . . . 19311.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20011.3 Detection lemmas for colimiting cocones . . . . . . . . . . . . . . 203

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III Pointed and stable derivators 211

12 Iterated cofibers, total cofibers, and iterated cones 213

12.1 Iterated cofibers of morphisms . . . . . . . . . . . . . . . . . . . 214

12.2 Cones of compositions . . . . . . . . . . . . . . . . . . . . . . . . 218

12.3 Total cofibers of squares . . . . . . . . . . . . . . . . . . . . . . . 222

12.4 Examples of total cofibers . . . . . . . . . . . . . . . . . . . . . . 224

12.5 Total cofibers as iterated cones . . . . . . . . . . . . . . . . . . . 227

12.6 More examples of total cofibers . . . . . . . . . . . . . . . . . . . 231

13 More on iterated cofibers and total cofibers 235

13.1 Colimits of punctured cubes . . . . . . . . . . . . . . . . . . . . . 235

13.2 Iterated cones and total cofibers, again . . . . . . . . . . . . . . . 238

13.3 Parametrized (co)exceptional inverse image functors . . . . . . . 241

14 Loop objects in pointed derivators 245

14.1 Loop objects as monoid objects . . . . . . . . . . . . . . . . . . . 245

14.2 Loop objects as group objects . . . . . . . . . . . . . . . . . . . . 251

14.3 Signs from loop squares . . . . . . . . . . . . . . . . . . . . . . . 254

14.4 Two-fold loop objects as abelian group objects . . . . . . . . . . 259

15 Basics on stable derivators 263

15.1 Definition and first properties . . . . . . . . . . . . . . . . . . . . 263

15.2 Examples of stable derivators . . . . . . . . . . . . . . . . . . . . 268

15.3 The additivity of stable derivators . . . . . . . . . . . . . . . . . 271

15.4 Strength of a derivator . . . . . . . . . . . . . . . . . . . . . . . . 275

15.5 Canonical triangulations in stable derivators . . . . . . . . . . . . 276

15.6 Negative canonical triangulations . . . . . . . . . . . . . . . . . . 282

16 Spectra, Barratt–Puppe sequences, and octahedra 283

16.1 Spectra in pointed derivators . . . . . . . . . . . . . . . . . . . . 284

16.2 Barratt–Puppe sequences . . . . . . . . . . . . . . . . . . . . . . 289

16.3 Barratt–Puppe sequences in stable derivators . . . . . . . . . . . 295

16.4 Octahedron diagrams . . . . . . . . . . . . . . . . . . . . . . . . . 297

16.5 Octahedron diagrams in stable derivators . . . . . . . . . . . . . 299

17 Unstable versions of canonical triangulations 303

18 Outlook 305

A Some category theory 307

A.1 Adjunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 307

A.2 Limits and colimits . . . . . . . . . . . . . . . . . . . . . . . . . . 311

A.3 Basic 2-categorical terminology . . . . . . . . . . . . . . . . . . . 315

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B Examples of derivators 317B.1 Represented derivators . . . . . . . . . . . . . . . . . . . . . . . . 317B.2 Opposite derivators . . . . . . . . . . . . . . . . . . . . . . . . . . 318B.3 Homotopy derivators of model categories . . . . . . . . . . . . . . 319

Bibliography 323

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Abstract

In this series of books we give an introduction to the theory and applicationsof derivators. Introduced independently by Grothendieck, Heller, and Franke,derivators provide a purely categorical approach to homological algebra andhomotopical algebra. While stable derivators are an enhancement of triangu-lated categories, the theory of derivators is more general in that it also appliesto unstable homotopy theories like the homotopy theory of topological spaces.More generally, arbitrary Quillen model categories have underlying homotopyderivators, and these homotopy derivators encode the Quillen equivalence typesof combinatorial Quillen model categories.

As a slogan, derivators are minimal extensions of the more classical derivedcategories of abelian categories or homotopy categories of model categories toa framework with a well-behaved calculus of homotopy limits, homotopy col-imits, and homotopy Kan extensions. In this approach these constructions arecharacterized by ordinary universal properties, making them accessible to ele-mentary categorical techniques. One main point about derivators is that thiscalculus encodes many interesting constructions arising in various areas of puremathematics.

Starting with a short review of homological algebra (including a discussionof derived categories) and Kan extensions, in this volume we give a thoroughmotivation for the definition of a derivator. We use a proof of the result that(strong) stable derivators canonically take values in triangulated categories asa pretext to study various basic aspects of derivators. As we shall see, thecalculus of homotopy Kan extensions specializes, for example, to (functorial)cone constructions, Barratt–Puppe sequences, (higher) octahedron diagrams,and total cofibers. Along the way we establish various convenient tools forthe calculus of (homotopy) Kan extensions, which are of constant use both inthis volume and in the sequels. We conclude this first volume by a discussion ofvariants of canonical triangulations for pointed derivators or additive derivators,thereby yielding canonical pretriangulations in the sense of Beligiannis.

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Preface

This book is the first of three volumes of a series of books providing an intro-duction to the basic theory of derivators. A detailed introduction describing themain goals and containing a rough outline of the content of this volume is givenin §1. While additional introductory remarks can also be found at the beginningof the respective chapters, in §18 we also briefly indicate the content of the nexttwo volumes. Before we turn to the detailed introduction to this volume, inthis preface we include a few general remarks concerning prerequisites for thisvolume, the philosophy of this series of books, closely related theories, the factthat this first volume is essentially self-contained, suggestions for a first reading,as well as acknowledgements.

Prerequisites for this volume. Basic acquaintance with homological algebraand/or homotopy theory, mostly for motivational purposes, is assumed, as isfamiliarity with the basic language from category theory. For convenience, werevisit some facts from category theory in Appendix A. However, backgroundknowledge from homotopical algebra and higher category theory is not assumed.

Philosophy of the series of books. This series of books gives an introduction tothe theory and techniques of derivators as well as to their applications. Whilesome of the proofs given in the first two volumes may seem rather detailed(in particular, since the arguments are often very similar), this is intended.One main goal of these volumes is precisely to show that the basic calculus ofderivators, once a few elementary tools are in place, allows for rather mechanicalproofs, using the same kind of reasoning over and over again.

While the audience I had in mind in writing these books is not disinclined tocategory theory, the typical reader was imagined as a working mathematicianin pure mathematics. Consequently, in the first two volumes of this series ofbooks I on purpose avoid more advanced notions from category theory (likeGrothendieck fibrations, ends, and coends) as well as techniques from homotopytheory and homotopical algebra (these will, however, show up in volume three).

Following this philosophy, in this first volume we minimized the amount of 2-categorical terminology. It was an additional goal to keep each individual volumeat a reasonable length, while still including many details. As a consequenceand in contrast to a more systematic treatment of the subject, a discussionof morphisms of derivators and natural transformations appears only at the

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10 CONTENTS

beginning of volume two [Gro16a]. As a general rule, even at the risk of resultingin a more lengthy exposition, in order to not disturb the flow of the rathersimple main ideas of the theory, I often tried to postpone the discussion of moretechnical aspects of the theory to a later stage.

Closely related theories. The theory of derivators belongs to the vast field ofabstract or axiomatic homotopy theory, homotopical algebra, or higher categorytheory. By now there is a plethora of different approaches to such a theory, allof them having their merits and drawbacks. A distinguishing feature of thetheory of derivators is that, while using elementary techniques from ordinarycategory theory only, it allows us to manipulate derived or homotopy-invariantconstructions.

Alternative, closely related, and very powerful approaches are given byQuillen model categories [Qui67, Hov99, DS95],∞-categories or, synonymously,quasi-categories or weak Kan complexes (see [Joy, Joy02, Joy08], [Lur11, Lur09],or [Gro10]), and the more classical triangulated categories ([Pup67, Ver96] or[Nee01, HJR10]). Moreover, besides ∞-categories there are many additionalaxiomatizations of a theory of (∞, 1)-categories (see [Ber07, Ber10, Cam13] formany references), and, conjecturally, nice such (∞, 1)-categories should haveunderlying homotopy derivators.

This volume is self-contained up to one and a half black boxes. While inthis volume we do not assume background knowledge on model categories, forreaders with such a background we include in §B.3 a detailed construction ofthe homotopy derivator of a combinatorial model category. Besides some ratherelementary reasoning only, for that purpose we use the existence of projectiveand injective model structures on diagram categories in combinatorial modelcategories. While the existence of projective model structures follows from afairly elementary lifting lemma for model structures [Hov99], the existence ofthe injective model structure is more complicated, and we refer the reader to[Lur11].

Suggestions for a first reading. This book relies mostly on basic languagefrom category theory only. In order to get acquainted with the basic notationused throughout this book, we suggest the reader to skim through §§A.1-A.2.

While this book can be read in a linear fashion from the first page to the lastpage, this is not necessarily the most rewarding way. After having read someof the introductory chapters or sections, we suggest the reader to jump moredirectly to sections she or he might be interested in and to only read about thebackground as needed. In particular, depending on the reader, we propose toskip the following parts.

(i) As already mentioned, in this book we do not assume background on aboutmodel categories. The readers without such a background are asked tosimply ignore the few results and examples related to homotopy derivatorsof model categories which are included in this book.

(ii) The more experienced reader is suggested to skip most of §§2-6, and to

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CONTENTS 11

directly begin with §7 (where derivators are defined).

(iii) On a first reading, we suggest the reader to skip §9.6 on (co)exceptionalinverse image functors.

(iv) We propose the reader to skip §11.1 on a first reading. Moreover, in§11.3 we suggest to first only take a look at Lemma 11.3.1 and its sampleapplication (Example 11.3.2).

(v) In §§12-13 we illustrate how to use the techniques established so far toset up a basic calculus of cubes and corresponding functors in pointedderivators. On a first reading, we suggest the reader to skip §§12.3-13.3.

(vi) ...

Acknowledgements. It is a great pleasure to begin by thanking my Ph.D. su-pervisor Stefan Schwede for having proposed to me to think about derivators inthe first place. My understanding of the subject was sharpened through variouscooperations, hence special thanks go to my coauthors Kate Ponto and MikeShulman [GPS14b, GPS14a], and Jan Stovıcek [GS14b, GS15b, GS14a, GS15a,GS16].

During the last years many colleagues and friends gave me (willingly ornot) the opportunity to discuss about derivators, and it is a pleasure to thankDimitri Ara, Peter Arndt, Denis-Charles Cisinski, Ivo Dell’Ambrogio, AndreJoyal, Georges Maltsiniotis, Justin Noel, Eric Peterson, George Raptis, UlrichSchlickewei, Timo Schurg, Mike Shulman, Greg Stevenson, and Jan Stovıcek.Moreover, I thank Martin Gallauer Alves de Souza, John Greenless, ThorgeJensen, Philipp Jung, Alexander Korschgen, Chrysostomos Psaroudakis, GregStevenson, and Jan Stovıcek for helpful comments on earlier versions of thisbook.

This first volume is based on various introductory talks on derivators, onseries of talks given at a summer school in Freiburg, at a workshop at theCRM in Bellaterra, and at a workshop in Barcelona, as well as on a courseon derivators given by the author at the University of Bonn during the winterterm 2014-2015. I thank the participants of all these activities for their interest,remarks, and questions.

While writing this volume, the author was first a postdoc at the MPIM inBonn, Germany, and then an assistent at the math department at the Universityof Bonn. Hence special thanks go to the Max-Planck-Gesellschaft and to theRheinische Friedrich-Wilhelms-Universitat. A good deal of the writing of thebook was done during a one-month visit at the CRM in Bellaterra, Spain, whichwas made possible through a research visiting grant from the CRM and theUniversity of Bielefeld. An additional two-weeks visit of Sabhal Mor Ostaigon the Isle of Skye, Scotland, was supported financially by the Anglo-Franco-German representation theory network. The author is very grateful for thisfinancial support as well as for the hospitality and the nice atmosphere at allthese places.

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Chapter 1

Introduction and overview

This book offers an introduction to the theory of derivators, providing one ofthe many different approaches to homological algebra and homotopical alge-bra. Starting with the context of homological algebra, one goal of derivatorsis to address the following problem: the passage from the category Ch(A) ofchain complexes in an abelian category A to the corresponding derived categoryD(A) [Ver96, Hap88] results in a loss of information. In the theory of derivators,this problem is addressed by also keeping track of derived categories of diagramcategories D(AA), A ∈ Cat , and various constructions between them, therebyenhancing the classical derived categories to a more flexible notion.

A very similar situation arises in homotopy theory where the passage fromthe category Top of topological spaces to the homotopy category Ho(Top) shouldbe suitably refined. In this case, this is achieved by also considering homotopycategories of diagram categories Ho(TopA), A ∈ Cat , together with suitable func-tors between them. Among these functors are the homotopy limit and homotopycolimit functors holimA,hocolimA : Ho(TopA)→ Ho(Top) [BK72, BV73, Vog73]as well as the more general homotopy Kan extension functors. These functorsencode various interesting constructions including (reduced) supensions, loopspaces, total cofibers, Borel constructions, homotopy fixed points, and spectri-fications of prespectrum objects.

It turns out that, besides these two central examples, there is a plethora ofadditional situations leading to such ‘systems of diagram categories’ A 7→ D(A).By definition a derivator simply axiomatizes key formal properties of such sys-tems, leading to a purely categorical framework to study typical situations aris-ing in algebra, geometry, and topology. In particular, both derived (co)limitsand derived Kan extensions as well as homotopy (co)limits and homotopy Kanextensions are characterized by ordinary universal properties, making them ac-cessible to elementary categorical techniques.

Derivators were introduced independently by Grothendieck [Gro], Heller[Hel88], Franke [Fra96]. We refer the reader to the webpage mentioned in [Gro]for a comprehensive bibliography on the subject. While this book offers an in-troductory account of basic aspects of the theory of derivators, we will come

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14 CHAPTER 1. INTRODUCTION AND OVERVIEW

back to a systematic treatment of additional aspects in the sequels.In the remainder of this introduction we try to fill the above paragraphs with

more life. This introduction is a rather informal account, while the precise math-ematics only begin in §2. We include some references in the remainder of thisintroduction; further references can be found, together with the correspondingdetails, in the appropriate sections.

1.1 From classical derived functors to derivedcategories

To begin with one typical input leading to derivators, we stick to the frameworkof homological algebra [Rot79, Wei94, GM03] and consider, for example, cate-gories of modules over rings, categories of abelian sheaves on topological spaces[God73, Ive86], or more general abelian categories. Classical homological alge-bra is the study of such abelian categories and additive functors between them.This includes tensor product and hom functors associated to bimodules overrings and direct and inverse image functors associated to maps between ringedspaces.

Let us recall that abelian categories essentially axiomatize key properties ofthe calculus of kernels, cokernels, and short exact sequences of module homo-morphisms. Correspondingly, there is the class of exact functors between abeliancategories, i.e., functors preserving short exact sequences (these functors are, inparticular, additive). However, typical examples of additive functors, like theones mentioned above, fail to be exact. In many cases the functors under con-sideration preserve exactness on one side, for example such a functor F : A → Bsends short exact sequences 0→ X ′ → X → X ′′ → 0 in A to exact sequences

FX ′ → FX → FX ′′ → 0

in B. Since the kernel of the morphism FX ′ → FX is in general non-trivial, weare interested in systematic tools to study such kernels, and dually for functorspreserving exactness on the other side.

Classical homological algebra offers various techniques, based on projective,injective, and more general F -adapted resolutions, to define (classical) left andright derived functors. In the case of a right exact functor F : A → B as abovethis leads to additive left derived functors

LkF : A → B, k ≥ 0, and isomorphisms L0F ∼= F. (1.1.1)

Together with suitable natural connecting homomorphisms δ = δk these functorsassociate to a short exact sequence 0 → X ′ → X → X ′′ → 0 in A a naturallong exact sequence

. . .→ (L1F )(X)→ (L1F )(X ′′)δ→ FX ′ → FX → FX ′′ → 0 (1.1.2)

in B. These long exact sequences are convenient for both calculational and the-oretical purposes. Moreover, they can be characterized by universal propertiesin terms of universal δ-functors [Gro57] as we recall in §2.

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1.1. FROM DERIVED FUNCTORS TO DERIVED CATEGORIES 15

Implicitly in the above construction we use the trivial observation that everyadditive functor F : A → B yields, by a levelwise application, an additive functorF : Ch(A) → Ch(B) between the corresponding categories of chain complexes.One easily checks that the original functor F : A → B is exact if and only ifthe induced functor F : Ch(A) → Ch(B) preserves quasi-isomorphisms (let usrecall that a chain map C → D is a quasi-isomorphism if the induced mapsHkC → HkD, k ∈ Z, between homology objects are isomorphisms). The pointof this different characterization is that it suggests the following perspectiveon derived functors: try to approximate the functor F : Ch(A) → Ch(B) in auniversal way by a functor which preserves quasi-isomorphisms.

The natural domain for such functors are the classical derived categories ofabelian categories [Ver96, Har66, KS06]. Ignoring set-theoretic issues for thetime being, let us recall that the derived category D(A) of A is obtained fromCh(A) by inverting the class of quasi-isomorphisms. The resulting localizationfunctor

γ : Ch(A)→ D(A) (1.1.3)

has the defining universal property that every other functor Ch(A)→ C whichsends quasi-isomorphisms to isomorphisms factors uniquely through γ.

To relate this to classical derived functors, we again consider our right exactfunctor F : A → B together with the induced functor F : Ch(A) → Ch(B). Ingeneral, it is not possible to find a dashed arrow in

Ch(A)F //

γ

=

Ch(B)

γ

D(A)@F// D(B)

making the above diagram commute. As we recall in §3, a left derived functoror left hyperderived functor is a functor

LF : D(A)→ D(B)

together with a universal natural transformation LF γ → γ F . These de-sired ‘exact approximations’ (and their duals) exist in typical situations andcan be constructed by means of suitable resolutions. The classical derived func-tors (1.1.1) are recovered by passing to homology objects and using stalk com-plexes, i.e., there are natural isomorphisms

(LkF )(X) ∼= (Hk LF )(X), k ∈ Z, (1.1.4)

where for X ∈ A we also denote the associated chain complex concentrated indegree zero by the same symbol.

As an illustration of such left derived functors, in §3 we establish the follow-ing example: given an arbitrary abelian category A, the classical constructionof the cone of a chain map (together with a suitable universal natural transfor-mation) simply amounts to constructing the left derived cokernel functor. Thus,

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denoting the abelian category of morphisms in A by A[1], we consider the rightexact cokernel functor F = cok: A[1] → A and obtain the cone functor as thecorresponding left derived functor,

Lcok = C : D(A[1])→ D(A). (1.1.5)

1.2 From derived categories to derivators

As we just recalled, the natural domain for derived functors are derived cate-gories, and these derived categories come with localization functors (1.1.3). Itis important to be aware of the fact that the categories Ch(A) and D(A) haverather different formal properties. While the category Ch(A) is again abelianand hence is, in particular, finitely complete and finitely cocomplete, the cat-egory D(A) is typically rather ill-behaved. Often the only limits and colimitswhich exist in D(A) are (finite) products and coproducts. More importantly,derived limits and derived colimits can neither be constructed nor characterizedusing the derived category D(A) only.

A classical way of adressing these bad properties of D(A) is by endowingthem with more structure. Derived categories are often considered as triangu-lated categories [Pup67, Ver96, Nee01, HJR10], hence together with a triangu-lation consisting of an equivalence Σ: D(A) → D(A) and a class of so-calleddistinguished triangles. These triangles are diagrams of the form

X → Y → Z → ΣX, (1.2.1)

and this structure is then subject to a certain list of axioms; see §5. A typ-ical slogan is that the distinguished triangles in D(A) are certain shadows ofshort exact sequences in Ch(A). It turns out that they essentially arise fromiterations of the derived cokernel construction. Moreover, by means of the iden-tifications (1.1.4), the distinguished triangles can be used to recover the longexact sequences (1.1.2).

Triangulated categories are very sucessful and have already been applied fora long time in various areas of pure mathematics, including algebra, algebraicgeometry, and topology, and this will certainly continue. Nevertheless, fromthe very beginning on (as can be seen already in the introduction to [Hel68])it was obvious that the axioms of a triangulated category have certain defects— reminiscent of the fact that the passage from Ch(A) to D(A) (via (1.1.3))results in a loss of information. We conclude §5 by a short discussion of some ofthese defects. One goal of this book is to show that these problems are avoidedby the theory of stable derivators.

As mentioned above, triangulations consist of additional structure imposedon certain additive categories, like derived categories of abelian categories. Thethird object in a distinguished triangle X → Y → Z → ΣX is often referred toas ‘the’ cone of the morphism X → Y . A crucial observation, however, is that ingeneral these cones do not depend functorially on the morphism in D(A), i.e.,in general, there is no cone functor D(A)[1] → D(A). But, as we already saw,

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1.2. FROM DERIVED CATEGORIES TO DERIVATORS 17

there is a functorial cone construction (1.1.5), and a more precise statement isthat this cone functor does not factor through D(A)[1],

D(A[1])C //

D(A)

D(A)[1].

@C

::

To put this in words: there is a functorial cone construction defined on thederived category of the morphism category but not on the morphism categoryof the derived category. As an upshot, this suggests that one should refine thepassage A 7→ D(A) by also keeping track of the category D(A[1]).

To continue building towards derivators we make the following trivial, seem-ingly picky observation. Let us recall that the cokernel of a chain map f : X → Yis some ‘colimit type construction’. In fact, while the cokernel is not the colimitof the diagram X → Y (which would be isomorphic to Y ), the cokernel cok(f)can be constructed as the pushout on the right in

Xf// Y

Xf//

Y

Xf

//

Y

0 0 // cok(f).

(1.2.2)

(One easily checks that this reduces to the usual universal property of the cok-ernel.) The first above step extends a morphism of chain complexes to a span ofchain complexes by adding the morphism X → 0, while the second step formsthe pushout square. From a more category theoretic perspective, this passageto the span and the passage to the pushout square (as opposed to the pushoutcorner only) are examples of Kan extensions [Bor94a, ML98]. In §6 we recallsome basics concerning Kan extensions, including a few examples and first keyproperties.

As an upshot, this suggests that in order to also encode a construction ofthe cone functor C : D(A[1]) → D(A) at the level of derived categories only,we should, using suggestive notation, also keep track of the derived categoriesD(Ap) and D(A) together with suitable functors between them. And formore sophisticated purposes it will be useful to also remember more generalderived categories of diagram categories, for example D(AA) for all small cate-gories A. (One can make a choice here and for example only consider suitablyfinite shapes.) Pursuing this more systematically, one is led to consider a 2-functor

DA : A 7→ D(AA), (1.2.3)

the derivator DA of the abelian category A, thereby also keeping track of derivedrestriction functors and induced transformations between them. The derivedcategory D(A) is recovered by considering diagrams of trivial shape, and onegoal of this book is to show that DA enhances D(A) to a more flexible notion.

DRAFT


As we discuss in §7, the notion of a derivator axiomatizes key propertiesof 2-functors like (1.2.3). For example, derived restriction functors have rightadjoints and left adjoints, given by derived limits, derived colimits, and derivedKan extension functors. It turns out that the more general derived Kan exten-sions can be calculated pointwisely in terms of certain derived limit and derivedcolimit expressions, and this key property is turned into one of the axioms of aderivator.

1.3 Derivators arising in algebra, geometry, andtopology

Before we expand in §§1.4-1.7 on the content of §§7-18, let us step back a littleand see what we discussed so far. Given an abelian category A, we essentiallystarted with the pair (Ch(A),WA) consisting of the category of chain complexestogether with the class of quasi-isomorphisms and refined the localization pro-cess

(Ch(A),WA) 7→ D(A) = Ch(A)[W−1A ].

As particularly interesting special cases, one can consider the category Mod(R)of modules over a ring R or the category Qcoh(X) of quasi-coherentOX -moduleson a scheme X. In those cases, (1.2.3) yields the derivator DR of the ring andthe derivator DX of the scheme, thereby enhancing the more classical derivedcategories D(R) [Hap87, Hap88, AHHK07] and D(X) [Cal05, Huy06, Lip09].

But pairs of the form (Ch(A),WA) are only one class of examples of cate-gories together with a chosen class of morphisms ‘which we would like to treatas isomorphisms’. For example, in homotopy theory [Whi78, Swi02, Str11] oneoften does not want to distinguish topological spaces as soon as they are con-nected by a zigzag of weak homotopy equivalences (let us recall that a continuousmap f : X → Y is a weak homotopy equivalence if π0(f) : π0(X) → π0(Y ) isbijective and if the induced maps πk(X,x0) → πk(Y, f(x0)), k ≥ 1, x0 ∈ X, arebijections). Denoting by WTop the class of all weak homotopy equivalences, thehomotopy category of spaces Ho(Top) is defined as the localization

(Top,WTop) 7→ Ho(Top) = Top[W−1Top].

Similarly to derived categories, also the category Ho(Top) lacks the existenceof most limits and colimits. Moreover, the powerful calculus of homotopy limits,homotopy colimits, and homotopy Kan extensions [BK72, Vog73] is not visible tothe homotopy category alone. Again, this can be fixed by refining the passage toHo(Top) by considering the (homotopy) derivator of spaces, a suitable 2-functor

HoTop : A 7→ Ho(TopA) = TopA[(WATop)−1]. (1.3.1)

Here, WATop denotes the class of natural transformations α : X → Y such that

all components αa : Xa → Ya, a ∈ A, are weak homotopy equivalences.There is a plethora of additional examples of such 2-functors arising in var-

ious areas of algebra, geometry, and topology. For example, stable homotopy

DRAFT

1.3. DERIVATORS ARISING IN NATURE 19

theory is the study of generalized cohomology theories and spectra. In stablehomotopy theory one wants to identify two spectra as soon as they are connectedby a zigzag of stable homotopy equivalences, i.e., maps inducing isomorphismson stable homotopy groups. Passing to the localization at the class WSp of allstable homotopy equivalences,

(Sp,WSp) 7→ SHC = Ho(Sp) = Sp[W−1Sp ],

one obtains the stable homotopy category SHC [Vog70, Ada74]. And, again, inthe background of this triangulated category there is a (homotopy) derivator ofspectra, a suitable 2-functor

Sp : A 7→ Ho(SpA) = SpA[(WASp)−1]. (1.3.2)

For additional explicit examples of such 2-functors we refer the reader to §7,§15.2, Appendix B, and the references given therein.

Before we comment on the common core of the above examples, we reiteratethat in this book we do not expect the reader to have background knowledge onQuillen model categories, ∞-categories, or higher category theory. The point ofthe following paragraphs is merely to put the theory of derivators into contextand to mention how it relates to these other theories. These remarks imply thatthere are many additional interesting examples of derivators.

Now, the common core of all the above-mentioned examples is that theyarise from a sufficiently well-behaved pair (C,W ) consisting of a category Cand a class W of weak equivalences, i.e., a class of morphisms we would like totreat as isomorphisms. In such situations we are mainly interested in homotopy-invariant constructions, i.e., constructions which send weakly equivalent inputdata to weakly equivalent output data. While ordinary category theory is avery powerful language, the study of categories with weak equivalences andhomotopy-invariant constructions requires more refined techniques.

The theory of Quillen model categories [Qui67, Hov99, DS95] provides onesuch set of tools. (From now on in this book by model category we alwaysmean Quillen model category.) The basic idea is that by also taking into ac-count suitable classes of cofibrations and fibrations one axiomatizes some formalproperties of obstruction theory. It turns out that for every model categoryMwith weak equivalences W the passage to the homotopy category

(M,W ) 7→ Ho(M) =M[W−1]

factors through the construction of the homotopy derivator

HoM : A 7→ Ho(MA) =MA[(WA)−1];

related to this see [CS02, Cis03] in the general case and [Gro13] in the combi-natorial case. Moreover, the Quillen equivalence type of a combinatorial modelcategory is encoded by its homotopy derivator [Ren09]; see Appendix B.

A different approach to deal with homotopy-invariant constructions is givenby the theory of ∞-categories or, synonymously, weak Kan complexes, quasi-categories, and quategories. Introduced by Boardman–Vogt [BV73] in their

DRAFT


study of homotopy-invariant algebraic structures on topological spaces, ∞-categories were recently intensively studied by Joyal [Joy08], Lurie [Lur09,Lur11], and others; see [Gro10] for an introduction emphasizing the philoso-phy of this approach. In [GPS14b] there is a sketch proof that every completeand cocomplete∞-category has an underlying homotopy derivator. And a com-bination of work of Renaudin [Ren09] and Lurie [Lur09] (both relying on earlierwork of Dugger [Dug01]) gives strong evidence for the fact that the equiva-lence type of a locally presentable ∞-category can be reconstructed from thehomotopy derivator.

Thus, model categories and ∞-categories give rise to derivators, and thereare hence plenty of explicit examples of derivators. Among these are derivatorsassociated to rings, schemes, differential-graded algebras [Hin97, SS00, Fre09],and ring spectra [HSS00, EKMM97, MMSS01]. Additional examples arise in sta-ble module theory [EJ00, Hov02, Bec14] as well as in equivariant stable [MM02,LMSM86], motivic stable [Voe98, MV99, Jar00], and parametrized stable homo-topy theory [MS06, ABG+09, ABG10]. (All these examples are actually stablederivators (see §1.6) and the results from abstract representation theory alludedto in §18 hence apply to all these examples.) If one focuses on one such exam-ple only, then passing to the homotopy derivator simply amounts to a differentpackaging of the given abstract homotopy theory.

As already mentioned, we do not assume the reader to have backgroundknowledge on model categories or∞-categories. Those readers are consequentlyasked to take the existence of the above examples as a black box result. Thefocus of this book is not on showing how derivators arise. Instead, the maingoal is to illustrate how one works with such derivators.

At the same time we also want to satisfy readers with some background fromhomotopical algebra. Since a detailed, reasonably self-contained proof that ar-bitrary model categories have homotopy derivators [Cis03] is rather involved werefrain from including such a proof here. As a compromise, in §B.3 we discussthe construction of homotopy derivators associated to combinatorial model cat-egories [Gro13]. This class of examples is more easily established and it alreadyincludes many interesting examples. In particular, all the above-mentioned ex-amples of derivators can be realized that way.

Besides these homotopical examples, there are also the following more classi-cal ones. Given a complete and cocomplete category C, we obtain the associatedrepresented derivator simply by forming diagram categories systematically,

y(C) : A 7→ CA. (1.3.3)

(Of course, these examples are subsumed by the homotopical examples if onepasses to the corresponding pair (C, IsoC) consisting of the category and the classof isomorphisms considered as weak equivalences.) While represented derivatorsare not the motivational examples for the theory of derivators, these examplesare helpful to check our intuition and to make plausibility checks for results.Moreover, as we see in this series of books, many of the classical concepts fromcategory theory have rather obvious variants in derivator theory, and in suchsituations one should make sure that the notions agree on the overlap.

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1.4. DERIVATORS AND THE CALCULUS OF KAN EXTENSIONS 21

1.4 Derivators and the calculus of Kan exten-sions

After this short detour we now continue with the outline of the content of thisbook. As already mentioned, in §7 we define derivators as suitable systems ofdiagram categories A 7→ D(A) coming with restriction functors. One key axiomof a derivator is that these restriction functors have adjoints on both sides,and these adjoints are referred to as Kan extension functors. As a special case,every derivator admits abstract limit and colimit functors. These Kan extensionfunctors are a common generalization of

(i) limits, colimits, and Kan extensions in ordinary complete and cocompletecategories,

(ii) derived limits, derived colimits, and derived Kan extensions between de-rived categories associated to abelian categories, and

(iii) homotopy limits, homotopy colimits, and homotopy Kan extensions be-tween homotopy categories associated to model categories or∞-categories,like the ones mentioned above.

Besides two rather obvious axioms, the remaining key axiom asks for point-wise formulas, thereby making precise that Kan extensions can be calculatedpointwisely using limits and colimits.

It turns out that these axioms suffice to extend a good deal of basic manipu-lation rules for limits and Kan extensions from ordinary category theory to thecontext of an arbitrary derivator. In particular, these rules then also apply inthe derived and the homotopical situation. The formalism behind these rulesresembles base change formulas or projection formulas arising in various areasof pure mathematics. More precisely, the formalism is governed by the notion ofa homotopy exact square, which is arguably the key technical tool in the theoryof derivators; see §8.

To illustrate the calculus of Kan extensions, in §9 we right away pass topointed derivators which are defined as derivators admitting a zero object, i.e.,an object which is simultaneously initial and final. Typical examples of pointedderivators are derivators of abelian categories (1.2.3), the derivator of pointedtopological spaces (defined as the obvious variant of (1.3.1)), and the derivatorof spectra (1.3.2). In every pointed derivator one can mimic the construction ofthe cokernel using Kan extensions (1.2.2) in order to define a cofiber functor,which, in the above examples, recovers the usual constructions from homologicalalgebra and homotopy theory. Minor variants lead to fibers, suspensions, loops,and cofiber sequences (see below), and we show that there are suspension-loopsadjunctions and cofiber-fiber adjunctions. We conclude §9 by a characterizationof pointed derivators by the existence of (co)exceptional inverse image functors.These are additional adjoints to Kan extension functors which exist for suitablefunctors in the context of pointed derivators.

DRAFT


With a view towards canonical triangulations, we would like to also collect afew lemmas about iterated cofiber constructions. Given a morphism (f : x→ y)in a pointed derivator, one can functorially associate to it a diagram lookinglike

xf//

y //

g

0

0 // zh //

x′

f ′

0 // y′,

(1.4.1)

such that the corners are populated by zero objects as indicated and such thatall squares are (homotopy) pushout squares. Taking the two squares on thetop only, we obtain a so-called (coherent) cofiber sequence. In the derivator ofpointed spaces this leads to the classical Barratt–Puppe sequences, while in thederivator of an abelian category this gives rise to the distinguished triangles asin (1.2.1).

As a preparation for the rotation axiom for canonical triangulations, wewould like to conclude that the morphism (f ′ : x′ → y′) in (1.4.1) is canonicallyisomorphic to Σf , which is to say that the threefold iteration of the cofiber cofis canonically isomorphic to the suspension. For this and for later applicationsit is convenient to establish two additional tools (see §§10-11).

(i) The identification Σf ∼−→ f ′ relies on a basic understanding of parametrizedKan extensions in derivators, which is a derivator version of the classicalcalculus of Kan extensions in functor categories.

(ii) In order to avoid repeating the same kind of homotopy (co)finality argu-ments again and again, it is useful to establish tools detecting pushouts,pullbacks, and more general (co)limiting (co)cones in larger diagrams.

The first of these tools is established in §10. We begin that chapter byrevisiting the classical counterpart of Kan extensions in functor categories andthen discuss some rudiments of the calculus of parametrized Kan extensions inderivators. This includes a construction of shifted derivators: for every derivatorD and small category B there is the derivator DB of coherent diagrams ofshape B in D . As we see in the sequel [Gro16a], this exponential constructionprovides the proper domains for parametrized Kan extensions, thereby definingKan extension morphisms of derivators.

The second of these tools is taken care of in §11. In that chapter we begin bysetting up the basic yoga of (co)cones and (co)limiting (co)cones in derivators.We then establish various related detection results which allow us to detectsuch (co)limiting (co)cones in larger diagrams. For example, we establish a‘workhorse proposition’ giving sufficient conditions that certain Kan extensionsprecisely amount to adding a (co)cartesian square, and we also obtain variants

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1.5. THE BASIC CALCULUS IN POINTED DERIVATORS 23

of this result. These detection results are often useful when it seems ‘obvious’that certain (co)cones are (co)limiting.

1.5 The basic calculus in pointed derivators

The tools established in §§10-11 are used a lot in later chapters and in the se-quels, and some first illustrations are provided in §§12-13. Based on (1.4.1), in§12 we begin with a short discussion of iterated cofibers of morphisms, show-ing that Σ ∼−→ cof3 in pointed derivators. We also consider coherent pairs ofcomposable morphisms and the associated fibers and cofibers, thereby obtaining‘unstable versions of the octahedron axiom for triangulated categories’. More

precisely, given a pair of composable morphisms (xf→ y

g→ z) ∈ D([2]) in apointed derivator D , we can functorially construct a coherent diagram lookinglike

Ff //

0

F (gf) //

Fg //

0

x

f//

yg

//

z

0 // Cf //

C(gf)

0 // Cg.

(1.5.1)

This diagram vanishes on the boundary points as indicated, the upper squaresare pullback squares, the lower squares are pushouts, and F and C denote thefiber and cone functors, respectively. In particular, there are cofiber squaresrelating Cf,C(gf), and Cg, as well as fiber squares relating Ff, F (gf), and Fg,and we hence establish results for fibers and cofibers which resemble the thirdNoether isomorphism theorem for quotients in abelian categories.

The largest part of §12 is devoted to a study of coherent squares in pointedderivators. Let D be a pointed derivator and let X ∈ D() be a coherent squarein D , i.e., a diagram looking like

xf//

g

y

g′

x′f ′// y′.

DRAFT


Given such a square we can form the pushout p = y tx x′ which comes with acanonical comparison map p→ y′. The total cofiber of X is

tcof(X) = C(p→ y′). (1.5.2)

Alternatively, we can also form partial cones in the horizontal or vertical direc-tion, leading to morphisms

C1(X) : Cf → Cf ′ and C2(X) : Cg → Cg′, (1.5.3)

and we can then pass to iterated cones C(C1X) and C(C2X). We show that inevery pointed derivator total cofibers and these iterated cones are canonicallyisomorphic. Moreover, we illustrate these functors by quite some examples. Onegoal of this chapter is to illustrate that, once a few lemmas are in place, it israther simple to develop calculi of diagrams of simple shapes in derivators.

In §13 we revisit these functors and the canonical isomorphisms betweenthem from a more systematic perspective. This includes a first discussion of(co)limits of punctured cubes and the observation that partial cones, iteratedcones, and total cofibers are instances of exceptional inverse image functors.The results of these two chapters belong to a fairly rich calculus of squares,cubes, and hypercubes in derivators, and a few additional steps of this calculuswill be developed in the sequel [Gro16a].

Preparing the ground for the proof of the additivity of stable derivators, in§14 we include a more systematic discussion of loop objects in pointed derivators.Recall the classical fact that the concatenation of loops turns the loop space ofa pointed topological space into a group object in the homotopy category ofpointed spaces. Moreover, these group object structures are abelian for two-fold loop spaces. Here we mimic these constructions in the context of a pointedderivator and show that certain abstractly defined concatenation maps turn loopobjects into group objects. Using the Eckmann–Hilton trick, we also show thatthe two possibly different group structures on twofold loop objects agree and areabelian. The results obtained here and the dual results for suspension objectsallow us to identify minus signs induced by certain canonical identifications,and this will be useful in the discussion of the rotation axiom of canonicaltriangulations.

1.6 Stable derivators and canonical triangula-tions

A pointed derivator is stable if it has the property that a square is a pullbacksquare if and only if it is a pushout square. In §15 we define stable derivatorsand collect a plethora of examples arising in algebra, geometry, and topology.This includes the derivator of an abelian category, the derivator of spectra, andthe ones mentioned close to the end of §1.3. While there are plenty of interestingexamples, the phenomenon of stability is invisible to ordinary category theory.

DRAFT

1.6. ENHANCING TRIANGULATED CATEGORIES 25

.... . .

. . .

. . .

ΩFf //

Ωx //

0

0 // Ωy //

Ff //

0

0 // x

f//

y //

0

0 // Cf //

Σx //

0

. . .

0 //

. . .

Σy //

. . .

ΣCf. . .

Figure 1.1: Barratt–Puppe sequence in a pointed derivator.

In fact, a represented derivator (1.3.3) is stable if and only if the representingcategory is trivial.

We begin the chapter by a few sanity checks and verify that some of theexpected properties are satisfied in stable derivators.

(i) Morphisms are isomorphisms if and only if the cones or the fibers vanish.

(ii) There is a 5-lemma for morphisms of cofiber squares.

(iii) The suspension-loops and the cofiber-fiber adjunctions are equivalences.

(iv) A square is cocartesian if and only if the partial cones (see (1.5.3)) areisomorphisms if and only if the total cofiber (see (1.5.2)) is trivial.

This implies that every object is a twofold loop object, hence an abelian groupobject by the discussion of loop objects in §14, thereby showing that the valuesof a stable derivator are additive categories.

Stable derivators provide an enhancement of triangulated categories. As afirst evidence of this, we show in §15 that the values of (strong) stable derivatorscan be endowed with the structure of canonical triangulations. (The terminology‘canonical’ will be justified in the sequel [Gro16a].) The distinguished trianglescome from cofiber sequences in stable derivators, i.e., coherent diagrams look-ing like the two top squares in (1.4.1). The proof of this result is essentially aconsequence of a careful discussion of the cofiber construction, including a dis-cussion of diagrams similar to the bottom half of (1.5.1). The proof also givesa conceptual explanation of the minus sign in the rotation axiom, which turnsout to be closely related to the loop inversion in loop objects. These canoni-cal triangulations reproduce the classical triangulations on derived categories ofabelian categories [Ver67] as well as the triangulations on homotopy categoriesof stable model categories [Hov99, §7] or stable ∞-categories [Lur11, §1].

DRAFT


. . .. . .

. . .

Ωx //

0

. . .

Ωy //

Ff //

0

. . .

Ωz //

F (gf) //

Fg //

0

0 // x

f//

yg//

z //

0

0 // Cf //

C(gf) //

Σx //

0

0 // Cg //

Σy //

ΣCf //

0

. . .

0 //

. . .

Σz //

. . .

ΣC(gf) //

. . .

ΣCg. . .

Figure 1.2: Octahedron diagram in a pointed derivator.

The language of stable derivators fixes some of the typical defects of triangu-lated categories, and a first illustration of this is provided in §16. For example,at the level of derivators, cone constructions enjoy universal properties and theyare functorial. Moreover, a coherent morphism is equivalently specified by theassociated (doubly-infinite, coherent) Barratt–Puppe sequence. A similar resultis true in pointed derivators, in which case the Barratt–Puppe sequence asso-ciated to a morphism f : x → y is a diagram as in Figure 1.1. These diagramsenjoy the indicated exactness properties, and they hence encode all iteratedcofibers and fibers of the corresponding morphisms.

More interestingly, there is a similar statement for composable pairs of mor-phisms, showing that such a pair is equivalent to a coherent diagram lookinglike Figure 1.2. By construction these diagrams encode all iterated cofibers andfibers of both morphisms and their composition. In the stable case these di-agrams provide a refinement of the classical octahedron diagrams at the leveltriangulated categories. In particular, they are a convenient device to orga-nize canonical isomorphisms between various expressions involving suspensions,loops, cofibers and fibers applied to the morphisms and their composition. Thisobservation extends to longer chains of composable morphisms and we will comeback to this in the sequel [Gro16a] in the discussion of canonical higher trian-gulations in stable derivators.

The canonical triangulations in stable derivators also admit suitable variantsAdapt once the chapter is

written. in pointed derivators. In this case one axiomatizes certain unstable distinguishedtriangles on pointed categories, and in typical examples this datum results fromcertain canonical actions of group objects and coactions of suspension objects

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1.7. OUTLOOK ON OUTLOOK AND APPENDICES 27

in pointed derivators; see §17.

1.7 Outlook on outlook and appendices

Finally, BLABLA. While we conclude this book by an outlook on such topics,Todo.

a detailed treatment will only appear in the sequels [Gro16a, Gro16b].

DRAFT


DRAFT

Part I

Motivation and background

29

DRAFT

Chapter 2

Abelian categories andclassical derived functors

We assume that the reader is familiar with basic homological algebra, includingthe construction and interesting examples of classical derived functors. Never-theless, mostly to set the stage and to smoothen the transition to the theoryof derived categories, here we quickly recall some basics. For more details andexamples we refer the reader to the literature; see for example [Rot79], the moreadvanced [Wei94, GM03, KS06], and the original [CE99, Gro57].

In §2.1 we recall basic definitions, emphasizing the distinction between prop-erties and structures. In §2.2 we sketch the construction of classical derivedfunctors and show that they yield universal δ-functors. In §2.3 we illustrate thenotion of derived limit and derived colimit functors by observing that groupcohomology and group homology, respectively, are special cases of these moregeneral notions.

2.1 Review of abelian categories

To put it as a slogan, homological algebra is the study of abelian categoriesand derived functors. We begin by recalling basic definitions concerning abeliancategories.

Definition 2.1.1. A preadditive category is a pointed category with finitebiproducts.

Thus, denoting the singleton by ∗, a preadditive category A satisfies thefollowing three defining axioms.

(i) The category A has a zero object, i.e., an object 0 ∈ A which is both finaland initial,

homA(X, 0) ∼= ∗ ∼= homA(0, Y ), X, Y ∈ A.

Such a category is also called a pointed category.

31

DRAFT

32 CHAPTER 2. CLASSICAL HOMOLOGICAL ALGEBRA

(ii) The category A has finite coproducts and finite products.

(iii) For any X,Y ∈ A the canonical map X tY → X×Y from the coproductto the product is an isomorphism.

Just to be completely specific, in matrix notation the canonical map inaxiom (iii) is given by (

1 0

0 1

): X t Y → X × Y.

Following standard notation, we write X ⊕ Y for any of X t Y ∼= X × Y andrefer to it as the biproduct or direct sum of X and Y .

Remark 2.1.2.

(i) All axioms of a preadditive category ask for properties. The only structureis the category itself.

(ii) Any preadditive category A can be canonically endowed with an enrich-ment in the category AbMon of abelian monoids. In fact, given morphismsf, g : X → Y , then the sum f + g : X → Y is the composition

f + g : X∆→ X ⊕X f⊕g→ Y ⊕ Y ∇→ Y,

where ∆ is the diagonal X → X×X and∇ is the fold map ∇ : Y tY → Y .We leave it to the reader to check that this defines an abelian monoidstructure on homA(X,Y ) with neutral element 0: X → 0 → Y and thatthe composition is bilinear.

(iii) A closely related perspective on this enrichment is as follows (we willcome back to this in §15). In a preadditive category, the fold map∇ : Y ⊕ Y → Y endows any Y ∈ A with the structure of an abelianmonoid object. We leave it as an exercise to verify that such an abelianmonoid structure is equivalent to specifying a lift of the represented func-tor homA(−, Y ) : Aop → Set against the forgetful functor AbMon→ Set,

AbMon

Aop

homA(−,Y )//

::

Set.

Dually, the diagonal map ∆: X → X ⊕ X endows any X ∈ A with thestructure of an abelian comonoid object. And one checks that such anabelian comonoid structure is equivalent to a lift of the corepresentedfunctor homA(X,−) : A → Set against the forgetful functor AbMon →Set.

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2.1. REVIEW OF ABELIAN CATEGORIES 33

Definition 2.1.3. A preadditive category A is additive if for every X ∈ A theshear map (

1 1

0 1

): X ⊕X → X ⊕X

is an isomorphism.

Lemma 2.1.4. The following are equivalent for a preadditive category A.

(i) The category A is additive.

(ii) Identity morphisms id : X → X have additive inverses in homA(X,X).

(iii) The abelian monoids homA(X,Y ) are abelian groups.

(iv) The fold map ∇ : Y ⊕ Y → Y endows Y ∈ A with the structure of anabelian group object.

(v) The diagonal map ∆: X → X ⊕X endows X ∈ A with the structure ofan abelian cogroup object.

Proof. This proof is left as an exercise.

Thus, to emphasize, from the defining ‘exactness properties’ of an additivecategory one can construct the structure of an enrichment in the category Abof abelian groups. By ‘exactness properties’ we mean the fact that certainlimits and colimits exist, that certain limits and colimits coincide or that certaincanonical maps are isomorphisms.

The good notion of functors between additive categories are additive func-tors, i.e., functors which preserve zero objects and finite direct sums. Given anadditive functor F : A → B it follows that the functor is automatically compat-ible with this enrichment, i.e., that the map homA(X,Y )→ homB(FX,FY ) isa group homomorphism for every X,Y ∈ A.

Finally, abelian categories are additive categories admitting kernels and cok-ernels and such that the first Noether isomorphism theorem is true. In moredetail, let us consider a morphism f : X → Y in an additive category withkernels and cokernels.

(i) The image im(f) of f is the kernel of the canonical map Y → cok(f) tothe cokernel, yielding the diagram

im(f)→ Y → cok(f).

(ii) The coimage coim(f) of f is the cokernel of the canonical map ker(f)→X from the kernel, hence there is the diagram

ker(f)→ X → coim(f).

DRAFT


One easily checks that f factors through a canonical map coim(f)→ im(f),

f : X → coim(f)→ im(f)→ Y.

In the case of the category of abelian groups A = Ab the map coim(f)→ im(f)is the usual isomorphism X/ ker(f)→ im(f) as guaranteed by the first Noetherisomorphism theorem.

Definition 2.1.5. An abelian category is an additive category satisfying thefollowing two properties.

(i) Every morphism has a kernel and a cokernel.

(ii) For every morphism f : X → Y the canonical map coim(f)→ im(f) is anisomorphism.

Thus, as a slogan, abelian categories axiomatize additive categories allowingfor the ‘usual calculus of short exact sequences’.

Examples 2.1.6.

(i) The category Ab and the category Mod(R) of left R-modules over a ring Rare abelian.

(ii) Given a topological space X, the categories of presheaves or sheaves on Xwith values in an abelian category are again abelian.

(iii) Given an abelian category A, the corresponding category Ch(A) of chaincomplexes is again abelian.

(iv) A category is abelian if and only if its opposite is abelian.

(v) IfAi, i ∈ I, are abelian categories, then so is the product category∏i∈I Ai.

In order to state a generalization of the final item in Examples 2.1.6 we recallthe following definition.

Definition 2.1.7. Let C be a category and let A be a small category. Thefunctor category or diagram category CA = Fun(A, C) has as objects thefunctors X : A→ C and as morphisms the natural transformations α : X → Y .

Lemma 2.1.8. Let A be an abelian category and let A be a small category. Thefunctor category AA = Fun(A,A) is again abelian.

Proof. This proof is left as an exercise, the only hint being that limits andcolimits are constructed pointwisely.

DRAFT

2.2. CLASSICAL DERIVED FUNCTORS 35

2.2 Classical derived functors

We now focus on additive functors between abelian categories. Let us recall thata functor F : A → B between additive categories is additive if it preserves zeroobjects and finite biproducts. As a consequence there is the following result.

Lemma 2.2.1. Any additive functor between abelian categories preserves splitshort exact sequences.


While, in general, short exact sequences are not preserved by additive func-tors, many functors showing up in nature preserve exactness ‘on one side’, mo-tivating the following definitions.

Definition 2.2.2. Let F : A → B be an additive functor between abelian cat-egories.

(i) The functor F is left exact if for every exact sequence 0 → X ′ → X →X ′′ in A also the sequence 0→ FX ′ → FX → FX ′′ is exact.

(ii) The functor F is right exact if for every exact sequence X ′ → X →X ′′ → 0 in A also the sequence FX ′ → FX → FX ′′ → 0 is exact.

(iii) The functor F is exact if it is left exact and right exact.

Examples 2.2.3. Let R,S be rings and let M be an R-S-bimodule.

(i) The tensor product M ⊗S − : Mod(S) → Mod(R) is right exact, but, ingeneral, not left exact.

(ii) The hom functor homR(M,−) : Mod(R) → Mod(S) is left exact, but, ingeneral, not right exact.

A formal reason for these examples is as follows.

Remark 2.2.4.

(i) By means of Proposition A.2.4 one checks that any abelian category isfinitely complete and that it is complete if and only if it admits products.There is a dual statement concerning colimits and coproducts.

(ii) Proposition A.2.11 implies that an additive functor between abelian cat-egories is left exact in the sense of homological algebra (Definition 2.2.2)if and only if it is left exact in the sense of category theory (Defini-tion A.2.10). Together with Lemma A.2.12 this yields a formal proofof the positive statements in Examples 2.2.3.

We next recall the classical construction of left derived functors. Thus, letA,B be abelian categories, let F : A → B be a right exact functor, and let us

DRAFT


for simplicity assume that A has enough projective objects. Hence, for everyobject X ∈ A we can find a projective resolution, i.e., an exact sequence

. . .→ P2 → P1 → P0ε→ X → 0

such that all Pi are projective. Since additive functors preserve zero objects, weobtain an induced chain complex FP ∈ Ch(B) which is given by

. . .→ FP2 → FP1 → FP0.

A final application of the homology functors Hn : Ch(B)→ B, n ≥ 0, concludesthe definition of the (classical) left derived functors LnF : A → B, i.e., weset

(LnF )(X) = Hn(FP ), n ≥ 0.

Since we assume the reader to be familiar with basic homological algebra,we content ourselves by claiming that the above construction is well-defined,that it can be extended to additive functors LnF : A → B, and that, using theright exactness of F , there is a canonical natural isomorphism L0F ∼= F .

As a first justification of the definition of classical left derived functor, let usrecall further that they measure systematically the deviation of F from beingleft exact. More precisely, as a consequence of the horseshoe lemma, every shortexact sequence 0→ X ′ → X → X ′′ → 0 in A gives rise to a long exact sequence

. . .→ (L1F )(X)→ (L1F )(X ′′)δ→ FX ′ → FX → FX ′′ → 0

in B. In particular, the image im(δ) : (L1F )(X ′′) → FX ′ of the connectinghomomorphism agrees with the kernel of FX ′ → FX.

An even better justification is given by the observation that classical leftderived functors yield the universal way to measure the deviation from beingleft exact (see Theorem 2.2.7). To make this precise we first recall the followingdefinitions.

Definition 2.2.5. Let A and B be abelian categories. A homological δ-functor T from A to B consists of

(i) additive functors Tn : A → B, n ≥ 0, and

(ii) of morphisms δn : Tn(X ′′) → Tn−1(X ′), n ≥ 1, associated to every shortexact sequence 0→ X ′ → X → X ′′ → 0 in A,

such that for every such short exact sequence we obtain a natural long exactsequence

. . .→ T1(X)→ T1(X ′′)δ1→ T0(X ′)→ T0(X)→ T0(X ′′)→ 0.

The naturality means that every morphism of short exact sequences givesrise to a commutative ladder in B. Thus, the connecting homomorphisms δn of(ii) assemble to suitable natural transformations.

DRAFT

2.2. CLASSICAL DERIVED FUNCTORS 37

Definition 2.2.6.

(i) A morphism of homological δ-functors α : S → T consists of naturaltransformations αn : Sn → Tn, n ≥ 0, such that for every short exactsequence 0→ X ′ → X → X ′′ → 0 in A the diagram

Sn(X ′′)δ //

αn

Sn−1(X ′)

αn−1

Tn(X ′′)δ// Tn−1(X ′)

commutes for all n ≥ 1.

(ii) A homological δ-functor T is universal if for every homological δ-functorS and every natural transformation α0 : S0 → T0 there is a unique mor-phism of homological δ-functors α : S → T extending the given α0.

Here is a justification of the seemingly adhoc construction of classical leftderived functors.

Theorem 2.2.7. Let F : A → B be a right exact functor between abelian cat-egories and let A have enough projective objects. The functors LnF, n ≥ 0,together with the connecting homomorphisms define a universal homological δ-functor.

Proof. We assume that the reader knows that the LnF together with the con-necting homomorphisms assemble to a homological δ-functor. To prove its uni-versality, let T be a homological δ-functor from A to B and let α0 : T0 → F bea natural transformation.

For n ≥ 1 we assume by induction that a partial morphism of δ-functorsconsisting of αj , 0 ≤ j < n, has already been constructed and that it is unique.We want to show that there is a unique way of extending it up to degree n.Given X ∈ A we choose a short exact sequence

0→ K → P → X → 0 (2.2.8)

such that P is a projective object. Since the left derived functors LnF, n ≥ 1,vanish on projective objects, by induction assumption we have a commutative(solid arrow) diagram with exact rows

TnXδ //

∃!αn

Tn−1K //

αn−1

Tn−1P

αn−1

0 // (LnF )Xδ// (Ln−1F )K // (Ln−1F )P.

An easy diagram chase shows that there is a unique dashed morphism αn suchthat the square on the left commutes. Thus, the compatibility with short exact

DRAFT


sequences of the form (2.2.8) implies that there is at most one way of definingthe desired αn : Tn → LnF .

As for the existence, we first show that the above morphism is independentof the choice of (2.2.8) and that these morphisms assemble into a natural trans-formation αn : Tn → LnF . To this end, let us consider a morphism f : X → Yin A. For any choice of short exact sequences (2.2.8) for X and Y (the rows inthe following diagram), we can extend f to a morphism of short exact sequences

0 // K //

P //

X

// 0

0 // L // Q // Y // 0.

Associated to this we obtain the following diagram in which all squares commutewith possibly the exception of the naturality square on the left,

TnXδ //

''

Tn−1K

((

TnY //

Tn−1L

(LnF )X

''

// (Ln−1F )K

((

(LnF )Yδ

// (Ln−1F )L.

A diagram chase shows that in that square the two possibly different compo-sitions TnX → (LnF )Y agree when composed with δ : (LnF )Y → (Ln−1F )L.Since again this morphism δ is a monomorphism, also the square on the left com-mutes. This shows that we constructed a natural transformation αn : Tn → LnFand that the result is independent of the choice of (2.2.8) (simply considerf = idX : X → X).

It remains to show that this natural transformation is compatible with arbi-trary short exact sequences (and not only the ones of the form (2.2.8)). For thispurpose, we note that for any short exact sequence 0 → X1 → X2 → X3 → 0and any short exact sequence 0 → K → P → X3 → 0 such that P is a projec-tive object we can extend the identity morphism id: X3 → X3 to a morphismof short exact sequences, yielding a commutative diagram

0 // K //

P //

X3

=

// 0

0 // X1// X2

// X3// 0.

DRAFT

2.3. GROUP COHOMOLOGY AS A DERIVED LIMIT 39

Similarly to the previous step, associated to this morphism we obtain a diagram

TnX3δ //

=

((

Tn−1K

))

TnX3//

Tn−1X1

(LnF )X3

= ''

// (Ln−1F )K

((

(LnF )X3δ

// (Ln−1F )X1

in which all squares with possibly the exception of the front square commute(by induction assumption and by the previous steps). Precomposing the twomorphisms TnX3 → (Ln−1F )X1 with id: TnX3 → TnX3, a short diagram chaseimplies that also the front face commutes, concluding the proof.

Dualizing these constructions we are led to right derived functors of left exactfunctors F : A → B. In this case, assuming the existence of enough injectiveobjects in A, every object X ∈ A has an injective resolution 0 → X → I0 →I1 → . . .. If we denote the truncated cochain complex by I, then the n-th(classical) right derived functor RnF is defined by

(RnF )(X) = Hn(FI), n ≥ 0.

These classical right derived functors assemble to universal cohomological δ-functors in the obvious sense.

For variants of these results and a discussion of examples and applicationswe refer the reader to the literature. Here we only mention that in the caseof Examples 2.2.3 we obtain the torsion products TorRn (M,N) and Ext-groupsExtnR(M,N) as typical examples of classical left derived and right derived func-tors, respectively.

2.3 Group cohomology as a derived limit

In this section we observe a close relation between group (co)homology andcertain derived limit and derived colimit functors. This is meant to be a firstillustration that derived (co)limit functors encode interesting constructions, andwe do not expect the reader to be familiar with group (co)homology.

Recall from Lemma 2.1.8 that for every abelian category A and every smallcategory A the functor category AA is again abelian. Assuming A to haveproducts in case A is not finite, the limit functor limA : AA → A exists and ispart of an adjunction

(∆A, limA) : A AA,

showing that limA is a left exact functor (see Lemma A.2.12 and Remark 2.2.4).If we, moreover, assume that AA has enough injective objects (which follows

DRAFT


for example from A having enough injective objects), then we can apply thetechniques of §2.2 in order to obtain (classical right) derived limit functors

RnlimA : AA → A, n ≥ 0.

Making dual assumptions onA, we obtain an adjunction (colimA,∆A) : AA Aand associated (classical left) derived colimit functors

LncolimA : AA → A, n ≥ 0.

Here we are only interested in the category A = Ab of abelian groups (butthere are obvious variants of what follows if we replace Ab by Mod(R), R acommutative ring). The category of abelian groups is complete and cocomplete.Moreover, it can be shown that all functor categories AbA have enough injectivesand projectives so that the derived (co)limit functors exist.

In our special situation we do not need this general statement since we canargue more directly. Given a discrete group G, there is an associated cate-gory, again denoted by G, which has one object ∗ only and such that G is thecorresponding endomorphism monoid,

homG(∗, ∗) = G.

We recall that the integral group ring ZG on G is defined as follows.The underlying abelian group is the free abelian group generated by the set G.Denoting by eg the generator associated to g ∈ G, the ring multiplication is thebilinear extension of the assignment

eg · eg′ = egg′ , g, g′ ∈ G.

The reader easily checks that there is an equivalence of categories

AbG ' Mod(ZG), (2.3.1)

so that AbG clearly has enough injectives and projectives. We will not distin-guish notationally objects corresponding to each other under this equivalence.

The following two constructions apply to modules M ∈ Mod(ZG) and arecentral to the theory of (co)homology of groups. Here we simplify notation andwrite gm = egm, g ∈ G,m ∈M .

(i) The abelian group of invariants is MG = m ∈ M | gm = m, g ∈ Gand this defines a functor (−)G : Mod(ZG)→ Ab.

(ii) The abelian group of coinvariants is MG = M/gm −m | g ∈ G,m ∈M, defining a functor (−)G : Mod(ZG)→ Ab.

One can check directly that the invariants functor is left exact and that thecoinvariants functor is right exact. A different way of seeing this is as follows.We observe that the integral group ring ZG comes with an augmentation map

ε : ZG→ Z :∑

riegi 7→∑

ri,

DRAFT


which is easily seen to be a ring homomorphism. Restriction of scalars on theleft and on the right applied to the left and right regular bimodule ZZZ yields

ZGZZ and ZZZG,

respectively. We refer to these G-actions as trivial G-actions.

Lemma 2.3.2. Let G be a discrete group.

(i) There is a natural isomorphism (−)G ∼= homZG(Z,−) : Mod(ZG) → Ab,where Z is endowed with the trivial left G-action.

(ii) There is a natural isomorphism (−)G ∼= Z⊗ZG− : Mod(ZG)→ Ab, whereZ is endowed with the trivial right G-action.


Together with Examples 2.2.3 this also yields the claimed exactness prop-erties. As a special case of Ext- and Tor-functors we can make the followingdefinitions (but refer the reader to almost any book on homological algebra or,for example, [Bro94] for more details).

Definition 2.3.3. Let G be a discrete group, let M ∈ Mod(ZG), and let n ≥ 0.

(i) The n-th group cohomology of G with coefficients in M is the abeliangroup

Hn(G;M) =(Rn(−)G

)(M) ∼= ExtnZG(Z,M).

(ii) The n-th group homology of G with coefficients in M is the abeliangroup

Hn(G;M) =(Ln(−)G

)(M) ∼= TorZGn (Z,M).

The equivalences (2.3.1) offer a different perspective on group (co)homology.To make this precise, we include the following definition, which is a ratherobvious weakening of the assumption that two functors are naturally isomorphic.

Definition 2.3.4. Let F1 : C1 → D1 and F2 : C2 → D2 be functors. The functorsF1, F2 are equivalent, in notation F1 ' F2, if there are equivalences φ : C1 'C2, ψ : D1 ' D2 and a natural isomorphism ψ F1

∼= F2 φ,

C1F1 //

φ '

∼=

D1

ψ'

C2F2

// D2.

The point is that equivalent functors are often equally good for many practi-cal purposes. Let us say a property P of a functor is invariant if for equivalentfunctors F1 ' F2 the functor F1 has the property P if and only F2 has theproperty P .

We recall that a functor F : C → D is conservative if a morphism f in C isan isomorphism as soon as Ff is an isomorphism (F reflects isomorphisms).

DRAFT


Lemma 2.3.5. The properties of being fully faithful, essentially surjective, con-servative, preserving (co)limits of a fixed shape and of being a left adjoint, a rightadjoint, or an equivalence are invariant properties.

Proof. This proof is left as an exercise to the reader.

Proposition 2.3.6. Let G be a discrete group.

(i) The invariants functor (−)G : Mod(ZG)→ Ab and limG : AbG → Ab areequivalent.

(ii) The coinvariants functor (−)G : Mod(ZG)→ Ab and colimG : AbG → Abare equivalent.

Proof. More precisely, we show that for every discrete group G the followingdiagram commutes up to natural isomorphisms

Mod(ZG)

(−)G

zz

(−)G

%%'

∼= ∼=Ab Ab.

AbGcolimG

dd

limG

99

In this diagram the vertical functor is the equivalence (2.3.1). The claim con-cerning the triangle on the right follows for example from the general construc-tion of limits in terms of equalizers and products; see (A.2.5). In fact, in thisparticular case, the general construction specializes to the equalizer of abeliangroups

limGM ∼= eq(M ⇒

∏g∈G

M),

where one map is the diagonal map while the other map is m 7→ (gm)g∈G.Clearly, this equalizer is naturally isomorphic to the invariants MG ∈ Ab.

Similarly, using the explicit construction of colimits in (A.2.6), one observesthat also the triangle on the left commutes up to a natural isomorphism.

Corollary 2.3.7. Let G be a discrete group and let n ≥ 0.

(i) The functors Hn(G;−) : Mod(ZG) → Ab and RnlimG : AbG → Ab areequivalent.

(ii) The functors Hn(G;−) : Mod(ZG)→ Ab and LncolimG : AbG → Ab areequivalent.

Proof. This is a purely formal consequence of Proposition 2.3.6 and the detailsare left to the reader.

DRAFT


Thus, there is a very close relation between group cohomology functorsHn(G;−) and derived limit functors RnlimG and, similarly, between group ho-mology functors Hn(G;−) and derived colimit functors LncolimG. We do notclaim that this different perspective simplifies the study of group (co)homology.Instead, the point of these observations was to illustrate that the seemingly veryabstract derived (co)limit functors specialize to well-known constructions whichare of independent interest.

In the remainder of this series of books we indicate that suitable combi-nations of derived limit functors (and, more generally, derived Kan extensionfunctors) encode interesting constructions, and that such constructions are con-veniently organized by means of derivators.

DRAFT


DRAFT

Chapter 3

Derived categories ofabelian categories

In this section we recall the construction of the derived category of an abeliancategory [Ver96]. These derived categories can be thought of as rather refinedinvariants of the given abelian categories [AHHK07]. Important special cases arederived categories of rings [Hap87, Hap88, AHHK07] and derived categories ofschemes [Cal05, Huy06, KS06, Lip09], which respectively show up prominentlyin representation theory and algebraic geometry. While in this section we con-sider derived categories as plain categories only (without additional structure),in §5 we briefly discuss classical triangulations on derived categories.

In §3.1 we observe that cokernel functors in general fail to be exact, butthat they are exact on monomorphisms. Following the typical reasoning fromhomological algebra, this suggest to ‘redefine’ the cokernel, thereby obtainingcone and cofiber functors (see §3.2). In §3.3 we recall the definition of derivedcategories and the corresponding notion of derived functors. In §3.4 we showthat derived categories, although defined as localizations, enjoy the strongeruniversal property of being 2-localizations. This allows us to conclude in §3.5that for every abelian category the cone functor is the left derived cokernelfunctor. In general, these cone functors do not factor through the morphismcategory of the derived category.

3.1 Towards derived cokernels

LetA be an abelian category and let A be a small category. In §2.3 we mentionedfirst examples of such categories such that the limit and colimit functors

limA, colimA : AA → A

exist but fail to be exact. The existence of the adjunction (colimA,∆A) impliesthat colimA is right exact, but, in general colimA is not left exact, and dually

45

DRAFT

46 CHAPTER 3. DERIVED CATEGORIES OF ABELIAN CATEGORIES

for limA. In this section we discuss exactness properties of (co)products and(co)kernels.

We already observed that abelian categories are finitely cocomplete and thatarbitrary finite colimits can be constructed from finite coproducts and cokernels,and dually. Hence we consider these two types of constructions independently.Let us recall that a category is discrete if all morphisms are identity morphisms.Given a discrete category S and an arbitrary category C, there is a canonicalisomorphism CS ∼=

∏s∈S C.

Lemma 3.1.1. Let A be an abelian category and let S be a finite discretecategory. The finite biproduct functor⊕

s∈S: AS → A

is exact.

Proof. Given an abelian category A, finite coproduct and finite product functorsare part of adjunctions

(∐s∈S

,∆S) : AS A and (∆S ,∏s∈S

) : A AS ,

showing that∐s∈S is right exact and

∏s∈S is left exact. The preadditivity of

abelian categories yields natural isomorphisms∐s∈S∼=⊕

s∈S∼=∏s∈S , imply-

ing that finite biproducts are exact.

Hence, finite coproducts and finite products in abelian categories are exact.Of course this lemma already applies to preadditive categories but here we focuson abelian categories.

Remark 3.1.2. Infinite (co)products in abelian categories are, in general, notexact. However, one can impose additional axioms on abelian categories guar-anteeing first of all the existence of infinite coproducts or infinite products and,second, that infinite coproducts or infinite products are exact. In this order, at-tributing credit to [Gro57], these assumptions are referred to as axioms (AB3),(AB3*), (AB4), and (AB4*). We refer the reader to the literature for more onthese axioms, including a discussion of examples of abelian categories satisfyingthese stronger axioms; see [Gro57] or [Fai73, §14].

We now turn to the case of (co)kernels which, in general, happen to fail tobe exact. To begin with, as a special case of Lemma 2.1.8, associated to anabelian category A there is the abelian category A[1] of morphisms in A. Here,[1] is the partially ordered set [1] = (0 < 1), considered as a category. Thus,objects in A[1] are morphisms X0 → X1 in A while a morphism X → Y is acommutative square

X0//

Y0

X1// Y1.

DRAFT

3.1. TOWARDS DERIVED COKERNELS 47

Since limits and colimits in A[1] are constructed levelwise (see again the proofof Lemma 2.1.8 which was left as an exercise), a short exact sequence in A[1]

corresponds precisely to a morphism of short exact sequences in A, i.e., a com-mutative diagram

0 // X0

f

// Y0//

g

Z0//

h

0

0 // X1// Y1

// Z1// 0,

(3.1.3)

such that the rows are short exact sequences.

Lemma 3.1.4. Let A be an abelian category.

(i) The kernel functor ker : A[1] → A is left exact.

(ii) The cokernel functor cok: A[1] → A is right exact.

(iii) In general, the functors ker, cok: A[1] → A are not exact.

Proof. Since both functors are additive, the first two statements are simply areformulation of the snake lemma. In fact, given a short exact sequence ofmorphisms as in (3.1.3), the snake lemma yields an exact sequence

0→ ker(f)→ ker(g)→ ker(h)→ cok(f)→ cok(g)→ cok(h)→ 0,

establishing the first two statements.To establish the third statement it suffices to consider any non-zero object

X ∈ A and the corresponding short exact sequence of morphisms

0 // 0

// Xid //

id

X //

0

0 // Xid// X // 0 // 0.

In fact, by the snake lemma we obtain an induced exact sequence

0→ 0→ 0→ X ∼−→ X → 0→ 0→ 0,

showing that the kernel is not right exact and that the cokernel is not leftexact.

We now reformulate the failure of exactness using a slightly different per-spective.

Definition 3.1.5. Let A be an abelian category and let f : X → Y be amorphism in Ch(A). The chain map f is a quasi-isomorphism, in notationf : X

∼→ Y, if it induces isomorphisms in homology,

Hk(f) : Hk(X)∼=→ Hk(Y ), k ∈ Z.

The class of all quasi-isomorphisms is denoted by W = WA.

DRAFT


We recall that any additive functor F : A → B induces an additive functorF : Ch(A) → Ch(B) between the corresponding categories of chain complexes,obtained by applying F in all degrees.

Lemma 3.1.6. An additive functor F : A → B between abelian categories isexact if and only if the induced functor F : Ch(A) → Ch(B) preserves quasi-isomorphisms.


This motivates the following definition.

Definition 3.1.7. Let A,B be abelian categories. A functor F : Ch(A) →Ch(B) is exact or homotopy-invariant if it preserves quasi-isomorphisms.

Lemma 3.1.6 is to be seen in contrast with the following result.

Lemma 3.1.8. Let F : A → B be an additive functor between abelian categories.The induced functor F : Ch(A) → Ch(B) preserves chain homotopies and, inparticular, chain homotopy equivalences.


For later applications of Lemma 3.1.6 to limit and colimit functors, it isconvenient to have a refined version of Lemma 2.1.8. In that refined versionwe consider the following isomorphism of categories. Given a diagram of chaincomplexes X : A→ Ch(A), one easily checks that

Yk(a) = X(a)k, k ∈ Z, a ∈ A,

defines a chain complex Y ∈ Ch(AA), and conversely. Moreover, this obviouslyyields an isomorphism of categories

Ch(A)A ∼= Ch(AA) : X ↔ Y. (3.1.9)

Definition 3.1.10. Let A be an abelian category, let A be a small category, andlet X ′, X ′′ : A → Ch(A) be functors. A natural transformation α : X ′ → X ′′

is a levelwise quasi-isomorphism if αa : X ′(a) → X ′′(a), a ∈ A, are quasi-isomorphisms. The class of all levelwise quasi-isomorphims is denoted by WA

A .

Lemma 3.1.11. Let A be an abelian category and let A be a small category.

(i) The functor category AA = Fun(A,A) is abelian.

(ii) The isomorphism of categories (3.1.9) identifies the class WAA of levelwise

quasi-isomorphisms with the class WAA of quasi-isomorphisms.


Let us recall that kernels and cokernels in Ch(A) are calculated degreewise.Thus, Lemma 3.1.4, Lemma 3.1.6, and Lemma 3.1.11 together imply the fol-lowing warning.

DRAFT

3.2. CONES AND COFIBERS 49

Warning 3.1.12. Let A be an abelian category. In general, the functors

cok, ker : Ch(A)[1] → Ch(A)

are not homotopy-invariant, i.e., they do not send levelwise quasi-isomorphismsto quasi-isomorphisms.

