Homotopy Limit Functors on Model Categories and Homotopical Categories

William G. Dwyer

Philip S. Hirschhorn

Daniel M. Kan

Jeffrey H. Smith

Department of Mathematics, University of Notre Dame, Notre Dame, IN 46556

E-mail address: dwyer.1@nd.edu

Department of Mathematics, Wellesley College, Wellesley, MA 02481

E-mail address: psh@math.mit.edu

Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139

Department of Mathematics, Purdue University, West Lafayette, IN 47907

E-mail address: jhs@math.purdue.edu

Draft: May 14, 2004

Contents

Preface v

Part I. Model Categories 1

Chapter I. An overview 3 1. Introduction 3 2. Slightly unconventional terminology 3 3. Problems involving the homotopy category 5 4. Problem involving the homotopy colimit functors 8 5. The emergence of the current monograph 11 6. A preview of part II 12

Chapter II. Model categories and their homotopy categories 17 7. Introduction 17 8. Categorical and homotopical preliminaries 20 9. Model categories 23 10. The homotopy category 27 11. Homotopical comments 30

Chapter III. Quillen functors 33 12. Introduction 33 13. Homotopical uniqueness 36 14. Quillen functors 38 15. Approximations 40 16. Derived adjunctions 42 17. Quillen equivalences 46 18. Homotopical comments 49

Chapter IV. Homotopical cocompleteness and completeness of model

categories 53

19. Introduction 53 20. Homotopy colimit and limit functors 57 21. Homotopical cocompleteness and completeness 60 22. Reedy model categories 63 23. Virtually cofibrant and fibrant diagrams 66 24. Homotopical comments 70

Part II. Homotopical Categories 73

Chapter V. Summary of part II 75 25. Introduction 75

iii Draft: May 14, 2004

26. Homotopical categories 76 27. The hom-sets of the homotopy categories 78 28. Homotopical uniqueness 80 29. Deformable functors 81 30. Homotopy colimit and limit functors and homotopical ones 83

Chapter VI. Homotopical Categories and Homotopical Functors 87 31. Introduction 87 32. Universes and categories 91 33. Homotopical categories 93 34. A colimit description of the hom-sets of the homotopy category 99 35. A Grothendieck construction 101 36. 3-arrow calculi 105 37. Homotopical uniqueness 110 38. Homotopically initial and terminal objects 113

Chapter VII. Deformable Functors and Their Approximations 117 39. Introduction 117 40. Deformable functors 121 41. Approximations 124 42. Compositions 128 43. Induced partial adjunctions 131 44. Derived adjunctions 136 45. The Quillen condition 141

Chapter VIII. Homotopy Colimit and Limit Functors and Homotopical Ones 145 46. Introduction 145 47. Homotopy colimit and limit functors 146 48. Left and right systems 150 49. Homotopical cocompleteness and completeness (special case) 157 50. Homotopical colimit and limit functors 159 51. Homotopical cocompleteness and completeness (general case) 164

Index 169

Bibliography 179

Preface

This monograph, which is aimed at the graduate level and beyond, consists of two parts.

In part II we develop the beginnings of a kind of “relative” category theory of what we will call homotopical categories. These are categories with a single distinguished class of maps (called weak equivalences) containing all the isomorphisms and satisfying one simple two out of six axiom which states that

(*) for every three maps r, s and t for which the two compositions sr and ts are defined and are weak equivalences, the four maps r, s, t and tsr are also weak equivalences,

which enables one to define “homotopical” versions of such basic categorical notions as initial and terminal objects, colimit and limit functors, adjunctions, Kan extensions and universal properties.

In part I we then use the results of part II to get a better understanding of Quillen’s so useful model categories, which are categories with three distinguished classes of maps (called cofibrations, fibrations and weak equivalences) satisfying a few simple axioms which enable one to “do homotopy theory”, and in particular to show that such model categories are homotopically cocomplete and homotopically complete in a sense which is much stronger than the existence of all small homotopy colimit and limit functors.

Both parts are essentially self-contained. A reader of part II is assumed to have some familiarity with the categorical notions mentioned above, while those who read part I (and especially the introductory chapter) should also know something about model categories. In the hope of increasing the local as well as the global readability of this monograph, we not only start each section with some introductory remarks and each chapter with an introductory section, but also each of the two parts with an introductory chapter, with the first chapter of part I as motivation for and introduction to the whole monograph and the first chapter of part II summarizing the main results of its other three chapters.

Draft: May 14, 2004

Part I

Model Categories

Draft: May 14, 2004

CHAPTER I An overview

1. Introduction

1.1. Summary. This monograph is essentially the result of an unsuccessful attempt to give an updated account of Quillen’s closed model categories, i.e. categories with three distinguished classes of maps (called weak equivalences, cofibrations and fibrations) satisfying a few simple axioms which enable one to “do homotopy theory”. That attempt however failed because, the deeper we got into the subject, and especially when we tried to deal with homotopy colimit and limit functors on arbitrary model categories, the more we realized that we did not really understand the role of the weak equivalences.

In this introductory chapter we will therefore try to explain the problems we ran into and the solutions we came up with, and how all this determined the content and the layout of this monograph.

1.2. Organization of the chapter. After fixing some slightly unconventional terminology (in §2), we discuss the problems we encountered in dealing with the homotopy category of a model category (in §3) and with the homotopy colimit functors on a model category (in §4) and explain (in §5) how all this led to the current two-part monograph, where in the first part we review some mostly known results on model categories, but do so whenever possible from the “homotopical” point of view which we develop in the second part, and in the second part we investigate what we call homotopical categories which are categories with only a single distinguished class of maps (called weak equivalences). The last section (§6) then contains some more details on this second part.

2. Slightly unconventional terminology

In order to be able to give a reasonably clear formulation of the above mentioned problems and their solutions, we slightly modify the customary meaning of the terms “model category” and “category” and introduce, just for use in this introductory chapter, the notion of a “category with weak equivalences” or “wecategory”.

2.1. Model categories. In [Qui67] Quillen introduced the notion of a model category, but then almost right away only concerned himself with the slightly more restricted but also more useful closed model categories which in [Qui69] he characterized by five simple axioms. However since then it has become clear that it is more convenient to restrict the definition even further by

(i) strengthening his limit axiom, which requires the existence of all finite colimits and limits, by requiring the existence of all small colimits and limits, and

3 Draft: May 14, 2004

(ii) strengthening his factorization axiom by requiring the therein mentioned

factorizations to be functorial,

and we will therefore use the term model category for a closed model category which satisfies these stronger versions of Quillen’s limit and factorization axioms.

2.2. Categories. We will use the term locally small category for what one usually calls a category, i.e. a category with small hom-sets, and reserve the term category for a more general notion which ensures that

(i) for every two categories C and D, the functors C → D and the natural transformations between them also form a category (which is usually denoted by D^{C}), and

(ii) for every category C and subcategory W ⊂ C, there exists a localization of C with respect to W, i.e. a category C[W^{−1}] together with a localization functor γ: C → C[W^{−1}] which is 1-1 and onto on objects and sends the maps of W to isomorphisms in C[W^{−1}], with the universal property that, for every category B and functor b: C → B which sends the maps of W to isomorphisms in B, there is a unique functor b_{γ}: C[W^{−1}] → B such that b_{γ}γ = b.

It is also convenient to consider, for use in this chapter only, the notion of

2.3. Categories with weak equivalences. By a category with weak equivalences or we-category we will mean a category C with a single distinguished class W of maps (called weak equivalences) such that

(i) W contains all the isomorphisms, and (ii) W has Quillen’s two out of three property that, for every pair of

maps f, g ∈ C such that gf exists, if two of f, g and gf are in W, then so is the third

which readily implies that

(iii) W is a subcategory of C. For such a we-category C one then can

(iv) define the homotopy category Ho C of C as the category obtained from C by “formally inverting” the weak equivalences, i.e. the category with the same objects as C in which, for every pair of objects X, Y ∈ C, the hom-set Ho C(X, Y) consists of the equivalence classes of zigzags in C from X to Y in which the backward maps are weak equivalences, by the weakest equivalence relation which puts two zigzags in the same class when one can be obtained from the other by (iv)′ omitting an identity map, (iv)′′ replacing two adjacent maps which go in the same direction by their

composition, or (iv)′′′ omitting two adjacent maps when they are the same but go in op-

posite directions, note that this homotopy category Ho C, together with the functor γ: C → Ho C which is the identity on the objects and which sends a map c: X → Y ∈ C to the class containing the zigzag which consists of the map c only, is a localization of C with respect to its subcategory of weak equivalences (2.2(ii)) and call C saturated whenever a map in C is a weak equivalence

Draft: May 14, 2004

iff its image under the localization functor C → Ho C is an isomorphism in Ho C, and

(v) given two we-categories C and D with localization functors γ: C → Ho C and γ′: D → Ho D, and a functor g: C → D (which is not required to preserve weak equivalences), define a total left (resp. right) derived functor of g as a pair (n, c) consisting of a functor n: Ho C → Ho D and a natural transformation

c: nγ ⟶ γ′g (resp. c: γ′g ⟶ nγ)

which is a terminal (resp. an initial) object in the category of all such pairs.

3. Problems involving the homotopy category

We now discuss four problems that came up when we started to review some of the classical results involving the homotopy categories of model categories. The first of these involved the notion of

3.1. Saturation. Our problem with saturation was that, while this notion was defined in terms of only the weak equivalences (2.3(iv)), the classical proof of the fact that

(i) every model category is a saturated we-category (2.3)

was very ad-hoc. We felt that it should have been possible to describe “reasonable” conditions on an arbitrary we-category C such that

(ii) these conditions implies the saturation of C

and then show that

(iii) these conditions held for every model category.

However at that time we did not have the faintest idea what such conditions might look like.

A similar problem concerned

3.2. The hom-sets. The usual description, for a cofibrant object X_{c} and a fibrant object Y_{f} in a model category M, of the hom-set Ho M(X_{c}, Y_{f}) (2.3(iv)) as a quotient of the hom-set M(X_{c}, Y_{f}) by a homotopy relation, readily implied that

(i) for every pair of objects X, Y ∈ M, the elements of Ho M(X, Y) are in a natural 1-1 correspondence with the equivalence classes of the zigzags in M of the form

\xymatrix{X & · \ar[r] \ar[l] & · & Y \ar[l]}

in which the two backward maps are weak equivalences, where two such zigzags are in the same class iff they are the top row and the bottom row in a commutative diagram in M of the form

\xymatrix{X & · \ar[l] \ar[r] & · & Y \ar[l] \\ X \ar[u]^{1} \ar[d]_{1} & · \ar[l] \ar[u] \ar[r] \ar[d] & · \ar[u] \ar[d] & Y \ar[l] \ar[u]_{1} \ar[d]^{1} \\ X & · \ar[l] \ar[r] & · & Y \ar[l] \\ X \ar[u]^{1} & · \ar[l] \ar[u] \ar[r] & · \ar[u] & Y \ar[l] \ar[u]_{1}}

in which the backward maps (and hence all the vertical maps) are weak equivalences.

This suggested a question similar to the one raised in 3.1, but in this case we knew the answer, which was provided by the results of [DK80a] which implied that

(ii) a sufficient condition on a we-category C (2.3) in order that (i) holds with C instead of M, is that C admits a 3-arrow calculus, i.e. that there exist subcategories U and V of the category W of the weak equivalences in C such that, in a functorial manner

(iii) for every zigzag A′ ⟵^{u} A ⟶^{f} B in C with u ∈ U, there exists a zigzag

A′ ⟶^{f′} B′ ⟵^{u′} B in C with u′ ∈ U such that u′f = f′u and in which u′ is an isomorphism whenever u is,

(iv) for every zigzag X ⟶^{g} Y ⟵^{v} Y′ in C which v ∈ V , there exists a zigzag

X ⟵^{v′} X′ ⟶^{g′} Y′ in C with v′ ∈ V such that gv′ = vg′ and in which v′ is an isomorphism whenever v is, and

(v) every map w ∈ W admits a factorization w = vu with u ∈ U and v ∈ V . and the fact that, in view of the assumption made in (2.1(ii))

(vi) every model category admits a 3-arrow calculus consisting of the subcategories of the trivial cofibrations and the trivial fibrations, i.e. the cofibrations and the fibrations which are also weak equivalences.

Next we considered

3.3. Certain subcategories of the homotopy category. A simpler problem arose from the fact that, if for a model category M one denotes its full subcategories spanned by the cofibrant and the fibrant objects respectively by M_{c} and M_{f} and their intersection by M_{cf}, then

(i) the inclusions of M_{cf}, M_{c}, M_{f} and M in each other induce a commutative diagram

\xymatrix{ & Ho M_{c} \ar@{-->}[dr] \\ _{Ho M_{cf}} \ar@{-->}[ur] \ar@{-->}[dr] &  & _{Ho M} \\  & _{Ho M_{f}} \ar@{-->}[ur]}

in which all the maps are equivalences of categories.

This made us, for an arbitrary we-category C (2.3), define

Draft: May 14, 2004

(ii) a left (resp. a right) deformation of C as a pair (r, s) consisting of a weak equivalence preserving functor r: C → C and a natural weak equivalence

s: r ⟶ 1_{C} (resp. s: 1_{C} ⟶ r)

and

(iii) a left (resp. a right) deformation retract of C as a full subcategory C_{0} ⊂ C for which there exists a left (resp. a right) deformation (r, s) of C into C_{0}, i.e. such that the functor r: C → C_{0} sends all of C into C_{0},

and note that

(iv) for every left (resp. right) deformation retract C_{0} ⊂ C, its inclusion

C_{0} → C induces an equivalence of categories Ho C_{0} → Ho C,

and that, in view of the assumption made in 2.1(ii)

(v) for every model category M there exist left and right deformations of M into M_{c} and M_{f} respectively (which are often referred to as functorial cofibrant and fibrant replacements,) so that M has M_{c} as a left deformation retract and M_{f} as a right deformation retract, and for similar reasons M_{cf} is a left deformation retract of M_{f} and a right deformation retract of M_{c}.

Our fourth and last problem involving the homotopy categories concerned the

3.4. Total derived functors of Quillen functors. Given a Quillen adjunction, i.e. an adjoint pair of functors f: M ↔ N :f′ between model categories of which the left adjoint f preserves cofibrations and trivial cofibrations (3.2(vi)) and the right adjoint f′ preserves fibrations and trivial fibrations,

(i) the left adjoint f has total left derived functors (2.3(v)) and the right adjoint f′ has total right derived functors, and

(ii) for every pair consisting of a total left derived functor (n, c) of f and a total right derived functor (n′, c′) of f′, the given adjunction induces a derived adjunction n: Ho M ↔ Ho N :n′,

which raised the problem of isolating the “weak equivalence properties” of Quillen adjunctions which ensured the truth of these two statements.

To deal with this we (iii) called a functor g: C → D (as in 2.3(v)) left (resp. right) deformable

if g was a weak equivalence preserving functor on a left (resp. a right) deformation retract of C (3.3(iii)) or equivalently, if there existed a left (resp. a right) g-deformation, i.e. a left (resp. a right) deformation (r, s) of C (3.3(ii)) such that g preserved weak equivalences on the full subcategory of C spanned by the image of r,

and noted that:

(iv) if g: C → D was such a left (resp. right) deformable functor, then g had total left (resp. right) derived functors (2.3(v)) (as in that case, for every left (resp. right) g-deformation (r, s), the pair (Ho gr, γ′gs) (2.3(v)) turned out to be such a total derived functor),

and that

Draft: May 14, 2004

(v) for every pair of adjoint functors g: C ↔ D :g′ between we-categories (2.3) of which the left adjoint g was left deformable and the right adjoint g′ was right deformable, its adjunction induced, for every pair consisting of a total left derived functor (n, c) of g and a total right derived functor (n′, c′) of g′, a derived adjunction n: Ho C ↔ Ho D :n′.

That (i) and (ii) above then indeed were special cases of (iv) and (v) followed readily from 3.3(v) and the observation that

(vi) as, for every Quillen adjunction f: M ↔ N :f′ the functor f preserved weak equivalences between cofibrant objects and the functor f′ preserved weak equivalences between fibrant ones, the functors f and f′ were respectively left and right deformable.

4. Problem involving the homotopy colimit functors

Very different problems came up when we tried to deal with homotopy colimit functors on arbitrary model categories. To explain these we first recall the various definitions of

4.1. Homotopy colimit functors. For most authors, given a model category M and a small category D,

(i) a homotopy D-colimit functor on M is a pair (k, a) consisting of a functor k: M^{D} → M which sends objectwise weak equivalences in (M_{c})^{D} (2.2(i) and 3.3) to weak equivalences in M and a natural transformation a: k → colim^{D} (where colim^{D} denotes an arbitrary but fixed D-colimit functor on M, i.e. an arbitrary but fixed left adjoint of the constant diagram functor M → M^{D}), which is obtained (as in [Hir03]) by choosing a so-called framing of M and then applying the formulas of [BK72],

but for other authors

(ii) a homotopy D-colimit functor on M is a pair (k′, a′) obtained from one of the above mentioned homotopy D-colimit functors by (objectwise) pre-composition with a functorial cofibrant replacement (3.3(v)), as a result of which the functor k′: M^{D} → M sends all objectwise weak equivalences to weak equivalences in M.

The first of these notions is the more practical one and is often used in calculations, but the second is more “homotopically correct” and was therefore of more interest to us, especially as we realized that

(iii) each of these homotopically correct homotopy D-colimit functors on M was of the form (colim^{D} r, colim^{D} s) for a special kind of left colim^{D}- deformation (3.4(iii))

and that moreover, if one considered these homotopy colimit functors as objects of the we-category (2.3)

(Fun_{w}(M^{D}, M) ↓ colim^{D})

which has as objects the pairs (k, a) consisting of an (objectwise) weak equivalence preserving functor k: M^{D} → M and a natural transformation a: k → colim^{D} and as maps and weak equivalences (k_{1}, a_{1}) → (k_{2}, a_{2}) the natural transformations and natural weak equivalences t: k_{1} → k_{2} such that a_{2}t = a_{1}, then (iv) any two of these homotopically correct homotopy D-colimit functors on M were weakly equivalent (i.e. could be connected by a zigzag of weak equivalences) and in fact were so in a “homotopically unique” manner.

Now we can formulate

4.2. The problem. The last two comments (4.1(iii) and (iv)) raised the question whether, for every left colim^{D}-deformation (r′, s′) (for instance one that was special as in 4.1(iii) with respect to some other model structure on M with the same weak equivalences), the pair (colim^{D} r′, colim^{D} s′) would be weakly equivalent to the pairs considered in 4.1(iii), and as we could not imagine that this was not the case, make us wonder whether there existed, for every we-category C (2.3) on which there existed D-colimit functors (4.1(i)) and for which we denoted by colim^{D} an arbitrary but fixed such functor, we could define a set of objects in the we-category (4.1(iii))

(Fun_{w}(C^{D}, C) ↓ colim^{D})

such that

(i) any two such objects were weakly equivalent in some homotopically unique manner, and

(ii) if C were the underlying we-category of a model category, then this class included all the homotopically correct homotopy D-colimit functors on this model category,

in which case these objects deserved to be called homotopy D-colimit functors on C.

It seemed that such objects would have to be (objectwise) weak equivalence preserving functors C^{D} → C over the functor colim^{D} which “approached colim^{D} as closely as possible from the left”. They could however not be required to be terminal objects in the category (Fun_{w}(C^{D}, C) ↓ colim^{D}), as in that case any two of them would be isomorphic, but could only be expected to be some kind of

4.3. Homotopically terminal objects. Given a we-category Y and an object Y ∈ Y , we denoted by 1_{Y} : Y → Y the identity functor of Y and by cst_{Y} : Y → Y the constant functor which sends all maps of Y to the identity map 1_{Y} of the object Y, and motivated by the fact that (see 38.1) Y would be a terminal object of Y iff there existed a natural transformation f: 1_{Y} → cst_{Y} such that the map fY : Y → Y was an isomorphism, we decided to call Y a homotopically terminal object of Y if there existed functors F_{0}, F_{1} : Y → Y and a natural transformation f: F_{1} → F_{0} such that

(i) F_{0} is naturally weakly equivalent (i.e. can be connected by a zigzag of natural weak equivalences) to cst_{Y}, (ii) F_{1} is naturally weakly equivalent to 1_{Y}, and (iii) the map fY : F_{1}Y → F_{0}Y is a weak equivalence.