Despite this warning, there are the following positive statements concerninglevelwise quasi-isomorphisms X → Y in Ch(A)[1], i.e., commutative squares ofchain complexes as in

X0f//

∼

X1

∼

Y0 g// Y1.

(3.1.13)

Lemma 3.1.14. Let A be an abelian category and let (3.1.13) be a levelwisequasi-isomorphism in Ch(A)[1].

(i) If f and g are monomorphisms, then the induced map cok(f) → cok(g)is a quasi-isomorphism.

(ii) If f and g are epimorphisms, then the induced map ker(f)→ ker(g) is aquasi-isomorphism.

Proof. We give a proof of (i) and observe that there is a morphism of shortexact sequences of chain complexes

0 // X0

∼

f// X1

//

∼

cok(f) //

0

0 // Y0 g// Y1

// cok(g) // 0.

The induced long exact sequences in homology and the 5-lemma allow us toconclude the proof of (i). The case of (ii) is dual.

Thus, the cokernel and the kernel functors are exact on certain morphisms ofchain complexes. The idea behind the construction of derived (co)kernel func-tors is now simply to quasi-isomorphically approximate arbitrary morphisms ofchain complexes by these good ones.

3.2 Cones and cofibers

In this section we recall some classical constructions from homological algebra.This includes the seemingly adhoc construction of the cone functor, a certain‘corrected version’ of the cokernel functor. While a first justification of thisconstruction is provided by Proposition 3.2.10, a more conceptual one will be

DRAFT


given in §3.5. Everything can be dualized to yield a similar ‘correction’ of thekernel functor.

To motivate these constructions we include the following different descriptionof the cokernel. Let f : X0 → X1 be a morphism in an abelian category A. Wenote that the cokernel of f can also be constructed by considering pushoutsquares

X0f//

X1

0 // cok(f).

(3.2.1)

In fact, the universal property of this pushout square is easily checked to reduceto the usual universal property of the cokernel. Thus, although the cokernelcok(f) is not the colimit of the diagram f : X0 → X1 (which would be isomorphicto X1) it is the colimit of the span (0 ← X0 → X1). We come back to thispassage to the span later.

Now, as recalled in Warning 3.1.12, in general the cokernel functor is notexact. At the level of chain complexes there is the following classical way ofadressing this problem. Using the above different description of the cokernel,we simply replace the morphism X0 → 0 in (3.2.1) quasi-isomorphically by anice inclusion, namely by the inclusion i : X0 → CX0 of X0 in its cone CX0. Wequickly recall this construction and, since we also have a use for further variantsof such constructions, we allow us to include a minor digression.

Digression 3.2.2. Let FZ denote the category of finitely generated, free abeliangroups and let Chb(FZ) be the category of bounded chain complexes in FZ. Werecall that for every additive category A there is a two-variable functor

⊗ : Chb(FZ)× Ch(A)→ Ch(A) (3.2.3)

which is additive in both variables separately.To begin with there is a similar biadditive action ⊗ : FZ × A → A which

is already determined up to canonical isomorphism by asking that there is anatural isomorphism Z ⊗X ∼= X,X ∈ A. Thus, for a free abelian group F ofrank r, there is a natural isomorphism F ⊗X ∼= r ·X, where r ·X denotes ther-fold direct sum X ⊕ . . .⊕X.

This action can be extended to chain complexes by the following ‘convolutiontype construction’. Given chain complexes F ∈ Chb(FZ) and X ∈ Ch(A) wedefine the underlying graded object of F ⊗X ∈ Ch(A) by

(F ⊗X)k =⊕p+q=k

Fp ⊗Xq, k ∈ Z.

(This direct sum is finite by the boundedness assumption on F .) The differ-entials (F ⊗ X)k → (F ⊗ X)k−1 are obtained from those on F and X usingthe usual Koszul sign convention. (Recall that this convention says that, indifferential graded algebra, ‘whenever the order of two homogeneous elementsof degree p and q is swapped, a sign (−1)pq has to appear’.)

DRAFT

3.2. CONES AND COFIBERS 51

We illustrate the action (3.2.3) by the following examples.

Examples 3.2.4.

(i) Let I = Ccell([0, 1]) ∈ Chb(FZ) be the reduced cellular chain complex ofthe interval endowed with the CW structure consisting of two 0-cells andone 1-cell only. More explicitly, I0 = I1 = Z, the differential I1 → I0 isthe identity, and all Ik, k 6= 0, 1, vanish. We call I the interval and itis easily verified that I is a contractible chain complex. For X ∈ Ch(A)there is a natural isomorphism

I ⊗X ∼= CX,

where C : Ch(A) → Ch(A) is the cone functor. (If you did not seea definition of C before, then you can simply take this construction asa definition and we suggest as an exercise to unravel it.) The notationis of course motivated from similar constructions for pointed topologi-cal spaces, where smashing with the interval yields the (reduced) coneconstruction.

(ii) Let S0 = Ccell(0, 1) ∈ Chb(FZ) be the reduced cellular chain complexof the 0-sphere endowed with the CW structure consisting of two 0-cellsonly. Thus, S0 is the stalk complex Z. For X ∈ Ch(A) there is a naturalisomorphism

S0 ⊗X ∼= X.

(iii) The continuous map 0, 1 → [0, 1] yields a map S0 → I in Chb(FZ)and hence a natural transformation S0⊗− → I⊗− of functors Ch(A)→Ch(A). Under the above natural isomorphisms, for X ∈ Ch(A) this yieldsthe usual natural monomorphism

(S0 ⊗X → I ⊗X) ∼= (i : X → CX).

(iv) Let S1 ∈ Chb(FZ) be the reduced cellular chain complex of the 1-sphereconsidered as a quotient CW complex [0, 1]/0, 1. Thus, S1 is the abeliangroup Z considered as a complex concentrated in degree 1. ForX ∈ Ch(A)there is a natural isomorphism

S1 ⊗X ∼= ΣX,

where Σ: Ch(A) → Ch(A) is the suspension functor, i.e., the usual‘shift against the differential’.

Having recalled the cone construction for chain complexes, we now replacethe vertical morphism on the left in (3.2.1) by X0 → CX0 and hence considerthe pushout diagram

X0f//

i

X1

cof(f)

CX0// Cf.

(3.2.5)

DRAFT


Definition 3.2.6. Let A be an abelian category and let f : X0 → X1 be inCh(A).

(i) The chain complex Cf in (3.2.5) is the cone of f .

(ii) The chain map cof(f) : X1 → Cf in (3.2.5) is the cofiber of f .

It is immediate that both the cone and the cofiber are functorial in f , i.e.,we have functors

C : Ch(A)[1] → Ch(A) and cof : Ch(A)[1] → Ch(A)[1].

Lemma 3.2.7. Let A be an abelian category and let X ∈ Ch(A).

(i) The cone CX is contractible.

(ii) The cone of an isomorphism is contractible.

Proof. Since the interval I is contractible and the construction (3.2.3) is additivein the first variable it follows from Lemma 3.1.8 that I⊗X ∼= CX is contractible.If f : X0 → X1 is an isomorphism, then the construction (3.2.5) yields an iso-morphism CX0

∼= Cf . Thus, by the first part, Cf is contractible.

To prove the homotopy invariance of the cone construction we collect thefollowing lemma.

Lemma 3.2.8. Let A be an abelian category and let us consider a square

X0i //

j

X1

p

X2 q// X

in A. The square is a pushout square if and only if the sequence

X0(i,j)t→ X1 ⊕X2

(p,−q)→ X → 0

is exact.


In the special case of the defining pushout square (3.2.5) of the cone, wehence obtain a short exact sequence

0→ X0 → X1 ⊕ CX0 → Cf → 0, (3.2.9)

since the first map is clearly also a monomorphism (see Examples 3.2.4(iii)). Thereader easily checks that this short exact sequence is functorial in the morphismf : X0 → X1. While this construction allows us to give a partial justificationof the cone construction (see again Lemma 3.1.6), a better one will be givenin §3.5 (see Theorem 3.5.6).

DRAFT

3.3. DERIVED CATEGORIES AND DERIVED FUNCTORS 53

Proposition 3.2.10. Let A be an abelian category. The cone C : Ch(A)[1] →Ch(A) sends levelwise quasi-isomorphisms to quasi-isomorphisms.

Proof. We again consider a diagram of chain complexes (3.1.13) such that thevertical maps are quasi-isomorphisms, i.e., a levelwise quasi-isomorphism f → g.Associated to such a levelwise quasi-isomorphism we obtain (by the functorialityof (3.2.9)) a morphism of short exact sequences

0 // X0//

∼

X1 ⊕ CX0//

Cf //

0

0 // Y0// Y1 ⊕ CY0

// Cg // 0.

Since the objects CX0, CY0 have trivial homology objects (Lemma 3.2.7), anychain map between them is a quasi-isomorphism. As finite direct sums of quasi-isomorphisms are again quasi-isomorphisms, in the above diagram also the sec-ond vertical morphism is a quasi-isomorphism. The long exact sequence inhomology and the 5-lemma imply that the morphism Cf → Cg is a quasi-isomorphism, concluding the proof.

3.3 Derived categories and derived functors

As a preparation for the definition of derived categories, let us reconsider theconstruction of classical derived functors between abelian categories. Thus, letF : A → B be a right exact functor between abelian categories and let A haveenough projective objects. The key steps in the construction of the LkF, k ≥ 0,consisted of choosing projective resolutions

. . .→ P2 → P1 → P0ε→ X (3.3.1)

for X ∈ A, applying F to the truncated chain complex

P = (. . .→ P2 → P1 → P0),

and then passing to homology. A different way of saying that (3.3.1) is a reso-lution is to say that the commutative diagram

...

...

P2

// 0

P1//

0

P0 ε// X

(3.3.2)

DRAFT


defines a quasi-isomorphism ε : P → X, where X also denotes the associatedcomplex concentrated in degree 0.

This simple trick of rewriting the horizontal diagram (3.3.1) in a verticalfashion as in (3.3.2) is an important step. In fact, it suggests that, secretely inthe construction of classical derived functors, we already passed from objectsin the abelian category (like modules over a ring) to chain complexes. From amore abstract perspective this amounts to passing from a good old category toa ‘homotopy theory’, namely to the ‘homotopy theory’ (Ch(A),WA) consistingof the category Ch(A) together with the class W = WA of quasi-isomorphisms.We say a bit more about ‘abstract homotopy theories’ in §7.3 and Appendix B.

The passage to stalk complexes defines a fully faithful functor A → Ch(A),suggesting that if we understand Ch(A) we also understand A. For variousreasons we would like to invert the quasi-isomorphisms WA in Ch(A). Forexample this would allow us to identify all projective resolutions of a fixedobject X ∈ A. For introductory accounts which motivate the study of derivedcategories see for example [Kel96, Kel98, Kel07, Tho01].

Definition 3.3.3. Let A be an abelian category. The derived categoryD(A) of A is the localization of the category Ch(A) at the class WA of quasi-isomorphisms,

D(A) = Ch(A)[W−1A ].

By the very definition, the derived category is hence a pair (D(A), γ) con-sisting of a category D(A) and a localization functor γ : Ch(A)→ D(A) whichsends quasi-isomorphisms to isomorphisms and which is initial with this prop-erty. Thus, every functor F : Ch(A) → C which sends quasi-isomorphisms toisomorphisms factors uniquely through D(A),

Ch(A)∀F //

γ

C, F (WA) ⊆ IsoC ,

D(A).

∃!

<<

Warning 3.3.4. There is a typical warning in the context of such localizations,namely, that, in general, such localizations do not exist (at least not in a fixeduniverse). A different way of saying this is that such localizations can always beconstructed (see for example [GZ67]), but, in general, the resulting categoriesare not locally small, i.e., there might be proper classes of morphisms betweencertain objects. There are various ways of dealing with this issue.

(i) One can ignore these issues (as it was done in the more classical literature),reassuring the reader every now and then by claiming that all steps canbe justified.

(ii) One can restrict attention to the large class of Grothendieck abelian cate-gories since in that case all these set-theoretic problems happen to disap-pear. See for example [Fai73, §14] for a discussion of Grothendieck abeliancategories.

DRAFT

3.3. DERIVED CATEGORIES AND DERIVED FUNCTORS 55

(iii) The language of model categories [Qui67, Hov99, DS95] is an abstractframework which, among many other things, allows us to deal with suchproblems. As soon as the pair (Ch(A),WA) can be extended to a modelcategory, these set-theoretic problems automatically disappear. And, infact, in the case of Grothendieck abelian categories such model structuresalways exist; see [Hov01].

(iv) If one imposes certain boundedness conditions on the chain complexes andassumes the existence of sufficently many injective or projective objects,then one can show that corresponding derived categories are equivalentto quotient categories of suitable categories of chain complexes (see, forexample, [GM03, Theorem III.5.21] or [Spa88]). In particular, these cat-egories are again locally small categories.

For an actual example of such set-theoretic problems we refer to an old exampleof Freyd (see the reprint [Fre03]) as discussed in [Kra10, Example 4.15].

For the remainder of this section we assume that the derived categories underconsideration exist. (Thus, we essentially follow approach (i).)

Examples 3.3.5.

(i) Let R be a ring and let Mod(R) be the Grothendieck abelian categoryof R-modules. The derived category of the ring R is the derivedcategory

D(R) = D(Mod(R)) = Ch(R)[W−1R ],

where WR = WMod(R) is the class of quasi-isomorphisms in Ch(R). Inparticular in the case of finite-dimensional algebras over fields, such de-rived categories have been studied a lot in representation theory; see forexample [Hap87, Hap88, AHHK07].

(ii) Let X = (X,OX) be a scheme, let Qcoh(X) be the category of quasi-coherent OX -modules, let Ch(X) = Ch(Qcoh(X)) be the correspondingcategory of unbounded cochain complexes, and let WX be the class ofquasi-isomorphisms in Ch(X). The derived category of the scheme Xis the derived category

D(X) = D(Qcoh(X)) = Ch(X)[W−1X ].

References in this case include [Cal05, Huy06, KS06, Lip09].

Corollary 3.3.6. Let A,B be abelian categories and let F : Ch(A) → Ch(B)be an exact functor. There is a unique functor F : D(A) → D(B) making thefollowing diagram commutative,

Ch(A)F //

γ

Ch(B)

γ

D(A)∃!F// D(B).

This applies, in particular, if F comes from an exact functor F : A → B.

DRAFT


Proof. Since the functor γ F : Ch(A) → D(B) sends quasi-isomorphisms toisomorphisms this is immediate from the defining universal property of the lo-calization functor γ : Ch(A) → D(A) (see Definition 3.3.3). The special casecoming from an exact functor A → B is simply Lemma 3.1.6.

A first trivial example is given by the homology functors. For every abeliancategory A, we denote by AZ the category of Z-graded objects in A. (Thus,we consider Z as a discrete category and AZ is the corresponding diagram cate-gory.) Taking all homology functors Hk : Ch(A)→ A, k ∈ Z, at once we obtaina functor H∗ : Ch(A)→ AZ.

Example 3.3.7. Let A be an abelian category. The functor H∗ : Ch(A) →AZ sends quasi-isomorphisms to isomorphisms and hence induces a functorH∗ : D(A)→ AZ.

The following examples are more interesting.

Examples 3.3.8. Let A be an abelian category.

(i) The cone C : Ch(A)[1] → Ch(A) induces by Proposition 3.2.10 as well asLemma 3.1.11 a functor

C : D(A[1])→ D(A).

(ii) The cofiber cof : Ch(A)[1] → Ch(A)[1] induces by the same results a func-tor

cof : D(A[1])→ D(A[1]).

Thus, to emphasize, there is a functorial cone construction defined on thederived category of the morphism category of an abelian category. We will getback to this later.

As recalled in §2, many additive functors showing up in nature are only leftexact or right exact. In such situations one has to work harder in order to obtaincertain universal induced functors at the level of derived categories.

Definition 3.3.9. Let A,B be abelian categories and let F : Ch(A) → Ch(B)be a functor. A left derived functor of F is a pair (LF, ε) consisting of afunctor LF : D(A)→ D(B) and a natural transformation ε : (LF )γ → γF suchthat for every other such pair (G : D(A) → D(B), α : Gγ → γF ) there is aunique natural transformation α′ : G→ LF such that

α = ε (α′γ) : Gγ → (LF )γ → γF.

Thus, a left derived functor can be depicted by a square populated by anatural transformation as in

Ch(A)F //

γ

Ch(B)

γ

D(A)LF// D(B)

@Hε

DRAFT

3.4. DERIVED CATEGORIES AS 2-LOCALIZATIONS 57

and the defining universal property of a left derived functor can be illustratedas follows

Ch(A)F //

γ

Ch(B)

γ

=

Ch(A)F //

γ

Ch(B)

γ

D(A)G// D(B)

@H∀

D(A)

LF**

G

33⇑∃! D(B).

@Hε

Remark 3.3.10.

(i) The idea behind the definition is that the left derived functor (LF, ε) is auniversal approximation of F by an exact functor.

(ii) Since derived functors are defined by a universal property, once one knowsthat derived functors exist, then they are unique up to a canonical naturalisomorphism. The existence of derived functors is often settled using suit-able resolutions; see for example [GM03, §III.6] for a classical reference.

(iii) There is the dual notion of a right derived functor consisting of a func-tor RF : D(A)→ D(B) and a universal natural transformation η : γF →(RF )γ such that (RF, η) is suitably initial.

(iv) We warn the reader that Definition 3.3.9 differs slightly from the onefound in typical books on homological algebra; see [Wei94, GM03]. Whilein those references in the definition of a left derived functor both F andLF are assumed to be exact functors of triangulated categories (see §5),here we define derived functors as it is typically done in the theory ofmodel categories [Hov99].

In §3.5 we show that the functor C : D(A[1])→ D(A) constructed in Exam-ples 3.3.8 together with a suitable natural transformation is a left derived functorof the cokernel cok: Ch(A[1]) → Ch(A) (see Theorem 3.5.6). As a preparationwe establish in §3.4 that derived categories enjoy a seemingly stronger universalproperty than that one of a localization.

3.4 Derived categories as 2-localizations

Derived categories of abelian categories are defined as suitable localizations(Definition 3.3.3) and as such they satisfy a universal property. In this section weshow that they actually enjoy a stronger universal property, namely that derivedcategories are 2-localizations. This seemingly technical result has interestingconsequences as we see here, in §3.5, and in §4.1.

We begin by collecting the following lemma.

Lemma 3.4.1. Let C,D be categories and let X,Y : C → D be functors. Thereare bijections between

(i) the natural transformations α : X → Y ,

DRAFT


(ii) the functors H : C × [1] → D such that H(−, 0) = X and H(−, 1) = Y ,and

(iii) the functors K : C → D[1] such that 0∗K = X and 1∗K = Y .


Let us consider an abelian category A and the associated localization functorγ : Ch(A) → D(A). For every category C, precomposition of functors with γyields a map

γ∗ : hom(D(A), C)→ hom(Ch(A), C).

If we denote by homW(Ch(A), C) ⊆ hom(Ch(A), C) those functors which sendquasi-isomorphisms to isomorphisms, then the defining universal property of γis that for all categories C precomposition with γ induces a bijection

γ∗ : hom(D(A), C)∼=→ homW(Ch(A), C).

It turns out that this can be refined to an isomorphism of categories. Forthis purpose, we denote by

FunW(Ch(A), C) ⊆ Fun(Ch(A), C)

the full subcategory of the functor category spanned by all functors which sendquasi-isomorphisms to isomorphisms.

Proposition 3.4.2. Let A be an abelian category. The localization γ : Ch(A)→D(A) is a 2-localization, i.e., for every category C precomposition with γ in-duces an isomorphism of categories

γ∗ : Fun(D(A), C)∼=→ FunW(Ch(A), C).

In particular, for every pair of functors F ′, G′ : D(A) → C there is a bijectionγ∗ : hom(F ′, G′)→ hom(F ′γ,G′γ).

Proof. By definition of the localization, γ∗ is bijective on objects and it hencesuffices to show that γ∗ is also fully faithful. Let F,G ∈ FunW(Ch(A), C) withcorresponding factorizations F = F ′γ, G = G′γ and let α : F → G be a naturaltransformation. By Lemma 3.4.1, α : F → G is equivalently specified by thefunctor α : Ch(A)→ C[1] defined by

α(X) = (αX : FX → GX).

If w : X → Y is a quasi-isomorphism, then, since F,G send quasi-isomorphismsto isomorphisms, the vertical morphisms in

FXα //

∼= Fw

GX

∼=Gw

FYα// GY,

DRAFT

3.4. DERIVED CATEGORIES AS 2-LOCALIZATIONS 59

are isomorphisms, showing that α ∈ FunW(Ch(A), C[1]). The universal property

of γ implies that there is a unique functor α′ : D(A)→ C[1] such that

Ch(A)α //

γ

C[1]

D(A)∃!α′

II

commutes. Using Lemma 3.4.1 again, α′ corresponds to a unique natural trans-formation α′ : F ′ → G′ defined by

α′(X) = (α′X : F ′X → G′X), X ∈ D(A).

The reader easily checks that this is the unique natural transformation α′ : F ′ →G′ satisfying α′γ = α.

Remark 3.4.3. The proof did not use anything specific about the context ofabelian categories and derived categories. In fact, we have shown the moregeneral result that if a localization of a category at a class of morphisms exists,then this localization already enjoys the universal property of a 2-localization.

As a first illustration of this 2-categorical universal property we collect thefollowing result. This result makes precise the slogan that ‘exact functors (Def-inition 3.1.7) do not have to be derived’; see again Corollary 3.3.6.

Corollary 3.4.4. Let A,B be abelian categories, let F : Ch(A) → Ch(B) beexact, and let F : D(A)→ D(B) be the unique induced functor such that

Ch(A)F //

γ

Ch(B)

γ

D(A)F// D(B)

commutes.

(i) The pair (F : D(A) → D(B), id : F γ → γ F ) is a left derived functorof F .

(ii) The pair (F : D(A) → D(B), id : γ F → F γ) is a right derived func-tor of F .

Proof. We take care of statement (i), the case of (ii) is dual. To this end, let

(G : D(A)→ D(B), α : G γ → γ F )

be an instance of a pair among which a left derived functor is supposed to beterminal (see Definition 3.3.9). Since γF = F γ, the natural transformation α

DRAFT


reads as α : G γ → F γ. By Proposition 3.4.2 there is a unique naturaltransformation α′ : G→ F such that

α = α′γ = id (α′γ),

i.e., we have

Ch(A)F //

γ

Ch(B)

γ

=

Ch(A)F //

γ

Ch(B)

γ

D(A)G// D(B)

@Hα

D(A) ⇑α′F))

G

55 D(B).

@Hid

Thus, (F, id) enjoys the universal property of a left derived functor.

Using that γ : Ch(A) → D(A) is a 2-localization, in the above proof wewere able to avoid the explicit construction of a natural transformation betweenfunctors defined on derived categories. This is convenient since, in general,morphisms in derived categories are a bit tricky to understand. We will see afurther instance of such a situation in the next section.

3.5 Cones as derived cokernels

We now take a closer look at the cone construction C : Ch(A[1]) → Ch(A) asdefined via the pushout square (3.2.5). Clearly, CX0 → 0 induces a morphism ofspans (CX0 ← X0 → X1)→ (0← X0 → X1), and hence an induced morphismof pushout diagrams

X0f

//

= ##

X1=

%%

X0//

X1

CX0

""

// Cf

$$

0 // cok(f).

The exact sequence of Lemma 3.2.8 is easily seen to be natural in the pushoutdiagram, hence we obtain a natural morphism of exact sequences

0 // X0//

= ∼

X1 ⊕ CX0

∼

// Cf //

0

X0f

// X1// cok(f) // 0,

(3.5.1)

and, in particular, a natural transformation

ϕ : Cf → cok(f). (3.5.2)

DRAFT

3.5. CONES AS DERIVED COKERNELS 61

Remark 3.5.3. We observe that the first two components of the natural mor-phism of exact sequences (3.5.1) define a natural levelwise quasi-isomorphism

X0//

= ∼

X1 ⊕ CX0

∼

X0f

// X1.

(3.5.4)

Moreover, the domain of this quasi-isomorphism is a monomorphism and weknow by Lemma 3.1.14 that the cokernel preserves quasi-isomorphisms betweenmonomorphisms. We refer to this by saying that (3.5.4) is a functorial reso-lution on Ch(A[1]) which is adapted to the functor cok: Ch(A[1])→ Ch(A).It follows that the morphism ϕ : Cf → cok(f) is a quasi-isomorphism for allmonomorphisms f .

As a preparation for the theorem we recall that, by Examples 3.3.8, thecone induces a functor C : D(A[1])→ D(A). The natural transformation (3.5.2)yields a natural transformation ε : C γ → γ cok given by

ε : C γ = γ C γϕ→ γ cok. (3.5.5)

We also collect the corresponding dual statement concerning the fiber functorF : D(A[1]) → D(A). Since its construction is dual to that of C : D(A[1]) →D(A), we leave the details to the reader.

Theorem 3.5.6. Let A be an abelian category.

(i) The cone functor C : D(A[1]) → D(A) together with ε defined in (3.5.5)is a left derived functor of the cokernel cok: Ch(A[1])→ Ch(A).

(ii) The fiber functor F : D(A[1])→ D(A) together with a dually defined η isa right derived functor of the kernel ker : Ch(A[1])→ Ch(A).

Proof. We have to show that (C, ε) satisfies the universal property of Def-inition 3.3.9. For this purpose let G : D(A[1]) → D(A) be a functor andlet α : G γ → γ cok be a natural transformation,

Ch(A[1])cok //

γ

Ch(A)

γ

D(A[1])G// D(A).

AIα

We claim that it is enough to show that there is a unique natural transformationψ : G γ → C γ such that

α = ε ψ : G γ → C γ → γ cok.

DRAFT


In fact, by Proposition 3.4.2 we then deduce that there is a unique naturaltransformation α′ : G → C such that ψ = α′γ, which immediately yields thedesired universal property of (C, ε).

In order to construct such a transformation ψ we consider (X0f→ X1) ∈

Ch(A[1]) and associate to it the following commutative diagram

Gγ(X0 → X1 ⊕ CX0)α //

∼=

γcok(X0 → X1 ⊕ CX0)

Cγ(X0 → X1 ⊕ CX0)ε∼=oo

∼=

Gγ(X0 → X1)α

// γcok(X0 → X1) Cγ(X0 → X1).ε

oo

In this diagram, the vertical morphisms are induced by the resolutions adaptedto the cokernel (see Remark 3.5.3) and it hence follows that the two outer verticalmorphisms are isomorphisms. Moreover, both squares commute by naturalityof α and ε, respectively. Finally, the top horizontal morphism

ε = γϕ : Cγ(X0 → X1 ⊕ CX0)→ γcok(X0 → X1 ⊕ CX0)

is an isomorphism since X0 → X1 ⊕ CX0 is a monomorphism; see again Re-mark 3.5.3. Thus, the above commutative diagram shows that there is at mostone such natural transformation ψ. And these morphisms Gγ(f) → Cγ(f) ac-tually assemble to a natural transformation Gγ → Cγ as one checks by showingthat the above diagram is natural in (f : X0 → X1).

Note that Proposition 3.4.2 again allowed us to avoid the explicit construc-tion of a natural transformation between functors defined on derived categories.

Remark 3.5.7. We include a short philosophical remark concerning the con-struction of the cone as a derived version of the cokernel. The cokernel isa categorical construction which amounts to ‘collapsing’ the image of a mor-phism. As observed in Warning 3.1.12 the cokernel is not an exact or homotopyinvariant construction in that levelwise quasi-isomorphisms are not always sentto quasi-isomorphisms.

One obtains the cone from the cokernel by, instead of collapsing the image,simply ‘adding the potential of collapsing the image’. This is made precise bythe defining pushout square (3.2.5). In fact, thinking geometrically, by glueinga cone on the image of the morphism we add the potential of collapsing it sincewe can now push it to the apex of the cone. Finally, the comparison map (3.5.2)simply collapses the cone which was glued to the image.

Similar comments also apply to other derived (co)limit constructions andthis picture also extends to the context of topological spaces and homotopy(co)limits (see [BK72, Dug08, Str11, MV15]). Moreover, also in that generalitythere are canonical comparison maps similar to (3.5.2) and we briefly come backto these maps in the sequel [Gro16a].

DRAFT

Chapter 4

Coherent versus incoherentdiagrams

Given an abelian category A and a small category A, there are two associatedcategories: the derived category D(AA) of the diagram category and the dia-gram category D(A)A of the derived category. The entire point of derivatorsis that these categories are honestly different categories. In general, there aremany interesting functorial constructions on D(AA) which do not allow for func-torial variants on D(A)A. For example, a functorial cone construction exists onD(A[1]) but in most cases not on D(A)[1].

To put this into context, in §4.1 we emphasize that one has to distinguishbetween derived categories of diagram categories and diagram categories of de-rived categories. These categories are related by an underlying diagram functors,which, in general, is far from being an equivalence. In §4.2 we illustrate theseunderlying diagram functors in the context of category algebras (like group al-gebras, path algebras, and incidence algebras) and observe that for semisimplecoefficients they are equivalent to homology functors. This shows that underly-ing diagram functors in general discard relevant information. In §4.3 we describeexplicitly some specific underlying diagram functors and observe that they failto be equivalences.

4.1 Underlying diagram functors

In this section we briefly discuss differences between coherent and incoherentdiagrams. The main point of the theory of derivators is that many interestingconstructions are available at the level of coherent diagrams while, in general,this is not the case for incoherent diagrams.

To already get used to the terminology from the theory of derivators, wemake the following definition. Let us recall that given an abelian category Bthe localization functor γ : Ch(B) → D(B) can be chosen to be the identity on

63

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64 CHAPTER 4. COHERENT VERSUS INCOHERENT DIAGRAMS

objects. Hence, objects in the derived category are again simply chain com-plexes.

Definition 4.1.1. Let A be an abelian category and let A be a small category.

(i) An object in D(AA) is a coherent diagram of shape A.

(ii) An object in D(A)A is an incoherent diagram of shape A.

Thus, a coherent diagram is simply a chain complex X ∈ Ch(AA), i.e., achain complex of functors A→ A. Under the isomorphism Ch(AA) ∼= Ch(A)A

from Lemma 3.1.11 this corresponds to a diagram X : A→ Ch(A). The adjec-tive ‘coherent’ is meant to indicate that this is a strictly commutative diagramin the sense that it preserves compositions and identies on the nose.

In contrast to this, an incoherent diagram is a functor X : A→ D(A). Let usthink of the derived category as being realized as a quotient category, i.e., objectsare suitably nice chain complexes and morphisms are chain homotopy classes.An incoherent diagram then is merely a homotopy-commutative diagram. Forexample, for every pair of composable morphisms f : a0 → a1 and g : a1 → a2

in A it follows that the triangle on the right in the diagram

a1

g

'

X(a1)

X(g)

$$

a0

f

??

gf// a2, X(a0)

X(f)::

X(gf)// X(a2),

commutes up to an unspecified chain homotopy. (In that triangle we of courseabused notation and wrote X(f), X(g), and X(gf) respectively for representa-tives of the corresponding chain homotopy classes.) Now, in general, it is notpossible to replace a homotopy-commutative diagram up to quasi-isomorphismby a strictly commutative one.

Put differently, the two categories D(AA) and D(A)A are honestly differ-ent categories. As an additional application of Proposition 3.4.2, we observenext that these categories are related by a functor D(AA) → D(A)A. Similarfunctors arise in the context of abstract prederivators, and, as a warm-up, theconstruction given here already uses a reasoning inspired by that more generalsituation. We illustrate this functor by some specific examples in §§4.2-4.3.

Let us denote by 1 the terminal category consisting of one object and itsidentity morphism only. For any category A the evaluation at this unique objectyields an isomorphism of categories

A1 ∼= A,

which we use implicitly in the rest of this book. Thus, given an object a ∈ Awe also write a : 1 → A for the corresponding functor pointing at that object.Similarly, any morphism f : a→ b in A yields a natural transformation

1

a''

b

77 A,

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4.1. UNDERLYING DIAGRAM FUNCTORS 65

which we again denote by f .

Lemma 4.1.2. Let A be an abelian category, let A be a small category, and leta ∈ A. The evaluation functor a∗ : Ch(AA)→ Ch(A) sends quasi-isomorphismsto quasi-isomorphisms. Hence, there is a unique evaluation functor

a∗ : D(AA)→ D(A)

such that the following diagram commutes,

Ch(AA)a∗ //

γ

Ch(A)

γ

D(AA)a∗// D(A).

Proof. This is immediate from Lemma 3.1.11 and Definition 3.3.3.

Given a morphism of coherent diagrams g : X → Y in D(AA) we simplifynotation by writing

ga : Xa → Ya, a ∈ A,for the image of g under the evaluation functor a∗ : D(AA)→ D(A).

Now, every morphism f : a → b in A induces a transformation f∗ : a∗ → b∗

between the corresponding evaluation functors a∗, b∗ : AA → A. A levelwiseapplication gives rise to a natural transformation f∗ : a∗ → b∗ between chain-level evaluation functors a∗, b∗ : Ch(AA)→ Ch(A).

Lemma 4.1.3. Let A be an abelian category, let A be a small category, and letf : a→ b be a morphism in A. There is a unique induced natural transformationf∗ : a∗ → b∗ between a∗, b∗ : D(AA)→ D(A) such that γf∗ = f∗γ,

Ch(AA) ⇓f∗a∗))

b∗55

γ

Ch(A)

γ

D(AA) ⇓f∗a∗))

b∗55 D(A).

Proof. This is immediate from Proposition 3.4.2 and Lemma 4.1.3.

Given a coherent diagram X ∈ D(AA), we denote the corresponding com-ponent of the natural transformation f∗ : a∗ → b∗ by

Xf : Xa → Xb,

which is a morphism in D(A). As always we see that uniqueness implies func-toriality. Given morphisms f : a → b and g : b → c, uniqeness implies theequations

Xg Xf = Xgf : Xa → Xc and Xida = idXa : Xa → Xa.

DRAFT


Thus, associated to every coherent diagram X ∈ D(AA) we obtain an incoherentdiagram diaA(X) : A→ D(A) defined by

diaA(X) : A→ D(A), a 7→ Xa, f 7→ Xf . (4.1.4)

We refer to diaA(X) as the underlying diagram of X.Finally, let us again consider a morphism g : X → Y in D(AA). Then

there is a natural transformation diaA(g) : diaA(X) → diaA(Y ) between thecorresponding underlying diagrams. In fact, the component of diaA(g) at anobject a ∈ A is simply the morphism

diaA(g)a = ga : Xa → Ya. (4.1.5)

We leave it to the reader to check that diaA(g) is a natural transformation. Thefunctoriality of this construction is summarized in the following proposition.

Proposition 4.1.6. Let A be an abelian category and let A be a small category.The assignments X 7→ diaA(X) (4.1.5) and g 7→ diaA(g) (4.1.6) define a functor

diaA : D(AA)→ D(A)A.


We refer to diaA : D(AA)→ D(A)A as the underlying diagram functor.

Remark 4.1.7. In the situation of the proposition, let us note that the functor

Ch(AA) ∼= Ch(A)AγAA→ D(A)A

sends quasi-isomorphisms to isomorphisms. Hence, this functor factors uniquelythrough the localization functor γAA : Ch(AA)→ D(AA), and we leave it to thereader to verify that this induced functor D(AA) → D(A)A is diaA. Here, weon purpose included the above more 2-categorical approach to diaA since thisis how versions of these functors are constructed for abstract derivators.

Warning 4.1.8.

(i) Just to re-emphasize, the domain and the target categories of diaA arevery different. While the domain is the derived category of a diagramcategory the target is a diagram category of the derived category. In gen-eral, diaA is not an equivalence and it is important to distinguish betweenthese two categories. Intuitively speaking, the functor diaA takes a strictdiagram to the underlying homotopy-commutative diagram, thereby dis-carding important information.

(ii) As a special case let us consider A = [1] and the associated underlyingdiagram functor dia[1] : D(A[1]) → D(A)[1] from the derived category ofthe morphism category to the morphism category of the derived category.One slogan concerning defects of triangulated categories (see §5) is thatcone constructions are not functorial at that level. In the case of derived

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4.2. THE CASE OF CATEGORY ALGEBRAS 67

categories, this statement takes the more precise form that the cone func-tor C : D(A[1]) → D(A) (see Examples 3.3.8) does not factor throughdia[1] : D(A[1])→ D(A)[1], i.e., in general, there is no dashed arrow

D(A[1])C //

dia[1]

D(A)

D(A)[1]

@C

::

making the diagram commute. In general, the functorial cone constructiononly exists at the level of coherent diagrams. We revisit this warningin §4.3 in a special case and again in §16.

(iii) In general, the question whether one can lift objects or morphisms againstthe underlying diagram functor diaA : D(AA) → D(A)A is difficult toanswer. For example in the case of a discrete group A = G this reducesto the question if we can replace ‘G-actions up to homotopy’ by strictG-action, and similarly in the case of morphism. In the general case,references addressing these questions include [?, ?, ?, ?].

This underlying diagram functor has a variant in the context of pointedtopological spaces, yielding diaA : Ho(TopA) → Ho(Top)A. Referencesconsidering the question if objects and morphisms in Ho(Top)A can belifted to Ho(TopA) include [DKS89, ?, ?].

In the following section we illustrate the construction of underlying diagramfunctors. In particular, this yields examples of underlying diagram functorswhich are far from being equivalences.

4.2 The case of category algebras

Since the construction of underlying diagram functors (Proposition 4.1.7) wasfairly abstract, in this section we want to identify them in more specific situa-tions. These special cases illustrate a few fairly typical features of underlyingdiagram functors.

To begin with we collect a proposition about the homology functor (Ex-ample 3.3.7) in the context of semisimple abelian categories, i.e., abeliancategories in which every short exact sequence splits.

Proposition 4.2.1. For every semisimple abelian category A the homologyfunctor H∗ : D(A)→ AZ is an equivalence of categories.

Proof. We refer the reader to [GM03, §III.2] for a discussion of this lemma.

As a special case, given a semisimple ring R (for example, R could be afield k), the homology functor is an equivalence of categories

H∗ : D(R) ' Mod(R)Z.

DRAFT


This observation will allow us to identify particular instances of underlyingdiagram functors with suitable homology functors.

Let A be a small category with finitely many objects only and let R be aring. The category algebra RA is the following R-algebra. The underlying R-module of RA is free with basis given by the morphisms in A,

RA =⊕

f : a0→a1

R.

Let us denote by ef ∈ RA the generator corresponding to the morphism f : a0 →a1. The multiplication is essentially the R-bilinear extension of the compositionin A, i.e., it is the bilinear extension of the assignment

(eg, ef ) 7→egf , s(g)=t(f),0 , otherwise.

It is easy to see that Σa∈Aeida ∈ RA is a neutral element for this multiplication(and this is the reason why we assumed A to have finitely many objects only).

Having a module over the category algebraRA is as good as having a diagramX : A → Mod(R). More precisely, given such a diagram X, the reader checksthat the direct sum

⊕a∈AX(a) ∈ Mod(R) actually can be turned into an

RA-module. Moreover, this construction extends to a functor and there is thefollowing statement about this functor.

Proposition 4.2.2. Let A be a category with finitely many objects only and letR be a ring. There is an equivalence of categories

Mod(R)A ' Mod(RA) : X 7→⊕a∈A

X(a).

Proof. This proof is left as an exercise. As a hint we suggest to take a look atthe idempotent elements eida ∈ RA, a ∈ A, in order to establish the essentialsurjectivity.

As general references for representations of category algebras and small cat-egories we mention [Mit72, Web07, Xu07]. Here we content ourselves by illus-trating the notion of a category algebra by the following examples.

Examples 4.2.3. Let R be a ring.

(i) Any discrete group G can be considered as a category with one objectonly (as in §2.3). In this case the category algebra is simply the groupalgebra RG and the proposition reproduces the equivalence of categories

Mod(R)G ' Mod(RG).

(ii) Let us recall that a quiver Q is simply an oriented graph, hence a quadru-pel Q = (Q0, Q1, s, t : Q1 → Q0) given by a set of vertices, a set of edges,and source and target maps. Abusing notation, we also denote by Q the

DRAFT

4.2. THE CASE OF CATEGORY ALGEBRAS 69

corresponding path category, i.e., the category freely generated by thequiver Q (this can be thought of as a many object version of the construc-tion of the free monoid on a set). Assuming Q0 to be finite, the categoryalgebra RQ is the path algebra of the quiver and the proposition spe-cializes to the equivalence

Mod(R)Q ' Mod(RQ).

(iii) Let P = (P0,≤) be a partially ordered set. We abuse notation and denoteby P the category with P0 as set of objects and morphisms given by

homP (x, y) =

∗ , x ≤ y,∅ , otherwise.

Under the assumption that P0 is finite, in this case the category alge-bra RP is the incidence algebra of P and the proposition yields anequivalence of categories

Mod(R)P ' Mod(RP ).

Proposition 4.2.2 describes an equivalence of abelian categories. Every suchequivalence induces a derived equivalence in the following sense.

Lemma 4.2.4. Let F : A ' B be an equivalence of abelian categories A,B.Then applying F levelwisely induces equivalences

F : AZ ' BZ, F : Ch(A) ' Ch(B), and F : D(A) ' D(B).


We want to apply this lemma to the situation of Proposition 4.2.2. Thus,let A be a small category with finitely many objects only and let R be a ring.The equivalence Mod(R)A ' Mod(RA) induces the two vertical equivalences inthe diagram

D(Mod(R)A)diaA //

'

D(R)AH∗ // (Mod(R)Z)A

∼= // (Mod(R)A)Z

'

D(RA)H∗

// Mod(RA)Z.

(4.2.5)

Lemma 4.2.6. Let A be a small category with finitely many objects only andlet R be a ring. The diagram (4.2.5) commutes up to a canonical natural iso-morphism.

Proof. Unraveling definitions, this is an immediate consequence of the fact thathomology is additive. In more detail, using the explicit description of the equiv-alence in Proposition 4.2.2 we see that if we trace X ∈ D(Mod(R)A) clockwisely

DRAFT


through (4.2.5) then we obtain⊕

a∈AH∗(Xa). Tracing the object X counter-clockwisely through (4.2.5) we get H∗(

⊕a∈AXa). Since homology commutes

with finite direct sums this shows that the two compositions are canonicallynaturally isomorphic.

In the semisimple case there is the following corollary (see again Defini-tion 2.3.4).

Corollary 4.2.7. Let A be a small category with finitely many objects onlyand let R be a semisimple ring. The functor diaA : D(Mod(R)A) → D(R)A isequivalent to the homology functor H∗ : D(RA) → Mod(RA)Z. In particular,diaA is an equivalence if and only if H∗ is an equivalence.

Proof. Since R is semisimple, the homology functor H∗ : D(R) → Mod(R)Z isan equivalence of categories by Proposition 4.2.1. Hence the same is true forthe induced functor H∗ : D(R)A → (Mod(R)Z)A and the first claim follows since(4.2.5) commutes up to a natural isomorphism (Lemma 4.2.6). Finally, it is easyto see that if two functors F1, F2 are equivalent in the sense of Definition 2.3.4then F1 is an equivalence if and only if F2 is an equivalence.

If we specialize to the case of a field and take up again Examples 4.2.3 thenwe obtain the following examples.

Examples 4.2.8. Let k be a field.

(i) For every discrete group G the functor diaG : D(Mod(k)G) → D(k)G isequivalent to the homology functor

H∗ : D(kG)→ Mod(kG)Z.

(ii) For every quiver Q with only finitely many objects the underlying dia-gram functor diaQ : D(Mod(k)Q)→ D(k)Q is equivalent to the homologyfunctor

H∗ : D(kQ)→ Mod(kQ)Z.

(iii) For every finite, partially ordered set P the underlying diagram functordiaP : D(Mod(k)P )→ D(k)P is equivalent to the homology functor

H∗ : D(kP )→ Mod(kP )Z.

These examples illustrate the following two points.

Remark 4.2.9.

(i) In general, underlying diagram functors are far from being equivalences.In fact, an abstract way to see this is as follows. If they were equivalences,then the derived categories of diagram categories were at the same timetriangulated categories (see §5) and abelian. But any such category isnecessarily semisimple — a property which, in general, is not enjoyedby group algebras, path algebras, and incidence algebras. (In §15.4 we

DRAFT

4.3. SOME EXPLICIT EXAMPLES 71

take a closer look at specific examples of underlying diagram functors,showing more explicitly that one has to distinguish between coherent andincoherent diagrams.)

(ii) Even if we are in one of the few cases in which derived categories admit(co)limits different from (co)products, then derived (co)limits and cate-gorical (co)limits of course do not match up to an application of underlyingdiagram functors.

As a specific example, let k be a field and let G be a discrete group. Sincethe derived category D(k) is equivalent to Mod(k)Z it admits colimits ofshape G. However, the diagram

D(Mod(k)G)diaG //

LcolimG

D(k)G' // (Mod(k)Z)G

colimG

D(Mod(k))H∗

// Mod(k)Z

does not commute up to a natural isomorphism. In fact, since LcolimG isequivalent to the group homology functor H∗(G;−) (see again §2.3), by

group homology relative to afield?Examples 4.2.3 the previous diagram commutes up to a natural isomor-

phism if and only if this is the case for

D(Mod(kG))H∗ //

H∗(G;−)

Mod(kG)Z

(−)G

D(Mod(k))H∗

// Mod(k)Z.

Now, consider any G and M ∈ Mod(kG) such that the group homologyHn(G;M) is non-trivial for some n > 0. Then (H∗M)G is the vectorspace MG concentrated in degree zero, while this is not the case for theimage of M under the remaining composition. In particular, neither ofthe above diagrams commutes up to a natural isomorphism.

4.3 Some explicit examples

DRAFT


DRAFT

Chapter 5

Derived categories astriangulated categories

Derived categories are obtained from categories of chain complexes by invertingthe class of quasi-isomorphisms. It is important to note that these two cate-gories (the category of chain complexes and the derived category) behave quitedifferently.

(i) Categories of chain complexes are again abelian and as such they arefinitely complete and finitely cocomplete. In contrast to this, in general,the only limits and colimits which exist in derived categories are finitebiproducts.

(ii) More importantly, at the level of chain complexes one can construct reso-lutions of diagrams leading to derived limits and derived colimits (see §2.3and §3.5 for special cases). However, this calculus of derived limits andderived colimits is not visible to the derived category alone.

While (i) is a matter of fact one has to live with, there are various ways oftrying to deal with (ii). One way is given by endowing derived categories withadditional structure, thereby turning them into triangulated categories. The the-ory of triangulated categories goes back to Verdier [Ver96] and Puppe [Pup67].In this section we only discuss some basics and we instead refer the reader tothe monographs [Nee01, HJR10] or to [GM03, §§III-IV], [Wei94, §10] for moredetails.

The basic idea is that triangulations essentially encode certain shadows ofiterated cofiber constructions, hence by §3.5 of iterated derived cokernel con-structions. This is classically achieved in two steps, first, by endowing thehomotopy category of an abelian category with a triangulation and, second,by showing that this induces a triangulation on the derived category. Here wesketch the construction of these classical triangulations, but many details areomitted. (The results of this section will not be applied in later sections; insteadthey only serve as motivation for later constructions.)

73

DRAFT

74 CHAPTER 5. D(A) AS A TRIANGULATED CATEGORY

In §5.1 we introduce the homotopy category of an abelian category. In §5.2we construct the class of distinguished triangles on the homotopy category.In §5.3 we define triangulated categories and collect a few basic results. In §5.4we sketch the classical construction of Verdier of a triangulation on derivedcategories of abelian categories. Finally, in §5.5 we summarize a few defects ofthe axioms of triangulated categories. We will later see how these problems areavoided by the theory of derivators.

5.1 The homotopy category of an abelian cate-gory

In this section we consider the homotopy category K(A) of an abelian cate-gory A. This category occurs as an intermediate step in the classical construc-tion of the derived category D(A) = Ch(A)[W−1

A ].The chain homotopy relation ' is a congruence relation on Ch(A), i.e.,

we have equivalence relations on all morphism sets homCh(A)(X,Y ) which arecompatible with compositions. Hence, we can make the following definition.

Definition 5.1.1. Let A be an abelian category. The homotopy categoryK(A) of A is the quotient category

K(A) = Ch(A)/ ' .

Thus, objects in K(A) are simply chain complexes and morphisms are chainhomotopy classes of morphisms. The composition is defined by choosing repre-sentatives and passing to the chain homotopy class of the corresponding com-position.

By definition there is a quotient functor

γ′ : Ch(A)→ K(A)

which is the universal example of a functor identifying chain homotopic maps.Since K(A) is a quotient category it is fairly simple to put ones hands on it.

Lemma 5.1.2. The homotopy category K(A) of an abelian category is additive.


Warning 5.1.3. Note however that, in general, the homotopy category of anabelian category is not abelian. We will come back to this warning in §5.5.

Note that a chain map f is a chain homotopy equivalence if and only if γ′(f)is an isomorphism. In particular, a chain complex X ∈ Ch(A) is contractible ifand only if γ′(X) ∈ K(A) is a zero object.

In order to relate the homotopy categoryK(A) to the derived categoryD(A),we observe that K(A) can be constructed as a localization of Ch(A). As apreparation for this, we recall the cylinder construction at the level of chaincomplexes. This construction is a further instance of the biadditive pairingdescribed in Digression 3.2.2 (see again Examples 3.2.4).

DRAFT

5.1. THE HOMOTOPY CATEGORY OF AN ABELIAN CATEGORY 75

Examples 5.1.4.

(i) Let I+ = Ccell([0, 1]+) ∈ Chb(FZ) be the reduced cellular chain complexof [0, 1]+ = [0, 1] t ∗ endowed with the CW structure consisting of three0-cells and one 1-cell only. More explicitly, we have I0 = Z ⊕ Z, I1 = Z,and Ik = 0, k 6= 0, 1 with non-trivial differential (−id, id) : Z⊕Z→ Z. ForX ∈ Ch(A) there is a natural isomorphism

I+ ⊗X ∼= cyl(X),

where cyl : Ch(A)→ Ch(A) is the cylinder functor. (Again, if you didnot see a definition of cyl before, then we suggest to take this constructionas a definition and to unravel it.)

The continuous map [0, 1] → ∗ induces a chain map I+ → S0 whichis easily seen to be a chain homotopy equivalence. It follows from Di-gression 3.2.2, Examples 3.2.4, and Lemma 3.1.8 that there is a naturalisomorphism

(I+ ⊗X → S0 ⊗X) ∼= (cyl(X)→ X)

and that this map is a natural chain homotopy equivalence.

(ii) Let S0+ = Ccell(0, 1+) ∈ Chb(FZ) be the reduced cellular chain complex

of 0, 1+ endowed with the CW structure consisting of three 0-cells only.Thus, S0

+∼= S0 ⊕ S0 and for X ∈ Ch(A) there is a natural isomorphism

S0+ ⊗X ∼= X ⊕X.

(Recall that the space 0, 1+ is the wedge of two copies of 0, 1.)

(iii) The continuous map 0, 1 → [0, 1] yields a map S0+ → I+ in Chb(FZ) and

hence a natural transformation S0+ ⊗ − → I+ ⊗ − of functors Ch(A) →

Ch(A). Under the above isomorphisms, for X ∈ Ch(A) this yields theusual natural inclusion

(S0+ ⊗X → I+ ⊗X) ∼= (X ⊕X → cyl(X)).

We denote the two resulting inclusions by i0, i1 : X → cyl(X).

The point of this cylinder construction is the following (which is a chain-levelversion of Lemma 3.4.1).

Lemma 5.1.5. Let f, g : X → Y be chain maps. There is a bijection betweenchain homotopies s : f → g and chain maps H : cyl(X)→ Y making the follow-ing diagram commute

X

i0 $$

f

cyl(X)H // Y.

X

i1::

g

@@

DRAFT



Thus, as in topology, the cylinder is the natural domain for homotopies. Inthe special case of the constant chain homotopy of the identity id : X → X onechecks that the resulting homotopy

X

i0 $$

id

cyl(X)HX // X

X

i1::

id

@@ (5.1.6)

is the chain homotopy equivalence cyl(X)→ X from Examples 5.1.4(i). Theseobservations can be reformulated as follows.

Proposition 5.1.7. Let A be an abelian category and let C be a category. Thefollowing are equivalent for a functor F : Ch(A)→ C.

(i) For every pair of chain homotopic maps f, g we have Ff = Fg.

(ii) The functor F sends chain homotopy equivalences to isomorphisms.

Proof. Clearly, (i) implies (ii). Conversely, (ii) implies that F sends the chainhomotopy equivalences HX in (5.1.6) to isomorphisms in C. Consequently, theinclusions i0, i1 : X → cyl(X) have the same images in C and statement (i) ishence an immediate consequence of Lemma 5.1.5.

The following corollary implies that the homotopy category indeed occursas an intermediate step in the construction of the derived category.

Corollary 5.1.8. Let A be an abelian category and let HEA be the class of chainhomotopy equivalences in Ch(A). The functor γ′ : Ch(A) → K(A) exhibitsK(A) as the localization of Ch(A) at the class of chain homotopy equivalences,

K(A) = Ch(A)[HE−1A ].

Proof. In view of Proposition 5.1.7 this is simply a reformulation of the universalproperty of the quotient functor γ′ : Ch(A)→ K(A).

5.2 Distinguished triangles in the homotopy cat-egory

As already mentioned, besides being additive, the category K(A) is, in general,rather ill-behaved. A classical way of adressing this is by endowing it with someadditional structure which can be roughly thought of as shadows of the existenceof short exact sequences in the category Ch(A).

DRAFT

5.2. TRIANGLES IN THE HOMOTOPY CATEGORY 77

To begin with, the suspension functor Σ: Ch(A) → Ch(A) clearly sendschain homotopy equivalences to chain homotopy equivalences. Hence, there isa unique suspension functor Σ: K(A)→ K(A) such that

Ch(A)Σ //

γ′

Ch(A)

γ′

K(A)∃!Σ

// K(A)

commutes. One easily checks that this induced functor is again an equivalenceof categories.

Definition 5.2.1. Let A be an abelian category.

(i) A triangle in K(A) is a diagram of the form X → Y → Z → ΣX.

(ii) A morphism of triangles is a commutative diagram

X //

f

Y //

g

Z //

h

ΣX

Σf

X ′ // Y ′ // Z ′ // ΣX ′.

Note that we insist that the vertical morphism to the very right is the sus-pension of the morphism to the very left. Correspondingly there is the notionof an isomorphism of triangles. These notions, of course, make sense in anycategory with an endofunctor (see [KV87, Kel90]).

We now single out a particular class of triangles on K(A). Recall from(3.2.5) and Definition 3.2.6 that associated to any chain map f : X → Y thereis the cone Cf which is endowed with a chain map cof(f) : Y → Cf . There isalso a canonical map Cf → ΣX. To construct it, we note that by the definitionof the cone CX of a chain complex there is a natural short exact sequence

0→ Xi→ CX

q→ ΣX → 0. (5.2.2)

(In fact, by means of Examples 3.2.4 this is induced by S0⊗− → I⊗− → S1⊗−.)The universal property of the defining pushout diagram (3.2.5) applied to q andthe zero map 0: Y → ΣX implies that there is a unique chain map Cf → ΣXmaking the following diagram

Xf//

i

Y

cof(f)

0

CX //

q //

Cf

∃!

""

ΣX

DRAFT


commutative. The resulting triangle

X → Y → Cf → ΣX (5.2.3)

in K(A) is the standard triangle of f . We say that a triangle in K(A) isdistinguished if it is isomorphic to a standard triangle.

We also include an alternative construction of the standard triangles whichgeneralizes more directly to other situations. This different description showsthat distinguished triangles essentially arise as iterated cofiber constructions.Given a chain map f : X → Y we consider the following commutative diagramof chain complexes

Xf//

iX

YiY //

g

CY

CX // Cfh′// Cg,

(5.2.4)

in which both squares are pushout squares. Thus, the squares are the cofibersquares of f and of g = cof(f), respectively (see again (3.2.5)). Now, since bothsquares are pushout squares the same is true for the composite square which isdepicted on the left in

X //

iX

CY

X //

iX

0

CX // Cg, CX // ΣX.

If we replace the cone CY by the trivial chain complex and consider the pushoutsquare on the right, then it is easy to see that we obtain ΣX (simply refor-mulate (5.2.2) using Lemma 3.2.8). Since the pushout functor is an additivefunctor, the chain homotopy equivalence CY → 0 induces by Lemma 3.1.8 achain homotopy equivalence

φ : Cg → ΣX.

Finally, if we combine the induced isomorphism φ : Cg ∼= ΣX in K(A) (Corol-lary 5.1.8) with the morphisms in (5.2.4), then we obtain an alternative descrip-tion of the standard triangle (5.2.3) as

Xf→ Y

g→ Cfφh′→ ΣX. (5.2.5)

Remark 5.2.6. By the above construction the distinguished triangles in thehomotopy category of an abelian category are obtained from two iterationsof the cofiber construction. But by the results of §3.5 this means that weconsider twofold iterations of the derived cokernel construction. It is importantto realize that we are working in the derived context. In fact, at the level ofchain complexes two iterations of the cokernel construction of course are trivialsince for every chain map f : X → Y the canonical map Y → cok(f) is anepimorphism so that its cokernel is trivial.

DRAFT

5.3. TRIANGULATED CATEGORIES 79

Two iterations of the cofiber construction lead to distinguished triangles.So, let us see what happens if we apply the construction once more, therebyextending (5.2.4) to the diagram

Xf

//

iX

YiY //

g

CY

CX // Cfh′

//

Cg,

f ′

C(Cf) // C(h′).

(5.2.7)

Similarly to the situation in (5.2.4), we note that C(Cf) is a contractible chaincomplex and that the chain homotopy equivalence C(Cf) ' 0 induces a chainhomotopy equivalence C(h′) ' ΣY . This shows that for every distinguishedtriangle (5.2.5) there is an additional distinguished triangle such that its firstmorphism agrees with the second one of the original triangle. Intuitively, wecan ‘rotate’ a distinguished triangle to obtain a new one.

This property together with a few additional properties are turned into thedefinition of a triangulated structure or triangulation on an additive category aswe recall in §5.3.

5.3 Triangulated categories

In this section we take a glimpse at the theory of triangulated categories. Herewe only include what is strictly necessary in this motivational section. Formore details we refer the reader to the established literature which includes[Nee01, HJR10].

We begin right away with the definition and then include a short discussionof the axioms.

Definition 5.3.1. Let T be an additive category with a self-equivalence Σ: T →T and a class of distinguished triangles X → Y → Z → ΣX. The pair consist-ing of Σ and the class of distinguished triangles defines a triangulation or atriangulated structure on T if the following four axioms are satisfied.

(T1) For every X ∈ T , the triangle Xid→ X → 0→ ΣX is distinguished. Every

morphism in T occurs as the first morphism in a distinguished triangle andthe class of distinguished triangles is replete, i.e., is closed under isomor-phisms.

(T2) A triangle Xf→ Y

g→ Zh→ ΣX is distinguished if and only if the rotated

triangle Yg→ Z

h→ ΣX−Σf→ ΣY is distinguished.

DRAFT


(T3) Given two distinguished triangles and a commutative solid arrow diagram

X //

u

Y //

v

Z //

∃≥1w

ΣX

Σu

X ′ // Y ′ // Z ′ // ΣX ′

there exists a dashed arrow w : Z → Z ′ as indicated such that the extendeddiagram commutes.

(T4) For every pair of composable arrows f3 : Xf1→ Y

f2→ Z there is a commuta-tive diagram in which the rows and columns are distinguished triangles:

Xf1 // Y

g1 //

f2

C1h1 //

ΣX

Xf3

// Z

g2

g3

// C3h3

//

ΣX

Σf1

C2

h2

C2

Σg1h2

h2

// ΣY

ΣYΣg1

// ΣC1

(5.3.2)

A triangulated category is an additive category together with a triangulatedstructure.

Motivated by the construction for the homotopy category K(A) of an abeliancategory in §5.2, given a distinguished triangle

Xf→ Y

g→ Zh→ ΣX,

the third object Z is referred to as ‘the’ cone of f . Inspired by similar construc-tions in the category of pointed topological spaces, Z is sometimes also called‘the’ mapping cone of f . A few comments about these axioms are in order.

Remark 5.3.3.

(i) One way to think of distinguished triangles is that they are some shadowsof certain derived cokernel or homotopy cokernel constructions on ‘a modelin the background’ (this will be justified in §5.4 and again in §15.5). Theaxioms (T1)-(T4) capture some compatibility properties satisfied by suchderived cokernel or homotopy cokernel constructions.

(ii) Axiom (T2) is the rotation axiom, axiom (T3) the (mapping) coneaxiom, and axiom (T4) the octahedron axiom. The name octahedronaxiom is motivated by a different way of drawing the diagram in axiom

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5.3. TRIANGULATED CATEGORIES 81

(T4); see for example [Wei94, p. 375]. The precise form of the octahe-dron axiom varies a bit depending on the author. For various equivalentversions of the axiom see, for example, [Nee91, Nee01, HJR10].

(iii) Triangulated categories were invented independently by Puppe [Pup67](motivated by algebraic topology) and Verdier (motivated by algebraicgeometry) in his 1967 thesis [Ver67]; see [Ver96] for a reprint. The differ-ence between these axioms is that Puppe triangulations are only asked tosatisfy axioms (T1)-(T3). (As of this writing, there is not a single exampleof a Puppe triangulated category which does not satisfy the octahedronaxiom.)

(iv) Note that axiom (T3) asks for a weak functoriality of the cone construc-tion: any ‘partial morphism’ of triangles can be extended to an actualmorphism of triangles. However, we do not ask for the uniqueness ofsuch an extension which is to say that the cone is not necessarily a func-torial construction T [1] → T . This lack of functoriality has importantimplications and we come back to this in §5.5.

(v) Using the interpretation of the third object in a distinguished triangle asa cone, we can think of the octahedron axiom as a triangulated categoryversion of the third Noether isomorphism theorem. In fact, while theNoether isomorphism theorem says that ‘the quotient of two quotients isagain a quotient’ the octahedron axiom asks that ‘a cone of two cones isagain a cone’. We try to provide more intuition for this axiom in §§5.4-5.5, §15.5, and §16.

We suggest the reader who just saw the definition of a triangulated categoryfor the first time to provide the proofs of the following statements. Alternatively,detailed proofs can be found, for example, in [Nee01, HJR10].

Lemma 5.3.4. Let T be a triangulated category and let Xf→ Y

g→ Zh→ ΣX

be a distinguished triangle. The compositions g f and h g vanish.


Proposition 5.3.5. Let T be a triangulated category and let W ∈ T . Forevery distinguished triangle X → Y → Z → ΣX the corepresented functorhomT (W,−) yields an exact sequence of abelian groups

homT (W,X)→ homT (W,Y )→ homT (W,Z).


Together with the rotation axiom (T2) we see that represented functors thussend distinguished triangles to long exact sequences. This is often referred toby saying that represented functors are homological.

Also the proof of the following 5-lemma is suggested as an exercise.

DRAFT


Proposition 5.3.6. Let T be a triangulated category and let

X //

u

Y //

v

Z //

w

ΣX

Σu

X ′ // Y ′ // Z ′ // ΣX ′

be a morphism of distinguished triangles. If two of the maps u, v, w are isomor-phisms then so is the third one.


Corollary 5.3.7. Let X → Y be a morphism in a triangulated category. Anytwo distinguished triangles extending the morphism are isomorphic.

Proof. This is immediate from axiom (T3) and the 5-lemma.

Warning 5.3.8. Thus, in particular, every morphism in a triangulated cate-gory has a cone and any two cones are isomorphic. In general, however, theseisomorphisms are not canonical. We will get back to this in §5.5.

Proposition 5.3.9. A morphism in a triangulated category is an isomorphismif and only if any of its cones is zero.


5.4 Exact morphisms and classical triangulations

In this section we again consider an abelian category A and sketch the con-struction of the classical triangulation on the derived category D(A). This isclassically achieved by first constructing a triangulation on the homotopy cat-egory K(A) and then showing that it induces a triangulation on the derivedcategory D(A).

Recall from §5.2 that we already defined a suspension functor Σ: K(A) →K(A) and a class of distinguished triangles on K(A). This structure defines atriangulation on the homotopy category. Instead of providing a detailed proof,we content ourselves by making plausible that the octahedron axiom (T4) issatisfied. Recall from (5.2.4) that the standard triangles are constructed by twoiterations of the cofiber construction. Now, for the octahedron axiom (5.3.2) weconsider two composable morphisms

f3 = f2 f1 : X → Y → Z

DRAFT

5.4. EXACT MORPHISMS AND CLASSICAL TRIANGULATIONS 83

together with three corresponding distinguished triangles as in

Xf1 // Y

g1 //

f2

C1h1 //

ΣX

Xf3

// Z

g2

g3

// C3h3

//

ΣX

Σf1

C2

h2

C2

h2

// ΣY

ΣYΣg1

// ΣC1.

The task then is to show that the cones themselves sit in a suitably compatibledistinguished triangle (as indicated by the above dashed arrows). Starting withthe pair of composable morphisms this is roughly achieved by passing to thefollowing diagram

Xf1 //

1

Yf2 //

2

Z //

3

0

0 // C(f1) //

4

C(f3) //

5

X ′ //

6

0

0 // C(f2) // Y ′ // C(f1)′.

(5.4.1)

This diagram is obtained step-by-step by adding the pushout squares 1 − 6and the zeros are placeholders for suitable contractible chain complexes (hencezero objects in K(A)). Using again that pushout squares glue, we observe that

a distinguished triangle for f1 can be obtained from the pushout squares 1

and 2 + 3 . Similarly, the pushout squares 2 + 4 and 3 + 5 yield adistinguished triangle for f2, while the one for f3 = f2 f1 is induced from thepushout squares 1 + 2 and 3 . The desired distinguished triangle, finally, is

obtained from considering the pushout squares 4 and 5 + 6 .A reasoning along these lines yields the following result.

Theorem 5.4.2. The homotopy category K(A) of an abelian category A to-gether with the equivalence Σ: K(A) → K(A) and the class of distinguishedtriangles defined in §5.2 is a triangulated category.

Proof. The category K(A) is additive by Lemma 5.1.2 and Σ: K(A) → K(A)is an equivalence of categories. Axiom (T1) is immediate and we gave a sketchproof for the rotation axiom (T2) and the octahedron axiom (T4). A detailedproof is a bit more involved and we refer the reader to [HJR10] or [Wei94,Proposition 10.2.4].

DRAFT


Note the sign in the rotation axiom (T2) of Definition 5.3.1. In the courseof the construction of canonical triangulations on stable derivators we will givea conceptual explanation of this sign; see §14.3 and §15.5.

Having settled the classical triangulation on K(A), we now turn to the in-duced triangulation on D(A). Recall from Corollary 5.1.8 that HEA denotes theclass of chain homotopy equivalences. There clearly is an inclusion HEA ⊆WAand, by that corollary, the localization functor γ : Ch(A)→ D(A) hence factorsuniquely through γ′ : Ch(A)→ K(A), yielding the canonical functor

γ′′ : K(A)→ D(A).

Passing through the homotopy category K(A) as an intermediate step, thederived category is obtained as the localization

D(A) = K(A)[W−1A ].

There are general techniques which allow us to conclude that such localizationsof triangulated categories are again triangulated and that the localization func-tor ‘is’ exact in the sense of the following definition; see for example [Wei94,§10] or [Kra10].

The naive idea behind the definition of a morphism of triangulated categoriesT and T ′ is that is should be an additive functor F : T → T ′ which sendsdistinguished triangles to distinguished triangles. To make this precise, however,

we have to be a bit more careful. In fact, given a distinguished triangle Xf→

Yg→ Z

h→ ΣX in T , an application of F yields the diagram

FXFf→ FY

Fg→ FZFh→ FΣX

in T ′. Note that, by the very definition of a triangle, this diagram can not be adistinguished triangle since the fourth object is not the suspension of the firstobject. In order to at least potentially obtain a distinguished triangle we needa map FΣX → ΣFX, and this is taken care of by the following definition.

Definition 5.4.3. Let T and T ′ be triangulated categories. An exact functorT → T ′ is a pair (F, σ) consisting of

(i) an additive functor F : T → T ′ and

(ii) a natural transformation σ : FΣ→ ΣF

such that for every distinguished triangle Xf→ Y

g→ Zh→ ΣX in T the image

triangle FXFf→ FY

Fg→ FZσFh→ ΣFX is distinguished in T ′.

Remark 5.4.4.

(i) Note that exactness of a functor of triangulated categories is not a prop-erty but that it amounts to specifying an additional structure. If we wantto emphasize this then we refer to σ : ΣF → FΣ as an exact structureon F .

DRAFT

5.5. BEYOND TRIANGULATED CATEGORIES 85

(ii) Given an exact functor (F, σ) it follows that the exact structure σ : FΣ→ΣF is necessarily a natural isomorphism.

The following theorem is a consequence of the more general Verdier local-ization theorem [Kra10, §4]; again we refer the reader to the literature.

Theorem 5.4.5. Let A be an abelian category. The derived category D(A) canbe turned into a triangulated category and the localization functor γ′′ : K(A)→D(A) into an exact functor. We call this triangulation the classical triangu-lation.

Remark 5.4.6. Given two abelian categories A and B, as already claimed, formany additive functors F : Ch(A) → Ch(B) arising in nature there are associ-ated left or right derived functors LF,RF : D(A) → D(B). It turns out thatthose functors admit exact structures (in the sense of Definition 5.4.3) withrespect to classical triangulations. In the sequel we will give a conceptual ex-planation of this phenomenon.

5.5 Beyond triangulated categories

The theory of triangulated categories originated in the 1960’s and is by nowa well-established theory. This theory has been successfully applied in variousareas of pure mathematics, including algebra, geometry, topology, and analyis;see the book [HJR10] for a few sample applications and additional references.

At the same time, despite all these successes, from the very beginning on(see already the introduction to [Hel68]) it was also apparent that the axiomsof a triangulated category come with certain defects. We conclude this sectionby a short discussion of some of these defects. One goal of this discussion is toclarify the axioms of a triangulated category.

(i) One crucial observation is that the cone construction at the level of tri-angulated categories is not functorial. For every morphism in T there isa distinguished triangle (by axiom (T1)), hence we can associate a coneobject to it. There are the following (closely related) drawbacks to thisdefinition of a cone.

(a) Cone objects are not characterized by a universal property.

(b) We already observed that any two cones of a fixed object are isomor-phic (Corollary 5.3.7), but only by means of non-canonical isomor-phisms.

(c) More generally, every morphism of morphisms in T yields an inducedmorphism of cones (by axiom (T3)), but, again, this morphism is notcanonical and it lacks functoriality.

Because of these remarks, it might be tempting to simply modify theaxioms of a triangulated category and to ask for a functorial cone con-struction T [1] → T . However, it turns out that this does not work since

DRAFT


every triangulated category with a functorial cone construction is neces-sarily semisimple ([Ver96, Proposition 1.2.13] or [Ste11]). This fact is tobe seen in contrast to Examples 3.3.8 guaranteeing the existence of a conefunctor C : D(A[1])→ D(A).

(ii) The axioms of a triangulated category ask for additional structure con-sisting of a suspension functor and a class of distinguished triangles. Thisdatum does not satisfy any universal property and is non-canonical. Toput it as a slogan, at the level of the triangulated category we do notsee anymore where this structure comes from. We illustrate this in thecontext of derived categories of abelian categories by the following points.The claims made in the following points will be justified in the remainderof this book and the sequels.

(a) We know that two consecutive morphisms in a distinguished trian-gle vanish (Lemma 5.3.4). However, the reason why this is true,namely, in the case of the derived category of an abelian category,the existence of derived pushout squares as in (5.2.4), is not visibleat the level of D(A). To put it in vague words, these derived pushoutsquares encode chain homotopies witnessing that the correspondingcompositions are null homotopic.

(b) The characterization of quasi-isomorphisms as those maps whichhave trivial cones (Proposition 5.3.9) is a shadow of the followingfacts.

i. Derived pushout and derived pullbacks of quasi-isomorphismsare again quasi-isomorphism.

ii. A square is a derived pushout square if and only if it is a derivedpullback square.

(c) Given a morphism in a triangulated category, axiom (T1), threeapplications of the rotation axiom (T2), and Corollary 5.3.7 implythat the cone and the suspension commute up to a non-canonicaland non-functorial isomorphism.

Again for derived categories of abelian categories, a reason for thisis given by the existence of diagrams of the form (5.2.7) consist-ing of derived pushout squares. In fact, let us start with an ac-tual chain map instead of a morphism in D(A). Then, denoting bycof : D(A[1]) → D(A[1]) the functor constructed in Examples 3.3.8,diagrams (5.2.7) can be used to show that there is a canonical iso-morphism

Σ ∼= cof3 : D(A[1])→ D(A[1]).

And this canonical isomorphism induces additional canonical isomor-phisms

cof Σ ∼= cof4 ∼= Σ cof.

However, diagram (5.2.7) is not visible to the derived category D(A)alone.

DRAFT


(d) The octahedron axiom tells us that for a pair of composable mor-phisms in D(A) together with their composition

f3 = f2 f1 : Xf1→ Y

f2→ Z

there are relations between arbitrarily chosen corresponding cone ob-jects C1, C2, and C3. For example, using suggestive notation, the oc-tahedron axiom (5.3.2) in combination with the rotation axiom (T2)yields isomorphisms

C1∼= Σ−1C(C3 → C2), (5.5.1)

C2∼= C

(C1 → C3

),

C3∼= Σ−1C

(C2 → ΣC1

).

Again, these isomorphisms are neither canonical nor natural if onekeeps track of the derived category D(A) only.

However, if one starts with pairs of composable chain maps insteadand if one also keeps track of the diagrams (5.4.1), then one canconstruct canonical such isomorphisms. To this end, let us denoteby [2] the partially ordered set (0 < 1 < 2), considered as a category,so that a diagram of shape [2] simply specifies two composable mor-phisms. One checks that the diagram (5.4.1) depends functorially onthe pair of composable morphisms in Ch(A). Moreover, the passageto these diagrams is a homotopy-invariant construction, i.e., it sendslevelwise quasi-isomorphisms to levelwise quasi-isomorphisms. Thus,as a first upshot, if B denotes the small category given by the shapeof (5.4.1), then we obtain a functor

D(A[2])→ D(AB),

which sends (X → Y → Z) ∈ D(A[2]) to (5.4.1). Moreover, in everysuch diagram all squares are derived pushout squares and derivedpullback squares. Hence, as a second upshot, such diagrams leadto canonical isomorphisms as in (5.5.1), thereby defining naturalisomorphisms of suitably defined functors

D(A[2])→ D(A).

Motivated by this, we think of the diagrams (5.4.1) as refinementsof the corresponding underlying octahedron diagrams (5.3.2).

(iii) A triangulation specifies some non-canonical structure on categories likederived categories of abelian categories. Since these structures are non-canonical, morphisms between triangulated categories have to be definedby means of additional structure as well. In order to make precise thatsuch a morphism is compatible with suspensions and distinguished trian-gles one has to fix an exact structure (Definition 5.4.3).

DRAFT


However, in the typical examples the fact that we have an exact morphismof triangulated categories reflects the idea that ‘in the background on themodels’ (like on categories of chain complexes or suitable stable model cat-egories) there is a morphism of ‘stable homotopy theories’ which has theproperty that it preserves certain finite homotopy (co)limit constructions(as we discuss in the sequel [Gro16a]).

As a consequence of the fact that exact morphisms of triangulated cat-egories are defined by means of exact structures, the good notion of anatural transformation between exact functors is that these are naturaltransformations which are compatible with the chosen exact structures.Morally, however, such a compatibility assumption should not be neces-sary for induced transformations, and we justify this claim in [Gro16a].

(iv) A further related observation is that the axioms of a triangulated cat-egory, while satisfied in many examples, might miss interesting aspectsof homological algebra or homotopy theory. As mentioned above, theaxioms encode certain shadows of iterated cofiber constructions togetherwith some basic compatibilities. The octahedron axiom (T4) goes onestep further and considers two composable morphisms together with thevarious associated distinguished triangles.

However, the axioms have nothing to say if one wants to consider longerstrings of composable arrows and all the related distinguished trianglesand octahedron diagrams. This line of thought was pursued further andleads to the notion of higher triangulations; see for example [BBD82, Re-mark 1.1.14] and [Mal05a]. The main idea behind higher triangulationsis that one also asks for the existence of suitable higher octahedron dia-grams. However, again, these structures are only asked to exist and theylack functoriality and universality. We will come back to this in [Gro16a],but already refer the reader to [GS14a].

(v) And, of course, besides finite chains of composable morphisms, there aremore general diagrams to be considered. Given an abelian category Aand a small category A, often one can construct corresponding derivedcolimit and limit functors

LcolimA : D(AA)→ D(A) and RlimA : D(AA)→ D(A).

A crucial observation however is that, in general, these functors do notfactor through the underlying diagram functor diaA : D(AA) → D(A)A

constructed in Proposition 4.1.7. Thus, in the case of colimits, in general,there is no dashed arrow

D(AA)Lcolim//

diaA

D(A)

D(A)A@

::

DRAFT


making the diagram commutative, and dually for limits. Related to this,there are the following closely related slogans and observations.

(a) A typical slogan referring to this fact is that ‘diagrams in triangulatedcategories do not carry enough information (to canonically determinetheir homotopy (co)limits)’. For certain shapes of diagrams corre-sponding notions of homotopy limits and homotopy colimits were de-fined at the level of triangulated categories only; see [BN93, May01].However, again, these constructions are neither characterized by uni-versal properties nor are they functorial in the diagrams.

(b) A different but related slogan is that an application of underlyingdiagram functors diaA : D(AA) → D(A)A amounts to passing fromstrict diagrams to homotopy commutative diagrams (see §§4.1-4.2),and that such homotopy commutative diagrams do not carry enoughinformation. (In fact, one would need homotopy coherent diagramsto construct associated homotopy (co)limits.)

(c) Finally, given a triangulated category T and a small category A,in general, there is no canonical triangulation on the diagram cate-gory T A.

In §§12-13 we illustrate the added flexibility of derivators by setting upa basic calculus of diagrams of the shape of a square. This belongs tofairly rich calculus of cubes and hypercubes, which will be developed abit further in the sequel [Gro16a].

(vi) There are similar remarks when it comes to compatibility axioms for cat-egories which are simultaneously endowed with a triangulated structureand a monoidal structure (aka. tensor triangulated categories). We willbriefly take this up in §18 and mention its relevance for example to theadditivity of Euler characteristics and traces. These additivity resultsare known to fail at the level of triangulated categories, and this failurewas part of the original motivation of Grothendieck to consider derivatorsinstead.

One goal of this series of books is to show that the above issues can besuccessfully adressed if one keeps track more systematically of the calculus ofderived limits, derived colimits, and derived Kan extensions. Derived Kan exten-sions are certain variants of derived (co)limits which implicitly already showedup in the construction of diagrams like (5.2.4), (5.2.7), and (5.4.1). Before weturn to derived Kan extensions and homotopy Kan extensions in §§7-18, in §6we first recall some basics on Kan extensions in the ‘underived context’, i.e., inordinary category theory.

DRAFT


DRAFT

Chapter 6

Kan extensions

In this section we briefly review some basics concerning the theory of Kan ex-tensions from ordinary category theory. Kan extensions are certain universalconstructions which can be thought of as ‘relative versions of limits and colim-its’. The basic idea behind a derivator is to axiomatize key formal propertieswhich are common to

(i) the calculus of limits, colimits, and Kan extensions in complete and co-complete categories,

(ii) the calculus of derived limits, derived colimits, and derived Kan extensionsin derived categories, and

(iii) the calculus of homotopy limits, homotopy colimits, and homotopy Kanextensions in model categories or complete and cocomplete ∞-categories.

While here we establish these formal properties for ordinary categories, in §7they are essentially turned into the definition of a derivator and then one es-tablishes the non-trivial results that abelian categories, model categories, and∞-categories yield derivators (see §7 and Appendix B for a discussion of theseresults).

In §6.1 we define Kan extensions and observe that they generalize (co)limits.In §6.2 we include a short discussion of final functors. In §6.3 we note thatin complete and cocomplete categories Kan extensions exist and can be con-structed pointwisely using the better known (co)limits. In §6.4 we collect a fewbasic results about ordinary Kan extensions which we later extend to arbitraryderivators. We also mention some first examples to illustrate the concept of aKan extension. In §6.5 we show that (co)kernels can be described in terms ofKan extensions.

6.1 Motivation and definition

To motivate the notion of a Kan extension, we recall from §3 (see, in particular,Theorem 3.5.6) that for every abelian categoryA the cone functor C : D(A[1])→

91

DRAFT

92 CHAPTER 6. KAN EXTENSIONS

D(A) is the derived cokernel. Given a morphism f : X0 → X1 in Ch(A), thecone Cf can be constructed as the pushout object in

X0f//

i

X1

cof(f)

CX0// Cf.

Thus, in all detail, the cone of a chain map is obtained by associating aparticular span to it, passing to the pushout square of the span, and thenevaluating the pushout square on the final vertex.

X0f// X1

X0f//

i

X1

X0f//

i

X1

cof(f)

CX0 CX0// Cf Cf

(6.1.1)Similarly to the case of cokernels, also for cones it is often important to keeptrack of the morphism cof(f) : X1 → Cf instead of the chain complex Cf only.And to obtain a detailed description of the construction of cof(f) it suffices toreplace the final step in (6.1.1) by the restriction to the vertical morphism onthe right.

Note that the fact that this construction of the cone is seemingly complicatedis not a consequence of working in the derived setting. In fact, this is alreadytrue for cokernels in ordinary category theory: given a morphism f in a finitelycocomplete category C admitting a zero object, the cokernel is not the colimit ofthe morphism (which would be isomorphic to its target) but it is an instance ofthe more general notion of a weighted colimit. Similarly to the derived setting,the cokernel of a morphism f : X0 → X1 in such a category C can be obtainedin the following steps.

X0f// X1

X0f//

X1

X0f//

X1

0 0 // cok(k) cok(k)

(6.1.2)The only difference between (6.1.1) and (6.1.2) is that in the derived setting wereplace the map X0 → 0 by the inclusion X0 → CX0 in order to fix the problemthat, in general, pushouts are not exact or homotopy-invariant.

As an upshot, a detailed construction of the cokernel of a morphism passesthrough intermediate steps, the first two being the extension of the morphismto a span and then the extension to a pushout square. The purely categoricalnotion which is in the background of these two intermediate steps is the notion ofa Kan extension as we define it next. (The diagrams (5.2.4), (5.2.7), and (5.4.1)indicate further examples of such constructions.)

DRAFT

6.1. MOTIVATION AND DEFINITION 93

To abstract from our specific situation, we consider the following extensionproblem. Let u : A→ B be a functor between small categories and let X : A→ Cbe a functor taking values in a not necessarily small category C. The aim is tofind ‘extensions of X along u’,

A

u

X // C

B.

∃?

>>

(6.1.3)

The notion of an ‘extension’ can be made precise in different ways, namely byasking that the diagram (6.1.3)

(i) commutes on the nose,

(ii) commutes up to a specified natural isomorphism, or

(iii) commutes up to a specified universal natural transformation.

The first condition is ‘against the philosophical principle from category theorythat one should not ask that functors are equal but only naturally isomorphic’.While satisfied in many specific situations, in general, the second condition turnsout to be too strict. The third condition is the one we would like to axiomatizeand, of course, there are two versions of such universal natural transformations,depending on whether we want the datum to be initial or final.

Definition 6.1.4. Let u : A → B be a functor between small categories, let Cbe a category, and let X : A→ C.

(i) A left Kan extension of X along u is a functor LKanu(X) : B → Ctogether with a natural transformation η : X → LKanu(X) u satisfyingthe following universal property. For every pair (Y : B → C, α : X → Y u)there is a unique β : LKanu(X)→ Y such that α = (βu) · η : X → Y u.

(ii) A right Kan extension of X along u is a functor RKanu(X) : B → Ctogether with a natural transformation ε : RKanu(X) u→ X satisfyingthe following universal property. For every pair (Y : B → C, α : Y u→ X)there is a unique β : Y → RKanu(X) such that α = ε · (βu) : Y u→ X.

These notions are dual to each other and in this section we will mostly focuson left Kan extensions. For convenience, we include the following diagramaticdescription of this definition. A left Kan extension (LKanu(X), η) can be de-picted by a triangle populated by a natural transformation as in

AX //

u

⇓η

C

B

LKanu(X)

??

DRAFT


and the universal property of the left Kan extension can be illustrated as follows:

AX //

u

⇓∀α

C

=

AX //

u

⇓ηC

B

Y

??

B

55

Y

II

#∃!β

To develop some intuition for this notion let us consider the following specialcase. Recall that we denote the terminal category by 1.

Example 6.1.5. Let A be a small category and let X : A → C be given. A leftKan extension of X along πA : A→ 1 consists of a functor LKanπA(X) : 1→ Cand a universal natural transformation η : X → LKanπA(X) πA. The func-tor LKanπA(X) : 1 → C amounts to picking an object c ∈ C and the naturaltransformation is of the form η : X → ∆A(c),

a

∀f

X(a)

X(f)

ηa // c

a′, X(a′),

ηa′

==

thereby specifying a cocone on X. Together with the initiality of this datum,we see that the left Kan extension (c, η) is an initial cocone on X, which is tosay that c = colimAX and that η is the colimiting cocone.

Thus, we just saw that the notion of a left Kan extension along πA : A→ 1reduces to the notion of a colimit. In particular, such Kan extensions exist assoon as the target category is cocomplete. The goal of §6.3 is to show thatthis is true more generally and that those more general Kan extensions can beconstructed pointwisely by certain colimits. Before we get to that, in §6.2 weinclude a short digression on final functors.

6.2 Final functors

Let C be a cocomplete category and let u : A→ B. For every diagram X : B → Cthere is a canonical comparison map from colimA(X u) to colimB X. A functoru is final if this comparison map is always an isomorphism, and here we recallsome basics concerning such functors.

We begin by defining the canonical comparison map and for this purposewe consider colimB X ∈ C together with its colimiting cocone η which hascomponents ηb : Xb → colimB X, b ∈ B. Precomposition along u yields thefunctor X u : A → C. Applying the same precomposition to the colimitingcocone η we obtain a cocone α on X u given by αa = ηua : (X u)(a) →

DRAFT

6.2. FINAL FUNCTORS 95

colimB X, a ∈ A. In fact, for every morphism f : a→ a′ in A the triangle

X(ua)

X(uf)

ηua // colimB X

X(ua′)

ηua′

99

commutes since η is a cocone. But also the diagram X u : A → C has acolimit colimA(X u) and α hence factors uniquely through the correspondingcolimiting cocone. Thus, there is a canonical morphism

colimA(X u)→ colimB X, (6.2.1)

which is compatible with the cocones, i.e., such that the diagram

X(ua)

αa=ηua // colimB X

colimA(X u)

∃!

77

commutes for all a ∈ A. (The unlabelled morphisms belong to the colimitingcocone of X u.)

Dually, if C is a complete category then for every X : B → C there is acanonical morphism

limBX → limA(X u) (6.2.2)

which is induced from the respective limiting cones.

Definition 6.2.3. Let u : A→ B be a functor between small categories.

(i) The functor u is final if the canonical map (6.2.1) is an isomorphism forevery cocomplete category C and every diagram X : B → C.

(ii) The functor u is cofinal if the canonical map (6.2.2) is an isomorphismfor every complete category C and every diagram X : B → C.

Since these notions are dual to each other we mostly focus on final functors.

Proposition 6.2.4. Right adjoint functors between small categories are final.

Proof. Let C be a cocomplete category and let (u, v) : A B be an adjunctionbetween small categories. It is easy to check that the associated precompositionfunctors define an adjunction

(v∗, u∗) : CA CB .

Recall that adjunctions can be composed. In our situation, if we combine theabove adjunction with the adjunction (colimB ,∆B) : CB C, then we see thatcolimB v∗ is left adjoint to u∗ ∆B . But since the functor u∗ ∆B is equal to∆A, the uniqueness of left adjoints implies that there is a canonical isomorphismcolimA

∼= colimB v∗. With a bit more care one checks that this isomorphismagrees with the desired canonical map (6.2.1).

DRAFT


Corollary 6.2.5. Let A be a small category admitting a final object ∗. Ev-ery category C has colimits of shape A and for X : A → C there is a naturalisomorphism

X(∗) ∼= colimAX.

Proof. This is immediate from Proposition 6.2.4 since (πA, ∗) : A 1 is anadjunction.

There is a combinatorial criterion which allows us to characterize final func-tors. To state the result we need the following definition. For later reference,we also make the dual definition.

Definition 6.2.6. Let u : A→ B be a functor between small categories and letb ∈ B be an object.

(i) The slice category (b/u) has as objects pairs (a ∈ A, f : b → u(a)) andas morphisms (a, f)→ (a′, f ′) morphisms a→ a′ in A such that

b

f

f ′

##

u(a) // u(a′)

commutes.

(ii) The slice category (u/b) has as objects pairs (a ∈ A, f : u(a) → b) andas morphisms (a, f)→ (a′, f ′) morphisms a→ a′ in A such that

u(a)

f

// u(a′)

f ′

b

commutes.

Proposition 6.2.7. Let u : A→ B be a functor between small categories.

(i) The functor u is final if and only if the categories (b/u), b ∈ B, are non-empty and connected.

(ii) The functor u is cofinal if and only if the categories (u/b), b ∈ B, arenon-empty and connected.

To be completely specific, u is final if and only if for each b ∈ B the followingtwo properties are satisfied.

(i) There is an object in (b/u), i.e., an object a ∈ A and a morphism b→ u(a)in B.

(ii) Any two objects in (b/u) can be connected by a finite zigzag of morphisms.

DRAFT

6.3. POINTWISE KAN EXTENSIONS 97

For a proof of Proposition 6.2.7 we refer the reader, for example, to [KS06, §2.5].Here we content ourselves by showing how this proposition can be used to re-establish Corollary 6.2.5 and Proposition 6.2.4 and by relating the propositionto a more classical statement about colimits of diagrams defined on partiallyordered sets (aka. posets).

Examples 6.2.8.

(i) Let B be a poset, let A ⊆ B be a subposet, and let i : A → B be thecorresponding inclusion functor. The inclusion is final if and only if forevery b ∈ B the category (b/i) is non-empty and connected. Thus, weask for the existence of a ∈ A such that b ≤ a. Moreover, any a1, a2 ∈ Asuch that b ≤ a1 and b ≤ a2 have to be connected by a finite zigzag ofmorphisms in (b/i). A typical sufficient condition is that A is directed,i.e., that any two elements in A have a common successor. Thus, a directedsubposet which is unbounded in the larger poset yields a final functor.

(ii) Let B be a small category, let ∗ ∈ B be a final object, and let us considerthe functor ∗ : 1→ B classifying the final object. For every b ∈ B the slicecategory (b/∗) is isomorphic to 1 and is hence non-empty and connected.This reproduces Corollary 6.2.5.

(iii) Let A,B be small categories and let v : B → A be a right adjoint functor.We want to show that for every a ∈ A the slice category (a/v) is non-empty and connected. The unit of the adjunction (u, v) : A B yieldsa morphism ηa : a → vu(a), i.e., an object (u(a), ηa) ∈ (a/v), therebyshowing that (a/v) is non-empty. The explicit description of the adjunc-tion isomorphism in terms of the unit (see (A.1.2)) actually shows that(u(a), ηa) is an initial object in (a/v), so that the slice category is, inparticular, connected. This recovers Proposition 6.2.4.

6.3 Pointwise Kan extensions

In this section we show that in cocomplete categories left Kan extensions alongfunctors between small categories always exist. Moreover, there is an explicitconstruction of such left Kan extensions using colimits over slice categories.Given the importance of these formulas to the theory of derivators, we includecomplete and fairly detailed proofs. And again, duality allows us to mostlyfocus on left Kan extensions.

For the remainder of this section we assume that C is a cocomplete categoryand that u : A → B is a functor between small categories. The defining uni-versal property of left Kan extensions (Definition 6.1.4) suggests the followingconstruction of them.

Let us consider a diagram X : A → C and let (Y, α) be a datum amongwhich the left Kan extension is supposed to be initial, i.e., we have a functorY : B → C and a natural transformation α : X → Y u. For every morphism

DRAFT


g : a→ a′ in A we obtain the naturality square

X(a)X(g)

//

αa

X(a′)

αa′

Y u(a)Y u(g)

// Y u(a′).

Now, for every object b ∈ B, the value Y (b) has to be compatible with thesenaturality squares in the following sense. Let us assume we can find morphismsf : u(a) → b and f ′ : u(a′) → b in B which are compatible with g in the sensethat f = f ′ u(g). For each such datum the above naturality square extends tothe following commutative diagram

X(a)X(g)

//

αa

X(a′)

αa′

Y u(a)Y u(g)

//

Y (f)

Y u(a′)

Y (f ′)zz

Y (b)

(6.3.1)

in C. If we ignore the line in the middle, then we see that Y (b) is part of acocone on a certain diagram. And the universal property of a left Kan extensionsuggests that we should form universal such cocones by passing to colimits. Itturns out that this indeed works and we now carry out the details.

Note first that in the above heuristics (6.3.1) the slice category (u/b) fromDefinition 6.2.6 came up.

Lemma 6.3.2. Let u : A→ B be a functor between small categories.

(i) For every b ∈ B there is a projection functor p = pb : (u/b) → A definedon objects by (a, f : u(a)→ b) 7→ a and on morphisms by g 7→ g.

(ii) For every morphism h : b → b′ in B there is a functor (u/h) : (u/b) →(u/b′) defined on objects by (a, f : u(a) → b) 7→ (a, h f : u(a) → b → b′)and on morphisms by g 7→ g.

(iii) For every morphism h : b→ b′ in B the diagram

(u/b)(u/h)

//

pb

(u/b′)

pb′zz

A

commutes.

Proof. This is obvious and is left to the reader.

DRAFT


Thus, for every object b ∈ B we can consider the diagram

X pb : (u/b)→ A→ C,

and let us note that in the above heuristics (6.3.1) we observed that Y (b) is partof a cocone on precisely this diagram. This suggests that we should make thedefinition

L(X)(b) = colim(u/b)X pb, b ∈ B. (6.3.3)

Moreover, as a special instance of the canonical morphisms (6.2.1), for everymorphism h : b→ b′ in B we use the relation pb′ (u/h) = pb from Lemma 6.3.2to obtain a canonical morphism

L(X)(h) : colim(u/b)X pb → colim(u/b′)X pb′ . (6.3.4)

The uniqueness of these canonical morphisms show that they are functorial,thereby concluding the definition of the functor L(X) : B → C.

Lemma 6.3.5. Let C be a cocomplete category, let u : A → B be a functor be-tween small categories, and let X : A→ C. The assignments (6.3.3) and (6.3.4)define a functor L(X) : B → C.

Proof. This is immediate from the above discussion.

In order to conclude the construction of a left Kan extension, we need anatural transformation η : X → L(X) u. For a ∈ A the pair (a, id : u(a) →u(a)) defines an object in (u/u(a)) and we let ηa be the corresponding canonicalmap belonging to the colimiting cocone of L(X)(u(a)) = colim(u/u(a))X pu(a),

ηa = ι(a,idu(a)) : Xa → colim(u/u(a))X pu(a). (6.3.6)

Given a morphism g : a→ a′ in A, let us consider the following diagram

Xa

ι(a,idu(a))

//

X(g)

ι′(a,u(g))

**

colim(u/u(a))X pu(a)

Xa′ι′(a′,id

u(a′))

// colim(u/u(a′))X pu(a′),

in which the respective colimiting cocones are denoted by ι and ι′. The uppertriangle commutes by definition of (6.3.4) as a particular instance of the canon-ical morphism (6.2.1), while the lower triangle commutes since ι′ is a cocone.Thus, the morphisms (6.3.6) define a natural transformation η : X → L(X) u.

Proposition 6.3.7. Let C be a cocomplete category, let u : A→ B be a functorbetween small categories, and let X : A→ C. The pair (L(X), η) consisting of thefunctor L(X) : B → C defined by (6.3.3),(6.3.4) and the natural transformationη : X → L(X) u defined by (6.3.6) is a left Kan extension of X along u.

DRAFT


Proof. By Definition 6.1.4 we have to show that for every Y : B → C the assign-ment

φ : homCB (L(X), Y )→ homCA(X,Y u) : β 7→ (βu) η

is a bijection. It follows from the explicit construction (6.3.3) and (6.3.6) thatthe component of φ(β) at an object a ∈ A is given by

φ(β)a = βua ι(a,id) : X(a)→ colimX pua → (Y u)(a).

We next construct the following map in the converse direction

ψ : homCA(X,Y u)→ homCB (L(X), Y ).

To this end, let α : X → Y u and let us fix an object b ∈ B. For every object(a, f : u(a) → b) ∈ (u/b) we form the map Y f αa : Xa → (Y u)(a) → Y (b).By (6.3.1) these maps define a cocone on X pb : (u/b)→ C and there is hencea unique map ψ(α)b : L(X)(b) = colimX pb → Y (b) such that

colimX pb∃! ψ(α)b

// Y (b)

X(a)

ι(a,f)

OO

αa// Y (ua)

Y f

OO

(6.3.8)

commutes. Here, the morphism ι(a,f) belongs to the colimiting cocone. We leaveit to the reader to check that for every morphism h : b→ b′ in B the diagram

colimX pb

L(X)(h)

ψ(α)b// Y (b)

h

colimX pb′ψ(α)b′

// Y (b′)

commutes, i.e., that ψ(α) : L(X)→ Y is a natural transformation.It remains to check that φ and ψ are inverse to each other. Let β : L(X)→ Y

and let α = φ(β) = (βu) η. For every b ∈ B and (a, f) ∈ (u/b) we considerthe diagram

colimX pbβb // Y (b)

X(a)

ι(a,f)

88

ηa// colimX pua

βua

//

L(X)(f)

OO

Y (ua),

Y f

OO

in which the square is a naturality square of β. Since ηa = ιa,idua (see (6.3.6))and ι(a,f) belong to the respective colimiting cocones, the definition of L(X)(f)(see (6.3.4)) as an instance of the canonical map (6.2.1) implies that also thetriangle commutes. But the uniqueness of the map ψ(α)b in the definition of ψ(see (6.3.8)) implies that βb = ψ(φ(β))b.

DRAFT


Conversely, let α : X → Y u be given and we want to show that (φψ)(α) = α,i.e., that (ψ(α)u) η = α. If we consider a ∈ A and the corresponding object(a, idu(a)) ∈ (u/a), then as a special case of (6.3.8) we see that the diagram

colimX puaψ(α)ua

// Y (ua)

X(a)

ι(a,idua)

OO

αa// Y (ua)

=

OO

commutes. Using once more the precise form of η (see (6.3.6)) we conclude thatφψ(α) = α, concluding the proof.

Definition 6.3.9. Let C be a category and let u : A→ B be a functor betweensmall categories. The assignments

u∗ : CB → CA : X 7→ X u, β 7→ βu

define a restriction functor or precomposition functor.

For later reference we summarize our findings concerning the existence andthe precise form of left Kan extensions in the following theorem.

Theorem 6.3.10. Let C be a cocomplete category and let u : A→ B be a functorbetween small categories.

(i) For every diagram X : A→ C a left Kan extension LKanu(X) exists andthis defines a functor LKanu : CA → CB which is left adjoint to u∗ : CB →CA,

(LKanu, u∗) : CA CB .

(ii) For every diagram X : A→ C and every object b ∈ B there is a canonicalisomorphism

colim(u/b)X pb ∼= LKanu(X)b.

Proof. Associated to u there is the restriction functor u∗ : CB → CA and a leftKan extension of X along u is precisely an initial object in the category (X/u∗).By Proposition 6.3.7 we know that such an initial object always exists, and fromthis it follows immediately that any choices of such left Kan extensions definea functor LKanu : CA → CB which is left adjoint to u∗ : CB → CA.

Moreover, any two initial objects are canonically isomorphic, implying thatany left Kan extension LKanu(X) is canonically isomorphic to the one con-structed in (6.3.3).

The corresponding result for right Kan extensions is dual. Given a functoru : A→ B between small categories and b ∈ B there is a functor

q = qb : (b/u)→ A (6.3.11)

defined on objects by (a, f : b→ u(a)) 7→ a and on morphisms by g 7→ g.

DRAFT


Theorem 6.3.12. Let C be a complete category and let u : A→ B be a functorbetween small categories.

(i) For every diagram X : A → C a right Kan extension RKanu(X) : B → Cexists and this defines a functor RKanu : CA → CB which is right adjointto u∗ : CB → CA,

(u∗,RKanu) : CB CA.

(ii) For every diagram X : A→ C and every object b ∈ B there is a canonicalisomorphism

RKanu(X)b ∼= lim(b/u)X qb.

Proof. This is the dual of Theorem 6.3.10.

Thus, in sufficiently (co)complete categories, Kan extensions always existand can be constructed using pointwise formulas, thereby sort of reducingthe more general notion of a Kan extension to the more well-known limits andcolimits.

Warning 6.3.13. There are examples of Kan extensions (necessarily in contextslacking certain limits or colimits) which cannot be calculated using pointwiseformulas. For a particular example we refer the reader to [Bor94a, §3]. SuchKan extensions will not play a role in this book.

Before we take up more interesting examples in §6.4, we revisit Exam-ple 6.1.5.

Example 6.3.14. Let C be a complete and cocomplete category, let A be a smallcategory, and let πA : A→ 1 be the unique functor.

(i) Under the isomorphism C1 ∼= C the adjunction (LKanπA , π∗A) : CA C1

gets identified with (colimA,∆A) : CA C.

(ii) Under the isomorphism C1 ∼= C the adjunction (π∗A,RKanπA) : C1 CAgets identified with (∆A, limA) : C CA.

This leads us to think of Kan extensions as ‘relative versions of limits andcolimits’, which are obtained by replacing πA : A→ 1 by more general functorsu : A→ B between small categories.

6.4 Basic properties and first examples

In this section we illustrate the notion of Kan extensions by collecting a fewexamples and also establish the useful property that Kan extensions along fullyfaithful functors are again fully faithful, thereby justifying the terminology Kanextensions.

Since Kan extensions will be used a lot in the remainder of this book, weintroduce the following short-hand notation.

DRAFT

6.4. BASIC PROPERTIES AND FIRST EXAMPLES 103

Notation 6.4.1. Let C be a complete and cocomplete category and let u : A→B be a functor between small categories. We also denote Kan extension functorsby

u! = LKanu : CA → CB and u∗ = RKanu : CA → CB .

In the special case that u = b : 1→ B classifies an object b ∈ B we thus write

b! = LKanb : C → CB and b∗ = RKanb : C → CB .

As first examples we already saw that Kan extensions along πA : A → 1,A ∈ Cat , reduce to limits and colimits (Example 6.3.14). The other extremecase is given by Kan extensions along functors 1 → B defined on the terminalcategory, and this leads to free and cofree diagrams.

Example 6.4.2. Let B be a small category, let b ∈ B, and let C be a completeand cocomplete category (actually, the existence of products and coproductssuffices). Here, we describe the Kan extension functors b!, b∗ : C → CB alongb : 1→ B.

(i) The left Kan extension functor b! : C → CB sends an object X ∈ C to thefree diagram b!(X) : B → C generated by X. Given an object b′ ∈ B,the slice category (b/b′) is simply the discrete category on homB(b, b′).The pointwise formula yields the first isomorphism in

b!(X)b′ ∼= colim(b/b′)X pb′ ∼=∐

homB(b,b′)

X.

The second isomorphism is a consequence of (b/b′) being a discrete cat-egory so that the colimit reduces to a coproduct. The defining universalproperty of the free diagram b!(X) is that there are natural isomorphisms

homCB (b!(X), Y ) ∼= homC(X,Y (b)), Y ∈ CB ,

i.e., b! is left adjoint to the evaluation functor b∗ : CB → C.

(ii) The right Kan extension functor b∗ : C → CB sends an object X ∈ C tothe cofree diagram b∗(X) : B → C generated by X. In this case thereare natural isomorphisms

b∗(X)b′ ∼= lim(b′/b)X qb′ ∼=∏

homB(b′,b)

X.

Remark 6.4.3.

(i) Note that this example shows that Kan extensions are, in general, notfully faithful. In fact, assuming that the respective Kan extensions exist,for a diagram X : A→ C the diagrams

LKanu(X) u,RKanu(X) u : A→ C

DRAFT


obtained by first extending and then restricting back are, in general, notnaturally isomorphic to X. For example, for free and cofree diagrams wehave isomorphisms

b!(X)b ∼=∐

homB(b,b)

X and b∗(X)b ∼=∏

homB(b,b)

X.

(ii) Let b : 1 → B classify an initial object. In this case the free diagramfunctor b! : C → CB essentially forms constant diagrams. In fact, thisfollows from the explicit description in Example 6.4.2.

We illustrate the concept of free diagrams by two more specific examples.

Examples 6.4.4. Let C be a complete and cocomplete category.

(i) Given a quiver Q and a vertex q0 ∈ Q we obtain the free diagram functor(q0)! : C → CQ. Specializing to Mod(R) for a ring R we obtain a functor(q0)! : Mod(R) → Mod(R)Q. The universal property of free diagramsyields natural isomorphisms

homMod(R)Q((q0)!(R), X) ∼= homR(R,X(q0)) ∼= Xq0 ,

i.e., (q0)!(R) corepresents the evaluation functor (q0)∗. It follows that(q0)!(R) is a projective object in the abelian category Mod(R)Q. More-over, (q0)!(R) is an indecomposable object since any non-trivial decom-position would, by the Yoneda lemma, be induced by a non-trivial idem-potent endomorphism of q0 ∈ Q which is impossible since Q is a freecategory.

To specialize this a bit, let us consider a field R = k so that (q0)!(k) isthe usual indecomposable projective representation P (q0) correspondingto the vertex q0 ∈ Q. In the case of the linearly oriented A3-quiver(1→ 2→ 3) this yields the indecomposable projective representations

P (1) = (kid→ k

id→ k), P (2) = (0→ kid→ k), and P (3) = (0→ 0→ k).

(ii) Let G be a discrete group, considered as a groupoid with one object,and let i : 1 → G classify the unique object. The free and the cofreediagram functors i!, i∗ : C → CG, respectively, send an object X ∈ C tothe G-objects

i!(X) ∼=∐g∈G

X and i∗(X) ∼=∏g∈G

X.

The G-actions simply permute summands and factors.

Specializing this to C = Mod(R) for a ring R and using implicitly theequivalence Mod(R)G ' Mod(RG), this recovers the induction and coin-duction functors Mod(R)→ Mod(RG), i.e., the respective adjoints of therestriction of scalar functor i∗ : Mod(RG)→ Mod(R).

DRAFT

6.4. BASIC PROPERTIES AND FIRST EXAMPLES 105

The second example allows for the following minor variant.

Example 6.4.5. Let C be a complete and cocomplete category, let G be a dis-crete group, and let H < G be a subgroup. The Kan extensions along thecorresponding functor i : H → G give rise to adjunctions

(i!, i∗) : CH CG and (i∗, i∗) : CG → CH .

The functors i! and i∗ are, respectively, referred to as induction and coin-duction, and they make perfectly well sense for not necessarily injective grouphomomorphisms. The reader is invited to use the pointwise formulas to obtaina more precise description of these functors and to specialize to the case ofmodules over a ring.

The following property of Kan extensions is important, and this result jus-tifies the terminology Kan extensions.

Proposition 6.4.6. Let C be a complete and cocomplete category and let u : A→B be a fully faithful functor between small categories.

(i) The functor u! : CA → CB is fully faithful and induces an equivalenceonto the full subcategory of CB spanned by all Y such that the counitε : (u! u∗)Y → Y is an isomorphism.

(ii) The functor u∗ : CA → CB is fully faithful and induces an equivalence ontothe full subcategory of CB spanned by all Y such that the unit η : Y →(u∗ u∗)Y is an isomorphism.

Proof. By duality it is enough to take care of the first statement, and it followsfrom Lemma A.1.4 that u! is fully faithful if and only if the unit η : 1→ u∗u! isa natural isomorphism. For X : A→ C and a ∈ A it follows from the pointwiseformula (Theorem 6.3.10) that u!(X)u(a)

∼= colim(u/u(a))Xpu(a). Note that thefully faithfulness of u implies that (a, idu(a) : u(a) → u(a)) is a terminal objectin (u/u(a)). Hence by Corollary 6.2.5 the functor (a, idu(a)) : 1 → (u/u(a)) isfinal, and, as a particular instance of (6.2.1), there is a canonical isomorphism

X(a) = X pu(a)(a, 1)→ colim(u/u(a))X pu(a).

Note that this canonical morphism is simply the map ι(a,1) belonging to thecolimiting cocone of colim(u/u(a))X pu(a). But this map agrees with the com-ponent ηa of the adjunction unit η : 1→ u∗u! by the above explicit constructionof left Kan extensions (see (6.3.6)). Since this is the case for all a ∈ A, weconclude that η indeed is a natural isomorphism, thereby establishing that u!

is fully faithful. The description of the essential image of u! is a special caseof Lemma A.1.4.

Thus, in the situation of the proposition for a diagram X : A → C the re-spective adjunction units and counits yield natural isomorphisms

η : X∼=→ u∗u!(X) and ε : u∗u∗(X)

∼=→ X.

In this precise sense Kan extensions along fully faithful functors u : A → B donot modify a given diagram X : A→ C on A.

DRAFT


6.5 Cokernels and kernels via Kan extensions

In this section we show that cokernel and kernel functors can be constructed assuitable combinations of Kan extensions (as indicated in (6.1.2)).

Given a diagram X : A→ C in a cocomplete category, let us recall that thecolimit of X is an initial cocone on X. Thus, it consists of the colimit objectcolimAX ∈ C together with the colimiting cocone ιa : Xa → colimAX, a ∈ A,i.e., a natural transformation ι : X → ∆A colimAX. As noted in Lemma 3.4.1the natural domain for ι is the cylinder A× [1], thereby yielding the functor

ι : A× [1]→ C.

Since the target of the transformation is a constant diagram, this induced func-tor factors over the cocone in the sense of the following definition. Of course,there is a dual statements for cones and, in particular, for limiting cones.

Definition 6.5.1. Let A be a small category.

(i) The cocone AB of A is the small category obtained by freely adjoining anew terminal object ∞ to A.

(ii) The cone AC on A is the small category obtained by freely adjoining anew initial object −∞ to A.

Let us be more specific about the construction of the cocone AB. Theobjects in AB are the objects in A and a unique new object ∞ ∈ AB. The setof morphisms is defined by

homAB(x, y) =

homA(x, y) , x, y ∈ A,∗ , y =∞,∅ , x =∞, y ∈ A.

And the composition law in AB is unique determined by the fact that there isa fully faithful inclusion functor A → AB : x 7→ x. The construction of AC isdual and there is also a fully faithful functor A→ AC.

Examples 6.5.2. Let C be a complete and cocomplete category and let A be asmall category.

(i) Since the inclusion functor i : A → AB is fully faithful, by Proposi-

tion 6.4.6 also the left Kan extension functor i! : CA → CAB

is fully faith-ful. One can check that the essential image of i! consists precisely ofthe colimiting cocones. In particular, for a diagram X : A → C, the dia-gram i!(X) : AB → C encodes both the object colimAX ∈ C and also thecolimiting cocone.

(ii) Dually, since the inclusion functor i : A → AC is fully faithful, the right

Kan extension functor i∗ : CA → CAC is fully faithful and the essentialimage consists precisely of the limiting cones. Starting with X : A → C,the diagram i∗(X) : AC → C encodes the object limAX ∈ C together withthe limiting cone.

DRAFT

6.5. COKERNELS AND KERNELS VIA KAN EXTENSIONS 107

Let us illustrate this abstract class of examples. To this end, we denote by = [1]× [1] the commutative square, i.e., the category

(0, 0) //

=

(1, 0)

(0, 1) // (1, 1).

Moreover, let ip : p→ and iy : y → be the full subcategories obtained byremoving the final object (1, 1) and the initial object (0, 0), respectively. Notethat the square is the cocone on p and the cone on y, hence Examples 6.5.2specializes to the following example.

Example 6.5.3. Let C be a finitely complete and finitely cocomplete category.The Kan extension functors

(ip)! : Cp → C and (iy)∗ : Cy → C

are fully faithful and the essential images consist precisely of the pushout squaresand the pullback squares, respectively.

To conclude this section, we briefly come back to the construction of kernelsand cokernels in terms of Kan extensions; see again the discussion of (6.1.2)and its dual. With Example 6.5.3 in mind it only remains to take care of thepassage from a morphism to the span in (6.1.2). For this purpose we considerthe functors

j : [1]→ p and k : [1]→y

classifying the horizontal morphisms (0, 0) → (1, 0) and (0, 1) → (1, 1), respec-tively.

Lemma 6.5.4. Let C be a category admitting a zero object and let X0 → X1 bea morphism in C, thereby determining a functor X : [1]→ C.

(i) The right Kan extension j∗(X) : p→ C is naturally isomorphic to the span

X0//

X1

0.

(ii) The left Kan extension k!(X) : y→ C is naturally isomorphic to the cospan

0

X0// X1.

DRAFT


Proof. By duality it is enough to take care of the first statement. The functorj : [1]→ p is fully faithful and hence so is j∗ : C[1] → Cp (Proposition 6.4.6). Inparticular, for X : [1]→ C, the counit ε : j∗j∗X → X is an isomorphism. Thus,it remains to show that the value of j∗(X) at the lower left corner (0, 1) ∈ p is azero object. By the pointwise formula for right Kan extensions (Theorem 6.3.12)we calculate

j∗(X)(0,1)∼= lim((0,1)/i0)X q(0,1)

= lim∅X q(0,1)

∼= 0.

In fact, it is obvious that the slice category ((0, 1)/i0) is empty and the limit ishence a terminal object. By assumption on C we thus obtain a zero object.

Thus, the intended passage from a morphism to the span in (6.1.2) amountsto forming the right Kan extension along j : [1] → p. Combining this withExamples 6.5.2 we obtain the following example.

Examples 6.5.5. Let C be a finitely complete and finitely cocomplete categorywith a zero object.

(i) The Kan extension functors C[1] j∗→ Cp (ip)!→ C send a map f : X0 → X1

to the following pushout square containing the cokernel,

X0f//

X1

0 // cok(f).

(ii) The Kan extension functors C[1] k!→ Cy (iy)∗→ C send a map f : X0 → X1

to the following pullback square containing the kernel,

ker(f) //

0

X0f// X1.

If one only cares about the context of ordinary categories, it is arguable howmuch is gained by the description of cokernels in terms of Kan extensions. Thepoint however is that for an abelian category A the corresponding derived Kanextensions yield a construction of the functors

cof : D(A[1])→ D(A[1]) and C : D(A[1])→ D(A)

at the level of derived categories only. And such a construction is useful if oneaims for a more careful study of these functors (see §§9-16).

DRAFT

6.5. COKERNELS AND KERNELS VIA KAN EXTENSIONS 109

Similar remarks are true for homotopy categories of model categories or ∞-categories and the corresponding homotopy Kan extensions. In fact, the abovedescription only relies on having a suitable framework for a ‘calculus of Kanextensions’, and precisely such a framework is axiomatized by the notion of aderivator. In §7 we define derivators and in §§8-18 we illustrate the notion bydiscussing some constructions available in such derivators (this includes gener-alizations of the diagrams (5.2.4), (5.2.7), and (5.4.1)).

DRAFT


DRAFT

Part II

The basic calculus inderivators

111

DRAFT

Chapter 7

Basics on derivators

Motivated by the discussion in §§1-6, in this chapter we introduce derivators —the main notion studied in this series of books. To put it as a slogan, derivatorscan be thought of as minimal, purely categorical extensions of derived categoriesof abelian categories to a framework with a powerful calculus of derived limitsand, more generally, derived Kan extensions. Similarly, the derivator of spacesis such an extension of the homotopy category of spaces to a framework witha calculus of homotopy limits and homotopy Kan extensions. It turns out thatmany constructions are captured by such a calculus of Kan extensions and wecollect some first examples in the remaining chapters of this book.

In §7.1 we introduce prederivators roughly as systems of diagram categoriescoming with restriction functors. In §7.2 we define derivators as suitable homo-topically complete and cocomplete prederivators. In §7.3 we collect key exam-ples of derivators coming from ordinary category theory, homological algebra,and homotopical algebra. With these examples in mind, in §7.4 we emphasizethe difference between categorical limits on the one hand side and derived limitsor homotopy limits on the other side. We illustrate this in §7.5 by some specificexamples in the context of topological surfaces. In §7.6 we collect some resultsrelated to the commutativity of limits.

7.1 Prederivators

Let A be an abelian category and let D(A) be the derived category. By The-orem 3.5.6 there is a functorial cone C : D(A[1]) → D(A) which, in general,does not factor through D(A)[1] (see the discussion in §5.5). More generally, asalready emphasized in §§4.1-4.2 and §5.5, it is important to distinguish betweenthe following two types of categories.

(i) Derived categories of diagram categories have as objects strict diagramsand these diagrams allow for many important constructions.

(ii) Diagram categories in derived categories are less well-behaved in that their

113

DRAFT

114 CHAPTER 7. BASICS ON DERIVATORS

objects do not carry enough information and many constructions can notbe performed at that level anymore.

These categories are related by functors diaA : D(AA) → D(A)A (Proposi-tion 4.1.7), which, in general, are far from being equivalences (see Warning 4.1.9and §4.2). Since the categories D(AA) can not be reconstructed from D(A),this suggests that the passage A 7→ D(A) should be refined by keeping track ofthe derived categories D(AA).

The formalization of this idea is as follows (see §A.3 for a review of basic2-categorical terminology). Let Cat denote the 2-category of small categories,functors, and natural transformations. Similarly, let CAT be the 2-category of(not necessarily small) categories, functors, and natural transformations. Onecan make a choice in the following definition and restrict attention to, for ex-ample, suitably finite shapes of diagrams only; see Remark 7.1.3.

Definition 7.1.1. A prederivator D is a strict 2-functor D : Catop → CAT .

The 2-category Catop is obtained from Cat by reversing the direction of thefunctors while the direction of the natural transformations is unchanged. Thus,given a prederivator D , for every small category A we obtain a category D(A).Associated to a functor u : A→ B in Cat , there is an induced functor

u∗ = D(u) : D(B)→ D(A).

Finally, given functors u, v : A→ B in Cat and a natural transformation α : u→v, we obtain an induced natural transformation α∗ : u∗ → v∗, as depicted in thediagram

A

u''

v

77 B, D(B)

u∗++

v∗33 D(A).

This datum is compatible with compositions and identities in a strict sense, i.e.,we have equalities of the respective expressions and not only coherent naturalisomorphisms between them. Given a prederivator D , we refer to D(1) asthe underlying category of D (recall that 1 denotes the terminal categoryconsisting of one object and its identity morphism only).

Examples 7.1.2.

(i) Every category C ∈ CAT gives rise to a represented prederivator y(C)defined by

y(C)(A) := CA, A ∈ Cat .

Given a functor u : A → B in Cat , we define y(C)(u) to be the precom-position functor u∗ : CB → CA : X 7→ X u (Definition 6.3.9). Finally,given two functors u, v : A → B and a natural transformation α : u → vthe reader easily checks that

α∗X = X α : X u→ X v, X ∈ CB ,

DRAFT

7.1. PREDERIVATORS 115

defines a natural transformation α∗ : u∗ → v∗, concluding the definitionof the 2-functor y(C). The underlying category of y(C) is canonicallyisomorphic to C itself,

y(C)(1) ∼= C.

(ii) Let A be an abelian category, let Ch(A) be the category of unboundedchain complexes, and for A ∈ Cat we denote by WA

A be the class oflevelwise quasi-isomorphisms (Definition 3.1.10). The homotopy pre-derivator DA : Catop → CAT is defined by the localizations

DA(A) := Ch(A)A[(WAA )−1], A ∈ Cat .

It is an immediate consequence of Lemma 3.1.11 that there are canonicalisomorphisms

DA(A) ∼= D(AA), A ∈ Cat ,

which we use from now on without further reference.

The restriction functor u∗ : Ch(A)B → Ch(A)A associated to a functoru : A → B in Cat clearly preserves levelwise quasi-isomorphisms. Hence,there is a canonical functor

u∗ : DA(B)→ DA(A)

which is compatible with the respective localization functors. And theuniqueness of these functors implies that the assignment u 7→ u∗ is func-torial.

Finally, every natural transformation α : u → v first induces a transfor-mation α∗ : u∗ → v∗ between the corresponding precomposition functorsu∗, v∗ : Ch(A)B → Ch(A)A (these transformations belong to the repre-sented prederivator y(Ch(A)) as constructed in (i)). Since the localiza-tion functors Ch(A)A → D(AA) are 2-localizations (Proposition 3.4.2),the reader easily concludes the definition of the prederivator

DA : Catop → CAT .

The underlying category of DA is canonically isomorphic to the derivedcategory D(A),

DA(1) ∼= D(A).

(iii) Let Top denote the category of topological spaces and continuous maps.We recall that a continuous map f : X → Y is a weak homotopy equiv-alence if the induced maps f∗ : πk(X,x0) → πk(Y, f(x0)), k ≥ 0, x0 ∈ Xare bijections. For every A ∈ Cat we denote by WA

Top the class of levelwise

weak homotopy equivalences in TopA. The prederivator of topologicalspaces is defined by the localizations

HoTop(A) = Ho(TopA) := TopA[(WATop)−1], A ∈ Cat .

DRAFT


Also the localization functors TopA → Ho(TopA) are 2-localizations (Re-mark 3.4.3) and, as in the previous case, this assignment extends to aprederivator

HoTop : Catop → CAT .

The underlying category of HoTop is canonically isomorphic to the ho-motopy category of spaces,

HoTop(1) ∼= Ho(Top).

(iv) We conclude by a broad class of examples induced by model categories;see the original [Qui67], the monographs [Hov99, Hir03] or the introduc-tory [DS95]. Let M be a model category with class of weak equiva-lences WM. For every small category A, the category MA comes withthe class WA

M of levelwise weak equivalences. Although WAM does not nec-

essarily belong to a model structure onMA, we can define the homotopyprederivator HoM of M by

HoM(A) = Ho(MA) := (MA)[(WAM)−1], A ∈ Cat .

(It can be shown that these categories are again locally small [Cis03].)Using Remark 3.4.3 the reader easily concludes the construction of theprederivator

HoM : Catop → CAT .

The underlying category of HoM is canonically isomorphic to the usualhomotopy category Ho(M),

HoM(1) ∼= Ho(M) =M[W−1].

These examples are the main motivation for the theory of derivators.

Remark 7.1.3.

(i) Our convention for (pre)derivators (which agrees with the one of Heller[Hel88] and Franke [Fra96]) is based on the idea that we want to modeldiagrams (covariant functors) in a fixed abstract homotopy theory. Thisresults in the use of Catop as domain of definition for prederivators.

There is an alternative but isomorphic approach (as used by Cisinski[Cis03, Cis08], Grothendieck [Gro], Maltsiniotis [Mal01, Mal07]) based onpresheaves (contravariant functors). In that case the 2-category Catop isreplaced by Catcoop, which is obtained from Cat by reversing the orienta-tions of both functors and natural transformations.

Although the resulting theories are equivalent, many statements differslightly since the directions of various natural transformations in eachconvention are reversed with respect to the other.

DRAFT

7.1. PREDERIVATORS 117

(ii) For the purpose of a slicker exposition, in this book we choose Cat asthe domain of definition for prederivators. However, often (for exampleunder certain finiteness conditions) it is convenient to have some flexibilitywith respect to the choice of the ‘admissible shapes’. There is the notionof a category of diagrams Dia, a full sub-2-category of Cat enjoyingcertain closure properties. And related to this there is the definition ofa prederivator of type Dia as a 2-functor Diaop → CAT . In thisintroductory book we will not get into this.

We establish some basic terminology related to prederivators D . Motivatedby Examples 7.1.2, we refer to an object X ∈ D(A) as a coherent, A-shapeddiagram in D (see again the discussion in §4.1). We also write X ∈ D if thereis a small category A such that X ∈ D(A).

Warning 7.1.4. We want to warn the reader that this is just terminology; foran abstract prederivator D , an object X ∈ D(A) is not a diagram in whateversense; see also Warning 7.1.7. However, every such object induces an ordinarydiagram of shape A as we discuss next.

Given a prederivator D and a functor u : A→ B in Cat , the induced functoru∗ : D(B) → D(A) is a precomposition functor or restriction functor.We keep using implicitly the canonical isomorphisms A ∼= A1 and hence writea : 1 → A for the functor classifying a ∈ A. Hence, as a special case of aprecomposition functor we obtain an evaluation functor a∗ : D(A) → D(1).Given a morphism g : X → Y in D(A) we will write ga : Xa → Ya for theinduced map in the underlying category D(1).

Similarly, every morphism f : a→ b in A yields a natural transformation ofthe corresponding functors 1→ A, and for every prederivator we hence obtain

1

a''

b

77 A, D(A)

a∗**

b∗44 D(1) .

Evaluated at X ∈ D(A) this defines a map Xf : Xa → Xb in D(1). We summa-rize the functoriality of this construction in the following lemma.

Lemma 7.1.5. Let D be a prederivator and let g : X → Y be a morphism inD(A).

(i) The assignment diaA(X) : A → D(1) : a 7→ Xa, f 7→ Xf , defines a func-tor diaA(X) : A → D(1), the underlying (incoherent) diagram ofX ∈ D(A).

(ii) There is a natural transformation diaA(g) : diaA(X) → diaA(Y ) : A →D(1) with components diaA(g)a = ga : Xa → Ya, a ∈ A.

(iii) The assignments X 7→ diaA(X) and g 7→ diaA(g) define a functor, theunderlying diagram functor

diaA : D(A)→ D(1)A.

DRAFT


Proof. The proof is left as an exercise. We suggest the reader to really do thisexercise in order to get used to the concept of 2-functoriality.

Examples 7.1.6. We take up again the prederivators from Examples 7.1.2 anddescribe the respective underlying diagram functors.

(i) For represented prederivators y(C), C ∈ CAT , the underlying diagramfunctors diaA : CA → (C1)A reduce to the canonical isomorphisms CA ∼=(C1)A.

(ii) In the case of the homotopy prederivator DA of an abelian category A,the functor diaA : DA(A)→ DA(1)A corresponds under DA(A) ∼= D(AA)to the functor diaA : D(AA) → D(A)A constructed in Proposition 4.1.7.As discussed in §4.2, in general, this functor is not an equivalence ofcategories.

(iii) The underlying diagram functor diaA : HoTop(A) → HoTop(1)A in the

prederivator of spaces is the usual functor Ho(TopA) → Ho(Top)A. Thecategory Ho(Top) can be described as the category of CW complexes andhomotopy classes of maps, and in this description diaA sends a strictdiagram to a homotopy commutative diagram.

(iv) Similarly, if (M,WM) comes from a model category, then the underly-ing diagram functor diaA : HoM(A) → HoM(1)A is the usual functorHo(MA) → Ho(M)A. If we use the description of Ho(M) as the cate-gory of cofibrant and fibrant objects and homotopy classes of morphisms,then this example makes precise that diaA sends a strict diagram to ahomotopy commutative diagram.

Warning 7.1.7. Let D be a prederivator and let A ∈ Cat . Given a coherentdiagram X ∈ D(A) we will often draw it as an ordinary, A-shaped diagram,which is to say that we draw the underlying diagram diaA(X) : A → D(1).However, it is important to be aware of the fact that, in general, X is notdetermined by diaA(X), even up to isomorphism.

More specifically, as we saw in Examples 7.1.6, the functor diaA : D(A) →D(1)A is usually not an equivalence, and one has hence to be careful and distin-guish between coherent diagrams and incoherent ones. The main point of thetheory of derivators is that many interesting constructions which are availableat the level of coherent diagrams can not be performed anymore at the levelof underlying incoherent diagrams. To put it more frankly, this suggests thatunderlying diagram functors should not be applied.

Prederivators allow us to describe systems of diagram categories and as-sociated restriction functors and similarly for derived categories or homotopycategories of such diagram categories. However, to be able to perform interest-ing constructions we impose additional axioms, guaranteeing the existence of awell-behaved calculus of limits, colimits, and Kan extensions.

DRAFT

7.2. DERIVATORS 119

7.2 Derivators

The basic idea behind derivators is to encode collections of derived categoriesor homotopy categories together with a well-behaved calculus of homotopy Kanextensions. For (pre)derivators we follow the established terminology for ∞-categories and simply speak of Kan extensions as opposed to homotopy Kanextensions (and similarly for related notions). This does not result in a risk ofambiguity since in the context of an abstract (pre)derivator the concept ‘Kanextension’ is meaningless.

Definition 7.2.1. Let D be a prederivator and let u : A→ B be in Cat .

(i) The prederivator D admits left Kan extensions along u if the restric-tion functor u∗ : D(B)→ D(A) has a left adjoint u! : D(A)→ D(B),

(u!, u∗) : D(A) D(B).

(ii) The prederivator D admits right Kan extensions along u if the restric-tion functor u∗ : D(B)→ D(A) has a right adjoint u∗ : D(A)→ D(B),

(u∗, u∗) : D(B) D(A).

Motivated by Example 6.3.14, in the case of left Kan extensions along func-tors πA : A→ 1 to the terminal category we also speak of colimits of shape Aand write

(πA)! = colimA : D(A)→ D(1).

Similary, right Kan extensions along πA : A→ 1 will be referred to as limits ofshape A and will occasionally be denoted by

(πA)∗ = limA : D(A)→ D(1).

Sometimes we want to emphasize notationally with respect to which prederiv-ator certain functors are considered. In that case, we will also write

u∗D , uD! , colimD

A , uD∗ , and limD

A .

Example 7.2.2. Let C be an ordinary category, let y(C) be the represented pre-derivator (Examples 7.1.2), and let u : A → B be in Cat . If C is complete andcocomplete, then by Theorem 6.3.10 and Theorem 6.3.12 there are adjunctions

(u!, u∗) : CA CB and (u∗, u∗) : CB CA,

which is to say that y(C) admits Kan extensions. Conversely, if y(C) admitsKan extensions then the representing category is by Example 6.3.14 necessarilycomplete and cocomplete.

DRAFT


As recalled in §6.3, a key formal property of Kan extensions in classicalcategory theory is that in complete and cocomplete categories they can be cal-culated pointwise (see Theorem 6.3.10 and Theorem 6.3.12). Given a completeand cocomplete category C and a functor u : A → B in Cat , then for everyX : A→ C the Kan extensions u!(X), u∗(X) : B → C exist. Moreover, for everyb ∈ B certain canonical natural transformations

colim(u/b)X p→ u!(X)b and u∗(X)b → lim(b/u)X q

are isomorphisms in C. Here, (u/b) and (b/u) are slice categories and p : (u/b)→A and q : (b/u) → A the corresponding canonical projections functors (seeLemma 6.3.2 and (6.3.11)).

Warning 7.2.3. In the represented case one can use these pointwise (co)limitexpressions to construct a diagram of shape B which will be a Kan extension.This is impossible for an abstract prederivator D since objects in D(B) arejust abstract objects, in particular, not ordinary diagrams (Warning 7.1.4 andWarning 7.1.7).

For a derivator we nevertheless ask that Kan extensions exist and can becalculated pointwise. To make this precise, we consider a functor u : A → B,and object b ∈ B, and the slice squares

(u/b)p

//

π(u/b)

A

u

1b

// B

and

(b/u)q

//

π(b/u)

A

u

1b

// B.

AI(7.2.4)

Let us focus on the square on the left, and consider an object (a, f : u(a)→ b) in(u/b), and trace it through the diagram. Passing through the upper right cornerwe obtain u(a) while the other path yields b. Thus, in general the square doesnot commute but we claim that a canonical (not necessarily invertible) naturaltransformation lives in that square, and dually.

Lemma 7.2.5. Let u : A→ B be in Cat and let b ∈ B.

(i) The assignment (a, f : u(a) → b) 7→ f defines a natural transformationu p→ b π.

(ii) The assignment (a, f : b → u(a)) 7→ f defines a natural transformationb π → u q.


Now, let us assume that D : Catop → CAT is a prederivator admitting thenecessary Kan extensions. Associated to the slice squares (7.2.4) we obtain thefollowing canonical mate-transformations

colim(u/b) p∗ = π!p

∗ η→ π!p∗u∗u! → π!π

∗b∗u!ε→ b∗u!, (7.2.6)

b∗u∗η→ π∗π

∗b∗u∗ → π∗q∗u∗u∗

ε→ π∗q∗ = lim(b/u)q

∗, (7.2.7)

DRAFT

7.2. DERIVATORS 121

where the transformations denoted η and ε are suitable adjunction units andcounits, respectively. The transformation (7.2.6) is hence defined as the follow-ing pasting

D(1)

ε

D(u/b)π!oo

D(A)p∗oo

η

D(1)

π∗

OO

id

YY

D(B)b∗

oo

u∗

OO

D(A).u!

oo

idjj

Similarly, the other canonical mate (7.2.7) is defined by the following pasting

D(1) D(b/u)π∗oo D(A)

q∗oo

D(1)

π∗

OO

id

YY@Hη

D(B)b∗

oo

u∗

OO@H

D(A).u∗oo

idjj

@Hε

(We will say a bit more about the passage to canonical mates in §8.1.) For everyobject X ∈ D(A) these canonical mates hence are morphisms

colim(u/b) p∗(X)→ u!(X)b and u∗(X)b → lim(b/u)q

∗(X)

in the underlying category D(1). It turns out that the good way to expressthat Kan extensions in D are pointwise is by asking these canonical mates tobe isomorphisms.

Definition 7.2.8. A prederivator D is a derivator if it has the following prop-erties.

(Der1) D : Catop → CAT takes coproducts to products, i.e., the canonicalfunctor D(

∐i∈I Ai) →

∏i∈I D(Ai) is an equivalence. In particular,

D(∅) is equivalent to the terminal category.

(Der2) For any A ∈ Cat , a morphism f : X → Y in D(A) is an isomorphismif and only if each fa : Xa → Ya is an isomorphism in D(1).

(Der3) The prederivator D admits Kan extensions, i.e., for every u : A → Bthere are adjunctions

(u!, u∗) : D(A) D(B) and (u∗, u∗) : D(B) D(A).

(Der4) For any functor u : A→ B and any object b ∈ B, the canonical mate-transformations (7.2.6) and (7.2.7) associated to (7.2.4)

colim(u/b) p∗ ∼−→ b∗u! and b∗u∗

∼−→ lim(b/u)q∗

are isomorphisms.

A few comments about these axioms are in order.

DRAFT


Remark 7.2.9.

(i) Axiom (Der1) simply says that a coherent diagram on a disjoint unionis canonically determined by its restrictions to the summands. Axiom(Der2) makes precise that a transformation of two coherent diagrams ofthe same shape is an isomorphism if and only if all components are. Thisaxiom is motivated by the following two facts.

(a) A natural transformation is an isomorphism if and only if all com-ponents are isomorphisms.

(b) For every abelian category A and every A ∈ Cat the isomorphismCh(AA) ∼= Ch(A)A (3.1.9) identifies the quasi-isomorphisms WAAwith the levelwise quasi-isomorphisms WA

A (Lemma 3.1.11).

(ii) Axiom (Der3) encodes the ‘homotopical completeness and cocompletenessproperty’, thereby guaranteeing that ‘homotopical versions’ of limits, col-imits, and Kan extensions exist. By definition, these functors are simplyadjoints to restriction functors and they hence enjoy ordinary categoricaluniversal properties, making them accessible to elementary categoricaltechniques. Finally, axiom (Der4) makes precise that we can calculatethese Kan extensions by means of pointwise formulas.

It turns out that these axioms (Der1)-(Der4) suffice to develop a good dealof the calculus of categorical Kan extensions, derived Kan extensions, andhomotopy Kan extensions in a uniform and completely formal way. Somefirst steps of this calculus are developed in the remainder of this book; see§§8-18.

(iii) The formalism behind axiom (Der4) — based on the calculus of canonicalmates as in (7.2.6) and (7.2.7) — is arguably the key tool in the theory ofderivators. We will study this formalism in more detail in §8. Although abit heavy on a first view, this formalism turns out to be a very convenienttool allowing for rather mechanical proofs. Often it allows us to show thatcertain ‘obvious maps which clearly are isomorphisms’ first of all exist andsecond turn out to be isomorphisms.

(iv) Note that axioms (Der1)-(Der4) are asking for properties while the onlyactual structure is the underlying prederivator. This is in contrast to moreclassical approaches including triangulated categories in which case somenon-canonical structure is part of the notion. We will get back to thisin §§15-17.

(v) Stepping back a little, the axioms of a derivator might seem a bit naive inthat they simply capture ‘rather obvious’ formal properties of the calculusof Kan extensions from ordinary category theory. So, one might wonder ifthe axioms have anything to say about homological algebra and homotopytheory. It turns out that this is indeed the case and here we already wantto mention the following two aspects of this claim.

DRAFT

7.2. DERIVATORS 123

First, one can show that typical situations in homological algebra and ho-motopy theory give rise to derivators. In particular, there are derivatorsassociated to nice abelian categories, to the category of topological spaces,and to the category of spectra. More generally, model categories and com-plete and cocomplete ∞-categories have underlying homotopy derivators(see §7.3, §15.2, and Appendix B). Thus, the axioms of a derivator aresatisfied in all these examples.

Second, and these are deeper results, the axioms of a derivator singleout the typical contexts of homotopy theory. To illustrate this slogan wemention the following two results.

(a) The derivator HoTop of topological spaces is the free derivator gen-erated by the singleton ∗ ∈ Ho(Top) = HoTop(1). In more detail, forevery derivator D and any object X ∈ D(1) there is an essentiallyunique colimit-preserving morphism of derivators F : HoTop → Dsuch that F (∗) ∼= X. There are variants of these results for thederivator of pointed topological spaces and the derivator of spectra.

(b) By definition, a derivator D is homotopically complete and cocom-plete in that for every A ∈ Cat there are limit and colimit functors,

limA : D(A)→ D(1) and colimA : D(A)→ D(1).

Despites these functors being defined as mere adjoints to restrictionfunctors π∗A : D(1) → D(A), it turns out that they admit a moreexplicit description. In fact, it follows from the axioms that they canbe calculated by means of the classical Bousfield–Kan formulas forhomotopy (co)limits in ordinary homotopy theory (see [BK72] forthe classical situation). More generally, there are similar formulasshowing that ends and coends in derivators can be calculated bythe usual two-sided bar constructions (see [May72, May75] for theclassical context and the more general [Mey84, Mey86]).

In this introductory book we do not include a detailed discussion of thesefacs. While in §18 we give a short outlook on these and related topics, weintend to only come back to these facts in [Gro16b].

(vi) Although already mentioned in Remark 7.1.3 we want to reemphasize thatthere is some flexibility with respect to the choice of allowable shapes inthe definition of a derivator. By the very definition a derivator encodesthe idea of having a complete and cocomplete homotopy theory (enjoyingsome nice properties). Under certain finiteness assumptions (like in K-theoretic contexts) one wants to restrict to suitably finite shapes only andthis can be done by means of categories of diagrams. Related to this,there are also variants of derivators satisfying only half of (Der3) and halfof (Der4), leading to left derivators and right derivators. We will not getinto these variants in this book.

DRAFT


In §7.3 we establish some examples of derivators. The goal in the remainderof the book then is to show that the structure of a derivator D enhances theunderlying category D(1) to a more flexible notion. Before we turn to examples,related to Warning 7.1.7 we include the following warning.

Warning 7.2.10. Note that we defined a derivator to be a prederivator whichis homotopically cocomplete and homotopically complete in that all restrictionfunctors u∗ : D(B) → D(A) have left adjoints and right adjoints. It is im-portant to be aware of the fact that, in general, this does not imply that thecategories D(A) are complete or cocomplete in the categorical sense. Thus, inthe context of an abstract derivator, in general, ordinary categorical (co)limitsand categorical Kan extensions do not exist.

We will expand on this warning in §7.4.

7.3 Examples of derivators

In this section we collect a few classes of examples of derivators, arising incategory theory, homological algebra, and homotopical algebra. This shows thatderivators provide an axiomatic framework to study simultaneously ordinaryKan extensions, derived Kan extensions, and homotopy Kan extensions. Specialcases of such derivators are derivators associated to rings and schemes, thederivator of spaces, the derivator of pointed spaces, and the derivator of spectra.

As mentioned in §1, detailed proofs that we actually obtain these derivatorsare more involved and we suggest the reader to treat these results as black boxeson a first reading. For convenience, the construction of homotopy derivators ofcombinatorial model categories is treated in detail in §B.3.

We begin with the examples originating from ordinary category theory.

Theorem 7.3.1. If C ∈ CAT is complete and cocomplete, then the representedprederivator y(C) is a derivator.

Proof. We have to verify that y(C) : Catop → CAT satisfies axioms (Der1)-(Der4). For every small family Ai ∈ Cat , i ∈ I, the inclusions ιj : Aj →∐iAi, j ∈ I, induce an isomorphism of categories

(ιi)∗ : C

∐i Ai ∼−→

∏i

CAi .

In particular, there is an isomorphism C∅ ∼= 1, and this settles (Der1). Axiom(Der2) reduces to the statement that a natural transformation α : X → Y inCA, A ∈ Cat , is a natural isomorphism if and only if all components αa : Xa →Ya, a ∈ A, are isomorphisms. As mentioned in Example 7.2.2, (Der3) is takencare of by Theorem 6.3.10 and Theorem 6.3.12. It remains to verify that (Der4)precisely amounts to saying that Kan extensions in C are constructed point-wise. This follows from unraveling the definition of the canonical mates (7.2.6)and (7.2.7) and is suggested as an exercise. A detailed proof is neverthelessincluded in §B.1.

DRAFT

7.3. EXAMPLES OF DERIVATORS 125

By Example 7.2.2 we know that also the converse is true. The class of rep-resented derivators is interesting when it comes to generalizing notions fromordinary category theory to derivator theory; in fact, the notions should be de-fined for derivators such that in the represented case they reduce to the classicalones.

We next discuss examples of derivators arising in homological algebra andhomotopical algebra. It turns out that various different approaches to homo-topical algebra forget to derivators, more precisely, that the typical passagesto homotopy categories factor through derivators. Here we discuss this forGrothendieck abelian categories, model categories, and ∞-categories.

We begin with the context of homological algebra.

Definition 7.3.2. Let A be an abelian category.

(i) A generator is an object G ∈ A such that f : X → Y in A is an isomor-phism if and only if f∗ : homA(G,X)→ homA(G, Y ) is a bijection.

(ii) The category A is Grothendieck abelian if it is cocomplete, if filteredcolimits in A are exact, and if it has a generator.

It can be shown that Grothendieck abelian categories are also complete; seefor example [KS06, §8.3]. To illustrate this notion we include the followingexamples.

Examples 7.3.3.

(i) The category Mod(R), R a ring, is Grothendieck abelian.

(ii) The category Qcoh(X) of quasi-coherent OX -modules on a scheme X isGrothendieck abelian [EE05].

(iii) If A is Grothendieck abelian and A ∈ Cat , then the diagram category AAis again Grothendieck abelian.

(iv) For every Grothendieck abelian category A the category Ch(A) of chaincomplexes in A is again a Grothendieck abelian category.

One can establish the following non-trivial result concerning the homotopyprederivator of a Grothendieck abelian category (see Examples 7.1.2).

Theorem 7.3.4. Let A be a Grothendieck abelian category and let WA be theclass of quasi-isomorphisms in Ch(A). The prederivator

DA : Catop → CAT : A 7→ Ch(A)A[(WAA )−1]

is a derivator, the homotopy derivator of the Grothendieck abelian cate-gory A.

We illustrate this theorem by the following examples.

Examples 7.3.5.

DRAFT


(i) The category Mod(R), R a ring, is Grothendieck abelian (Examples 7.3.3).Thus, denoting by WR the quasi-isomorphisms in Ch(R), we obtain aderivator

DR : Catop → CAT : A 7→ Ch(R)A[(WAR )−1],

the derivator of the ring R. In particular, we have the derivator DZ ofthe integers and the derivator Dk of a field k.

These derivators enhance the more classical derived categories and, evenin the case of a field, a lot of interesting information is encoded by thederivator.

(a) Given a discrete group G there is an isomorphism DR(G) ∼= D(RG),i.e., DR keeps track of derived categories of all group algebras RG.

(b) Given a quiver Q with finitely many vertices only, there is an isomor-phism DR(Q) ∼= D(RQ), which is to say that the derived categoriesof path algebras RQ of quivers Q are encoded by DR.

(c) Given a finite poset P , there is an isomorphism DR(P ) ∼= D(RP ),i.e., DR also encodes derived categories of incidence algebras RP offinite posets P .

(d) More generally, for every A ∈ Cat with finitely many objects only,there are isomorphisms DR(A) ∼= D(RA), i.e., DR knows about thederived categories of the category algebras.

We want to emphasize that, by the very definition of the derivator DR, allthe values DR(A), A ∈ Cat , are considered as plain categories. Neverthe-less, the derived Kan extension functors can be used to construct the clas-sical triangulations on all the derived categories DR(A) ∼= D(Mod(R)A)(see §15). Similarly, as we discuss in [Gro16a], also higher triangulationscan be constructed that way.

Besides these triangulations, the derived Kan extension functors encode agood deal of additional information. For example, as special cases of de-rived limit and colimit functors, DR encodes group cohomology and grouphomology for arbitrary discrete groups as well as their versions for chaincomplexes (see §2.3). To mention different examples, certain classical de-rived equivalences from representation theory (like reflection functors fortrees, acyclic quivers, and even more general categories) can also be con-structed from the derivator DR only (see [GS14b, GS15b, GS14a, GS15a]).

(ii) The category Qcoh(X) of quasi-coherent OX -modules on a scheme X isGrothendieck abelian (Examples 7.3.3). Thus, if WX denotes the class ofquasi-isomorphisms in Ch(X) = Ch(Qcoh(X)), then we obtain a derivator

DX : Catop → CAT : A 7→ Ch(X)A[(WAX )−1],

the derivator of the scheme X. The underlying category DX(1) isisomorphic to the more classical derived category D(X).

DRAFT


Again, the calculus of derived Kan extensions can be used to construct theclassical triangulation on DX(1) ∼= D(X) and, more generally, on DX(A)for A ∈ Cat (see §15). In fact, we even obtain higher triangulations onthese categories (see [GS14b, GS15b, GS14a, GS15a]).

Moreover, the description of the above-mentioned classical derived equiv-alences from representation theory relies only on having a well-behavedcalculus of derived Kan extensions. As a consequence, the derived equiv-alences which were classically considered over fields only for example alsoextend to similar equivalences over arbitrary schemes; see [GS14b, GS15b,GS14a, GS15a].

The existence of homotopy derivators of Grothendieck abelian categories(Theorem 7.3.4) follows from the following more general results. (As of thiswriting there is no more direct proof of Theorem 7.3.4, but the author thinksthat such a proof would be interesting in that it makes precise how classicalconstructions from homological algebra relate to the more abstract derived Kanextensions.)

Theorem 7.3.6. The category Ch(A),A a Grothendieck abelian category, ad-mits a combinatorial model structure with the class of quasi-isomorphisms asclass of weak equivalences.

A proof of this result can be found in [Bek00, Hov01] (see also [CD09]).Let us recall that a model category [Qui67, Hov99, DS95] is combinatorialif the model structure is cofibrantly generated and if the underlying categoryis locally presentable. A short discussion of combinatorial model categories aswell as references to the literature can be found in Appendix B. Theorem 7.3.6together with the following result then provides a proof of Theorem 7.3.4.

Theorem 7.3.7. Let (M,WM) be a combinatorial model category. The pre-derivator

HoM : Catop → CAT : A 7→ MA[(WAM)−1]

is a derivator, the homotopy derivator of M.