Given a we-category C as in 4.2, the homotopically terminal objects of the we-category (Fun_{w}(C^{D}, C) ↓ colim^{D}) then had the property mentioned in 4.2(ii), but in order to see whether they also satisfied 4.2(i) we first had to decide what exactly should be meant by

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4.4. Homotopical uniqueness. Homotopical uniqueness had to be a kind of we-category version of the categorical notion of uniqueness up to a (unique) isomorphism (which we like to refer to as categorical uniqueness) of certain objects in a category Y , by which one meant that those objects (for instance because they had a common universal property) had the property that, for any two of them, Y_{1} and Y_{2}, there was exactly one map Y_{1} → Y_{2} ∈ Y and that moreover this unique map was an isomorphism. Another way of saying that was that

(i) given a non-empty set of objects in a category Y , these objects were categorically unique or canonically isomorphic if the full subcategory G ⊂ Y spanned by these objects and all isomorphic ones was categorically contractible in the sense that G was a non-empty groupoid in which there was exactly one isomorphism between any two objects, or equivalently that, for one (and hence every) object G ∈ G, the identity functor 1_{G}: G → G and the constant functor cst_{G}: G → G (4.3) were naturally isomorphic.

We therefore similarly said that

(ii) given a non-empty set of objects in a we-category Y , these objects were homotopically unique or canonically weakly equivalent if the full subcategory G ⊂ Y spanned by these objects and all weakly equivalent ones (4.1(iv)) was homotopically contractible in the sense that, for one (and hence every) object G ∈ G, the functors 1_{G}: G → G and cst_{G}: G → G were naturally weakly equivalent (4.3(i)).

At this point we had hoped, in order to settle the question that we raised at the end of 4.2, that just as

(iii) in every category, the terminal objects, if they existed, would be canoni-

cally isomorphic (i),

it would also be true that

(iv) in every we-category, the homotopically terminal objects (4.3) would be

canonically weakly equivalent (ii).

However it turned out that this was not the case, unless one put some restrictions on the we-category considered, and after much trial and error we concluded that the minimal such restriction would be to require a property which was only slightly stronger than the “two out of three” property (2.3) and which, for want of a better name, we called

4.5. The two out of six property. We said that a we-category C has the two out of six property if

(i) for every three maps r, s and t ∈ C for which the two compositions sr and ts were weak equivalences, the four maps r, s, t and tsr also were weak equivalences

and as (as we mentioned at the end of 4.4)

(ii) the presence of the two out of six property implied that the homotopically terminal objects (4.3), if they existed, were canonically weakly equivalent (4.4)

we could now finally answer the question which we raised at the end of 4.2 by defining, for every we-category as in 4.2 which had the two out of six property, and every small category D, (iii) a homotopy D-colimit functor on C as a homotopically terminal

object (4.3) of the we-category

(Fun_{w}(C^{D}, C) ↓ colim^{D})

(which clearly inherited the two out of six property from C)

and note that in view of (ii)

(iv) such homotopy D-colimit functors on C, if they existed, were homotopi-

cally unique

and that

(v) a sufficient condition for their existence was that the functor colim^{D}: C^{D} → C be left deformable (3.4(iii)) as in that case, for every left colim^{D}- deformation (r, s), the pair (colim^{D} r, colim^{D} s) was a homotopy Dcolimit functor on C.

5. The emergence of the current monograph

Ever since the discovery of simplicial localizations [DK80a, DK80b, DK80c] we had wondered whether there might not exist some reasonable theory of wecategories (2.3) which would on the one hand be a relative version of ordinary category theory and on the other hand provide a framework for homotopical algebra. These thoughts caused us, some ten years later, to ask the questions which we discussed in the preceding two sections and the answers we came up with suggested that there might indeed be such a theory, however not of we-categories, but of what we decided to call

5.1. Homotopical categories. We defined a homotopical category as a category C with a single distinguished class of maps (called weak equivalences) such that

(i) these weak equivalences had the two out of six property that (4.5(i)), for every three maps r, s and t ∈ C for which the two compositions sr and ts were weak equivalences, the four maps r, s, t and tsr were also weak equivalences, and

(ii) for every object C ∈ C its identity map 1_{C}: C → C was also a weak

equivalence

which readily implied, by considering the cases in which sr and ts are both identity maps or in which at least one of r, s and t is an identity map, that

(iii) all isomorphisms in C were weak equivalences, and C had the two out

of three property, so that (iv) C was a we-category (2.3)

However we still were not sure that this was the right setting until we realized that

(v) every homotopical category which admits a 3-arrow calculus (3.2) is sat-

urated ( 2.3(iv)),

which, in view of 3.2(vi), finally provided an answer to the rather basic question which we raised in 3.1.

At this point we decided to give up on our original plan (see §1) and write instead the current monograph which consists of two essentially self contained parts, of which the first deals with model categories and the second with homotopical categories.

Part I which, apart from this introductory chapter, consists of the chapters II–IV, deals mainly with the model category results which we mentioned before (in §3 and §4), but does so, whenever possible, from the “homotopical” point of view which we develop in part II. More precisely, in the chapters II and III we introduce model categories, their homotopy categories and Quillen functors and discuss the results we mentioned in §3, while in chapter IV we not only prove the existence, on every model category, of homotopy colimit and limit functors (see §4), but also the considerably stronger result that (in an appropriate sense) every model category is homotopically cocomplete and complete.

Part II is concerned with homotopical categories only and consists of the four chapters V–VIII, of which the first is just a summary of the other three. These three chapters, of which we will give a preview in the next (and last) section of this introductory chapter, parallel the chapters II–IV and deal respectively with homotopical categories and their homotopy categories, deformable functors (3.4(iii)) and homotopy colimit and limit functors and more generally homotopical colimit and limit functors and the associated notions of homotopical cocompleteness and completeness.

6. A preview of part II

We end this chapter with a brief outline of the contents of part II, i.e. of the chapters VI–VIII.

6.1. Homotopical categories and homotopical functors. In the first part of chapter VI we

(i) motivate the somewhat unconventional use of the term category mentioned in 2.2,

(ii) define homotopical categories (5.1) and homotopical functors (i.e.

weak equivalence preserving functors) between them, and (iii) discuss a few immediate consequences of these definitions.

6.2. The homotopy category of a homotopical category. The middle part of chapter VI is concerned with the homotopy category Ho C of a homotopical category C (2.3(iv)) and in it we

(i) give an alternate description of Ho C as the category which has the same objects as C in which, for every pair of objects X, Y ∈ C, the hom-set Ho C(X, Y) consists of the equivalence classes of zigzags in C from X to Y in which the backward maps are weak equivalences, with respect to the weakest equivalence relation which puts two such zigzags in the same class

(i)′ when one can be obtained from the other by omitting identity maps, (i)′′ when one can be obtained from the other by replacing adjacent maps

which go in the same direction by their composition, and (i)′′′ when they are the top and the bottom row in a rectangular diagram in

C of the form

[figure: X; 1; ·; ···; ·; Y; 1; X; ·; ·; Y]

in which in every column the two horizontal maps go in the same direction

and use this description to show that, if C admits a 3-arrow calculus (3.2(ii)), then

(ii) the hom-sets of Ho C admit the 3-arrow description given in 3.2(i), which in turn implies that

(iii) C has the rather useful property (see 6.5(iii) and 6.7 below) of being saturated (2.3(iv)).

6.3. Homotopically terminal and initial objects. In the last part of chapter VI we

(i) discuss in more detail the notions of homotopically terminal and initial objects (4.3) and homotopical uniqueness (4.4), and

(ii) note that just as, given two (not necessarily homotopical) functors between homotopical categories

p: R ⟶ P  and  q: R ⟶ Q

one can define the right (or as we will say terminal) Kan extensions of q along p as the terminal objects of the homotopical category (−p ↓ q) which has as objects the pairs (r, e) consisting of a (not necessarily homotopical) functor r: P → Q and a natural transformation e: rp → q, and as maps and weak equivalences t: (r_{1}, e_{1}) → (r_{2}, e_{2}) the natural transformations and natural weak equivalences t: r_{1} → r_{2} such that e_{2}(tp) = e_{1}, one can define homotopically terminal Kan extensions of q along p as the homotopically terminal objects of its full subcategory (−p ↓ q)_{w} ⊂ (−p ↓ q) spanned by the objects (r, e) for which r is a homotopical (6.1) functor, and

(iii) define initial Kan extension and homotopically initial Kan extensions in a dual manner.

6.4. Deformable functors. In chapter VII we investigate left and right deformable functors (3.4(iii)). These are not necessarily homotopical functors between homotopical categories which have homotopical meaning because they have (homotopically unique (4.4(ii))) left and right approximations respectively, where we

(i) define, for a (not necessarily homotopical) functor f: X → Y between homotopical categories, a left approximation of f as a homotopically terminal Kan extension of f along 1_{X} (6.3(ii)), i.e. a homotopically terminal object (4.3) of the “homotopical category of homotopical functors X → Y over f” (cf. 4.1(iii))

(Fun_{w}(X, Y ) ↓ f) .

They have the properties that

(ii) such left approximations of f, if they exist, are homotopically unique

(4.4(ii) and 4.5(ii)), (iii) a sufficient condition for their existence is that f be left deformable

(3.4(iii)) as in that case for every left f-deformation (r, s) (3.4(iii)), the pair (fr, fs) is a left approximation of f, (iv) for every left approximation (k, a) of f, the pair (Ho k, γ′a) (where γ′: Y →

Ho Y denotes the localization functor of Y (2.2(ii))) is a total left derived functor of f (2.3(v)), and

(v) for every composable pair of left deformable functors

f_{1} : X ⟶ Y  and  f_{2} : Y ⟶ Z ,

a sufficient condition in order that their composition f_{2}f_{1} : X → Z is also left deformable and every (appropriately defined) composition of their left approximations is a left approximation of their composition, is that the pair (f_{1}, f_{2}) is left deformable in the sense that

(vi) there exist left deformation retracts X_{0} ⊂ X and Y_{0} ⊂ Y (3.3(iii)) such that f_{1} and f_{2}f_{1} are both homotopical (6.1) on X_{0} and f_{2} is homotopical on Y_{0}, and

(vi)′ f_{1} sends all of X_{0} into Y_{0} which turns out to be equivalent to requiring that

(vii) there exist left deformations (r_{1}, s_{1}) of X and (r_{2}, s_{2}) of Y (3.3(ii)) such that (r_{1}, s_{1}) is both a left f_{1}- and a left f_{2}f_{1}-deformation (3.4(iii)) and (r_{2}, s_{2}) is a left f_{2}-deformation, and

(vii)′ the natural transformation

f_{2}s_{2}f_{1}r_{1} : f_{2}r_{2}f_{1}r_{1} ⟶ f_{2}f_{1}r_{1}

is a natural weak equivalence. Dual results of course hold for right deformable functors.

6.5. Deformable adjunctions. We also note in chapter VII that deformable functors often are part of a deformable adjunction, i.e. an adjunction f: X ↔ Y :f′ between homotopical categories of which the left adjoint is left deformable (3.4(iii)) and the right adjoint is right deformable.

Such deformable adjunctions have derived adjunctions (3.4(v)). However, as we noted that in dealing with these derived adjunctions we only used the total derived functors mentioned in 6.4(iv), it seemed to make more sense to describe derived adjunctions in terms of approximations instead. Instead of 3.4(v) we therefore

(i) show that every deformable adjunction f: X ↔ Y :f′ induces, for every pair consisting of a left approximation (k, a) of f and a right approximation (k′, a′) of f′, an adjunction (called derived adjunction)

Ho k: Ho X ⟷ Ho Y :Ho k′

which is natural in (k, a) and (k′, a′).

We also note that (ii) for every two composable deformable adjunctions

f_{1}: X ⟷ Y :f′_{1}  and  f_{2}: Y ⟷ Z :f′_{2}

a sufficient condition in order that their composition f_{2}f_{1} : X ↔ Z : f′_{1}f′_{2} is also an adjoint pair of deformable functors and the compositions of their derived adjunctions are derived adjunctions of their composition is that the pairs (f_{1}, f_{2}) and (f′_{2}, f′_{1}) are respectively left and right deformable (6.4(v))

and that (iii) for every two composable deformable adjunctions between saturated (2.3(iv))

homotopical categories

f_{1}: X ⟷ Y :f′_{1}  and  f_{2}: Y ⟷ Z :f′_{2}

for which the pairs (f_{1}, f_{2}) and (f′_{2}, f′_{1}) are locally left and right deformable in the sense that 6.4(vi) (or equivalently 6.4(vii)) holds but not necessarily 6.4(vi)′ (or 6.4(vii)′) the pair (f_{1}, f_{2}) is actually left deformable (6.4(v)) iff the pair (f′_{2}, f′_{1}) is actually right deformable,

This last result is rather useful as it often allows one, for a composable pair of left or right deformable functors between saturated homotopical categories, to verify the left deformability of this pair by only verifying its local left or right deformability in the sense of (iii).

6.6. A Quillen condition. At the end of chapter VII we note that, given a deformable adjunction (6.5) f: X ↔ Y :f′, a left deformation retract X_{0} ⊂ X on which f is homotopical (3.3(iii) and 6.1) and a right deformation retract Y_{0} ⊂ Y on which f′ is homotopical

(i) for one (and hence every) pair consisting of a left approximation (k, a) of f and a right approximation (k′, a′) of f′, the homotopical functor

k: X ⟶ Y  and  k′: Y ⟶ X

are (inverse) homotopical equivalences of homotopical categories in the sense that the two compositions k′k and kk′ are naturally weakly equivalent (4.3(i)) to the identity functors of X and Y respectively

iff the Quillen condition holds, i.e.

(ii) for every pair of objects X_{0} ∈ X_{0} and Y_{0} ∈ Y_{0}, a map fX_{0} → Y_{0} ∈ Y is a weak equivalence iff its adjunct X_{0} → f′Y_{0} ∈ X is so.

6.7. Homotopy colimit and limit functors and homotopical ones. In the last chapter, chapter VIII, we note that, just as in 4.5(iii) we defined, for every small category D, a homotopy D-colimit functor on a homotopical category as a left approximation (6.4(i)) of an arbitrary but fixed D-colimit functor colim^{D}: X^{D} → X on X, one can, for every functor u: A → B between not necessarily small categories, define a homotopy u-colimit functor on X as a left approximation of an arbitrary but fixed u-colimit functor colim^{u}: X^{A} → X^{B} on X (by which we mean an arbitrary but fixed left adjoint of the induced diagram functor u^{∗}: X^{B} → X^{A}). We then show that

(i) such homotopy u-colimit functors on X, if they exist, are homotopically unique (4.4(ii)),

(ii) a sufficient condition for their existence is the existence of a left de-

formable (3.4(iii)) u-colimit functor on X,

and that, for every composable pair of functors u: A → B and v: B → D between small categories

(iii) a sufficient condition in order that there exist homotopy u-colimit and vcolimit functors on X and that every (appropriately defined) composition of such a homotopy u-colimit functor with such a homotopy v-colimit functor is a homotopy vu-colimit functor on X is that there exist ucolimit and v-colimit functors on X, and

(iii)′ the pair (colim^{u}, colim^{v}) is left deformable (6.4(v)) which in view of 6.5(iii) is in particular the case if

(iii)′′ X is saturated (2.3(iv)) and the pair (colim^{u}, colim^{v}) is locally left deformable in the sense of 6.5(iii).

We also note that there is a corresponding notion of homotopical cocompleteness of a homotopical category X which is considerably stronger than the requirement that, for every small category D, there exist homotopy D-colimit functors on X and that

(iv) a sufficient condition for such homotopical cocompleteness of a homotopical category X is that X is cocomplete (which implies that, for every functor u: A → B between small categories, there exist u-colimit functors on X) and that there exists, for every small category D, a left deformation retract (X^{D})_{0} ⊂ X^{D} (3.3(iii)) such that, for every functor u: A → B between small categories, the u-colimit functor colim^{u}

(iv)′ is homotopical on (X^{A})_{0}, and (iv)′′ sends all of (X^{A})_{0} into (X^{B})_{0},

which last restriction is superfluous if X is saturated (2.3(iv)).

We end the chapter with the observation that, while the above homotopy ucolimit functors are only defined for homotopical categories on which there exist u-colimit functors (and even then in addition require a choice of such a functor)

(v) there is also the notion of what we will call a homotopical u-colimit functor which, as we define such a functor as a homotopically terminal Kan extension (6.3(ii)) of the identity functor 1_{X}^{B} along the induced diagram functor u^{∗}: X^{B} → X^{A}, is defined for arbitrary homotopical categories and which “generalizes” the notion of a homotopy u-colimit functor in the sense that

(vi) if there exist u-colimit functors on X, then (vi)′ there exist homotopical u-colimit functors on X iff there exist homotopy

u-colimit functors on X, and (vi)′′ in this case “composition with colim^{u}” yields a 1-1 correspondence be-

tween the homotopy u-colimit functors on X and the homotopical ucolimit functors on X.

Of course dual results hold for homotopy limit functors, homotopical completeness and homotopical limit functors.

CHAPTER II Model categories and their homotopy categories

7. Introduction

7.1. Summary. In this chapter we

(i) introduce a notion of model category (i.e. a category M with three distinguished classes of maps, weak equivalences, cofibrations and fibrations, satisfying a few simple axioms which enable one to “do homotopy theory in M”), which is a slightly strengthened version of Quillen’s notion of a closed model category ([Hir03], [Hov99]), and

(ii) discuss the homotopy category of such a model category M, i.e. the category obtained from M by “formally inverting” its weak equivalences.

In more detail:

7.2. Model Categories. In [Qui67] Quillen introduced a notion of model category and called such a model category closed whenever any two of the three distinguished classes of maps (7.1) determined the third. Furthermore he characterized these closed model categories by five particularly nice axioms [Qui69] and noted that the requirement that a model category be closed is not a serious one. In fact he showed that every model category, in which (as always seems to be the case) the class of the weak equivalences is closed under retracts, can be turned into a closed model category just by closing the other two classes under retracts.

The first and the fifth of Quillen’s five axioms for a closed model category however are weaker than one would expect. The first axiom assumes the existence of finite colimits and limits, but not of arbitrary small ones and the fifth axiom assumes the existence of certain factorizations, but does not insist on their functoriality. This allows for the inclusion of various categories of finitely generated chain complexes, but also considerably complicates much of the theory.

In view of all this we therefore throughout this monograph use the term model category for a closed model category in the sense of Quillen, which satisfies the above suggested stronger versions of this first and fifth axioms.

7.3. The second and third axioms. Quillen’s five axioms, with the first and the fifth axioms strengthened as indicated above (7.2) are perfectly adequate. However, for reasons which we will explain in 7.4 and 7.5 below, we prefer the equivalent (9.3) set of axioms obtained by

(i) strengthening the second axiom which requires the weak equivalences to have the two out of three property (i.e. if f and g are maps such that gf is defined and two of f, g and gf are weak equivalences, then so is the third) by requiring them to have the two out of six property that, for every three maps r, s and t for which the two compositions sr and ts are defined and are weak equivalences, the four maps r, s, t and tsr are also weak equivalences (which by restriction to the cases in which both sr and ts are identity maps or at least one of r, s and t is an identity map, implies that all isomorphisms are weak equivalences, as well as the two out of three property), and

(ii) compensating for this by weakening the third axiom which requires that the classes of the weak equivalences, the cofibrations and the fibrations are all three closed under retracts, by omitting this requirement for the class of the weak equivalences.

These changes are suggested by the following observations on

7.4. The role of the weak equivalences. A closer look at the notion of a model category reveals that the weak equivalences already determine its “homotopy theory”, while the cofibrations and the fibrations provide additional structure which enables one to “do” homotopy theory, in the sense that, while many homotopy notions involved in doing homotopy theory can be defined in terms of the weak equivalences, the verification of many of their properties (e.g. their existence) requires the cofibrations and/or the fibrations. Moreover in dealing with model categories one often uses auxiliary categories which are not model categories, but which still have obvious “weak equivalences” induced by the weak equivalences of the model categories involved.