A detailed proof of this result can be found in Appendix B. Obviously thesetwo theorems immediately imply Theorem 7.3.4.

Remark 7.3.8. Given a combinatorial model category M, one can pass to theunderlying homotopy derivator HoM, and one might wonder how much infor-mation is lost by this passage M 7→ HoM. Renaudin [Ren09] has shown thatthe Quillen equivalence type of M can be reconstructed from HoM; see Ap-pendix B for a more precise statement.

Before we turn to more specific examples, we collect two additional largeclasses of examples of derivators. The first result is due to Cisinski [Cis03] andit is a rather deep result. Related to this see also [CS02].

DRAFT


Theorem 7.3.9. Let (M,WM) be a model category. The homotopy prederiv-ator

HoM : Catop → CAT : A 7→ MA[(WAM)−1]

is a derivator, the homotopy derivator of M.

Similarly, ∞-categories give rise to derivators. We recall that ∞-categorieswere introduced by Boardman–Vogt [BV73] in their study of homotopy-invariantalgebraic structures on topological space. Recently, ∞-categories were studieda lot by Joyal [Joy08], Lurie [Lur09, Lur11], and others; see [Gro10] for anintroduction. A sketch proof of the following result can be found in [GPS14b].Recall that the nerve NA of a small category A is the simplicial set whichin simplicial degree n is given by the set of strings of n composable arrowsa0 → a1 → . . .→ an.

Theorem 7.3.10. Let C be a complete and cocomplete ∞-category. There is aderivator

HoC : Catop → CAT : A 7→ Ho(CNA),

the homotopy derivator of the ∞-category C.

Thus, derivators encode key formal aspects of the calculus of ordinary Kanextensions (Theorem 7.3.1), derived Kan extensions (Theorem 7.3.4), and ho-motopy Kan extensions (Theorem 7.3.7, Theorem 7.3.9, and Theorem 7.3.10).These theorems yield many interesting examples of derivators arising in variousfields of mathematics. Here we collect the following key examples from topologybut additional ones will be mentioned in §15.2.

Examples 7.3.11.

(i) The category Top of topological spaces can be endowed with the Serremodel structure [Hov99] such that the weak equivalences are the weakhomotopy equivalences. Denoting by WA

Top, A ∈ Cat , the class of levelwise

weak homotopy equivalences in TopA, there is a derivator

HoTop : Catop → CAT : A 7→ TopA[(WATop)−1],

the derivator of topological spaces.

An alternative description of this derivator is based on the Kan–Quillenmodel structure [Hov99] on simplicial sets. Let us recall that the geometricrealization and the singular complex functor define an adjunction

(|-|,Sing) : sSet Top.

The weak equivalences WsSet in the Kan–Quillen model structure arethose maps such that the geometric realization is a weak homotopy equiv-alence of topological spaces. Correspondingly, there is the derivator

HosSet : Catop → CAT : A 7→ sSetA[(WAsSet)

−1],

DRAFT


the derivator of simplicial sets.

There are various additional ways to define an equivalent derivator (seefor example [Tho80], [Gro83, Mal05b, Cis06]), and we refer to any of theseas the derivator of spaces. All these derivators enjoy the same universalproperty, namely as ‘free derivators generated by the singleton’; see §18.

In these derivators the limit and colimit functors specialize to the clas-sical homotopy limit and homotopy colimit functors. Special instancesof these functors already occured in [Pup58, Mat76, Seg68, tD71], whilemore systematic classical accounts are in [BK72, BV73, Vog73].

(ii) Let us recall the notion of a spectrum in the sense of topology. A spec-trum X consists of pointed topological spaces Xn, n ≥ 0, and pointedstructure maps ΣXn → Xn+1. By means of Brown representability andthe suspension isomorphism theorem, spectra arise as representing objectsfor generalized cohomology theories [Vog70, Ada74, Ada78]. A morphismf : X → Y of spectra is a collection of pointed maps fn : Xn → Yn com-patible with the structure maps.

The category Sp of spectra can be endowed with the stable Bousfield–Friedlander model structure [BF78]. Denoting by WA

Sp, A ∈ Cat , the corre-sponding levelwise stable equivalences, we obtain an associated homotopyderivator,

Sp : Catop → CAT : A 7→ SpA[(WASp)−1], (7.3.12)

the derivator of spectra. The underlying category of Sp is the stablehomotopy category Ho(Sp) = SHC (see [Vog70] and [Ada74, Part III]).

Again, there are various alternative descriptions of the derivator of spec-tra, based on Quillen equivalent model categories [HSS00, EKMM97,Sch01, MMSS01]. All of these derivators enjoy the universal propertyof a ‘free stable derivator generated by the sphere spectrum’; see §18.Starting in §15, we introduce stable derivators and collect a few featuresof the calculus of homotopy Kan extensions which are particular to thestable picture. For example, the derived equivalences from representa-tion theory mentioned in Examples 7.3.5 also extend to the derivator ofspectra.

We conclude this section by two operations which apply to derivators.

Example 7.3.13. Let D and E be prederivators. The product prederivatorD × E is defined by

(D × E )(A) = D(A)× E (A).

More precisely, D × E is defined as the composition

Catop ∆→ Catop × Catop → CAT × CAT×→ CAT

in which the undecorated 2-functor is the external product of D and E , whilethe remaining ones are diagonal and product 2-functors, respectively.

DRAFT


Lemma 7.3.14. If D and E are derivators, then D × E is a derivator, theproduct derivator.


The proof shows that for every u : A→ B in Cat the Kan extension functorsin D ,E , and D × E are related by the formula

uD×E! = uD

! × uE! and uD×E

∗ = uD∗ × uE

∗ .

Recall the duality principle from ordinary category theory which allows oneoften, for example, to formally deduce results concerning colimits from similarresults concerning limits. This duality principle is based upon the passage toopposite categories. In order to obtain a similar principle for derivators weinclude the following example.

Example 7.3.15. Given a prederivator D , the opposite prederivator Dop isdefined to send A ∈ Cat to

Dop(A) = D(Aop)op.

The axioms of a derivator (Definition 7.2.8) are self-dual, and this is reflectedby the following result. We suggest the proof as an exercise to the reader.Because of its importance, we nevertheless include a detailed proof in §B.2.

Proposition 7.3.16. A prederivator D is a derivator if and only if the oppositeDop is a derivator.

Proof. The proof is left as an exercise.

The details of the proof show that for every u : A → B in Cat the Kanextension functors in D and Dop are related by the formula

uDop

! =((uop)D

∗)op

and uDop

∗ =((uop)D

!

)op. (7.3.17)

7.4 Limits versus homotopy limits

In this short section we expand a bit on Warning 7.2.10 and emphasize thedifference between

(i) completeness and cocompleteness, i.e., the existence of categorical limits,categorical colimits, and categorical Kan extensions, and

(ii) homotopical completeness and homotopical cocompleteness, i.e., the ex-istence of homotopy (co)limits and homotopy Kan extensions.

In this section only, we use the expressions categorical (co)limits and categoricalKan extensions in order to refer to the usual constructions from category theoryand the terms homotopy (co)limits and homotopy Kan extensions to refer toadjoints to restriction functors in a derivator D .

DRAFT

7.4. LIMITS VERSUS HOMOTOPY LIMITS 131

Given a derivator D and a small category A, one might wonder whether thecategory D(A) admits categorical (co)limits of shape B ∈ Cat . By definitionof categorical (co)limits these would be adjoint functors to the diagonal functor∆B : D(A)→ D(A)B which sends X ∈ D(A) to the constant B-shaped diagramwith value X. The point is that, in general, the axioms of a derivator do notimply the existence of such adjoints.

However, there are adjoints to a closely related functor. Let πB : A×B → Abe the projection away from B with restriction functor π∗B : D(A)→ D(A×B).By definition of a derivator, π∗B admits a left adjoint (πB)! : D(A×B)→ D(A)and a right adjoint (πB)∗ : D(A×B)→ D(A). Note that the functors

π∗B : D(A)→ D(A×B) and ∆B : D(A)→ D(A)B (7.4.1)

have the same domain but that the targets, in general, are essentially differentcategories.

(i) Objects in D(A×B) are diagrams which are coherent in the A-directionand the B-direction.

(ii) Objects in D(A)B are diagrams which are coherent in the A-direction butincoherent in the B-direction.

The functors in (7.4.1) are related by the following partial underlying diagramfunctors which make diagrams of two variables partially incoherent.

Construction 7.4.2. Let D be a prederivator and let A,B ∈ Cat . Evaluating onobjects and morphisms in B only we obtain a partial underlying diagramfunctor

diaA,B : D(A×B)→ D(A)B . (7.4.3)

In formulas, associated to b ∈ B there is the functor idA×b : A ∼= A×1→ A×B,and given X ∈ D(A×B) we set

diaA,B(X)(b) = (idA × b)∗X ∈ D(A). (7.4.4)

Similarly to the case of underlying diagram functors (Lemma 7.1.5), thereader easily concludes the definition of (7.4.3) and verifies the following.

Lemma 7.4.5. Let D be a prederivator and let A,B ∈ Cat.

(i) There is a partial underlying diagram functor diaA,B : D(A×B)→ D(A)B

defined by (7.4.4).

(ii) The underlying diagram functor diaB : D(B)→ D(1)B is canonically iso-morphic to dia1,B : D(1×B)→ D(1)B.

(iii) The functor diaA,B satisfies ∆B = diaA,B π∗B : D(A)→ D(A)B.


DRAFT


Thus, given a derivator D and A,B ∈ Cat , there is the diagram

D(A×B)diaA,B

//

(πB)!

(πB)∗

D(A)B

D(A)

π∗B

OO

id// D(A).

∆B

OO

We emphasize that, in general, also the functors diaB,A, which make dia-grams partially incoherent, are far from being equivalences.

Lemma 7.4.6. Let D be a derivator and let A,B ∈ Cat. If the partial underly-ing diagram functor diaA,B : D(A×B)→ D(A)B is an equivalence, then D(A)has categorical limits and colimits of shape B.

Proof. By Lemma 7.4.5 the assumption on diaA,B implies that the functors π∗Band ∆B are equivalent in the sense of Definition 2.3.4. In particular, since π∗Badmits adjoints on both sides, the same is true for ∆B (Lemma 2.3.5), which isto say that D(A) has categorical (co)limits of shape B.

Proposition 7.4.7. Let D be a derivator and let A ∈ Cat.

(i) The category D(A) admits a terminal object ∗ and an initial object ∅.

(ii) The category D(A) admits products and coproducts.

Proof. Let S be a set which we consider as a discrete category, i.e., a categorywith identity morphisms only. As a special case of axiom (Der1), the canonicalfunctor D(

∐s∈S A) ∼−→

∏s∈S D(A) induced from the inclusions is an equiva-

lence. One easily checks that the diagram

D(∐s∈S A)

' //

∼=

∏s∈S D(A)

∼=

D(A× S)diaA,S

// D(A)S .

commutes where the vertical maps are the canonical identifications, i.e., thatdiaA,S is equivalent to D(

∐s∈S A) ∼−→

∏s∈S D(A). Thus, for discrete cate-

gories S, the functors diaA,S are equivalences as well. By Lemma 7.4.6 weconclude that D(A) hence has S-fold products and coproducts. Specializing toS = ∅ we deduce the existence of terminal and initial objects.

Remark 7.4.8.

(i) We just saw that categorical coproduct functors and homotopy coproductfunctors both exist and are equivalent (in the sense of Definition 2.3.4)in arbitrary derivators. A related result which also takes into accountthe (homotopy) coproduct cocones is included as Proposition 11.2.2. Ofcourse, there is a dual statement for (homotopy) products.

DRAFT

7.5. THE CASE OF TOPOLOGICAL SURFACES 133

(ii) These statements are particular to products and coproducts in the follow-ing two senses.

(a) In general, D(A) does not admit categorical limits or categoricalcolimits of more general shapes. We will take up examples alongthese lines in §7.5.

(b) Moreover, even if D(A) admits (co)limits of shape B for some smallcategory B, then, in general, these are not compatible with the ho-motopical versions. We already discussed such an instance in thecontext of group actions; see Remark 4.2.9.

(iii) Using categories of diagrams one can also include examples of derivatorssuch that only finite (co)products exist (this comment is related to Re-mark 7.1.3(ii)).

7.5 The case of topological surfaces

In this section we illustrate by some examples that the homotopy categoryFix pointedness issues

Ho(Top∗) of pointed topological spaces is not cocomplete. We also show that thelocalization functor γ : Top∗ → Ho(Top∗), in general, does not send homotopypushout squares to pushout diagrams. All these examples are variants of thesame idea and they arise in the context of topological surfaces.

For n ≥ 0 we denote by Sn, Dn ∈ Top∗ the n-sphere and the n-disc, respec-tively, and let Sn → Dn+1 be the corresponding boundary inclusion. We beginwith an example related to the real projective plane.

Example 7.5.1. Let us recall that the real projective plane RP 2 admits a CWstructure with a unique cell in dimensions 0, 1, 2 and no further cells. This CWstructure arises from the following pushout square

S1

2 // S1

D2 // RP 2

(7.5.2)

in Top∗, in which the top horizontal map is any degree two map. Since D2

is contractible, the localization functor γ : Top∗ → Ho(Top∗) sends the abovesquare to the following square on the left

S1

2 // S1

S1

2 // S1

0 // RP 2, 0 // P.

We claim that this induced square is not a pushout square. More generally, weshow that the map 2: S1 → S1 has no cokernel in Ho(Top∗), i.e., that there isno pushout square as indicated on the right.

DRAFT


Let us assume that such a pushout exists and let us consider the reducedcohomology functors Hk(−;A) : Ho(Top∗)

op → Ab for suitable A ∈ Ab. Sincethese cohomology functors are representable,

Hk(−;A) ∼= [−, H(k,A)]∗,

we would obtain induced pullback squares

Hk(P ;A)

// Hk(S1;A)

2

0 // Hk(S1;A),

which is to say that Hk(P ;A) is the kernel of 2 : Hk(S1;A) → Hk(S1;A). ForS1 one calculates the following table on the left,

Hk(S1;A) A = Z A = Z/2Z

k = 0 0 0

k = 1 Z Z/2Z

k ≥ 2 0 0

Hk(P ;A) A = Z A = Z/2Z

k = 0 0 0

k = 1 0 Z/2Z

k ≥ 2 0 0

and for P we hence obtain the corresponding table on the right. To obtain theintended contradiction, we consider the long exact sequence in reduced coho-mology of P associated to the short exact sequence 0 → Z → Z → Z/2Z → 0.By the above table this would look like

0→ 0→ 0→ Z/2Z→ 0→ 0→ 0→ 0,

which is impossible.Hence, the map 2: S1 → S1 has no cokernel in Ho(Top∗). In particular, the

localization functor does not send the homotopy pushout square (7.5.2) in Top∗to a pushout square in Ho(Top∗).

There is a similar example related to the Kleinian bottle.Nicer with reduced cylinders?

Example 7.5.3. Let us recall that the Kleinian bottle K can be defined by thefollowing pushout square

S1 t S1

(1,−1)

(1,1)// S1 × [0, 1]

S1 × [0, 1] // K

(7.5.4)

in Top∗. In this diagram, the top horizontal map is the usual boundary inclusion,while the left vertical map is the boundary inclusion twisted by −1: S1 → S1 on

DRAFT

7.5. THE CASE OF TOPOLOGICAL SURFACES 135

one side. The localization functor γ : Top∗ → Ho(Top∗) sends the above squareto the following square on the left

S1 t S1

(1,−1)

(1,1)// S1

S1 t S1

(1,−1)

(1,1)// S1

S1 // K, S1 // P.

(7.5.5)

We claim that this induced square is not a pushout square, and, again, moregenerally, we show that there is no such pushout square as shown on the right.

Passing to reduced singular cohomology, such a pushout square would inducepullback squares

Hk(P ;A)

// Hk(S1;A)

(1,1)∗

Hk(S1;A)(1,−1)∗

// Hk(S1 t S1;A)

of abelian groups. Hence, Hk(P ;A), k ≥ 1, is the kernel of the map[1 −11 1

]: Hk(S1;A)⊕ Hk(S1;A)→ Hk(S1;A)⊕ Hk(S1;A).

Based on the tables in Example 7.5.1, this implies that, for specific choices ofA ∈ Ab, the reduced cohomology Hk(P ;A) is given by the following table:

Hk(P ;A) A = Z A = Z/2Z

k = 0 0 0

k = 1 0 Z/2Z

k ≥ 2 0 0

But we already observed in Example 7.5.1 that this is impossible, and we henceobtain the desired contradiction.

Thus, there is no pushout square in Ho(Top∗) as indicated on the rightin (7.5.5). In particular, the localization functor does not send the homotopypushout square (7.5.4) in Top∗ to a pushout square in Ho(Top∗).

As a final, closely related example, let us recall that there is the connectedsum operation # on topological surfaces. Given two surfaces M1,M2, theconnected sum forms a new surface

M1#M2 =(M1 −

D2)tS1

(M2 −

D2).

The construction relies on choices of embeddings of small disks D2 → M1,M2,considering the resulting maps

S1 →M1 −D2 and S1 →M2 −

D2,

DRAFT


and passing to the defining pushout square

S1 //

M1 −D2

M2 −D2 // M1#M2.

Related to the Kleinian bottle there is the following example.

Example 7.5.6. Recall for example from [Mas91] that the Kleinian bottle K canbe written as the connected sum RP 2#RP 2, i.e., that there is a pushout square

S1 //

RP 2 −D2

RP 2 −D2 // K

(7.5.7)

in Top∗. The space RP 2 −D2 is homotopy equivalent to S1, and under this

homotopy equivalence the inclusion S1 → RP 2−D2 corresponds to 2: S1 → S1.

Thus, the localization functor γ : Top∗ → Ho(Top∗) sends the square (7.5.7) tothe following square on the left

S1

2

2 // S1

S1

2

2 // S1

S1 // K, S1 // P.

(7.5.8)

We claim that this induced square is not a pushout square. In fact, we show,more generally, that there is no pushout square as shown on the right in whichP is a finite type space.

Really needed?Assuming such a pushout square to exist, we would obtain the following

pullback square in reduced cohomology

Hk(P ;A)

// Hk(S1;A)

2

Hk(S1;A)2// Hk(S1;A)

for any A ∈ Ab. This exhibits Hk(P ;A) as the kernel of

(2,−2) : Hk(S1;A)⊕ Hk(S1;A)→ Hk(S1;A).

Using again the tables in Example 7.5.1, this allows us to calculate the reducedcohomology Hk(P ;A) for A = Z,Z/2Z, and the result is shown in the following

DRAFT

7.6. COMMUTING LIMITS 137

table:

Hk(P ;A) A = Z A = Z/2Z

k = 0 0 0

k = 1 Z Z/2Z⊕ Z/2Z

k ≥ 2 0 0

But the universal coefficient theorem in cohomology for finite type spaces impliesthat there would be an isomorphism

H1(P ;Z/2Z) ∼= H1(P ;Z)⊗ Z/2Z⊕ Tor(H2(P ;Z),Z/2Z),

which is impossible by the above calculation.Thus, this shows that there is no pushout square in Ho(Top∗) as indicated

on the right in (7.5.8) in which P is a finite type space. As a special case, thelocalization functor does not send the homotopy pushout square (7.5.7) in Top∗to a pushout square in Ho(Top∗).

(Let us note that this example also shows the following. Clearly, the mor-phism 2: S1 → S1 is a continuous surjection and hence an epimorphism inTop∗. However, the morphism 2: S1 → S1 is not an epimorphism in Ho(Top∗).In fact, if this map were an epimorphism in Ho(Top∗), then the reader easilychecks that the square

S1 2 //

2

S1

=

S1=// S1

would be a pushout square. But since S1 is a finite type space, this is impossibleby the above discussion.)

These examples show that Ho(Top∗) is not cocomplete, since, in general,pushouts do not exist. Since Ho(Top∗) is the underlying category of the deriva-tor of pointed spaces HoTop∗ (Theorem 7.3.9), by Lemma 7.4.6 these examplesalso imply that the underlying diagram functor

diap : Ho(Topp∗)→ Ho(Top∗)p

is not an equivalence of categories. However, it can be shown that this functoris full and essentially surjective, and we will come back to this in the discussionof strong derivators; see §15.4.

7.6 Commuting limits

In this short section we collect a few immediate manipulation rules for thecalculus of Kan extensions.

DRAFT


Lemma 7.6.1. Let D be a derivator, let u : A → B, v : B → C be in Cat, andlet X ∈ D(A). There are canonical isomorphisms

(v u)!(X) ∼= v!(u!(X)) and (idA)!(X) ∼= X.

Proof. This is immediate from the uniqueness of adjoints (Lemma A.1.11) andthe relation (v u)∗ = u∗v∗ : D(C) → D(A). In the same way we obtain acanonical isomorphism (idA)!

∼= idD(A).

There are the following immediate consequences.

Corollary 7.6.2. Let D be a derivator, let u : A→ B in Cat, and let X ∈ D(A).There are canonical isomorphisms

colimAX ∼= colimB u!(X).

Proof. We simply apply Lemma 7.6.1 to πA = πB u : A→ 1.

A variant is the following Fubini theorem, saying that left Kan extensionsin unrelated variables commute.

Corollary 7.6.3. Let D be a derivator, let u : A→ A′ and v : B → B′, and letX ∈ D(A×B). There are canonical isomorphisms

(u× id)!(id× v)!X ∼= (u× v)!X ∼= (id× v)!(u× id)!X.

Proof. Considering the naturality square

A×B u×id//

id×v

A′ ×B

id×v

A×B′u×id

// A′ ×B′,

this is immediate from Lemma 7.6.1.

Using suggestive notation, the Fubini theorem specializes as follows.

Corollary 7.6.4. Let D be a derivator, let A,B ∈ Cat , and let X ∈ D(A×B).There are canonical isomorphisms

colimA colimB X ∼= colimA×B X ∼= colimB colimAX.

Proof. This is the special case of the previous results applied to πA : A→ 1 andπB : B → 1.

Thus, colimits in unrelated variables commute in every derivator. A classicalreference for such a result can already be found in [Vog77], hence the title of thissection. The results of this section of course dualize to yield similar statementsfor right Kan extensions and limits (Proposition 7.3.16).

The mixed situation, i.e., the question whether limits and colimits in unre-lated variables commute, is more subtle, and we will get back to a special caseof it in [Gro16a].

DRAFT

Chapter 8

Homotopy exact squares

In this chapter we discuss in more detail the formalism behind axiom (Der4)in the definition of a derivator (Definition 7.2.8), making precise that Kan ex-tensions in derivators are pointwise. This formalism relies on the calculus ofmates, a certain calculus which applies to natural transformations. The pointof this formalism is the following. While working with derivators, one often runsinto the situation that outputs of certain universal constructions ‘obviously areisomorphic’. In many cases, the formalism of homotopy exact squares allowsone to actually conclude this by, first, providing canonical maps between suchgadgets and, second, guaranteeing that these maps are isomorphisms.

We begin in §8.1 with a review of the calculus of mates and illustrate it bysome basic examples from ordinary category theory. In §8.2 we define homotopyexact squares, arguably the main technical tool in the theory of derivators. Thisis used to establish some first facts about the calculus of Kan extensions inderivators which will be exploited in later chapters.

8.1 The calculus of mates

In this section we review the pasting of natural transformations and the calculusof canonical mates. While these are general 2-categorical concepts, here we focuson the special case of the 2-category CAT .

Refs!The pasting operation is a composition operation which applies to natural

transformations living in ‘larger diagrams’. Here we refrain from giving a formaldefinition of such diagrams and instead only mention a few relevant examples.

Example 8.1.1. Let us consider the diagram

C1

α1

C2

α2

v∗1oo C3v∗2oo

D1

u∗1

OO

D2

u∗2

OO

w∗1

oo D3,w∗2

oo

u∗3

OO

(8.1.2)

139

DRAFT

140 CHAPTER 8. HOMOTOPY EXACT SQUARES

which consists of categories, functors, and transformations α1 : v∗1u∗2 → u∗1w

∗1

and α2 : v∗2u∗3 → u∗2w

∗2 . We define the natural transformation α1 α2 as

α1 α2 = (α1w∗2) (v∗1α2) : v∗1v

∗2u∗3 → v∗1u

∗2w∗2 → u∗1w

∗1w∗2 ,

and refer to it as the horizontal pasting of the squares in (8.1.2). Similarly,we define the vertical pasting of squares.

The pasting operation also applies to more complicated such diagrams aslong as all natural transformations ‘point in the same direction’ (like in (8.1.7)).As an additional simple example we include the following one.

Example 8.1.3. Let (L,R) : C D be an adjunction with unit η : id→ RL andcounit ε : LR→ id. We recall from §A.1 that the unit and the counit are subjectto the triangular identities

LLη//

id ++

LRL

εL

RηR//

id ++

RLR

Rε

L, R.

Using the pasting operation we can rewrite the triangular identity concerningthe left adjoint L as

C L //

id ++

D

R

id

=

C

L

CL//

;Cη

D

;Cε

D,

where the line segment on the right is meant to indicate the identity transfor-mation of L. There is a similar picture for the other triangular identity.

The point of this rewriting is that it shows that units and counits canceleach other in larger pasting diagrams.

As a special case of this pasting operation we now define mates. Let usconsider a natural transformation α : p∗u∗ → v∗q∗ living in a square of (notnecessarily small) categories

C1 oop∗

OO

v∗ α

C2OOu∗

D1ooq∗D2.

(8.1.4)

Having later applications to derivators in mind, in this section functors areoften denoted like precomposition functors u∗ and left adjoints and right adjointsto such functors are denoted by u! and u∗, respectively. If the necessary adjointfunctors exist, then associated to a natural transformation α as in (8.1.4) there

DRAFT

8.1. THE CALCULUS OF MATES 141

are two additional natural transformations, the associated canonical mates,namely

α! : v!p∗ η→ v!p

∗u∗u!α→ v!v

∗q∗u!ε→ q∗u! and (8.1.5)

α∗ : u∗q∗η→ p∗p

∗u∗q∗α→ p∗v

∗q∗q∗ε→ p∗v

∗. (8.1.6)

Here, η denotes the units of the adjunctions (u!, u∗) and (p∗, p∗), and ε the

counits of the adjunctions (v!, v∗) and (q∗, q∗), respectively. Thus, α! is defined

as the following pasting on the left, while α∗ is defined as the following pastingon the right,

C2

η

D1

ε

C1v!oo

α

C2p∗oo

η

C1

p∗

OO

α

C2p∗oo

=kk

D1

v∗

OO

=

UU

D2q∗oo

u∗

OO

C2,u!

oo

=kk

D1

v∗

OO

ε

D2q∗oo

u∗

OO

D1.

q∗

OO

=

UU

(8.1.7)

In the remainder of this section, we always assume that the necessary adjointfunctors exist.

Remark 8.1.8.

(i) Note that the construction of canonical mates depends on choosing cer-tain adjoint functors, adjunction units, and adjunction counits. However,adjoint functors are unique up to unique isomorphisms in a way thatthe isomorphisms are compatible with units and counits (Lemma A.1.11).This implies that the canonical mates are well-defined up to some pastingwith certain canonical natural isomorphisms. In particular, the questionif a canonical mate is an isomorphism does not depend on these choices.This motivates us to refer to the transformations (8.1.5) and (8.1.6) asthe canonical mates.

(ii) Let us again consider the square (8.1.4) and let us assume that the fourfunctors have adjoints on both sides. Using ‘conjugations by units andcounits from opposite sides’ the two natural transformations (8.1.5) and(8.1.6) are the only ones which make sense for arbitrary α. Thus, thenotation α 7→ α! and α 7→ α∗ is unambiguous.

Examples 8.1.9.

(i) Associated to a functor u∗ : D → C there is the identity transformation

DRAFT


u∗ → u∗, which populates the two commutative squares

C

| id

Du∗oo C

id

Cidoo

D

u∗

OO

D,

id

OO

idoo C

id

OO

D.

u∗

OO

u∗oo

If u∗ admits a left adjoint u! : C → D, then the canonical mates (8.1.5) ofthe two squares yield the counit ε : u!u

∗ → id and the unit η : id→ u∗u!,respectively. Similarly, if u∗ admits a right adjoint u∗ : C → D, then thecanonical mates (8.1.6) of these two squares are the unit η : id → u∗u

∗

and the counit ε : u∗u∗ → id, respectively.

(ii) Let us consider functors u∗, v∗ : D → C and a natural transformationu∗ → v∗ which we want to rewrite as two different squares

C

|

Cidoo C

Du∗oo

D

v∗

OO

D,

u∗

OO

idoo C

id

OO

D.

id

OO

v∗oo

populated by this transformation. If u∗ and v∗ admit left adjoints u!

and v!, respectively, then the canonical mate (8.1.5) of the square on theleft is the conjugate transformation or total mate v! → u! in thesense of (A.1.8). Similarly, if u∗ and v∗ admit right adjoints u∗ and v∗,respectively, then the canonical mate (8.1.6) of the square on the right isthe total mate v∗ → u∗ as defined in (A.1.7).

A few additional classical examples for this calculus of mates appear in §B.1.In this section we content ourselves by showing that the calculus of mates enjoysa functoriality with respect to pasting (Lemma 8.1.10) and that, if both mates(8.1.5) and (8.1.6) are defined, then one of them is an isomorphism if and only ifthe other is (Lemma 8.1.14). In view of part (i) of Remark 8.1.8, such statementshave to be read as ‘the following mates can be chosen as we claim now’.

Lemma 8.1.10. The passages to canonical mates are compatible with respectto horizontal and vertical pasting as expressed by the formulas

(α1 α2)! = (α2)! (α1)! and (α1 α2)∗ = (α2)∗ (α1)∗.

Proof. Let us consider the horizontal pasting α1 α2 of two natural transfor-mations α1 : v∗1u

∗2 → u∗1w

∗1 and α2 : v∗2u

∗3 → u∗2w

∗2 ,

C1

α1

C2

α2

v∗1oo C3v∗2oo

D1

u∗1

OO

D2

u∗2

OO

w∗1

oo D3.w∗2

oo

u∗3

OO

DRAFT

8.1. THE CALCULUS OF MATES 143

Assuming that the functors u∗1, u∗2, and u∗3 have left adjoints, the natural trans-

formation

(α2)! (α1)! : (u1)!v∗1v∗2 → w∗1(u2)!v

∗2 → w∗1w

∗2(u3)!

is by definition given by the pasting

D1

ε1

C1

α1

(u1)!oo C2

η2

v∗1oo

D1

=

UU

u∗1

OO

D2

ε2

w∗1

oo

u∗2

OO

C2

α2

=kk

(u2)!

oo C3

η3

v∗2oo

D2

=

UU

u∗2

OO

D3w∗2

oo

u∗3

OO

D3;

=kk

(u3)!

oo

see (8.1.7). Note that the two triangles in the middle cancel out by a triangularidentity (Example 8.1.3), and we are hence left with

D1

ε1

C1

α1

(u1)!oo C2

α2

v∗1oo C3v∗2oo

η3

D1

=

UU

u∗1

OO

D2w∗1

oo

u∗2

OO

D3

u∗3

OO

w∗2

oo C3,(u3)!

oo

=kk

which is to say (α1 α2)! as intended. The remaining cases are similar.

Let us consider a natural transformation (8.1.4) such that the canonical mate(8.1.5) is defined. This mate α! can then also be written as

D1oov!

OO

q∗ α!

C1OOp∗

D2oou!C2,

(8.1.11)

and obviously the horizontal functors now have right adjoints. The canonicalmate (8.1.6) of α! is hence defined, and this mate (α!)∗ can be chosen to be α.

Lemma 8.1.12. The two different formations of mates α 7→ α! and α 7→ α∗are inverse to each other, i.e., we have α = (α!)∗ and α = (α∗)!.


We now consider a natural transformation (8.1.4) such that all four functorshave left adjoints. The canonical mate α! can again be written as in (8.1.11),and by our assumption it has a further canonical mate (α!)! : q!v! → u!p!. Recallfrom (A.1.7) and (A.1.8) the notion of conjugate natural transformations.

DRAFT


Lemma 8.1.13. Let α be a natural transformation (8.1.4) such that all fourfunctors have left adjoints. The canonical mate (α!)! : q!v! → u!p! is conjugateto α : p∗u∗ → v∗q∗.


Finally, as an application of the fact that conjugation preserves and reflectsnatural isomorphisms (Corollary A.1.9) we obtain the following intended result.

Lemma 8.1.14. Let α be a natural transformation as in (8.1.4) such that thecanonical mates α! : v!p

∗ → q∗u! and α∗ : u∗q∗ → p∗v∗ exist. Then these mates

α!, α∗ are conjugate. In particular, α! is an isomorphism if and only if α∗ is anisomorphism.

Proof. The canonical mate α! : v!p∗ → q∗u! can be written as in (8.1.11) and

all four functors in (8.1.11) have right adjoints. By Lemma 8.1.12 we henceconclude that the canonical mate α∗ : u∗q∗ → p∗v

∗ is given by ((α!)∗)∗, andthe dual version of Lemma 8.1.13 then implies that α! and α∗ are conjugate.Finally, the second claim is guaranteed by Corollary A.1.9.

Also the passage to inverse natural transformations and opposite naturaltransformations are functorial with respect to pasting.

Lemma 8.1.15. Let α1, α2 be invertible natural transformations as in (8.1.2).some refs: loop, continuity

The pasting α1 α2 is again invertible and the inverse is given by

(α1 α2)−1 = α−12 α

−11 .

Proof. This is immediate from the definition of horizontal pasting.

Lemma 8.1.16. For natural transformations α1, α2 as in (8.1.2) there is thepasting relation

(α1 α2)op = αop2 α

op1 .

Proof. This is immediate from the definition of horizontal pasting and the con-travariance of the passage to opposite natural transformations.

Of course there is a similar result for the vertical pasting. We now continuewith applications of this calculus of canonical mates and again turn to derivators.

8.2 Homotopy exact squares and Kan extensions

The reason for us to consider the calculus of mates in this book is that it allows usto systematically encode ‘manipulation rules for Kan extensions in derivators’.In this section we collect a few first such rules, while additional ones can befound in later sections.

DRAFT

8.2. HOMOTOPY EXACT SQUARES AND KAN EXTENSIONS 145

To make these manipulation rules specific, let D be a derivator and let usconsider a square of small categories

Cp//

v

| α

A

u

Dq// B

(8.2.1)

populated by a natural transformation α : up → qv. If we apply the 2-functorD : Catop → CAT to α, then we obtain an induced natural transformation

D(C) oop∗

OO

v∗ α∗

D(A)OO

u∗

D(D) ooq∗

D(B).

As summarized in §8.1, such a transformation has canonical mates

α! : v!p∗ η→ v!p

∗u∗u!α∗→ v!v

∗q∗u!ε→ q∗u! and (8.2.2)

α∗ : u∗q∗η→ p∗p

∗u∗q∗α∗→ p∗v

∗q∗q∗ε→ p∗v

∗. (8.2.3)

(Again we generically write η for adjunction units and ε for adjunction counits.)Thus, α! is a natural transformation of functors D(A)→ D(D) and α∗ a naturaltransformation of functors D(D) → D(A), and it follows from Lemma 8.1.14that α! is an isomorphism if and only if α∗ is an isomorphism.

Definition 8.2.4. A square (8.2.1) of small categories populated by a naturaltransformation α is homotopy exact if the canonical mate α! (8.2.2) or, equiv-alently, the canonical mate α∗ (8.2.3) is an isomorphism for every derivator D .

Remark 8.2.5. Note again that the canonical mates (8.2.2) and (8.2.3) dependon choices of adjunction data. However, as a consequence of Remark 8.1.8, thequestion whether these canonical mates are natural isomorphisms is independentfrom these choices.

Lemma 8.2.6. Let u : A → B be a functor between small categories and letb ∈ B. The slice squares (7.2.4) are homotopy exact.

Proof. This amounts to unravelling definitions only. In fact, in the case of theslice square

(u/b)p//

π

~

A

u

1b// B,

DRAFT


the canonical mate (8.2.2) is given by

π!p∗ η→ π!p

∗u∗u! → π!π∗b∗u!

ε→ b∗u!.

But this is precisely the natural transformation (7.2.6), which is hence an iso-morphism by (Der4).

Thus, (Der4) is precisely saying that slice squares are homotopy exact. Bydefinition, every homotopy exact square gives rise to canonical isomorphisms(8.2.2) and (8.2.3), which ressemble base change theorems, projection formulas,or push-pull formulas. Sometimes one also refers to such situations by sayingthat the square satisfies the Beck–Chevalley condition.

The point of Definition 8.2.4 is that there is a battery of additional ex-amples of homotopy exact squares (see also the discussion in Remark 8.2.25).These squares allow us to show that in the context of abstract derivators —like homotopy derivators of Grothendieck abelian categories, model categoriesor complete and cocomplete ∞-categories — certain canonical maps exist andare isomorphisms, thereby simultaneously establishing ‘manipulation rules forthe calculus of ordinary Kan extensions, derived Kan extensions, and homotopyKan extensions’.

For example, one ‘of course’ expects that given a category A ∈ Cat admittinga terminal object t ∈ A then for every derivator D and diagram X ∈ D(A) thereshould be a canonical isomorphism Xt

∼= colimAX in D(1). A convenient way toestablish such results is by means of the calculus of homotopy exact squares. Bythe duality principle (Proposition 7.3.16), it suffices to take care of statementsabout left Kan extensions.

Proposition 8.2.7. The class of homotopy exact squares is closed under hori-zontal and vertical pasting.

Proof. This is immediate from Lemma 8.1.10.

Proposition 8.2.8. Let v : B → A a right adjoint functor between small cate-gories. The square

Bv //

πB

| id

A

πA

1id// 1

(8.2.9)

is homotopy exact, i.e., for every derivator D and X ∈ D(A) there is a canonicalisomorphism

colimB v∗X ∼−→ colimAX.

Proof. Let (u, v, η, ε) : A B be an adjunction with unit η : id→ vu and counitε : uv → id. As it is the case for every 2-functor, an application of D yields aninduced adjunction

(v∗, u∗, η∗, ε∗) : D(B) D(A).

DRAFT


Thus, u∗ is a model for the right Kan extension functor v∗. By definition of ahomotopy exact square we hence have to show that

η∗π∗A : π∗A → u∗v∗π∗A

is an isomorphism for every derivator D . In fact, in this situation the canonicalmate (8.2.3) simplifies accordingly. But this natural transformation is given by

η∗π∗A = (πAη)∗ = id∗ = id,

and hence clearly is an isomorphism.

This proposition makes precise that colimits of coherent diagrams are notaffected by restrictions along right adjoint functors. It is a derivator version ofthe ‘finality of right adjoints’ (Proposition 6.2.4).

Corollary 8.2.10. Let A ∈ Cat admit a terminal object t. For every deriva-tor D and X ∈ D(A) there is a canonical isomorphism

Xt∼−→ colimAX.

Proof. This is immediate from Proposition 8.2.8 since t : 1→ A is right adjointto πA : A→ 1.

Warning 8.2.11. Let us consider a natural isomorphism α : up→ qv living in asquare (8.2.1) of small categories. Passing to the inverse natural isomorphism,we hence obtain two such squares

Cp//

v

| α

A

u

Cp//

v

A

u

Dq// B, D

q// B.

<Dα−1

It can (and will) happen that in such a situation one of these squares is homotopyexact but not the other, and it is hence important to keep track of the orientationof such invertible natural transformation.

To illustrate this phenomenon, we again consider a right adjoint v : B → A,and the corresponding commutative square (8.2.9). This square can be writtenas a square populated by an identity transformation in two ways,

Bv //

πB

| id

A

πA

Bv //

πB

A

πA

1id// 1, 1

id// 1,

<Did

and we just observed that the square on the left is homotopy exact (Proposi-tion 8.2.8). However, the canonical mate (8.2.3) associated to the square onthe right is a map

limBX → limAv∗X, X ∈ D(B),

DRAFT


and one easily writes down an explicit example in which this map is not anisomorphism (for example, in the represented case).

As a consequence, even at the risk of being a bit picky, in the context ofa commutative square (8.2.1) we will often populate it by an identity transfor-mation with a chosen orientation (as already done in Proposition 8.2.8). If wedrop the identity transformation and claim that the square is homotopy exact,then this implicitly means that this is true for both orientations.

Definition 8.2.12. A functor u : A → B in Cat is homotopy final if thesquare

Au //

πA

| id

B

πB

1id// 1

(8.2.13)

is homotopy exact.

Thus, u : A → B is homotopy final if and only if for every derivator D andX ∈ D(B) the canonical mate

colimA u∗X ∼−→ colimB X

is an isomorphism.

Examples 8.2.14.

(i) Identity functors are homotopy final. The class of homotopy final functorsis closed under composition (Proposition 8.2.7).

(ii) Right adjoint functors are homotopy final functors (Proposition 8.2.8). Inparticular, isomorphisms and equivalences are homotopy final.

Remark 8.2.15. Let y(C) be a represented derivator, let u : A → B, and letX ∈ y(C)(B), i.e., X : B → C. Unravelling definitions one easily checks that in

Promise.this case the canonical mate associated to (8.2.13) is the canonical map

colimAX u→ colimB X

from (6.2.1); see §B.1 for details. It follows that every homotopy final functor(Definition 8.2.12) is also final in the sense of Definition 6.2.3. In particular,such a functor satisfies the criterion of Proposition 6.2.7 that for every b ∈ Bthe slice categories (b/u) are non-empty and connected.

However, since a homotopy final functor in the sense of Definition 8.2.12also induces isomorphisms in homotopy derivators of abelian categories, modelcategories, and∞-categories (see §7.3), Definition 8.2.12 is, a priori, much morerestrictive. And, in fact, for such functors there are stronger statements concern-ing the slice categories (b/u), b ∈ B, and we will come back to this in [Gro16b].

As in ordinary category theory, the following result justifies that we speakof Kan extensions. Since this is the first proof of this kind in this book, we givea fairly detailed proof.

DRAFT


Proposition 8.2.16. If u : A→ B is fully faithful, then the square

Aid //

id

A

u

Au// B

(8.2.17)

is homotopy exact. This is equivalent to saying that for every derivator D thefunctors u!, u∗ : D(A)→ D(B) are fully faithful.

Proof. The canonical mate (8.2.2) associated to this square is the adjunctionunit η : 1 → u∗u! while the canonical mate (8.2.3) is the counit ε : u∗u∗ → 1.Thus, the square (8.2.17) is homotopy exact if and only if the Kan extensionfunctors u!, u∗ are fully faithful (Lemma A.1.4).

We now show that (8.2.17) is homotopy exact by showing that η : 1→ u∗u!

is an isomorphism. Since isomorphisms can be detected pointwise (by axiom(Der2)) it is enough to show that all components ηa, a ∈ A, are isomorphisms.To express this differently, let us consider the following pasting

(A/a)p//

π(A/a)

~

A

id

id //

id

A

u

1a// A

u// B,

(8.2.18)

in which the square on the left is a slice square (7.2.4). The functoriality ofmates with respect to pasting implies that the canonical mate of (8.2.18) is

(π(A/a))!p∗ ∼−→ a∗

ηa→ a∗u∗u!,

where the first map is an isomorphism by (Der4) (see Lemma 8.2.6). Thus, ηais an isomorphism if and only if the canonical mate of the pasting (8.2.18) is anisomorphism.

Since u : A→ B is fully faithful, the reader easily verifies that the functor

(A/a)→ (A/u(a)) : (a′, f : a′ → a) 7→ (a′, u(f) : u(a′)→ u(a))

is an isomorphism of categories. Moreover, using this isomorphism, we canrewrite (8.2.18) as the pasting

(A/a)∼= //

π(A/a)

id

(A/u(a))p//

π

A

u

1id

// 1u(a)

// B

(8.2.19)

in which the square on the right is a slice square (7.2.4). Since isomorphisms arehomotopy final (see Examples 8.2.14) the square on the left is homotopy exact,

DRAFT


as is the square on the right by (Der4) (see Lemma 8.2.6). By Proposition 8.2.7the pasting (8.2.19)=(8.2.18) is homotopy exact, and ηa, a ∈ A, is hence anisomorphism.

Thus, Kan extensions along fully faithful functors u : A→ B are fully faith-ful. In particular, an object X ∈ D(B) lies in the essential image of u! or u∗if and only if the adjunction counit ε : u!u

∗(X) → X or the adjunction unitη : X → u∗u

∗(X) is an isomorphism, respectively. By (Der2) it is enough tocheck this for every object b ∈ B. The point of the following proposition is thatit suffices to take care of the objects which do not lie in the image of u.

Proposition 8.2.20. Let D be a derivator, let u : A→ B be fully faithful, andlet X ∈ D(B).

(i) The functor u! : D(A)→ D(B) is fully faithul and X lies in the essentialimage of u! if and only if the adjunction counit εb : u!u

∗(X)b → Xb is anisomorphism for all b ∈ B − u(A).

(ii) The functor u∗ : D(A)→ D(B) is fully faithful and X lies in the essentialimage of u∗ if and only if the adjunction unit ηb : Xb → u∗u

∗(X)b is anisomorphism for all b ∈ B − u(A).

Proof. By duality it suffices to take care of the case of left Kan extensions. ByProposition 8.2.16, u! is fully faithful and X lies in the essential image of u! ifand only if the adjunction counit

ε : u!u∗(X)→ X (8.2.21)

is an isomorphism. By (Der2) this is the case if and only if all components εb,b ∈ B, are isomorphisms, establishing one direction. The converse directionfollows from the triangular identity

id = u∗ε ηu∗ : u∗ → u∗u!u∗ → u∗;

see (A.1.3). In fact, since id and ηu∗ both are isomorphisms (Proposition 8.2.16),this is also the case for u∗ε, and (8.2.21) is an isomorphism if and only if this isthe case for all components at objects which do not lie in the image of u.

This proposition will be useful in later constructions, for example, in thecontext of pointed derivators in §9.

We include a short discussion of free and cofree diagrams in derivators;compare to Example 6.4.2.

Definition 8.2.22. Let D be a derivator, B ∈ Cat , and X ∈ D(B).

(i) The diagram X is free if it lies in the essential image of b! : D(1)→ D(B)for some b ∈ B.

(ii) The diagram X is cofree if it lies in the essential image of b∗ : D(1) →D(B) for some b ∈ B.

DRAFT


(iii) The diagramX is constant if diaB(X) : B → D(1) is essentially constant,i.e., all structure morphisms in diaB(X) are isomorphisms.

The following is a derivator version of statement (ii) in Remark 6.4.3.

Proposition 8.2.23. Let D be a derivator and let B ∈ Cat admit a terminalobject t. The functor t∗ : D(1)→ D(B) is fully faithful and the essential imageconsists precisely of the constant diagrams.

Proof. By Proposition 8.2.16 the functor t∗ : D(1)→ D(B) is fully faithful andX ∈ D(B) lies in the essential image of t∗ if and only if the unit η : X → t∗t

∗Xis an isomorphism. By (Der2) this is the case if and only if ηb is an isomorphismfor every b ∈ B. To reformulate this, let us consider the pasting on the left in

1 id //

id

1 t //

t

B

id

=

1 t //

id

B

id

1b// B

id//

;C

B

<Did

1b// B.

<D

In this pasting the square to the left is given by the unique morphism b → tin B. The reader easily identifies this square as a slice square, which is hencehomotopy exact by (Der4). The functoriality of mates with respect to past-ing (Lemma 8.1.10) shows that ηb : Xb → t∗t

∗(X)b is an isomorphism if andonly if the canonical mate of this pasting is an isomorphism on X. Since theabove pasting trivially agrees with the square to the very right, we deduce thatηb : Xb → t∗t

∗(X)b, b ∈ B, is an isomorphism if and only if Xb → Xt, b ∈ B,is an isomorphism. But, this is easily seen to be equivalent to the fact thatXb → Xb′ is an isomorphism for all b→ b′ in B, i.e., that X is constant.

Example 8.2.24. Let D be a derivator. The functors 0, 1: 1 → [1] induce fullyfaithful Kan extension functors

0! : D(1)→ D([1]) and 1∗ : D(1)→ D([1]).

In both cases, the essential image consists precisely of the coherent morphismsX ∈ D([1]) such that X0 → X1 is an isomorphism. We abuse terminology andrefer to such coherent morphisms as isomorphisms.

This special case as well as Proposition 8.2.23 applied to coherent squareswill have its uses in §9. We conclude this chapter by a short discussion of thecalculus of homotopy exact squares.

Remark 8.2.25.

(i) The calculus of homotopy exact squares provides one of the main technicaltools in the theory of derivators. In a more systematic treatment of thetheory, one might want to begin with a more systematic study of theclass of these squares, including a discussion of formal closure propertiesas well as of plenty of additional examples (see [Ayo07a, Ayo07b, Mal12]and [Gro13, GPS14b, GPS14a, GS14b, GS15b, GS14a]).

DRAFT


In this introductory account instead, besides the homotopy exact squaresconsidered in this chapter, we only have a use for a few additional exam-ples. For pedagogic reasons, these additional homotopy exact squares areonly established as needed (see for example §10).

(ii) By the very definition, the notion of a homotopy exact square (Defini-tion 8.2.4) seems to be specific to the theory of derivators — since suchsquares provide manipulation rules for Kan extensions in arbitrary deriva-tors. However, it turns out that this is only seemingly case. In fact, as

Promise.we indicate in §18, the notion is more fundamental in that it only dependson the classical homotopy theory of topological spaces or simplicial sets.Moreover, it turns out that one obtains the same notion of homotopy ex-act squares if one works with model categories or ∞-categories instead(see §18). We will get back to this in [Gro16b].

(iii) Even in the case of classical category theory, a corresponding calculusof squares encoding manipulation rules for Kan extensions is very con-venient. If in the definition of homotopy exact squares (Definition 8.2.4)one only asks the conclusion to be true for represented derivators, thenone reproduces the classical notion of Guitart exact squares; see forexample [Gui80, Gui14] as well as [Mal12] for additional references.

DRAFT

Chapter 9

Basics on pointed derivators

In this section we introduce pointed derivators as derivators admitting zero ob-jects. Typical examples of pointed derivators are homotopy derivators of abeliancategories, the derivator of pointed topological spaces, and, more generally ho-motopy derivators of pointed model categories or pointed ∞-categories. Weextend a few classical constructions from homological algebra and homotopytheory to the context of an arbitrary pointed derivator and establish some firststeps of the calculus of these constructions.

In §9.1 we define pointed derivators and show that suitable Kan extensionfunctors are ‘extensions by zero’. This yields a convenient tool allowing us to‘add zero objects to diagrams where desired’, and we illustrate this tool in §9.2in the construction of suspension, loop, cofiber, and fiber functors in pointedderivators. In §9.3 we extend a few well-known results concerning pushoutsquares in ordinary categories to cocartesian squares in derivators. This al-lows us in §9.4 to collect a few sample applications and in §9.5 to introducecofiber sequences in pointed derivators. Finally, in §9.6 we characterize pointedderivators as derivators admitting (co)exceptional inverse image functors.

9.1 Definition and first properties

Let us recall from Proposition 7.4.7 that, given a derivator D and a smallcategory A, the category D(A) admits an initial object ∅ and a final object ∗.

Definition 9.1.1. A derivator D is pointed if the underlying category D(1)is pointed, i.e., has a zero object.

Thus, we ask axiomatically that the unique map ∅ → ∗ in D(1) is an iso-morphism. Following standard conventions, any zero object will be denoted by0 ∈ D(1). We again take up the examples from §7.3.

Examples 9.1.2.

(i) A represented derivator y(C) is pointed if and only if the representingcategory C is pointed.

153

DRAFT

154 CHAPTER 9. BASICS ON POINTED DERIVATORS

(ii) Homotopy derivators of Grothendieck abelian categories are pointed, hence,in particular, derivators of fields, rings, and schemes are pointed.

(iii) The derivator HoTop∗ of pointed topological spaces and the derivator Spof spectra are pointed.

(iv) More generally, homotopy derivators of pointed model categories andpointed ∞-categories are pointed.

Lemma 9.1.3.

(i) A derivator D is pointed if and only if Dop is pointed.

(ii) If D is a pointed derivator, then D(A), A ∈ Cat , are pointed categories.

(iii) If D is a pointed derivator and u : A → B, then u∗ : D(B) → D(A) andu!, u∗ : D(A)→ D(B) preserve zero objects.


We know from Proposition 8.2.20 that Kan extensions along fully faithfulfunctors u : A→ B are fully faithful and that the essential images can be char-acterized by the components of the adjunction (co)units lying in B− u(A). Forspecial classes of fully faithful functors these characterizations of the essentialimages admit a particularly simple form.

In fact, we already used a particular instance of such a situation. Recallfrom §6.5 that in sufficiently complete, cocomplete, and pointed categories Cthe cokernel functor can be described as

C[1] j∗→ Cp (ip)!→ C.

Here, the first step is a right Kan extension which precisely amounts to pass-ing to a span by adding a zero object (Lemma 6.5.4). The fact that the firststep precisely adds zero objects is a particular instance of a more general phe-nomenon.

Definition 9.1.4. Let u : A→ B be a fully faithful functor.

(i) The functor u is a cosieve if for every a ∈ A and every morphism u(a)→ bin B it follows that b lies in the image of u.

(ii) Dually, u is a sieve if for every a ∈ A and every morphism b → u(a) inB it follows that b lies in the image of u.

The following proposition will be of constant use in later sections, for exam-ple in the construction of suspensions, cofibers, and cofiber sequences in pointedderivators; see §9.2 and §9.5. To indicate its natural generality, we formulatethe result for arbitrary derivators.

Proposition 9.1.5. Let D be a derivator and let u : A→ B be a functor.

DRAFT

9.1. DEFINITION AND FIRST PROPERTIES 155

(i) If u is a cosieve, then u! : D(A) → D(B) is fully faithful and X ∈ D(B)lies in the essential image of u! if and only if Xb

∼= ∅ for all b ∈ B−u(A).

(ii) If u is a sieve, then u∗ : D(A) → D(B) is fully faithful and X ∈ D(B)lies in the essential image of u∗ if and only if Xb

∼= ∗ for all b ∈ B−u(A).

Proof. We give a proof of the first statement. Proposition 8.2.20 guaranteesthat u! : D(A)→ D(B) is fully faithful and that X ∈ D(B) lies in the essentialimage of u! if and only if the counit εb : u!u

∗(X)b → Xb is an isomorphism forall b ∈ B − u(A). By (Der4) there are canonical isomorphisms

u!u∗(X)b ∼= colim(u/b) p

∗u∗(X), b ∈ B.

We note that if b ∈ B − u(A) and u : A → B a cosieve, then the slice category(u/b) is empty. By (Der1) the category D(∅) is equivalent to the terminalcategory and any object therein is hence a zero object. Since left adjoint functorspreserve initial objects, we deduce that for b ∈ B−u(A) there are isomorphisms

u!u∗(X)b ∼= colim(u/b) p

∗u∗(X)

= colim∅ p∗u∗(X)

= colim∅ 0∼= ∅.

Thus, for b ∈ B − u(A) the map εb : u!u∗(X)b → Xb is an isomorphism if and

only if Xb∼= ∅, concluding the proof.

In the case of pointed derivators the two characterizations of the essentialimages agree. We say that X ∈ D(B) vanishes at an object b ∈ B if Xb

∼= 0,and we similarly speak of diagrams which vanish on a subcategory of B.

Corollary 9.1.6. Let D be a pointed derivator and let u : A→ B be a functor.

(i) If u is a cosieve, then u! : D(A) → D(B) is fully faithful and inducesan equivalence onto the full subcategory of D(B) spanned by all diagramswhich vanish on B − u(A).

(ii) If u is a sieve, then u∗ : D(A) → D(B) is fully faithful and induces anequivalence onto the full subcategory of D(B) spanned by all diagramswhich vanish on B − u(A).

Proof. This is immediate from Proposition 9.1.5.

In the situation of the corollary, we refer to u! as left extension by zeroand to u∗ as right extension by zero. These results already allow us tocarry out some interesting constructions in arbitrary pointed derivators, as weillustrate in §9.2.

DRAFT


9.2 Suspensions, loops, cofibers, and fibers

In this section we define suspensions, loops, cofibers, and fibers for arbitrarypointed derivators, thereby generalizing the classical constructions from homo-logical algebra and homotopy theory. The following definition is motivated byExample 6.5.3. Let us recall that we denote by = [1] × [1] the commutativesquare, i.e., the category

(0, 0) //

=

(1, 0)

(0, 1) // (1, 1),

and by ip : p→ and iy : y → the full subcategories obtained by removingthe final object (1, 1) and the initial object (0, 0), respectively.

Definition 9.2.1. Let D be a derivator.

(i) A (coherent) square in D is an object in D().

(ii) A square X in D is cocartesian if X ∼= (ip)!(Y ) for some Y ∈ D(p).

(iii) A square X in D is cartesian if X ∼= (iy)∗(Y ) for some Y ∈ D(y).

For every derivator D , the category D(p) of (coherent) spans is equivalentto the category of cocartesian squares, and dually for (coherent) cospans andcartesian squares. Duality allows us to focus on cocartesian squares.

Lemma 9.2.2. Let D be a derivator.

(i) A square X ∈ D() is cocartesian if and only if the adjunction counitε : (ip)!i

∗p(X) → X is an isomorphism if and only if the component ε(1,1)

is an isomorphism on X.

(ii) The functor (ip)! : D(p)→ D() is fully faithful and it induces an equiv-alence onto the full subcategory D()cocart ⊆ D() spanned by the co-cartesian squares,

D(p) ' D()cocart.

Proof. Since ip : p→ is fully faithful this follows from Proposition 8.2.20.

Examples 9.2.3.

(i) A square in a represented derivator is cocartesian if and only if it is apushout square in the classical sense (Example 6.5.3).

(ii) A square in the derivator of a Grothendieck abelian category is cocartesianif and only if it is a derived pushout square.

(iii) Similarly, a square in the homotopy derivator of a model category is co-cartesian if and only if it is a homotopy pushout square.

DRAFT

9.2. SUSPENSIONS, LOOPS, COFIBERS, AND FIBERS 157

Let us recall from §3.5 that cones are derived cokernels. Moreover, in §6.5we observed that cokernels can be described as suitable combinations of Kanextensions. We now combine these two results and turn them into a definitionof cones and cofibers in pointed derivators. Related to the span and the cospanwe have the fully faithful functors

i : [1]→ p and k : [1]→y (9.2.4)

classifying the horizontal morphism (0, 0) → (1, 0) and the vertical morphism(1, 0) → (1, 1), respectively. Combining these functors with the fully faithfulinclusions ip : p→ and iy : y→ we obtain the fully faithful functors

i′ = ip i : [1]→ p→ and k′ = iy k : [1]→y→ . (9.2.5)

Definition 9.2.6. Let D be a pointed derivator.

(i) The cofiber functor cof : D([1])→ D([1]) is defined as

cof : D([1])i∗→ D(p)

(ip)!→ D()(k′)∗→ D([1]).

A further evaluation at 1 ∈ [1] yields C = 1∗ cof : D([1])→ D(1).

(ii) The fiber functor fib : D([1])→ D([1]) is defined as

fib : D([1])k!→ D(y)

(iy)∗→ D()(i′)∗→ D([1]).

A final evaluation at 0 ∈ [1] yields F = 0∗ fib : D([1])→ D(1).

Note that i : [1]→ p is a sieve while k : [1]→y is a cosieve (Definition 9.1.4).As a special case of Corollary 9.1.6 it follows that i∗ is right extension by zeroand k! left extension by zero. Given f ∈ D([1]) with underlying diagram x→ ythere are thus a cocartesian square and a cartesian square in D with respectiveunderlying diagrams

xf//

y

cof(f)

Fffib(f)

//

x

f

0 // Cf, 0 // y.

(9.2.7)


(i) A square X ∈ D() is a cofiber square if it is cocartesian and X0,1∼= 0.

We denote by D()cof ⊆ D() the full subcategory spanned by the cofibersquares.

(ii) A square X ∈ D() is a fiber square if it is cartesian and X0,1∼= 0.

We denote by D()fib ⊆ D() the full subcategory spanned by the fibersquares.

DRAFT


Lemma 9.2.9. Let D be a pointed derivator.

(i) The functor i∗ : D([1]) → D(p) is fully faithful and it induces an equiva-lence onto the full subcategory D(p)ex ⊆ D(p) spanned by all X such thatX0,1

∼= 0,

D([1]) ' D(p)ex.

(ii) The functors i∗ : D([1])→ D(p) and (ip)! : D(p)→ D() induce an equiv-alence of categories

D([1]) ' D()cof .

Proof. Since the functors i : [1] → p and ip : p→ are fully faithful, the sameis true for i∗ and (ip)! (Proposition 8.2.16). Note that i is a sieve, so thati∗ : D([1]) → D(p) is right extension by zero (Corollary 9.1.6), thereby estab-lishing (i). By Lemma 9.2.2 the functor (ip)! shows that D(p) ' D()cocart.Moreover, the fully faithfulness of (ip)! implies that the equivalence restrictsto an equivalence D(p)ex ' D()cof . Together with (i) we obtain the desiredequivalence

D([1]) ' D(p)ex ' D()cof ,

thereby establishing (ii).

Thus, using coherent formulations, a morphism is simply ‘as good as a(co)fiber square’.

Examples 9.2.10.

(i) In the case of a represented derivator, the constructions in Lemma 9.2.9yield a refinement of Examples 6.5.5.

(ii) In the derivator of a Grothendieck abelian category, these constructionsrecover the derived cokernel construction (Theorem 3.5.6).

(iii) In the derivator of pointed topological spaces, these constructions special-ize to the classical (mapping) cocone construction.

While the above definitions (see Definition 9.2.6) make perfectly well sensefor arbitrary derivators, we need the pointedness assumption to obtain the fol-lowing result.

Proposition 9.2.11. For every pointed derivator D there is an adjunction

(cof, fib) : D([1]) D([1]).

Proof. By definition, the functors cof and fib respectively factor as indicated in

D([1])i∗ // D(p)

(ip)!//

i∗oo D()

(iy)∗//

(ip)∗

oo D(y)k∗ //

(iy)∗

oo D([1]);k∗

oo

DRAFT


see (9.2.4) and (9.2.5). Thus, there are four adjunctions of which the outer ones‘point in the bad direction’. In order to deal with this issue, let us denote by

D(p)ex ⊆ D(p), D(y)ex ⊆ D(y), and D()ex ⊆ D()

the respective full subcategories spanned by the coherent diagrams vanishing atthe lower left corner (0, 1). As a special case of Corollary 9.1.6 we obtain adjointequivalences of categories

(i∗, i∗) : D([1]) ' D(p)ex and (k∗, k!) : D(y)ex ' D([1]).

Moreover, the fully faithfulness of Kan extensions along fully faithful functors(Proposition 8.2.16) implies that all four functors (ip)!, (ip)

∗ and (iy)∗, (iy)∗

preserve the vanishing condition at the lower left corner (0, 1). As an upshot,the functors cof and fib respectively factor as

D([1])i∗ // D(p)ex

(ip)!//

i∗oo D()ex

(iy)∗//

(ip)∗

oo D(y)exk∗ //

(iy)∗

oo D([1]).k∗

oo

Thus, (cof, fib) is obtained by composing four adjunctions (two of which actuallyare adjoint equivalences), concluding the proof.

In a similar way we can define suspension and loop functors in pointedderivators. Naively, for a pointed derivator D and an object x ∈ D(1) we wouldlike to set

Σx = C(x→ 0) and Ωx = F (0→ x).

This can be made precise using the functors

0∗ : D(1)→ D([1]) and 1! : D(1)→ D([1]).

In fact, since 0: 1 → [1] is a sieve and 1: 1 → [1] is a cosieve, the above twoKan extension functors are extensions by zero (Corollary 9.1.6).

To get a bit more used to the calculus of Kan extensions, we include adetailed, equivalent construction of these functors. Let us consider the fullyfaithful functors

i = (0, 0) : 1→ p and k = (1, 1) : 1→y (9.2.12)

classifying the initial and the final object, respectively. Postcomposition withthe functors ip : p→ and iy : y→ yields the fully faithful functors

i′ = ip i : 1→ p→ and k′ = iy k : 1→y→ . (9.2.13)


(i) The suspension functor Σ: D(1)→ D(1) is defined as

Σ: D(1)i∗→ D(p)

(ip)!→ D()(k′)∗→ D(1).

DRAFT


(ii) The loop functor Ω: D(1)→ D(1) is defined as

Ω: D(1)k!→ D(y)

(iy)∗→ D()(i′)∗→ D(1).

We note that i : 1 → p is a sieve and that k : 1 →y is a cosieve. Hence, byCorollary 9.1.6, i∗ is right extension by zero and k! left extension by zero. As asummary of these constructions, for x ∈ D(1) there is a cocartesian square anda cartesian square in D with respective underlying diagrams

x //

0

Ωx //

0

0 // Σx, 0 // x.

(9.2.15)

Remark 9.2.16. Note that Σ and Ω are trivial in the represented case. In fact,let y(C) be a pointed, represented derivator so that, by Examples 9.2.3, thesquares in (9.2.15) are ordinary pushout and pullback squares in C, respectively.Obviously, in that case, Σx and Ωx are always zero objects.

However, in the pointed derivator of an abelian category the constructionrecovers the typical shift operation on chain complexes. Similarly, in the deriva-tor of pointed topological spaces this abstractly defined suspension recovers theclassical (reduced) suspension.


(i) A square X ∈ D() is a suspension square if it is cocartesian and ifX1,0

∼= X0,1∼= 0. We denote by D()susp ⊆ D() the full subcategory

spanned by the suspension squares.

(ii) A square X ∈ D() is a loop square if it is cartesian and if X1,0∼=

X0,1∼= 0. We denote by D()loop ⊆ D() the full subcategory spanned

by the loop squares.

Lemma 9.2.18. Let D be a pointed derivator and let i = (0, 0) : 1→ p.

(i) The functor i∗ : D(1)→ D(p) is fully faithful and it induces an equivalenceonto the full subcategory spanned by all diagrams vanishing at (1, 0) and(0, 1).

(ii) The functors i∗ : D(1)→ D(p) and (ip)! : D(p)→ D() induce an equiv-alence

D(1) ' D()susp.


Proposition 9.2.19. For every pointed derivator D there is an adjunction

(Σ,Ω): D(1) D(1).

DRAFT


Proof. It follows from the definition as well as from (9.2.12) and (9.2.13) thatthe functors Σ and Ω respectively factor as

D(1)i∗ // D(p)

(ip)!//

i∗oo D()

(iy)∗//

(ip)∗

oo D(y)k∗ //

(iy)∗

oo D(1).k∗

oo

Again, there are four adjunctions of which the outer ones ‘point in the baddirection’. We denote by


the respective full subcategories spanned by the coherent diagrams vanishing at(0, 1) and (1, 0). By Lemma 9.2.18 and its dual, there are adjoint equivalences

(i∗, i∗) : D(1) ' D(p)ex and (k∗, k!) : D(y)ex ' D(1).

Since the functors (ip)!, (ip)∗ and (iy)

∗, (iy)∗ preserve the vanishing conditionsat the corners (0, 1), (1, 0), the functors Σ and Ω respectively factor as

D(1)i∗ // D(p)ex

(ip)!//

i∗oo D()ex

(iy)∗//

(ip)∗

oo D(y)exk∗ //

(iy)∗

oo D(1).k∗

oo

As a composition of two adjoint equivalences and two adjunctions, we concludethat (Σ,Ω) is an adjunction.

For later reference we collect the following construction.

Construction 9.2.20. Let D be a pointed derivator and let X ∈ D() suchthat X1,0

∼= X0,1∼= 0. If i : 1 → p is the sieve classifying the initial object

(0, 0), then (ip)∗X lies in the essential image of i∗ : D(1) → D(p), i.e., the

unit η : (ip)∗X → i∗i

∗(ip)∗X is an isomorphism. This allows us to consider the

following morphism of squares

(ip)!i∗X0,0 = (ip)!i∗i∗(ip)

∗Xη−1

→ (ip)!(ip)∗X

ε→ X.

Since Σ = (1, 1)∗(ip)!i∗ : D(1)→ D(1), an evaluation of the above morphism ofsquares at (1, 1) yields the canonical morphism

φX : Σ(X0,0)→ X1,1. (9.2.21)

Lemma 9.2.22. Let D be a pointed derivator and let X ∈ D() such thatX1,0

∼= X0,1∼= 0. The square X is a suspension square if and only if the

canonical map φX : Σ(X0,0)→ X1,1 (9.2.21) is an isomorphism.

Proof. By definition of suspension squares (Definition 9.2.17) and our assump-tion on X, it is obvious that X is a suspension square if and only if it iscocartesian if and only if ε : (ip)!(ip)

∗(X) → X is an isomorphism at (1, 1)(Lemma 9.2.2). It follows from Construction 9.2.20 that this is the case if andonly if (9.2.21) is an isomorphism.

DRAFT


In a similar way we obtain a corresponding result for cofiber squares.

Construction 9.2.23. Let D be a pointed derivator and let X ∈ D() suchthat X0,1

∼= 0. If i : [1] → p is the sieve classifying the horizontal morphism(0, 0) → (1, 0), then (ip)

∗X lies in the essential image of i∗ : D([1]) → D(p)and the unit η : (ip)

∗X → i∗i∗(ip)

∗X is hence an isomorphism. Passing to theinverse of the unit, we obtain morphisms of squares

(ip)!i∗i∗(ip)

∗Xη−1

→ (ip)!(ip)∗X

ε→ X.

Using the defining equation C = (1, 1)∗(ip)!i∗ : D([1])→ D(1), an evaluation ofthe above moprhisms at (1, 1) yields the canonical morphism

φ : C(i∗X)→ X1,1.

Lemma 9.2.24. Let D be a pointed derivator and let X ∈ D() such thatX0,1

∼= 0. The square X is a cofiber square if and only if the canonical mapφ : Ci∗X → X1,1 (9.2.21) is an isomorphism.


Our next goal is to define and study cofiber sequences in pointed derivators.For this purpose, however, we first need some basic results on cartesian squaresand cocartesian squares, and these will be established in §9.3.

9.3 Cartesian and cocartesian squares

In this section we extend a few well-known results concerning pushout andpullback squares in ordinary categories to cartesian and cocartesian squares inarbitrary derivators. This includes results concerning constant squares as wellas a glueing and cancellation property for (co)cartesian squares.

We begin by establishing some terminology. Let D be a derivator and let usconsider a square X ∈ D() with underlying diagram

X(0,0)//

X(1,0)

X(0,1)// X(1,1).

Definition 9.3.1. Let D be a derivator.

(i) A square X ∈ D() is vertically constant if X(0,0) → X(0,1) andX(1,0) → X(1,1) are isomorphisms.

(ii) A square X ∈ D() is horizontally constant if X(0,0) → X(1,0) andX(0,1) → X(1,1) are isomorphisms.

(iii) A square X ∈ D() is constant if it is vertically constant and horizon-tally constant.

DRAFT

9.3. CARTESIAN AND COCARTESIAN SQUARES 163

We now consider the horizontal inclusion (id[1] × 0) : [1]→ classifying themorphism (0, 0)→ (1, 0). This inclusion factors as

(id[1] × 0) = ip i : [1]→ p→ . (9.3.2)

Lemma 9.3.3. Let D be a derivator. The functor i! : D([1]) → D(p) is fullyfaithful and induces an equivalence onto the full subcategory of D(p) spanned bythose X such that X(0,0) → X(0,1) is an isomorphism.

Proof. By Proposition 8.2.20 the functor i! is fully faithful with essential imagethose X such that ε : i!i

∗(X) → X is an isomorphism at (0, 1). To reformulatethis let us consider the pasting on the left

1∼= //

id

id

(i/(0, 1)) //

[1]

id

i //

i

p

id

=

1(0,0)

//

id

z

p

id

1id

// 1(0,1)

// pid// p 1

(0,1)// p,

in which the square in the middle is a slice square and the square to the left isgiven by an isomorphism. Since these two squares are homotopy exact (Exam-ples 8.2.14), the functoriality of canonical mates with pasting implies that ε(0,1)

is an isomorphism on X if and only if the canonical mate associated to thispasting is an isomorphism at X. Since this pasting agrees with the square tothe right, given by the morphism (0, 0)→ (0, 1) in p, the canonical mate (8.2.2)associated to this square is X(0,0) → X(0,1), concluding the proof.

Corollary 9.3.4. Let D be a derivator.

(i) The functor (id[1] × 0)! : D([1]) → D() is fully faithful and induces anequivalence onto the full subcategory of D() spanned by the verticallyconstant squares.

(ii) The functor (0, 0)! : D(1)→ D() is fully faithful and induces an equiva-lence onto the full subcategory of D() spanned by the constant squares.

Proof. Since id[1] × 0 is fully faithful, so is (id[1] × 0)! and X ∈ D() liesin the essential image if and only if ε : (id[1] × 0)!(id[1] × 0)∗X → X is anisomorphism at (0, 1), (1, 1) (Proposition 8.2.20). The proof of Lemma 9.3.3shows that ε(0,1) is an isomorphism if and only if X0,0 → X0,1 is an isomorphism.Using essentially the same proof, we leave it to the reader to show that ε(1,1) isan isomorphism if and only if X1,0 → X1,1 is an isomorphism. These two factssettle the first statement. The second statement is a special case of (the dualof) Proposition 8.2.23.

The relation of suitably constant squares to (co)cartesian squares is as fol-lows.

DRAFT


Proposition 9.3.5. Let D be a derivator and let X be a square in D such thatX(0,0) → X(0,1) is an isomorphism. The square X is cocartesian if and onlyif X(1,0) → X(1,1) is an isomorphism, i.e., the square is vertically constant. Inparticular, constant squares are cocartesian.

Proof. The inclusion id[1] × 0: [1] → factors as in (9.3.2). Hence, there is acanonical isomorphism (id[1] × 0)!