All this points to the desirability of having a better understanding of “categories with weak equivalences”. But when one tries to develop such a “relative” category theory, and in particular one which allows for reasonable “relative” versions of such basic categorical notions as initial and terminal objects, colimit and limit functors and universal properties, one realizes that the weak equivalences should have some closure properties and that it is convenient to restrict oneself to what we will call

7.5. Homotopical categories. A homotopical category will be a category with a single distinguished class of maps, called weak equivalences, which

(i) has the two out of six property (7.3) and (ii) contains all the identity maps

which (see 7.3(i)) implies that it also (iii) has the two out of three property (7.3), and (iv) contains all the isomorphisms.

These homotopical categories will be investigated in Part II of this monograph where we define for them such notions as homotopically initial and terminal objects and homotopy and homotopical colimit and limit functors, which can be characterized by homotopically universal properties which ensure that such notions, if they exist, are homotopically unique in the sense that they form a homotopically contractible category, i.e. a non-empty homotopical category in which all maps are weak equivalences and for which its identity functor can be connected by a (finite) zigzag of natural transformations to a constant functor, i.e. functor which sends all maps to a single identity map.

Model categories and the auxiliary categories referred to in 7.4 are all homotopical categories and the various homotopy notions one encounters when working with model categories are homotopically unique in the strong sense mentioned above. To emphasize this strong connection between model categories and homotopical categories we therefore prefer the modifications of the second and third axioms suggested in 7.3. Moreover, as the modified retract axiom no longer involves the weak equivalences, this modification of the axioms also emphasizes the fact that the weak equivalences play a different role than the cofibrations and fibrations.

7.6. The homotopy category. We also discuss the homotopy category of a model category M, i.e. the category Ho M obtained from M by “formally inverting” the weak equivalences and in particular note that

(i) this homotopy category Ho M is equivalent to the “classical homotopy

category” M_{cf}/∼ of M,

which is the quotient of the full subcategory M_{cf} ⊂ M spanned by the cofibrant fibrant objects by a homotopy relation ∼ on the maps of M which, on the maps of M_{cf} is an equivalence relation which is compatible with the composition. Moreover this category M_{cf} has the property that

(ii) a map in M_{cf} is a weak equivalence iff its image in Ho M is an isomor-

phism. The proofs of these results are rather long and technical and as several good

versions have recently been published in [DS95], [GJ99], [Hir03] and [Hov99], we refer the reader to those sources.

7.7. Homotopical comments. From the results we just mentioned in 7.6(i) and (ii) one can deduce that

(i) every model category M is saturated in the sense that a map in M is a weak equivalence iff its image in the homotopy category Ho M is an isomorphism, and

(ii) for every pair of objects X and Y in a model category M, the set of the maps X → Y ∈ Ho M can be considered as the set of the equivalence classes of the zigzags in M of the form

\xymatrix{X & · \ar[l] \ar[r] & · & Y \ar[l]}

in which the backward maps are weak equivalences, where two such zigzags are in the same class iff they are the top row and the bottom row in a commutative diagram in M of the form

\xymatrix{X & · \ar[l] \ar[r] & · & Y \ar[l] \\ X \ar[u]^{1} \ar[d]_{1} & · \ar[u] \ar[r] \ar[d] \ar[l] & · \ar[u] \ar[d] & Y \ar[l] \ar[u]_{1} \ar[d]^{1} \\ X & · \ar[l] \ar[r] & · & Y \ar[l] \\ X \ar[u]^{1} & · \ar[l] \ar[u] \ar[r] & · \ar[u] & Y \ar[l] \ar[u]_{1}}

in which the backward maps (and hence the vertical maps) are weak equivalences.

As both these statements involve only the weak equivalences and not the cofibrations and the fibrations, it should be possible to find some reasonable properties of the weak equivalences which imply these two results and we will in fact show that both these results can be deduced from the observation that (iii) M is a homotopical category (7.5), and (iv) this homotopical category admits a so-called 3-arrow calculus.

7.8. Organization of the chapter. There are four more sections. The first of these (§8) is devoted to some categorical and homotopical preliminaries. The next two sections deal with model categories (§9) and their homotopy categories (§10) , while the last section (§11) elaborates on the homotopical comments of 7.7.

8. Categorical and homotopical preliminaries

In this section we

(i) indicate what exactly we will mean by small categories, locally small categories and just plain categories, and

(ii) recall from chapter VI (in part II of this monograph) some homotopical

notions and results which will be needed in this chapter. We thus start with a brief discussion of

8.1. Categories, locally small categories and small categories. In order to avoid set theoretical difficulties one often defines categories in terms of a universe, i.e. (see §32)

(i) one assumes that every set is an element of some universe, where one defines a universe as a set U of sets (called U-sets) satisfying a few simple axioms which imply that U is closed under the usual operations of set theory and that every U-set is a subset of U, but that the set U itself and many of its subsets are not U-sets, and

(ii) one then defines a U-category as a category of which the hom-sets are U-sets and the set of objects is a subset of U and calls such a U-category small if the set of its objects is actually a U-set.

It turns out that the notion of a small U-category is indeed a convenient one, in the sense that any “reasonable” operation, when applied to small U-categories yields again a small U-category, but that the notion of a U-category is not. For instance one readily verifies that

(iii) for every two small U-categories C and D, the diagram or functor category Fun(C, D) or D^{C} (which has as objects the functors C → D and as maps the natural transformations between them) is also a small U-category, and that

(iv) for every small U-category C and subcategory W ⊂ C, there exists a localization of C with respect to W, i.e. a small U-category C[W^{−1}], together with an often suppressed localization functor γ: C → C[W^{−1}] which is 1-1 and onto on objects, sends the maps of W to isomorphisms in C[W^{−1}] and has the universal property that, for every category B and functor b: C → B which sends the maps of W to isomorphisms in B, there is a unique functor b_{γ}: C[W^{−1}] → B such that b_{γ}γ = b.

However both these statements are no longer true if one drops the requirement that the U-categories involved are small. Still one cannot avoid the notion of U-category as many of the categories one is interested in are not small. A way to get around this problem is to note that

(v) for every universe U, there exists a unique successor universe U^{+}, i.e. the unique smallest universe U^{+} such that the set U is a U^{+}-set.

Draft: May 14, 2004

This implies that every U-category is a small U^{+}-category so that any “reasonable” operation when applied to U-categories yields a small U^{+}-category.

In view of all this we choose an arbitrary but fixed universe U and use the term

small category  for small U-category, and
locally small category  for U-category

(instead of the customary use of the term category for U-category) and reserve, unless the context clearly indicates otherwise, the term

category  for small U^{+}-category.

Similarly we will usually use the term

set  for U^{+}-set, and
small set  for U-set.

We then define as follows

8.2. Homotopical categories. A homotopical category will be a category C (8.1) with a distinguished set W (8.1) of maps (called weak equivalences) such that

(i) W contains all the identity maps of C, and (ii) W has the two out of six property that, for every three maps r, s and t ∈ C for which the two composition sr and ts exist and are in W, the four maps r, s, t and tsr are also in W

which one readily verifies (by assuming that both sr and ts are identity maps or that at least one of r, s and t is an identity map) is equivalent to requiring that

(i)′ W contains all the identity maps of C,

(ii)′ W has the weak invertibility property that every map s ∈ C for which there exist maps r and t ∈ C such that the compositions sr and ts exist and are in W, is itself in W (which, together with (i)′, implies that all isomorphisms of C are in W), and

(iii)′ W has the two out of three property that, for every two maps f and g ∈ C for which gf exists and two of f, g and gf are in W, so is the third (which implies that W is actually a subcategory of C).

and two objects of C will be called weakly equivalent whenever they can be connected by a zigzag of weak equivalences. Furthermore we will refer to the category W as the homotopical structure of C and consider every subcategory C′ ⊂ C as a homotopical category with W ∩ C′ as its homotopical structure.

There is of course the corresponding notion of

8.3. Homotopical functors. Given two homotopical categories C and D, a homotopical functor C → D will be a functor C → D which preserves weak equivalences.

One can also consider not necessarily homotopical functors C → D and call a natural transformation between two such functors a natural weak equivalence if it sends the objects of C to weak equivalences in D. The homotopical structure

Draft: May 14, 2004

on D (8.2) then gives rise to a homotopical structure on the diagram or functor category (8.1(iii))

Fun(C, D)  or  D^{C}

in which the weak equivalences are the natural weak equivalences, as well as (8.2) on its full subcategory spanned by the homotopical functors which we will denote by

Fun_{w}(C, D)  or  (D^{C})_{w} .

Furthermore, given two homotopical categories C and D, we

(i) call two homotopical functors C → D naturally weakly equivalent if they can be connected by a zigzag of natural weak equivalences, and (ii) call a homotopical functor f: C → D a homotopical equivalence of homotopical categories if there exists a homotopical functor f′: D → C (called a homotopical inverse of f) such that the compositions f′f and ff′ are naturally weakly equivalent to the identity functors 1_{C} and 1_{D} respectively

and given a homotopical category C and a subcategory C_{0} ⊂ C, we

(iii) call C_{0} a left (or a right) deformation retract of C if there exists a left (or a right) deformation of C into C_{0}, i.e. a pair (r, s) consisting of a homotopical functor r: C → C_{0} and a natural weak equivalence

s: r ⟶ 1_{C} (or s: 1_{C} ⟶ r)

such that rC ∈ C_{0} for every object C ∈ C.

We end with discussing

8.4. The homotopy category of a homotopical category. Given a homotopical category C, its homotopy category Ho C will be the category obtained from C by “formally inverting” the week equivalences, i.e. the category with the same objects as C in which, for every pair of objects X, Y ∈ C, the hom-set Ho C(X, Y) consists of the equivalence classes of zigzags in C from X to Y in which the backward maps are weak equivalences, by the weakest equivalence relation which puts two zigzags in the same class when one can be obtained from the other by

(iv)′ omitting an identity map, (iv)′′ replacing two adjacent maps which go in the same direction by their

composition, or (iv)′′′ omitting two adjacent maps when they are the same but go in opposite

directions,

and this homotopy category Ho C, together with the functor γ: C → Ho C (called localization functor) which is the identity on the objects and which sends a map c: X → Y ∈ C to the class containing the zigzag which consists of the map c only, is a localization of C with respect to its subcategory of weak equivalences (8.1(iv)).

In view of the terminological assumptions make in 8.1, such homotopy categories always exist (see 33.8), although the homotopy category of a locally small homotopical category need not also be locally small.

As localizations were defined by means of a universal property (8.1(iv)), one readily verifies that

Draft: May 14, 2004

(i) for every two homotopical categories C and D with localization functors γ: C → Ho C and γ′: D → Ho D, and homotopical functor f: C → D, there is a unique functor Ho f: Ho C → Ho D such that (Ho f)γ = γ′f, and

(ii) for every two such functors f_{1}, f_{2} : C → D and natural transformations (resp. natural weak equivalences) h: f_{1} → f_{2}, there is a unique natural transformation (resp. natural isomorphism) Ho h: Ho f_{1} → Ho f_{2} such that (Ho h)γ = γ′h

which implies that

(iii) for every two homotopical functors f_{1}, f_{2} : C → D which are naturally weakly equivalent ( 8.3(i)), the functors Ho f_{1}, Ho f_{2} : Ho C → Ho D are naturally isomorphic, and

(iv) if a functor f: C → D is a homotopical equivalence of homotopical categories ( 8.3(ii)), then the functor Ho f: Ho C → Ho D is an equivalence of categories.

We end with mentioning that we will call a homotopical category C saturated when a map c ∈ C is a weak equivalence iff the map γc ∈ Ho C is an isomorphism, and noting (see 33.9(v)) that this definition readily implies that

(v) if C is a saturated homotopical category, then so are, for every homotopical category D, the homotopical functor categories (8.3)

Fun(D, C)  and  Fun_{w}(D, C) .

9. Model categories

We now

(i) define model categories, as mentioned in 7.2, as closed model categories in the sense of Quillen, but satisfying a strengthened version of his first and fifth axioms in which, as explained in 7.3, we modify his second and third axioms in a manner which does not affect the resulting notion of model category,

(ii) discuss a few immediate consequences of this definition, and (iii) describe a few rather obvious examples.

We thus start with defining

9.1. Model categories. A model category will be a locally small category M (8.1) with a model structure, i.e. three sets (8.1) of maps, weak equivalences, cofibrations and fibrations (which are often denoted by →^{∼}, ↣ and ↠ respectively) satisfying the following five axioms:

MC1 Limit axiom: The category M is cocomplete and complete, i.e. there exists, for every small category D (8.1) a D-colimit and a D-limit functor M^{D} → M on X (i.e. a left and a right adjoint of the constant diagram functor M → M^{D}).

MC2 Two out of six axiom: The weak equivalences have the two out of six property, i.e. if r, s and t are maps in M such that the two compositions sr and ts are defined and are weak equivalences, then the four maps r, s, t and tsr are also weak equivalences (8.2).

MC3 Retract axiom: The cofibrations and the fibrations are closed under retracts, i.e. if in a commutative diagram in M of the form

\xymatrix{A \ar[r]^{p} \ar[d]_{a} & X \ar[r]^{q} \ar[d]_{x} & A \ar[d]_{a} \\ A′ \ar[r]_{p′} & X′ \ar[r]_{q′} & A′}

in which qp = 1_{A} and q′p′ = 1_{A′}, the map x: X → X′ is a cofibration or a fibration, then so is the map a: A → A′.

MC4 Lifting axiom:

(i) Every cofibration has the left lifting property with respect to every trivial fibration (i.e. a fibration which is also a weak equivalence and which therefore often will be indicated by ↠^{∼}), and

(ii) every fibration has the right lifting property with respect to every trivial cofibration (i.e. a cofibration which is also a weak equivalence and which therefore often will be indicated by ↣^{∼}),

by which one means that (iii) for every commutative solid arrow diagram in M

[figure: A; f; i; X; p; B; g; k; Y]

in which i is a cofibration and p is a fibration, there exists a dotted arrow k: B → X such that ki = f and pk = g whenever at least one of i and p is a weak equivalence.

MC5 Factorization axiom: The maps f ∈ M admit functorial factorizations

(i) f = qi, where i is a cofibration and q is a trivial fibration, and (ii) f = pj, where p is a fibration and j is a trivial cofibration.

One then has

9.2. Proposition. Every model category M is a homotopical category (8.2).

Proof. It suffices to prove that, for every object X ∈ M, the identity map 1_{X}: X → X ∈ M is a weak equivalence and one does this by noting that the map 1_{X} admits (in view of MC5) a factorization 1_{X} = qi in which q is a trivial fibration, and then applying the two out of three property (8.2(iii)) to the equality 1_{X}q = q.

Next we deal with the

9.3. Comparison with Quillen’s closed model categories. To show that the above notion of a model category agrees with that of a closed model category in the sense of Quillen [Qui69] with his first and last axiom strengthened as in MC1 and MC5, one has to verify that

(i) the axioms MC1–5 imply that the weak equivalences are closed under retracts, and

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(ii) the axioms MC1, MC3, MC4 and MC5, together with the closure of the weak equivalences under retract and the two out of three property (8.2(iii)) imply axiom MC2, i.e. the two out of six property (8.2(i)).

Such a verification of (i) will be given in 9.4 below, while (ii) follows from proposition 10.6 below (which states that every model category is saturated (8.4)) and the observation that its proof, for which we refer the reader to [DS95], [GJ99],

[Hir03] or [Hov99], uses only the assumptions made in (ii).

It thus remains to prove

9.4. Proposition. The weak equivalences in a model category are closed under retracts.

Proof. We have to show that, if in the diagram of MC3 the map x is a weak equivalence

then so is the map a, and we do this by noting

(i) that, in view of MC5, the diagram of MC3 gives rise to a commutative diagram of the form (9.1)

[figure: A; p; a′; X; q; x′; A; ∼; j; B; r; A′′; p′′; a′′; ∼; X′′; q′′; x′′; ∼; A′′; ∼; a′′; A′; p′; X′; q′; A′]

in which a′′a′ = a, x′′x′ = x, q′′p′′ = 1_{A′′} and rj = a′ (and of course qp = 1_{A} and q′p′ = 1_{A′}),

(ii) that in view of the two out of three property and the assumption on the

map x, the map x′ is a weak equivalence, (iii) that therefore (MC4) there exists a map k: X′′ → B such that

rk = q′′  and  kx′ = jq

which implies that

rkp′′ = q′′p′′ = 1_{A′′}  and  kp′′a′ = kx′p = jqp = j

(iv) and that, as (9.2) the map 1_{A′′} is a weak equivalence, application of the two out of six property to the triple (a′, kp′′, r) yields that a′ and r are weak equivalences, and so is therefore, in view of the two out of three property the map a.

Using this proposition we now prove the following useful

9.5. Closure properties. Any two of the sets of the weak equivalences, the cofibrations and the fibrations in a model category (9.1) determine the third. In fact

(i) a map is a cofibration iff it has the left lifting property with respect to all trivial fibrations,

(i)′ a map is a fibration iff it has the right lifting property with respect to all trivial cofibrations, (ii) a map is a trivial cofibration iff it has the left lifting property with respect

to all fibrations, (ii)′ a map is a trivial fibration iff it has the right lifting property with respect

to all cofibrations, and (iii) a map is a weak equivalence iff it admits a factorization into a trivial

cofibration followed by a trivial fibration.

9.6. Corollary.

(i) The cofibrations and the trivial cofibrations form subcategories which are closed under pushouts, and

(ii) the fibrations and the trivial fibrations form subcategories which are closed under pullbacks.

Proof of 9.5. Let f be a map which has the left lifting property with respect to all fibrations and (MC5) factor f into a trivial cofibration j followed by a fibration p. The lifting axiom (MC4) then yields a commutative diagram of the form

[figure: ·; ∼; j; f; ·; p; ·; 1; ·]

This readily implies that f is a retract of j and hence (MC3 and 9.4) a trivial cofibration, proving (ii).

The proofs of (i), (i)′ and (ii)′ are similar, while (iii) is a consequence of the factorization axiom (MC5) and the two out of three property (MC2).

We end with a few rather straightforward

9.7. Examples.

(i) If M is a model category, then so is its opposite M^{op}, with the “opposites” of the weak equivalences as weak equivalences, the “opposites” of the fibrations as cofibrations and the “opposites” of the cofibrations as fibrations.

(ii) For every object X in a model category M, the under category (X ↓ M) and the over category (M ↓ X) inherit from M a model structure in which a map is a weak equivalence, a cofibration or a fibration iff its image in M under the forgetful functor is so.

(iii) Every product of model categories, indexed by a small set (8.1), admits a model structure in which a map is a weak equivalence, a cofibration or a fibration iff all its projections onto the given model categories are so.

(iv) Every cocomplete and complete category admits a (unique) minimal model structure in which the weak equivalences are the isomorphisms and all maps are cofibrations as well as fibrations.

(v) Every cocomplete and complete category also admits maximal model structures, i.e., model structures in which all maps are weak equivalences. In one such model structure all maps are cofibrations, while the fibrations are the isomorphisms or dually one could require all maps to be fibrations and take for the cofibrations the isomorphisms. But there might be other maximal model structures as well. For instance in the

Draft: May 14, 2004

category of small sets (8.1) one could take as cofibrations the monomorphisms and as fibrations the epimorphisms, or conversely as cofibrations the epimorphisms and as fibrations the monomorphisms.

10. The homotopy category

With every model category M one can associate its homotopy category Ho M (i.e. the category obtained from M be “formally inverting” the weak equivalences) and our main aim in this section is to obtain a more explicit description of this homotopy category by noting that

(i) the homotopy category Ho M is equivalent to the homotopy category Ho M_{cf} of the full subcategory M_{cf} ⊂ M spanned by the cofibrant fibrant objects, and

(ii) the homotopy category Ho M_{cf} is canonically isomorphic to the quotient

M_{cf}/∼ of M_{cf} by a homotopy relation ∼.