∼= (ip)! i!, and, as a consequence of Corol-lary 9.3.4, vertically constant squares are thus cocartesian. Conversely, if Xis a cocartesian square such that X(0,0) → X(0,1) is an isomorphism, then, byLemma 9.3.3, X lies in the essential image of (ip)! i! ∼= (id[1] × 0)! and is byCorollary 9.3.4 vertically constant.

One implication of the proposition is a derivator version of the classical factthat isomorphisms are stable under cobase change.

We now turn to the composition and cancellation property of cocartesiansquares.

Notation 9.3.6. Let [2] be the poset [2] = (0 < 1 < 2) considered as acategory, i.e., the category which corepresents pairs of composable morphisms.For 0 ≤ i ≤ j ≤ 2 there is the functor ιi,j : [1] → [2] which sends 0 to i and1 to j. The product category [2] × [1] will be denoted by . Given an objectX ∈ D(),

X(0,0)//

X(1,0)//

X(2,0)

X(0,1)// X(1,1)

// X(2,1),

by restriction we obtain the left square ι∗01X ∈ D(), the right square ι∗12X ∈D(), and the composite square ι∗02X ∈ D(). (Here we abused notation andsimply wrote ιi,j instead of ιi,j × id[1].)

As a preparation for Proposition 9.3.10 we establish Lemma 9.3.8. We in-clude rather detailed proofs of the lemma and the proposition in order to illus-trate some typical argumentations based on the calculus of mates and homotopyexact squares. In later cases of such standard arguments we allow ourselves tobe more sketchy and to leave some details to the reader (as we already do inthe final step of the proof of Proposition 9.3.10).

Let A ⊆ be the full subcategory obtained by removing (1, 1), (2, 1) andlet B ⊆ be the full subcategory obtained by removing the object (2, 1) only.Related to these subcategories there are fully faithful inclusion functors

k = j i : A i→ Bj→ . (9.3.7)

Lemma 9.3.8. Let D be a derivator.

(i) The functor i! : D(A)→ D(B) is fully faithful and induces an equivalenceonto the full subcategory spanned by all X ∈ D(B) such that ι∗01(X) ∈D() is cocartesian.

DRAFT


(ii) The functor j! : D(B)→ D() is fully faithful and induces an equivalenceonto the full subcategory spanned by all X ∈ D() such that ι∗12(X) ∈D() is cocartesian.

(iii) The functor k! : D(A)→ D() is fully faithful and induces an equivalenceonto the full subcategory spanned by all diagrams X ∈ D() such that bothsquares ι∗01X, ι

∗12X are cocartesian.

Proof. We begin with statement (i). Since i : A→ B is fully faithful, i! is fullyfaithful and X ∈ D(B) lies in the essential image if and only if the counitε : i!i

∗(X) → X is an isomorphism at (1, 1) (Proposition 8.2.20). Using thehomotopy exactness of slice squares and the functoriality of mates with respectto pasting, this is the case if and only if the canonical mate associated to thepasting on the left in

pι01 //

π

A

| id

i //

i

B

id

=

pid //

π

p

| id

ip //

ip

| idid

ι01 // B

id

1(1,1)

// Bid// B 1

(1,1)//

id//

ι01

// B

is an isomorphism on X. In the above pasting on the right, the square to theleft is a slice square. Since the above two pastings coincide, using again thefunctoriality of mates with pasting, the homotopy exactness of slice squares,and Lemma 9.2.2, we deduce that X lies in the essential image of i! if and onlyif ι∗01(X) is cocartesian.

As for statement (ii), we note that the essential image of j! : D(A)→ D(B)consists precisely of those X ∈ D(B) such that the counit ε : j!j

∗(X)→ X is anisomorphism at (2, 1) (Proposition 8.2.16). To re-express this differently let usconsider the pasting on the left in

p

id

ι12 //

π

B

π

id //

B

id

j//

j

id

=

pid //

π

p

id

ip //

ip

| idid

ι12 //

id

1id// 1

(2,1)//

id// 1

(1,1)//

id//

ι12

// ,

and for now let us focus on the two squares to the right. Proposition 8.2.16, thehomotopy exactness of slice squares, and the compatibility of mates with pastingimply that X lies in the essential image of j! if and only if the canonical mateof the pasting of those two squares is an isomorphism on X. We next observethat the functor ι01 : p→ B is a right adjoint. In fact, the functor s0 : [2]→ [1]determined by 0, 1 7→ 0 and 2 7→ 1 induces a left adjoint to ι01 : p→ B. Hence, byProposition 8.2.8 the corresponding square is homotopy exact and we concludethat X lies in the essential image of j! if and only if the canonical mate of thepasting to the left is an isomorphism on X. Since that pasting can be rewrittenas the above pasting on the right, we can now conclude as in the case of (ii).

One easily combines these two statements to deduce (iii).

DRAFT


Remark 9.3.9. Some readers certainly judge that this proof of Lemma 9.3.8 israther lengthy. While here we include on purpose an elementary proof, in §11.3we develop techniques which allow us to ‘make such proofs more mechanical’.

The composition and cancelation property of cocartesian squares is as fol-lows.

Proposition 9.3.10. Let D be a derivator and let X ∈ D() such that ι∗01Xis cocartesian. Then ι∗12X is cocartesian if and only if ι∗02X is.

Proof. We consider X ∈ D() such that the square ι∗01(X) is cocartesian. ByLemma 9.3.8 and Proposition 8.2.20 for such a diagram X also the square ι∗12(X)is cocartesian if and only if X lies in the essential image of k! : D(A) → D()if and only if the counit ε : k!k

∗(X) → X is an isomorphism at (2, 1). Inmore detail, by Proposition 8.2.20 the essential image of k! is characterized byε : k!k

∗ → 1 being an isomorphism at (1, 1) and (2, 1). Under our assumptionthe component ε1,1 is always an isomorphism as the counit factors as

k!k∗(X) ∼= j!i!i

∗j∗(X)ε→ j!j

∗(X)ε→ X

and the component at (1, 1) of the first morphism is an isomorphism by ourassumption on X (use Proposition 8.2.16 and Lemma 9.3.8) while the corre-sponding component of the second morphism is always an isomorphism (Propo-sition 8.2.20).

To reformulate the fact that ε2,1 is an isomorphism on X we consider thepasting on the left in

p

id

ι02 //

π

A

π

id //

A

id

k //

k

id

=

pid //

π

p

id

ip //

ip

id

ι02 //

| id

id

1id// 1

(2,1)//

id// 1

(1,1)//

id//

ι02

// .

Here, the functor ι02 : p→ A is a right adjoint so that the square to the veryleft is homotopy exact by Proposition 8.2.8. In fact, a left adjoint is induced bythe functor s1 : [2]→ [1] determined by 0 7→ 0 and 1, 2 7→ 1. Together with thehomotopy exactness of slice squares and the functoriality of mates with pastingthis shows that for our diagram X ∈ D() the square ι∗12(X) is cocartesian ifand only if the canonical mate of the pasting on the left is an isomorphism on X.In the pasting on the right, the square to the very left is a slice square and onenotes that the above two pasting coincide. Using standard arguments we thusconclude that ι∗12(X) is cocartesian if and only if ι∗02(X) is cocartesian.

Remark 9.3.11. For represented derivators this recovers the classical composi-tion and cancelation property for pushout squares in ordinary categories. Bymeans of Examples 9.2.3 this proposition establishes a corresponding result forderived pushout squares and homotopy pushout squares.

DRAFT


In §9.5 we apply this composition property to twofold iterations of cofiberconstructions, which is to say (coherent) cofiber sequences. We conclude thissection by showing how twofold coproducts can be described by means of co-cartesian squares. More precisely, we generalize to derivators the following triv-ial result from ordinary category theory. Given a pushout square

∅ //

Y

X // W

in a cocomplete category such that the upper left corner is populated by aninitial object, the object W is isomorphic to the coproduct XtY , more precisely,the cospan X →W ← Y is a coproduct cocone for the pair (X,Y ).

To extend this to derivators let us consider the functor

k = ((1, 0), (0, 1)) : 1 t 1→ (9.3.12)

which factors as compositions of fully faithful functors

1 t 1i→ p ip→ and 1 t 1

j→y iy→ . (9.3.13)

Definition 9.3.14. Let D be a derivator. A coherent cospan X ∈ D(y) is acoproduct cocone if it lies in the essential image of j! : D(1 t 1)→ D(y).

Notation 9.3.15. Let D be a derivator. We denote by D()copr ⊆ D() thefull subcategory spanned by the cocartesian squares X such that X(0,0)

∼= ∅,and refer to any object in D()copr as a coproduct square.

The justification for this notation is provided by the following lemma.

Lemma 9.3.16. For every derivator D the left Kan extension along k (9.3.12)induces an equivalence D(1 t 1) ' D()copr. Moreover, a square X lies inD()copr if and only if X(0,0)

∼= ∅ and if the restriction i∗y(X) ∈ D(y) is acoproduct cocone.

Proof. The functor (9.3.12) is fully faithful hence so is k! : D(1 t 1) → D()by Proposition 8.2.16. Since k factors as indicated in (9.3.13), there are naturalisomorphisms

k!∼= (ip)!i! ∼= (iy)!j!.

All of these functors are fully faithful and these factorizations yield two differentdescriptions of the essential image. Using the natural isomorphism k!

∼= (ip)!i!we see that X ∈ D() lies in the essential image of k! if and only if X iscocartesian and X(0,0)

∼= ∅. In fact, since i is a cosieve this follows from Propo-sition 9.1.5 and Proposition 8.2.16. Similarly, using the isomorphism k!

∼= (iy)!j!and the fact that iy is a cosieve it follows from the same two propositions thatthe essential image of k! consists precisely of those X with X(0,0)

∼= ∅ and suchthat (iy)

∗(X) is a coproduct cocone.

DRAFT


9.4 First applications of cocartesian squares

In this section we collect a few sample applications of the calculus of cocartesiansquares. As a first application of Proposition 9.3.5 we obtain a derivator versionof the classical fact that the cone of a quasi-isomorphism or weak equivalenceis acyclic.

Corollary 9.4.1. Let D be a pointed derivator and let f ∈ D([1]). If f is anisomorphism then Cf ∼= 0 and Ff ∼= 0.

Proof. By duality it is enough to take care of the cone Cf . However, in this casethere is a defining pushout square as in (9.2.7). Since x→ y is an isomorphism,so is 0→ Cf by Proposition 9.3.5.

Here is a second application of Proposition 9.3.5, which allows us to describeisomorphisms as cofibers and fibers (see again Example 8.2.24).

Proposition 9.4.2. Let D be a pointed derivator and let x ∈ D(1).

(i) There are natural isomorphisms cof(1!x) ∼= 0!(x) ∼= 1∗(x). In particular,there is a natural isomorphism C(1!x) ∼= x.

(ii) There are natural isomorphisms fib(0∗x) ∼= 0!(x) ∼= 1∗(x). In particular,there is a natural isomorphism F (0!x) ∼= x.

Proof. By duality it suffices to prove the first statement. The functor 1: 1→ [1]is a cosieve and 1! : D(1) → D([1]) is hence left extension by zero (Corol-lary 9.1.6). If we pass from 1!x ∈ D([1]) to the corresponding cofiber square,then we obtain a cocartesian square

0 //

x

0 // C(1!x).

Since the left vertical map is clearly an isomorphism, the same is true for thevertical map on the right (Proposition 9.3.5). The first part of the statementhence follows from Example 8.2.24. The second part is immediate from thiscofiber square and the fully faithfulness of Kan extensions along fully faithfulfunctors (Proposition 8.2.16).

As a third application of Proposition 9.3.5 we establish the following ob-servation. In the proof of that result we consider the functors ι0, ι1 : [1] → classifying the vertical morphisms (0, 0) → (0, 1) and (1, 0) → (1, 1), respec-tively. The horizontal morphisms in define a unique natural transformationα : ι0 → ι1. Hence, in every prederivator D we obtain an induced transformation

α∗ : ι∗0 → ι∗1 : D()→ D([1]). (9.4.3)

(This is a special case of the construction of partial underlying diagram functors;see Lemma 7.4.5.)

DRAFT

9.4. FIRST APPLICATIONS OF COCARTESIAN SQUARES 169

Proposition 9.4.4. Let D be a pointed derivator.

(i) There is a natural isomorphism 0∗ ∼= cof 0! : D(1) → D([1]) and hencean adjunction

(0∗, F ) : D(1) D([1]).

(ii) There is a natural isomorphism fib 1∗ ∼= 1! : D(1) → D([1]) and hencean adjunction

(C, 1!) : D([1]) D(1).

Proof. Since 0 ∈ [1] is initial, the left Kan extension 0! : D(1)→ D([1]) is fullyfaithful with essential image the isomorphisms (Example 8.2.24). It follows fromthe construction of cof : D([1])→ D([1]) that for x ∈ D(1) there is a cocartesiansquare Q ∈ D() looking like

x0!x

∼=//

x′

cof(0!x)

0 ∼=// 0,

(9.4.5)

in which the top horizontal morphism is 0!x ∈ D([1]). In fact, by Proposi-tion 9.3.5 also the bottom horizontal morphism is an isomorphism so that theremaining corner is populated by a zero object. Since the left vertical morphismι∗0(Q) vanishes at its target and since 0: 1→ [1] is a sieve, there is a canonicalisomorphism ι∗0(Q) ∼= 0∗x (Corollary 9.1.6). Hence, combined with the natu-ral transformation (9.4.3) applied to the horizontally constant square (9.4.5),we obtain the desired natural isomorphism 0∗ ∼= cof 0!. Finally, the relationF = 0∗ fib and Proposition 9.2.11 imply that F is right adjoint to 0∗. Theproof of (ii) is dual.

Let us put the first part of the first statement in words. Given a pointedderivator D and x ∈ D(1), the cofiber of an isomorphism (x ∼−→ x′) ∈ D([1]) isnaturally isomorphic to (x→ 0) ∈ D([1]).

Remark 9.4.6. Let D be a pointed derivator.

(i) The adjunction (0∗, F ) : D(1) D([1]) says that for every x ∈ D(1) and(f : y → z) ∈ D([1]) there is a natural bijection between morphisms

(x0∗x //

0)

x

(yf// z), Ff,

which reminds us of the universal property of a kernel. This correspon-dence can also be expressed ‘in a more coherent fashion’.

DRAFT


(ii) We just observed that the right Kan extension functor 0∗ : D(1)→ D([1])has an additional right adjoint, given by F : D([1]) → D(1). This is aspecial case of the more general result that in a pointed derivator D rightKan extension functors along sieves have an additional right adjoint, anddually. These adjoints are referred to as (co)exceptional inverse imagefunctors, and we come back to these functors in §9.6.

Finally, as a first application of Proposition 9.3.10, we establish a derivatorversion of the following result from topology. Given a null-homotopic mapf : X → Y of pointed spaces, then the (homotopy) fiber Ff is weakly equivalentto X × ΩY . To extend this to pointed derivators D , given a coherent pair ofcomposable morphisms

(x→ 0→ y) ∈ D([2])

which vanishes in the middle we simply write (x0→ y) ∈ D([1]) for the restriction

along d1 = ι02 : [1]→ [2].

Proposition 9.4.7. Let D be a pointed derivator and (x → 0 → y) ∈ D([2]).There is a canonical isomorphism

F (x0→ y) ∼= x× Ωy.

Proof. Let u : [2] → classify the morphisms (0, 1) → (1, 1) → (2, 1). Then ufactors as a composition of fully faithful functors

u = j i : [2]→ A→

where A ⊆ is the full subcategory obtained from the image of u by addingthe object (2, 0). The corresponding Kan extension functors

D([2])i!→ D(A)

j∗→ D()

are fully faithful (Proposition 8.2.16). Since i is a cosieve, i! is left extension byzero (Corollary 9.1.6). Moreover, the dual of Lemma 9.3.8 says that j∗ preciselyforms two cartesian squares. As an upshot, the above Kan extensions send(x→ 0→ y) ∈ D([2]) to a coherent diagram Q ∈ D() looking like

Q0,0//

Q1,0//

0

x // 0 // y.

The square on the right is a loop square and hence yields a canonical isomor-phism Q1,0

∼= Ωy. Hence, by Lemma 9.3.16, the square on the left impliesthat Q0,0

∼= x × Ωy. By Proposition 9.3.10 also the composite square ι∗02(Q)is cartesian, and by construction of the functor F : D([1])→ D(1) (see (9.2.7))

there is hence a canonical isomorphism Q0,0∼= F (x

0→ y). Combining these

isomorphisms, we obtain the intended isomorphism F (x0→ y) ∼= x× Ωy.

DRAFT

9.5. FIBER AND COFIBER SEQUENCES 171

9.5 Fiber and cofiber sequences

In this section we define coherent and incoherent cofiber sequences in pointedderivators and we show that coherent morphisms in pointed derivator give riseto such cofiber sequence. In the stable case, these incoherent cofiber sequencesdefine canonical triangulations (as we discuss in §15).

Definition 9.5.1. A cofiber sequence in a pointed derivator D is a diagramX ∈ D() such that the following two conditions are satisfied.

(i) The squares ι∗01(X), ι∗12(X) ∈ D() are cocartesian.

(ii) The diagram vanishes at (2, 0) and (0, 1), X2,0∼= X0,1

∼= 0.

We denote by D()cof ⊆ D() the full subcategory spanned by the cofibersequences.

Thus, given a cofiber sequence X, we obtain an underlying incoherent dia-gram dia(X) : → D(1) looking like

xf//

y //

g

0

0 // z

h// w.

(9.5.2)

By Proposition 9.3.10 also the composite square ι∗02(X) is cocartesian and it hasunderlying diagram

x //

0

0 // w.

Thus, ι∗02(X) is a suspension square (Definition 9.2.17), and by Lemma 9.2.22there is hence a canonical isomorphism

φ : Σx ∼−→ w. (9.5.3)

Construction 9.5.4. Let [3] be the poset (0 < 1 < 2 < 3) and let k : [3]→ bethe ‘staircase embedding‘ pointing at the objects (0, 0) < (1, 0) < (1, 1) < (2, 1).Given a cofiber sequence X ∈ D()cof , we can pass to the underlying diagram ofk∗(X) ∈ D([3]) which is an ordinary incoherent diagram [3]→ D(1). Combiningthis diagram with the canonical isomorphism (9.5.3) we obtain, as an upshot,for every cofiber sequence X as in (9.5.2) an underlying incoherent cofibersequence

xf

−→ yg

−→ zφ−1X h

−→ Σx, (9.5.5)

which is an ordinary diagram [3]→ D(1). Clearly, this construction is functorialin X ∈ D()cof , thereby defining a functor

D()cof → D(1)[3].

DRAFT


Remark 9.5.6. In Construction 9.5.4 we applied the underlying diagram functordia[3] : D([3]) → D(1)[3] and this step results in a loss of information. Theseincoherent cofiber sequences are some non-canonical shadows of certain universalconstructions which were applied to coherent morphisms. In particular, givenan incoherent cofiber sequence (9.5.5) it is true that the compositions x → zand y → Σx are zero morphisms. But at that level we do not remember thereason why these compositions are zero, namely that they belong to certaincofiber squares in the sense of Definition 9.2.8; see (9.5.2).

As a related remark, in general, it is not possible to canonically reconstructa coherent version of the morphism f : x → y starting with the given inco-herent cofiber sequence. This is to be seen in contrast to the following result(Proposition 9.5.8); for an expanded discussion of this see §16.

We now show that every coherent morphism can be extended to a cofibersequence. For this purpose, as in Lemma 9.3.8, let A ⊆ denote the fullsubcategory obtained by removing the objects (1, 1), (2, 1) and let k : A → be the corresponding inclusion. Moreover, let i : [1]→ A classify the horizontalmorphism (0, 0)→ (1, 0). For every derivator D there are related Kan extensionfunctors

D([1])i∗→ D(A)

k!→ D(). (9.5.7)

Proposition 9.5.8. For every pointed derivator D the functors (9.5.7) inducean equivalence of categories

D([1]) ' D()cof .

Proof. The functors i : [1] → A and k : A → are both fully faithful andhence the same is true for the Kan extension functors i∗ : D([1]) → D(A),k! : D(A) → D(). Both functors restrict to equivalences onto their essentialimages and we deal with both cases individually.

Since i is a sieve, i∗ is right extension by zero (Corollary 9.1.6) with essentialimage D(A)ex ⊆ D(A) spanned by all X ∈ D(A) which vanish at (2, 0) and(0, 1). As for k! : D(A) → D() we already know by Lemma 9.3.8 that theessential image of k! consists precisely of those X ∈ D() such that both squaresι∗01(X), ι∗12(X) are cocartesian. Since k! is a fully faithful Kan extension functorit respects the vanishing conditions at (2, 0), (0, 1) ∈ A, and it restricts to anequivalence k! : D(A)ex ' D()cof .

As an upshot, we see that i∗ and k! respectively restrict to equivalences

D([1]) ' D(A)ex ' D()cof ,

concluding the proof.

Thus, as a variant of Lemma 9.2.9, if we stay at the level of coherent dia-grams, then a morphism in a pointed derivator is simply as good as a cofibersequence.

Remark 9.5.9. Dualization yields a similar notion of fiber sequences. Given apointed derivator D , a diagram X ∈ D() is a fiber sequence if the squares

DRAFT

9.5. FIBER AND COFIBER SEQUENCES 173

ι∗01(X), ι∗12(X) are cartesian and X vanishes at (2, 0), (0, 1), X2,0∼= X0,1

∼= 0.There is an equivalence of categories D([1]) ' D()fib.

As a special case we obtain split cofiber sequences as follows.

Example 9.5.10. Let D be a pointed derivator and let x, y ∈ D(1). Startingwith a corresponding coproduct square X ∈ D()copr (Notation 9.3.15 andLemma 9.3.16), we construct a coherent square looking like

0 //

y

x //

x t y //

0

0 // y0// x′

and making all squares cocartesian.In fact, let B ⊆ [2] × [2] be the full subcategory obtained by removing the

upper right corner (2, 0). The inclusion i : → [2] × [2] classifying the upperleft square factors as

i = i2 i1 : → B1 → B,

where B1 is obtained from the image of i by adding the objects (2, 1) and (0, 2).The functor i1 is a sieve so that (i1)∗ is right extension by zero (Corollary 9.1.6).Using obvious homotopy finality arguments the reader verifies that (i2)! amountsprecisely to forming two cocartesian squares. Thus, starting with a coproductsquare associated to the objects x, y ∈ D(1), the functors

D()(i1)∗→ D(B1)

(i2)!→ D(B)

construct a coherent diagram which looks objectswise as intended. Since thecomposition of the vertical morphism on the left is an isomorphism, it followsfrom Proposition 9.3.5 and Proposition 9.3.10 that also (y → y) ∈ D([1]) is anisomorphism. Moreover, since in the underlying category the following compo-sitions agree

(y ∼−→ y → x′) = (y → x t y → 0→ x′),

we conclude that the morphism y → x′x is trivial. If we restrict the abovediagram to the two bottom cocartesian squares, then we obtain the split cofibersequence associated to x, y ∈ D(1). The corresponding incoherent cofibersequence can be identified with

x→ x t y → y0→ Σx.

In the case of stable derivators we will see in §15.5 that the incoherentcofiber sequences from Construction 9.5.4 define triangulations. In particular,the rotation axiom hence has to be verified. While two iterations of the cofiber

DRAFT


construction were enough to define cofiber sequences, for the rotation axiomwe need a third iteration. Given a coherent morphism f : x → y in a pointedderivator, as we discuss in detail in §12.1, one can construct a coherent diagramlooking like

xf//

1

y //

g

2

0

0 // zh //

3

x′

f ′

0 // y′.

In this diagram, there are three zero objects and all squares are cocartesian.Hence, by Proposition 9.3.10 the squares 1 + 2 and 2 + 3 are suspensionsquares, thereby yielding canonical isomorphisms x′ ∼= Σx and y′ ∼= Σy. Wewould like to conclude that f ′ gets identified with Σf , but already for thisstatement to make sense we have to define a suspension functor

Σ: D([1])→ D([1]). (9.5.11)

Moreover, we need tools to understand such suspension functors with parame-ters. A systematic tool for this and similar purposes is given by the calculus ofKan extensions with parameters, and this calculus is the content of §10.

9.6 Exceptional inverse image functors

In this section we show that pointed derivators can be characterized by theexistence of so-called (co)exceptional inverse image functors. These functors areright adjoints to right Kan extensions and left adjoints to left Kan extensionswhich happen to exist in pointed derivators in the case of Kan extensions along(co)sieves (Proposition 9.6.7). We suggest the reader to skip this section on afirst reading.

Construction 9.6.1. Let u : A→ B be a functor and let ik : A→ A×[1], k = 0, 1,be the inclusions a 7→ (a, k). Associated to this datum, there are the followingtwo versions of the mapping cylinder of u, which are respectively defined bythe pushout squares

Au //

i0

B

s

Au //

i1

B

s

A× [1] // cyl0(u), A× [1] // cyl1(u).

These mapping cylinder constructions corepresent the following two functors.proof-read!!!

(i) A functor cyl0(u) → C is equivalently specified by functors F : A → C,G : B → C and a natural transformation α : F → G u.

DRAFT

9.6. EXCEPTIONAL INVERSE IMAGE FUNCTORS 175

(ii) A functor cyl0(u) → C is equivalently specified by functors F : A → C,G : B → C and a natural transformation α : G u→ F .

In the case of cyl0(u) there is an inclusion i : Ai1→ A× [1]→ cyl0(u). More-

over, the universality of the pushout square in

Au //

i0

B

s

id

A× [1] //

uπ))

cyl0(u)

""B

induces a unique dashed functor q : cyl0(u)→ B making the diagram commute.By construction, the equations q i = u and q s = idB are satisfied, and thereare similar results in the case of cyl1(u).

We will typically apply this construction in the case where u is fully faithful,for example when u is a sieve or a cosieve.

Lemma 9.6.2. For every sieve u : A→ B the following commutative square ishomotopy exact

Ai //

id

id

cyl0(u)

q

Au// B.

(9.6.3)

Proof. To reformulate the homotopy exactness of the square we consider thefollowing pasting on the left

1 a //

id

Ai //

id

id

cyl0(u)

q

=

1(ia,id)

//

id

~ id

(q/ua) //

cyl0(u)

q

1a// A

u// B 1 // 1

ua// B

in which the constant square to the left obviously is homotopy exact. It followsfrom (Der2) and the functoriality of canonical mates with pasting that it sufficesto show that the above pasting on the left is homotopy exact for all a ∈ A.

We note that this pasting agrees with the pasting on the right in which theright square is a slice square and in which the functor 1 → (q/ua) classifiesthe object (ia, id : qia → ua). The reader easily verifies that (ia, id) ∈ (q/ua)is a terminal object under the assumption that u is a sieve. Hence, the corre-sponding square in the above pasting is homotopy exact by Corollary 8.2.10.The homotopy exactness of slice squares and the pasting property of homotopyexact squares concludes the proof.

DRAFT


The dual version of the lemma guarantees the homotopy exactness of thecommutative square

Ai //

id

cyl1(u)

q

Av// B

?Gid

under the assumption that v : A→ B is a cosieve.

Notation 9.6.4. Let D be a pointed derivator and let u : A→ B be a functor(typically the inclusion of a subcategory). We denote by D(B,A) ⊆ D(B) thefull subcategory spanned by all X ∈ D(B) such that u∗(X) ∈ D(A) vanishes,u∗(X) ∼= 0.

Lemma 9.6.5. Let D be a pointed derivator and let u : A→ B be a functor.

(i) If u : A→ B is a sieve, then D(B,A) ⊆ D(B) is a reflective subcategory,i.e., the inclusion admits a left adjoint.

(ii) If u : A→ B is a cosieve, then D(B,A) ⊆ D(B) is a coreflective subcate-gory, i.e., the inclusion admits a right adjoint.

Proof. We begin with the first statement and consider the mapping cylinder con-struction cyl0(u) together with the commutative square (9.6.3). Since this squareis homotopy exact (Lemma 9.6.2), the canonical mate i∗ → u∗q! is an isomor-phism. This implies that the Kan extension functor q! : D(cyl0(u))→ D(B) re-stricts to a functor q! : D(cyl0(u), A)→ D(B,A). Obviously, also the restrictionfunctor q∗ restricts correspondingly and the adjunction (q!, q

∗) : D(cyl0(u)) D(B) hence restricts to an adjunction

(q!, q∗) : D(cyl0(u), A) D(B,A).

Note that the functor s : B → cyl0(u) is a sieve. By Corollary 9.1.6 the adjunc-tion (s∗, s∗) : D(cyl0(u)) D(B) induces an adjoint equivalence

(s∗, s∗) : D(cyl0(u), A) D(B).

Combining these two adjunctions we obtain the intended adjunction

(q!, q∗) (s∗, s

∗) : D(B) ' D(cyl0(u), A) D(B,A).

In fact, since s∗q∗ = id restricts to the inclusion D(B,A) ⊆ D(B) this concludesthe proof of the first statement.

The second statement follows by duality. In fact, if u : A → B is a cosieve,then one verifies that

(s∗, s!) (q∗, q∗) : D(B,A) ' D(cyl1(u), A) D(B)

defines the intended coreflection.

DRAFT


Lemma 9.6.6. Let u : A→ B be a fully faithful functor and let v : B−u(A)→B be the fully faithful inclusion. The functor u is a sieve if and only if thefunctor v is a cosieve.

Proof. This is immediate from Definition 9.1.4.

We now obtain the intended characterization of pointed derivators.

Proposition 9.6.7. The following are equivalent for a derivator D .

(i) The derivator D is pointed.

(ii) For every sieve u : A → B the right Kan extension functor u∗ : D(A) →D(B) has a right adjoint u! : D(B)→ D(A).

(iii) For every cosieve u : A → B the left Kan extension functor u! : D(A) →D(B) has a left adjoint u? : D(B)→ D(A).

Proof. By duality it is enough to show that (i) and (ii) are equivalent. Let usassume that (ii) is satisfied and consider the empty sieve ∅ : ∅ → 1. The rightKan extension functor ∅∗ : D(∅)→ D(1) admits a right adjoint, and is hence anadjoint on both sides. As a such, ∅∗ preserves zero objects. Note that (Der1)implies that D(∅) is a trivial category and any object x ∈ D(∅) is hence a zeroobject. Thus, ∅∗(x) ∈ D(1) is a zero object, showing that D is pointed.

Let us now assume that D is pointed and let u : A→ B be a sieve. The corre-sponding cosieve v : B−u(A)→ B defines a coreflective subcategory D(B,B−u(A)) ⊆ D(B). Combining this with the equivalence (u∗, u∗) : D(B,B−u(A)) 'D(A) induced by the sieve u (Corollary 9.1.6), we obtain a composite adjunction

(u∗, u!) : D(A) ' D(B,B − u(A)) D(B).

Since the left adjoint is simply u∗ : D(A)→ D(B) this concludes the proof.

Definition 9.6.8. Let D be a pointed derivator and let u : A→ B be a functor.

(i) If u is a sieve, then any right adjoint u! : D(B) → D(A) to u∗ : D(A) →D(B) is an coexceptional inverse image functor.

(ii) If u is a cosieve, then any left adjoint u? : D(B) → D(A) to u! : D(A) →D(B) is a exceptional inverse image functor.

Remark 9.6.9. The original definition of pointed derivators of Maltsiniotis (see[Mal07]) was as derivators such that coexceptional inverse image functors alongsieves and exceptional inverse image functors along cosieves exist. By Proposi-tion 9.6.7 this is equivalent to the definition given here (Definition 9.1.1).

The above proofs yield explicit formulas for the (co)exceptional inverse imagefunctors, which we collect here for later reference.

Construction 9.6.10. Let D be a pointed derivator and let u : A → B be afunctor.

DRAFT


(i) Let u : A→ B be a sieve and let v : B − u(A)→ B be the correspondingcosieve. Using the above formulas we conclude that the coexceptionalinverse image functor u! : D(B)→ D(A) is given by

u! = u∗q∗s! : D(B)→ D(cyl1(v), B−u(A))→ D(B,B−u(A))→ D(A),

where we kept using he notation of the construction of cyl1(v) (Construc-tion 9.6.1). In fact, it follows from the above proofs that there are thefollowing three adjunctions defining u!,

D(A) ' D(B,B − u(A)) D(cyl1(v), B − u(A)) ' D(B).

The outer equivalences are induced by the right Kan extension along thesieve u : A → B and the left Kan extension along the cosieve s : B →cyl1(u), respectively, while the remaining adjunction is induced by theright Kan extension along q : cyl1(u)→ B.

(ii) Dually, let u : A → B be a cosieve and let v : B − u(A) → B be thecorresponding sieve. The exceptional inverse image functor u? : D(B) →D(A) is given by

u? = u∗q!s∗ : D(B)→ D(cyl0(v), B−u(A))→ D(B,B−u(A))→ D(A).

In fact, the above proofs imply that u? is defined by the following threeadjunctions,

D(B) ' D(cyl0(v), B − u(A)) D(B,B − u(A)) ' D(A),

The outer equivalences are induced by the right Kan extension along thesieve s : B → cyl0(v) and the left Kan extension along the cosieve u : A→B, respectively, while the remaining adjunction is induced by the left Kanextension along q : cyl0(v)→ B.

We illustrate these inverse image functors and the above explicit formulasby the following example.

Examples 9.6.11. Let D be a pointed derivator.

(i) Let us consider the sieve u = 0: 1 → [1] and let us make precise theconstruction of 0? : D([1]) → D(1) as in Construction 9.6.10. The cor-responding cosieve is v = 1: 1 → [1] and the functor s can hence beidentified with the functor s : [1] →y classifying the vertical morphism(1, 0) → (1, 1). Since s is a cosieve, s! extends (f : x → y) ∈ D([1]) byzero to a cospan. The functor q : y→ [1] collapses the horizontal intervalto a point and it factors as q = πiy : y→ → [1]. This yields a canonicalisomorphism q∗ ∼= π∗ (iy)∗ : D(y)→ D([1]) and it is easy to observe thatπ∗ is naturally isomorphic to the functor which restricts to the verticalmorphim (0, 0)→ (0, 1). Thus, at the level of coherent diagrams the first

DRAFT


two steps hence form the diagrams

x

f

x

f

Ff //

x

f

Ff

y, 0 // y, 0 // y, 0,

where the square is a fiber square. A final application of the functor 0∗

hence yields Ff , showing that there is a natural isomorphism

0! ∼= F : D([1])→ D(1),

i.e., that F is a coexceptional inverse image functor.

(ii) The exceptional inverse image functor associated to the cosieve 1: 1→ [1]is naturally isomorphic to C : D([1])→ D(1),

1? ∼= C : D([1])→ D(1).

In fact, this can be obtained by dualizing the above steps and the detailsare left to the reader.

Corollary 9.6.12. For every pointed derivator D there are natural isomor-phisms

Σ ∼= 1? 0∗ : D(1)→ D(1) and Ω ∼= 0! 1! : D(1)→ D(1).

Proof. This is immediate from the definition of Σ,Ω and Examples 9.6.11.

Lemma 9.6.13. Let D be a pointed derivator and let A,B,C ∈ Cat.

(i) The identity functor idA : A → A is a sieve and there is a canonicalisomorphism id!

A∼= id: D(A)→ D(A).

(ii) If u : A→ B and v : B → C are sieves then so is v u : A→ C and thereare canonical isomorphisms (v u)! ∼= u! v! : D(C)→ D(A).

(iii) The identity functor idA : A → A is a cosieve and there is a canonicalisomorphism id?

A∼= id: D(A)→ D(A).

(iv) If u : A → B and v : B → C are cosieves then so is v u : A → C andthere are canonical isomorphisms (v u)? ∼= u? v? : D(C)→ D(A).

Proof. It is immediate that identities are (co)sieves and that (co)sieves are closedunder composition. The remaining statements follow immediately from theuniqueness of adjoints. In fact, if u, v are sieves, then there is a canonicalisomorphism (v u)∗ ∼= v∗ u∗. Passing to the conjugate transformation of thisisomorphism (see (A.1.7)) we obtain the intended isomorphism (v u)! ∼= u! v!

(Corollary A.1.9).

DRAFT


DRAFT

Chapter 10

Parametrized Kanextensions

In this chapter we study the calculus of parametrized Kan extensions in deriva-tors and collect some first examples. The classical counterpart of parametrizedKan extensions in ordinary category theory is given by Kan extensions in dia-gram categories; given a complete and cocomplete category C and a small cate-gory A, the diagram category CA is again complete and cocomplete and henceadmits Kan extensions which are referred to as Kan extensions with parameters(in A).

One main point of the theory of derivators is that many constructions in var-ious areas of pure mathematics arise as parametrized Kan extensions in deriva-tors. First examples of such constructions are collected in §12, while manyadditional ones show up in later chapters.

In §10.1 we review the classical case of Kan extensions in ordinary diagramcategories. In §10.2 we show that derivators admit exponentials, i.e., that given aderivator D and A ∈ Cat there is a derivator DA of coherent A-shaped diagramsin D . In §10.3 we show that Kan extension and restriction functors in unrelatedvariables commute in derivators, and deduce some consequences.

10.1 Parametrized Kan extensions in categories

In this section we discuss the construction and basic properties of parametrizedKan extensions, i.e., Kan extensions in diagram categories. The notions andresults collected in this section serve as motivation for corresponding notionsand statements for derivators; see §§10.2-10.3. In this section we do not provideany proofs since these results can be obtained from corresponding results in§§10.2-10.3 by specializing to represented derivators.

Again, by duality, it is enough to focus on left Kan extensions. To beginwith, we recall the following observation from ordinary category theory.

181

DRAFT

182 CHAPTER 10. PARAMETRIZED KAN EXTENSIONS

Lemma 10.1.1. If C is a cocomplete category and A ∈ Cat, then the diagramcategory CA is cocomplete.

A refined statement gives a description of colimits in CA which turn out tobe constructed coordinatewise. More precisely, for every small category B ∈ Catand diagram X : B → CA, the colimit colimb∈B X(b) ∈ CA exists and for everya ∈ A there is an isomorphism(

colimb∈B X(b))(a) ∼= colimb∈B

(X(b)(a)

).

A typical proof of the cocompleteness is given by showing that the right handside of the above isomorphism defines a diagram A → C and that it, togetherwith a suitably defined cocone, yields a colimit colimB X ∈ CA.

We sometimes refer to colimits in CA as colimits with parameters (inA) or parametrized colimits. It follows from this construction of colimits indiagram categories that evaluation functors preserve colimits.

Proposition 10.1.2. If C is a cocomplete category and A ∈ Cat, then thediagram category CA is cocomplete. For every B ∈ Cat and a ∈ A the diagram

(CA)BcolimB //

a∗

CA

a∗

CBcolimB

// C

>F∼=

commutes up to a canonical isomorphism.

More generally, as a cocomplete category, CA admits left Kan extensions(Theorem 6.3.10), hence the following definition makes sense.

Definition 10.1.3. Given a cocomplete category C, a small category A, andu : B → B′ in Cat , the left adjoint

u! : (CA)B → (CA)B′

to u∗ : (CA)B′ → (CA)B forms left Kan extensions with parameters (in A)

or parametrized left Kan extensions.

It turns out that also these parametrized Kan extensions can be constructedpointwise. In more detail, given a diagram X : B → CA, the parametrized leftKan extension u!(X) : B′ → CA exists. Moreover, the diagram a∗X : B → C, a ∈A, also admits a left Kan extension u!(a

∗X) : B′ → C and for b′ ∈ B′ there areisomorphisms (

u!(X)(b′))(a) ∼= u!(a

∗X)(b′).

A less cryptic way of putting this is as follows.

DRAFT

10.1. PARAMETRIZED KAN EXTENSIONS IN CATEGORIES 183

Theorem 10.1.4. For every cocomplete category C, small category A, and func-tor u : B → B′ in Cat , the parametrized left Kan extension functor

u! : (CA)B → (CA)B′

exists. Moreover, for every a ∈ A, the diagram

(CA)Bu! //

a∗

(CA)B′

a∗

CBu!

// CB′

@H∼=


We illustrate the concept of parametrized Kan extensions by the followingexample. Similar types of constructions, carried out in abstract derivators, areused in later sections.

Example 10.1.5. Let C be a cocomplete category admitting a zero object andlet us consider a morphism of morphisms in C,

xf//

g

y

g′

x′f ′// y′,

i.e., an object X ∈ C ∼= (C[1])[1]. Here, we want to read X ∈ (C[1])[1] as amorphism

X = (f, f ′) : g → g′,

so that the outer variable is drawn horizontally. If we apply the parametrizedKan extensions

(C[1])[1] i∗→ (C[1])p(ip)!→ (C[1])

to X, then we obtain a commutative cube Q : [1] × → C. This cube is builtby first adding the zero objects as in the diagram on the left in

xf

//

g

yg′

x′f ′

//

y′

0

0,

xf

//

g

yg′

%%

x′f ′

//

y′

0

// cokf

%%

0 // cokf ′,

and then passing to the cube by forming two pushout squares.

DRAFT


As a final generalization of the above results, we can pass from evaluationfunctors to general restriction functors.

Corollary 10.1.6. Let C be a cocomplete category and let v : A→ A′, u : B →B′ be functors in Cat. The diagram

(CA′)B u! //

v∗

(CA′)B′

v∗

(CA)Bu!

// (CA)B′

@H∼=


Proof. The proof of Theorem 10.3.1 shows that this is indeed a corollary toTheorem 10.1.4.

In §§10.2-10.3 we establish corresponding results for arbitrary derivators,hence leading to a calculus of parametrized derived Kan extensions and parametrizedhomotopy Kan extensions.

10.2 Exponentials for derivators

In this section we show that derivators admit exponentials, i.e., that for everyderivator D and A ∈ Cat there is a derivator DA of coherent, A-shaped diagramsin D .

To motivate the construction of DA, let us recall how we motivated theconcept of a derivator in the first place. Starting with an abelian category A,the key idea was to refine the passage

A 7→ D(A)

by also keeping track of derived diagram categories, thereby constructing thederivator

DA : B 7→ DA(B) ∼= D(AB).

If we now fix a category A ∈ Cat , then we would like to consider a similarrefinement of the value DA(A) ∼= D(AA). Thus, given the abelian category AA,we want to consider derived categories of diagram categories (AA)B , B ∈ Cat .By the categorical exponential law (AA)B ∼= AA×B , this amounts to consideringthe 2-functor

B 7→ DA(A×B) ∼= D(AA×B).

Abstracting from this special case there is the following construction.

Construction 10.2.1. Given a small category A, there is the associated 2-functor

A×− : Cat → Cat : B 7→ A×B.

DRAFT

10.2. EXPONENTIALS FOR DERIVATORS 185

Hence, if D : Catop → CAT is a prederivator, then we can define the shiftedprederivator DA : Catop → CAT by setting

DA(B) = D(A×B), B ∈ Cat .

More precisely, the shifted prederivator is defined as the 2-functor

DA : Catop A×−→ Catop D→ CAT .

In particular, given u : B → B′ in Cat , the corresponding restriction functor inthe shifted prederivator DA is hence calculated by

u∗DA = (idA × u)∗D .

The underlying category of DA is canonically isomorphic to D(A).

We illustrate the shifting operation by a few examples.

Examples 10.2.2.

(i) Let y(C) be a represented derivator. Then for A,B ∈ Cat the exponentiallaw yields canonical isomorphisms

y(C)A(B) = y(C)(A×B)

= CA×B

∼= (CA)B

= y(CA)(B),

showing that the shifting operation reproduces the usual exponentials.

(ii) Let DA be the homotopy derivator of a Grothendieck abelian category Aand let A,B ∈ Cat . A combination of Lemma 3.1.11 with the categoricalexponential law yields canonical isomorphisms

DAA(B) = DA(A×B)

= Ch(A)A×B [(WA×BA )−1]

∼= Ch(AA)B [(WBAA)−1]

= DAA(B).

Thus, the shifting operation is compatible with the exponential mentionedin Examples 7.3.3.

(iii) Let M be a combinatorial model category with weak equivalences WM.The projective model structure on MA, A ∈ Cat , with levelwise weakequivalences WMA = WA

M defines a further combinatorial model category

DRAFT


(Lemma B.3.9). It follows that for B ∈ Cat there are canonical isomor-phisms

HoAM(B) = HoM(A×B)

=MA×B [(WA×BM )−1]

∼= (MA)B [(WBMA)−1]

= HoMA(B).

Again, this shows that the shifting operation is compatible with the onefor combinatorial model categories. And a similar result can also be es-tablished for complete and cocomplete ∞-categories.

In the above examples we saw instances of derivators such that the associ-ated shifted prederivators again are derivators. As a further application of thecalculus of mates and homotopy exact squares (see §8), we now show that thisis always the case.

Theorem 10.2.3. For a derivator D and A ∈ Cat, the prederivator DA isagain a derivator, the shifted derivator or the derivator of (coherent) A-shaped diagrams in D . The Kan extension functors in DA along u : B → B′

are calculated by

uDA

! = (idA × u)D! and uDA

∗ = (idA × u)D∗ .

Proof. The verification of the axioms (Der1)-(Der3) for DA is left to the reader.It remains to check the pointwise formulas encoded by axiom (Der4). Thus,given a functor u : B → B′ and b′ ∈ B′, let us consider the square

A× (u/b′)id×p

//

id×π

A×B

id×u

A× 1id×b′

// A×B′,

which is obtained from the slice square (7.2.4) by forming the product with A.We have to show that the canonical mate (id×π)!(id× p)∗ → (id× b′)∗(id×u)!

is an isomorphism of functors D(A× B)→ D(A× 1). Since isomorphisms aredetected pointwise by (Der2), it is enough to show that all components of themate are isomorphisms. For a ∈ A we consider the following pasting diagram

((id× u)/(a, b′))

id

φ

∼=//

π

((id× π)/a)p′//

π

A× (u/b′)id×p

//

id×π

A×B

id×u

1 =// 1

a// A× 1

id×b′// A×B′

(10.2.4)

in which the square in the middle is the slice square (7.2.4) associated to thefunctor 1 × π and a ∈ A ∼= A × 1. The square on the left is induced from the

DRAFT

10.2. EXPONENTIALS FOR DERIVATORS 187

isomorphisms of categories

((id× π)/a) ∼= (A/a)× (u/b′) ∼= ((id× u)/(a, b′)), (10.2.5)

where (a, b′) : 1 → A × B′. The functoriality of mates with respect to pasting(Lemma 8.1.10) implies that the canonical mate associated to (10.2.4) factorsas

π!φ∗(p′)∗(id×p)∗ ∼= π!(p

′)∗(id×p)∗ ∼= a∗(id×π)!(id×p)∗ → a∗(id×b′)∗(id×u)!.

Here, the second arrow is an isomorphism by (Der4) applied to D and thecorresponding slice square. The first arrow is also an isomorphism becausethe isomorphism (10.2.5) is homotopy final; see Examples 8.2.14. In order toconclude the proof it suffices to show that (10.2.4) is homotopy exact for everya ∈ A. Since the pasting (10.2.4) agrees with the slice square (7.2.4) associatedto the functor id × u : A × B → A × B′ and the object (a, b′) ∈ A × B′, thisfollows by a further application of (Der4) to D .

Remark 10.2.6. Let us assume that we want to prove a statement about arbi-trary values of suitable derivators. The following two observations then oftenallow us to focus on the underlying category.

(i) Given a derivator D , all its values D(A), A ∈ Cat are underlying categoriesof derivators. In fact, this follows from Theorem 10.2.3 since there arecanonical isomorphisms

DA(1) ∼= D(A).

(ii) Many classes of derivators are closed under the shifting operation, as wesee this in this book for pointed and stable derivators. In fact, the secondstatement in Lemma 9.1.3 says that pointed derivators are closed undershifting, and in §15 we see that the same is true for stable derivators.

While this restriction to underlying categories is of course not mathemati-cally neccessary, at times it might be easier to focus on this case. To illustratethis, in §9.5 we defined incoherent cofiber sequences in the underlying categoryD(1) of a pointed derivator D . By the closure property we implicitly also de-fined incoherent cofiber sequences in D(A), A ∈ Cat (and, in particular, thefunctor (9.5.11)). For a better understanding of these cofiber sequences, how-ever, we need better control over Kan extension functors in shifted derivators,and this is the goal of §10.3.

Remark 10.2.7. By definition of derivators, slice squares are homotopy exact.The proof of Theorem 10.2.3 essentially consisted of showing that products ofslice squares with fixed categories are again homotopy exact.

More generally, Theorem 10.2.3 implies the following.

DRAFT


Corollary 10.2.8. The class of homotopy exact squares is closed under theformation of products with fixed categories, i.e., if the square on the left in

Cp//

v

α

A

u

E × Cid×p

//

id×v

id×α

E ×A

id×u

Dq// B, E ×D

id×q// E ×B,

is homotopy exact and if E ∈ Cat, then also the square on the right is homotopyexact.

Proof. Given a derivator D , we have to show that the canonical mate associatedto id×α is an isomorphism in D . By construction of Kan extension functors inshifted derivators (Theorem 10.2.3), this mate can be chosen to be the canonicalmate associated to α in the derivator DE . Since the square populated by α ishomotopy exact, this canonical mate is an isomorphism.

We conclude this section by the following remark, which offers a justificationof the title of this section. The content of this remark will be discussed in detailin the sequel to this book.

Remark 10.2.9. As we discuss in [Gro16a], derivators, morphisms of derivators,and natural transformations assemble to a 2-category DER of derivators. Simi-larly, there is 2-category PDER of prederivators. Both 2-categories are cartesianmonoidal, by means of the product defined in Lemma 7.3.14. It can be shownthat these symmetric monoidal structures are closed in the bicategorical sense,i.e., for prederivators D ,D ′, and D ′′, there is a prederivator HOM(D ′,D ′′) ofmorphisms and there are pseudo-natural equivalences of categories

HomPDER(D ×D ′,D ′′) ' HomPDER(D ,HOM(D ′,D ′′)).

It turns out that, as in ordinary category theory, HOM(D ′,D ′′) inherits allexactness properties of D ′′. In particular, HOM(D ′,D ′′) is a derivator as soonas this is the case for D ′′.

The justification for the terminology in Theorem 10.2.3, in which we referto DA as the derivator of diagrams of shape A in D , is given by the existenceof equivalences of derivators

DA ' HOM(y(A),D).

10.3 Parametrized Kan extensions in derivators

In this section we study basic aspects of parametrized Kan extensions in deriva-tors, i.e., of Kan extensions in shifted derivators. The key result here is thatKan extensions and restrictions in unrelated variables commute up to canonicalisomorphisms. This commutativity is made precise by the following theorem,which is a derivator version of Corollary 10.1.6.

DRAFT

10.3. PARAMETRIZED KAN EXTENSIONS IN DERIVATORS 189

Theorem 10.3.1. For functors v : A → A′ and u : B → B′ between smallcategories the naturality square

A×B v×id//

id×u

A′ ×B

id×u

A×B′v×id// A′ ×B′

(10.3.2)

is homotopy exact, i.e., in every derivator the canonical mate transformations

(id×u)!(v×id)∗ → (v×id)∗(id×u)! and (v×id)∗(id×u)∗ → (id×u)∗(v×id)∗

are isomorphims.

Proof. We begin by reducing to the special case of u = πB : B → 1. For thispurpose, we consider the following two pastings

A× (u/b′)id×p//

π

A×B v×id//

id×u id

A′ ×Bid×u

=

A× (u/b′)v×id//

π id

A′ × (u/b′)id×p//

π

A′ ×Bid×u

Aid×b′

// A×B′v×id// A′ ×B′ A

v// A′

id×b′// A′ ×B′.

The square to the very left and the square to the very right are obtained fromslice squares by forming the product with a fixed category and are hence homo-topy exact (Remark 10.2.7). Since the above two pastings agree, we concludeby (Der2) and the functoriality of mates with respect to pasting, that (10.3.2) ishomotopy exact as soon as the second square from the right is homotopy exact.This completes the reduction to the case of u = πB : B → 1.

As an additional reduction we show that it is enough to consider evaluationfunctors instead of more general restriction functors. To this end, let us considerthe pasting on the left in

Bt×id//

π

id

(A/a)×Bp×id//

π

A×B v×id//

π

id

A′ ×Bπ

=

Bva×id//

π id

A′ ×Bπ

1id

// 1a

// Av

// A′ 1va// A′.

(10.3.3)

We have to show that in that pasting the right square is homotopy exact. As aslice square, we know that the square in the middle is homotopy exact. More-over, the square on the left is induced from the functor t : 1 → (A/a) classify-ing the terminal object (a, id : a → a) ∈ (A/a), and is hence homotopy exact(Proposition 8.2.8). By the functoriality of mates with pasting and (Der2) ithence suffices to show that the pasting on the left is homotopy exact. This past-ing is easily seen to agree with the commutative square on the right in (10.3.3),and this concludes the reduction to evaluation functors.

DRAFT


Finally, let us consider an object a′ ∈ A′. Our remaining task is to showthat the commutative square on the left in

Ba′×id

//

π

id

A′ ×B

π

=

Bt×id//

π

id

(A′/a′)×B

π

p×id//

A′ ×B

π

1a′

// A′ 1id

// 1a′

// A′

is homotopy exact. In the pasting to the right, the right square is a slice squareand hence homotopy exact. The remaining square is induced by t : 1→ (A′/a′)classifying the terminal object (a′, id : a′ → a′). Since the pasting on the rightis easily seen to agree with the square to the very left, the homotopy finalityof right adjoints (Proposition 8.2.8) and the functoriality of homotopy exactsquares with respect to pasting (Proposition 8.2.7) conclude the proof.

Remark 10.3.4. In the case of represented derivators, Theorem 10.3.1 specializesto Theorem 10.1.4 and Corollary 10.1.6. As always, since Theorem 10.3.1 is astatement about arbitrary derivators, this also establishes similar results forderived Kan extensions and homotopy Kan extensions.

Notation 10.3.5. Let D be a prederivator, A,B ∈ Cat , a ∈ A, b ∈ B, andX ∈ D(A×B). In this situation we abuse notation and write

Xa = (a× id)∗(X) ∈ D(B) and Xb = (id× b)∗(X) ∈ D(A).

Using this notation, the following is immediate.

Corollary 10.3.6. Let D be a derivator, A,B ∈ Cat , X ∈ D(A × B), anda ∈ A. There are canonical isomorphisms

colimB(Xa) ∼−→ (colimB X)a and (limBX)a∼−→ limB(Xa).

Proof. This is a special case of Theorem 10.3.1, applied to u = a : 1 → A andv = πB : B → 1.

For Kan extensions along fully faithful functors there is the following conse-quence.

Corollary 10.3.7. Let D be a derivator, let u : B → B′ be fully faithful, andlet A ∈ Cat. A coherent diagram X ∈ D(A× B′) lies in the essential image of(id × u)! if and only if each Xa ∈ D(B′), a ∈ A, lies in the essential image ofu!.

Proof. Let us consider the following pasting of commutative squares in whichthe square to the very left is homotopy exact as an instance of (10.3.2),

B

u

a×id//

id

A×B

id×u

id×u//

id

A×B′

id

=

B

u

u //

| id

B′

id

a×id//

id

A×B′

id

B′a×id

// A×B′id// A×B′ B′

id// B′

a×id// A×B′.

DRAFT

10.3. PARAMETRIZED KAN EXTENSIONS IN DERIVATORS 191

Since id×u is fully faithful, X ∈ D(A×B′) lies in the essential image of (id×u)! ifand only if the adjunction counit (id×u)!(id×u)∗ → id is an isomorphism on X(Proposition 8.2.16). We note that this counit is the canonical mate associatedto the second square from the left. Thus, (Der2), the homotopy exactness of thesquare to the very left (Theorem 10.3.1), and the functoriality of mates withpasting imply that X lies in the essential image of (id × u)! if and only if foreach a ∈ A the canonical mate of the pasting on the left is an isomorphism.Since the above two pastings agree, similar arguments show that this is the caseif and only if for every a ∈ A the counit u!u

∗ → id applied to Xa ∈ D(B′) isan isomorphism if and only if Xa ∈ D(B′), a ∈ A, lie in the essential image ofu! : D(B)→ D(B′).

Remark 10.3.8. Let D be a derivator and let A,B ∈ Cat . A coherent diagramX ∈ D(A × B′) can be considered as an object in DA(B′), while the objectsXa, a ∈ A, all live in D(B′). Using the explicit construction of Kan extensionsin shifted derivators (Theorem 10.2.3), Corollary 10.3.7 says that the essentialimage of a fully faithful Kan extension functor in a shifted derivator DA can,by means of the evaluations at all a ∈ A, be detected by the correspondingessential image in D .

We illustrate this result by a few special cases.

Corollary 10.3.9. Let D be a derivator and let A ∈ Cat. A diagram X ∈ D(A)is an initial object if and only if Xa ∈ D(1), a ∈ A, are initial objects.

Proof. It follows from the construction of initial objects in Proposition 7.4.7that an object in D(A) is initial if and only if it lies in the essential image of theleft Kan extension functor along the (fully faithful) empty functor ∅ : ∅ → A.Rewriting the empty functor as idA×∅ : A×∅ → A× 1 the result is immediatefrom Corollary 10.3.7.

We now turn to a less exotic special case.

Corollary 10.3.10. Let D be a derivator and let A ∈ Cat.

(i) A square X ∈ DA() is cocartesian if and only if Xa ∈ D() is cocarte-sian for all a ∈ A.

(ii) In a pointed derivator D , a square X ∈ DA() is a suspension square ora cofiber square if and only if this is the case for Xa ∈ D(), a ∈ A.

Proof. The first statement is an immediate application of Corollary 10.3.7 tothe fully faithful functor ip : p→ . The second part follows from this andCorollary 10.3.9 (see Definition 9.2.17 and Definition 9.2.8).

We leave it to the reader to establish a similar result for coherent cofibersequences (see Definition 9.5.1).

Recall that a coherent morphism X ∈ D([1]) is an isomorphism if the un-derlying diagram X0 → X1 is an isomorphism in the usual sense. By Exam-ple 8.2.24, coherent isomorphisms can be obtained by suitable Kan extensions.

DRAFT


Corollary 10.3.11. Let D be a derivator and let A ∈ Cat. The following areequivalent for a coherent morphism X ∈ DA([1]).

(i) The morphism X is an isomorphism.

(ii) The morphisms Xa ∈ D([1]), a ∈ A, are isomorphisms.

(iii) The morphism X lies in the essential image of 0! : DA(1)→ DA([1]).

Proof. This follows from Example 8.2.24 and (Der2). Alternatively, we can alsoinvoke Example 8.2.24 and Corollary 10.3.7.

Having the basic calculus of parametrized Kan extensions under control, weinvite the reader to already take a look at §12.1. In that section we apply thiscalculus to threefold iterations of the cofiber construction, thereby establishing aresult which is closely related to the rotation axiom for triangulated categories.

In §12 we also collect a few additional results concerning iterated cone andcofiber constructions. For these and similar purposes it is convenient to havebasic tools to study (co)limiting (co)cones in derivators, and the goal of §11 isto provide such tools.

DRAFT

Chapter 11

The yoga of colimitingcocones

In this chapter we include a more systematic discussion of cocones and colimitingcocones in derivators. This includes a construction of canonical comparisonmaps between colimiting cocones and general cocones. As in ordinary categorytheory, colimiting cocones are precisely the cocones such that these comparisonmaps are isomorphisms. These comparison maps show up again in the discussionof total cofibers (see §12.5) and strongly cocartesian n-cubes (see [Gro16a]).

On a more technical side, we also establish results along the following lines.Often, in more advanced constructions with derivators, one runs into the situ-ation that in certain larger diagrams some squares ‘obviously’ are cocartesian.More generally, often certain cocones in larger diagrams ‘obviously’ are colim-iting. Here we establish detection results, which actually allow us to concludethis.

In §11.1 we study cocones and colimiting cocones in more detail, and weillustrate these notions in §11.2 by specializing to specific examples. In §11.3we obtain detection results for colimiting cocones.

11.1 Colimiting cocones in derivators

In this section we discuss colimiting cocones in derivators and the constructionof canonical comparison maps from colimiting cocones to arbitrary cocones.

Let us recall from Definition 6.5.1 the construction of the cocone AB and thecone AC on a category A which, respectively, come with fully faithful inclusionfunctors

iA : A→ AB and iA : A→ AC.

The unique objects which are not hit by these fully faithful inclusion functorsare ∞ ∈ AB and −∞ ∈ AC, respectively. The following definition generalizesDefinition 9.2.1.

193

DRAFT

194 CHAPTER 11. THE YOGA OF COLIMITING COCONES

Definition 11.1.1. Let D be a derivator and let A ∈ Cat .

(i) A diagram X ∈ D(AB) is a cocone (with base i∗A(X) ∈ D(A)).

(ii) A cocone X ∈ D(AB) is colimiting if it lies in the essential image of(iA)! : D(A)→ D(AB).

Cones and limiting cones are defined dually. In this section and theremainder of the chapter we mostly focus on (colimiting) cocones, and this isjustified by the following construction.

Construction 11.1.2. Let A ∈ Cat and let iA : A→ AB be the inclusion functor.As a special case of the compatibility of the join construction and the formationof opposite categories, there is an isomorphism of categories φ : (AB)op ∼= (Aop)C

such that the diagram on the right in

A

iA

Aop

(iA)op

iAop

%%

AB (AB)op∼=

φ// (Aop)C

1,

∞A

OO

1

(∞A)op

OO

−∞Aop

99

commutes.Given a derivator D and A ∈ Cat , associated to the left Kan extension

functor (iA)! : D(A)→ D(AB) there is the opposite functor (iA)op! . By definition

of Kan extension functors in opposite derivators (7.3.17), this opposite functorsits in the commutative square in the diagram

D(A)op

id

((iA)D! )op

// D(AB)op

id

Dop(Aop)((iA)op)Dop

∗ //

(iAop )Dop

∗ ))

Dop((AB)op)

φDop

∗∼=

Dop((Aop)C).

Moreover, the bottom triangle commutes by Lemma 7.6.1 up to a canonicalnatural isomorphism.

In particular, given a diagram X ∈ D(AB) we may consider it as an objectin D(AB)op and apply φDop

∗ to it. Abusing notation, we denote the resultingcone in Dop still by X ∈ Dop((Aop)C).

Lemma 11.1.3. Let D be a derivator and A ∈ Cat. A cocone X ∈ D(AB) iscolimiting if and only if the corresponding cone X ∈ Dop((Aop)C) is limiting.

Proof. This is immediate from the details of Construction 11.1.2.

DRAFT

11.1. COLIMITING COCONES IN DERIVATORS 195

This observation is the duality principle for (co)cones.

Proposition 11.1.4. For A ∈ Cat the following squares are homotopy exact,

Aid //

πA

A

iA

Aid //

πA

A

iA

1 ∞// AB, 1

−∞// AC.

=E

Proof. By duality it is enough to take care of the square on the left. Note thatthis square is isomorphic to the slice square associated to iA : A → AB and∞ ∈ AB, which is hence homotopy exact.

More explicitly, the proposition says that for a derivator D and X ∈ D(A)there are canonical isomorphisms

colimAX∼−→ (iA)!(X)∞ and (iA)∗(X)−∞

∼−→ limAX.

In fact, such isomorphisms characterize colimiting cocones and limiting cones,as made precise by the following proposition, in which we consider the squares

AiA //

πA

AB

id

AiA //

πA

AC

id

1 ∞// AB, 1

−∞// AC.

=E(11.1.5)

Proposition 11.1.6. Let D be a derivator and let A ∈ Cat. The left Kanextension functor (iA)! : D(A) → D(AB) is fully faithful and X ∈ D(AB) liesin the essential image if and only if the canonical mate

colimA i∗A(X)→ X∞ (11.1.7)

associated to the left square in (11.1.5) is an isomorphism.

Proof. Since iA is fully faithful, so is (iA)! : D(A) → D(AB) and the essentialimage consists precisely of those X such that ε : (iA)!i

∗A(X)→ X is an isomor-

phism at ∞ (Proposition 8.2.20). Note that the left square in (11.1.5) can alsobe written as the pasting

Aid //

πA

AiA //

iA

~ id

AB

id

1 ∞// AB

id// AB.

Since the left square in this pasting is homotopy exact (Proposition 11.1.4),the functoriality of mates with pasting allows us to conclude that X lies in theessential image of (iA)! if and only if the canonical mate colimA i

∗A(X) → X∞

is an isomorphism.

DRAFT


Thus, a cocone is colimiting in the sense of Definition 11.1.1 if and only ifthe apex of it is canonically the colimit of the restriction to the base, justifyingthe terminology.

The canonical morphism colimA i∗A(X) → X∞, X ∈ D(AB), also admits a

different description which is inspired by the following trivial observation fromordinary category theory. Let C be a cocomplete category and let G : AB → Cbe a cocone on F = GiA : A → C. It is immediate from the definition of acolimit as an initial cocone, that there is always a comparison map from thecolimiting cocone on F to G and that this comparison map is an isomorphismif and only if G is also a colimiting cocone.

To extend this to derivators we make the following construction, which leadsto a derivator version of these comparison maps (see (11.1.16)). We then show inProposition 11.1.17 that colimiting cocones in derivators can be characterized bythese comparison maps being isomorphisms. Special cases of these comparisonmaps show up again in §12.5 and in [Gro16a].

Construction 11.1.8. We note that the cone construction A 7→ AB is functorialin A, thereby defining a functor (−)B : Cat → Cat . Moreover, the fully faithfulinclusions iA : A→ AB define a natural transformation

i : idCat → (−)B : Cat → Cat .

If A is a small category, then we can iterate the cocone construction andobtain the category (AB)B. This category is obtained from AB by adding anew terminal object∞+1. In particular, there is thus a morphism∞→∞+1.(Similarly, in (AC)C there is a morphism −∞− 1→ −∞.)

The category (AB)B corepresents morphisms of cocones. In more detail,related to this category there are the following two functors.

(i) The functor sA = iAB : AB → (AB)B is the component of the naturaltransformation i at AB. Thus, the behavior of sA on objects is given bya 7→ a and ∞ 7→ ∞ and, given a morphism of cocones, restriction alongsA yields the source of this morphism.

(ii) In a similar way we also have the functor tA = iBA : AB → (AB)B obtainedfrom iA : A → AB by an application of the cocone functor. On objectsthe functor tA is given by a 7→ a and ∞ 7→ ∞+ 1 and, given a morphismof cocones, restriction along tA yields the target of this morphism.

(An alternative description of the category (AB)B is as the join A ∗ [1] of Aand [1] = (0 < 1). In that description, the above functors are induced by0, 1: 1→ [1], i.e., we have sA = idA ∗ 0 and tA = idA ∗ 1.)

If D is a derivator, then we refer to D((AB)B) as the category of mor-phisms of cocones (with base A). The category comes with a source functors∗A : D((AB)B)→ D(AB) and a target functor t∗A : D((AB)B)→ D(AB).

The functor tA = iBA : AB → (AB)B induces a fully faithful left Kan extensionfunctor (tA)! : D(AB) → D((AB)B) which sends a cocone X ∈ D(AB) to a

DRAFT


morphism of cocones with target t∗A(tA)!(X) ∼= X. To identify its source, weconsider the naturality squares

AiA //

iA

id

AB

tA

AiA //

iA

AC

tA

ABsA// (AB)B, AC

sA// (AC)C.

@Hid (11.1.9)

The following two lemmas are a preparation for Proposition 11.1.13.

Lemma 11.1.10. For A ∈ Cat the squares (11.1.9) are homotopy exact.

Proof. We take care of the square on the left. Since the vertical functors arefully faithful, it suffices by Lemma 11.1.12 to show that for every derivatorthe canonical mate (iA)!i

∗A → s∗A(tA)! is an isomorphism when evaluated at

∞ ∈ AB. For this purpose we consider the following pasting on the left

Aid //

π

|

AiA //

iA

id

AB

tA

=

AiA //

π

AB

tA

1 ∞// AB

sA// (AB)B 1 ∞

// (AB)B,

in which the square on the left is homotopy exact by Proposition 11.1.4. Sincethe pasting agrees with the square to the very right, by functoriality of canonicalmates with respect to pasting, it is enough to show that that square is homotopyexact. As that square is isomorphic to a slice square, we conclude by (Der4).

To conclude the proof of Lemma 11.1.10 it remains to establish the followinglemma which is also of independent interest. In that lemma we consider anatural isomorphism living in a square of small categories

A′j//

u′

∼=

A

u

B′k// B.

(11.1.11)

The point of the following result is that it suffices to check objects in B′−u′(A′).

Lemma 11.1.12. Let (11.1.11) be a natural isomorphism in Cat such that uand u′ are fully faithful. The square (11.1.11) is homotopy exact if and only if forall derivators D the canonical mate βb′ : (u′!j

∗)b′ → (k∗u!)b′ is an isomorphismfor all b′ ∈ B′ − u′(A′).

Proof. By (Der2) we have to show that, under the above assumptions, thecanonical mate is always an isomorphism at objects of the form u′(a′), a′ ∈ A′.

DRAFT


To this end, we consider the pasting on the left in

1 //

id

id

(u′/u′(a′)) //

π

A′j//

u′

| ∼=

A

u

1 //

id

id

(u/uj(a′)) //

π

A

u

1id

// 1u′(a′)

// B′k// B, 1

id// 1

uj(a′)

// B,

in which the square in the middle is a slice square and hence homotopy ex-act by (Der4). The morphism 1 → (u′/u′(a′)) classifies the terminal object(a′, id : u′(a′)→ u′(a′)) (using that u′ is fully faithful, this is a terminal object),and the square on the left is hence also homotopy exact (Proposition 8.2.8).The functoriality of mates with pasting implies that βu′(a′) is an isomorphismif and only if the canonical mate associated to the pasting on the left is anisomorphism.

In the pasting on the right, the square on the right is a slice square and theleft square is induced by the functor 1 → (u/uj(a′)) classifying the terminalobject (j(a′), id : uj(a′) → uj(a′)) (using this time that u is fully faithful). Upto a vertical pasting with the component of the isomorphism (11.1.11) at a′,the above two pasting agree. Similar arguments as above hence show that thepasting on the right and thus also the pasting on the left is homotopy exact,thereby concluding the proof.

Lemma 11.1.10 allows us to canonically identify the source s∗A(tA)!(X) of themorphism of cocones (tA)!(X) as a colimiting cocone on i∗AX. And this alreadycharacterizes the essential image of (tA)!.

Proposition 11.1.13. Let D be a derivator and let A ∈ Cat. The functor(tA)! : D(AB) → D((AB)B) is fully faithful and Y ∈ D((AB)B) lies in theessential image if and only if the source cocone s∗AY ∈ D(AB) is colimiting.

Proof. Since tA is fully faithful, so is (tA)! : D(AB) → D((AB)B) and the es-sential image consists precisely of those Y such that ε : (tA)!t

∗A(Y ) → Y is an

isomorphism at ∞ (Proposition 8.2.20). To reformulate this we consider thepasting on the left in

AiA //

πA

~

ABtA //

tA

id

(AB)B

id

1 ∞// (AB)B

id// (AB)B.

(11.1.14)

In this pasting the square on the left is a slice square and hence homotopy exact.Thus, the functoriality of mates with pasting implies that Y lies in the essentialimage of (tA)! if and only if the canonical mate of this pasting is an isomorphism

DRAFT


on Y . Note that this pasting agrees with

Aid //

πA

|

AiA //

iA

id

ABsA //

id

id

(AB)B

id

1 ∞// AB

id// AB

sA// (AB)B.

Using similar arguments including Proposition 11.1.4 this time, the canonicalmate of this pasting is an isomorphism if and only if the source cocone s∗AY iscolimiting.

Definition 11.1.15. Let D be a derivator, A ∈ Cat , and X ∈ D(AB). The(cocone) comparison map is the coherent morphism

(tA)!(X)∞ → (tA)!(X)∞+1. (11.1.16)

For every cocone X ∈ D(AB), the underlying incoherent morphism of thecomparison map can be identified with a map

colimA i∗A(X)→ X∞.

In fact, this follows from the fact that, as a slice square, the left square in(11.1.14) is homotopy exact and since tA is fully faithful. The proof of the fol-lowing proposition shows that this morphism can be chosen to be the canonicalmate (11.1.7). The morphism (11.1.16) is a derivator version of the compari-son map between a cocone and the associated colimiting cocone from classicalcategory theory.

Proposition 11.1.17. Let D be a derivator and let A be small category. Acocone X ∈ D(AB) is colimiting if and only if the comparison map (11.1.16) isan isomorphism.

Proof. By Proposition 11.1.6, X ∈ D(AB) is colimiting if and only if the canon-ical mate of the left square in (11.1.5) is an isomorphism on X. To reformulatethis we consider the pasting on the left in

AiA //

πA

|

ABid //

id

id

AB

tA

1 ∞// AB

tA// (AB)B.

The square on the right is homotopy exact (Proposition 8.2.16) and, by thefunctoriality of mates with pasting, X is hence colimiting if and only if thecanonical mate of the above pasting is an isomorphism on X. We note that this

DRAFT


pasting can also be rewritten as the vertical pasting

AiA //

πA

AB

tA

1 ∞//

id

~

(AB)B

id

1∞+1

// (AB)B,

in which the bottom square is given by the morphism ∞ → ∞ + 1. Thefunctoriality of mates with respect to pasting and the homotopy exactness ofthe top slice square imply that the canonical mate of the vertical pasting factorsas

colimA i∗A(X) ∼−→ (tA)!(X)∞ → (tA)!(X)∞+1

and that the first morphism is an isomorphism. Thus, X ∈ D(AB) is colimitingif and only if (tA)!(X)∞ → (tA)!(X)∞+1 is an isomorphism.

We note that the proof of this proposition shows that for X ∈ D(AB) thecanonical mate (11.1.7) and the comparison map (tA)!(X)∞ → (tA)!(X)∞+1 sitin a commutative square

colimA i∗A(X) //

∼=

X∞

∼= η∞

(tA)!(X)∞ // (tA)!(X)∞+1,

in which the vertical isomorphisms are the canonical mate associated to thehomotopy exact square on the left in (11.1.5) and the adjunction unit, respec-tively.

By the following lemma the results of this section of course generalize to‘cocones with parameters’.