These results readily imply that

(iii) the homotopy category Ho M is, just like the category M itself, locally

small, and (iv) the maps in Ho M can be considered as equivalence classes of zigzags in

M of the form

\xymatrix{· & · \ar[l] \ar[r] & · & · \ar[l]}

in which the backward maps are weak equivalences, by an equivalence relation which can be formulated in terms of the weak equivalences.

We also note that

(v) a map in M_{cf} is a weak equivalence iff its image in M_{cf}/∼ is an isomorphism which implies that

(vi) every model category M is saturated, i.e. (8.4) a map in M is a weak

equivalence iff its image in Ho M is an isomorphism. We start with recalling from 8.4

10.1. The homotopy category. Given a model category M (9.1), its homotopy category Ho M is the homotopy category (8.4) of its underlying homotopical category (9.2), i.e. the category with the same objects as M in which, for every pair of objects X, Y ∈ M, the hom-set Ho M(X, Y) consists of equivalence classes of zigzags in M from X to Y in which the backward maps are weak equivalences. It comes with a localization functor γ: M → Ho M which is the identity on the objects and which sends a map f: X → Y ∈ M to the class containing the zigzag which consists of the map f only.

In order to obtain a more explicit description of this homotopy category of M one first considers the full subcategories of M spanned by its

10.2. Cofibrant and fibrant objects. Given a model category M, an object X ∈ M will be called cofibrant if the unique map to X from an initial object of M (which exists in view of the limit axiom (9.1)) is a cofibration, and dually an object Y ∈ M will be called fibrant if the unique map from Y to a terminal object is a fibration. Clearly (9.5)

(i) if X ∈ M is cofibrant and f: X → Y ∈ M is a cofibration, then Y is cofibrant, and dually

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(ii) if Y ∈ M is fibrant and f: X → Y ∈ M is a fibration, then X is fibrant. Let M_{c}, M_{f} and M_{cf} = M_{c} ∩ M_{f} denote the full subcategories of M spanned

respectively by the cofibrant objects, the fibrant objects and the cofibrant fibrant ones, i.e. the objects which are both cofibrant and fibrant. Then the inclusions of M_{cf}, M_{c}, M_{f} and M in each other induce (8.4) functors between their homotopy categories Ho M_{cf}, Ho M_{c}, Ho M_{f} and Ho M and (i) and (ii) above together with the factorization axiom (9.1) readily imply

10.3. Proposition. Let M be a model category (9.1). Then

(i) M_{c} (10.2) is a left deformation retract of M (i.e. ( 8.3(iii)) M_{c} is a full subcategory of M for which there exists a pair (r, s) consisting of a weak equivalence preserving functor r: M → M_{c} and a natural weak equivalence (8.3) s: r → 1_{M}), and dually

(ii) M_{f} is a right deformation retract of M, while (iii) M_{cf} is a left deformation retract of M_{f} and a right deformation retract

of M_{c}.

10.4. Corollary. The inclusions of M_{cf}, M_{c}, M_{f} and M in each other induce ( 8.3(iii), 8.4(iv) and 10.3) a commutative diagram of functors

\xymatrix{ & Ho M_{c} \ar@{-->}[dr] \\ _{Ho M_{cf}} \ar@{-->}[ur] \ar@{-->}[dr] &  & _{Ho M} \\  & _{Ho M_{f}} \ar@{-->}[ur]}

in which all functors are equivalences of categories.

To obtain a more explicit description of Ho M_{cf} (and hence of Ho M) one next considers various

10.5. Homotopy relations. Given a model category M one can consider the following homotopy relations on the maps of M:

(i) Left homotopic maps. Two maps f, g: B → X ∈ M will be called left homotopic if there exists a factorization

[figure: B ⨿ B; i_{0}⨿i_{1}; B′; ∼; B]

of the folding map ∇: B ⨿ B → B into a cofibration followed by a weak equivalence, together with a map h: B′ → X ∈ M such that hi_{0} = f and hi_{1} = g.

(ii) Right homotopic maps. Dually two maps f, g: B → X ∈ M will be called right homotopic if there exists a factorization

\xymatrix{X \ar[r]^{∼} & X′ \ar@{->>}[r]^{p_{0}×p_{1}} & X × X}

of the diagonal map Δ: X → X × X into a weak equivalence followed by a fibration, together with a map k: B → X′ ∈ M such that p_{0}k = f and p_{1}k = g.

(iii) Homotopic maps. Two maps f, g: B → X ∈ M will be called homotopic (denoted by f ∼ g) if they are both left and right homotopic and a map f: B → X ∈ M will be called a homotopy equivalence if there exists a map f′: X → B ∈ M (called a homotopy inverse of f) such that f′f ∼ 1_{B} and ff′ ∼ 1_{X}.

The usefulness of these homotopy relations is due to the fact that not only clearly

(iv) if two maps f, g: B → X ∈ M are homotopic, left homotopic or right homotopic, then (10.1) γf = γg ∈ Ho M,

but that, under suitable restrictions on B and X, these homotopy relations are equivalence relations for which the opposite of (iv) also holds. More precisely:

10.6. Proposition. Let M be a model category. Then, for every pair of objects B ∈ M_{c} and X ∈ M_{f} (10.2)

(i) the above (10.5) three homotopy relations agree on the set M(B, X) and are an equivalence relation on this set (which we will denote by ∼), and

(ii) this equivalence relation induces a map ( 10.5(iv))

M(B, X)/∼ ⟶ Ho M(B, X)

which is 1-1 and onto.

Moreover

(iii) on the subcategory M_{cf} ⊂ M, this equivalence relation is compatible with the composition and thus induces a functor

M_{cf}/∼ ⟶ Ho M_{cf}

which is an isomorphism of categories, and (iv) a map in M_{cf} is a weak equivalence iff it is a homotopy equivalence

( 10.5(iii)) or equivalently (iii) iff its image in Ho M_{cf} is an isomorphism.

In view of 9.2, 10.3, 10.4 and 10.5 this result implies the following corollaries.

10.7. Corollary. For every model category M,

(i) its homotopy category Ho M (10.1) is equivalent to its “classical homotopy category” M_{cf}/∼ ( 10.6(iii)), and hence

(ii) its homotopy category Ho M is, just like M itself (9.1), a locally small category (8.1).

10.8. Corollary. Let M be a model category. Then M is a saturated homotopical category (8.4).

10.9. Corollary. Let M be a model category and γ: M → Ho M its localization functor. Then for every pair of objects X, Y ∈ M,

(i) the function which assigns to every zigzag in M of the form

\xymatrix{X & · \ar[l]_{p}^{∼} \ar[r]^{q} & · & Y \ar[l]_{r}^{∼}}

in which the backward maps are weak equivalences, the map

(γr)^{−1}(γg)(γp)^{−1} : X ⟶ Y ∈ Ho M

induces a 1-1 correspondence between the maps X → Y ∈ Ho M and the equivalence classes of the above zigzags, in which two such zigzags are in the same class whenever they are the top row and the bottom row in a commutative diagram in M of the form

\xymatrix{X & · \ar[l]_{∼} \ar[r] & · & Y \ar[l]_{∼} \\ X \ar[u]^{1} \ar[d]_{1} & · \ar[l]_{∼} \ar[u]_{∼} \ar[r] \ar[d]^{∼} & · \ar[u]^{∼} \ar[d]_{∼} & Y \ar[l]_{∼} \ar[u]_{1} \ar[d]^{1} \\ X & · \ar[l]_{∼} \ar[r] & · & Y \ar[l]_{∼} \\ X \ar[u]^{1} & · \ar[l]_{∼} \ar[u]_{∼} \ar[r] & · \ar[u]^{∼} & Y \ar[l]_{∼} \ar[u]_{1}}

in which all the backward maps (and hence all the vertical maps) are weak equivalences, and

(ii) for every map f: X → Y ∈ M, the equivalence class corresponding to the map γf: X → Y ∈ Ho M is the class which contains the zigzag

\xymatrix{X & X \ar[l]_{1} \ar[r]^{f} & Y & Y \ar[l]_{1}}

It thus remains to prove proposition 10.6. However, as the proof of this proposition is rather long and technical and as good versions of it have recently been published in [DS95], [GJ99], [Hir03] and [Hov99], we refer the interested reader to those sources.

11. Homotopical comments

We ended the preceding section with two corollaries 10.8 and 10.9 in which we established two properties of model categories of which the formulation involved only the weak equivalences and not the cofibrations and the fibrations, and our aim in this section is to point out that there is a relatively simple property of the weak equivalences in a model category which implies these two properties.

To do this we start with recalling from 36.1 the notion of

11.1. 3-arrow calculi. Given a homotopical category C with category of weak equivalences W (8.2), C is said to admit a 3-arrow calculus {U, V } if there exist subcategories U, V ⊂ W such that in a functorial manner

(i) for every zigzag A′ ⟵^{u} A ⟶^{f} B in C which u ∈ U, there exists a zigzag

A′ ⟶^{f′} B′ ⟵^{u′} B in C with u′ ∈ U such that

u′f = f′u and

u′ is an isomorphism whenever u is

(e.g. if C is closed under pushouts and every pushout of a map in U is again in U),

Draft: May 14, 2004

(ii) for every zigzag X ⟶^{g} Y ⟵^{v} Y′ in C with v ∈ V , there exists a zigzag

X ⟵^{v′} X ⟶^{g′} Y in C with v′ ∈ V such that

gv′ = vg′ and

v′ is an isomorphism whenever v is

(e.g. if C is closed under pullbacks and every pullback of a map in V is again in V ), and

(iii) every map w ∈ W admits a factorization w = vu with u ∈ U and v ∈ V . Note that (iii) implies that U and V contain all the objects of W and hence

of C.

We then showed in 36.3

11.2. Proposition. Let C be a homotopical category (8.2) which admits a 3arrow calculus (11.1) and let γ: C → Ho C be its localization functor (8.4). Then, for every pair of objects X, Y ∈ C,

(i) the function which assigns to every zigzag in C of the form

\xymatrix{X & · \ar[l]^{∼}_{p} \ar[r]^{q} & · & Y \ar[l]^{∼}_{r}}

in which the backward maps are weak equivalences, the map

(γr)^{−1}(γq)(γp)^{−1} : X ⟶ Y ∈ Ho C

induces a 1-1 correspondence between the maps X → Y ∈ Ho C and the equivalence classes of the above zigzags, in which two such zigzags are in the same class whenever they are the top row and the bottom row in a commutative diagram in C of the form

\xymatrix{X & · \ar[l]_{∼} \ar[r] & · & Y \ar[l]_{∼} \\ X \ar[u]^{1} \ar[d]_{1} & · \ar[l]_{∼} \ar[u]_{∼} \ar[r] \ar[d]^{∼} & · \ar[u]^{∼} \ar[d]_{∼} & Y \ar[l]_{∼} \ar[u]_{1} \ar[d]^{1} \\ X & · \ar[l]_{∼} \ar[r] & · & Y \ar[l]_{∼} \\ X \ar[u]^{1} & · \ar[l]_{∼} \ar[u]_{∼} \ar[r] & · \ar[u]^{∼} & Y \ar[l]_{∼} \ar[u]_{1}}

in which all the backward maps (and hence all the vertical maps) are weak equivalences, and

(ii) for every map f: X → Y ∈ C, the equivalence class corresponding to the map γf: X → Y ∈ Ho C is the class which contains the zigzag

\xymatrix{X & X \ar[l]_{1} \ar[r]^{f} & Y & Y . \ar[l]_{1}}

Furthermore, using this result, we prove in 36.4

11.3. Proposition. Every homotopical category which admits a 3-arrow calculus is saturated (8.4).

Draft: May 14, 2004

Corollaries 10.8 and 10.9 then follow immediately from 11.3 and 11.2 and the observation that 9.1, 9.2 and 9.6 readily imply

11.4. Proposition. Every model category admits a 3-arrow calculus (11.1) consisting of the subcategories of the trivial cofibrations and the trivial fibrations.

CHAPTER III Quillen functors

12. Introduction

12.1. Summary. We now consider a useful kind of “morphism between model categories” which is not, as one might expect, a functor which is compatible with the model structures, but an adjoint pair of functors, each of which is compatible with one half of the model structures involved. Neither of these two functors (which are called left and right Quillen functors) is required to be homotopical, i.e. (8.3) to preserve (all the) weak equivalences, but they still have “homotopical meaning”: The left Quillen functor has left approximations, i.e. homotopical functors which, in a homotopical sense, are closest to it from the left, and which are homotopically unique, and the right Quillen functors have similar right approximations. Moreover these approximations have the convenient property that the functors between the homotopy categories induced by the left approximations are left adjoints of those induced by the right approximations.

In more detail:

12.2. Quillen functors. Given two model categories M and N, a Quillen adjunction between them will be an adjunction

f: M ⟷ N :f′

such that the left adjoint f (the left Quillen functor) preserves cofibrations and trivial cofibrations (9.1), while the right adjoint f′ (the right Quillen functor) preserves fibrations and trivial fibrations. Thus neither functor is required to preserve weak equivalences. However they have the crucial property that

(i) the left Quillen functor f preserves weak equivalences between cofibrant objects, i.e., f restricts to a homotopical functor (8.3) on the left deformation retract M_{c} of M (10.2 and 10.3), and dually

(ii) the right adjoint f′ preserves weak equivalences between fibrant objects, i.e. f′ restricts to a homotopical functor on the right deformation retract M_{f} of M,

which ensures the existence of

12.3. Approximations. The usefulness of Quillen functors is that they have approximations which are homotopically unique. More precisely, a left Quillen functor f: M → N has left approximations, i.e. pairs (k, a) consisting of a homotopical functor k: M → N and a natural transformation a: k → f, which in a homotopical sense are closest to f from the left. They are homotopically unique,

33 Draft: May 14, 2004

in the sense that they form a homotopical category which is homotopically contractible, i.e. for which the (unique) functor to the terminal (homotopical) category [0] (which consists of only one object and its identity map) is a homotopical equivalence of homotopical categories (8.3).

Furthermore given two composable left Quillen functors f_{1} : M → N and f_{2} : N → P, their composition f_{2}f_{1} : M → P is also a left Quillen functor and for every pair of left approximations (k_{1}, a_{1}) of f_{1} and (k_{2}, a_{2}) of f_{2}, their composition (k_{2}k_{1}, a_{2}a_{1}) in which a_{2}a_{1} is the diagonal of the commutative diagram

\xymatrix{k_{2}k_{1} \ar[r]^{a_{2}} \ar[d]_{a_{1}} & f_{2}k_{1} \ar[d]^{a_{1}} \\ k_{2}f_{1} \ar[r]^{a_{2}} & f_{2}f_{1}}

is a left approximation of f_{2}f_{1}.

Of course dually every right Quillen functor has right approximations with similar properties.

12.4. Derived adjunctions. A convenient property of the approximations of Quillen functors (12.3) is that, given a Quillen adjunction f: M ↔ N :f′, its adjunction induces, for every left approximation (k, a) of f and right approximation (k′, a′) of f′, a derived adjunction between the homotopy categories of M and N

Ho k: Ho M ⟷ Ho N :Ho k′

which is natural in (k, a) and (k′, a′) and which is compatible with compositions in the sense that, for every two composable Quillen adjunctions

f_{1}: M ⟷ N :f′_{1}  and  f_{2}: N ⟷ P :f′_{2}

the compositions of their derived adjunctions are derived adjunctions of their composition.

We end with a brief discussion of Quillen adjunctions whose derived adjunctions are equivalences of categories and which are called

12.5. Quillen equivalences. It turns out that, given a Quillen adjunction f: M ↔ N :f′, the following six statements are equivalent:

(i) for one (and hence every) left approximation (k, a) of f, the functor k: M → N is a homotopical equivalence of homotopical categories (8.3), (i)′ for one (and hence every) right approximation (k′, a′) of f′, the functor k′: N → M is a homotopical equivalence of homotopical categories, (ii) for one (and hence every) pair consisting of a left approximation (k, a) of f and a right approximation (k′, a′) of f′, the functors k: M → N and k′: N → M are homotopically inverse homotopical equivalences of homotopical categories (8.3),

(iii) for one (and hence every) left approximation (k, a) of f, the functor

Ho k: Ho M → Ho N is an equivalence of categories, (iii)′ for one (and hence every) right approximation (k′, a′) of f′, the functor

Ho k′: Ho N → Ho M is an equivalence of categories, and

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(iv) for one (and hence every) pair consisting of a left approximation (k, a) of f and a right approximation (k′, a′) of f′, the functors Ho k: Ho M → Ho N and Ho k′: Ho N → Ho M are inverse equivalences of categories.

Thus a Quillen adjunction with these properties induces a kind of “equivalence of homotopy theories” and one therefore calls the left adjoint of such a Quillen adjunction a left Quillen equivalence and the right adjoint a right Quillen equivalence.

A useful necessary and sufficient condition in order that a Quillen adjunction f : M ↔ N : f′ be an adjunction of Quillen equivalences is the Quillen condition that, for every pair of objects X ∈ M_{c} and Y ∈ N_{f} (10.2), a map X → f′Y ∈ M is a weak equivalence iff its adjunct fX → Y ∈ N is so.

12.6. Homotopical comments. Most of the results of this chapter are special cases of more general results on homotopical categories, some of which we will need in chapter IV.

(i) Left and right Quillen functors (12.2) are special cases of left and right deformable functors, i.e. not necessarily homotopical functors between homotopical categories which are homotopical on a left or a right deformation retract (8.3) of the domain category and such left and right deformable functors have left or right approximations which are homotopically unique (12.3).

(ii) Given two composable left or right deformable functors f_{1} : X → Y and f_{2} : Y → Z, their composition f_{2}f_{1} : X → Z need not again be left or right deformable, nor need the compositions of their left or right approximations be left or right approximations of their composition. However a sufficient condition for this to happen is that the pair (f_{1}, f_{2}) is what we will call left or right deformable.

(iii) Deformable functors often are part of a deformable adjunction (i.e. an adjunction f: X ↔ Y :f′ of which the left adjoint f is left deformable and the right adjoint f′ is right deformable) and the adjunction of such a deformable adjunction induces, for every pair consisting of a left approximation (k, a) of f and a right approximation (k′, a′) of f′, a derived adjunction Ho k: Ho X ↔ Ho Y :Ho k′ (8.4).

(iv) Given two composable deformable adjunctions f_{1} : X ↔ Y :f′_{1} and f_{2} : Y ↔ Z :f′_{2}, a sufficient condition in order that their composition f_{2}f_{1}: X ↔ Z :f′_{1}f′_{2} is also a deformable adjunction and that the compositions of their derived adjunctions are derived adjunctions of their composition is that

(iv)′ the pair (f_{1}, f_{2}) is left deformable (ii) and the pair (f′_{2}, f′_{1}) is right deformable,

which in particular is the case if

(iv)′′ the category X is saturated (8.4), the pair (f_{1}, f_{2}) is left deformable and the pair (f′_{2}, f′_{1}) is locally right deformable (which is a weaker requirement than being right deformable)

or dually if

(iv)′′′ the category Z is saturated, the pair (f′_{2}, f′_{1}) is right deformable and the pair (f_{1}, f_{2}) is just locally left deformable.

Draft: May 14, 2004

(v) We end with noting that, for a deformable adjunction f: X ↔ Y :f′, one can formulate a Quillen condition which implies the six statements of 12.5 and which, if the categories X and Y are saturated (8.4) is in turn implied by each of them.

12.7. Organization of the chapter. We start (in §13) with a brief review of the notion of homotopical uniqueness. Next we introduce Quillen functors (in §14), their approximations (in §15), their derived adjunctions (in §16) and Quillen equivalences (in §17), and in the last section (§18) we elaborate on the homotopical comments of 12.6.

13. Homotopical uniqueness

In this section we recall from chapter VI (§37 and §38) below some of the definitions and results on homotopical uniqueness, which are the homotopical analog of the categorical notion of uniqueness up to a unique isomorphism which we will call

13.1. Categorical uniqueness. This is the kind of uniqueness produced by universal properties and can be described as follows.