Lemma 11.1.18. Let D be a derivator and let A,B ∈ Cat. A cocone X ∈DA(BB) = D(A×BB) is colimiting if and only if the cocones Xa ∈ D(BB) forall a ∈ A are colimiting.

Proof. This is an immediate application of Corollary 10.3.7 to the fully faithfulfunctor iA : A→ AB.

11.2 Examples

In this section we illustrate the notion of colimiting cocones by specializingto two typical examples. We also show that a coherent cocone on a discrete

DRAFT

11.2. EXAMPLES 201

category is a coproduct cocone if and only if this is the case for the underlyingincoherent cocone.

Let us begin by recalling that the values of a derivator admit (co)products(Proposition 7.4.7).

Example 11.2.1. Let D be a derivator and let S be a discrete category. Thefunctor (iS)! : D(S)→ D(SB) is fully faithful with essential image the coproductcocones. Moreover, a cocone X ∈ D(SB) is a coproduct cocone if and only ifthe coherent comparison map ∐

s∈SXs → X∞

in t!(X) ∈ D((SB)B) is an isomorphims. (Here, t : SB → (SB)B again denotesthe target inclusion.)

In the special case of the discrete category 2 = 1t1 on two objects we have2B =y. The left Kan extension along the target inclusion t : y →yB extendsa cospan X ∈ D(y) as shown below to a coherent diagram t!(X) ∈ D(yB)encoding the canonical comparison map,

x

x

"" ))z t!7→ x t y // z.

y

@@

y

<<66

In this case, X is a coproduct cocone if and only if xty → z is an isomorphism.

Slightly imprecisely and in the terminology of §7.4, we now show that ho-motopy coproducts and categorical coproducts coincide in every derivator.

Proposition 11.2.2. Let D be a derivator and let S be a discrete category. Acocone X ∈ D(SB) is a coproduct cocone if and only if the underlying diagram

diaSB(X) ∈ D(1)SB

is a coproduct cocone, i.e., it exhibits X∞ as the coproductof Xs, s ∈ S.

Proof. Let us consider the following pasting diagram

D(1)SB i∗S // D(1)S

colimS

D(SB)i∗S //

diaSB

OO

id

D(S)

' diaS

OO

(πS)!//

∼=

D(1)

D(SB)∞∗

//

id

OO

D(1)id

GG

π∗S

OO

ε

DRAFT


in which the canonical isomorphism in the top triangle follows from Propo-sition 7.4.7, the bottom triangle is the counit, the top square commutes andthe bottom square is populated by the natural transformation induced by thesquare (11.1.5) (in the special case of A = S). Let us recall that diaS π∗S = ∆S

(Lemma 7.4.5). Thus, the vertical pasting of the triangles evaluated on anobject y ∈ D(1) is the fold map

∇ : colimS ∆S(y) =∐s∈S

y → y.

An evaluation of the vertical pasting of the squares at X ∈ D(SB) is the naturaltransformation i∗SdiaSB(X) → ∆S(X∞) induced by the structure maps of X.Thus, the total pasting applied to X yields the map

∐s∈S Xs → X∞ induced

from the underlying diagram diaSB(X), i.e., the map detecting if diaSB(X) is acoproduct cocone in the usual sense. Since the upper two natural transforma-tions are invertible, this is the case if and only if the pasting of the lower twonatural transformations is an isomorphism on X. Note that this latter pastingis the canonical mate associated to the square on the left in (11.1.5) (in thespecial case of A = S), which by Proposition 11.1.6 is an isomorphism if andonly if X ∈ D(SB) is a coproduct cocone.

We now turn to examples related to squares and cocartesian squares.

Example 11.2.3. Let us consider A = p so that AB = and (AB)B = P is givenby

(0, 0) //

(1, 0)

(0, 1) //

))

(1, 1)

$$

(2, 2).

Let t = tp : → P be the ‘target inclusion’ tp = iBp sending (1, 1) to (2, 2).

Let D be a derivator. Using the usual notation for pushout objects, theequivalence (ip)! : D(p) ∼−→ D()cocart (Lemma 9.2.2) sends a span to the cor-responding cocartesian square,

x //

y

7→

x //

y

z z // y tx z.

Moreover, for every X ∈ D() looking like the square on the left in the following

DRAFT

11.3. DETECTION LEMMAS FOR COLIMITING COCONES 203

diagramx //

y

x //

y

z // w t!7→ z //

))

y tx z

##w

there is the coherent diagram t!(X) ∈ D(P ) as shown on the right. This dia-gram encodes the canonical comparison morphism y tx z → w, and the originalsquare X ∈ D() is cocartesian if and only if this comparison morphism is anisomorphism.

Example 11.2.4. As a special case of the previous example, let D be a pointedderivator and let X ∈ D() be a square vanishing at (0, 1),

xf//

y

xf//

y

cof(f)

0 // z t!7→ 0 //

((

Cf

!!z.

The square X is a cofiber square if and only if the comparison map Cf → z int!(X) is an isomorphism.

Specializing further to a square X ∈ D() which vanishes at (1, 0) and (0, 1),

x //

0

x //

0

0 // y t!7→ 0 //

''

Σx

y,

we deduce that X is a suspension square if and only if the comparison mapΣx→ y in t!X is an isomorphism.

Remark 11.2.5. It can be shown that the canonical comparison maps con-structed in Example 11.2.4 are coherent refinements of the maps constructedin Lemma 9.2.22 and Lemma 9.2.24; see [Gro16a].

11.3 Detection lemmas for colimiting cocones

In this section we establish criterions which allow us to detect cocartesiansquares and, more generally, colimiting cocones in larger diagrams. These cri-

DRAFT


terions enable us to avoid using the same kind of homotopy finality argumentsagain and again.

For example, in the discussion of cofiber sequences we verified by hand thatcertain squares in larger diagrams are cocartesian. Here we will revisit thisconstruction to illustrate the criterion, and there will be additional applicationsin this book (for example in the discussion of iterated cofiber constructions§12.1, total cofibers §12.5, octahedron diagrams §15.5, Barratt–Puppe sequences§16.2, refined octahedron diagrams §16.4, and the doubly-infinite WaldhausenS•-construction in [Gro16a]).

Despite its technical character, the following lemma is used a lot when inthe context of ‘larger’ coherent diagrams it seems ‘obvious’ that certain squaresare cocartesian. As always, there is also a dual version of this result.

Lemma 11.3.1. Let D be a derivator and let u : A→ B, v : → B be functors.Suppose that there is a full subcategory B′ ⊆ B such that

(i) u(A) ⊆ B′,

(ii) v(p) ⊆ B′ and v(1, 1) /∈ B′, and

(iii) the functor p→ (B′/v(1, 1)) induced by v is a right adjoint.

Then for any X ∈ D(A) the diagram v∗u!(X) is a cocartesian square.

Proof. By assumption, u factors as j u′ : A→ B′ → B, and the situation canbe summarized by the commutative diagram

(B′/v(1, 1))

p

p

99

//

ip

B′

j

v

// B A.u

oo

u′ee

To be completely specific, the induced functor p→ (B′/v(1, 1)) sends x ∈ p to(v(x), v(x→ (1, 1))

).

We begin by reformulating that v∗Y ∈ D() is cocartesian for a coherentdiagram Y ∈ D(B). Using the functoriality of mates with pasting, the homotopyexactness of slice squares, and Proposition 8.2.20, the reader easily observes thatv∗Y is cocartesian if and only if the canonical mate of the pasting

pid //

π

pip //

ip| id

v //

id

| id

B

id

1(1,1)//

id//

v// B

DRAFT


is an isomorphism on Y . This pasting agrees with the following pasting, in whichthe square on the left is homotopy exact by assumption (Proposition 8.2.8),

p //

π

id

(B′/v(1, 1))p//

π

B′j//

j

| id

B

id

1id

// 1v(1,1)

// Bid// B.

The square in the middle is a slice square, which is hence also homotopy exact.Using again the functoriality of mates with pasting and Proposition 8.2.20 weconclude that v∗Y is cocartesian as soon as Y lies in the essential image ofj! : D(B′) → D(B). To conclude the proof, it remains to observe that forX ∈ D(A) there are canonical isomorphisms u!(X) ∼= j!u

′!(X), hence showing

that v∗u!(X) is cocartesian.

This detection lemma admits refinements and we include some of them inthis section. But we first illustrate the lemma and revisit the construction ofcofiber sequences, thereby partially adressing the claims made in Remark 9.3.9.

Example 11.3.2. Let k = j i : A → B → be the functor as in (9.3.7). Wewant to apply the criterion to show that in every diagram in the essential imageof k! : D(A)→ D() both squares are cocartesian, and this is done by applyingthe criterion first to i and then to j.

(i) In the first step, we let v = ι01 be the inclusion of the square on the leftand choose B′ to be the image i(A). Clearly, the first two assumptionsof Lemma 11.3.1 are satisfied. As for the remaining assumption, in thiscase the functor induced by v = ι01 is even an isomorphism, and we hencededuce that ι∗01i!(X) is cocartesian for every X ∈ D(A).

(ii) In the second step, we choose v = ι12 to be the inclusion of the squareon the right and let B′ be the image j(B) (the reader easily resolves thisoverload of notation). The first two assumptions of Lemma 11.3.1 areobviously satisfied and one checks that the functor p→ −(2, 1) inducedby v = ι12 is a right adjoint (for example by means of Lemma 11.3.3). Weconclude that ι∗12j!(X) is cocartesian for every X ∈ D(B).

These two steps are combined to yield the statement about k!.

In slightly more complicated applications of the detection lemma, the fol-lowing trivial lemma is helpful in constructing adjunctions between posets.

Lemma 11.3.3. Let P be a poset and let ι : P ′ → P be the inclusion of a fullsubposet such that

(i) for every x ∈ P there is a unique minimal Lx ∈ P ′ such that x ≤ Lx and

(ii) for x ≤ y in P it follows that Lx ≤ Ly.

The functor L : P → P ′ : x 7→ Lx is left adjoint to ι : P ′ → P .

DRAFT


Proof. By assumption we obtain a functor L : P → P ′ satisfying Lι = idP ′ andx ≤ ιLx, x ∈ P. Since P ′, P are posets, this defines a transformation η : id→ ιL.Moreover, again since P ′, P are posets the triangular identities (see (A.1.3)) for(L, ι, η, id) are automatic.

While the detection lemma for cocartesian squares is used more frequently,it immediately extends to general colimiting cocones.

Lemma 11.3.4. Let D be a derivator, let C ∈ Cat, and let u : A → B andv : CB → B be functors. Suppose that there is a full subcategory B′ ⊆ B suchthat

(i) u(A) ⊆ B′,

(ii) v(C) ⊆ B′ and v(∞) /∈ B′, and

(iii) the functor C → (B′/v(∞)) induced by v is a right adjoint.

Then for any X ∈ D(A) the cocone v∗u!(X) ∈ D(CB) is colimiting.

Proof. We leave it as an exercise to the reader to verify that the proof ofLemma 11.3.1 can easily be adapted to this more general situation.

Remark 11.3.5. Is is obvious from the proofs of Lemma 11.3.1 and Lemma 11.3.4that in both cases one can weaken the third assumption to assuming that thefunctors p→ (B′/v(1, 1)) and C → (B′/v(∞)) induced by v, respectively, arehomotopy final (Definition 8.2.12).

The above criterions show that diagrams in the essential image of suitableKan extension functors enjoy certain exactness properties, i.e., make certain(co)cones (co)limiting. In the remainder of this section, we include similar re-sults which allow us to characterize the essential image of suitable Kan extensionfunctors. We suggest the reader to skip the details on a first reading, and toreturn to this section with applications at hand.

We begin by collecting sufficient conditions such that a left Kan extensionfunctor precisely adds a colimiting cocone.

Lemma 11.3.6. Let D be a derivator, let C ∈ Cat , and let u : A→ B be a fullyfaithful functor between small categories such that the following two conditionsare satisfied.

(i) The complement B − u(A) consists of precisely one object b0.

(ii) There is a small category C and a homotopy exact square

C

iC

j//

id

A

u

CBk// B

(11.3.7)

such that k satisfies k(∞) = b0.

DRAFT


The functor u! : D(A)→ D(B) is fully faithful and induces an equivalence ontothe full subcategory spanned by all X ∈ D(B) such that k∗(X) ∈ D(CB) is acolimiting cocone.

Proof. Since u : A → B is fully faithful, it follows that u! : D(A) → D(B) isfully faithful and that X ∈ D(B) lies in the essential image of u! if and onlyif the counit ε : u!u

∗(X) → X is an isomorphism if and only if the componentεb0 : u!u

∗(X)b0 → Xb0 is an isomorphism (Proposition 8.2.20). By our assump-tion on k and (Der2), this is the case if and only if k∗ε : k∗u!u

∗(X)→ k∗(X) isan isomorphism.

To re-express this differently, let us consider the pasting on the left in

Cj//

iC

id

Au //

u

id

B

id

=

CiC //

iC

~ id

CBk //

id

id

B

id

CBk// B

id// B CB

id// CB

k// B.

Using the functoriality of mates with respect to pasting and the assumed ho-motopy exactness of the square to the very left, we observe that X ∈ D(B)lies in the essential image of u! if and only if the canonical mate associated tothe pasting on the left is an isomorphism on X. Since the above two pastingsagree this is the case if and only if the canonical mate of the pasting on theright is an isomorphism on X. Using that iC is fully faithful we conclude theproof by observing that this is by Proposition 8.2.16 the case if and only ifk∗(X) ∈ D(CB) lies in the essential image of (iC)! : D(C) → D(CB), i.e., is acolimiting cocone.

Remark 11.3.8. Lemma 11.3.6 admits the following generalization in which thesquare (11.3.7) is replaced by a homotopy exact square

Cj//

w

| id

A

u

C ′k// B

such that the following two conditions are satisfied.

(i) The functors u : A→ B and w : C → C ′ are fully faithful.

(ii) The objects in B − u(A) lie in k(C ′ − w(C)).

Under these assumptions the functor u! : D(A) → D(B) is fully faithful andinduces an equivalence onto the full subcategory spanned by all X ∈ D(B) suchthat k∗(X) lies in the essential image of v! : D(C) → D(C ′). In fact, the proofof Lemma 11.3.6 also works under these more general assumptions.

DRAFT


We are also interested in the special case of Lemma 11.3.6 in which C = p,i.e., in which we consider a square of small categories

p

ip

j//

| id

A

u

k// B

(11.3.9)

The following two propositions give sufficient conditions guaranteeing thatassumption (ii) of Lemma 11.3.6 is satisfied. Although a bit technical on a firstview, this result often applies and will be used a lot when it is ‘obvious thatcertain left Kan extensions precisely amount to adding a colimiting cocone’. Weagain begin with the special case of cocartesian squares.

Proposition 11.3.10 (Workhorse Proposition, special case). Let (11.3.9)be a square of small categories such that the following conditions are satisfied.

(i) The functor u : A → B is fully faithful and the complement B − u(A)consists of precisely one object b0.

(ii) The functor k : → B sends (1, 1) to b0.

(iii) The induced functor r : p→ (A/b0) : c 7→ (j(c), k(c → (1, 1))) is a rightadjoint.

Under these assumptions, for every derivator D the functor u! : D(A)→ D(B)is fully faithful and induces an equivalence onto the full subcategory spanned byall X ∈ D(B) such that k∗(X) ∈ D() is a cocartesian square.

Proof. By Lemma 11.3.6 it suffices to show that (11.3.7) is homotopy exact. Itfollows from Lemma 11.1.12 that this is the case if and only if for every derivatorD the canonical mate (ip)!j

∗ → k∗u! is an isomorphism at (1, 1). To re-expressthis differently, we consider the following pasting on the left in

pid //

π

pj//

ip

id

A

u

=

pr //

π

~ id

(A/b0)p//

π

A

u

1(1,1)

// k// B 1

id// 1

b0

// B

in which the left square is a slice square. Using the functoriality of mates withpasting and the homotopy exactness of slice squares, we deduce that (11.3.7) ishomotopy exact if and only if the pasting on the left is homotopy exact. Notethat that pasting can also be written as the pasting on the right in which theright square is a slice square. In that pasting also the square on the left ishomotopy exact since r is by assumption a right adjoint (Proposition 8.2.8),and the functoriality of mates with pasting hence concludes the proof.

DRAFT


We invite the reader to revisit Example 11.3.2 and Example 11.3.2 with thisproposition at hand.

Proposition 11.3.11 (Workhorse Proposition, general case). Let (11.3.7)be a square of small categories such that the following conditions are satisfied.

(i) The functor u : A → B is fully faithful and the complement B − u(A)consists of precisely one object b0.

(ii) The functor k : CB → B sends ∞ to b0.

(iii) The induced functor r : C → (A/b0) : c 7→ (j(c), k(c → ∞)) is a rightadjoint.

Under these assumptions, for every derivator D the functor u! : D(A)→ D(B)is fully faithful and induces an equivalence onto the full subcategory spanned byall X ∈ D(B) such that k∗(X) ∈ D(CB) is a colimiting cocone.

Proof. We leave it as an exercise to the reader to adapt the proof of Proposi-tion 11.3.10 so that it also covers this more general case.

In [Gro16a] we also collect parametrized versions of these two propositions,which will be formulated in terms of Kan extension morphisms of derivators.

Todo: Section on coherent(co)units in this case (generalKan extensions in volume II)

DRAFT


DRAFT

Part III

Pointed and stablederivators

211

DRAFT

Chapter 12

Iterated cofibers, totalcofibers, and iterated cones

In this chapter we include a short discussion of iterated cofiber and fiber con-structions. This is meant to indicate that, once a few basic lemmas about Kanextensions are in place, it is rather straightforward to develop a convenientcalculus of coherent morphisms and coherent squares in pointed derivators. Ad-ditional bits of such a calculus will be developed in §13 and [Gro16a].

We begin this chapter by a discussion of iterated cofibers of coherent mor-phisms, showing that the cube of the cofiber is the suspension. Next, we turnto relations between the various cones associated to pairs of composable mor-phisms. These observations are closely related to the rotation axiom and theoctahedron axiom as enjoyed by the canonical triangulations in stable deriva-tors.

We then discuss various cofiber and fiber constructions which apply to co-herent squares in pointed derivators. Parametrized Kan extensions can be usedto form cones in two different directions, which composed with the usual coneconstruction lead to two iterated cone constructions. It turns out that these arenaturally isomorphic and that they are also naturally isomorphic to the totalcofiber of a square. (While the commutativity of cone operations is satisfied inpointed derivators, the commutativity of cone and fiber operations turns out tocharacterize stable derivators; see §15 and [Gro16a].)

In §12.1 we study iterated cofiber constructions of morphisms and in §12.2we study relations between the various cones associated to pairs of composablemorphisms. In §12.3 we introduce total cofibers of squares and illustrate thisnotion by some examples in §12.4. In §12.5 we discuss iterated cofiber construc-tions on squares and show these to be naturally isomorphic to total cofibers.This allows us in §12.6 to obtain additional formulas for specific total cofibers.

213

DRAFT

214 CHAPTER 12. ITERATED COFIBER CONSTRUCTIONS

12.1 Iterated cofibers of morphisms

Let us recall that in §9.5 we defined cofiber sequences by two iterations of thecofiber construction. Here we show that if we iterate the cofiber constructionones more then we obtain the suspension morphism up to natural isomorphism.

Construction 12.1.1. Let B ⊆ [2] × [2] be the full subcategory obtained byremoving (0, 2),

(0, 0) //

(1, 0) //

(2, 0)

(0, 1) // (1, 1) //

(2, 1)

(1, 2) // (2, 2).

We want to extend a coherent morphism f ∈ D([1]) to a diagram of the aboveshape by first adding certain zero objects and then adding cocartesian squares.

To this end, let A ⊆ B be the full subcategory spanned by (0, 0), (1, 0) and(0, 1), (2, 0), (1, 2). Related to this there is the fully faithful functor i : [1] → Aclassifying the horizontal morphism (0, 0)→ (1, 0) and the obvious fully faithfulinclusion j : A → B. At the level of coherent diagrams in a derivator D weobtain the corresponding Kan extension functors

D([1])i∗→ D(A)

j!→ D(B). (12.1.2)

Lemma 12.1.3. Let D be a pointed derivator and let D(B)cof ⊆ D(B) be thefull subcategory spanned by all X ∈ D(B) satisfying the following conditions.

(i) The diagram X vanishes at (0, 1), (2, 0), and (1, 2).

(ii) The restrictions of X to the left square, to the top right square, and tothe bottom right square are cocartesian.

The functors (12.1.2) induce an equivalence of categories

D([1]) ' D(B)cof .

Proof. Since the proof is similar to the one of Proposition 9.5.8 we allow our-selves to be a bit sketchy. Both functors i : [1]→ A and j : A→ B are fully faith-ful, hence the same is true for the Kan extension functors i∗ : D([1]) → D(A)and j! : D(A) → D(B). The functor i : [1] → A is a sieve and i∗ is hence rightextension by zero (Corollary 9.1.6) and as a such it induces an equivalence ontothe full subcategory D(A)ex of D(A) spanned by all diagrams which vanish at(0, 1), (2, 0), and (1, 2).

We next analyze the functor j : A→ B and begin by observing that it factorsas the composition of fully faithful inclusions

j : Aj1→ A1

j2→ A2j3→ B

DRAFT

12.1. ITERATED COFIBERS OF MORPHISMS 215

where the individual steps add the objects (1, 1), (2, 1), and (2, 2), respectively.The left Kan extension functor j! : D(A) → D(B) is accordingly naturally iso-morphic to the composition

D(A)(j1)!→ D(A1)

(j2)!→ D(A2)(j3)!→ D(B);

see Lemma 7.6.1. Based on the detection lemmas of §11.3, one now observesthat each of these three left Kan extensions precisely amounts to adding a newcocartesian square in the obvious sense. To illustrate this tool, we includecomplete details.

(i) In order to understand (j1)! we apply Proposition 11.3.10 to u = j1 andthe functor k : → A1 classifying the square. The first two assumptionsof the proposition are obviously satisfied. As for the third one, it remainsto show that the induced functor p→ (j1/(1, 1)) is a right adjoint, but it iseven an isomorphism in this case. Thus, (j1)! : D(A)→ D(A1) induces anequivalence onto the full subcategory spanned by all X ∈ D(A1) makingthe square commutative.

(ii) We now apply Proposition 11.3.10 to j2 and the functor k : → A2

classifying the square on the right. Again, the first two assumptions of theproposition are obvious, while the right adjointness of the induced functorp→ (j2/(2, 1)) is immediate from Lemma 11.3.3. Thus, also the essentialimage of (j2)! is characterized by the cocartesianness of the square on theright.

(iii) Finally, in order to understand (j3)! we apply Proposition 11.3.10 to itand the functor k : → B classifying the lower right square. The firstassumptions again being trivial, it remains to verify that the inducedfunctor p→ (j3/(2, 2)) is a right adjoint. This again follows immediatelyfrom Lemma 11.3.3, showing that (j3)! precisely adds a cocartesian square.

As an upshot, the functor j! hence amounts precisely to adding three cocartesiansquares. Since j! is fully faithful, combined with the first part of the proof weare thus done.

In later applications of the detection results from §11.3 (like for exampleProposition 11.3.10) we allow ourselves to be more sketchy and to leave moredetails to the reader.

The equivalence D([1]) ' D(B)cof of Lemma 12.1.3 sends a coherent mor-phism (f : x→ y) ∈ D([1]) to a coherent diagram looking like

xf//

y //

g

02

01// z

h //

x′

f ′

03// y′

(12.1.4)

DRAFT


and making all three squares cocartesian. In this diagram, the objects 01, 02, 03

denote zero objects in D(1) and the subscripts can be ignored for now. We nextshow that the vertical morphism (f ′ : x′ → y′) is naturally isomorphic to Σf .

To make this precise, let us recall from Lemma 9.1.3 that pointed derivatorsare stable under shifting. In particular, if D is pointed then so is D [1], andthe pointed derivator D [1] comes with a suspension functor Σ: D([1])→ D([1]).The classical case of the following result is [Pup58, Satz 4] in the context ofpointed topological spaces.

Proposition 12.1.5. For every pointed derivator D there is a natural isomor-phism

cof3 ∼= Σ: D([1])→ D([1]).

Proof. Let B and D(B)cof ⊆ D(B) be as in Lemma 12.1.3, and let us considerthe functors k : [1]→ B and l : [1]→ B classifying the morphisms

k(0→ 1) = (0, 0)→ (1, 0) and l(0→ 1) = (2, 1)→ (2, 2),

respectively. Given Y ∈ D(B)cof , it is immediate from Lemma 12.1.3 that thethree squares in Y are cofiber squares (Definition 9.2.8), and we hence obtain anatural isomorphism

ψ : cof3(k∗Y ) ∼−→ l∗Y.

The main step consists of showing that for every Y ∈ D(B)cof there also is anatural isomorphism

φ : Σ(k∗Y ) ∼−→ l∗Y.

To this end, let Y ∈ D(B)cof with underlying diagram (12.1.4). There is aunique functor q : [1] × → B such that q∗(Y ) ∈ D([1] × ) has underlyingdiagram looking like

x //

f

02

y //

02

01

// x′

f ′

03// y′.

(12.1.6)

In this diagram, the [1]-coordinate is drawn diagonally while the -coordinateis as before. The decoration of the objects implies that there is at most onesuch functor since the categories under consideration are posets. We leave it tothe reader that such a functor q : [1]×→ B actually exists, and we note thatit satisfies

q(−, (0, 0)) = k : [1]→ B and q(−, (1, 1)) = l : [1]→ B.

The diagram Z = q∗(Y ) ∈ D([1] × ) = D [1]() has the property thatboth squares Z0, Z1 ∈ D() are cocartesian. In fact, Z0 is cocartesian as a

DRAFT

12.1. ITERATED COFIBERS OF MORPHISMS 217

horizontal pasting of two cocartesian squares and Z1 as a vertical pasting of twococartesian squares (Proposition 9.3.10). Thus, by Corollary 10.3.10, the squareZ ∈ D [1]() is cocartesian, when considered as an object of D [1]. Since thiscocartesian square vanishes at (1, 0), (0, 1), it is a suspension square and hencequalifies for the construction of Σ: D [1](1) → D [1](1), i.e., there is a canonicalisomorphism

φZ : Σ(Z0,0) ∼−→ Z1,1;

see Lemma 9.2.22. Unraveling definitions we see that

Z0,0 = (0, 0)∗q∗Y = k∗Y and Z1,1 = (1, 1)∗q∗Y = l∗Y,

and φ is hence a natural isomorphism φ : Σ(k∗Y ) ∼−→ l∗Y.If we now start with a coherent morphism f ∈ D([1]), then by Lemma 12.1.3

there is a diagram Y = Y (f) ∈ D(B)cof and a natural isomorphism k∗Y ∼= f .Hence we obtain a chain of natural isomorphisms

cof3(f) ∼= cof3(k∗Y )ψ∼= l∗Y

φ−1

∼= Σ(k∗Y ) ∼= Σf,

concluding the proof.

We will see in §15 that (strong) stable derivators canonically take values intriangulated categories. Recall the rotation axiom (T2) for triangulated cate-gories (Definition 5.3.1) by which a distinguished triangle can be rotated back-wards. At the level of coherent morphisms this amounts to passing to the cofiberof the coherent morphism. Note that the coherent diagram (12.1.4) encodes acofiber sequence for the coherent morphism f and its cofiber cof(f). Ignoringsigns for the moment, Proposition 12.1.5 hence is the reason for the rotationaxiom.

We conclude this section by the following refined version of Corollary 9.4.1,which is easily seen to have a dual form.

Corollary 12.1.7. Let D be a pointed derivator and let f ∈ D([1]).

(i) If f is an isomorphism, then Cf ∼= 0.

(ii) If Cf ∼= 0, then Σf ∈ D([1]) is an isomorphism.

Proof. The first statement follows from Corollary 9.4.1. Let us now assume thatCf ∼= 0 and let us again consider a coherent diagram as in (12.1.4). By the abovediscussion the object z is hence trivial. Moreover, since the bottom square iscocartesian and isomorphisms are stable under cobase change (Proposition 9.3.5)we conclude that also f ′ is an isomorphism. But this morphism is naturallyisomorphic to Σf (see the proof of Proposition 12.1.5), thereby concluding theproof.

Warning 12.1.8. Let D be a pointed derivator and let f ∈ D([1]). In general,neither of the two conclusions in Corollary 12.1.7 admits a converse. In fact,there are the following counter-examples (which are based on Remark 9.2.16).

DRAFT


(i) It is, in general, not true that the triviality of the cone, Cf ∼= 0, impliesthat f would be an isomorphism. To obtain a simple counter-example, letus consider the derivator y(C) represented by a complete, cocomplete, andpointed category. In this case, the cone functor reduces to the cokernelconstruction. It is immediate that in such a category the cokernel of anepimorphism is the zero object. Hence, we obtain a counter-example byany such category which admits an epimorphism which is not an isomor-phism.

(ii) Also it is in general not true, that if Σf is an isomorphism, then Cfis necessarily trivial. To obtain a simple counter-example, let us againconsider a represented, pointed derivator y(C). In this case, the suspensionfunctor Σ: C[1] → C[1] is the constant functor which takes as value theidentity morphism of the zero object. In particular, Σf is an isomorphismfor every f : [1]→ C. Since the cone functor is the cokernel, any morphismwith non-trivial cokernel provides a counter-example.

In the case of stable derivators, however, these three statements are equiva-lent; see §15.

12.2 Cones of compositions

In this section we include a short discussion of cones of compositions of coherentmorphisms. By duality, there are similar results for fibers. In the case of stablederivators, the content of this discussion is classically reflected by the octahedronaxiom of triangulated categories; see §15.5 and §16.4.

We consider the poset [2] = (0 < 1 < 2) as a category, so that this categorycorepresents two composible morphisms. Given a pointed derivator D and a

coherent pair of composable morphisms X = (xf→ y

g→ z) ∈ D([2]) we write

gf = (ι02)∗(X) = (x→ z) ∈ D([1])

for the restriction along the functor ι02 : [1] → [2] classifying the composition0→ 2 in [2]. Similarly, let [4] be the poset (0 < 1 < 2 < 3 < 4).

Construction 12.2.1. Let B ⊆ [2] × [4] be the full subcategory obtained byremoving (2, 0) and (0, 4); see Figure 12.1. There is a fully faithful embeddingi : [2]→ B : x 7→ (x, 2) and this embedding factors as

i : [2]i1→ A1

i2→ A2i3→ A3

i4→ B,

where the full subcategories A1, A2, A3 ⊆ B are defined as follows and thefunctors ik, 1 ≤ k ≤ 4, are the obvious inclusions.

(i) The category A1 is obtained from the image of i by also adding the objects(0, 3), (1, 4).

(ii) The category A2 also contains the objects (1, 3), (2, 3), (2, 4).

DRAFT

12.2. CONES OF COMPOSITIONS 219

(0, 0) //

(1, 0)

(0, 1) //

(1, 1) //

(2, 1)

(0, 2) //

(1, 2) //

(2, 2)

(0, 3) // (1, 3) //

(2, 3)

(1, 4) // (2, 4).

Figure 12.1: The shape of B ∈ Cat .

(iii) The category A3 is obtained from A2 by adding the objects (2, 1), (1, 0).

For every derivator D there are associated fully faithful Kan extension functors

D([2])(i1)∗→ D(A1)

(i2)!→ D(A2)(i3)!→ D(A3)

(i4)∗→ D(B) (12.2.2)

At the level of pointed derivators the above construction does the following.

Proposition 12.2.3. For every pointed derivator D the functors (12.2.2) inducean equivalence D([2]) ' D(B)ex onto the full subcategory D(B)ex ⊆ D(B)spanned by all X ∈ D(B) satisfying the following three exactness properties.

(i) The diagram X vanishes at (0, 3), (1, 4) and (2, 1), (1, 0).

(ii) The diagram X makes the three lower squares cocartesian.

(iii) The diagram X makes the three upper squares cartesian.

Proof. In order to prove this statement it suffices to obtain a good understandingof each of the functors showing up in (12.2.2). We describe these functors oneby one.

(i) Note that the funcor i1 is a sieve, and that (i1)∗ is consequently rightextension by zero (Corollary 9.1.6), thereby inducing an equivalence withessentially image determined by the vanishing at (0, 3), (1, 4).

(ii) The functor i2 factors through three intermediate step, each step obtainingthe next object in the lexicographic order. We leave it to the reader toapply Proposition 11.3.10 to each of these steps in order to verify thateach step precisely amounts to adding a new cocartesian square. In thelast two cases it might be useful to apply Lemma 11.3.3 in order to checkthe third assumption of Proposition 11.3.10.

DRAFT


(iii) We note that the funcor i3 is a cosieve, and that (i3)! hence is left exten-sion by zero, the essential image being characterized by the vanishing at(2, 1), (1, 0).

(iv) Finally, the functor i4 factors through three intermediate steps and thereader easily applies Proposition 11.3.10 and Lemma 11.3.3 in order toshow that (i4)∗ precisely amounts to adding three cartesian squares.

Using once more the fully faithfulness of Kan extensions along fully faithfulfunctors (Proposition 8.2.16), we can assemble these four steps to conclude theproof.

Thus, given X = (xf→ y

g→ z) ∈ D([2]) in a pointed derivator D , under theequivalence we obtain a coherent diagram Y ∈ D(B)ex looking like

u //

0

v //

w //

0

x

f//

yg//

z

0 // u //

v

0 // w

(12.2.4)

and enjoying the exactness properties as indicated. We can easily identify theobjects in such a diagram.

Proposition 12.2.5. Let D be pointed derivator, let X = (xf→ y

g→ z) ∈D([2]), and let Y ∈ D(B)ex be the corresponding diagram looking like (12.2.4).

(i) There are canonical isomorphisms Cf ∼= u, C(gf) ∼= v, and Cg ∼= w.

(ii) There is a coherent map (Cf → C(gf)) ∈ D([1]) and a canonical isomor-phism C(Cf → C(gf)) ∼−→ Cg.

(iii) There are canonical isomorphisms u ∼= Ff , v ∼= F (gf), and w ∼= Fg.

(iv) There is a coherent map (F (gf) → Fg)) ∈ D([1]) and a canonical iso-morphism Ff ∼−→ F (F (gf)→ F (g)).

Proof. The proof follows from a diagram chase in (12.2.4). By duality it sufficesto settle the first two statements. Considering the square with f as top horizon-tal edge, there is a canonical map Cf → u which happens to be an isomorphism

DRAFT

12.2. CONES OF COMPOSITIONS 221

since the square is a cofiber square (Lemma 9.2.24). Since horizontal past-ings of cocartesian squares are again cocartesian (Proposition 9.3.10), a similarargument applied to the obvious pasting square in (12.2.4) yields a canonicalisomorphism C(gf) ∼= v. The case of w follows similarly by a vertical pasting.For the second statement it suffices to consider the very bottom square and tonote that it is a cofiber square.

In the case of a stable derivator, we shall soon see that the two parts of theabove proposition fit together very nicely. Moreover, we leave it to the inter-ested reader to also come up with formulas, which, at the price of introducingsome suspensions, allow us to express the cones C(gf) and Cf in terms of therespective remaining cone objects, and similarly for fibers. For example, weinvite the reader to verify that there are canonical isomorphisms

C(C(gf)→ Cg) ∼= ΣCf and C(Cg → ΣCf) ∼= ΣC(gf).

Again, in the stable case, these formulas admit variants expressing Cf andC(gf) in terms of the remaining two morphisms only; see §16.

Here instead, to conclude this section we content ourselves by drawing someimmediate consequences from the assumption that any of the morphisms f, g, gfis an isomorphism.

Proposition 12.2.6. Let D be a pointed derivator and let (xf→ y

g→ z) ∈D([2]).

(i) If f is an isomorphism, then Cf ∼= Ff ∼= 0 and there are canonicalcoherent isomorphisms

C(gf) ∼−→ Cg and F (gf) ∼−→ Fg.

(ii) If g is an isomorphism, then Cg ∼= Fg ∼= 0 and there are canonicalcoherent isomorphisms

Cf ∼−→ C(gf) and Ff → F (gf).

(iii) If gf is an isomorphism, then C(gf) ∼= F (gf) ∼= 0 and there are canonicalcoherent isomorphisms

ΣCf ∼−→ Cg and Ff ∼−→ ΩFg.

Proof. The proof is again a diagram chase in (12.2.4). In the first case, we knowfrom Corollary 9.4.1 that Cf ∼= Ff ∼= 0. As for the first isomorphism, let usconsider the bottom cofiber square in (12.2.4). Since the left vertical morphismin that square is an isomorphism and since isomorphisms are stable under cobasechange (Proposition 9.3.5), we conclude that C(gf) → Cg is an isomorphism.Moreover, let us now consider the cartesian square in (12.2.4) with f as bottomhorizontal morphism. Using the stability of isomorphism under base change

DRAFT


(Proposition 9.3.5) we deduce that F (gf)→ Fg is an isomorphism. The secondcase is obtained in the same way as the reader easily verifies.

Finally, let us assume that gf is an isomorphism, so that C(gf) ∼= F (gf) ∼= 0by Corollary 9.4.1. This implies that the very bottom and the very top squaresin (12.2.4),

Cf //

0

Ff //

0

0 // Cg, 0 // Fg,

are suspension and loop squares, respectively. These yield by Lemma 9.2.22 theintended canonical isomorphisms ΣCf ∼−→ Cg and Ff ∼−→ ΩFg.

12.3 Total cofibers of squares

In this section we define total cofibers and total fibers of coherent squares inpointed derivators. We illustrate these notions by quite some examples.

The results obtained in this section and the remainder of this chapter belongto a fairly rich calculus of squares, cubes, and hypercubes in pointed derivators,and some additional bits will be discussed in [Gro16a]. The reader less interestedin these results can continue with §14 as the content of the remainder of thischapter as well as §13 will only briefly come up again later.

As a preparation for the definition of the total cofiber, recall from §11.1the construction of canonical comparison maps between cocones and colimitingcocones. As spelled out in Example 11.2.3, in the special case in which we startwith A = p, one is lead to consider the category P = B and the associatedinclusions of the source and target square

s = sp : → P = B, t = tp : → P = B.

Proposition 12.3.1. Let D be a derivator and let s, t : → P = B be theinclusions of the source and target squares.

(i) The functor t! : D() → D(P ) is fully faithful and Y ∈ D(P ) lies in theessential image of t! if and only if the source square s∗Y is cocartesian.

(ii) A square X ∈ D() is cocartesian if and only if the following canonicalcomparison map is an isomorphism,

can = can(X) : t!(X)1,1 → t!(X)∞. (12.3.2)

Proof. This is immediate from Proposition 11.1.13 and Proposition 11.1.17 ap-plied to A = p.

Definition 12.3.3. Let D be a pointed derivator and let X ∈ D(). The totalcofiber of X is the cone of the comparison map can(X) ∈ D([1]) (see (12.3.2)),in formulas

tcof(X) = C(can(X)) ∈ D(1).

DRAFT

12.3. TOTAL COFIBERS OF SQUARES 223

The definition of the total fiber tfib(X) ∈ D(1) is dual.

It is immediate from the above construction that there are functors

tcof : D()→ D(1) and tfib : D()→ D(1).

Lemma 12.3.4. In every pointed derivator D the functor tcof : D()→ D(1)is a left adjoint and the functor tfib : D()→ D(1) is a right adjoint.

Proof. Let j : [1] → P = B be the functor classifying the morphism (1, 1) →∞. It follows from the definition of the total cofiber (Definition 12.3.3) thattcof : D()→ D(1) is the composition

D()t! // D(P )

j∗// D([1])

cof // D([1])1∗ // D(1).

The functors t!, j∗, and 1∗ are obviously left adjoint functors as is cof by Propo-

sition 9.2.11. (Alternatively, we can also invoke Examples 9.6.11 to concludethat the composition of the last two functors is a left adjoint.)

It can be shown that a right adjoint to tcof is given by (1, 1)! : D(1)→ D(),i.e., by the left extension by zero functor which sends x ∈ D(1) to a coherentsquare looking like

0 //

0

0 // x.

Note that this means that tcof : D() → D(1) is an exceptional inverse imagefunctor (Definition 9.6.8) and dually for tfib. We will get back to this in §13.2.

The total cofiber is an obstruction against a square being cocartesian.

Lemma 12.3.5. For every pointed derivator D and every cocartesian squareX ∈ D() the total cofiber tcof(X) vanishes, tcof(X) ∼= 0.

Proof. By Proposition 12.3.1 a square X ∈ D() is cocartesian if and only ifthe canonical morphism can(X) ∈ D([1]) is an isomorphism. Hence we canconclude by Corollary 9.4.1.


(i) Let x ∈ D(1) and let (0, 0)!(x) ∈ D() be the corresponding constantsquare,

x //

x

x // x.

The total cofiber tcof((0, 0)!x) is trivial. In fact, this is immediate fromLemma 12.3.5 to Corollary 9.3.4.

DRAFT


(ii) Let (f : x→ y) ∈ D(1) and let (id[1]×0)!(f) ∈ D() be the correspondingvertically constant square,

xf//

y

x

f// y.

The total cofiber tcof((id×0)!f) is trivial. In fact, this is immediate fromLemma 12.3.5 to Corollary 9.3.4.

(iii) Let x, y ∈ D(1) and let X ∈ D() be the corresponding coproduct square,

0 //

x

y // x t y;

see Lemma 9.3.16. The total cofiber tcof(X) is trivial.

12.4 Examples of total cofibers

We now collect a few examples of coherent squares with non-trivial total cofibers.In the first case we consider x ∈ D(1) in a pointed derivators and the coherentsquare (1, 1)!(x) ∈ D(). Note that (1, 1) : 1 → is a cosieve and that (1, 1)!

is hence left extension by zero, the square (1, 1)!x hence looking like

0 //

0

0 // x;

see Corollary 9.1.6.

Proposition 12.4.1. For a pointed derivator D and x ∈ D(1) there is a naturalisomorphism

tcof((1, 1)!x) ∼= x.

Proof. To calculate tcof((1, 1)!x) we have to extend the zero span i∗p((1, 1)!x) to apushout square which is again constant to zero (for example by Corollary 9.3.4).Hence there is a natural isomorphism tcof((1, 1)!x) ∼= C(0 → x) ∼= C(1!x), andwe conclude by Proposition 9.4.2 which describes isomorphisms as cofibers.

We next consider total cofibers of squares supported at (1, 0) only. In moredetail, left extension by zero 1! : D(1) → D([1]) followed by right extension byzero (id × 0)∗ : D([1]) → D() induces an equivalence between D(1) and the

DRAFT

12.4. EXAMPLES OF TOTAL COFIBERS 225

full subcategory of D() spanned by the diagrams supported at (1, 0) ∈ ; seeCorollary 9.1.6. Under this equivalence, x ∈ D(1) corresponds to a coherentsquare X ∈ D() looking like

0 //

x

0 // 0.

Proposition 12.4.2. Let D be a pointed derivator and let X ∈ D() be sup-ported at (1, 0) only. There is a natural isomorphism

tcof(X) ∼= Σ(X1,0).

Proof. To see this, in order to calculate the total cofiber we have restrictX to thespan i∗pX and extend it to a cocartesian square. Since the vertical morphism inin the span is an isomorphism, the same is true by the right vertical morphismin (ip)!i

∗p(X) (Proposition 9.3.5). Hence, tcof(X) is naturally isomorphic to

C(x → 0) ∼= C(0!x). By construction of the suspension in pointed derivators,there is a natural isomorphism C(0!x) ∼= Σx.

We generalize the previous two examples and consider (f : x → y) ∈ D([1])in a poined derivator. The cosieve 1 × id : [1] → induces a left extension byzero functor (1 × id)! : D([1]) → D(), and it restricts to an equivalence ontothe full subcategory spanned by all squares X ∈ D() looking like

0 //

x

f

0 // y;

see Corollary 9.1.6.

Proposition 12.4.3. Let D be a pointed derivator and let f = (x → y) ∈D([1]). There is a natural isomorphism

tcof((id× 1)!f) ∼= Cf.

Proof. The proof is exactly the same as in the previous case.

Examples 12.4.4. Let D be a pointed derivator and let f = (x → x′) andg = (y → y′) be coherent morphisms in D .

(i) Suitable combinations of Kan extensions together with the categoricalcoproduct operation in D() (Proposition 7.4.7) allow us to construct acoherent square X ∈ D() looking like

0 //

x

(f,0)

=

0 //

x

f

t

0

// 0

y(0,f ′)

// x′ t y′ 0 // x′ yf ′// y′.

DRAFT


Since the functor tcof : D() → D(1) is a left adjoint (Lemma 12.3.4),it preserves coproducts and Proposition 12.4.3 together with its obviousvaiant yields a natural isomorphism

tcof(X) ∼= C(f) t C(f ′).

(ii) The total cofiber of a square X ∈ D() which is supported at (1, 0) and(0, 1) only,

0 //

X1,0

X0,1// 0,

is given by tcof(X) ∼= ΣX1,0tΣX0,1. This is immediate from the previouscase and the natural isomorphism Σ ∼= C0∗ : D(1)→ D(1).

An additional class of simple examples is given by the following result.

Proposition 12.4.5. Let D be a pointed derivator and let X ∈ D() such thatX1,1

∼= 0. There is a natural isomorphism

tcof(X) ∼= Σ colimp i∗p(X).

Proof. This is immediate from the definition of the total cofiber and the naturalisomorphism Σx ∼= C(x→ 0) ∼= C(0∗x) for x ∈ D(1).

There are the following special cases of this proposition, obtained by spe-cializing to the following two types of coherent squares,

x //

0

xf//

y

0 // 0, 0 // 0.

Such squares are precisely the ones lying in the essential image of right exten-sions by zero along the sieves (0, 0)∗ : D(1) → D() and (id × 0)∗ : D([1]) →D(), respectively (Corollary 9.1.6).

Examples 12.4.6. Let D be a pointed derivator, x ∈ D(1), and f = (x→ y) bein D([1]).

(i) There is a natural isomorphism tcof((0, 0)∗x) ∼= Σ2x.

(ii) There is a natural isomorphism tcof(id× 0)∗f) ∼= ΣCf.

In the case of a general X ∈ D() such that X1,1∼= 0 we will obtain an

alternative formula for tcof(X) ∈ D(1) in §12.6. And this relies on a descriptionof the total cofiber as an iterated cone construction; see §12.5.

DRAFT

12.5. TOTAL COFIBERS AS ITERATED CONES 227

To conclude this section we consider the generic case of squares vanishing atthe initial corner, i.e., of squares in pointed derivator looking like

0 //

x

f

y

g// z.

(12.4.7)

Proposition 12.4.8. Let D be a pointed derivator, let X ∈ D() with van-ishing upper left corner X0,0

∼= 0, i.e., X is as in (12.4.7). There is a naturalisomorphism

tcof(X) ∼= C((f, g) : x t y → z

).

Proof. To calculate tcof(X) we have to extend i∗p(X) to a cocartesian square.Since the span vanishes at the initial object, the resulting cocartesian squareis a coproduct square (Lemma 9.3.16), and we obtain the intended naturalisomorphism.

In §12.6 we reformulate the expression C((f, g) : x t y → z

), and, again,

this reformulation relies on a description of the total cofiber as an iterated coneconstruction established in §12.5.

12.5 Total cofibers as iterated cones

As an application of parametrized Kan extensions, in this section we discussiterated cone constructions on squares in pointed derivators. We show thatthese iterated cone constructions are naturally isomorphic to total cofibers.

Let D be a pointed derivator and let X ∈ D() be a coherent square lookinglike

xf//

g

y

g′

x′f ′// y′.

(12.5.1)

Given such a square, we can form the cone first in one direction and then in theremaining direction, or conversely. The details are as follows.

Construction 12.5.2. Given a square in a pointed derivator, we can consider itas a morphism of morphisms in two ways, and hence pass to the cone in thefirst or the second coordinate. Thus, we consider the Kan extensions

(ip × id)! (i× id)∗ : D([1]× [1])→ D(p×[1])→ D(× [1])

and

(id× ip)! (id× i)∗ : D([1]× [1])→ D([1]× p)→ D([1]×),

DRAFT


where i : [1]→ p again classifies the horizontal morphism (0, 0)→ (1, 0). Givena square X ∈ D() as in (12.5.1), the images of X under the above compositionsare cubes which respectively look like

xf

//

g

yg′

##

x′f ′

//

y′

0

// Cf

""

0 // Cf ′,

xf

//

g

yg′

!!

x′f ′

//

y′

0

// 0

!!

Cg // Cg′.

(12.5.3)

Let us focus on the cube on the left and describe its construction. First, us-ing that i is a sieve, for example a combination of Corollary 9.1.6 and Corol-lary 10.3.7 shows that the right Kan extension adds two zero objects. Second,the cube is then obtained by left Kan extension and in that cube the back andthe front faces are cocartesian by Corollary 10.3.10. The construction of theother cube is similar and in this case the left and the right faces are cocartesian.

As an upshot, given a square X ∈ D() as in (12.5.1), there are canonicalcoherent maps C1(X), C2(X) ∈ D([1]),

C1(X) : Cf → Cf ′ and C2(X) : Cg → Cg′. (12.5.4)

Associated to these we obtain the iterated cones C(C1(X)), C(C2(X)) ∈ D(1).

The following is now immediate.

Proposition 12.5.5. Let D be a derivator and let X ∈ D() be as in (12.5.1).If the square X is cocartesian, then the canonical coherent maps (12.5.4) areisomorphisms.

Proof. In order to show that C1(X) is an isomorphism, let us consider the leftcube in (12.5.3). By construction the back and front faces are cocartesian asis the top face by assumption on X. It follows from Proposition 9.3.10 thatalso the bottom face is cocartesian. Since isomorphisms are stable under cobasechange (Proposition 9.3.5), the map C1(X) : Cf → Cf ′ is an isomorphism.Similar arguments show that C2(X) : Cg → Cg′ is an isomorphism.

Remark 12.5.6. Let C be a complete, cocomplete, and pointed category and lety(C) be the pointed derivator. In this case Proposition 12.5.5 reduces to thestatement that given a pushout square

xf//

g

y

g′

x′f ′// y′

DRAFT

12.5. TOTAL COFIBERS AS ITERATED CONES 229

in C, then the induced maps cok(f) → cok(f ′) and cok(g) → cok(g′) are iso-morphisms; see Examples 6.5.5 and Example 10.1.5.

Corollary 12.5.7. Let D be a pointed derivator and let X ∈ D() be cocarte-sian. There are isomorphisms C(C1(X)) ∼= 0 ∼= C(C2(X)) in D(1).

Proof. By Proposition 12.5.5 we know that C1(X), C2(X) are isomorphisms assoon as X is cocartesian. Hence it suffices to apply Corollary 9.4.1 to concludethat the iterated cones are trivial.

In particular, for cocartesian squares we just observed that iterated cones areisomorphic. It turns out that these iterated cones are isomorphic even withoutthe assumption that X ∈ D() is cocartesian. Here we give an adhoc proofwhich also shows the relation to total cofibers. An additional proof based oniterations of exceptional inverse image functors can be found in §13.2. And in[Gro16a] we give an alternative proof using cofiber hypercubes.

We now use parametrized Kan extensions to show that total cofibers andthe iterated cones from (12.5.2) are canonically isomorphic.

Theorem 12.5.8. Let D be a pointed derivator and let X ∈ D(). There arecanonical isomorphisms

tcof(X) ∼= C(C1X) ∼= C(C2X),

where C1(X), C2(X) ∈ D([1]) are as in (12.5.4).

Proof. Let X ∈ D() be a square looking like (12.5.1). We show that suitablecombinations of Kan extensions can be used to construct a coherent diagramas in Figure 12.2. Let B ⊆ [1]× [2]× [2] the full subcategory given that figure.There is a fully faithful functor i : → B given by

(0, 0) 7→ (0, 0, 0), (1, 0) 7→ (1, 0, 0), (0, 1) 7→ (0, 1, 0), (1, 1) 7→ (1, 2, 0).

As we describe next, this functor factors as a composition of fully faithful func-tors

i : i1→ B1

i2→ B2i3→ B3

i4→ B4i5→ B5

i6→ B,

where all intermediate categories are obtained by adding certain objects to theimage of i. The corresponding Kan extension functors

D()(i1)!→ D(B1)

(i2)!→ D(B2)(i3)∗→ D(B3)

(i4)!→ D(B4)(i5)∗→ D(B5)

(i6)!→ D(B)

then build the desired diagram. Since these Kan extension functors are fullyfaithful (Proposition 8.2.16), it suffices to understand the individual steps.

(i) The category B1 is obtained by adding the object (1, 1, 0). Obviously, thefunctor i1 : → B1 is isomorphic to the functor t from Example 11.2.3.Hence, by Proposition 12.3.1, (i1)! adds a cocartesian square and thecomparison map.

DRAFT


xf

//g~~

y

g

x′f

//=

p

canzz

x′f ′

//

y′

01//

Cf

∼=

02//

Cf

03// Cf ′

04

05

zz

c(X).

Figure 12.2: Total cofiber versus iterated cofiber

(ii) The category B2 also contains the object (0, 2, 0). To calculate the Kanextension functor (i2)! : D(B1) → D(B2), we invoke (Der4) and observethat the slice category under consideration is isomorphic to [1]. Since thiscategory admits 1 ∈ [1] as terminal object, we can mimic the proof ofExample 8.2.24 to conclude that (i2)! essentially adds the identity idx′

and forms the composition f ′ = c f .

(iii) The categoryB3 is obtained by adding the objects (0, 0, 1), (0, 1, 1), (0, 2, 1).Note that the inclusion i3 : B2 → B3 is isomorphic to

k × id : [1]× [2]→ p×[2],

where k classifies the horizontal morphism. Since k is a sieve, so is thefunctor i3 and (i3)∗ is hence right extension by zero (Corollary 9.1.6).Thus, this Kan extension adds the zero objects 01, 02, 03.

(iv) The category B4 also contains the objects (1, 0, 1), (1, 1, 1), (1, 2, 1), hencethe inclusion i4 is isomorphic to the functor

ip × id : p×[2]→ × [2].

By Corollary 10.3.10 the functor (i4)! : D(B3)→ D(B4) hence forms threecocartesian squares. Applied to the essential image of the previous Kanextension functor, it thus forms three cofiber squares, thereby addingthe objects Cf,Cf, and Cf ′. Since the square populated by x-y-x′-p iscocartesian, the map Cf → Cf is an isomorphism (Proposition 12.5.5).

(v) The category B5 is obtained by adding the objects (1, 0, 2) and (1, 1, 2).Since the inclusion i5 : B4 → B5 is a sieve, (i5)∗ is right extension by zeroand it hence adds the zero objects 04, 05.

DRAFT

12.6. MORE EXAMPLES OF TOTAL COFIBERS 231

(vi) The final step adds the remaining object (1, 2, 2), and it suffices to un-derstand (i6)! : D(B5) → D(B). It follows from the basic form of thedetection lemma (Lemma 11.3.1) that (i6)! forms a cocartesian squareCf -Cf ′-05-c(X). In fact, an application of that lemma to A = B5, u = i6,and B′ = B5 implies that we only have to show that the functor p→ B5

classifying the span 111-121-112 is a right adjoint functor. And this fol-lows easily from Lemma 11.3.3.

This concludes the construction of a diagram Q ∈ D(B) as in Figure 12.2. Thepoint of this diagram is that it allows us to identify c(X) ∈ D(1) in the followingtwo ways.

(i) As already observed, in this diagram the square Cf -Cf ′-05-c(X) is co-cartesian. Moreover, also the square p-y′-Cf -Cf ′ is cocartesian. Infact, this is from the fact that x′-x′-02-03 is cocartesian as a horizon-tally constant square (Proposition 9.3.5) and two applications of Proposi-tion 9.3.10. Proposition 9.3.10 then implies that the square p-y′-05-c(X)is cocartesian, and this cofiber square shows that there is a canonicalisomorphism c(X) ∼= C(can: p→ y′) = tcof(X).

(ii) Moreover, the square Cf -Cf -04-05 is horizontally constant and hence co-cartesian (Proposition 9.3.5). As already mentioned, Cf -Cf ′-05-c(X) isalso cocartesian, and a further application of Proposition 9.3.10 impliesthat Cf -Cf ′-04-c(X) is cocartesian. This cofiber square yields a canonicalisomorphism c(X) ∼= C(Cf → Cf ′) = C(C1(X)).

Taking these two steps together, we obtain tcof(X) ∼= C(C1(X)), as intended.By symmetry we also obtain canonical isomorphisms tcof(X) ∼= C(C2(X)).

12.6 More examples of total cofibers

In this short section we use the identification of the total cofiber as an inter-ated cone in order to obtain additional formulas for specific examples of totalcofibers. More specifically, we take up again the case of generic squares whichvanish either at the final or at the initial object (see Proposition 12.4.5 andProposition 12.4.8) and describe their total cofibers in elementary terms.

In the case of a square vanishing at the source we hence, in particular,consider two coherent morphisms (f1 : x1 → y), (f2 : x2 → y) ∈ D([1]) in apointed derivator. Let

x1f1 //

y

g1

x2f2 //

y

g2

0 // Cf1, 0 // Cf2

(12.6.1)

be cofiber squares associated to f1, f2. In these diagrams and in the followingproposition we implicitly identify lower right corners of cofiber squares with

DRAFT


cones of the upper horizontal morphisms (Lemma 9.2.24). The justification forthis is provided by Proposition 12.2.6.

Proposition 12.6.2. Let D be a pointed derivator and let X ∈ D() be suchthat X0,0

∼= 0, i.e., X looks like

0 //

x1

f1

x2

f2

// y,

and let (12.6.1) be coherent cofiber squares associated to f1 and f2. There arecanonical isomorphisms

tcof(X) ∼= C((f1, f2) : x1 t x2 → y)∼= C(g2f1 : x1 → Cf2)∼= C(g1f2 : x2 → Cf1).

Proof. The existence of the first isomorphism follows immediately from the def-inition of tcof; see Proposition 12.4.8. By Theorem 12.5.8 there are canonicalisomorphisms

tcof(X) ∼= C(C1(X)) ∼= C(C2(X)).

In order to calculate these respective iterated cone constructions, let us considerthe following cubes, which are obtained by passing to cofiber squares in therespective directions,

0 //

x1f1

##

x2f2

//

y

g2

0

// x1

""

0 // Cf2,

0 //

x1f1

##

x2f2

//

y

g1

0

// 0

##

x2// Cf1.

The cube on the left yields an isomorphism tcof(X) ∼= C(g2f1 : x1 → Cf2),while the other cube shows tcof(X) ∼= C(g1f2 : x2 → Cf1).

We now turn to the case of a square vanishing at the sink and we hence have,in particular, two coherent morphisms (f1 : x→ y1), (f2 : x→ y2) ∈ D([1]) in apointed derivator. Let

xf1 //

y1

g1

// 0

x2f2 //

y

g2

// 0

0 // Cf1h1

// Σx, 0 // Cf2h2

// Σx

DRAFT

12.6. MORE EXAMPLES OF TOTAL COFIBERS 233

be cofiber sequences associated to f1, f2. Again, here and in the statement of thefollowing proposition we make implicitly some identifications, the justificationbeing provided by Lemma 9.2.22, Lemma 9.2.24, and Proposition 12.2.6.

Proposition 12.6.3. Let D be a pointed derivator and let X ∈ D() be suchthat X1,1

∼= 0, i.e., X looks like

xf1 //

f2

y1

y2// 0.

and let (12.6.1) be coherent cofiber sequences associated to f1 and f2. There arecanonical isomorphisms

tcof(X) ∼= Σ colimp i∗pX

∼= C(Σf2 h1 : Cf1 → Σy2)∼= C(Σf1 h2 : Cf2 → Σy1)

Proof. The existence of the first isomorphism is precisely Proposition 12.4.5.By symmetry it suffices to establish an isomorphism

tcof(X) ∼= C(Σf2 h1 : Cf1 → Σy2).

To this end, we consider X ∈ D() = D([1]× [1]) and we pass to parametrizedcofiber sequences (with parameters from the second copy of [1]). This extensionsends X to a coherent diagram Q ∈ D(× [1]) looking like

xf1 //

f2

y1

##

g1

// 0

##

y2//

0

// 0

0

// Cf1

##

h1 // ΣxΣf2

""

0 // Σy2// Σy2.

By Theorem 12.5.8 there is an isomorphism tcof(X) ∼= C(C1(X)), and hencetcof(X) ∼= C(Cf1 → Σy1) where the unlabelled morphism is from the abovecube on the left. In order to identify this map, let us also consider the abovecube on the right. The bottom right face implies that the morphism agrees withΣ(f2)h1, thereby showing tcof(X) ∼= C(Σ(f2)h1 : Cf1 → Σy2).

DRAFT


DRAFT

Chapter 13

More on iterated cofibersand total cofibers

In this short chapter we collect a few additional observations concerning partialcones and total cofibers of squares in pointed derivators. We exhibit thesefunctors and their duals as examples of (co)exceptional inverse image functors.Since (co)exceptional inverse image functors are closed under composition, thisyields a conceptual reason why iterated cones and total cofibers are naturallyisomorphic.

For the identification of the total cofiber as an exceptional inverse imagefunctor, it is convenient to include a short discussion of punctured cubes and toestablish a formula for colimits of punctured cubes. The calculus developed herebelongs to a fairly rich calculus of cubes and hypercubes in pointed derivators,and some additional steps will be developed in [Gro16a].

In §13.1 we obtain a formula for colimits of punctured cubes, which weapply in §13.2 to express partial and total cofibers as exceptional inverse imagefunctors. Finally, in §13.3 we take a glimpse at parametrized (co)exceptionalinverse image functors.

13.1 Colimits of punctured cubes

In this short section we provide a formula for colimits of punctured cubes inderivators, resulting in an alternative description of the total cofiber. Here wedeal with the case of 3-cubes, but similar formulas apply to higher dimensionsand also to more general shapes [Gro16a].

Notation 13.1.1. Let [1]3≤2 ⊆ [1]3 be the full subcategory of the cube obtainedby removing the final vertex (1, 1, 1). The notation is motivated by the following:the cube [1]3 can be realized as the power set of a set with three elements, andthe punctured cube [1]3≤2 is the full subcategory spanned by all subsets ofcardinality at most two. Reading the cube as × [1], there are fully faithful

235

DRAFT

236 CHAPTER 13. MORE ITERATED AND TOTAL COFIBERS

functors

jk : p→ [1]3≤2 : t 7→ (t, k), k = 0, 1.

Correspondingly, for every derivator D and every punctured cube X ∈ D([1]3≤2),

x0//

!!

y0

x′0 //

y′0

x1

!!

// y1

x′1,

(13.1.2)

there are coherent spans j∗0 (X), j∗1 (X) ∈ D(p). Note that, using obvious termi-nology, p is the punctured square.

Moreover, since is the cocone on p, for every coherent punctured cubeX ∈ D([1]3≤2), Definition 11.1.15 yields a canonical coherent map

(can: colimp j∗0 (X)→ X(1,1,0)) ∈ D([1]).

In the notation of (13.1.2) this canonical map is a morphism y0 tx0 x′0 → y′0.

Dually, there is the punctured cube [1]3≥1 ⊆ [1]3 obtained by removing theinitial object (0, 0, 0), and this punctured cube comes with similar embeddingsy→ [1]3≥1.

Proposition 13.1.3. Let D be a derivator and let X ∈ D([1]3≤2) be a coher-ent punctured cube. There is a cocartesian square Q ∈ D() with underlyingdiagram

colimp j∗0X

can //

X(1,1,0)

colimp j∗1X // colim[1]3≤2

X.

(13.1.4)

Proof. This result follows from a factorization of the colimit colim[1]3≤2using

Lemma 7.6.1. In more detail, let t : → P = B be the target cocone inclusion,i.e., t = (ip)

B. The restriction of t× id : × [1]→ P × [1] to [1]3≤2 factors as

× [1]t×id// P × [1]

[1]3≤2 t1//

⊆

OO

B.

⊆ t2

OO

Here, B ⊆ P × [1] is the full subcategory obtained by removing the final object(∞, 1) and t1, t2 are the obvious fully faithful factorizations of the restriction of

DRAFT

13.1. COLIMITS OF PUNCTURED CUBES 237

t× id to [1]3≤2 (see the following diagrams for an illustration). By Lemma 7.6.1there is a canonical isomorphism

colim[1]3≤2

∼= colimP×[1](t2)!(t1)!.

Now, let X ∈ D([1]3≤2) be a coherent punctured cube as in (13.1.2), then(t1)!(X) ∈ D(B) looks like the diagram on the left in

x0//

y0

!!

x′0 //

p0

%%x1

// y1

y′0

x′1 // p1,

x0//

y0

x′0 //

p0

can

%%x1

// y1

y′0

x′1 // p1

&&c.

A routine application of the Special Workhorse Proposition (Proposition 11.3.10together with Lemma 11.3.3) implies that (t1)! simply adds two pushout squares,the pushout corners being denoted by p0, p1. In particular, the morphism p0 →y′0 is the canonical morphism

can: colimp j∗0X → X(1,1,0)

and there is an isomorphism p1∼= colimp j

∗1X (Proposition 11.1.6).

In order to understand (t2)!, we again use the Special Workhorse Proposition,thereby concluding that (t2)!(t1)!(X) is obtained from (t1)!(X) simply by addingan additional cocartesian square Q ∈ D(),

p0//

y′0

p1

// c.

In fact, the assumptions of Proposition 11.3.10 are satisfied, since the functorp→ B classifying the span (1, 1, 1) ← (1, 1, 0) → (∞, 0) is a right adjoint (thisfollows easily using Lemma 11.3.3).

Finally, since (∞, 1) ∈ P × [1] is a final object, by Corollary 8.2.10 there isa canonical isomorphism colimP×[1]

∼= (∞, 1)∗, showing that

c ∼= colimP×[1](t2)!(t1)!X ∼= colim[1]3≤2X,

and thereby concluding the proof.

In particular, if D is a pointed derivator and colimp j∗1X∼= 0, then the

cocartesian square (13.1.4) exhibits colim[1]3≤2X as the cofiber of the canonical

DRAFT


map, i.e., as the total cofiber of j∗0 (X). Here we abuse notation and denote byj0 the functor

j0 : → [1]3≤2 : t 7→ (t, 0).

Corollary 13.1.5. Let D be a pointed derivator and let j0 : → [1]3≤2 be theinclusion t 7→ (t, 0), t ∈ . There is a natural isomorphism

tcof ∼= colim[1]3≤2(j0)∗ : D()→ D(1).

Proof. Since j0 is a sieve, (j0)∗ is right extension by zero (Corollary 9.1.6).In particular, for every X ∈ D(), the restriction j∗1 (j0)∗(X) ∈ D(p) of thepunctured cube (j0)∗(X) vanishes, hence also the colimit colimp j

∗1 (j0)∗X is

trivial. In this situation, using the fully faithfulness of (j0)∗, the cocartesiansquare (13.1.4) is a cofiber square

colimp i∗pX

can //

X1,1

0 // colim[1]3≤2(j0)∗X,

thereby exhibiting colim[1]3≤2(j0)∗X as the total cofiber tcof(X).

By symmetry, there are of course similar formulas if we use the embeddings→ [1]3≤2 defined by setting the first or the second coordinate equal to zero.

13.2 Iterated cones and total cofibers, again

In this section we revisit iterated cones and total cofibers from the perspectiveof (co)exceptional inverse image functors. Let us recall that associated to theuniversal examples of sieves and cosieves,

10→ [1]

1← 1,

there are the adjunctions

(Σ,Ω) ∼= (C, 1!) (0∗, F ) : D(1) D([1]) D(1).

This exhibits F,C as (co)exceptional inverse image functors,

F = 0! : D([1])→ D(1) and C = 1? : D([1])→ D(1),

see Examples 9.6.11.

In this section we discuss some aspects of the corresponding picture in di-mension two. Associated to the category = [1] × [1], there are the following

DRAFT

13.2. ITERATED CONES AND TOTAL COFIBERS, AGAIN 239

fully faithful functors,

1 0 //

0

[1]

id×0

11oo

0

[1]0×id

// [1]1×idoo

1

1

OO

0// [1]

id×1

OO

1.

1

OO

1oo

In this commutative diagram, the functors pointing to the right or to the bot-tom are sieves, while the other ones are cosieves. Hence, the first class givesrise to coexceptional inverse image functors, and dually for the second class(Proposition 9.6.7).

Proposition 13.2.1. For every pointed derivator D there are natural isomor-phisms of functors D()→ D([1]),

(0× id)! ∼= F1, (id× 0)! ∼= F2, (1× id)? ∼= C1, and (id× 1)? ∼= C2.

Proof. By duality and symmetry it suffices to construct a natural isomorphism(0× id)! ∼= F1, and, similarly to Examples 9.6.11, this follows from the explicitconstruction of exceptional inverse image functors (Construction 9.6.10). Givena coherent square X ∈ D() looking like

xf//

g

y

g′

x′f ′// y′,

the first step in the construction of (0 × id)!(X) consists of forming the leftextension by zero as depicted on the left in

xg

f

x′

f ′

0

// y

g′

0 // y′,

Ff //

""

xg

f

Ff ′ //

x′

f ′

0

""

// y

g′

0 // y′.

As in Examples 9.6.11, the reader verifies that the remaining two functors arenaturally isomorphic to the right Kan extension forming the above cube on theright followed by the restriction to the upper left diagonal morphism. Since thiscomposition is precisely F1(X), this yields the intended natural isomorphism(0× id)! ∼= F1 : D()→ D([1]).

DRAFT


Thus, the functors F1, F2 : D() → D([1]) are respectively right adjoint tothe functors which send (x→ y) ∈ D([1]) to

x //

0

y // 0,

x //

y

0 // 0.

There is a similar description of the right adjoints of C1, C2 : D()→ D([1]).

Proposition 13.2.2. For every pointed derivator D there are natural isomor-phisms of functors D()→ D(1),

(0, 0)! ∼= tfib and (1, 1)? ∼= tcof.

Proof. In this proof we construct a natural isomorphism (1, 1)? ∼= tcof, andfor this purpose we again use the explicit description of (1, 1)? given in Con-struction 9.6.10. Unraveling definitions, the first step consists of the following.Using again Notation 13.1.1, let [1]3≤2 ⊆ [1]3 = × [1] be the punctured cube

and let j0 : → [1]3≤2 be the inclusion of the top square. The first step in the

construction of (1, 1)? amounts to forming (j0)∗ : D()→ D([1]3≤2).

The next two steps of Construction 9.6.10 consist of first forming the left Kanextension along q = π[1] : [1]3≤2 → and then applying (1, 1)∗ : D() → D(1).Since (1, 1) ∈ is terminal, there is a natural isomorphism (1, 1)∗ ∼= colim(Corollary 8.2.10). Consequently, we have to compose (j0)∗ with

(1, 1)∗q!∼= colim q!

∼= colim[1]3≤2: D([1]3≤2)→ D(1);

see Corollary 7.6.2. But by Corollary 13.1.5 there is a natural isomorphismcolim[1]3≤2

(j0)∗ ∼= tcof, thereby identifying tcof as exceptional inverse image

functor (1, 1)?.

Similarly to Proposition 13.2.1, a right adjoint of tfib : D() → D(1) anda left adjoint of tcof : D() → D(1), respectively, send x ∈ D(1) to coherentsqares looking like

x //

0

0 // 0,

0 //

0

0 // x.

As an immediate consequence we can now re-establish Theorem 12.5.8.

Corollary 13.2.3. For every pointed derivator D there are natural isomor-phisms of functors D()→ D(1),

tfib ∼= F F1∼= F F2 and tcof ∼= C C1

∼= C C2.

DRAFT

13.3. PARAMETRIZED (CO)EXCEPTIONAL INVERSE IMAGE FUNCTORS241

Proof. By duality and symmetry it suffices to show that there are natural iso-morphisms tfib ∼= F F1. Proposition 13.2.2 together with Lemma 9.6.13 appliedto the composition of sieves (0, 0) = (0×id)0: 1→ yield the first two naturalisomorphisms in

tfib ∼= (0, 0)! ∼= 0! (0× id)! ∼= F F1.

Finally, an application of Proposition 13.2.1 and Examples 9.6.11 establishesthe remaining natural isomorphism.

There are similar results in larger dimensions as we will see in [Gro16a].

13.3 Parametrized (co)exceptional inverse im-age functors

In this section we include a short discussion of (co)exceptional inverse imagefunctors with parameters. This is mainly to illustrate the techniques developedso far. There will be a more systematic approach in [Gro16a] and we suggestthe reader to skip this section on a first reading.

To begin with there is the following result.

Lemma 13.3.1. Let u : A → B be a sieve and let C ∈ Cat. The functoru× idC : A× C → B × C is again a sieve.

Proof. This is immediate from the definition of sieves.

There is a dual result for cosieves.

Construction 13.3.2. Let u : A → B be a sieve and let v be the correspond-ing cosieve. Recall from Construction 9.6.10 that in this situation we use thecategory cyl1(v) and the functors sv : B → cyl1(v), qv : cyl1(v) → B in orderto construct u!. Now, if we also consider C ∈ Cat , then there is the sieveu×id : A×C → B×C and the corresponding cosieve v×id. In the construction of(u×id)! we hence consider the category cyl1(v×id) and the functors sv×id, qv×id.As a left adjoint the functor −×C : Cat → Cat preserves pushouts, and there ishence a canonical isomorphism of categories φ : cyl1(v)×C ∼= cyl1(v× idC). It iseasy to verify that φ makes the two triangles in the following diagram commute.

B × Csv×id

//

sv×id&&

cyl1(v × idC)qv×id

// B × C A× Cu×idoo

cyl1(v)× C

φ ∼=

OO

qv×id

88

Bsv

//

id×c

OO

cyl1(v)

id×c

OO

qv// B

id×c

OO

Au

oo

id×c

OO

(13.3.3)

This diagram summarizes the functors relevant to the construction of u! and(u× id)!. Note that by Theorem 10.3.1 the three rectangles in the diagram arehomotopy exact.