Call a category G categorically contractible if

(i) the unique functor G → [0] (where [0] denotes the category consisting of a single object 0 and its identity map) is an equivalence of categories, or equivalently

(ii) G is a non-empty groupoid in which there is exactly one isomorphism

between any two of its objects. Given a category Y and a non-empty set I, we then say that objects Y_{i} ∈ Y

(i ∈ I) are canonically isomorphic or categorically unique if

(iii) the full subcategory of Y spanned by these objects or equivalently

(iv) the categorically full subcategory o Y spanned by these objects, i.e. the full subcategory of Y spanned by these objects and all isomorphic ones

is categorically contractible. Moreover as

(v) for every category, its full subcategory spanned by its initial or terminal

objects is either empty or categorically contractible

one can often verify the categorical uniqueness of a non-empty set of objects in a category Y by noting that these objects are initial or terminal objects of Y . This is for instance what happens when one makes the somewhat imprecise statement that

(vi) some objects in a category X are unique up to a unique isomorphism because they have a certain initial (or terminal) universal property,

as what one really means is that

(vii) these objects together with some, often not explicitly mentioned, additional structure are initial (or terminal) objects in the category of all objects of X with such additional structure.

However not every categorical uniqueness results in this manner. For instance, the coproducts of an objects A with a product of objects B and C are neither initial nor terminal objects of the category of diagrams of the form

\xymatrix{A \ar[d] & B \\ · & · \ar[l] \ar[u] \ar[r] & C}

Similarly we now define

13.2. Homotopical uniqueness. We call a homotopical category G (8.2) homotopically contractible if

(i) the unique functor G → [0] (13.1(i)) is a homotopical equivalence of

homotopical categories (8.3),

which in view of the two out of three property (8.2) readily implies that

(ii) every map in G is a weak equivalence and every two objects of G are

weakly equivalent (8.2). Given a homotopical category Y and a non-empty set I, we then say that

objects Y_{i} (i ∈ I) are canonically weakly equivalent or homotopically unique if

(iii) the homotopically full subcategory of Y spanned by these objects, i.e. the full subcategory of Y spanned by these objects and all weakly equivalent (8.2) ones is homotopically contractible.

However in order to formulate the corresponding homotopically universal properties we first have to define appropriate

13.3. Homotopically initial and terminal objects. Given a homotopical category Y , an object Y ∈ Y will (motivated by 38.1) be called homotopically initial (resp. homotopically terminal) if there exists a zigzag of natural transformations

cst_{Y} ⋯ F_{0} ⟶^{f} F_{1} ⋯ 1_{Y} (resp. 1_{Y} ⋯ F_{1} ⟶^{f} F_{0} ⋯ cst_{Y})

in which

(i) cst_{Y} : Y → Y denotes the constant functor which sends every map of Y to the identity map of the object Y,

(ii) 1_{Y} : Y → Y denotes the identity functor of Y ,

(iii) the ·’s denote (possibly empty) zigzags of natural weak equivalences, and (iv) the map fY ∈ Y is a weak equivalence

which readily implies that, as one would expect

(v) for every homotopically initial or terminal object of Y its image in Ho Y

(8.4) is initial or terminal. Furthermore, using the two out of six property (8.2) one then (see 38.3 and

38.5) can prove that

(vi) if Y is a homotopically initial or terminal object of Y , then an object Y′ ∈ Y is also homotopically initial or terminal iff Y′ is weakly equivalent (8.2) to Y,

and from this it is not difficult to deduce that

Draft: May 14, 2004

(vii) for every homotopical category, its full subcategory spanned by its homotopically initial or terminal objects is either empty or homotopically contractible

which in turn implies that, as in the categorical case (13.1), one can often verify the homotopical uniqueness (13.2) of a non-empty set of objects in a category Y by noting that these objects are homotopically initial or homotopically terminal objects of Y . It is therefore sometimes convenient to make the somewhat imprecise statement that

(viii) some objects in a homotopical category X are homotopically unique because they have a certain homotopically initial (or terminal) homotopically universal property,

as a shorthand for saying that

(ix) these objects together with some, not necessarily explicitly mentioned, additional structure are homotopically initial (or terminal) objects in the category of all objects of X with such additional structure.

We end with noting that an easily verifiable and sometimes useful application of this is

13.4. Proposition. Given a homotopical category X, subcategories X_{1}, X_{2} ⊂ Xof which X_{1} (resp. X_{2}) is categorically contractible (13.1), and objects X_{1} ∈ X_{1} and X_{2} ∈ X_{2},

(i) a map X_{1} → X_{2} ∈ X is a homotopically initial (resp. terminal) object of the category (X_{1} ↓ X_{2}) (which has as objects the maps Y_{1} → Y_{2} ∈ X with Y_{1} ∈ X_{1} and Y_{2} ∈ X_{2}), iff

(ii) it is a homotopically initial (resp. terminal) object of its subcategory (X_{1} ↓ X_{2}) (resp. (X_{1} ↓ X_{2})).

14. Quillen functors

We now introduce a useful notion of “morphism between model categories” which is not, as one might expect, a functor which is compatible with the model structures in the sense that it preserves cofibrations, fibrations and weak equivalences, but an adjoint pair of functors of which the left adjoint is compatible with one half of the model structures in the sense that it preserves cofibrations and trivial cofibrations, while the right adjoint is compatible with the other half and preserves fibrations and trivial fibrations. Neither adjoint is thus required to preserve weak equivalences.

14.1. Quillen functors. Given two model categories M and N (9.1), a Quillen adjunction will be an adjunction

f: M ⟷ N :f′

of which

(i) the left adjoint is a left Quillen functor, i.e. a functor which preserves cofibrations and trivial cofibrations, and

(ii) the right adjoint is a right Quillen functor, i.e. a functor which pre-

serves fibrations and trivial fibrations.

They have the 14.2. Elementary properties.

(i) The identity functor of a model category is both a left and a right Quillen functor.

(ii) Every right adjoint of a left Quillen functor is a right Quillen functor and every left adjoint of a right Quillen functor is a left Quillen functor. (iii) The opposite of a left Quillen functor is a right Quillen functor and the

opposite of a right Quillen functor is a left Quillen functor. (iv) The composition of two left Quillen functors is a left Quillen functor and the composition of two right Quillen functors is a right Quillen functor.

Proof. Parts (i) and (iv) are obvious, part (iii) follows from 9.7(i) and part (ii) can be verified using 9.5.

14.3. Remark. Probably the most useful of the above properties is 14.2(ii) as, for an adjoint pair of functors f: M ↔ N :f′ between model categories, of the two equivalent statements

(i) f preserves cofibrations and trivial cofibrations, and

(ii) f′ preserves fibrations and trivial fibrations

one is often much easier to verify than the other.

Less obvious is the following

14.4. Key property of Quillen functors. Let f: M ↔ N :f′ be a Quillen adjunction (14.1). Then ( 8.3, 10.2 and 10.3)

(i) f is homotopical on the left deformation retract M_{c} of M, i.e. f preserves weak equivalences between cofibrant objects, and dually

(ii) f′ is homotopical on the right deformation retract N_{f} of N, i.e. f′ preserves weak equivalences between fibrant objects.

Proof. This follows readily from

14.5. Ken Brown’s lemma. Let C be a homotopical category and let M be a model category. Then every functor M → C which sends trivial cofibrations (resp. trivial fibrations) to weak equivalences in C, sends weak equivalences in M_{c} (resp. M_{f}) (10.2) to weak equivalences in C, i.e. is homotopical (8.3) on M_{c} (resp. M_{f}).

Proof (of the cofibration half). Given a weak equivalence f: A → B ∈ M_{c}, factor the map (f, 1_{B}): A ⨿ B → B ∈ M_{c} into a cofibration j: A ⨿ B ↣ C and a trivial fibration p: C↠^{∼}B. Then the composite cofibrations

[figure: A; i_{1}; A ⨿ B; j; C; and; B; i_{2}; A ⨿ B; j; C]

have the property that p(ji_{1}) = f and p(ji_{2}) = 1_{B} and therefore are, in view of the two out of three property, trivial cofibrations and the desired result now follows readily using again the two out of three property.

Draft: May 14, 2004

15. Approximations

The usefulness of left and right Quillen functors is due to the fact that they have left and right approximations, i.e. homotopically unique (13.2) homotopical functors which, in a homotopical sense, are closest to them from the left or from the right.

To define these approximations it is convenient to consider certain

15.1. Auxiliary categories. With a (not necessarily homotopical) functor f: X → Y between homotopical categories (8.2 and 8.3) one can associate the homotopical category of the homotopical functors X → Y over (resp. under) f, i.e. the homotopical category

(Fun_{w}(X, Y ) ↓ f)  (resp. (f ↓ Fun_{w}(X, Y )))

which has

(i) as objects the pairs (k, a), where k is a homotopical functor k: X → Y and a is a natural transformation

a: k ⟶ f (resp. a: f ⟶ k)

and which has (ii) as maps and weak equivalences t: (k_{1}, a_{1}) → (k_{2}, a_{2}) respectively the natural transformations and natural weak equivalences t: k_{1} → k_{2} such that

a_{2}t = a_{1} (resp. ta_{1} = a_{2}).

15.2. Approximations of Quillen functors. Given a Quillen adjunction f: M ↔ N :f′

(i) a left approximation of the left Quillen functor f will be a homotopically terminal object (13.3) of (Fun_{w}(M, N) ↓ f) (15.1), and

(ii) a right approximation of the right Quillen functor f′ will be a homo-

topically initial object of (f′ ↓ Fun_{w}(N, M)).

Thus (13.3(viii))

(iii) the left approximations of f and the right approximations of f′, if they

exist, are homotopically unique (13.2)

and it remains to verify their

15.3. Existence. For every Quillen adjunction f: M ↔ N : f′ (14.1), left deformation (r, s) of M into M_{c} (8.3 and 10.2) and right deformation (r′, s′) of N into N_{f}, the pairs

(fr, fs)  and  (f′r′, f′s′)

are respectively a left approximation of f (15.2) and a right approximation of f′.

Proof (of the left half). In view of 14.4 the pair (fr, fs) is an object of (Fun_{w}(M, N) ↓ f) and to show that this object is homotopically terminal (13.3) one notes that every object (k, a) ∈ (Fun_{w}(M, N) ↓ f) gives rise to a zigzag in (Fun_{w}(M, N) ↓ f)

\xymatrix{(k, a) & (kr, as) \ar[l]_{ks} \ar[r]^{ar} & (fr, fs)}

where as denotes the diagonal of the commutative square

\xymatrix{kr \ar[r]^{ar} \ar[d]_{ks} & fr \ar[d]^{fs} \\ k \ar[r]^{a} & f}

which has the property that (i) this zigzag is natural in (k, a), (ii) the map ks is a weak equivalence, and (iii) if (k, a) = (fr, fs), then ar = fsr is also a weak equivalence.

As one would like, approximations are compatible with compositions, and to prove this we first consider two

15.4. Composition functors. Given two composable (not necessarily homotopical) functors

f_{1} : X ⟶ Y and f_{2} : Y ⟶ Z

between homotopical categories (8.2 and 8.3), one can consider the composition functor (15.1)

(Fun_{w}(X, Y ) ↓ f_{1}) × (Fun_{w}(Y , Z) ↓ f_{2}) ⟶ (Fun_{w}(X, Z) ↓ f_{2}f_{1})

which sends a pair of objects (k_{1}, a_{1}) and (k_{2}, a_{2}) to their composition

(k_{2}, a_{2})(k_{1}, a_{1}) = (k_{2}k_{1}, a_{2}a_{1})

where a_{2}a_{1} denotes the diagonal of the commutative diagram

\xymatrix{k_{2}k_{1} \ar[r]^{a_{2}} \ar[d]_{a_{1}} & f_{2}k_{1} \ar[d]_{a_{1}} \\ k_{2}f_{1} \ar[r]^{a_{2}} & f_{2}f_{1}}

and the composition functor

(f_{1} ↓ Fun_{w}(X, Y )) × (f_{2} ↓ Fun_{w}(Y , Z)) ⟶ (f_{2}f_{1} ↓ Fun_{w}(X, Z))

which sends a pair of objects (k_{1}, a_{1}) and (k_{2}, a_{2}) to their composition

(k_{2}, a_{2})(k_{1}, a_{1}) = (k_{2}k_{1}, a_{2}a_{1})

where a_{2}a_{1} denotes the diagonal of the commutative diagram

\xymatrix{f_{2}f_{1} \ar[r]^{a_{2}} \ar[d]_{a_{1}} & k_{2}f_{1} \ar[d]_{a_{1}} \\ f_{2}k_{1} \ar[r]^{a_{2}} & k_{2}k_{1}}

and note that

(i) these composition functors are both homotopical (8.3) and hence (13.3(vi))

(ii) if they send one homotopically terminal (or initial) object to a homotopically terminal (or initial) one, then they do so for every homotopically terminal (or initial) object.

Draft: May 14, 2004

Now we can state

15.5. Proposition. Given two composable Quillen adjunctions

f_{1}: M ⟷ N :f′_{1}  and  f_{2}: N ⟷ P :f′_{2}

(i) their composition

f_{2}f_{1}: M ⟷ P :f′_{1}f′_{2}

is also a Quillen adjunction, (ii) every composition of a left approximation of f_{1} with a left approximation

of f_{2} is a left approximation of f_{2}f_{1}, and (iii) every composition of a right approximation of f′_{2} with a right approxima-

tion of f′_{1} is a right approximation of f′_{1}f′_{2}.

Proof (of the left half). Part (i) is an immediate consequence of 14.2(iv) and in view of 15.4(ii) it suffices to verify (ii) for only one choice of left deformation. To do this, let (r_{1}, s_{1}) and (r_{2}, s_{2}) be left deformations of M into M_{c} and N into N_{c} respectively (8.3 and 10.2). Then the map (15.3 and 15.4)

(f_{2}r_{2}, f_{2}s_{2})(f_{1}r_{1}, f_{1}s_{1}) ⟶^{f_{2}s_{2}f_{1}r_{1}} (f_{2}f_{1}r_{1}, f_{2}f_{1}s_{1}) ∈ (Fun_{w}(M, P) ↓ f_{2}f_{1})

is readily verified to be a weak equivalence (15.1) and hence (13.3(vi)) the composition (f_{2}r_{2}, f_{2}s_{2})(f_{1}r_{1}, f_{1}s_{1}) (15.4) is just like (f_{2}f_{1}r_{1}, f_{2}f_{1}s_{1}) a left approximation of f_{2}f_{1}.

16. Derived adjunctions

A useful property of the approximations of Quillen functors is that

(i) for every Quillen adjunction f : M ↔ N : f′, its adjunction induces, for every pair consisting of a left approximation (k, a) of f and a right approximation (k′, a′) of f′ (15.2), a derived adjunction

Ho k: Ho M ⟷ Ho N :Ho k′

which is natural in (k, a) and (k′, a′), and has the property that (ii) for every two composable Quillen adjunctions, the compositions of their derived adjunctions are derived adjunctions of their composition.

Before precisely formulating this result we first note the existence of

16.1. An induced partial adjunction. Let f: M ↔ N : f′ be a Quillen adjunction. Then, for every pair of objects X ∈ M_{c} and Y ∈ N_{f} (10.2), the adjunction isomorphism

N(fX, Y) ≈ M(X, f′Y)

is compatibly with the homotopy relation ∼ ( 10.5(ii)) and hence (10.6) induces a partial adjunction isomorphism (10.1)

Ho N(fX, Y) ≈^{ϕ} Ho M(X, f′Y)

which is natural in X and Y.

Draft: May 14, 2004

Moreover, for every two composable Quillen adjunctions f_{1}: M ↔ N :f′_{1} and f_{2} : N ↔ P :f′_{2} and pair of objects X ∈ M_{c} and Z ∈ P_{f}, the diagram of partial adjunction isomorphisms

\xymatrix{Ho P (f_{2}f_{1}X, Z) \ar[rr]^{ϕ}_{≈} \ar[dr]^{≈}_{ϕ} &  & Ho M(X, f′_{1}f′_{2}Z) \\  & Ho N(f_{1}X, f′_{2}Z) \ar[ur]^{≈}_{ϕ}}

commutes.

Proof. This follows from the observation that, in view of 10.5, the given adjunction sends right homotopic maps fX → Y ∈ N to right homotopic maps X → f′Y ∈ M and left homotopic maps X → f′Y ∈ M to left homotopic maps fX → Y ∈ N, and the second part then follows readily from the first.

We also need certain

16.2. Derived natural isomorphisms. Let f : M ↔ N : f′ be a Quillen adjunction. Then there are unique functions u and u′ which assign to every pair of left approximations (k_{1}, a_{1}) and (k_{2}, a_{2}) of f and every pair of right approximations (k′_{1}, a′_{1}) and (k′_{2}, a′_{2}) of f′, natural isomorphisms (called derived natural isomorphisms)

u(k_{1}, k_{2}): Ho k_{1} ⟶ Ho k_{2}  and  u′(k′_{2}, a′_{2}): Ho k′_{2} ⟶ Ho k′_{1}

such that (i) u and u′ are natural in both variables, and (ii) for every map

t: (k_{1}, a_{1}) ⟶ (k_{2}, a_{2}) ∈ (Fun_{w}(M, N) ↓ f)

and every map

t′: (k′_{2}, a′_{2}) ⟶ (k′_{1}, a′_{1}) ∈ (f′ ↓ Fun_{w}(N, M))

one has

u(k_{1}, k_{2}) = Ho t  and  u′(k′_{2}, k′_{1}) = Ho t′ .

Proof (of the left half). The composition

\xymatrix{(Fun_{w}(M, N) ↓ f) \ar[r]^{j} & Fun_{w}(M, N) \ar[r]^{Ho} & Fun(Ho M, Ho N)}

in which j denotes the forgetful map and Ho is as in 8.4, sends weak equivalences to isomorphisms and hence (8.4) admits a unique factorization

\xymatrix{(Fun_{w}(M, N) ↓ f) \ar[r]^{γ′} & Ho (Fun_{w}(M, N) ↓ f) \ar[r] & Fun(Ho M, Ho N)}

in which γ′ denotes the localization functor. One then readily verifies that the function which sends a pair ((k_{1}, a_{2}), (k_{2}, a_{2})) to the image in Fun(Ho M, Ho N) of the (in view of 13.3(v)) unique map

γ′(k_{1}, a_{1}) ⟶ γ′(k_{2}, a_{2}) ∈ Ho (Fun_{w}(M, N) ↓ f)

has the desired properties.

Using these derived natural isomorphisms we now implicitly define

16.3. Derived adjunctions. Given a Quillen adjunction f: M ↔ N : f′, there is a unique function which associates with every pair consisting of a left approximation (k, a) of f (15.2) and a right approximation (k′, a′) of f′, a derived adjunction

Ho k: Ho M ⟷ Ho N :Ho k′

such that

(i) for every two left approximations (k_{1}, a_{1}) and (k_{2}, a_{2}) of f and right approximations (k′_{1}, a′_{1}) and (k′_{2}, a′_{2}) of f′, the derived natural isomorphisms (16.2)

u(k_{1}, k_{2}): Ho k_{1} ⟶ Ho k_{2}  and  u′(k′_{2}, k′_{1}): Ho k′_{2} ⟶ Ho k′_{1}

are conjugate (39.10) with respect to the associated derived adjunctions

Ho k_{1}: Ho M ⟷ Ho N :Ho k′_{1}  and  Ho k_{2}: Ho M ⟷ Ho N :Ho k′_{2}

and (ii) for every left deformation (r, s) of M into M_{c} (8.3) and right deformation (r′, s′) of N into N_{f}, the adjunction isomorphism of the derived adjunction

Ho(fr): Ho M ⟷ Ho N :Ho(f′r′)

associated with (fr, fs) and (f′r′, f′s′) (15.3) assigns to every pair of objects X ∈ M and Y ∈ N the composition

Ho N(frX, Y) ≈ Ho N(frX, r′Y) ≈^{ϕ} Ho M(rX, f′r′Y) ≈ Ho M(X, f′r′Y)

in which ϕ is the partial adjunction isomorphism (16.1) and the other isomorphisms are induced by s′ and s.

16.4. Corollary. Let f: M ↔ N :f′ be a Quillen adjunction and let (k, a) and (k′, a′) respectively be a left approximation of f and a right approximation of f′. Then the functor Ho k: Ho M → Ho N is an equivalence of categories iff the functor Ho k′: Ho N → Ho M is so.

We also note that the above (16.3) derived adjunctions are compatible with

16.5. Compositions. Let

f_{1}: M ⟷ N :f′_{1}  and  f_{2}: N ⟷ P :f′_{2}

be two composable Quillen adjunctions and let

(k_{1}, a_{1}) and (k_{2}, a_{2})  and  (k′_{1}, a′_{1}) and (k′_{2}, a′_{2})

respectively be left approximations of f_{1} and f_{2} and right approximations of f′_{1} and f′_{2}. Then the composition of the associated derived adjunctions

Ho k_{1}: Ho M ⟷ Ho N :Ho k′_{1}  and  Ho k_{2}: Ho N ⟷ Ho P :Ho k′_{2}

is exactly the derived adjunction

Ho(k_{2}k_{1}): Ho M ⟷ Ho P :Ho(k′_{1}k′_{2})

associated with their compositions ( 15.4 and 15.5)

(k_{2}k_{1}, a_{2}a_{1})  and  (k′_{1}k′_{2}, a′_{1}a′_{2})

It thus remains to give a proof of 16.3 and 16.5.

16.6. Proof of 16.3. As (39.10)

(i) given an adjunction g_{1} : X ↔ Y : g′_{1}, a pair of functors g_{2} : X → Y and g′_{2} : Y → X and a pair of natural isomorphisms h: g_{1} → g_{2} and h′: g′_{2} → g′_{1}, there is a unique adjunction g_{2} : X ↔ Y :g′_{2} such that h and h′ are conjugate with respect to these two adjunctions

it suffices, in view of 16.2, to show that

(ii) for every two left deformations (r_{1}, s_{1}) and (r_{2}, s_{2}) of M into M_{c} and right deformations (r′_{1}, s′_{1}) and (r′_{2}, s′_{2}) of N into N_{f}, the derived natural isomorphisms (16.2)

u(fr_{1}, fr_{2}): Ho(fr_{1}) ⟶ Ho(fr_{2})  and  u′(f′r′_{2}, f′r′_{1}): Ho(f′r′_{2}) ⟶ Ho(f′r′_{1})

associated with the approximations (15.3)

(fr_{1}, fs_{1}) and (fr_{2}, fs_{2})  and  (f′r′_{2}, f′s′_{2}) and (f′r′_{1}, f′s′_{1})

are conjugate with respect to the derived adjunctions

Ho(fr_{1}) : Ho M ⟷ Ho N : Ho(f′r′_{1})  and  Ho(fr_{2}) : Ho M ⟷ Ho N : Ho(f′r′_{2})

associated, as in 16.3(ii), with the approximations

(fr_{1}, fs_{1}) and (f′r′_{1}, f′s′_{1})  and  (fr_{2}, fs_{2}) and (f′r′_{2}, f′s′_{2}) .

To do this one notes that the commutativity of the diagrams

\xymatrix{r_{2}r_{1} \ar[r]^{r_{2}s_{1}} \ar[dd]_{s_{2}r_{1}} & r_{2} \ar[dd]^{s_{2}} &  & 1_{N} \ar[r]^{s′_{1}} \ar[dd]_{s′_{2}} & r′_{1} \ar[dd]^{s′_{2}r′_{1}} \\  &  & and \\ r_{1} \ar[r]_{s_{1}} & 1_{M} &  & r′_{2} \ar[r]_{r′_{2}s′_{1}} & r′_{2}r′_{1}}

implies that

(r_{2}r_{1}, s_{2}s_{1})  and  (r′_{2}r′_{1}, s′_{2}s′_{1})

where s_{2}s_{1} and s′_{2}s′_{1} denote their diagonals, are a left deformation of M into M_{c} and a right deformation of N into N_{f}, from which in turn one can deduce that (15.3 and 16.2)

Ho(fr_{2}s_{1}) Ho(fs_{2}r_{1})^{−1} = u(fr_{1}, fr_{2}): Ho(fr_{1}) ⟶ Ho(fr_{2})

and

Ho(f′s′_{2}r′_{1})^{−1} Ho(f′r′_{2}s′_{1}) = u′(f′r′_{2}, f′r′_{1}): Ho(f′r′_{2}) ⟶ Ho(f′r′_{1})

Draft: May 14, 2004

and the desired result then follows from the observation that every pair of objects X ∈ M and Y ∈ N gives rise to a commutative diagram of isomorphisms

\xymatrix{Ho N(fr_{2}X, Y) ≈ ⋯ ≈^{ϕ} ⋯ ≈ Ho M(X, f′r′_{2}Y) \ar[d]_{(fr_{2}s_{1})^{∗}} \ar[d]^{(f′r′_{2}s′_{1})_{∗}} \\ Ho N(fr_{2}r_{1}X, Y) ≈ ⋯ ≈^{ϕ} ⋯ ≈ Ho M(X, f′r′_{2}r′_{1}Y) \\ Ho N(fr_{1}X, Y) ≈ ⋯ ≈^{ϕ} ⋯ ≈ Ho M(X, f′r′_{1}Y) \ar[u]_{(f′s′_{2}r′_{1})_{∗}} \ar[u]^{(fs_{2}r_{1})^{∗}}}

in which the horizontal sequences are as in 16.3(ii).

16.7. Proof of 16.5. In view of 16.3(i) it suffices to prove 16.5 for only one choice of the left and right approximations. To do this let

(r_{1}, s_{1}) and (r_{2}, s_{2})  and  (r′_{1}, s′_{1}) and (r′_{2}, s′_{2})

respectively be left approximations of M into M_{c} and N into N_{c} and right approximations of N into N_{f} and P into P_{f}. Then the desired result follows readily from the observation that, for every pair of objects X ∈ M and Z ∈ P, in view of 16.1(ii) and the naturality of the maps induced by s_{1}, s_{2}, s′_{1} and s′_{2}, the following diagram of isomorphisms commutes.

\xymatrix{Ho P (f_{2}r_{2}f_{1}r_{1}X, Z) \ar[r]^{(s′_{2})_{∗}} \ar[dd]_{(s_{2}^{∗})^{−1}} & Ho P (f_{2}r_{2}f_{1}r_{1}X, r′_{2}Z) \ar[dd]_{(s_{2}^{∗})^{−1}} \ar[dr]_{ϕ} \\  &  & Ho N(r_{2}f_{1}r_{1}X, f′_{2}r′_{2}Z) \ar[dd]_{(s_{2}^{∗})^{−1}} \\ Ho P (f_{2}f_{1}r_{1}X, Z) \ar[r]^{(s′_{2})_{∗}} & Ho P (f_{2}f_{1}r_{1}X, r′_{2}Z) \ar[dr]_{ϕ} \ar[dd]_{ϕ} \\  &  & Ho N(f_{1}r_{1}X, f′_{2}r′_{2}Z) \ar[dl]_{ϕ} \ar[dd]_{(s′_{1})_{∗}} \\ Ho M(X, f′_{1}f′_{2}r′_{2}Z) \ar[dd]_{(s′_{1})_{∗}} & Ho M(r_{1}X, f′_{1}f′_{2}r′_{2}Z) \ar[l]_{(s_{1}^{∗})^{−1}} \ar[dd]_{(s′_{1})_{∗}} \\  &  & Ho N(f_{1}r_{1}X, r′_{1}f′_{2}r′_{2}Z) \ar[dl]_{ϕ} \\ Ho M(X, f′_{1}r′_{1}f′_{2}r′_{2}Z) & Ho M(r_{1}X, f′_{1}r′_{1}f′_{2}r′_{1}Z) \ar[l]_{(s_{1}^{∗})^{−1}}}

17. Quillen equivalences

We now briefly discuss Quillen equivalences, i.e. Quillen adjunctions f: M ↔ N :f′ (14.1) with the property that, for every left approximation (k, a) of f (15.2)

Draft: May 14, 2004

and every right approximation (k′, a′) of f′, the functors

k: M ⟶ N  and  k′: N ⟶ M

are inverse homotopical equivalences of homotopical categories (8.3).

We start with noting that corollary 16.4 generalizes to

17.1. Proposition. Let f : M ↔ N : f′ be a Quillen adjunction (14.1). Then the following six statements are equivalent:

(i) for one (and hence every) left approximation (15.2) (k, a) of f and one (and hence every) right approximation (k′, a′) of f′, the homotopical functors k and k′ are homotopically inverse homotopical equivalences of categories (8.3)

(i)′ for one (and hence every) left approximation (k, a) of f, the homotopical functor k is a homotopical equivalence of homotopical categories, (i)′′ for one (and hence every) right approximation (k′, a′) of f′ the homotopical functor k′ is a homotopical equivalence of homotopical categories, (ii) for one (and hence every) left approximation (k, a) of f and one (and hence every) right approximation (k′, a′) of f′, the functors Ho k and Ho k′ are inverse equivalences of categories

(ii)′ for one (and hence every) left approximation (k, a) of f, the functor Ho k

is an equivalence of categories, and (ii)′′ for one (and hence every) right approximation (k′, a′) of f′, the functor

Ho k′ is an equivalence of categories.

A Quillen adjunction with these properties thus induces some kind of “equivalence of homotopy theories”, and such a pair is therefore called an adjunction of

17.2. Quillen equivalences. A Quillen adjunction with the six equivalent properties of (17.2) will be called an adjunction of Quillen equivalences and one refers to its left Quillen functor as a left Quillen equivalence and its right Quillen functor as a right Quillen equivalence.

A useful necessary and sufficient condition in order that a Quillen adjunction be an adjunction of Quillen equivalences is the so-called

17.3. Quillen condition. A Quillen adjunction f: M ↔ N : f′ is an adjunction of Quillen equivalences (17.2) iff it satisfies the Quillen condition that, for every pair of objects X ∈ M_{c} and Y ∈ N_{f} (10.2), a map fX → Y ∈ N is a weak equivalence iff its adjoint X → f′Y ∈ M is so.

17.4. Corollary. If for three Quillen adjunctions

f_{1}: X ⟷ Y :f′_{1},  f_{2}: Y ⟷ Z :f′_{2}  and  f_{3}: Z ⟷ T :f′_{3}

the two compositions

f_{2}f_{1}: X ⟷ Z :f′_{1}f′_{2}  and  f_{3}f_{2}: Y ⟷ T :f′_{2}f′_{3}

are adjunctions of Quillen equivalences, then so are the original three pairs.

Draft: May 14, 2004

It remains to give

Proofs of 17.1 and 17.3. In view of the fact that a functor is an equivalence of categories iff it is part of an adjunction of which the unit and the counit are natural isomorphisms, 17.1 and 17.3 follow readily from 16.4 and the following

17.5. Proposition. Let f: M ↔ N : f′ be a Quillen adjunction and let (r, s) and (r′, s′) respectively be a left deformation of M into M_{c} and a right deformation of N into N_{f} ( 8.3 and 10.2). Then the following three statements are equivalent:

(i) for every pair of objects X ∈ M_{c} and Y ∈ N_{f}, a map X → f′Y ∈ M is a weak equivalence if (resp. only if) its adjunct fX → Y ∈ N is so

(ii) the zigzag of natural transformations

\xymatrix{1_{M} & r \ar[l]_{s} \ar[r] & f′r′fr}  (resp. \xymatrix{frf′r′ \ar[r] & r′ & 1_{N} \ar[l]_{s′}} )

in which the unnamed map is the adjunct of the natural weak equivalence

\xymatrix{fr \ar[r]^{s′fr} & r′fr}  (resp. \xymatrix{rf′r′ \ar[r]^{sf′r′} & f′r′} )

is a zigzag of natural weak equivalences, and (iii) the unit (resp. counit) (39.10)

1_{Ho M} ⟶ Ho(f′r′) Ho(fr) (resp. Ho(fr) Ho(f′r′) ⟶ 1_{Ho N})

of the associated derived adjunction ( 16.3(ii))

Ho(fr): Ho M ⟷ Ho N :Ho(f′r′)

is a natural isomorphism.

Proof (of the first half). Clearly (i) implies (ii) and a straightforward calculation using 16.1, 16.3(ii) and 39.10 yields that the image of the zigzag 1_{M} ⟵^{s} r → f′r′fr in Ho X is exactly the unit 1_{Ho M} → Ho(f′r′) Ho(fr) of the derived adjunction, so that (ii) implies (iii).

It thus remains to show that (iii) implies (i) and we do this by successively noting that, if p: X → f′Y ∈ M is a map such that its adjunct q: fX → Y ∈ N is a weak equivalence, then

(i) the map q(fsX): frX → Y ∈ N is also a weak equivalence and hence its image γ′(q(fsX)) ∈ Ho N under the localization functor γ′: N → Ho N is an isomorphism,

(ii) in view of 16.1 and 16.3(ii), the adjunct of γ′(q(fsX)) is the image γ((f′s′Y)p) ∈ Ho M of the map (f′s′Y)p: X → f′r′Y ∈ M under the localization functor γ: M → Ho M,

(iii) this adjunct γ((f′s′Y)p) is the composition of the unit (which is assumed to be an isomorphism) with the image of γ′(q(fsX)) under the functor Ho(f′r′) and hence is also an isomorphism, and

(iv) in view of the saturation of M (10.8), the map (f′s′Y)′ is thus a weak equivalence and so is therefore the original map p: X → fY ∈ M.

18. Homotopical comments

In this last section we note that the main results of this chapter are special cases of more general results on homotopical categories which we will obtain in chapter VII, some of which we will need in chapter IV.

We start with observing that, in view of 14.4, Quillen functors (14.1) are

18.1. Deformable functors. Given two homotopical categories X and Y (8.2), a (not necessarily homotopical) functor f: X → Y is called left (resp. right) deformable if (40.2) f is homotopical (8.3) on a left (resp. a right) deformation retract of X (8.3).

Similarly Quillen adjunctions (14.1) are

18.2. Deformable adjunctions. Given two homotopical categories X and Y , an adjoint pair of functors f: X ↔ Y :f′ is called a deformable adjunction if (43.1) the left adjoint f is left deformable (18.1) and the right adjoint is right deformable.

Just like Quillen functors (15.2 and 15.3), deformable functors have

18.3. Approximations. Given a (not necessarily homotopical) functor f: X → Y between homotopical categories, a left (resp. a right) approximation of f is (41.1) a homotopically terminal (resp. initial) object (13.3) of the “homotopical category of homotopical functors X → Y over (resp. under) f”

(Fun_{w}(X, Y ) ↓ f)  (resp. (f ↓ Fun_{w}(X, Y )))

(which is defined as in 15.1), and thus (13.3(viii))

(i) such approximations of f, if they exist, are homotopically unique (13.2).

The existence of approximations of Quillen functors (15.2 and 15.3) then is a special case of 41.2, which is proven exactly as 15.3, and which states that

(ii) a sufficient condition in order that a (not necessarily homotopical) functor f: X → Y between homotopical categories has left (resp. right) approximations is that f be left (resp. right) deformable (18.1),

as in that case

(iii) for every left (resp. right) deformation retract X_{0} ⊂ X (8.3) on which f is homotopical and every left (resp. right) deformation (r, s) of X into X_{0}, the pair

(fr, fs) ∈ (Fun_{w}(X, Y ) ↓ f) (resp. (fr, fs) ∈ (f ↓ Fun_{w}(X, Y )))

is a left (resp. a right) approximation of f.

Next we consider

Draft: May 14, 2004

18.4. Compositions of deformable functors. Deformable functors are not as well behaved under composition as Quillen functors. For instance, the composition of two deformable functors need not be deformable, nor if it is, are the compositions of their approximations necessarily approximations of their composition. To deal with this we call, for two composable (not necessarily homotopical) functors between homotopical categories

f_{1} : X ⟶ Y  and  f_{2} : Y ⟶ Z

the pair (f_{1}, f_{2}) locally left (resp. right) deformable if (42.3) there exist left (resp. right) deformation retracts X_{0} ⊂ X and Y_{0} ⊂ Y (8.3) such that

(i) f_{1} is homotopical on X_{0} (and hence f_{1} is left (resp. right) deformable), (ii) f_{2} is homotopical on Y_{0} (and hence f_{2} is left (resp. right) deformable),

and (iii) f_{2}f_{1} is homotopical on X_{0} (and hence f_{2}f_{1} is left (resp. right) de-

formable),

and call such a pair left (resp. right) deformable if (42.3)

(iv) f_{1} sends all of X_{0} into Y_{0} (which together with (i) and (ii) implies (iii)), and note that (42.3(v)) such left (resp. right) deformability is equivalent to the requirement that

(v) there exist left (resp. right) deformations (r_{1}, s_{1}) on X and (r_{2}, s_{2}) on Y such that f_{1} and f_{2} are homotopical on the full subcategories spanned by the images of r_{1} and r_{2} respectively and the natural transformation

f_{2}s_{2}f_{1}r_{1} : f_{2}r_{2}f_{1}r_{1} ⟶ f_{2}f_{1}r_{1} (resp. f_{2}s_{2}f_{1}r_{1} : f_{2}f_{1}r_{1} ⟶ f_{2}r_{2}f_{1}r_{1})

is a natural weak equivalence

as in that case f_{2} is homotopical on the full subcategory of Y spanned by the images of r_{2} and f_{1}r_{1}.

This last result implies that the behavior of approximations of Quillen functors under composition (15.5) is a special case of 42.4, which is proven exactly as in 15.5, and which states that

(vi) if, for two composable functors

f_{1} : X ⟶ Y  and  f_{2} : Y ⟶ Z

the pair (f_{1}, f_{2}) is left (resp. right) deformable, then their composition is left (resp. right) deformable and the compositions (as defined in 15.4) of their left (resp. right) approximations are left (resp. right) approximations of their composition.

Moreover, one can often verify the deformability of two composable functors, by noting that (42.6)

(vii) if, for two composable deformable adjunctions (18.2)

f_{1}: X ⟷ Y :f′_{1}  and  f_{2}: Y ⟷ Z :f′_{2}

between saturated (8.4) homotopical categories, the pair (f_{1}, f_{2}) is locally left deformable and the pair (f′_{2}, f′_{1}) is locally right deformable, then the pair (f_{1}, f_{2}) is left deformable iff the pair (f′_{2}, f′_{1}) is right deformable.

Next we note that, just like Quillen adjunctions, deformable adjunctions (18.2) give rise to

18.5. Partial adjunctions and derived adjunctions. The existence of the induced partial adjunction of a Quillen functor (16.1) is a special case of 43.2 which states in part that

(i) given a deformable adjunction (18.2) f: X ↔ Y :f′, a left deformation retract X_{0} ⊂ X on which f is homotopical and a right deformation retract Y_{0} ⊂ Y on which f′ is homotopical, its adjunction induces a partial adjunction which to every pair of objects X ∈ X_{0} and Y ∈ Y_{0} assigns the unique partial adjunction isomorphism (8.4)

Ho Y (fX, Y) ≈ Ho X(X, f′Y)

which (ii) is natural in X and Y, and (iii) is compatible with the given adjunction in the sense that the diagram

[figure: Y (fX, Y); ≈; X(X, f′Y); Ho Y (fX, Y); ≈; Ho X(X, f′Y)]

in which the top map is the given adjunction isomorphism and the vertical maps are induced by the localization functors (8.4), commutes.

Using these partial adjunctions and the observation that (44.1)

(iv) the results of 16.2 remain valid if one replaces everywhere the Quillen adjunction f: M ↔ N :f′ by a deformable adjunction f: X ↔ Y :f′ (18.2)

one then can show, using the same arguments as were used in proving 16.3 and 16.5 that (44.2)

(v) the results of 16.3 remain valid if one replaces everywhere the Quillen adjunction f: M ↔ N :f′ by a deformable adjunction f: X ↔ Y :f′ and the deformation retracts M_{c} and N_{f} by a left deformation retract X_{0} of X on which f is homotopical and a right deformation retract Y_{0} of Y on which f′ is homotopical

and that (44.4)

(vi) the results of 16.5 remain valid if one replaces the composable Quillen adjunctions

f_{1}: M ⟷ N :f′_{1}  and  f_{2}: N ⟷ P :f′_{2}

by composable deformable adjunctions

f_{1}: X ⟷ Y :f′_{1}  and  f_{2}: Y ⟷ Z :f′_{2}

for which the pairs (f_{1}, f_{2}) and (f′_{1}, f′_{2}) are respectively left and right deformable (18.4).