DRAFT


Proposition 13.3.4. Let D be a pointed derivator, let u : A → B a functor,and let C ∈ Cat.

(i) If u is a sieve, then the following square commutes up to a natural iso-morphism of functors

D(B × C)(u×id)!

//

c∗

∼=

D(A× C)

c∗

D(B)u!

// D(A).

(ii) If u is a cosieve, then the following square commutes up to a naturalisomorphism of functors

D(B × C)(u×id)?

//

c∗

∼=

D(A× C)

c∗

D(B)u?

// D(A).

Proof. By duality it suffices to take care of the first statement, and we hence con-sider a sieve u : A→ B and C ∈ Cat . In the notation of Construction 13.3.2, adiagram chase along (13.3.3) yields the following chain of natural isomorphisms,

(id× c)∗(u× idC)! = (id× c)∗(u× id)∗(qv×id)∗(sv×id)!

= u∗(id× c)∗(qv×id)∗(sv×id)!

∼= u∗(id× c)∗(qv × id)∗φ∗(sv×id)!

∼= u∗(qv)∗(id× c)∗φ∗(sv×id)!

∼= u∗(qv)∗(id× c)∗(sv × id)!

∼= u∗(qv)∗(sv)!(id× c)∗

= u!(id× c)∗.

Here, two of the isomorphisms are instances of Theorem 10.3.1, and the remain-ing two isomorphisms follow from the homotopy finality and homotopy cofinalityof isomorphisms (Examples 8.2.14).

To put this result in formulas, for X ∈ D(B ×C) and a sieve or a cosieve uthere are respectively natural isomorphisms(

(u× idC)!(X))c∼= u!(Xc) and

((u× idC)?(X)

)c∼= u?(Xc).

In the above situation, given a sieve u, we refer to (u× id)! as a parametrizedcoexceptional inverse image functor. Similarly, if u is a cosieve, then wecall (u× id)? a parametrized exceptional inverse image functor.

DRAFT

13.3. PARAMETRIZED INVERSE IMAGES 243

The following examples follow from Proposition 13.3.4, but they are of coursealso an immediate consequence of the construction of Ck : D() → D([1]) andFk : D()→ D([1]); see (12.5.3).

Examples 13.3.5. Let D be a pointed derivator, let X ∈ D(), and let k = 0, 1.

(i) There are natural isomorphisms

C1(X)k ∼= C((id× k)∗X) and C2(X)k ∼= C((k × id)∗X).

(ii) There are natural isomorphisms

F1(X)k ∼= F ((id× k)∗X) and F2(X)k ∼= F ((k × id)∗X).

It can be shown that there are similar natural isomorphisms as in Proposi-tion 13.3.4 if we replace c : 1 → C by more general functors C ′ → C. More-over, the isomorphisms can be constructed such they satisfy suitable coherenceaxioms. Instead of directly verifying these statemens, we will do this more con-ceptually in [Gro16a], once we have a basic understanding of adjunctions ofderivators.

DRAFT


DRAFT

Chapter 14

Loop objects in pointedderivators

In this chapter we include a more detailed discussion of loop objects in pointedderivators. Mimiking constructions from classical homotopy theory, we showthat loop objects in pointed derivators can be endowed with natural groupstructures, given by abstractly defined concatenation of loop morphisms. Simi-larly, using the classical Eckmann–Hilton trick, we prove that for two-fold loopobjects this leads to abelian group structures. The results of this chapter allowus to conclude in §15 that stable derivators are additive and to identify the mi-nus sign in the rotation axiom for canonical triangulations in stable derivators(see §15.5).

The construction of these concatenation morphisms relies on the Segalianapproach to coherently associative and unital multiplications, based on specialsimplicial objects in the case of monoid structures. If we change the combina-torics from finite totally ordered sets to finite pointed sets, then we obtain aconvenient way to encode group structures.

In §14.1 we construct the concatenation maps and show that they functori-ally endow loop objects with monoid structures. In §14.2 we observe that thesemonoid structures actually are group structures. In §14.3 we identify certainsigns resulting from loop squares. Finally, in §14.4 we use the Eckmann–Hiltontrick to show that the two possibly different group structures on two-fold loopobjects agree and are abelian.

14.1 Loop objects as monoid objects

In this section we show that loop objects in pointed derivators can be naturallyturned into monoid objects. The constructions are motivated from the topology,in which case the multiplication is given by concatenation of loops. There is, ofcourse, a dual result for suspensions, leading to natural comonoid structures.

To begin with we recall the notion of special simplicial objects. Let C be a

245

DRAFT

246 CHAPTER 14. LOOP OBJECTS IN POINTED DERIVATORS

category with finite products and let X : ∆op → C be a functor, i.e., a simplicialobject in C. Here, ∆ denotes the category of finite ordinals

[k] = (0 < . . . < k), k ≥ 0,

together with the order-preserving morphisms. The category ∆ admits a de-scription by generators and relations, the generators being the coface and code-generacy maps. The coface map di : [n − 1] → [n], 0 ≤ i ≤ n, is the uniqueinjection which avoids i and the codegeneracy map sj : [n+ 1]→ [n], 0 ≤ j ≤ n,is the unique surjection which hits j twice.

In ∆ there are also the maps ιk : [1] → [n] for 1 ≤ k ≤ n determined byιk(0) = k − 1 and ιk(1) = k. These maps taken together induce the Segalmaps

σn = (ι∗1, . . . , ι∗n) : Xn → X1 × . . .×X1. (14.1.1)

In dimension n = 0 the Segal map is defined as the unique map σ0 : X0 → ∗.

Definition 14.1.2. Let C be a category with finite products.

(i) A simplicial object X : ∆op → C is reduced if X0∼= ∗.

(ii) A simplicial object X : ∆op → C is special if all Segal maps are isomor-phisms.

We denote by Fun(∆op, C)sp ⊆ Fun(∆op, C) the full subcategory spanned by thespecial simplicial objects.

One point of this notion is recalled in the following proposition, in whichwe denote by Mon(C) the category of monoid objects in a category C withfinite products.

Proposition 14.1.3. For every category C with finite products there is an equiv-alence of categories

Mon(C) ' Fun(∆op, C)sp.

Proof. This is standard and we only sketch the constructions (for more detailswe refer the reader to [Erg15]). Given a special simplicial object X : ∆op → C itsuffices to truncate in low dimensions. The underlying object of the associatedmonoid M is X1. The multiplication is defined as

µ : M ×M ∼= X2d1→ X1 = M

where the first map is an inverse to the Segal map and the second map is Xevaluated on d1 : [1] → [2]. Similarly, the unique map s0 : [1] → [0] togetherwith the fact that X is reduced can be used to define a unit

η : ∗ ∼= X0 → X1.

It is instructive to verify that this defines a monoid object in C, and we leavethis task to the reader. The reader also verifies that given a morphism of specialsimplicial objects then evaluation at [1] yields a morphism of monoid objects.

DRAFT

14.1. LOOP OBJECTS AS MONOID OBJECTS 247

Conversely, given M ∈ Mon(C) the associated simplicial X : ∆op → C isdefined as follows. On objects we set

Xn = M×n = M × . . .×M.

The coface morphisms are sent to suitable multiplication maps while the code-generacy maps are sent to suitable inclusions of units. Since we will not needthe details of this construction, we leave it to the reader to come up with explicitformulas and to cary out the remaining steps.

Remark 14.1.4. The point of this lemma is, of course, not to encode a simple no-tion (that of a monoid object) in a seemingly unnecessarily complicated way (bymeans of special simplicial objects). Historically, this notion came up in topol-ogy in the study of topological spaces which are endowed with multiplicationmaps which are associative and unital up to coherent homotopies.

The prototypical example is that of a loop space of a pointed topologicalspace which comes with a multiplication map given by concatenation of loops.If we model a loop as a path defined on the unit interval [0, 1], then it is well-known that the usual concatenation is not strictly associative and unital, butthat these properties are satisfied up to homotopy. But much more is true:the homotopies witnessing the associativity up to homotopy themselves satisfycertain relations but again only up to homotopy, and this extends to higherdimensions.

One way of making this precise is by saying that loop spaces are algebrasover A∞-operads. A different way of encoding such a structure is by meansof special simplicial spaces where being special in this case essentially meansthat the Segal maps (14.1.1) are weak homotopy equivalences (as opposed toisomorphisms). For more details we refer to the classical literature including[Ada78, BV73, May72, Seg74, Sta63].

Given a pointed derivator D , we now want to apply this to the categoryD(1) which admits finite products by Proposition 7.4.7. We construct certainspecial simplicial objects in D(1), which induce natural monoid structures onloop objects.

Construction 14.1.5. Let yk ∈ Cat be the cocone on (k + 1) · 1, the discretecategory on k + 1 objects ej , 0 ≤ j ≤ k. Thus yk has an additional object ∞and the non-identity morphisms in yk are of the form ej →∞, 0 ≤ j ≤ k. Thecategories yk in dimensions k = 0, 1, 2 look like

e1

e1

!!

e2

e0

// ∞, e0// ∞, e0

// ∞.

In each case the inclusion of the cocone point∞ : 1→yk is a cosieve, so that∞!

is left extension by zero. Given a pointed derivator D , we consider the functors

ωk : D(1)∞!→ D(yk)

lim→ D(1), [k] ∈ ∆. (14.1.6)

DRAFT


For k = 0 we note that e0 ∈y0 ∼= [1] is an initial object. Hence, for x ∈ D(1)we obtain by Corollary 8.2.10 a canonical isomorphism

ω0x = limy0∞!x ∼= e∗0∞!x ∼= 0.

For k = 1 and x ∈ D(1) it follows from Definition 9.2.14 and the canonicalisomorphism limy ∼= lim(iy)∗ (Corollary 7.6.2) together with the homotopycofinality of (0, 0) : 1 → (Corollary 8.2.10) that there are canonical isomor-phisms

ω1x = limy∞!x ∼= lim

(iy)∗∞!x ∼= (0, 0)∗(iy)∗∞!x ∼= Ωx.

Thus, in these low dimensions there are canonical isomorphisms

ω0∼= 0: D(1)→ D(1) and ω1

∼= Ω: D(1)→ D(1). (14.1.7)

More generally, we claim that ωk is canonically isomorphic to the k-fold powerof the loop functor Ω, ωk ∼= Ω×k : D(1) → D(1). This will be achieved byshowing that there is a special simplicial object in the background of (14.1.6).

To begin with we show that the assignment [k] 7→ ωk extends to a simplicialobject ω : ∆op → Fun(D(1),D(1)) in the category of endomorphisms of D(1).Given a morphism f : [k] → [l] in ∆ there is an induced functor yf : yk →yldefined by ej 7→ ef(j) and ∞ 7→ ∞. Clearly, these functors make the uppersquare in

1 id //

∞k

id

1

∞l

yk

yf//

πk

yl

πl

1id// 1

=Eid

(14.1.8)

commutative while the remaining square commutes for trivial reasons. More-over, this construction is functorial in [k].

Note that the top square is homotopy exact. In fact, by Lemma 11.1.12 it isenough to check that the canonical mate (∞k)! → (yf )∗(∞l)! is an isomorphismat ej , 0 ≤ j ≤ k, which is trivial since the source and target vanish at thoseobjects. This observation allows us to define ωf : ωl → ωk as

ωf : (πl)∗(∞l)! → (πk)∗(yf )∗(∞l)!∼−→ (πk)∗(∞k)!, (14.1.9)

where the first morphism is the canonical mate of the bottom square and thesecond one is the inverse of the canonical mate of the top square. For identitymorphisms id: [k]→ [k] we choose ωid to be the identity. Hence, since both thepassage to canonical mates and to inverse natural transformations are functorialwith respect to pasting, we conclude that (14.1.6) and (14.1.9) define a simplicialobject

ω : ∆op → Fun(D(1),D(1)). (14.1.10)

DRAFT

14.1. LOOP OBJECTS AS MONOID OBJECTS 249

We next verify that this defines the desired special simplicial object. Notethat the category Fun(D(1),D(1)) admits finite products which are constructedpointwise.

Proposition 14.1.11. For every pointed derivator D the simplicial object ωin (14.1.10) is special. In particular, for x ∈ D(1) there are canonical isomor-phisms

σk : ωkx∼−→ ω1(x)× . . .× ω1(x), k ≥ 0.

Moreover, there are canonical isomorphisms ω1∼= Ω: D(1)→ D(1).

Proof. The canonical isomorphism ω1∼= Ω was already established in (14.1.7)

as was the isomorphism ω0∼= 0. It remains to show that the Segal maps σk are

isomorphisms for k ≥ 2. The reader easily checks that it suffices to show thefollowing claim.

Claim: For k ≥ 2, 0 < k′ < k, and k′′ = k − k′ we consider the inclusions of‘convex’ subsets ik′ : [k′] → [k] : l 7→ l and ik′′ : [k′′] → [k] : l 7→ k′ + l. Then forevery x ∈ D(1) the induced map

(i∗k′ , i∗k′′) : ωkx→ ωk′x× ωk′′x (14.1.12)

is an isomorphism.

To reformulate this claim, let us consider the following diagram

1 t 1 ∇ //

∞k′t∞k′′

1

∞k

yk′tyk′′

π

(yik′,yik′′

)// yk

π

1id

// 1.

@H(14.1.13)

Using that left Kan extensions along cosieves are left extensions by zero (Corol-lary 9.1.6), we see that the upper square is homotopy exact (Lemma 11.1.12).One checks that in order for (14.1.12) to be an isomorphism it suffices toshow that the canonical mate of the bottom square is an isomorphism on(∞k)!(x) ∈ D(yk) for x ∈ D(1).

To verify this, let Pk′ be the poset obtained from yk by adjoining two newelements α0, α1. The order relation is generated by the relation on yk and therelations

α0 ≤ el, 0 ≤ l ≤ k′, and α1 ≤ el, k′ ≤ l ≤ k.

Considering Pk as a category, there is a fully faithful inclusion jk′ : yk → Pk′ .

DRAFT


The bottom square in (14.1.13) can be rewritten as follows

yk′tyk′′

πtπ

// yk

jk′

1 t 1

((0,1),(1,0))

y

Lk′//

π

Pk′

π

@H

1id

// 1.

@H

(14.1.14)

In this diagram, the functor Lk′ : y→ Pk′ is given by

Lk′(0, 1) = α0, Lk′(1, 0) = α1, and Lk′(1, 1) = ek′ ,

and the upper square is populated by the unique natural transformation.We begin by observing that the functor Lk′ is a left adjoint, and this is a

consequence of the dual of Lemma 11.3.3. In fact, for every x ∈ Pk′ there areunique maximal elements R(x) ∈ Lk′(y) such that Rx ≥ x. Considering theseas objects in y by means of the identification y ∼= Lk′(y), these elements aregiven by R(α0) = (0, 1), R(α1) = (1, 0), R(ek′) = (1, 1), R(∞) = (1, 1),

R(el) = (0, 1), 0 ≤ l < k′, and R(el) = (1, 0), k′ < l ≤ k.

Since the assignment x 7→ Rx is order-preserving, we conclude by Lemma 11.3.3that Lk′ is a left adjoint. It follows from Proposition 8.2.8 that the bottomsquare in (14.1.14) is homotopy exact.

The functoriality of mates with pasting implies that it remains to showthat the canonical mate of the top square in (14.1.14) is an isomorphism on(∞k)!(x) ∈ D(yk), x ∈ D(1). By (Der2) it is enough to check that this is thecase at all objects, which we now do for the three objects individually.

To begin with, let us consider the object (0, 1) ∈y and let us reformulate thetask using the following diagram,

yk′ //

yk′tyk′′

// yk

jk′

=

yk′

// yk

jk′

1(0,1)

// yLk′

//

@H

Pk′

@H

1α0

// Pk′ .

>F

In the pasting on the left, the right square is the square under considerationwhile the left square is a slice square which is hence homotopy exact. Note thatthe pasting of these two squares agrees with the square on the right, which isagain a slice square and hence homotopy exact. Hence, using the functorialityof mates, the canonical mate of the top square in (14.1.14) is unconditionally

DRAFT

14.2. LOOP OBJECTS AS GROUP OBJECTS 251

an isomorphism at (0, 1) ∈y. In the same way one shows that this is the case atthe object (1, 0).

The remaining case is given by (1, 1) ∈y. If we again paste the top squarein (14.1.14) with the corresponding slice square, then we obtain the followingdiagram

∅ //

yk′tyk′′

// yk

jk′

=

∅

// yk

jk′

1(1,1)

// yLk′

//

?G

Pk′

@H

1ek′// Pk′ .

<D

In fact, in this case the slice category is empty. Using the functoriality of matesand the homotopy exactness of slice squares, we see that the canonical mate ofthe top square in (14.1.14) is an isomorphism for X ∈ D(yk) and at the object(1, 1) if and only if (jk′)∗(X) vanishes at ek′ if and only ifX vanishes at ek′ . Sincethis is the case for all diagrams in the essential image of (∞k)! : D(1)→ D(yk),this concludes the proof.

As a consequence we obtain the intended natural monoid structures.

Corollary 14.1.15. Let D be a pointed derivator. The special simplicial ob-ject (14.1.10) induces a lift of Ω: D(1) → D(1) against the forgetful functorMon(D(1))→ D(1), i.e., such that the following diagram commutes,

Mon(D(1))

D(1)Ω

//

Ω

99

D(1).

Proof. This is immediate from Proposition 14.1.3 and Proposition 14.1.11.

The corresponding multiplication map is denoted by ∗ : Ωx×Ωx→ Ωx andis referred to as the concatenation map. If we want to be very precise, thenwe denote the concatenation map on Ωx, x ∈ D(1), by ∗x.

14.2 Loop objects as group objects

In this section we show that the natural monoid structures on loop objects inpointed derivators (see §14.1) are actually group structures. To motivate theconstruction of the inversion map, we briefly recall the situation in topology.

Digression 14.2.1. Let (X,x0) be a pointed topological space. Then a typicalmodel for the loop space is given by the space of maps [0, 1] → X which sendthe boundary points to x0. This can be considered as the homotopy pullback

DRAFT


of the cospan on the left in the following diagram,

∗x0

ΩX //

PX

ev1

∗x0

// X, ∗x0

// X.

If one wants to calculate this homotopy pullback with respect to the usualSerre model structure, then it suffices to replace one of the maps by a weaklyequivalent Serre fibration and then to calculate the categorical pullback. Thestandard example of such a Serre fibration is given by the path space PX ofpaths [0, 1]→ X starting at x0 endowed with the evaluation map ev1 : PX → X.This space is weakly contractible since all such paths are homotopic to theconstant path at x0. Calculating the corresponding pullback leads to the typicaldescription of the loop space.

A different model, however, is obtained if we replace both maps by this Serrefibration and then calculate the pullback

ΩXp2 //

p1

PX

ev1

PXev1

// X.

(14.2.2)

This model of the loop space has as points paths [−1, 1]→ X which send −1, 1to x0, and in this model the inversion of loops ι : ΩX → ΩX is given by areparametrization via the reflection at the origin 0 ∈ [−1, 1].

The point of this lengthy discussion is the observation that this reparametriza-tion can be obtained by interchanging the two copies of PX in the pullbacksquare (14.2.2). More formally, the outer commutative square in the diagram

ΩXp1

))p2

""

ΩXp2

//

p1

PX

ev1

PXev1

// X

induces by the universal property of the pullback square a canonical dashedmorphism making everything commute. The reader easily checks that this mor-phism is the above reparametrization map ι.

This suggests that a similar trick will construct an inversion map in thecontext of abstract pointed derivators. To this end we would like to considerthe swap symmetry

σ : y→y : (1, 0)↔ (0, 1).

DRAFT

14.2. LOOP OBJECTS AS GROUP OBJECTS 253

Since the swap symmetry is not induced by an order preserving map in ∆ wehave however to ‘change the combinatorics’ and allow for maps which do notpreserve the order. The details are as follows.

Let Setfin be the skeleton of the category of non-empty finite sets spannedby the sets n = 0, . . . , n together with morphisms of sets. There is functor

i : ∆→ Setfin : [n] 7→ n

which forgets the ordering. Since this functor is injective on objects and faithful,we do not distinguish notationally a morphism in ∆ and its image in Setfin. Inparticular, there are again maps ιk : 1 → n given by 0 7→ k − 1 and 1 7→ k.And for a diagram X : Setop

fin → C in a category with finite products there areassociated Segal maps as in (14.1.1).

Definition 14.2.3. Let C be a category with finite products.

(i) A diagram X : Setopfin → C is reduced if X0

∼= ∗.

(ii) A diagram X : Setopfin → C is special if all Segal maps (14.1.1) are isomor-

phisms.

We denote by Fun(Setopfin, C)sp ⊆ Fun(Setop

fin, C) the full subcategory spanned bythe special diagrams.

As in the previous case, the point for us to recall this notion is given by thefollowing proposition. Given a category C with finite products, we denote byGrp(C) the category of group objects in C.

Proposition 14.2.4. For every category C with finite products there is an equiv-alence of categories

Grp(C) ' Fun(Setopfin, C)

sp.

Moreover, the restriction functor i∗ : Fun(Setopfin, C)sp → Fun(∆op, C)sp is equiv-

alent to the forgetful functor Grp(C)→ Mon(C).

Proof. The details of this proof are left to the reader. We only mention thatthe inversion is induced by the swap symmetry σ : 1 → 1 which interchanges 0and 1.

Remark 14.2.5. Let D be a pointed derivator. We note that Construction 14.1.5extends from ∆op to Setop

fin. In fact, by the same formulas (14.1.6) and (14.1.9)we constuct a diagram

ω : Setopfin → Hom(D(1),D(1)). (14.2.6)

Moreover, this diagram is special since this is the case for the underlying sim-plicial object (Proposition 14.1.11).

As an upshot we obtain the following theorem.

DRAFT


Theorem 14.2.7. Let D be a pointed derivator. The special diagram (14.2.6)induces a lift of Ω: D(1) → D(1) against Grp(D(1)) → D(1), i.e., such thatthe following diagram commutes,

Grp(D(1))

D(1)Ω

//

Ω

99

D(1).

Proof. This is immediate from Remark 14.2.5 and Proposition 14.2.4.

Thus, for every x ∈ D(1) the concatenation of loops ∗ : Ωx × Ωx → Ωxdefines a natural group structure. In the case of the derivator HoTop∗ of pointedtopological spaces this reduces to the classical concatenation group structure onloop spaces ΩX ∈ Ho(Top∗).

Notation 14.2.8. Let D be a pointed derivator and let y ∈ D(1). We denoteby

(−)−1 : Ωy → Ωy

the loop inversion morphism. Given an additional object x ∈ D(1), theset homD(1)(x,Ωy) inherits a group structure, which we will denote additively.Thus, given morphisms f, g : x→ Ωy, we make the definition

f + g = ∗y (f, g) : x(f,g)→ Ωy × Ωy

∗y→ Ωy

and

−f = (−)−1 f : xf→ Ωy

(−)−1

→ Ωy.

Dually, we use the same notation for the group structure on homD(1)(Σx, y)induced by the natural cogroup structure on Σx (using the dual of Theo-rem 14.2.7).

14.3 Signs from loop squares

Let D be a pointed derivator and let X ∈ D() such that X1,0∼= X0,1

∼= 0, i.e.,X looks like

x //

0

0 // x′.

(14.3.1)

By Construction 9.2.20 there is a canonical morphism φX : x → Ωx′. Denotingby σ : → the swap symmetry interchanging the objects (1, 0) and (0, 1),there is an additional square σ∗X satisfying the above vanishing conditions.Hence, also the canonical map φσ∗X : x → Ωx′ is defined, and the goal of this

DRAFT

14.3. SIGNS FROM LOOP SQUARES 255

section is to show that φX and φσ∗X agree up to a loop inversion (Proposi-tion 14.3.5).

The proof of this result essentially consists of unraveling definitions, and wesuggest the reader to skip this section on a first reading.

Note that both in the definition of the canonical morphisms φX (Construc-tion 9.2.20) and in the construction of the loop inversion (−)−1 : Ωx′ → Ωx′

(Construction 14.1.5) we used that certain canonical mates are invertible, al-lowing us to pass to associated fractions. As a preparation for the proof ofProposition 14.3.5, we make very precise the definition of the loop inversionmorphism. We begin by considering the commutative square

σ //

π

π

1 // 1,

(14.3.2)

which we consider as populated by an identity transformation in either direction.

Lemma 14.3.3. The square (14.3.2) is homotopy exact and the correspondingcanonical mate can be chosen to be

id : (0, 0)∗ → (0, 0)∗σ∗.

Proof. Since σ is an isomorphism and hence both a left or a right adjoint, thehomotopy exactness with either orientation is immediate from Proposition 8.2.8.Here we are interested in the precise description of the canonical mate, hencewe give a more detailed proof.

Since (0, 0) ∈ is an initial object, it follows from Corollary 8.2.10 thatthere is a canonical isomorphism lim ∼= (0, 0)∗. More precisely, there is anadjunction ((0, 0), π, η, ε) : 1 such that η = id: id → π(0, 0). Moreover,ε : (0, 0)π → id is given by the unique morphisms from the initial object. Inparticular, ε(0,0) is also the identity.

If we apply a derivator D to this adjunction, then we obtain an inducedadjunction

(π∗, (0, 0)∗, η∗, ε∗) : D(1) D(),

exhibiting (0, 0)∗ as a right Kan extension functor. Using this precise adjunction,we can pass to the canonical mate (8.2.3) associated to the square (14.3.2) inorder to obtain the composition

(0, 0)∗η∗→ (0, 0)∗π∗(0, 0)∗

id→ (0, 0)∗σ∗π∗(0, 0)∗ε∗→ (0, 0)∗σ∗.

Since the first two morphisms are identities, the canonical mate in this casereduces to (0, 0)∗σ∗ε∗ = (0, 0)∗ε∗ = (ε(0,0))

∗ = id∗ = id.

Construction 14.3.4. Let D be a pointed derivator and let (−)−1 : Ω → Ω bethe loop inversion (Theorem 14.2.7). Recall from Construction 14.1.5 that thismorphism is obtained by forming a certain fraction of canonical mates associated

DRAFT


to (14.1.8). In the special case of the swap symmetry σ : y →y, the diagram(14.1.8) factors as

1 id //

(1,1)

| id

1

(1,1)

y σ //

iy

y

iy

σ //

π

π

<Did

1id// 1.

<Did

The functoriality of canonical mates with pasting and Lemma 14.3.3 imply thatthe loop inversion map (−)−1 : Ω→ Ω is given by

Ω = (0, 0)∗(iy)∗(1, 1)!

= (0, 0)∗σ∗(iy)∗(1, 1)!

→ (0, 0)∗(iy)∗σ∗(1, 1)!

∼= (0, 0)∗(iy)∗(1, 1)!

= Ω.

Here, the morphism is the canonical mate associated to the square in the middle,while the isomorphism is the inverse of the canonical mate associated to the topsquare. Since the inversion morphism is an isomorphism we conclude that alsothe first morphism in the above composition is an isomorphism (related to thissee also Lemma 14.3.7).

With this preparation the proof of the following result is essentially givenby diagram chasing.

Proposition 14.3.5. Let D be a pointed derivator and let X ∈ D() be asin (14.3.1). There is the relation φσ∗X = −φX in homD(1)(x,Ωx

′), i.e., thefollowing diagram commutes

xφX //

φσ∗X !!

Ωx′

(−)−1

Ωx′.

Proof. Let us recall that since X vanishes at (1, 0), (0, 1), the adjunction counit

ε : (1, 1)!(1, 1)∗i∗y(X)→ i∗y(X)

is an isomorphism (Corollary 9.1.6), and the canonical morphism φX is givenby

(0, 0)∗Xη// (0, 0)∗(iy)∗i

∗yX (0, 0)∗(iy)∗(1, 1)!(1, 1)∗i∗yX.

ε∼=oo

DRAFT

14.3. SIGNS FROM LOOP SQUARES 257

Plugging in the explicit description of the loop inversion morphism in Construc-tion 14.3.4, we now consider the following solid arrow diagram, consisting offunctors and natural transformations. (In that diagram, let us for the momentignore the dashed morphisms.)

(0, 0)∗η

//

id

(0, 0)∗(iy)∗i∗y

id

(0, 0)∗(iy)∗(1, 1)!(1, 1)∗i∗y

ε

(∼=)oo

id

(0, 0)∗σ∗ η//

η

(0, 0)∗σ∗(iy)∗i∗y

η

(0, 0)∗σ∗(iy)∗(1, 1)!(1, 1)∗i∗y

ε

(∼=)oo

η

(0, 0)∗(iy)∗i∗yσ

∗ η//

id

(0, 0)∗(iy)∗i∗yσ

∗(iy)∗i∗y

id

(0, 0)∗(iy)∗i∗yσ

∗(iy)∗(1, 1)!(1, 1)∗i∗y

ε

(∼=)oo

id

(0, 0)∗(iy)∗σ∗i∗y

η//

id((

(0, 0)∗(iy)∗σ∗i∗y(iy)∗i

∗y

ε

(0, 0)∗(iy)∗σ∗i∗y(iy)∗(1, 1)!(1, 1)

∗i∗yε

(∼=)

oo

ε

(0, 0)∗(iy)∗σ∗i∗y (0, 0)∗(iy)∗σ

∗(1, 1)!(1, 1)∗i∗y

ε

(∼=)

oo

(0, 0)∗(iy)∗(1, 1)!(1, 1)∗σ∗i∗y

ε (∼=)

OO

(0, 0)∗(iy)∗(1, 1)!(1, 1)∗σ∗(1, 1)!(1, 1)

∗i∗yε

(∼=)oo

ε∼=

OO

(0, 0)∗(iy)∗(1, 1)!(1, 1)∗i∗y

id

OO

(0, 0)∗(iy)∗(1, 1)!(1, 1)∗(1, 1)!(1, 1)

∗i∗yε

(∼=)oo

id

OO

(0, 0)∗(iy)∗(1, 1)!(1, 1)∗i∗y

η∼=

OO

id

kk

In this diagram, the column to the very right is the loop inversion morphism(see Construction 14.3.4). In particular, in that column the composition ofthe three upwards pointing morphisms is invertible. Moreover, the horizontalmorphisms pointing to the left are adjunction counits and they are invertible onall X ∈ D() which vanish at (1, 0), (0, 1). Thus, if we evaluate the path fromthe upper left corner to the lower right corner which passes through the upperright corner on our square X ∈ D() we precisely obtain the composition

(−)−1 φX : x→ Ωx′.

Now, in the above diagram all squares commute as naturality squares. More-over, the two triangles containing the dashed arrows as one of their sides alsocommute by the triangular identities (see A.1.3). We conclude that (−)−1 φXhence agrees with the remaining boundary path from the upper left to the lowerright corner. Note that the adjunction counit occuring in that path is simply

ε : (0, 0)∗(iy)∗(1, 1)!(1, 1)∗i∗yσ∗ → (0, 0)∗(iy)∗i

∗yσ∗,

DRAFT


which is to say that this boundary path is φσ∗X : x→ Ωx′.

For later reference, we also make explicit the dual statement for canonicalmorphisms from suspensions.

Proposition 14.3.6. Let D be a pointed derivator and let X ∈ D() be asin (14.3.1). There is the relation φσ∗X = −φX in homD(1)(Σx, x

′), i.e., thefollowing diagram commutes

ΣxφX //

(−)−1

x′

Σx.

φσ∗X

==

Proof. This is the dual of Proposition 14.3.5.

The above two results apply, in particular, to loop and suspension squares,and this will prove handy in the discussion of the rotation axiom for canonicaltriangles (see §15.5).

We conclude this section by showing that the swap symmetry σ : → preserves (co)cartesian squares.

Lemma 14.3.7. Let σ : → be the symmetry interchanging (1, 0) and (0, 1),and let σ : p→ p and σ : y →y be the corresponding restrictions. The followingsquares are homotopy exact,

pσ //

ip

| id

p

ip

yσ //

iy

y

iy

σ// ,

σ// .

<Did

Proof. We give a proof for the square on the left, the other case is similar. ByLemma 11.1.12 it suffices to show that the canonical mate (ip)!σ

∗ → σ∗(ip)! isan isomorphism at (1, 1) ∈ . To reformulate this, we consider the pasting onthe left in the following diagram,

pid //

π

pσ //

ip

id

p

ip

=

pσ //

π

z id

pid //

π

p

ip

1(1,1)

// σ// 1 // 1

(1,1)// ,

in which the square on the left is a slice square. Note that the above two pastingsagree. Hence, the homotopy exactness of slice squares (Lemma 8.2.6) and thefunctoriality of canonical mates with pasting imply that it suffices to show thatthe above pasting on the right is homotopy exact. But in that pasting the squareon the right was already observed to be homotopy exact. Moreover, the squareon the left is homotopy exact since σ is an isomorphism and hence a right adjoint(Proposition 8.2.8), and we can conclude the proof by Proposition 8.2.7.

DRAFT

14.4. TWO-FOLD LOOP OBJECTS AS ABELIAN GROUP OBJECTS 259

Corollary 14.3.8. For every derivator D the functor σ∗ : D() → D() pre-serves cocartesian and cartesian squares. In particular, if D is a pointed deriva-tor, then σ∗ preserves suspension and loop squares.

Proof. The homotopy exactness guaranteed by Lemma 14.3.7 yields a canonicalisomorphism

(ip)!σ∗ ∼−→ σ∗(ip)!,

showing that σ∗ preserves the essential image of (ip)! which is to say cocartesiansquares. Moreover, by definition of σ, σ∗ clearly also preserves the defininingvanishing conditions for suspension squares. In a similar way one shows that σ∗

preserves cartesian squares and loop squares.

14.4 Two-fold loop objects as abelian group ob-jects

Abstracting a classical result concerning the homotopy category Ho(Top∗) ofpointed topological spaces, in this section we show that two-fold loop objectsin pointed derivators are abelian group objects. This result is an application ofthe Eckmann–Hilton argument.

We begin by recalling the classical Eckmann–Hilton argument [EH62].

Lemma 14.4.1. Let M ∈ C be an object in a category with finite products andlet ∗1, ∗2 : M ×M →M be monoid structures on M . If the following diagram

M ×M ×M ×M id×s×id//

∗2×∗2

M ×M ×M ×M ∗1×∗1 // M ×M∗2

M ×M ∗1// M.

commutes (where s is the swap symmetry), then both monoid structures agreeand are abelian.

Proof. We give the proof in the case that C is the category of sets and that wehence work with monoids in the usual sense, and we leave it to the reader topass to represented presheaves in order to adapt the proof to the general case.Given elements x, y, u, v ∈M , by assumption we have

(x ∗2 y) ∗1 (u ∗2 v) = (x ∗1 u) ∗2 (y ∗1 v).

If e1, e2 ∈M are the neutral elements of the respective monoid structures, thenthese elements agree since

e1 = e1 ∗1 e1 = (e1 ∗2 e2) ∗1 (e2 ∗2 e1) = (e1 ∗1 e2) ∗2 (e2 ∗1 e1) = e2 ∗2 e2 = e2.

We now write e = e1 = e2 for the common neutral element. Moreover, the twomonoid structures agree by the equalities

x ∗1 y = (x ∗2 e) ∗1 (e ∗2 y) = (x ∗1 e) ∗2 (e ∗1 y) = x ∗2 y.

DRAFT


Finally, together with the calculation

x ∗1 y = (e ∗2 x) ∗1 (y ∗2 e) = (e ∗1 y) ∗2 (x ∗1 e) = y ∗2 x

we conclude that the monoid structures agree and are abelian.

We want to apply the Eckmann–Hilton trick to the following monoid struc-tures on two-fold loop objects in pointed derivators.

Construction 14.4.2. Let D be a pointed derivator and let x ∈ D(1). Thetwo-fold loop object Ω2x ∈ D(1) comes with two (potentially different) groupstructures.

First, writing Ω2x as Ω(Ωx), by Theorem 14.2.7 a group structure on Ω2xgiven by

∗1 = ∗Ωx : Ω2x× Ω2x→ Ω2x. (14.4.3)

Second, since there is an adjunction (Σ,Ω): D(1) D(1), we deduce thatΩ preserves products, and there is hence an induced functor

Ω: Grp(D(1))→ Grp(D(1)).

In more detail, if we apply this to Ωx ∈ Grp(D(1)) endowed with the groupstructure ∗x from Theorem 14.2.7, then we obtain an induced group structure

∗2 : Ω2x× Ω2x ∼−→ Ω(Ωx× Ωx)Ω(∗x)→ Ω2x. (14.4.4)

Here, the isomorphism is the inverse of the canonical isomorphism expressingthat Ω preserves finite products.

Lemma 14.4.5. Let D be a pointed derivator, let x ∈ D(1), and let ∗1, ∗2 beas in (14.4.3) and (14.4.4), respectively. The following diagram commutes,

Ω2x× Ω2x× Ω2x× Ω2xid×s×id

//

∗2×∗2

Ω2x× Ω2x× Ω2x× Ω2x∗1×∗1 // Ω2x× Ω2x

∗2

Ω2x× Ω2x ∗1// Ω2x.

Proof. Unraveling definitions we have to show that the following solid arrowdiagram commutes

Ω2x× Ω2x× Ω2x× Ω2xid×s×id

// Ω2x× Ω2x× Ω2x× Ω2x∗Ωx×∗Ωx// Ω2x× Ω2x

Ω(Ωx× Ωx)× Ω(Ωx× Ωx)

Ω(∗x)×Ω(∗x)

∼=

OO

∗Ωx×Ωx// Ω(Ωx× Ωx)

Ω(∗x)

∼=

OO

Ω2x× Ω2x ∗Ωx// Ω2x.

DRAFT

14.4. TWO-FOLD LOOP OBJECTS AS ABELIAN GROUP OBJECTS 261

First, since Ω: D(1) → D(1) preserves finite products, the canonical mapΩ(Ωx × Ωy) → Ω2x × Ω2y, x, y ∈ D(1), is an isomorphism. Moreover, sinceΩ takes values in group objects, this isomorphism is actually an isomorphismof group objects. Specializing to x = y this implies that the upper rectanglecommutes. Finally, the commutativity of the lower rectangle simply expressesthat Ω(∗x) is a morphism of group objects, which we know to be true for Ωffor all morphisms f in D(1); see Theorem 14.2.7.

Corollary 14.4.6. Let D be a pointed derivator and let x ∈ D(1). The twogroup structures (14.4.3) and (14.4.4) on Ω2x agree and are abelian.

Proof. This is immediate from Lemma 14.4.1 and Lemma 14.4.5.

This construction is clearly functorial in x ∈ D(1). Given a category Cwith finite products, then we denote by AbGrp(C) the category of abeliangroup objects in C. As an upshot, for every pointed derivator D , the aboveconstructions yield a lift of Ω2 : D(1) → D(1) against AbGrp(D(1)) → D(1),i.e., such that the following diagram commutes,

AbGrp(D(1))

D(1)Ω2

//

Ω288

D(1).

Example 14.4.7. Let X be a pointed topological space which we consider as anobject in Ho(Top∗), i.e., in the underlying category of the pointed derivator ofpointed spaces. The loop space ΩX = Map∗(S

1, X) is a group object by theconcatenation of loops. If we write the two-fold loop space as

Ω2X = Map∗(S1,Map∗(S

1, X)) ∼= Map∗(S1 ∧ S1, X),

then one can check that the maps

∗1, ∗2 : Ω2X × Ω2X → Ω2X

given by (14.4.3) and (14.4.4) are the concatenations with respect to the firstand the second sphere coordinate, respectively. These two maps are well-knownto coincide and to define abelian group structures.

DRAFT


DRAFT

Chapter 15

Basics on stable derivators

Stable derivators are obtained from pointed derivators by imposing an additionallinearity condition. The defining assumption is that a square is cocartesian if andonly if it is cartesian. While this linearity property is enjoyed by a plethora ofexamples of derivators arising in algebra, geometry, and topology, it is a purelyhigher categorical phenomenon. More precisely, stable represented derivatorsare trivial and this notion is hence not visible to ordinary category theory.

In this chapter as well as in later chapters in this book we develop some firstaspects of a fairly rich calculus available in stable derivators. Here we beginby extending a few well-known results from homological algebra and stablehomotopy theory to stable derivators. As a consequence of the discussion ofloop objects in §14 we conclude that the values of stable derivators are additivecategories. We also construct canonical triangulations on the values of (strong)stable derivators, thereby providing a first illustration that stable derivatorsare an enhancement of triangulated categories. Additional aspects of stablederivators are discussed in §16 and in the sequels [Gro16a, Gro16b].

In §15.1 we define stable derivators and deduce some immediate consequencesfrom the axioms (characterization of isomorphisms by vanishing of cones orfibers, 2-out-of-3 property for bicartesian squares, 5-lemma, characterization ofbicartesian squares via vanishing of iterated cones and total cofibers, suspensionand cofiber functors are equivalences). In §15.2 we indicate the ubiquity of stablederivators by collecting a few explicit examples. In §15.3 we show that the valuesof stable derivators are additive categories. In §15.4 we discuss the strength ofa derivator. In §15.5 we construct canonical triangulations in strong, stablederivators. Finally, in §15.6 we discuss negative canonical triangulations.

15.1 Definition and first properties

In this section we define stable derivators and deduce a few direct consequences.Similarly to the theory of model categories and ∞-categories, we obtain stablederivators from pointed ones by imposing the following linearity condition.

263

DRAFT

264 CHAPTER 15. BASICS ON STABLE DERIVATORS

Definition 15.1.1.

(i) A square in a derivator is bicartesian if it is cartesian and cocartesian.

(ii) A pointed derivator is stable if the classes of cartesian and cocartesiansquares coincide.

We include a short list of examples. Additional examples together with moredetails and references can be found in §15.2.

Examples 15.1.2.

(i) Homotopy derivators of Grothendieck abelian categories are stable. Inparticular, derivators associated to fields, rings, and schemes are stable.

(ii) The derivator of spectra is stable. In a certain precise sense this is theuniversal example of a stable derivator, namely the free stable derivatorgenerated by the sphere spectrum; see §18 for more details.

(iii) More generally, homotopy derivators of stable model categories and stable,complete and cocomplete ∞-categories are stable.

Lemma 15.1.3. A derivator is stable if and only if its opposite is stable.

Proof. This is immediate from the definition.

Thus, the duality principle extends to stable derivators.

Lemma 15.1.4. If D is a stable derivator and B ∈ Cat, then also DB is stable.

Proof. By Lemma 9.1.3 the shifted derivator DB is pointed, thus it remainsto show that a square in DB is cocartesian if and only if it is cartesian. Thisfollows from our discussion of parametrized Kan extensions in §10.3. In fact, byCorollary 10.3.10 and its dual we can use the stability of D to see that a squareX ∈ DB() = D(B × ) is cocartesian if and only if all Xb ∈ D(), b ∈ B,are cocartesian if and only if all Xb ∈ D(), b ∈ B, are cartesian if and only ifX ∈ DB() = D(B ×) is cartesian. Thus, DB is also stable.

We begin by a few sanity checks, indicating that Definition 15.1.1 cap-tures some of the expected phenomenons. The following is a derivator versionof Proposition 5.3.9.

Proposition 15.1.5. Let D be a stable derivator. A morphism f ∈ D([1]) isan isomorphism if and only if Cf ∼= 0 if and only if Ff ∼= 0.

Proof. It follows from the construction of C : D([1]) → D(1) that for everymorphism f ∈ D([1]) there is a cocartesian square

xf//

y

0 // Cf.

DRAFT


If f is an isomorphism then the same is true for 0 → Cf by Proposition 9.3.5,establishing one direction. Conversely, let us assume that Cf ∼= 0. Thus, thebottom horizontal morphism in the above cocartesian square is an isomorphism.Since D is stable, the square is also cartesian and it follows from the dualof Proposition 9.3.5 that f is an isomorphism. By duality this concludes theproof.

Proposition 15.1.6. Let D be a stable derivator and let X ∈ D(). If two ofthe squares ι∗01(X), ι∗12(X), and ι∗02(X) are bicartesian then so is the third one.

Proof. This is immediate from Proposition 9.3.10 and its dual.

We now establish a derivator version of the 5-lemma (Proposition 5.3.6). Inthat result we consider two cofiber squares X,X ′ ∈ D()cof looking like

x //

y

x′ //

y′

0 // z, 0 // z′

and a morphism X → X ′ with components f : x→ x′, g : y → y′, and h : z → z′.

Proposition 15.1.7. Let D be a stable derivator and let F : X → X ′ be amorphism of cofiber squares with components f : x → x′, g : y → y′, h : z → z′.If two of the morphisms f, g, and h are isomorphisms, then so is the third one.

Proof. Let us assume that f and g are isomorphisms. It follows from axiom(Der2) that the restriction i∗p(F ) is an isomorphism. The fully faithfulness of(ip)! implies that also F ∼= (ip)!i

∗p(F ) is an isomorphism hence so is F(1,1) = h.

Using stability, the dual proof shows that if g, h are isomorphisms then so is f .We attack the remaining case and assume that f and h are isomorphisms.

A right Kan extension by zero followed by a left Kan extension allows us toextend X and X ′ to cofiber sequences X and X ′,

x //

y

// 0

x′ //

y′

// 0

0 // z // w, 0 // z′ // w′,

and to obtain an induced morphism of cofiber sequences F : X → X ′. Therestriction ι∗0,2(F ) to the cocartesian composite squares is an isomorphism. Infact, it is an isomorphism if we restrict it further along ip : p→ since f isan isomorphism so that axiom (Der2) allows us to conclude. By the first partof this proof it follows that ι∗0,2(F ) is an isomorphism. Thus, the restriction

ι∗1,2(F ) to the squares on the right has the property that the two components at(0, 1), (1, 1) are isomorphisms. Using again that D is stable, we conclude thatg is an isomorphism.

DRAFT


Proposition 15.1.8. For every stable derivator D there is an equivalence

(cof, fib) : D([1]) ' D([1]).

Proof. We note again that the functors cof and fib respectively factor as indi-cated in

D([1])i∗ // D(p)

(ip)!//

i∗oo D()

(iy)∗//

(ip)∗

oo D(y)k∗ //

(iy)∗

oo D([1]).k∗

oo

These four adjunctions restrict to equivalences as follows. Let us denote by


the respective full subcategories spanned by the coherent diagrams satisfying thefollowing exactness conditions. In the first two cases we impose the vanishingcondition at (0, 1) while in the third case we consider only the squares which arebicartesian and satisfy this vanishing condition. It follows from Corollary 9.1.6and Proposition 8.2.16 that the functors cof and fib respectively factor as

D([1])i∗ // D(p)ex

(ip)!//

i∗oo D()ex

(iy)∗//

(ip)∗

oo D(y)exk∗ //

(iy)∗

oo D([1]).k∗

oo

and that each individual step is an equivalence.

As already mentioned, we will see in §15.5 that stable derivators D canoni-cally take values in triangulated categories. In particular, the underlying cate-gory D(1) can be endowed with a triangulated structure. The proposition aboveestablishes that we can rotate triangles back and forth without a loss of infor-mation. And the following proposition gives rise to the suspension equivalencesΣ: D(1)→ D(1) belonging to the canonical triangulations.

Proposition 15.1.9. For every stable derivator D there is an equivalence

(Σ,Ω): D(1) ' D(1).

Proof. The proof is very similar to the previous case, the main modificationbeing that we impose an additional vanishing condition at the upper right corner(1, 0). The details are left as an exercise to the reader.

It turns out that stability actually can be characterized by these properties;see [Gro16a]. We conclude this section by a version of Proposition 12.5.5 forstable derivators, in which we consider a square X ∈ D(), looking like

xf//

g

y

g′

x′f ′// y′.

Let us recall that associated to such squares there are canonical coherent mor-phisms C1(X), C2(X), and similarly for fibers; see (12.5.4).

DRAFT


Proposition 15.1.10. The following are equivalent for a square X in a stablederivator.

(i) The square X is bicartesian.

(ii) The canonical map C1X : Cf → Cf ′ is an isomorphism.

(iii) The canonical map C2X : Cg → Cg′ is an isomorphism.

(iv) The canonical map F1X : Ff → Ff ′ is an isomorphism.

(v) The canonical map F2X : Fg → Fg′ is an isomorphism.

Proof. By Proposition 12.5.5, symmetry, and duality it remains to show thatif C1X is an isomorphism then X is bicartesian. Let us recall from Construc-tion 12.5.2 that there is a coherent cube looking like

xf

//

g

yg′

##

x′f ′

//

y′

0

// Cf

C1X ""

0 // Cf ′,

such that the back and the front face are bicartesian. If C1X is an isomor-phism, then the bottom face is bicartesian (Proposition 9.3.5). It follows fromProposition 15.1.6 that also the top face X is bicartesian.

By means of the results about total (co)fibers in §12.5, there is the followingadditional characterization.

Proposition 15.1.11. The following are equivalent for a square X in a stablederivator.

(i) The square X is bicartesian.

(ii) The total cofiber tcof(X) is trivial, tcof(X) ∼= 0.

(iii) The total fiber tfib(X) is trivial, tfib(X) ∼= 0.

Proof. By duality it is enough to show that the first two statements are equiv-alent. But a square X in a stable derivator is bicartesian if and only C1(X)is an isomorphism (Proposition 15.1.10) if and only if C(C1(X)) ∼= 0 (Proposi-tion 15.1.5). The isomorphism tcof(X) ∼= C(C1(X)) (Theorem 12.5.8) concludesthe proof.

DRAFT


15.2 Examples of stable derivators

A good deal of the remainder of this section and of §16 develop first aspectsof the calculus of stable derivators. In this section we indicate the ubiquity ofstable derivators and include a fairly long list of examples of stable derivators,so that in later sections we have specific examples in mind.

Let us, however, begin by the following remark.

Remark 15.2.1. We want to emphasize that a represented derivator is stableif and only if the representing category is trivial. In fact, if C is a completeand cocomplete category admitting a zero object, then the cone constructionC : C[1] → C is the usual cokernel functor cok: C[1] → C (see §6.5 and §9.2). itfollows that Σ: C → C sends every object to a zero object. Hence, by Proposi-tion 15.1.9, a represented derivator y(C) is stable if and only if C ' 1.

It is also easy to show that many of the consequences of stability mentionedin §15.1 do not hold in general pointed, complete, and cocomplete categories.

(i) The existence of epimorphisms which are not isomorphisms shows thatProposition 15.1.5 fails in general.

(ii) The 5-lemma (Proposition 15.1.7) does not hold in this form in ordinarycategories. In fact, given a short exact sequence 0→ A′ → A→ A′′ → 0in a complete and cocomplete abelian category A, it suffices to considerany non-zero object K ∈ A and the diagram

A′i //

Ap//

id

A′′ //

id

0

K ⊕A′(0,i)// A

p// A′′ // 0.

The reader easily comes up with additional counterexamples along these lines.

One of the main sources of stable derivators is the following.

Theorem 15.2.2. Homotopy derivators of stable model categories are stable.

Proof. Let M be a stable model category. The homotopy derivator HoM ispointed since this is the case for the homotopy category HoM(1) = Ho(M).Let us recall that a model category M is stable if and only if the suspensionfunctor Σ: HoM(1)→HoM(1) is an equivalence. Since a pointed derivator isstable as soon as the suspension is an equivalence (see [GPS14b, Thm. 7.1] or[Gro16a]), this concludes the proof.

Remark 15.2.3.

(i) Taking for granted the existence of homotopy derivators of Grothendieckabelian categories, [GPS14b, Thm. 7.1] can be invoked to establish theirstability without reference to model categories. In fact, in that case theabstractly defined suspension functor is the usual shift functor Σ: D(A)→D(A), which is well-known to be an equivalence.

DRAFT

15.2. EXAMPLES OF STABLE DERIVATORS 269

(ii) Besides stable model categories, also further typical approaches to ax-iomatic stable homotopy theory forget to derivators. For example, thereare stable derivators associated to stable ∞-categories [Lur11] or stablecofibration categories [Sch13] (possibly only after restricting the class ofallowable shapes of diagrams). Conjecturally, under suitable stability andcompleteness assumptions, this is also true for any of the many axioma-tizations of a theory of (∞, 1)-categories (see [Ber07, Ber10, Cam13] formany references).

(iii) The reader who does not like that the proof of Theorem 15.2.2 relies on[GPS14b, Thm. 7.1] is invited to instead invoke [Hov99, Rmk. 7.1.12]. Infact, this result shows that homotopy derivators are stable in the sense ofDefinition 15.1.1.

We now turn to relevant specific examples of stable derivators, arising invarious areas of pure mathematics.

Examples 15.2.4.

(i) Let R be an ordinary, not necessarily commutative ring. Then there is thestable model category Ch(R) of unbounded chain complexes of R-modules(with the projective model structure, see [Hov99, §2.3]). The associatedhomotopy derivator DR = DCh(R) is stable, and its underlying categoryis the ordinary derived category D(R). As observed in Examples 7.3.5,even in the case of a field k, the derivator Dk encodes many interestingcategories, like derived categories of group algebras, path algebras, andincidence algebras.

(ii) The previous example generalizes to Grothendieck abelian categories A.In fact, the category Ch(A) of unbounded (co)chain complexes admits theso-called injective model structure in which the weak equivalences are thequasi-isomorphisms (see [Bek00, Hov01, CD09] or [Lur11, Chapter 1]).This model structure is stable, thereby giving rise to the stable derivatorDA. In particular, given a scheme X, the category of quasi-coherent OX -modules is Grothendieck abelian, and we obtain the stable derivator DX

of the scheme.

(iii) An additional generalization is obtained by passing from abelian cate-gories to exact categories in the sense of Quillen ([Qui73] or [Buh10]).Also in that case we obtain stable derivators DE = DCh(E) (at least de-

fined on suitably finite shapes); see [Mal07, Appendice] and [Gil11, Sto14].

(iv) There are variants of the first example in the context of differential-gradedalgebra (see for example [Hin97, SS00, Fre09]). Given a differential-gradedalgebra A over an arbitrary ground ring, we can consider the categoryMod(A) of differential-graded modules over A. This category admits theprojective model structure with quasi-isomorphsms as weak equivalences.Since this model structure is stable, the same is true for the derivator

DRAFT


DA = DMod(A) of differential-graded modules over A. This derivatorencodes, for example, derived categories of differential-graded representa-tions of groups, quivers, and posets.

(v) An additional algebraic context leading to stable derivators is given bystable module theory and representation theory of groups. In this context,let R be a quasi-Frobenius ring or, more generally, an Iwanaga–Gorensteinring. That is, R is supposed to be left and right noetherian and of finite leftand right self-injective dimension [EJ00, §9.1]. Then there are two modelstructures on Mod(R), the so-called Gorenstein projective and Goren-stein injective model structures [Hov02, Theorem 8.6]. These Quillenequivalent model categories give rise to the same derivator DGor

R . Themodel structures are stable ([Bec14, Corollary 1.1.16]), hence the same istrue for the derivator DGor

R . The underlying categories DGorR (1) and its

subcategories have been studied as stable categories of Gorenstein pro-jective (or maximal Cohen–Macaulay) and Gorenstein injective modules;see [AB69, Buc87, EJ00, Hol04] and the references therein.

(vi) As a special case of the previous case we can consider a discrete group G,a field k, and the associated group algebra R = kG. (It turns out thatkG has a Hopf algebra structure which is compatible with the Gorensteinprojective model structure, and it follows that DGor

kG becomes a monoidalstable derivator; we come back to monoidal derivators in the sequel.) Thestructure of the underlying category DGor

kG (1) has been carefully studiedand references include [BCR97, Ric97, BIK11].

(vii) The derivator of spectra is the homotopy derivator associated to the sta-ble Bousfield–Friedlander model structure on spectra [BF78] (see Exam-ples 7.3.11). Since the model structure is stable, the same is true forthe homotopy derivator Sp. It turns out that this stable derivator is theuniversal example of a stable derivators (it is the free stable derivatorgenerated by the sphere spectrum; see §18 for more details).

(viii) As a variant of the previous example, we can consider any of the modern,monoidal model categories of spectra (see [HSS00, EKMM97, MMSS01]or [Sch07]), leading to a framework for the study of brave new algebra.Choosing one of these approaches, let E be a symmetric ring spectrum.Then the category of E-module spectra can be endowed with a stablemodel structure [HSS00] and there is hence the associated stable deriva-tor DE of E-module spectra. As special values, DE yields, for example,homotopy categories of spectral representations of groups, quivers, andposets.

The sphere spectrum S is a symmetric ring spectrum and in this case thereis an equivalence of derivators DS ' Sp. Moreover, if R is an ordinaryring, then the derivator DR of the ring is equivalent to the derivator DHR

of the corresponding Eilenberg-MacLane ring spectrum HR (see [SS03,

DRAFT

15.3. THE ADDITIVITY OF STABLE DERIVATORS 271

Thm. 5.1.6]). In general, however, DE is not equivalent to the derivatorof a ring.

(ix) The ubiquity of stable model categories and stable ∞-categories impliesthat there are many additional stable derivators. In particular, stablederivators arise in parametrized stable homotopy theory [MS06, ABG+09,ABG10], in motivic stable homotopy theory [Voe98, MV99, Jar00], inequivariant stable homotopy theory [MM02, LMSM86] or [Sch14], and inglobal stable homotopy theory [Sch15].

Additional examples of stable model categories and stable ∞-categoriesarising in various areas of algebra, geometry, and topology can be found in[SS03, Lur11].

15.3 The additivity of stable derivators

In this section we show that stability implies additivity, i.e., that the values ofa stable derivator are additive categories.

For later reference, we provide an independent proof of the preadditivityof stable derivators. As a preparation for the proof, recall the description of(co)products by (co)cartesian squares (Lemma 9.3.16).

Proposition 15.3.1. Let D be a stable derivator and let u : A→ B in Cat.

(i) The category D(A) is preadditive.

(ii) The functors u∗ : D(B) → D(A) and u!, u∗ : D(A) → D(B) are additive,i.e., preserve zero objects and finite direct sums.

Proof. Let D be a stable derivator and A ∈ Cat . The derivator DA is againstable (Lemma 15.1.4) and there is an isomorphism of categories DA(1) ∼= D(A).Thus, it is enough to show that the underlying category D(1) is preaddive.

The category D(1) admits small (co)products (Proposition 7.4.7) and a zeroobject (D is pointed). Hence, in order to conclude, it is enough to show that forx, y ∈ D(1) the canonical map xty → x×y is an isomorphism (Definition 2.1.1).To this end, we begin by observing that (Der1) yields a canonical equivalence ofcategories D(1t1) ' D(1)×D(1). For x, y ∈ D(1) we write (x, y) ∈ D(1t1) forany object corresponding to the pair (x, y) ∈ D(1)×D(1) under this equivalence.

Associated to (x, y) ∈ D(1 t 1) we construct a coherent diagram of shape[2] × [2]. Here, [2] is again the poset (0 < 1 < 2) and the product category

DRAFT


[2]× [2] hence looks like

(0, 0) //

(1, 0) //

(2, 0)

(0, 1) //

(1, 1) //

(2, 1)

(0, 2) // (1, 2) // (2, 2).

The fully faithful functor 1 t 1 → [2] × [2] classifying the objects (1, 2), (2, 1)factors throught the following full subcategories

1 t 1i→ A

j→ Bk→ [2]× [2].

(i) The category A ⊆ [2]× [2] is the full subcategory spanned by the objects(2, 0), (2, 1) and (0, 2), (1, 2). (Thus, A is isomorphic to [1] t [1].) Thefunctor i : 1 t 1→ A is the cosieve classifying (1, 2) and (2, 1).

(ii) The category B ⊆ [2] × [2] is the full subcategory obtained from A byadding the object (2, 2) and the functor j : A → B is the obvious fullyfaithful inclusion. Note that j is a sieve.

(iii) Finally, the functor k : B → [2]× [2] is the obvious fully faithful inclusion.

Associated to these fully faithful functors there are by Proposition 8.2.16 fullyfaithful Kan extension functors

D(1 t 1)i!→ D(A)

j∗→ D(B)k∗→ D([2]× [2]). (15.3.2)

Moreover, since i is a cosieve and j a sieve, it follows from Corollary 9.1.6 thati! and j∗ are left and right extension by zero, respectively. To understand theremaining functor k∗ we note that k : B → [2]× [2] factors as a composition offour fully faithful inclusions,

k : Bk1→ B1

k2→ B2k3→ B3

k4→ [2]× [2],

which are obtained by adding the objects (1, 1), (1, 0), (0, 1), and (0, 0), oneby one in turn. The reader easily applies the dual of the detection lemma(Proposition 11.3.10) together with Lemma 11.3.3 in order to conclude thateach of the corresponding right Kan extension functors precisely amounts toadding a new cartesian square.

As an upshot, we have shown that (15.3.2) induces an equivalence betweenD(1 t 1) and the full subcategory D([2] × [2])ex ⊆ D([2] × [2]) spanned by alldiagrams Q ∈ D([2]× [2]) satisfying the following exactness properties.

(i) The diagram Q vanishes at the three corners (0, 2), (2, 0), and (2, 2).

DRAFT

15.3. THE ADDITIVITY OF STABLE DERIVATORS 273

(ii) The diagram Q makes the four squares cartesian.

Now, let (x, y) ∈ D(1 t 1) and let Q ∈ D([2] × [2])ex be the correspondingcoherent diagram. The diagram Q looks like

z //

1

x′ //

2

0

y′ //

3

b //

4

y

0 // x // 0.

(15.3.3)

By Proposition 15.1.6 the composite square 2 + 4 is also cartesian and themap x′ → x is hence an isomorphism (Proposition 9.3.5). Similarly, also thestructure map y′ → y is an isomorphism. If we consider the composite square1 − 4 obtained by glueing all four squares, then there are two reasons that

the upper left corner in (15.3.3) is also populated by a zero object z ∼= 0.

An application of (the dual of) Lemma 9.3.16 to the square 4 implies that b

is the product of x, y. Finally, since D is stable the square 1 is also cocartesianand Lemma 9.3.16 shows that b is the coproduct of x′, y′. A final combinationof these observations with the isomorphisms x′ ∼= x and y′ ∼= y concludes theproof of the first statement.

As of the second statement, given a functor u : A→ B, there are adjunctions

(u!, u∗) : D(A)→ D(B) and (u∗, u∗) : D(B) D(A)

between preadditive categories, and u∗, u!, u∗ are hence additive.

Remark 15.3.4.

(i) The proof also works for pointed derivators in which cartesian squares arecocartesian or conversely. We did not use that the classes actually agree.

(ii) The category D([2] × [2])ex ⊆ D([2] × [2]) is one model for the categoryof two-fold biproduct diagrams in a stable derivator. The first part ofthe proof shows that having two objects in D(1) is equivalent to havinga coherent biproduct diagram.

Corollary 15.3.5. Let D be a stable derivator and A ∈ Cat. Every objectX ∈ D(A) is canonically an abelian monoid object.

Proof. Since D(A) is preadditive, this is immediate from Remark 2.1.2.

This is a typical feature of stability. Given a stable model category, althoughat the point-set level there might not be an obvious way to form the sum ofmorphisms, this is always the case at the level of homotopy categories.

Examples 15.3.6.

DRAFT


(i) Let A be a Grothendieck abelian category and let DA be the associatedstable derivator. The underlying category DA(1) is the derived categoryD(A) and in this case the localization functor

γ : Ch(A)→ D(A)

preserves finite biproducts. Hence, the abelian monoid structures aresimply induced from the corresponding structures at the level of chaincomplexes.

(ii) Let Sp be the category of spectra, endowed with the stable Bousfield–Friedlander model structure. In this case, the category Sp is not pread-ditive. However, since the derivator Sp of spectra is stable, all its valuesSp(A), A ∈ Cat , are preadditive categories. In particular, every spectrumX ∈ SHC ∼= Sp(1) is uniquely an abelian monoid object

We now take care of the additivity of stable derivators which is immediatefrom our discussion of loop objects and two-fold loop objects. In fact, it suf-fices to combine this with the fact that loop functors are equivalences in stablederivators.

Theorem 15.3.7. Let D be a stable derivator and let u : A→ B in Cat.

(i) The category D(A) is additive.

(ii) The functors u∗ : D(B)→ D(A) and u!, u∗ : D(A)→ D(B) are additive.

Proof. For the first statement, by Lemma 15.1.4, it is enough to show that theunderlying category D(1) is additive. The category D(1) has finite coprod-ucts and finite products (Proposition 7.4.7). Moreover, since Ω: D(1) → D(1)is an equivalence of categories (Proposition 15.1.9), there are natural isomor-phisms x ∼−→ Ω2Σ2x, x ∈ D(1), showing that every x is an abelian group ob-ject (Corollary 14.4.6). Moreover, the fully faithfulness of Ω2 shows that everymap in D(1) is a map of abelian group objects. Hence, the forgetful functorAbGrp(D(1)) → D(1) is an equivalence of categories, and we conclude thatD(1) is additive. As for the second statement, the adjunctions (u!, u

∗) and(u∗, u∗) show that these functors are additive.

Remark 15.3.8. This proof is independent of the proof of the preadditivity ofstable derivators (Proposition 15.3.1). However, since the constructions in theproof of Proposition 15.3.1 play some role in applications of stable derivators toabstract representation theory (see [GS14b, GS15b, GS14a, GS15a]), we never-theless decided to include that proof in this book.

Remark 15.3.9. We note that the above proof of the additivity of stable deriva-tors only uses that D is pointed and that the suspension Σ: D(1) → D(1) isan equivalence. This observation is useful in [Gro16a] where we show that suchderivators are already stable.

DRAFT

15.4. STRENGTH OF A DERIVATOR 275

15.4 Strength of a derivator

In this section we discuss the strength of derivators, a property enjoyed by deriva-tors coming from ordinary categories, abelian categories, model categories, orcomplete and cocomplete ∞-categories. The point of this property is that itallows us to use exactness properties of certain derivators (like stability or addi-tivity) to construct structure on the values of such derivators (like triangulations,higher triangulations, or pretriangulations).

Let us recall from §7.4 the construction of partial underlying diagram func-tors in prederivators (see (7.4.4)). For every prederivator D and categoriesA,B ∈ Cat there is the partial underlying diagram functor

diaA,B : D(A×B)→ D(A)B ,

making a coherent diagram incoherent in the B-direction. In general, thesefunctors are far from being equivalences. For example, in §7.5 we saw that, inthe derivator of pointed topological spaces, the underlying diagram functor

diap : DTop∗(p)→ DTop∗(1)p

is not an equivalence for categories. However, the following is satisfied in exam-ples arising in nature.

Definition 15.4.1. A derivator D is strong if for every A ∈ Cat and everyfinite, free category F the partial underlying diagram functor

diaA×F : D(A× F )→ D(A)F

is full and essentially surjective.

Note that we do not ask these functors to be faithful since this propertyis not satisfied by most derivators arising in homological algebra and abstracthomotopy theory. An explicit example illustrating this is discussed in §4.3.

Examples 15.4.2.

(i) Represented derivators are strong. In fact, in that case all partial under-lying diagram functors are even isomorphisms.

(ii) Homotopy derivators of Grothendieck abelian categories are strong.

(iii) Homotopy derivators of model categories are strong (see §B.3 for a proof).Thus, typical derivators arising in homological algebra and homotopy the-ory — including the ones mentioned in §7.3 and §15.2 — are strong. Simi-larly, one can show that homotopy derivators of complete and cocomplete∞-categories are strong.

We include a short discussion of this property.

Remark 15.4.3.

DRAFT


(i) If D is a strong derivator, then dia[1] : D([1])→ D(1)[1] is full and essen-tially surjective. The essential surjectivity allows us to do the following:given an incoherent morphism in D , up to isomorphism we can replace itby the underlying diagram of a coherent morphism. The point is that wecan apply Kan extension functors to this coherent approximation, and thiswill be used in the construction of canonical triangulations in §15.5. Sim-ilarly, the fullness allows us to lift a morphism of incoherent morphismsto a morphism of coherent morphisms, and this also plays a role in §15.5.

(ii) By (Der2) every partial underlying diagram functor is conservative, i.e.,it reflects isomorphisms. As a consequence, in a strong derivator, thefunctors diaA,F , A ∈ Cat , are full, essentially surjective, and conservative.This implies that the approximations of incoherent morphisms by coherentones are unique up to non-canonical isomorphism.

(iii) In some references on derivators the property of being strong is referred toas axiom (Der5) and is taken as part of the definition of a derivator. Here,we exclude this axiom from the definition. As of this writing, the strengthof derivators is used only to relate exactness properties of derivators tostructure on its values (see for example §15.5). So far it does not play animportant role in the theory of derivators itself.

Lemma 15.4.4.

(i) If D is a strong derivator and B ∈ Cat, then DB is strong.

(ii) A derivator D is strong if and only if the opposite Dop is strong.

(iii) If D and E are strong derivators, then also D × E is strong.

Proof. This proof is left to the reader.

15.5 Canonical triangulations in stable deriva-tors

In this section we construct canonical triangulations on the values of strongstable derivators. It turns out that restriction functors and Kan extension func-tors of such derivators can be turned into exact functors with respect to thesetriangulations, and we will come back to this in [Gro16a].

As a preparation we recall the construction of (coherent) cofiber sequencesin pointed derivators D (see §9.5). By Proposition 9.5.8 there is an equivalenceof categories

D([1]) ' D()cof

sending a morphism to its cofiber sequence. Every coherent cofiber sequencehas an underlying incoherent cofiber sequence, obtained by an application ofthe functor D()cof → D(1)[3] defined in (9.5.5). Recall from §9.5 that the

DRAFT

15.5. CANONICAL TRIANGULATIONS IN STABLE DERIVATORS 277

construction of this functor relies on the existence of canonical isomorphismsassociated to suspension squares; see Construction 9.2.20.

In the case of a stable derivator D we write

tria : D([1]) ' D()cof → D(1)[3]

for the resulting composition and refer to tria(f), f ∈ D([1]), as the standardtriangle associated to f . A triangle x→ y → z → Σx in D(1) is distinguishedif it is naturally isomorphic to a standard triangle.

Passing to shifted derivators this also defines a class of distinguished trianglesin D(A), A ∈ Cat . In fact, with D also the shifted derivator DA is stable(Lemma 15.1.4) and the underlying category of DA is canonically isomorphicto D(A).

Theorem 15.5.1. Let D be a strong, stable derivator and let A ∈ Cat. The sus-pension functor Σ: D(A)→ D(A) and the above class of distinguished trianglesdefine a triangulation on D(A).

Proof. By the above discussion, passing to shifted derivators we may assumethat A = 1. The category D(1) is additive (Theorem 15.3.7) and the suspensionfunctor Σ: D(1) → D(1) is an equivalence (Proposition 15.1.9). It remains toshow that the above class of distinguished triangles satisfies axioms (T1)-(T4)of Definition 5.3.1.

(T1): Let x ∈ D(1) and let us consider the corresponding identity morphismid: x → x as an object in D(1)[1]. Since D is strong, the underlying diagramfunctor D([1]) → D(1)[1] is essentially surjective and we can find a coherentmorphism f ∈ D([1]) and an isomorphism dia[1](f) ∼= id (one can also avoidusing the strength here by restricting along π : [1] → 1 or by invoking Exam-ple 8.2.24). To see that x → x → 0 → Σx is distinguished the reader easilychecks that it suffices to show that the third object in tria(f) is a zero object,and this is the case by Proposition 15.1.5.

By definition the class of distinguished triangles is closed under isomor-phisms, and for axiom (T1) it hence remains to show that every morphismx → y in D(1) extends to a distinguished triangle. By assumption the dia-gram functor D([1]) → D(1)[1] is essentially surjective and we can hence findf ∈ D([1]) and a natural isomorphism dia[1](f) ∼= (x → y). The reader checksthat this isomorphism together with the distinguished triangle tria(f) yields adistinguished triangle extending x→ y.

(T2): We show that if xf→ y

g→ zh→ Σx is distinguished then also the

rotated triangle yg→ z

h→ Σx−Σf→ Σy is distinguished. The converse implication

is similar and is left to the reader. As indicated in §12.1, for the rotationaxiom it is convenient to consider coherent diagrams encoding three iterationsof the cofiber construction. We suggest the reader to recall the statement ofLemma 12.1.3 as well as the statement and the proof of Proposition 12.1.5.

By this lemma, every coherent morphism (f : x → y) ∈ D([1]) can be ex-

DRAFT


tended to a coherent diagram looking like (12.1.4),

xf//

y //

g

02

01// z

h //

x′

f ′

03// y′,

(15.5.2)

vanishing as indicated and making all squares cocartesian (we can ignore thesubscripts of the objects 01, 02, 03 for now). Note that the cofiber sequenceconsisting of the two top squares in (15.5.2) is used to construct the standardtriangle tria(f).

In more detail, we consider the composition of these two squares X ∈ D(),looking like

x //

02

01// x′,

(15.5.3)

in order to obtain a suspension square. Hence, the morphism φX : Σx ∼−→ x′

from Construction 9.2.20 is an isomorphism in this case (Lemma 9.2.22), andthe corresponding standard triangle is given by

xf

−→ yg

−→ zφ−1X h−→ Σx. (15.5.4)

We next collect a detailed description of the standard triangle tria(g). Tobegin with we restrict (15.5.2) to the two squares on the right and then restrictfurther along the isomorphism [2]× [1] ∼= [1]× [2]. By Corollary 14.3.8 this yieldsan additional cofiber sequence as shown on the left in

yg//

z

h

// 03

y //

03

02// x′

f ′// y′, 02

// y′.

(15.5.5)

This cofiber sequence can hence be used to calculate tria(g). In fact, the sus-pension square Y ∈ D() on the right in (15.5.5), obtained by composing thecocartesian squares on the left, yields an identification φY : Σy ∼−→ y′, and tria(g)hence looks like

yg

−→ zh−→ x′

φ−1Y f

′

−→ Σy. (15.5.6)

It remains to show that this triangle is isomorphic to the rotation of (15.5.4).To this end, we recall that f ′ is naturally isomorphic to Σf (Proposition 12.1.5).

DRAFT


Since we need some details about this identification, we repete from the proofof this proposition that there is a coherent cube Z ∈ D([1]3) = D [1]() (see(12.1.6)),

x //

f

02

y //

02

01

// x′

f ′

03// y′.