We end with a comment on

18.6. The Quillen condition. A deformable adjunction (18.2) f: X ↔ Y : f′ is (45.1) said to satisfy the Quillen condition if

(i) there exist a left deformation retract X_{0} ⊂ X on which f is homotopical and a right deformation retract Y_{0} ⊂ Y on which f′ is homotopical, such that, for every pair of objects X ∈ X_{0} and Y ∈ Y_{0}, a map fX → Y ∈ Y is a weak equivalence iff its adjunct X → f′Y ∈ X is so,

which is readily verified to be equivalent to the requirement that

(ii) for every left deformation retract X_{0} ⊂ X on which f is homotopical and every right deformation retract Y_{0} ⊂ Y on which f′ is homotopical, for every pair of objects X ∈ X_{0} and Y ∈ Y_{0}, a map fX → Y ∈ Y is a weak equivalence iff its adjunct X → f′Y ∈ X is so.

The results of 17.1 and 17.3 for Quillen adjunctions then are a special case of the essentially identical results for deformable adjunctions obtained in the last sentence of 45.3 (where we assume that the categories involved are saturated (8.4)).

CHAPTER IV Homotopical cocompleteness and completeness of model categories

19. Introduction

19.1. Summary. In this chapter we consider the “homotopically correct” homotopy colimit and limit functors on a model category, i.e. the homotopy colimit and limit functor which, for all diagrams indexed by a small category (and not merely the objectwise cofibrant or fibrant ones), yield the “correct” homotopy type, and show in particular that every model category M not only (by definition) is cocomplete and complete; but also is homotopically cocomplete and complete in a sense which is considerably stronger than the requirement that there exist homotopy D-colimit and D-limit functors on M for every small category M.

In more detail (in the colimit case only):

19.2. Homotopy colimit functors. Let cat denote the category of small categories (8.1). Given a model category M we then define, for every object D ∈ cat,

(i) a homotopy D-colimit functor on M as a left approximation (18.3) of an arbitrary but fixed left adjoint

colim^{D}: M^{D} ⟶ M

of the constant diagram functor M → M^{D},

and more generally, for every map u: A → B ∈ cat,

(ii) a homotopy u-colimit functor on M as a left approximation of an arbitrary but fixed left adjoint

colim^{u}: M^{A} ⟶ M^{B}

of the induced diagram functor

u^{∗} = M^{u}: M^{B} ⟶ M^{A}

and show that

(iii) such homotopy u-colimit functors exist and are homotopically unique

(13.2), (iv) every such homotopy u-colimit functor (k, a) on M comes with a derived

adjunction (16.3 and 18.5(v))

Ho k: Ho X^{A} ⟷ X^{B} :Ho u^{∗}

and that

(v) these homotopy colimit functors on M are composable. We also note that

53 Draft: May 14, 2004

(vi) left Quillen functors are homotopically compatible with homotopy colimit functors in the sense that, for every left Quillen functor f: M → N and map u: A → B ∈ cat

(vi)′ the compositions ( 15.4 and 18.4) of a homotopy u-colimit functor on M with a left approximation (18.3) of the functor f^{B}: M^{B} → N^{B}, and (vi)′′ the compositions of a left approximation of the functor f^{A}: M^{A} → N^{A}

with a homotopy u-colimit functor on N,

are respectively left approximations of the (see 19.6(v) below) canonically (naturally) isomorphic compositions

M^{A} ⟶^{colim^{u}} M^{B} ⟶^{f^{B}} N^{B}  and  M^{A} ⟶^{f^{A}} N^{A} ⟶^{colim^{u}} N^{B}

and hence (13.4) canonically weakly equivalent (13.2).

19.3. Homotopical cocompleteness. We show that every model category M is homotopically cocomplete in a sense which is stronger than the requirement that there exist homotopy D-colimit functors on M for every object D ∈ cat (19.2) or even homotopy u-colimit functors on M for every map u: A → B ∈ cat, as it requires the existence of what we will call a homotopy colimit system on M, i.e. the existence of a function which assigns

(i) to every map u: A → B ∈ cat, a homotopy u-colimit functor (k_{u}, a_{u}) on M, and

(ii) to every composable pair of maps u: A → B and v: B → D ∈ cat, a weak equivalence from the composition of the chosen homotopy u- and v-colimit functors to the chosen homotopy vu-colimit functor, which is associative

We also note that (iii) such homotopy colimit systems on M are homotopically unique (13.2).

19.4. The proofs. The results mentioned in 19.2 and 19.3 will be obtained by combining some of the statements in §18 (the proofs of which are in Part II) with the deformability result (20.5) that

(i) there exists, for every model category M and object D ∈ cat (19.2), a left deformation retract M_{vc}^{D} ⊂ M^{D} (8.3) such that

(ii) for every map u: A → B ∈ cat, the functor colim^{u}: M^{A} → M^{B} sends objectwise weak equivalences in M_{vc}^{A} to objectwise weak equivalences in M^{B}, and

(iii) for every left Quillen functor f: M → N and object D ∈ cat, the functor f^{D}: M^{D} → N^{D} sends objectwise weak equivalences in M_{vc}^{D} to objectwise weak equivalences in N_{vc}^{D}.

The proof of this result relies heavily on the fact that, for every object D ∈ cat, the category of simplices Δ D of D and its opposite Δ^{op} D have the property that, for every model category M, the model structure of M induces a Reedy model structure on the diagram categories M^{Δ D} and M^{^{Δ^{op}} D}.

19.5. Organization of the chapter. After fixing some notation and terminology involving colimit and limit functors (in the remainder of this section) we devote two sections to homotopy colimit and limit functors (§20) and homotopical cocompleteness and completeness (§21) and then deal in the last two sections with

Draft: May 14, 2004

Reedy model structures (§22) and a proof of the deformability result (§23). In the last section (§24) we note that the main results of this chapter, i.e. those obtained in §20 and §21 are special cases of more general results on homotopical categories which we discuss in chapter VIII.

It thus remains to discuss

19.6. Colimit and limit functors. Given a category X and a category D,

(i) a D-colimit (resp. D-limit) functor on X will be a left (resp. a right) adjoint X^{D} → X of the constant diagram functor c^{∗}: D → X^{D} (which sends every object of X to the corresponding constant D-diagram) and, if such adjoints exist, we denote an arbitrary but fixed such adjoint by

colim^{D}  (resp. lim^{D})

and more generally, given a functor u: A → B

(ii) a u-colimit (resp. a u-limit) functor on X will be a left (resp. a right) adjoint X^{A} → X^{B} of the induced diagram functor

X^{u} = u^{∗}: X^{B} ⟶ X^{A}

(which sends every functor B → X to its composition with u) and, if such adjoints exist, we denote an arbitrary but fixed such adjoint by

colim^{u}  (resp. lim^{u})

and similarly

(iii) if, for two composable functors u: A → B and v: B → D there exist u-colimit (resp. u-limit) functors and v-colimit (resp. v-limit) functors (and hence vu-colimit (resp. vu-limit) functors) on X we denote by

colim^{(v,u)} : colim^{v} colim^{u} ⟶ colim^{vu} (resp. lim^{(v,u)} : lim^{vu} ⟶ lim^{v} lim^{u})

the conjugate (see 39.10) of the identity natural transformation of the functor

(vu)^{∗} = u^{∗}v^{∗}: X^{D} ⟶ X^{A} .

These definitions readily imply that colimit and limit functors are composable, i.e. that

(iv) for every composable pair of functors u: A → B and v: B → D, every composition of a u-colimit (resp. u-limit) functor on X with a v-colimit (resp. v-limit) functor on X, if these both exist, is a vu-colimit (resp. vu-limit) functor on X,

and that left and right adjoints are compatible with respectively colimit and limit functors, in the sense that

(v) for every adjunction f : X ↔ Y : g and functor u: A → B and every pair of u-colimit (resp. u-limit) functors s on X and t on Y , the compositions

f^{B}s and tf^{A}: X^{A} ⟶ Y^{B} (resp. g^{B}t and sg^{A}: Y^{A} ⟶ X^{B})

are left (resp. right) adjoints of the composition

X^{u}g^{B} = g^{A}Y^{u}: Y^{B} ⟶ X^{A} (resp. Y^{u}f^{B} = f^{A}X^{u}: X^{B} ⟶ Y^{A})

and hence canonically naturally isomorphic (13.1).

We end with some comments on the related notions of

19.7. Cocompleteness and completeness. One calls a category X cocomplete (resp. complete) if

(i) for every object D ∈ cat (the category of small categories (8.1)), there exist D-colimit (resp. D-limit) functors on X (19.6(i)),

and such cocompleteness (resp. completeness) implies that

(ii) for every map u: A → B ∈ cat, there exist u-colimit (resp. u-limit) functors on X ( 19.6(ii)),

for instance the functor which sends an object T ∈ X^{A} to the functor B → X which associates with each object B ∈ B the object (19.6)

colim^{(u↓B)} j^{∗}T ∈ X (resp. lim^{(B↓u)} j^{∗}T ∈ X)

where j denotes the forgetful functor

j: (u ↓ B) ⟶ A (resp. j: (B ↓ u) ⟶ A) .

However, in spite of 19.6(iv), it is in general not possible to find for each map u ∈ cat a u-colimit (resp. a u-limit) functor on X such that these functors form a diagram indexed by the category cat, but only what we will call a colimit (resp. a limit) system on X, and by which we mean a function F which assigns

(iii) to every object D ∈ cat the diagram category X^{D}, (iv) to every map u: A → B ∈ cat, a u-colimit (resp. u-limit) functor

F u: F A → F B, and

(v) to every composable pair of maps u: A → B and v: B → D ∈ cat the conjugate (see 39.10)

F (v, u): (F v)(F u) ⟶ F (vu) (resp. F (v, u): F (vu) ⟶ (F v)(F u))

of the identity natural transformation

(vu)^{∗} = u^{∗}v^{∗}: F D ⟶ F A .

Clearly

(vi) such a colimit (resp. limit) system on X exists iff X is cocomplete (resp. complete).

In fact in that case a for our purposes convenient example of such a colimit (resp. limit) system on X is the function

colim^{(cat)}  (resp. lim^{(cat)})

which assigns to every map u: A → B ∈ cat the u-colimit (resp. u-limit) functor (19.6(ii))

colim^{(cat)} u = colim^{u} (resp. lim^{(cat)} u = lim^{u})

and to every composable pair of maps u: A → B and v: B → D ∈ cat, the natural isomorphism (19.6(iii))

colim^{(cat)}(v, u) = colim^{(v,u)} (resp. lim^{(cat)}(v, u) = lim^{(v,u)}) .

20. HOMOTOPY COLIMIT AND LIMIT FUNCTORS 57

20. Homotopy colimit and limit functors

Given a model category M, we now

(i) define, for every functor u: A → B between small categories (8.1) homotopy u-colimit and u-limit functors on M,

(ii) prove their existence, homotopical uniqueness (13.2), composability and

homotopical compatibility with Quillen functors, and (iii) note that they give rise to derived adjunctions which are compatible with

composition. We thus start with defining

20.1. Homotopy colimit and limit functors. Given a model category M and an object D ∈ cat (the category of small categories (8.1)),

(i) a homotopy D-colimit (resp. D-limit) functor on M will be a left (resp. a right) approximation (18.3) of the functor colim^{D}: M^{D} → M (resp. lim^{D}: M^{D} → M) (19.6(i)),

and more generally, given a map u: A → B ∈ cat,

(ii) a homotopy u-colimit (resp. u-limit) functor on M will be a left (resp. a right) approximation of the functor colim^{u}: M^{A} → M^{B} (resp. lim^{u}: M^{A} → M^{B}) (19.6(ii))

and hence (18.3(i))

(iii) such homotopy colimit (resp. limit) functors on M, if they exist, are

homotopically unique (13.2).

Moreover

20.2. Existence. For every model category M and map u: A → B ∈ cat (i) there exist homotopy u-colimit (resp. u-limit) functors on M, and (ii) every homotopy u-colimit (resp. u-limit) functor (k, a) on M comes with

a derived adjunction Ho k: Ho M^{A} ⟷ Ho M^{B} :Ho u^{∗} (resp. Ho u^{∗}: Ho M^{B} ⟷ Ho M^{A} :Ho k)

associated ( 16.3 and 18.5(v)) with the pair of approximations

((k, a), (u^{∗}, 1_{u_{∗}})) (resp. ((u^{∗}, 1_{u^{∗}}), (k, a))) which has as its counit (resp. unit) the natural transformation

Ho(e(au^{∗})): (Ho k)(Ho u^{∗}) ⟶ 1_{Ho}_{M}^{B} (resp. Ho((au^{∗})e): 1_{Ho}_{M}^{B} ⟶ (Ho k)(Ho u^{∗}))

where

e: colim^{u} u^{∗} ⟶ 1_{M}^{B} (resp. e: 1_{M}^{B} ⟶ lim^{u} u^{∗})

denotes the counit (resp. the unit) of the adjunction

colim^{u}: M^{A} ⟷ M^{B} :u^{∗} (resp. u^{∗}: M^{B} ⟷ M^{A} :lim^{u}) .

Draft: May 14, 2004

20.3. Composability. For every model category M, composable pair of maps u: A → B and v: B → D ∈ cat and homotopy u-colimit (resp. u-limit) functor (k_{u}, a_{u}) and homotopy v-colimit (resp. v-limit) functor (k_{v}, a_{v}) on M,

(i) their “composition”

(k_{v}k_{u}, colim^{(v,u)} a_{v}a_{u}) (resp. (k_{v}k_{u}, a_{v}a_{u} lim^{(v,u)}))

in which colim^{(v,u)} (resp. lim^{(v,u)}) is as in 19.6(ii) and a_{v}a_{u} denotes the diagonal of the commutative diagram

\xymatrix{k_{v}k_{u} \ar[r]^{a_{v}} \ar[d]_{a_{u}} & colim^{v} k_{u} \ar[d]^{a_{u}} \\ k_{v} colim^{u} \ar[r]_{a_{v}} & _{colim_{v} colim_{u}}}  (resp. \xymatrix{lim^{v} lim^{u} \ar[r]^{a_{v}} \ar[d]_{a_{u}} & k_{v} lim^{u} \ar[d]^{a_{u}} \\ lim^{v} k_{u} \ar[r]_{a_{v}} & k_{v}k_{u}} )

is a homotopy vu-colimit (resp. vu-limit) functor on M, and (ii) the composition of their derived adjunctions (20.2) is the derived adjunc-

tion of their “composition”.

We also note their compatibility with Quillen functors, i.e.

20.4. Homotopical compatibility with Quillen functors. For every Quillen adjunction f: M ↔ N :g (14.1) and map u: A → B ∈ cat, the left Quillen functor f is homotopically compatible with the homotopy u-colimit functors and right Quillen functor g is homotopically compatible with the homotopy u-limit functors on M and N in the sense that

(i) the compositions of a homotopy u-colimit functor on M with a left approximation (18.3) of f^{B} and of a left approximation of f^{A} with a homotopy u-colimit functor on N are respectively left approximations of the canonically (naturally) isomorphic ( 19.6(v)) compositions

M^{A} ⟶^{colim^{u}} M^{B} ⟶^{f^{B}} N^{B}  and  M^{A} ⟶^{f^{A}} N^{A} ⟶^{colim^{u}} N^{B}

and hence are all canonically weakly equivalent (13.4), and dually (ii) the compositions of a homotopy u-limit functor on N with a right ap-

proximation of g^{B} and of a right approximation of g^{A} with a homotopy u-limit functor on M are respectively right approximations of the canonically (naturally) isomorphic compositions

N^{A} ⟶^{lim^{u}} N^{B} ⟶^{g^{B}} M^{B}  and  N^{A} ⟶^{g^{A}} M^{A} ⟶^{lim^{u}} M^{B}

and hence are all canonically weakly equivalent.

The proof of these three propositions will use the following result which we will prove in 23.4.

20.5. Deformability result. In 23.3 below we define, for every model category M and object D ∈ cat, a full subcategory

(M^{D})_{vc} ⊂ M^{D} (resp. (M^{D})_{vf} ⊂ M^{D})

and then show in 23.4 below that

(i) for every model category M and object D ∈ cat, (M^{D})_{vc} (resp. (M^{D})_{vf}) is a left (resp. a right) deformation retract (8.3) of M^{D},

Draft: May 14, 2004

(ii) for every Quillen adjunction f: M ↔ N :g (14.1) and object D ∈ cat, the functor

f^{D}: M^{D} ⟶ N^{D} (resp. g^{D}: N^{D} ⟶ M^{D})

sends objectwise weak equivalences in (M^{D})_{vc} (resp. (N^{D})_{vf}) to objectwise weak equivalences in (N^{D})_{vc} (resp. (M^{D})_{vf}), and

(iii) for every model category M and map u: A → B ∈ cat, the functor (19.6)

colim^{u}: M^{A} ⟶ M^{B} (resp. lim^{u}: M^{A} ⟶ M^{B})

sends objectwise weak equivalences in (M^{A})_{vc} (resp. (M^{A})_{vf}) to objectwise weak equivalences in M_{c}^{B} (resp. M_{f}^{B}) (10.2).

20.6. Proofs (of the colimit halves) of 20.2, 20.3 and 20.4. The first part of 20.2 follows from 18.3(ii) and the fact that, in view of 20.5(i) and (iii), the functor colim^{u} is left deformable (18.1), the first part of 20.3 is a consequence of 18.4, the saturation of M and hence of M^{A} and M^{B} (8.4 and 10.2), the right deformability (18.4) of the pair (M^{v}, M^{u}) and, in view of 20.5(iii), the local left deformability of the pair (colim^{u}, colim^{v}), and if (r, s) is a left deformation (8.2) of M^{A} into M_{vc}^{A} (20.5), then the resulting commutative diagram

\xymatrix{(kr)u^{∗} \ar[r]^{(ks)u^{∗}} \ar[d]_{(ar)u^{∗}} & ku^{∗} \ar[dr]^{e(au^{∗})} \ar[d]_{au^{∗}} \\ (colim^{u} r)u^{∗} \ar[r]^{(colim^{u} s)u^{∗}} & colim^{u} u^{∗} \ar[r]^{e} & 1_{M}^{B}}

gives rise to the commutative diagram

\xymatrix{(Ho kr)(Ho u^{∗}) \ar[r] \ar[d] & (Ho k)(Ho u^{∗}) \ar[d]^{Ho(e(au^{∗}))} \\ (Ho colim^{u} r)(Ho u^{∗}) \ar[r] & 1_{Ho}_{M}^{B}}

and the second part of 20.2 now follows from the observation that, in view of 16.3 and 18.5(v), the natural transformation (Ho k)(Ho u^{∗}) → 1_{Ho}_{M}^{B} obtained by going counter clockwise around this diagram is exactly the desired counit.

To prove the second part of 20.3 one notes that in the commutative diagram

\xymatrix{k_{v}k_{u}u^{∗}v^{∗} \ar[r]^{a_{u}} \ar[dd]_{1} & k_{v} colim^{u} u^{∗}v^{∗} \ar[r]^{e_{u}} \ar[d]_{a_{v}} & k_{v}v^{∗} \ar[r]^{a_{v}} \ar[d]_{a_{v}} & colim^{v} v^{∗} \ar[r]^{e_{v}} & 1_{M}^{B} \ar[d]^{1} \\  & colim^{v} colim^{u} u^{∗}v^{∗} \ar[r]^{e_{u}} \ar[d]_{_{colim_{(v,u)}}} & colim^{v} v^{∗} \ar[rr]^{e_{v}} &  & 1_{M}^{B} \ar[d]^{1} \\ k_{v}k_{u}(vu)^{∗} \ar[r] & colim^{vu}(vu)^{∗} \ar[rrr]^{e_{vu}} &  &  & 1_{M}^{B}}

in which e_{u}, e_{v} and e_{vu} are (induced by) the relevant counits, the top row, in view of 16.5 and 18.5(vi), induces the counit of the adjunction associated with the composition, in the sense of 15.4, of (k_{u}, a_{u}) and (k_{v}, a_{v}), while the bottom row,

Draft: May 14, 2004

in view of the first half of the proposition, induces the counit of the adjunction associated with their “composition”, in the sense of 20.3(i).