Considered as a square in the shifted derivator D [1], this square is a suspensionsquare, and we hence obtain a canonical isomorphism φZ : Σf ∼−→ f ′.

To conclude the proof it suffices to analyze φZ in more detail. It is straigh-forward to verify that the formation of the canonical morphism from Construc-tion 9.2.20 is compatible with evaluations. In our case this amounts to sayingthat φZ has components

(φZ)0 = φ(Z0) : Σx ∼−→ x′ and (φZ)1 = φ(Z1) : Σy ∼−→ y′.

In more detail, the respective components are induced from the cocartesiansquares

x //

02

y //

02

01// x′, 03

// y′.

(15.5.7)

Since the square on the left is precisely the square (15.5.3), Z0 = X, we deducethat (φZ)0 = φX : Σx → x′ is the morphism showing up in (15.5.4). However,if we compare the square on the right in (15.5.7) to the square on the right in(15.5.5), then we see that they differ by a restriction along the swap symmetryσ : ∼−→ interchanging (1, 0) and (0, 1). It follows from Proposition 14.3.6that the morphism (φZ)1 = φ(Z1) = φσ∗Y is −φY , hence the negative of theidentification showing up in (15.5.6).

To summarize, the natural isomorphism φZ : Σf ∼−→ f ′ has components

ΣxΣf//

φX ∼=

Σy

−φY∼=

x′f ′// y′,

DRAFT


and this finally gives us the desired isomorphism of triangles

y

=

g// z

=

φ−1X h // Σx

φX

−Σf// Σy

=

yg// z

h// x′

φ−1Y f

′// Σy

from the rotation of tria(f) to tria(g); see again (15.5.6) and (15.5.4).

(T3): This is similar to the verification of (T1) and follows from the assump-tion that D is strong. The details are left as an exercise to the reader.

(T4): It remains to establish the octahedron axiom, saying that for every pairof composable morphisms in D(1) there is an associated octahedron diagram.We first show that every coherent pair of composable morphisms X ∈ D([2])

with underlying diagram f3 = f2 f1 : xf1→ y

f2→ z gives rise to such a diagram.Choosing a slightly different notation than in Definition 5.3.1, we have to showthat there is a commutative diagram

xf1 //

=

yg1 //

f2

0

uh1 //

f4

1

Σx

=

xf3 // z

g2

g3 //

2

vh3 //

g4

3

Σx

Σf1

w

h2

=//

4

w

h4

h2

// Σy

ΣyΣg1

// Σu

(15.5.8)

such that the rows and colums are distinguished triangles, i.e., such that

(i) xf1→ y

g1→ uh1→ Σx,

(ii) yf2→ z

g2→ wh2→ Σy,

(iii) xf3→ z

g3→ vh3→ Σx, and

(iv) uf4→ v

g4→ wh4→ Σu

are distinguished triangles. Thus, these four distinguished triangles have to beconstructed such that all the squares 0 , . . . , 4 in (15.5.8) commute.

We achieve this by a variant of Construction 12.2.1, namely by passing to

DRAFT


coherent diagrams looking like

xf1 //

yf2 //

z //

0

0 // u //

v //

x′ //

0

0 // w // y′ // u′.

(15.5.9)

In more detail, let B ⊆ [4] × [2] be the full subcategory obtained by removingthe objects (4, 0), (0, 2). The functor [2] → B classifying the composable mor-

phisms (0, 0) → (1, 0) → (2, 0) factors as a composition [2]i→ A

j→ B of fullyfaithful functors where A ⊆ B contains the objects (0, 0), (1, 0), (2, 0) and thefour objects populated by zeros in (15.5.9). Since i is a sieve, i∗ : D([2])→ D(A)is right extension by zero (Corollary 9.1.6). Four applications of the detectionlemma (Proposition 11.3.10) imply that j! : D(A) → D(B) precisely adds forcocartesian squares (again the assumptions of the detection lemma are verified

using Lemma 11.3.3). As an upshot, the functors D([2])i∗→ D(A)

j!→ D(B)induce an equivalence

D([2]) ' D(B)cof , (15.5.10)

where D(B)cof ⊆ D(B) is the full subcategory spanned by all diagrams

(i) which vanish at the four objects (0, 1), (3, 0), (1, 2), (4, 1) and

(ii) which make all squares cocartesian.

Using implicitly the identification of cof3 with Σ (see §12.1), the equivalence(15.5.10) shows that every (f3 = f2 f1 : x → y → z) ∈ D([2]) gives rise to acoherent diagram Q ∈ D(B)cof with underlying diagram

xf1 //

yf2 //

g1

z //

g3

0

0 // uf4 //

vh3 //

g4

Σx //

Σf1

0

0 // wh2

// ΣyΣg1

// Σu.

(15.5.11)

A diagram chase in (15.5.9) combined with the identifications made in (15.5.11)yields the desired four distinguished triangles satisfying the above five relations0 − 4 . To begin with, if one considers the top three cocartesian squares

and glues the two to the right, then one obtains the first distinguished triangletogether with the relation h3f4 = h1, settling equation 1 . Let us next consider

DRAFT


the copy of [2]× [2] in the middle of the diagram and compose vertically. Thisgives the second distinguished triangle together with the relation g2 = g4g3,hence taking care of 2 . If we again consider the top three cocartesian squaresbut this time glue the two squares to the left, then we construct the thirddistinguished triangle. For the remaining distinguished triangle we considerthe bottom three cocartesian squares and compose the two to the right. Thisalso yields the relation h4 = (Σg1)h2, hence making 4 commute. Finally, in

order to show that also the squares 0 and 3 commute it suffices to considerthe middle square in the top and bottom row of (15.5.9), respectively. Thiscompletes the construction of an octahedron diagram for a coherent pair ofcomposable morphisms.

Finally, let us consider a pair of composable morphisms x → y → z in theunderlying category D(1). Since D is strong we can find a coherent diagramX ∈ D([2]) approximating x → y → z. The reader easily checks that this canbe used to construct an octahedron diagram for x→ y → z.

The triangulations of the theorem are referred to as canonical triangula-tions. We come back to a partial justification for this terminology in [Gro16a].

As a special case, the theorem yields canonical triangulations on underlyingcategories of strong, stable derivators. We illustrate this by a few examples.

Examples 15.5.12.

(i) If A is an abelian category, such that the homotopy derivator DA ex-ists (for example, if A is a Grothendieck abelian category; see Defini-tion 7.3.2), then the underlying category DA(1) ∼= D(A) is a triangulatedcategory. This reproduces the classical triangulation from Verdier’s 1967thesis [Ver67]; for a reprint see [Ver96]. Taking up again Examples 7.3.3,the theorem reproduces the classical triangulations on derived categoriesof fields, rings, and schemes.

(ii) In the case of the derivator of spectra (7.3.12) we recover the classicaltriangulation on the stable homotopy category SHC. This triangulationseems to go back to the 1964 thesis of Boardman [Boa64]; see [Vog70] foran early account.

(iii) More generally, homotopy derivators of stable model categories and com-plete, cocomplete, stable ∞-categories are strong, stable derivators. Ap-plied to such examples, Theorem 15.5.1 reproduces the triangulations onhomotopy categories of stable model categories ([Hov99, §7]) and homo-topy categories of stable ∞-categories ([Lur11, §1]).

15.6 Negative canonical triangulations

Todo!

DRAFT

Chapter 16

Spectra, Barratt–Puppesequences, and octahedra

In this chapter we revisit some of our earlier results concerning the calculusof functors like suspensions, cofibers, cofibers of compositions, and similar suchfunctors. Convenient organizational tools for this calculus are given by Barratt–Puppe sequences in the case of morphisms and (coherent) octahedron diagramsin the case of pairs of composable morphisms. This discussion is also meant toshed some additional light on the axioms of triangulated categories.

We show that a coherent morphism in a pointed derivator is equivalentlyspecified by a coherent Barratt–Puppe sequence, a coherent diagram encodingall iterated cofibers and fibers of the given morphism. In the case of a stablederivator the cofiber and the fiber part fit together particularly nicely, andwe illustrate this by establishing some basic formulas concerning the above-mentioned calculus.

More interestingly, a coherent pair of composable morphisms in a pointedderivator can be equivalently specified by means of a coherent octahedron di-agram, encoding all iterated cofibers and fibers of both morphisms and theircomposition. In particular, in the case of stable derivators this is again a conve-nient calculational tool, and we illustrate this by setting up a few basic formulas.

Similarly, there are such diagrams for longer strings of composable mor-phisms and we come back to this in [Gro16a]. Here instead, as a toy case, wealso briefly consider the most basic case, showing that an object in a pointedderivator is equivalently specified by a spectrum object which is a suspensionspectrum in positive degrees and an Ω-spectrum in negative degrees.

In §16.1 we discuss a few basics concerning spectra in pointed derivators. In§16.2 we encode morphisms by means of Barratt–Puppe sequences, which we seein §16.3 to be particularly well-behaved in the stable case. Similarly, in §16.4we discuss octahedra diagrams associated to pairs of composable morphisms,and in §16.5 we establish a few related formulas in the stable case.

283

DRAFT

284 CHAPTER 16. CANONICAL TRIANGULATIONS

16.1 Spectra in pointed derivators

In this section we discuss some basics concerning spectrum objects in pointedderivators. As a motivation for the remaining sections in this chapter, we showthat an object in a pointed derivator is equivalently specified by a spectrumobject which is a suspension spectrum in positive dimensions and an Ω-spectrumin negative dimensions.

Construction 16.1.1. Let M1 ⊆ Z×Z be the full subposet given by the followingdoubly-infinite shape:

. . .. . .

. . .

(−2,−2) //

(−1,−2)

(−2,−1) // (−1,−1) //

(0,−1)

(−1, 0) // (0, 0) //

(1, 0)

(0, 1) // (1, 1) //

(2, 1)

. . .

(1, 2) //

. . .

(2, 2).. .

(A justification for the notation of the poset will be given in §??.) Associatedto this poset, there is the functor (0, 0) : 1 → M1 which classifies the object(0, 0) ∈M1. In this section we will have a use for the following factorization of(0, 0) through fully faithful functors

(0, 0) : 1i1→ B1

i2→ B2i3→ B3

i4→M1.

The categories B1, B2, B3 are full subposets of M1, which are respectively ob-tained from (0, 0) by also adding

(i) the objects (j − 1, j), j ≥ 1, and (j + 1, j), j ≥ 0,

(ii) the above and the objects (j, j), j ≥ 1,

(iii) the above and the objects (j − 1, j), j ≤ 0 and (j + 1, j), j ≤ −1.

We refer to the full subposet spanned by the objects (j − 1, j), (j + 1, j) forarbitrary j ∈ Z as the boundary stripes. The functors ik, k = 1, . . . , 4, arethe obvious factorization of (0, 0) : 1→M1.

Note that i1 is a sieve and that i3 is a cosieve. In every pointed derivator Dwe obtain associated fully faithful Kan extension functors

D(1)(i1)∗→ D(B1)

(i2)!→ D(B2)(i3)!→ D(B3)

(i4)∗→ D(M1). (16.1.2)

DRAFT

16.1. SPECTRA IN POINTED DERIVATORS 285

Given a coherent diagram X ∈ D(M1), we simplify notation by writing

Xn = Xn,n = (n, n)∗X ∈ D(1).

Definition 16.1.3. Let D be a pointed derivator. A spectrum (in D(1)) isan object in D(M1) which vanishes on the boundary stripes. We denote bySp(D)(1) ⊆ D(M1) the full subcategory spanned by the spectra in D .

Remark 16.1.4. Given a pointed derivator D , there is an associated prederiv-ator Sp(D) defined by Sp(D)(A) = Sp(DA)(1), A ∈ Cat . In this introductorysection on spectra in the language of derivators we will not have a use for this.Here we mostly introduce some basic language in order to relate spectra toBarratt–Puppe sequences and octahedron diagrams as discussed in the remain-der of this chapter. However, we intend to come back to a more systematictreatment elsewhere, including the relation to the stabilization procedure forpointed derivators.

Given a spectrum X ∈ Sp(D)(1) we refer to Xn ∈ D(1), n ∈ Z, as the n-thlevel of the spectrum. The underlying diagram of such a spectrum X thuslooks as follows:

. . .. . .

. . .

X−2//

0

0 // X−1//

0

0 // X0//

0

0 // X1//

0

. . .

0 //

. . .

X2. . .

Remark 16.1.5. Let D be a pointed derivator and X ∈ Sp(D)(1). For eachn ∈ Z there is a coherent diagram looking like

Xn−1//

0

0 // Xn//

0

0 // Xn+1.

The vanishing conditions imply that there are canonical morphisms

σn−1 : ΣXn−1 → Xn and σn : Xn → ΩXn+1;

DRAFT


see Construction 9.2.20 and its dual. We refer to these morphisms as structuremorphisms of the spectrum. One can check that σn and σn correspond toeach other under the adjunction (Σ,Ω): D(1) D(1).

Remark 16.1.6. Classically, a spectrum in the sense of topology is a sequenceof pointed spaces Xn, n ≥ 0, together with pointed maps σn : ΣXn → Xn+1

for n ≥ 0. The early literature on this subject includes [SW53, Lim60, Whi62,Pup67, Vog70].

In the coherent formulation using pointed derivators D , having a spectrumamounts to asking for a coherent diagram X ∈ D(M ′1) which vanishes on theboundary stripes. Here, M ′1 ⊆M1 is the full subposet of Z× Z looking like:

(0, 0) //

(1, 0)

(0, 1) // (1, 1) //

(2, 1)

. . .

(1, 2) //

. . .

(2, 2).. .

Note that the obvious inclusion functor j : M ′1 → M1 is a cosieve, hence thefunctor j! : D(M ′1) → D(M1) is left extension by zero (Corollary 9.1.6). More-over, the fully faithfulness of j! implies that spectrum objects in the classicalsense give rise to spectrum objects in the sense of Definition 16.1.3.

In this chapter we follow Heller [Hel97, §8] and Lurie [Lur11, §1] using theZ-graded version of spectrum objects.

Definition 16.1.7. Let D be a pointed derivator, let X ∈ Sp(D)(1), and letk ∈ Z.

(i) The spectrum X is a suspension spectrum above degree k if themaps σn : ΣXn → Xn+1 are isomorphisms for all n ≥ k.

(ii) The spectrum X is an Ω-spectrum below degree k if the structuremaps σn : Xn → ΩXn+1 are isomorphisms for all n < k.

Spectrum objects which are suspension spectra in positive dimensions andΩ-spectra in negative suspensions should be determined by their 0-th levels, andthis is the content of the following theorem.

Theorem 16.1.8. Let D be a pointed derivator. The functors (16.1.2) are fullyfaithful and they induce an equivalence

D(1) ' D(M1)ex,

where D(M1)ex ⊆ D(M1) is the full subcategory spanned by the spectrum objectswhich are suspension spectra in positive dimensions and Ω-spectra in negativedimensions.

DRAFT

16.1. SPECTRA IN POINTED DERIVATORS 287

Proof. We describe the effect of each of the functors in (16.1.2). To begin with,since i1 is a sieve, the functor (i1)∗ is right extension by zero, and hence inducesan equivalence onto the full subcategory of D(B1) spanned by all diagramsvanishing on the boundary stripes.

In order to understand the functor (i2)!, we note that i2 : B1 → B2 factors asa countable composition of fully faithful functors such that each of the functorsis obtained by adding ‘the next object in the positive direction’. In more detail,let jk : B1,k → B2, k ≥ 0, be the full subcategory spanned by B1 and all objects(j, j) with 0 ≤ j ≤ k. There is the following sequence of fully faithful inclusionfunctors

B1 = B1,0j1,0→ B1,1

j2,1→ B1,2 → . . . ,

such that jk,k−1 simply adds the object (k, k), and we denote the correspondingfinite compositions by

jk2,k1: B1,k1

→ B1,k2, 0 ≤ k1 ≤ k2.

It is immediate from these definitions that there are the relations

i2 = jk jk,0 : B1 = B1,0 → B1,k → B2, k ≥ 0,

and these subcategories and inclusion functors exhibit B2 as the union of thesubcategories B1,k ⊆ B2, k ≥ 0.

In order to describe the essential image of (i2)! : D(B1) → D(B2) we be-gin with a description of (jk,k−1)!. By a standard application of the detectionlemma (Proposition 11.3.10 in combination with Lemma 11.3.3) we concludethat each of the functors (jk,k−1)! : D(B1,k−1)→ D(B1,k) adds precisely a newcocartesian square. Hence, a finite induction implies that the fully faithfulKan extension functors (jk,0)! : D(B1) = D(B1,0) → D(B1,k), k ≥ 0, preciselyamount to adding k new cocartesian squares. Finally, from Lemma 16.1.9 weconclude that (i2)! simply adds countably many cocartesian squares in the pos-itive direction.

Since i3 is a cosieve, the functor (i3)! is left extension by zero. Finally, weleave it to the reader to follow the strategy of the discussion of (i2)! to show thatthe right Kan extension (i4)∗ : D(B3) → D(M2) precisely amounts to addingcountably many cartesian squares in the negative direction. Putting these stepstogether, we obtain the intended equivalence D(1) ' D(M1)ex.

To conclude the proof of the theorem we have to establish the followinglemma which is also of independent interest.

Lemma 16.1.9. Let A ∈ Cat be the union of full subcategories Ak, k ≥ 0,and let ik : Ak → A, k ≥ 0, and ik2,k1

: Ak1→ Ak2

, 0 ≤ k1 ≤ k2, be the cor-responding fully faithful inclusion functors. For every derivator D the functor(i0)! : D(A0)→ D(A) is fully faithful and X ∈ D(A) lies in the essential imageof (i0)! if and only if i∗kX ∈ D(Ak) lies in the essential image of (ik,0)! for everyk ≥ 0.

DRAFT


Proof. Since all the functors ik, ik2,k1 are fully faithful, the same is true forthe corresponding left Kan extensions functors. Moreover, for every k ≥ 0there is the relation i0 = ik ik,0 : A0 → A and hence a canonical isomorphism(i0)!

∼= (ik)! (ik,0)! (Lemma 7.6.1). Let X ∈ D(A) be in the essential image of(i0)!, i.e., there is an isomorphism X ∼= (i0)!(Y ) for some Y ∈ D(A0). Then weobtain an isomorphism X ∼= (ik)!(ik,0)!Y and hence isomorphisms

i∗kX∼= i∗k(ik)!(ik,0)!Y ∼= (ik,0)!Y,

showing that i∗kX lies in the essential image of (ik,0)! : D(A0)→ D(Ak).

Conversely, let us consider X ∈ D(A) such that i∗kX lies in the essentialimage of (ik,0)! : D(A0)→ D(Ak) for all k ≥ 0. Since ik,0 is fully faithful this isto say that the adjunction counits εk,0 : (ik,0)!(ik,0)∗ → id are isomorphisms oni∗kX for all k ≥ 0. We want to conclude that X lies in the essential image of i!,i.e., that the counit ε : i!i

∗X → X is an isomorphism. By Proposition 8.2.20 thisis the case if and only if the component εa : (i!i

∗X)a → Xa is an isomorphismfor all a ∈ A − i(A0). For every fixed such a, there is an index k0 > 0 suchthat a ∈ Ak0 . Associated to the factorization i = ik0 ik0,0 there is a canonicalisomorphism i! ∼= (ik0)!(ik0,0)! which is compatible with the adjunction counits.Thus, using the obvious notation, the counit ε : i!i

∗ → id factors as

ε : i!i∗ ∼= (ik0

)!(ik0,0)!(ik0,0)∗i∗k0

εk0,0→ (ik0)!i∗k0

εk0→ id;

see Example A.1.14. Since a ∈ Ak0 and ik0 : Ak0 → A is fully faithful, weconclude by Proposition 8.2.20 that for every coherent diagram Y ∈ D(A) thecounit εk0

: (ik0)!i∗k0Y → Y is an isomorphism at a. As already observed, by

assumption on X, the counit εk0,0 is an isomorphism on i∗k0X. Hence, the above

factorization shows that εa is an isomorphism on X, thereby concluding theproof.

Thus, given a pointed derivator and a diagram X ∈ D(M1)ex, the underlyingdiagram looks up to canonical isomorphisms like as follows:

. . .. . .

. . .

Ω2X0//

0

0 // ΩX0//

0

0 // X0//

0

0 // ΣX0//

0

. . .

0 //

. . .

Σ2X0. . .

DRAFT

16.2. BARRATT–PUPPE SEQUENCES 289

In the remainder of this chapter we discuss similar types of diagrams, startingwith a coherent morphism or a coherent pair of composable morphisms instead.These diagrams are convenient tools to organize the calculus of suspensions,iterated cones, iterated cones of compositions and so on. In [Gro16a] we will alsodiscuss similar diagrams associated to longer strings of composable morphisms,and describe the connection to higher triangulations in the sense of Maltsiniotis.

16.2 Barratt–Puppe sequences

In this section we discuss (coherent, doubly-infinite) Barratt–Puppe sequences.These are certain coherent diagrams which provide a convenient, organizationaldevice for the calculus of suspension, loop, cofiber, and fiber functors in pointedderivators. Some of the natural isomorphisms we establish in the remainderof this chapter follow more systematically from a discussion of morphisms ofderivators and their compatibility with Kan extensions. While such a discussionis in [Gro16a], here we already obtain some firsts result in a more adhoc fashion.

Construction 16.2.1. Let M2 ⊆ Z×Z be the full subposet given by the following,doubly-infinite shape:

.. .. . .

. . .

(−1,−2) //

(0,−2)

. . .

(−1,−1) //

(0,−1) //

(1,−1)

(−1, 0) // (0, 0) //

(1, 0) //

(2, 0)

(0, 1) // (1, 1) //

(2, 1) //

(3, 1)

. . .

(1, 2) //

. . .

(2, 2) //

. . .

(3, 2).. .

(16.2.2)

Let us consider the functor i : [1]→M2 classifying the morphism (0, 0)→ (1, 0).This functor factors as a composition of fully faithful functors

i : [1]i1→ B1

i2→ B2i3→ B3

i4→M2,

as we describe now. The categories B1, B2, B3 are full subcategories of M2 whichare respectively obtained from the image of i by adding


(ii) the above and the objects (j, j), (j + 1, j), j ≥ 1,


DRAFT


The functors ik, k = 1, . . . , 4, are the obvious factorization of i : [1] → M2. Wenote that i1 is a sieve and that i3 is a cosieve. In every pointed derivator D weobtain associated fully faithful Kan extension functors

D([1])(i1)∗→ D(B1)

(i2)!→ D(B2)(i3)!→ D(B3)

(i4)∗→ D(M2). (16.2.3)

We again refer to the subposet of M2 spanned by the objects (j − 1, j) and(j + 2, j) for all j ∈ Z as the boundary stripes.

Definition 16.2.4. Let D be a pointed derivator and let M2 ∈ Cat be as inConstruction 16.2.1. A diagram X ∈ D(M2) is a (coherent, doubly-infinite)Barratt–Puppe sequence if it has the following exactness properties.

(i) The diagram X vanishes on the boundary stripes.

(ii) The diagram X makes all squares below i([1]) cocartesian.

(iii) The diagram X makes all squares above i([1]) cartesian.

We denote by D(M2)ex ⊆ D(M2) the full subcategory spanned by the Barratt–Puppe sequences.

Construction 16.2.1 leads to the following theorem.

Theorem 16.2.5. For every pointed derivator D the functors (16.2.3) inducean equivalence

D([1]) ' D(M2)ex.

Proof. Since i1 is a sieve, the functor (i1)∗ is right extension by zero, and sincei3 is a cosieve, the functor (i3)! is left extension by zero (Corollary 9.1.6). Inorder to conclude it remains to understand the functors (i2)! and (i4)∗. Asfor (i2)!, similarly to the proof of Theorem 16.1.8, we note that i2 : B1 → B2

factors as a countable composition of fully faithful functors such that each of thefunctors is obtained by adding ‘the next object in the positive direction’. By astandard application of the detection lemma (Proposition 11.3.10 in combinationwith Lemma 11.3.3) we conclude that each of these individual steps amountsprecisely to adding a new cocartesian square. Hence, a finite induction yieldsa similar statement for finite compositions of these. Finally, Lemma 16.1.9allows us to also carry out the countable induction, showing that (i2)! preciselyadds cocartesian squares in the positive direction. We leave it to the reader toconclude the proof by applying similar arguments in the case of (i4)∗.

Given a morphism (f : x → y) ∈ D([1]) in a pointed derivator, we referto the corresponding X = X(f) ∈ D(M2)ex guaranteed by Theorem 16.2.5 asthe Barratt–Puppe sequence generated by f . The underlying diagram ofX(f) takes the following form: All objects 0j are zero objects in D(1) and thesubscripts can be ignored for now.

Remark 16.2.6. Let D be a pointed derivator and let X be a Barratt–Puppesequence as in (16.1).

DRAFT


.... . .

. . .

. . .

z2//

x //

0−3

0−4

// y //

z2//

0−1

0−2

// xf//

y //

02

01// z1

//

x′ //

04

. . .

03//

. . .

y′ //

. . .

z′1. . .

Figure 16.1: Towards Barratt–Puppe sequences

(i) There is a canonical isomorphism Cf ∼−→ z1 (Lemma 9.2.24).

(ii) There is a canonical isomorphism z2∼−→ Ff (Lemma 9.2.24).

(iii) Combining these observations with Construction 9.2.20, we obtain canon-ical morphisms

Ff → ΩCf and ΣFf → Cf. (16.2.7)

Given a Barratt–Puppe sequence as in (16.1) we next show that the primedpart of the diagram is canonically isomorphic to the suspension of cofiber squareof the original morphism. The proof of this result follows the lines of the proofof Proposition 12.1.5, and as a preparation we consider the following two em-beddings

i1, i2 : →M2. (16.2.8)

The embedding i1 : →M2 classifies the square which in (16.2.2) are decoratedby the objects shown on the left in

(0, 0) //

(1, 0)

(2, 1) //

(3, 1)

(0, 1) // (1, 1), (2, 2) // (3, 2),

while i2 : →M2 classifies the above square on the right. Recall that σ : → denotes the symmetry interchanging (1, 0) and (0, 1).

Proposition 16.2.9. Let D be a pointed derivator and let i1, i2 : → M2 beas in (16.2.8). For every diagram X ∈ D(M2) which vanishes on the boundary

DRAFT


stripes there is a canonical morphism Σi∗1X → σ∗i∗2X in D(). This morphismis an isomorphism on Barratt–Puppe sequences,

Σi∗1X∼−→ σ∗i∗2X, X ∈ D(M2)ex.

Proof. Let X ∈ D(M2) vanish on the boundary stripes, so that the under-lying diagram hence looks like (16.1), possibly without making the squares(co)cartesian. We associate to X a coherent diagram Q = Q(X) ∈ D( × )taking the form

x //

y

02//

02

//

01// z1 04

// 04

01//

03

x′ //

y′

//

01// 03 04

// z′1.

In this diagram, the first square coordinate is drawn as an external coordinateand the second one internally. To obtain such a diagram Q, we note that thedecoration of the objects in Q determines a unique functor q : ×→M2 suchthat q∗ sends X to the above hypercube. In fact, since we are working withposets there is at most one such functor and the reader easily verifies that qexists, allowing us to set Q = q∗X ∈ D(×). We note the following about qand the diagram Q = q∗(X).

(i) The partial functor q((0, 0),−) : →M2 is i1 and we obtain the relationq((0, 0),−)∗(X) = i∗1X.

(ii) The squares q((1, 0),−)∗(X) and q((0, 1),−)∗(X) are trivial.

(iii) The partial functor q((1, 1),−) : → M2 agrees with i2 up to a flip, i.e.,we have q((1, 1),−) = i2 σ. This implies the relation q((1, 1),−)∗X =σ∗i∗2X.

Thus, if we consider Q ∈ D( × ) as an object Q ∈ D() by treatingthe second variable as a parameter, then the square vanishes at (1, 0), (0, 1). ByConstruction 9.2.20 there is a canonical morphism ΣQ((0, 0),−)→ Q((1, 1),−).By definition of Q as q∗X this is the intended morphism

Σi∗1X → σ∗i∗2X.

It remains to conclude that this morphism is an isomorphism on Barratt–Puppe sequences. By Lemma 9.2.22 it suffices to show that Q = q∗X ∈ D()is a suspension square if X ∈ D(M2)ex. Since the vanishing condition at(1, 0), (0, 1) was already verified, it remains to show that the square Q is co-cartesian. By Corollary 10.3.10 it is enough to show that this true for each

DRAFT


parameter value, i.e., that q(−, a)∗X ∈ D() is cocartesian for all a ∈ . Fora = (0, 0), (1, 0), (0, 1), (1, 1) the respective squares q(−, a)∗X look like

x //

02

y //

02

01//

04

z //

04

01// x′, 03

// y′, 01// 04, 03

// z′.

It follows from the exactness properties of Barratt–Puppe sequences that thefirst, the second, and the forth of these squares are obtained by horizontalor vertical pasting of cocartesian squares and are hence cocartesian (Proposi-tion 9.3.10). Moreover, the third square is constant and thus also cocartesian(Proposition 9.3.5). Thus, Q ∈ D() is a suspension square, and the canonicalmorphism Σi∗1X

∼−→ σ∗i∗2X is an isomorphism (Lemma 9.2.22).

Corollary 16.2.10. Let D be a pointed derivator, let (f : x→ y) ∈ D([1]) andlet X ∈ D(M2)ex be the Barratt–Puppe sequence generated by f . The underlyingdiagram of X takes up to canonical isomorphisms the following form:

......

...

...

ΩFf //

Ωx //

0

0 // Ωy //

Ff //

0

0 // x

f//

y //

0

0 // Cf //

Σx //

0

...

0 //

...

Σy //

...

ΣCf...

Proof. The statement concerning the part below the original horizontal mor-phism is immediate from Remark 16.2.6 and Proposition 16.2.9. Moreover, sim-ilarly to Proposition 16.2.9 one shows that the upper part contains besides thefiber square of the morphisms also positive loop powers of this square, possiblyup to flips. This yields the intended description of the upper part.

Example 16.2.11. In the case of the derivator HoTop∗ of pointed topologicalspaces, the Barratt–Puppe sequences from Definition 16.2.4 reproduce the clas-sical ones from []. These Barratt–Puppe sequences are the space-level origin of

Ref!many long exact sequences in algebraic topology.

Remark 16.2.12. One can also choose different ‘fundamental domains’ in theBarratt–Puppe sequences as a basic pattern which essentially repeat countablymany times in both directions.

DRAFT


(i) For example, one can choose a cofiber sequence associated to f which thenrepeats up to flips and suspensions, the individual copies overlapping at acorner. In the negative direction one could take a fiber sequence associatedto f as fundamental domain.

(ii) Alternatively, the fundamental domain could be chosen as the diagram

0 // x //

y

// 0

0 // z //

x′

0 // y′

0,

which repeats up to flips and suspensions in the positive direction, thecopies intersecting this time in diagrams of shape [3]. A similar patterncan be used for the negative direction.

The proofs of these statements are straightforward modifications of the proofof Proposition 16.2.9, the advantage of these versions being that all morphismsin the Barratt–Puppe sequence are explained by this description. Of coursethis can also be obtained from variants of Proposition 16.2.9 based on otherembeddings [1]→M2.

Besides being conceptually beautiful, these Barratt–Puppe sequences canbe used to visualize the calculus of the functors Σ, cof, C and Ω, fib, F in thatcalculating simply amounts to ‘traveling in the Barratt–Puppe sequence’. Weillustrate this by the following examples which of course can also be obtainedwithout Barratt–Puppe sequences.


(i) There are canonical isomorphims Σ C ∼= C Σ: D([1]) → D(1) andΣ cof ∼= cof Σ: D([1])→ D([1]).

(ii) There are canonical isomorphisms Σ ∼= cof3 : D([1])→ D([1]).

(iii) There are canonical isomorphisms Σ0∗ ∼= 0∗ Σ: D([1])→ D(1) and forexample also 0∗ cof3 ∼= 1∗ cof2 ∼= C cof.

There are dual formulas for loops and fibers, and we invite the reader tocome up with additional examples. However, if we merge these two groupsof functors, then instead of canonical isomorphisms we obtain certain canonicalmorphisms (as in (16.2.7)). In fact, as reflected by the two types of fundamentaldomains in Remark 16.2.12, in pointed derivators the upper and the lower partof Barratt–Puppe sequences to not interact perfectly well. This is different inthe context of stable derivators as we discuss next.

DRAFT

16.3. BARRATT–PUPPE SEQUENCES IN STABLE DERIVATORS 295

16.3 Barratt–Puppe sequences in stable deriva-tors

In this short section we discuss Barratt–Puppe sequences in stable derivators.We begin with the following variant of Corollary 16.2.10

Corollary 16.3.1. Let D be a stable derivator, let (f : x→ y) ∈ D([1]) and letX ∈ D(M2)ex be the Barratt–Puppe sequence generated by f . The underlyingdiagram of X takes up to canonical isomorphisms the following form:

......

...

...

ΩFf //

Ωx //

0

0 // Ωy //

Ff //

0

0 // x

f//

y //

0

0 // Cf //

Σx //

0

...

0 //

...

Σy //

...

ΣCf...

(16.3.2)

Proof. This is immediate from Corollary 16.2.10 since cocartesian and cartesiansquares coincide in stable derivators.

In these diagrams all squares are bicartesian and the two parts hence matchperfectly well together. In particular, in contrast to general pointed derivators,there are the following two immediate consequences.

Lemma 16.3.3. Let D be a stable derivator and let (f : x→ y) ∈ D([1]). Thereare canonical isomorphisms

Cf ∼= ΣFf ∼= FΣf and Ff ∼= ΩCf ∼= CΩf.

Proof. By duality it is enough to establish the first two isomorphisms, whichwe easily obtain by restricting a Barratt–Puppe sequence (16.3.2) to a coherentdiagram

Ff //

1

0

x //

2

y

//

3

0

0 // Cf

//

4

Σx

0 // Σy

DRAFT


consisting of bicartesian squares and vanishing as indicated. By the proofof Proposition 12.1.5 we know that the vertical morphism on the right in 4

is canonically isomorphic to Σf . Since the squares 2 and 4 are bicartesian,we obtain canonical isomorphisms Cf ∼= FΣf (Lemma 9.2.24). Similarly, an

application of Proposition 15.1.6 to the squares 1 and 2 yields a canonicalisomorphism Cf ∼= ΣFf (Lemma 9.2.22).

There are similar results about cof and fib.

Lemma 16.3.4. Let D be a stable derivator and let f ∈ D([1]). There arecanonical isomorphisms

coff ∼= Σfib2f ∼= fib2Σf and fibf ∼= Ωcof2f ∼= cof2Ωf.

Proof. We invite the reader to prove this result traveling through a Barratt–Puppe sequence (16.3.2). Alternatively, this also follow from Proposition 12.1.5and Proposition 15.1.8.

Remark 16.3.5. In contrast to pointed derivators, in stable derivators we onlyneed one fundamental domain in Barratt–Puppe sequences (Remark 16.2.12).In fact, the Barratt–Puppe sequence essentially consists of the fundamentaldomain, all its suspensions, and all its loops.

One way to think of Barratt–Puppe sequences in stable derivators is asrefinements of distinguished triangules. We conclude this section by a shortdiscussion of this perspective and illustrate how some of the defects mentionedin §5.5 are avoided that way.

Remark 16.3.6. Let D be a stable derivator. We summarize some of our earlierfindings concerning coherent diagrams.

(i) There are equivalences (Σ,Ω): D(1) ' D(1).

(ii) There are equivalences (cof, fib) : D([1]) ' D([1]).

(iii) There are equivalences D([1]) ' D()cof ' D(M2)ex.

(iv) A morphism f ∈ D([1]) is an isomorphism if and only if C(f) ∼= 0.

(v) There are canonical isomorphisms

cof3 ∼= Σ: D([1])→ D([1]) and fib3 ∼= Ω: D([1])→ D([1]).

In fact, these are the results Proposition 15.1.9, Proposition 15.1.8, Proposi-tion 9.5.8, Theorem 16.2.5, Proposition 15.1.7, and Proposition 12.1.5.

At the level of incoherent diagrams, i.e., at the level of triangulated categoriesthese results are closely related to the following facts.

• Every morphism sits in a distinguished triangle. The cone is weakly func-torial. Any two cones or cofibers of the same morphism are non-canonicallyisomorphic.

DRAFT

16.4. OCTAHEDRON DIAGRAMS 297

• Distinguished triangles can be rotated backwards and forwards.

• The composition of consecutive morphisms in distinguished triangles aretrivial.

• The suspension and the cone commute up to a non-canonical isomorphism.

• A morphism in a triangulated category is invertible if and only if any coneof it is trivial.

From the perspective of strong, stable derivators these differences are a con-sequence of the fact that incoherent morphisms can be lifted against dia[1] ina non-functorial way only. The relation between incoherent morphisms, coher-ent morphisms, cofiber sequences, Barratt–Puppe sequences, and distinguishedtriangles is summarized by the following diagram in which the vertical functorsdiscard relevant information,

D([1])' //

tria

%%

dia[1]

D()cof ' //

∼=

D(M2)ex

dia

D(1)M2

D(1)[1] D(1)[3].

16.4 Octahedron diagrams

We now turn to a discussion of octahedron diagrams in pointed derivators. Theseare the analogues of Barratt–Puppe sequences if we start with a pair of com-posable morphisms instead of a morphism only.

The following construction is a variant of Construction 16.2.1.

Construction 16.4.1. Let M3 ⊆ Z × Z be the poset given by the shape of thefollowing, doubly-infinite diagram:

.. .. . .

. . .. . .

. . .(0,−1) // (−1,−1) //

(0,−1) //

(1,−1) //

(2,−1)

(−1, 0) // (0, 0) //

(1, 0) //

(2, 0) //

(3, 0)

(0, 1) //

. . .

(1, 1) //

. . .

(2, 1) //

. . .

(3, 1) //

. . .

(4, 1).. .

The functor i : [2]→M3 classifying (0, 0)→ (1, 0)→ (2, 0) factors through fullyfaithful functors

i : [1]i1→ B1

i2→ B2i3→ B3

i4→M3,

DRAFT


which are obtained from i([2]) by adding


(ii) the above and the objects (j, j), (j + 1, j), (j + 2, j), j ≥ 1,


Again, i1 is a sieve and i3 is a cosieve, so that, for every pointed derivator D ,two of the following fully faithful Kan extension functors

D([1])(i1)∗→ D(B1)

(i2)!→ D(B2)(i3)!→ D(B3)

(i4)∗→ D(M3) (16.4.2)

are extensions by zero.

Definition 16.4.3. Let D be a pointed derivator and let M3 ∈ Cat as inConstruction 16.4.1. A diagram X ∈ D(M3) is a (coherent, doubly-infinite)octahedron diagram if it satisfies the following exactness properties.

(i) The diagram X vanishes on the boundary stripes.

(ii) The diagram X makes all squares below i([2]) cocartesian.

(iii) The diagram X makes all squares above i([2]) cartesian.

We denote by D(M3)ex ⊆ D(M3) the full subcategory spanned by all octahedrondiagrams.

Every (xf→ y

g→ z) ∈ D([2]) extends to an octahedron diagram. In fact,there is the following stronger result.

Theorem 16.4.4. Let D be a pointed derivator. The functors (16.4.2) are fullyfaithful and they induce an equivalence

D([2]) ' D(M3)ex.

Proof. The proof is similar to the proofs of Theorem 16.1.8 and Theorem 16.2.5,and the details are left to the reader.

Given an octahedron diagram X ∈ D(M3)ex, we say that it is associatedto or generated by its restriction i∗X ∈ D([2]).

Lemma 16.4.5. Let D be a pointed derivator, and let X ∈ D(M3)ex be the

octahedron diagram generated by (xf→ y

g→ z) ∈ D([2]). There are canonicalisomorphisms

Cf ∼−→ X1,1, C(gf) ∼−→ X2,1, and Cg ∼−→ X2,2.

Proof. This follows by a diagram chase, similar to the proof of the first statementof Proposition 12.2.5.

DRAFT

16.5. OCTAHEDRON DIAGRAMS IN STABLE DERIVATORS 299

Remark 16.4.6. Similarly to the case of Barratt–Puppe sequences, one showsthat there are two types of fundamental domains in octahedron diagrams whichrespectively repeat up to flips and suspensions in the positive direction and upto flips and loops in the negative direction. Precise statements and proofs areleft to the reader, since they are completely parallel to Proposition 16.2.9 andRemark 16.2.12.

Corollary 16.4.7. Let D be a pointed derivator and let X ∈ D(M3)ex be

the octahedron diagram generated by (xf→ y

g→ z) ∈ D([2]). The underlyingdiagram of X takes up to canonical isomorphisms the following form:

......

...

Ωx //

0

...

Ωy //

Ff //

0

...

Ωz //

F (gf) //

Fg //

0

0 // x

f//

yg//

z //

0

0 // Cf //

C(gf) //

Σx //

0

0 // Cg //

Σy //

ΣCf //

0

...

0 //

...

Σz //

...

ΣC(gf) //

...

ΣCg...

(16.4.8)

Proof. This is immediate from Lemma 16.4.5 and Remark 16.4.6.

Similarly to Barratt–Puppe sequences, the lower part of octahedron dia-grams as in (16.4.8) can be used to organize a calculus relating various sus-pensions, cones, and cofibers. The upper parts organize a similar calculus forthe dual constructions. These two parts and the corresponding constructionsinteract very nicely in the case of stable derivators.

16.5 Octahedron diagrams in stable derivators

In this short section we discuss octahedron diagrams in stable derivators. ByTheorem 16.4.4 every pair of composable morphisms

(xf→ y

g→ z) (16.5.1)

in a stable derivator is equivalently specified by the associated coherent octahe-dron diagram X ∈ D(M3)ex. It follows from the above discussion that, up to

DRAFT


canonical isomorphisms, the underlying diagram of X looks as follows:

. . .. . .

. . .

Ωx //

0

. . .

Ωy //

Ff //

0

. . .

Ωz //

F (gf) //

Fg //

0

0 // x

f//

yg//

z //

0

0 // Cf //

C(gf) //

Σx //

0

0 // Cg //

Σy //

ΣCf //

0

. . .

0 //

. . .

Σz //

. . .

ΣC(gf) //

. . .

ΣCg. . .

(16.5.2)

In (16.5.2) all squares are bicartesian, and this diagram allows us to visualizethe calculus of loops, suspensions, fibers and cofibers of both morphisms andthe their composition. In fact, establishing such canonical isomorphisms simplyamounts to traveling through octahedron diagrams.

We illustrate this by three examples. Let us recall that already in everypointed derivator and for every (16.5.1) we can exhibit Cg as C(Cf → C(gf))(Proposition 12.2.5). Thus, slightly imprecisely, this expresses Cg in terms of fand gf only. In the stable case there are similar formulas for C(gf) and C(f)in terms of the respective remaining morphisms only, and these are canonicalversions of (5.5.1).

Examples 16.5.3. Let D be a stable derivator and let (16.5.1) be a coherent pairof composable morphisms in D .

(i) There are canonical isomorphisms

Cf ∼= F (C(gf)→ Cg)∼= C(F (gf)→ Fg).

(ii) There are canonical isomorphisms

C(gf) ∼=F (Cg → ΣCf)∼=C(Fg → Cf)

(iii) There are canonical isomorphisms

Fg ∼= F (Cf → C(gf))∼= C(Ff → F (gf)).

DRAFT

16.5. OCTAHEDRON DIAGRAMS IN STABLE DERIVATORS 301

We invite the reader to establish these examples by traveling through thecorresponding octahedron diagram. The reader easily comes up with additionalsuch formulas and verifies them in a similar way.

We conclude this section by a short discussion of the relation between co-herent pairs of composable morphisms, coherent octahedron diagrams, and thecorresponding incoherent versions.

Remark 16.5.4. Let D be a strong, stable derivator. We begin by recalling themain steps of the proof of the octahedron axiom (T4) for canonical triangula-tions in Theorem 15.5.1.

(i) Let X ∈ D([2]) be a coherent pair of composable morphisms. Associatedto X we obtain a coherent diagram (15.5.9) satisfying certain exactnessand vanishing properties. More precisely, there is an equivalence of cat-egories (15.5.10), and the target category D(B)cof hence provides onemodel for the category of coherent octahedron diagrams.

(ii) Given a coherent octahedron diagram X ∈ D(B)cof , making some iden-tifications as indicated in (15.5.11) we obtain an incoherent octahedrondiagram looking like (15.5.8), and this step amounts to a loss of informa-tion since we pass to underlying diagrams. Note that this is a functorialconstruction of an incoherent octahedron diagram associated to a coher-ent pair of composable morphisms. Denoting by O ∈ Cat the shape of anincoherent octahedron diagram (15.5.8) we thus obtain a functor

octa : D([2]) ' D(B)cof → D(1)O.

(iii) Finally, starting with a pair of composable morphisms in the underlyingcategory D(1), we begin by lifting it to an object X ∈ D([2]) and thenapply the previous two steps.

Together with Theorem 16.4.4 the situation can hence be summarized by thefollowing diagram in which the vertical functors discard relevant information,

D([2])' //

octa

%%

dia[2]

D(B)cof ' //

∼=

D(M3)ex

dia

D(1)M3

D(1)[2] D(1)O.

Generalizations of Barratt–Puppe sequences and coherent octahedron dia-Todo: relate octahedron to totalcofibers: one edge iso!grams to longer chains of composable morphisms are obtained from coherent,

doubly infinite versions of the Waldhausen S•-construction [Wal85]. We willcome back to this in the discussion of canonical higher triangulations in [Gro16a].

DRAFT


DRAFT

Chapter 17

Unstable versions ofcanonical triangulations

303

DRAFT

304CHAPTER 17. UNSTABLE VERSIONS OF CANONICAL TRIANGULATIONS

DRAFT

Chapter 18

Outlook

FINAL LIST OF TODO’s:

(i) spell-check and specific refs!!!

(ii) good notation for represented prederivators (yC?), mention the Yonedaembedding

(iii) Check all refs and citations, fix problems resulting from splitting in vol-umes

(iv) use uniform quotation marks

(v) §§15.4-4.3 and §15.6, little changes in §15

(vi) §§17-18, §A, and §B

(vii) adapt intro to include §17

(viii) spell check, biblio, draft, Konsolenausgabe

(ix) check all links; warning: environments starting with enumerations to bedealt with hack as in tilting1.tex

(x) classical references for derived limits (in case of derivator of a ring?),add to section on examples, more classical refs for htpy limits in spaces?,homotopy Kan extensions or derived Kan extensions?

(xi) Cohen–Moore–Neisendorfer in book? Yes, in volume II

(xii) refs concerning attemps to lift again diagram functors: rectifications, stric-tifications, Dwyer, Kan, Drinfeld center, H∞?,...

(xiii) Convention for shifting: first variable is the shift, generic name: A

305

DRAFT

306 CHAPTER 18. OUTLOOK

DRAFT

Appendix A

Some category theory

In this section we review some basics from category theory and refer the readerto [ML98, Bor94a, Bor94b] for more details.

A.1 Adjunctions

Let us begin by recalling that an adjunction (L,R) : C D between two cate-Replace 1 by id.

gories C,D consists of two functors L : C → D and R : D → C together with thechoice of an isomorphism

φ = φc,d : homD(Lc, d) ∼= homC(c,Rd) (A.1.1)

which is natural in c ∈ C and d ∈ D. There are various equivalent ways ofencoding such a datum, one of them uses the notion of adjunction (co)units.Specializing (A.1.1) to objects d = Lc and applying φ to the identity morphisms1: Lc→ Lc we obtain a natural transformation η : 1→ RL, the (adjunction)unit. Dually, applying φ−1 to identity morphisms 1: Rd → Rd we obtain anatural transformation ε : LR→ 1, the (adjunction) counit.

These two transformations are not unrelated since they are constructed usingthe isomorphisms (A.1.1) and their inverses. First, let us note that we canreconstruct φ from R and η. In fact, the naturality of φ implies that for everyf : Lc→ d there is a commutative diagram

homD(Lc, Lc)φ//

f∗

homC(c,RLc)

(Rf)∗

homD(Lc, d)φ// homC(c,Rd).

Tracing 1: Lc → Lc through the diagram and using the definition of η we seethat φ(f) = (Rf)∗(ηc) = Rf ηc. Similarly, the inverse isomorphism φ−1 can

307

DRAFT

308 APPENDIX A. SOME CATEGORY THEORY

be reconstructed from L and ε, and we have the commutative diagram

homC(RLc,Rd)

η∗

((

homD(Lc, d)

R

66

φ//homC(c,Rd)

φ−1

oo

Lvv

homD(Lc, LRd).

ε∗

hh(A.1.2)

In particular, starting with any morphism Lc → d or c → Rd an applicationof the four maps in the clockwise direction gives us back the same morphism.If we specialize this to the identity morphisms id: Lc → Lc and id: Rd → Rd,then we deduce the following two relations between η and ε,

LLη//

id ++

LRL

εL

RηR//

id ++

RLR

Rε

L, R,

(A.1.3)

called the triangular identities. It can be shown that an adjunction is de-termined by the datum of two functors L : C → D, R : D → C and two naturaltransformations η : id→ RL, ε : LR→ id such that the triangular identities aresatisfied.

In this book we generically denote adjunction units by η and adjunctioncounits by ε. If we want to emphasize that there are different adjunctions ina given context, then we use obvious notational variations like η′, ε′. We recallthe following fact about adjunctions.

Lemma A.1.4. Let (L,R) : C D be an adjunction.

(i) The left adjoint L is fully faithful if and only if the unit η : 1 → RL is anatural isomorphism. An object d ∈ D lies in the essential image of L ifand only if the counit εd : LRd→ d is an isomorphism.

(ii) The right adjoint R is fully faithful if and only if the counit ε : LR→ 1 isa natural isomorphism. An object c ∈ C lies in the essential image of Rif and only if the unit ηc : c→ RLc is an isomorphism.

(iii) The adjunction (L,R) is an adjoint equivalence if and only if L and Rare fully faithful if and only if η and ε are natural isomorphisms.

It is immediate from (A.1.1) that adjunctions can be composed. Given ad-junctions (L′, R′) : C D and (L′′, R′′) : D E then composing the respectiveadjunction isomorphisms we obtain an adjunction (L,R) : C E with L = L′′L′

DRAFT

A.1. ADJUNCTIONS 309

and R = R′R′′. The adjunction unit and counit of this composite adjunctioncan be calculated as

η : 1η′→ R′L′

R′η′′L′→ R′R′′L′′L′ and ε : L′′L′R′R′′L′′ε′R′′→ L′′R′′

ε′′→ 1.

Natural transformations between left adjoint functors are closely related tonatural transformations between associated right adjoint functors.

Lemma A.1.5. Let (L,R), (L′, R′) : C D be adjunctions and let α : L→ L′ bea natural transformation. There is a unique natural transformation α′ : R′ → Rsuch that the following diagram commutes

homD(Lc, d)∼=φ//

OO

α∗

homC(c,Rd)OO

(α′)∗

homD(L′c, d) ∼=

φ′// homC(c,R

′d).

(A.1.6)

This defines a bijection between natural transformations α : L→ L′ and naturaltransformations α′ : R′ → R.

Proof. Let us choose c = R′d and trace the counit ε′d : L′R′d → d through thediagram. This implies α′d = φ(ε′d αR′d), showing that there is at most onesuch natural transformation. Note that using the description of the adjunctionisomorphism φ in terms of R and η as in (A.1.2) we obtain α′ = Rε′RαR′ηR′and this shows that we actually constructed a natural transformation α′.

Natural transformations α and α′ making (A.1.6) commute are conjugateor total mates. The proof and its converse show that conjugate natural trans-formations α, α′ determine each other by means of the relations

α′ = Rε′ RαR′ ηR′ : R′ → RLR′ → RL′R′ → R (A.1.7)

andα = εL′ Lα′L′ Lη′ : L→ LR′L′ → LRL′ → L′. (A.1.8)

The uniqueness of conjugate transformations implies that the construction iscompatible with compositions and identity transformations.

Corollary A.1.9. If α : L → L′ and α′ : R′ → R are conjugate natural trans-formations, then α is a natural isomorphism if and only if α′ is a natural iso-morphism.

A more systematic way of putting this is as follows. Given two categoriesC,D, let LAdj(C,D) be the category of left adjoint functors from C to D. Anobject in LAdj(C,D) is a left adjoint L : C → D and morphisms are natural trans-formations between left adjoints. Dually, we define the category RAdj(C,D).

Corollary A.1.10. Let C and D be categories. There is an equivalence ofcategories LAdj(C,D) ' RAdj(D, C)op.

DRAFT


Proof. We describe a functor LAdj(C,D)→ RAdj(D, C)op. For each left adjointfunctor L : C → D we choose a right adjoint R : D → C. Given these choices, forevery pair of left adjoint functors L,L′ the passage to conjugate transformationsdefines a bijection between natural transformations L→ L′ and R′ → R, which,as already observed, is compatible with compositions and identities. Thus, weobtain a fully faithful functor LAdj(C,D)→ RAdj(D, C)op. Clearly this functoris also essentially surjective and hence an equivalence.

Thus, the specification of a left adjoint is essentially equivalent to the specifi-cation of a right adjoint. We will later need the following more precise statementalong these lines.

Lemma A.1.11. Let (L,R, η, ε) : C D and (L′, R, η′, ε′) : C D be adjunc-tions with the same right adjoint R. There is a unique natural isomorphismα : L→ L′ making the following diagrams commute

1η//

η′ ))

RL

∼= Rα

LRε //

αR ∼=

1

RL′, L′R.ε′

JJ

Proof. The existence of a unique natural isomorphism α making the diagramon the left commute is immediate from the definition of units as certain initialobjects. In order to check that also the remaining triangle commutes it isenough to show that the two compositions have the same image under thenatural isomorphism φ : homD(LRd, d) ∼= homC(Rd,Rd). By definition of theadjunction counit ε we have to show that φ(ε′ αR) = 1. By (A.1.2) we knowthat φ(ε′ αR) is given by

RηR→ RLR

RαR→ RL′RRε′→ R. (A.1.12)

Since we already showed that Rα η = η′, we see that φ(ε′ αR) = Rε′ η′Rwhich is the identity 1: R→ R by a triangular identity (A.1.3).

Remark A.1.13. We note that the proof of Lemma A.1.11 shows the following.The unique natural isomorphism α : L ∼−→ L′ guaranteed by that lemma is thetotal mate or the conjugate transformation of idR : R→ R. In fact, this followsfrom (A.1.12), (A.1.7), and Lemma A.1.5.

We illustrate Lemma A.1.11 by the following example.

Example A.1.14. Let L = L2 L1 : C1 → C2 → C3 be the composition of twofunctors admitting right adjoints and let (L1, R1, η1, ε1), (L2, R2, η2, ε2), and(L = L2L1, R, η, ε) be some choices of adjunction data. There is a uniqueisomorphism R ∼= R1 R2 such that the following diagrams commute,

LR = L2L1R∼= //

ε

L2L1R1R2

ε1

RL = RL2L1

∼= // R1R2L2L1

id L2R2,ε2oo id

η1

//

η

OO

R1L1.

η2

OO

DRAFT

A.2. LIMITS AND COLIMITS 311

A.2 Limits and colimits

We assume the reader to have some basic familiarity with limits and colimits,including a discussion of terminal objects, products, pullbacks and the dualnotions of initial objects, coproducts, and pushouts. Nevertheless we recallsome key definitions and results about (co)limits, mainly to establish notationand to prepare the ground for homotopical generalizations as discussed in themain body of the text.

Definition A.2.1. Let A be a small category, let C be a category, and letX : A→ C be a functor. A cone on X is a pair (l, α) consisting of an object l ∈ Cand morphisms αa : l → Xa, a ∈ A, such that for every morphism f : a→ a′ inA the diagram

lαa //

αa′

Xa

f∗

Xa′

commutes. A morphism of cones (l, α) → (l′, α′) is a morphism l → l′ in Csuch that

l

αa

l′α′a

// Xa

commutes for every a ∈ A.

With the obvious composition law and identity morphisms this defines thecategory of cones on X. Passing to the formal dual, there is the notion of acocone (c, β) on X : A→ C. In this case the maps βa : Xa→ c are compatiblein the sense that

Xaβa //

f∗

c

Xa′βa′

>>

commutes for every morphism f : a→ a′. Defining morphisms of cocones in theobvious way, associated to a functor X : A→ C there are the two categories

cone(X), cocone(X) ∈ CAT ,

both coming with obvious forgetful functors cone(X)→ C and cocone(X)→ C.Universal examples of (co)cones deserve particular names.

Definition A.2.2. Let A be a small category, let C be a category, and letX : A→ C be a functor.

(i) A limit of X is a terminal object limAX of cone(X).

DRAFT


(ii) A colimit of X is an initial object colimAX of cocone(X).

Thus, to emphasize, such a limit is a pair consisting of an underlying object,also denoted by limAX, together with a universal cone which we also referto as a limiting cone. Similarly, we abuse notation and write colimAX forthe underlying object of a colimit and refer to the corresponding cocone as acolimiting cocone.

As always with universal constructions, if (co)limits exist, then they areunique up to unique isomorphisms and this uniqueness implies functoriality. Inthe following proposition we write ∆ = ∆A : C → CA for the diagonal functorwhich associates to x ∈ C the constant diagram ∆(x) : A→ C with value x.

Proposition A.2.3. Let A be a small category and let C be a category.

(i) If every X : A→ C has a limit, then the assignment X 7→ limAX extendsto a limit functor limA : CA → C which is right adjoint to ∆A,

(∆A, limA) : C CA.

(ii) If every X : A → C has a colimit, then the assignment X 7→ colimAXextends to a colimit functor colimA : CA → C which is left adjoint to ∆A,

(colimA,∆A) : CA C.

For particular choices of small categories A, there are special names for theresulting limits and colimits, namely,

(i) final objects and initial objects if A is empty,

(ii) products and coproducts if A is discrete (all morphisms are identities),

(iii) equalizers and coequalizers if A = (0⇒ 1), and

(iv) pullbacks for limits over A = ((0, 1) → (1, 1) ← (1, 0)) and pushoutsfor colimits over A = ((0, 1)← (0, 0)→ (1, 0)).

We next recall that these are the basic building blocks for arbitrary (co)limitsin the following precise sense. A category is complete or cocomplete if itadmits limits or colimits of all small diagrams, respectively.

Proposition A.2.4.

(i) A category is complete if and only if it admits equalizers and (small)products.

(ii) A category is cocomplete if and only if it admits coequalizers and (small)coproducts.

DRAFT

A.2. LIMITS AND COLIMITS 313

Proof. By duality it is enough to take care of the first statement. The taskis to show that an arbitrary limit can be obtained by combining products andequalizers. To this end, let A be a small category and X : A → C such that Cadmits small products and equalizers. We consider the following pair of parallelmorphisms

d0, d1 :∏a∈A

Xa→∏

f : a0→a1

Xa1

in C. Using the universal property of the product∏f : a0→a1

Xa1, in order todefine the morphisms d0, d1 it suffices to specify the respective compositionswith the projections onto the factor Xa1 associated to an arbitrary f : a0 → a1.In the case of d0 this composition is chosen to be∏

a∈AXa→ Xa0

f∗→ Xa1

while in the case of d1 we simply take the projection∏a∈A

Xa→ Xa1.

The underlying object of the limit limAX is now defined to be the equalizer ofthese two morphisms,

limAX = eq( ∏a∈A

Xa⇒∏

f : a0→a1

Xa1

). (A.2.5)

Using the universal cone belonging to this equalizer, we can define the universalcone of limAX to have components

limAX →∏a∈A

Xa→ Xa′, a′ ∈ A,

where the second morphism is the projection onto the a′-component. We leaveit to the reader to verify that this is a limiting cone for X.

In the case of colimits the key step consists of forming the coequalizer

colimAX = coeq( ∐f : a0→a1

Xa0 ⇒∐a∈A

Xa)

(A.2.6)

of two similarly defined morphisms d0, d1 :∐f : a0→a1

Xb →∐a∈AXa. The

details are left to the reader.

This proposition admits a few variants, one of which we include here. Acategory is finitely complete if it admits finite limits, i.e., limits of diagramsdefined on finite categories (recall that a category is finite if it has finitely manyobjects and morphisms only).

Corollary A.2.7. The following are equivalent for a category C.

DRAFT


(i) The category C is finitely complete.

(ii) The category C admits finite products and equalizers.

(iii) The category C admits terminal objects and pullbacks.

Proof. The equivalence of the first two statements follows as above, so that itis suffices to show that terminal objects and pullbacks generate finite productsand equalizers. Since equalizers are special cases of pullbacks it remains toconstruct finite products. The reader easily checks that products of two objectsare obtained by forming pullbacks over terminal objects, and the case of finiteproducts hence follows by induction.

We conclude this section by a short discussion of the preservation of (certain)limits by functors. Let us consider a functor X : A → C admitting a colimitcolimAX and let F : C → D be a functor such that also F X : A→ D admitsa colimit colimA F X. If we apply F to the colimiting cocone of X, then forevery f : a→ a′ in A there is a commutative diagram

F (Xa)

// F (colimAX)

F (Xa′)

88

in D, which is to say that we obtain a cocone on F X. The universality of thecolimiting cocone of F X implies that there is a unique morphism

colimA(FX)→ F (colimAX) (A.2.8)

compatible with these two cocones. We refer to this morphism and a duallydefined morphism

F (limAX)→ limA(FX) (A.2.9)

as the canonical morphisms.Let F : C → D be a functor between categories admitting colimits of shape

A. We say that F preserves colimits of shape A if for every X : A→ C thecanonical morphism colimA(FX)→ F (colimAX) is an isomorphism.

Definition A.2.10.

(i) A functor between complete categories is continuous if it preserves alllimits.

(ii) A functor between cocomplete categories is cocontinuous if it preservesall colimits.

(iii) A functor between finitely complete categories is left exact if it preservesfinite limits.

DRAFT

A.3. BASIC 2-CATEGORICAL TERMINOLOGY 315

(iv) A functor between finitely cocomplete categories is right exact if it pre-serves finite colimits.

(v) A functor between finitely complete and finitely cocomplete categories isexact if it is left exact and right exact.

Proposition A.2.4 and Corollary A.2.7 have the following variants for func-tors.

Proposition A.2.11.

(i) A functor between complete categories is continuous if and only if it pre-serves equalizers and products.

(ii) A functor between finitely complete categories is left exact if and only if itpreserves equalizers and finite products if and only if it preserves pullbacksand terminal objects.

The following lemma provides a large class of (co)continuous functors.

Lemma A.2.12.

(i) A right adjoint functor between complete categories is continuous andhence left exact.

(ii) A left adjoint functor between cocomplete categories is cocontinuous andhence right exact.

Proof. TODO. Include sketch proof, mention that one has to identify the re-sulting map with the canonical one.

A.3 Basic 2-categorical terminology

Will be added soon.TODO:

(i) 2-categories, and illustrate by examples:

(a) categories, functors, and natural transformations

(b) pointed categories, pointed functors, and all transformations

(c) additive categories, additive functors, and all transformations

(d) triangulated categories, exact functors, and exact transformations

(e) monoidal categories, monoidal functors, and monoidal transforma-tions

(ii) 2-functors

(iii) Mention that there are 2-natural transformations, weakening by pseudo-natural transformations, modifications, obtain the resulting 2-categories,but refer to second volume

DRAFT


DRAFT

Appendix B

Examples of derivators

This appendix will change a lot.B.1 Represented derivators

TODO: subcourse: reformulation of classical facts in terms of canonical matesIn this section we show that prederivators represented by complete and co-

complete categories are derivators. Moreover, we verify that suitable canonicalmaps which were defined for arbitrary derivators in the main body of the bookspecialize to the corresponding canonical morphisms considered in category the-ory.

Troughout this section we assume that C and D are complete and cocompletecategories. To begin with, given a small category A, we again take a closer lookat the adjunctions

(colimA,∆A) : CA C and (∆A, limA) : C CA.

(i) Let A be a small category and let F : C → D be a functor. We abusenotation and simply write F : CA → DA for the induced functor. By thevery definition we have the two commutative squares

DA F // CA DA F // CAYa

D

∆A

OO

F//

!

C,

∆A

OO

D

∆A

OO

F// C.

∆A

OO

Assuming the existence of the necessary (co)limits, the canonical mate(8.1.5) of the square on the left and the canonical mate (8.1.6) of thesquare on the right, respectively, are the canonical maps

colimA F → F colimA and F limA → limA F,

measuring whether F : C → D preserves (co)limits of shape A; see (A.2.8)and (A.2.9).

317

DRAFT

318 APPENDIX B. EXAMPLES OF DERIVATORS

(ii) Let C be a complete and cocomplete category, let u : A→ B be a functorbetween small categories, and let b ∈ B. In the background of Theo-rem 6.3.10 and Theorem 6.3.12 there were the slice squares

(u/b)p

//

π(u/b)

A

u

1b

// B

and

(b/u)q

//

π(b/u)

A

u

1b

// B.

AI

Here, the component of the natural transformation in the square on theleft at the object (a, f : u(a) → b) is given by f , and similarly in thesquare on the right. Passing to diagram categories we obtain inducednatural transformations

C(u/b)

CAp∗

oo

C

π∗(u/b)

OO

CBb∗

oo

u∗

OO

and

C(b/u) CAq∗

oo

C

π∗(b/u)

OO

CB .b∗

oo

AIu∗

OO

The canonical mate (8.1.5) of the square on the left and the canonicalmate (8.1.6) of the square on the right are the natural isomorphisms

colim(u/b) p∗ → b∗ LKanu and b∗ RKanu → lim(b/u) q∗

of Theorem 6.3.10 and Theorem 6.3.12, expressing that Kan extension insufficiently (co)complete categories are pointwise.

TODO:

(i) Represented derivators

(ii) For u : A→ B the canonical mate colimA u∗ → colimB specializes to the

classical one (see (6.2.1)).

(iii) For F : C → D the canonical mate colimA FX → F (colimAX) specializesas intended. Hence colimit preserving maps the same; refer to (??) and(A.2.8).

(iv) Make also precise what we obtain in the case of Kan extensions alongfunctors u : A→ B with B 6= 1.

B.2 Opposite derivators

TODO:

(i) digression: various opposited for 2-categories (already referred to!!!)

DRAFT

B.3. HOMOTOPY DERIVATORS OF MODEL CATEGORIES 319

(ii) review of duality in category theory: iso of cats of cones and cocones

(iii) detailed proof that Dop is a derivator

(iv) continuity versus cocontinuity

Because of its importance we will give a few more details about this example.Recall first that a 2-category D can be dualized in various ways; one can changethe orientation of 1-cells to obtain Dop, the orientation of 2-cells to get Dco, orcombine the two and hence obtain Moreover, a 2-functor between 2-categoriesinduces three more 2-functors between the various dualizations which will bedenoted by the same symbol. For example the passage to opposite categoriesis covariant in functors and contravariant in natural transformations, and thereare thus, in particular, dualization 2-functors

Since this example is rather important let us be more precise. To beginwith let us recall that the passage to opposite categories (−)op : C 7→ Cop iscovariant in functors and contravariant in natural transformations. Hence, weobserve that the dual prederivator Dop is defined to make the following diagramcommutative,

The following result formalizes the self-duality of the axioms of a derivator.It leads to the important duality principle. In more detail, many statementsin the theory of derivators have dual versions and by the proposition it will besufficient to state and prove only one of them.

B.3 Homotopy derivators of model categories

We do not include a discussion of locally presentable categories here and insteadonly mention that chain complexes over a ring (more generally, chain complexesin Grothendieck abelian categories) and simplicial sets yield examples of locallypresentable categories. For an introduction to locally presentable categories werefer to [Gro10, §3.1] while detailed accounts can be found in [GU71, MP89,AR94] and [Bor94a, Bor94b].

In this section we include a detailed proof that combinatorial model cate-gories have underlying homotopy derivators. The proof relies on the non-trivialresults that diagram categories in combinatorial model categories can be en-dowed with both the projective and the injective model structures.

LetM be a model category and let A ∈ Cat . In general, it is not true that wecan endow the diagram categoryMA with a model structure such that the weakequivalences are precisely the levelwise weak equivalences, i.e., those naturaltransformations f : X → Y : A → M such that all components fa : Xa →Ya, a ∈ A, are weak equivalences in M. However, if one imposes additionalconditions on M, then there are two such model structures for arbitrary A ∈Cat . We begin by the one which is adapted to the study of homotopy colimitsand homotopy left Kan extensions.

Definition B.3.1. LetM be a model category, let A ∈ Cat , and let X,Y : A→M.

DRAFT


(i) A morphism f : X → Y is a projective fibration if it is levelwise fibra-tion.

(ii) A morphism f : X → Y is a projective weak equivalence if it is alevelwise weak equivalence.

(iii) A morphism f : X → Y is a projective cofibration if it has the LLPwith respect to acyclic projective fibrations.

If these three classes define a model structure on MA, then we refer to itas the projective model structure. For the notion of a cofibrantly gener-ated model structure we refer the reader to [Hov99]. Let us only mention thatmost model structure showing up in nature (like homotopy theory, homologicalalgebra, and higher category theory) enjoy this property, so that the followingresult often applies.

Theorem B.3.2. Let M be a cofibrantly generated model category and let A ∈Cat. The projective model structure exists on MA.

The category MA endowed with the projective model structure will be de-noted by MA

proj. Recall that the category of simplicial sets can be endowedwith the cofibrantly generated Kan–Quillen model structure. In this case, thecorresponding projective model structures on diagram categories were first es-tablished in [].

Projective model structures have good functorial properties. For the studyof homotopy left Kan extensions we observe the following.

Proposition B.3.3. Let M be a cofibrantly generated model category and letu : A → B be in Cat. The adjunction (u!, u

∗) : MAproj MB

proj is a Quillen

adjunction. In particular, (colimA,∆A) : MAproj M is a Quillen adjunction.

Proof. We have to show that u∗ : MB →MA preserves projective fibrations andacyclic projective fibrations which is immediate since both classes are definedlevelwise.

Passing to left derived functors, this proposition yields homotopy left Kanextensions and homotopy colimits. Dualizing Definition B.3.1 we obtain thefollowing.

Definition B.3.4. LetM be a model category, let A ∈ Cat , and let X,Y : A→M.

(i) A morphism f : X → Y is an injective cofibration if it is levelwisecofibration.

(ii) A morphism f : X → Y is an injective weak equivalence if it is alevelwise weak equivalence.

(iii) A morphism f : X → Y is an injective fibration if it has the RLP withrespect to acyclic injective cofibrations.

DRAFT

B.3. HOMOTOPY DERIVATORS OF MODEL CATEGORIES 321

As in the projective case, if the above three classes define a model structureon MA, then we refer to it as the injective model structure and denote theresulting model category byMA

inj. One of the first examples was established byHeller in [Hel88] where he shows that diagram categories in simplicial sets admitthe injective model structure. In general, as of this writing, we only know thatinjective model structures exist on diagram categories in combinatorial modelcategories. Let us recall that a model category is combinatorial if

(i) the model structure is cofibrantly generated and if

(ii) the underlying category is locally presentable.

We refer the reader to [Gro10, §3.2] for a short introduction to combinatorialmodel categories and additional references. Here, we only mention that manymodel categories arising in nature are combinatorial.

Theorem B.3.5. Let M be a combinatorial model category and let A ∈ Cat.The injective model structure exists on MA.

Injective model structures enjoy the following dual functorial properties.

Proposition B.3.6. Let M be a combinatorial model category and let u : A→B be in Cat. The adjunction (u∗, u∗) : MB

inj MAinj is a Quillen adjunction.

In particular, (∆A, limA) : MMAinj is a Quillen adjunction.

Proof. The functor u∗ : MB →MA preserves injective cofibrations and acyclicinjective cofibrations since both classes are defined levelwise.

Thus, for combinatorial model categories we can use the projective modelstructures of Theorem B.3.2 to establish the existence of homotopy left Kanextensions and the injective model structures of Theorem B.3.5 to obtain ho-motopy right Kan extensions. To relate the domains of the respective functorwe make the following observation.

Proposition B.3.7. LetM be a combinatorial model category and let A ∈ Cat.The identity adjunction (id, id) : MA

proj MAinj is a Quillen equivalence.

Proof.

The main goal of this section is to show that combinatorial model categorieshave homotopy derivators. The proof that homotopy Kan extensions can becalculated pointwise needs some additional preparation. For this purpose, letus consider u : A→ B, b ∈ B, and the associated slice squares

(u/b)p

//

π(u/b)

A

u

1b

// B

and

(b/u)q

//

π(b/u)

A

u

1b

// B.

AI

DRAFT


Proposition B.3.8. Let M be a model category, let u : A→ B in Cat, and letb ∈ B.

(i) If M is a cofibrantly generated model category, then the restriction func-

tors p∗ : MAproj →M

(u/b)proj and b∗ : M→MB

proj are left Quillen functors.

(ii) If M is a combinatorial model category, then the functors q∗ : MAinj →

M(b/u)inj and b∗ : M→MB

inj are right Quillen functors.

Proof. By Theorem B.3.2 and Theorem B.3.5 the respective model structuresexist and it suffices by duality to take care of (i). We begin by showing thatb∗ : MB

proj →M is a left Quillen functor. Since b∗ preserves weak equivalences,it suffices to show that b∗ preserves projective cofibrations. Considering theadjunction (b∗, b∗) : MB M this is equivalent to b∗ preserving acyclic projec-tive fibrations, i.e., levelwise acyclic fibrations. Let us recall from the explicitconstruction of cofree diagrams in Example 6.4.2 that there are natural isomor-phisms

b∗(X)b′ ∼=∏

homB(b′,b)

X, X ∈M, b′ ∈ B.

Since products of acyclic fibrations are again acyclic fibration, b∗ sends acyclicfibrations to acyclic projective fibrations.

We now show that p∗ : MAproj →M

(u/b)proj also is a left Quillen functor. TODO

Lemma B.3.9. Combinatorial model structures and shifting

Proof.

Left over stuff

DRAFT

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