21. Homotopical cocompleteness and completeness

In this section we show that every model category M is homotopically cocomplete and complete in the sense that there exist homotopy colimit and limit systems on M, which we define as a kind of left and right approximation of the colimit and limit systems colim^{(cat)} and lim^{(cat)} (19.7).

To do this it is convenient to first introduce the notions of

21.1. Left and right cat-systems. Given a model category M, a left (resp. a right) cat-system on M will be a function F which assigns

(i) to every object D ∈ cat the diagram category F D = M^{D}, (ii) to every map u: A → B ∈ cat a (not necessarily homotopical) functor

fu: F A → F B, and (iii) to every composable pair of maps u: A → B and v: B → D ∈ cat, a

natural weak equivalence (called composer)

F (v, u): (F v)(F u) ⟶ F (vu) (resp. F (v, u): F (vu) ⟶ (F v)(F u))

which is associative in the sense that

(iv) for every three composable maps u: A → B, v: B → D and x: D → E ∈ cat, the following diagram commutes

[figure: (F x)(F v)(F u); (F x)F (v,u); F (x,v)(F u); (F x)F (vu); F (x,vu); F (xv)(F u); F (xv,u); F (xvu); (F x)F (vu); (F x)F (v,u); (F x)(F v)(F u); F (xvu); F (x,vu); (resp.; F (xv,u); F (xv)(F u); F (x,v)(F u); )]

and such a cat-system will be called homotopical if, for every map u: A → B ∈ cat, the functor F u: F A → F B is homotopical.

Furthermore a map h: F → G between two such left (resp. right) cat-systems F and G will be a function h which assigns to every map u: A → B ∈ cat a natural transformation hu: F u → Gu, which commutes with the composers in the sense that

(v) for every composable pair of maps u: A → B and v: B → D ∈ cat, the following diagram commutes:

[figure: (F v)(F u); F (v,u); (hv)(hu); F (vu); h(vu); (Gv)(Gu); G(v,u); G(vu); F (vu); F (v,u); (resp.; h(vu); (F v)(F u); (hv)(hu); ); G(vu); G(v,u); (Gv)(Gu)]

and such a map will be called a weak equivalence whenever, for every map u ∈ cat, the natural transformation hu is a natural weak equivalence.

We denote the resulting homotopical category of left (resp. right) cat-systems by

cat_{L}-syst  (resp. cat_{R}-syst)

Draft: May 14, 2004

and its full subcategory spanned by the homotopical cat-systems (iv) by

cat_{L}-syst_{w}  (resp. cat_{R}-syst_{w}) .

21.2. Example. For every model category M, the colimit (resp. the limit) systems on M (19.7) are left (resp. right) cat-systems on M, and so is in particular the colimit system colim^{(cat)} (resp. the limit system lim^{(cat)}) (19.7).

Now we can define

21.3. Homotopy colimit and limit systems. Given a model category M, a homotopy colimit (resp. limit) system on M will be a pair (K, a) consisting of an object (21.1)

K ∈ cat_{L}-syst_{w} (resp. K ∈ cat_{R}-syst_{w})

and a map (21.1 and 21.2)

a: K ⟶ colim^{(cat)} ∈ cat_{L}-syst (resp. a: lim^{(cat)} ⟶ K ∈ cat_{R}-syst)

such that

(i) for every map u ∈ cat, the pair (Ku, au) is a homotopy u-colimit (resp. u-limit) functor on M (20.1), and

(ii) (K, a) is a homotopically terminal (resp. initial) object of the “homotopical category of homotopical left (resp. right) cat-systems on M over colim^{(cat)} (resp. under lim^{(cat)})

(cat_{L}-syst_{w} ↓ colim^{(cat)}) (resp. (lim^{(cat)} ↓ cat_{R}-syst_{w}))

which has as objects the above pairs (K, a) and, for every two such pairs (K_{1}, a_{1}) and (K_{2}, a_{2}) as maps and weak equivalences (K_{1}, a_{1}) → (K_{2}, a_{2}) the maps and weak equivalences t: K_{1} → K_{2} (21.1) such that a_{2}t = a_{1} (resp. ta_{1} = a_{2}).

It then follows from 13.2(ii) and 13.3(vii) that (iii) if one object of

(cat_{L}-syst_{w} ↓ colim^{(cat)}) (resp. (lim^{(cat)} ↓ cat_{R}-syst_{w}))

satisfies both (i) and (ii), then every object which satisfies one of (i) and (ii) also satisfies the other

and hence (13.3(vii))

(iv) the homotopy colimit (resp. limit) systems on M, if they exist, are ho-

motopically unique (13.2). It thus remains to prove the promised

21.4. Homotopical cocompleteness and completeness of model categories. Let M be a model category. Then

(i) M is homotopically cocomplete and complete in the sense that (ii) there exist homotopy colimit and limit systems on M (21.3).

Moreover 21.3(iii)

Draft: May 14, 2004

(iii) every object of

(cat_{L}-syst_{w} ↓ colim^{(cat)}) (resp. (lim^{(cat)} ↓ cat_{R}-syst_{w}))

which satisfies either 21.3(i) or 21.3(ii) satisfies both and hence is such a homotopy colimit (resp. limit) system.

Proof (of the colimit half). Let (r, s) be a pair of functions which assign to every object D ∈ cat a left deformation (r_{D}, s_{D}) of M^{D} into M_{vc}^{D} (8.3 and 20.5(i)), and for every left cat-system K on M (21.1), let Kr denote the function which assigns

(i) to every map u: A → B ∈ cat, the functor

(Kr)u = (Ku)r_{A}: M^{A} ⟶ M^{B}

and

(ii) to every composable pair of maps u: A → B and v: B → D ∈ cat, the composition (Kr)(v, u) of the natural transformation

(Kr)v(Kr)u = (Kv)r_{B}(Ku)r_{A} ⟶^{(Kv)s_{B}(Ku)r_{A}} (Kv)(Ku)r_{A}

with the natural weak equivalence

(Kv)(Ku)r_{A} ⟶^{K(v,u)r_{A}} K(vu)r_{A} = (Kr)(vu)

and let Ks denote the function which assigns (iii) to every map u: A → B ∈ cat, the natural transformation

(Ks)u = (Ku)s_{A}: (Ku)r_{A} ⟶ Ku .

Clearly (iv) if K is a homotopical left cat-system (21.1), then so is Kr and the func-

tion Ks is a weak equivalence Kr → K ∈ cat_{L}-syst_{w} (21.1).

Furthermore the saturation of M, together with 18.3(iii), 18.4 and 20.5(iii) and the fact that, for every map u: A → B ∈ cat, the functor M^{u}: M^{B} → M^{A} is homotopical, implies that

(v) F r is a homotopical left cat-system and the function F s is a map F r →

F ∈ cat_{L}-syst.

In view of the naturality in K of the functions Kr and Ks, it therefore follows that

(vi) every object (K, a) ∈ (cat_{L}-syst_{w} ↓ colim^{(cat)}) ( 21.2 and 21.3) gives rise to a zigzag in (cat_{L}-syst_{w} ↓ colim^{(cat)})

(K, a) ⟵^{Ks} (Kr, as) ⟶^{ar} (colim^{(cat)} r, colim^{(cat)} s)

in which as denotes the diagonal of the commutative diagram

\xymatrix{Kr \ar[r]^{ar} \ar[d]_{Ks} & colim^{(cat)} r \ar[d]^{colim^{(cat)} s} \\ K \ar[r]^{a} & _{colim_{(cat)}}}

which is natural in (K, a) and in which ar: Kr → colim^{(cat)} r is the map given by (ar)u = (au)r_{A} for every map u: A → B ∈ cat and 21.4(ii) now is a consequence of 13.3, 21.3 and the observation that, as for every map u: A → B ∈ cat the natural transformation

((colim^{(cat)} s)r)u = (colim^{(cat)} s)ur_{A} = colim^{u} s_{A}r_{A}

is a natural weak equivalence, the map

(colim^{(cat)} s)r: (colim^{(cat)} r)r ⟶ colim^{(cat)} r ∈ cat_{L}-syst_{w}

is a weak equivalence.

22. Reedy model categories

In preparation for the next section, where we will give a proof of 20.5, we

(i) review the notion of a Reedy category and, for every Reedy category B and model category M, the resulting Reedy model structure on the diagram category M^{B} [Hir03] and [Hov99], and

(ii) discuss briefly the rather useful special case of a Reedy category with fibrant or cofibrant constants, i.e. [Hir03] a Reedy category B with the property that, for every model category M the adjoint pair of functors (19.6)

colim^{B}: M^{B} ⟷ M :c^{∗}  or  c^{∗}: M ⟷ M^{B} :lim^{B}

are the left and right adjoints of a Quillen adjunction. We thus start with

22.1. Reedy category. A Reedy category consists of a small category B (8.1), together with two subcategories \overrightarrow{B} and \overleftarrow{B} such that

(i) there exists a degree function which assigns to every object B ∈ B a non-negative integer deg B such that all non-identity maps of \overrightarrow{B} raise the degree and all non-identity maps of \overleftarrow{B} lower it, and

(ii) every map b ∈ B admits a functorial factorization b = \overrightarrow{b}\overleftarrow{b} with \overrightarrow{b} ∈ \overrightarrow{B}

and \overleftarrow{b} ∈ \overleftarrow{B}. Such Reedy categories give rise to

22.2. Reedy model structures. For every model category M and Reedy category B (22.1) the Reedy model structure on the diagram category M^{B} will be the model structure in which the weak equivalences, the cofibrations and the fibrations (which will sometimes be referred to as Reedy weak equivalences, Reedy cofibrations and Reedy fibrations) are as in the last part of the following

22.3. Proposition. For every model category M and Reedy category B (22.1),

(i) the diagram category M\overrightarrow{B} admits a model structure in which the weak equivalences and the fibrations are the objectwise ones,

(ii) the diagram category M\overleftarrow{B} admits a model structure in which the weak

equivalences and the cofibrations are the objectwise ones, and (iii) the diagram category M^{B} admits a model structure in which a map is a

weak equivalence, a cofibration or a fibration iff its restriction to both \overrightarrow{B} and \overleftarrow{B} is so in the sense of (i) and (ii).

22.4. Corollary. If f : M ↔ N : g is a Quillen adjunction (14.1), then so is, for every Reedy category B, the induced adjunction

f^{B}: M^{B} ⟷ N^{B} :g^{B} .

To give a more explicit description (than 22.3) of the (trivial) cofibrations and the (trivial) fibrations in these Reedy model structures it is convenient to first introduce the notions of

22.5. Latching and matching objects. Given a model category M, a Reedy category B and an object X ∈ M^{B} one can, for every object B ∈ B consider the maximal subcategories

∂(\overrightarrow{B} ↓ B) ⊂ (\overrightarrow{B} ↓ B)  and  ∂(B ↓ \overleftarrow{B}) ⊂ (B ↓ \overleftarrow{B})

which do not contain the identity maps 1_{B} of B and define the latching object LXB and the matching object MXB of X at B as the object (19.6)

LXB = colim^{∂(\overrightarrow{B}↓B)} j^{∗}X  and  MXB = lim^{∂(B↓\overleftarrow{B})} j^{∗}X

where j denotes the forgetful functor.

In terms of these latching and matching objects one then can formulate the following

22.6. Explicit description of the Reedy model structure. Let M be a model category and B a Reedy category (22.1). In the Reedy model structure on M^{B} (22.3) a map f: X → Y ∈ M^{B}

(i) is a weak equivalence iff it is an objectwise weak equivalence, (ii) is a (trivial) cofibration if, for every object B ∈ B, the induced map

(22.5)

XB ⨿_{LXB} LY B ⟶ Y B ∈ M

is a (trivial) cofibration, and (iii) is a (trivial) fibration iff, for every object B ∈ B, the induced map

XB ⟶ Y B Π_{MY B} MXB ∈ M

is a (trivial) fibration.

For a proof of 22.3 and 22.6 we refer the reader to [Hir03] and [Hov99].

A rather useful kind of Reedy category are the so-called

22.7. Categories with fibrant or cofibrant constants. A category with fibrant (resp. cofibrant) constants will be a Reedy category B such that, for every model category M, the adjunction (19.6)

colim^{B}: M^{B} ⟷ M :c^{∗} (resp. c^{∗}: M ⟷ M^{B} :lim^{B})

is a Quillen adjunction (14.1).

A necessary and sufficient condition for this to happen is provided by

22.8. Proposition. A Reedy category B has fibrant (resp. cofibrant) constants (22.7) iff the subcategory \overleftarrow{B} (resp. \overrightarrow{B}) is a disjoint union of categories with a terminal (resp. an initial) object.

Draft: May 14, 2004

Proof (of the cofibrant half). The proof of the “if” part is straightforward and if one endows the category set of small sets (8.1) with the model structure (9.7(v)) in which the cofibrations are the monomorphisms and the fibrations are the epimorphisms, then the requirement that the diagram c^{∗}[0] ∈ set^{B} (13.1) be Reedy cofibrant readily implies the “only if” part.

22.9. Corollary. If M is a model category, B a Reedy category with fibrant (resp. cofibrant) constants (22.7) which has an initial (resp. a terminal) object B_{0}, and X ∈ M^{B} is a Reedy cofibrant (resp. fibrant) diagram in which all maps are weak equivalences, then the induced map

XB_{0} ⟶ colim^{B} X ∈ M (resp. lim^{B} X ⟶ XB_{0} ∈ M)

is also a weak equivalence.

Important examples of categories with fibrant or cofibrant constants are the

22.10. Categories of simplices and their opposites. For every integer n ≥ 0 let [n] denote the category which has as objects the integers 0, …, n and which has exactly one map i → j whenever i ≤ j. For every small category D (8.1) its category of simplices Δ D then will be the category which has as objects the functors [n] → D (n ≥ 0) and which has as maps

(E_{1} : [n_{1}] → D) ⟶ (E_{2} : [n_{2}] → D)

the commutative triangles of the form

\xymatrix{[n_{1}] \ar[rr] \ar[dr]_{E_{1}} &  & [n_{2}] \ar[dl]^{E_{2}} \\  & D}

The opposite of Δ D will be denoted by Δ^{op} D.

Clearly

(i) Δ D and Δ^{op} D are Reedy categories with fibrant and cofibrant constants respectively (for which the subcategories

\overrightarrow{Δ D} ⊂ Δ D  and  \overleftarrow{Δ^{op} D} ⊂ Δ^{op} D

consist of the above triangles in which the horizontal map is 1-1 and the subcategories

\overleftarrow{Δ D} ⊂ Δ D  and  \overrightarrow{Δ^{op} D} ⊂ Δ^{op} D

consist of those in which this map is onto)

and hence ( 14.4 and 22.7)

(ii) for every model category M, the functor

colim^{Δ D}: M^{Δ D} ⟶ M (resp. lim^{^{Δ^{op}} D}: M^{^{Δ^{op}} D} ⟶ M)

sends Reedy weak equivalences between Reedy cofibrant (resp. fibrant) diagrams to weak equivalences between cofibrant (resp. fibrant) objects. Moreover one also readily verifies that, in view of the fact that left (resp. right)

adjoints preserve colimits (resp. limits)

Draft: May 14, 2004

(iii) every functor u: A → B between small categories gives rise to Quillen adjunctions (14.1)

(Δ u)^{∗}: M^{Δ B} ⟷ M^{Δ A} :lim^{Δ u}

(resp. colim^{Δ^{op} u}: M^{^{Δ^{op}} A} ⟷ M^{^{Δ^{op}} B} :(Δ^{op} u)^{∗}).

We end with generalizing (ii) to a similar result for colimit and limit functors involving the following

22.11. Projection functors. Given a small category D, it terminal projection functor p_{t} and its initial projection functor p_{i} will be the functors

p_{t} : Δ D ⟶ D  and  p_{i} : Δ^{op} D ⟶ D

which send an object E: [n] → D (22.7) to the objects En and E0 ∈ D respectively. A straightforward calculation then yields that

(i) for every object D ∈ D,

(p_{t} ↓ D) = Δ(D ↓ D)  and  (D ↓ p_{i}) = Δ^{op}(D ↓ D)

and more generally that

(ii) for every functor u: A → B between small categories and object B ∈ B,

(up_{t} ↓ B) = Δ(u ↓ B)  and  (B ↓ up_{i}) = Δ^{op}(B ↓ u)

and these two observations, together with 19.7(ii) and 22.10(ii) and (iii) then readily imply

22.12. Proposition. Let M be a model category. Then, for every small category D, the functor (19.6)

colim^{p_{t}} : M^{Δ D} ⟶ M^{D} (resp. lim^{p_{i}} : M^{^{Δ^{op}} D} ⟶ M^{D})

and more generally, for every functor u: A → B between small categories, the functor

colim^{up_{t}} : M^{Δ A} ⟶ M^{B} (resp. lim^{up_{i}} : M^{^{Δ^{op}} A} ⟶ M^{B})

sends Reedy weak equivalences between Reedy cofibrant (resp. fibrant) diagrams to objectwise weak equivalences between objectwise cofibrant (resp. fibrant) diagrams.

23. Virtually cofibrant and fibrant diagrams

Given a model category M and a small category D we now introduce the notions of virtually cofibrant and virtually fibrant D-diagrams in M and show that the full subcategories spanned by them

(M^{D})_{vc} ⊂ M^{D}  and  (M^{D})_{vf} ⊂ M^{D}

have the properties mentioned in 20.5.

We start with considering the auxiliary notion of

23.1. Restricted Δ D- and Δ^{op} D-diagrams. Given a model category M and a small category D, we call an object X ∈ M^{Δ D} (resp. M^{^{Δ^{op}} D}) (22.10) restricted if, for every map h ∈ Δ D (resp. Δ^{op} D) for which the associated functor [n_{1}] → [n_{2}] (22.10) sends n_{1} to n_{2} (resp. 0 to 0), the map Xh ∈ M is a weak equivalence and we denote by

(M^{Δ D})_{res} ⊂ M^{Δ D} (resp. (M^{^{Δ^{op}} D})_{res} ⊂ M^{^{Δ^{op}} D})

the full subcategory spanned by these restricted diagrams.

The usefulness of these restricted diagram categories is due to the following proposition which we will prove in 23.5.

23.2. Proposition. Let M be a model category and D a small category. Then (22.12)

(i) the adjunction

colim^{p_{t}} : (M^{Δ D})_{res} ⟷ M^{D} :p_{t}^{∗} (resp. p_{i}^{∗}: M^{D} ⟷ (M^{^{Δ^{op}} D})_{res} :lim_{i}^{p})

is a deformable adjunction (18.2), which (ii) satisfies the Quillen condition (18.6) that, for every Reedy cofibrant (resp.

fibrant) object X ∈ (M^{Δ D})_{res} (resp. (M^{^{Δ^{op}} D})_{res}) and every object Y ∈ M^{D}, a map

X ⟶ p_{t}^{∗}Y ∈ (M^{Δ D})_{res} (resp. p_{i}^{∗}Y ⟶ X ∈ (M^{^{Δ^{op}} D})_{res})

is a Reedy weak equivalence iff its adjoint

colim^{p_{t}} X ⟶ Y ∈ M^{D} (resp. Y ⟶ lim^{p_{i}} X ∈ M^{D})

is an objectwise weak equivalence, and hence (18.6) (iii) these functors are inverse homotopical equivalences of homotopical cate-

gories (8.3).

This result in turn suggests the promised notions of

23.3. Virtually cofibrant and fibrant diagrams. Given a model category M and a small category D, we

(i) call an object Y ∈ M^{D} virtually cofibrant if there exists a Reedy cofibrant object X ∈ (M^{Δ D})_{res} such that Y is isomorphic to colim^{p_{t}} X,

(ii) call an object Y ∈ M^{D} virtually fibrant if there exists a Reedy fibrant

object X ∈ (M^{^{Δ^{op}} D})_{res} such that Y is isomorphic to lim^{p_{i}} X, and (iii) denote by

(M^{D})_{vc} ⊂ M^{D}  and  (M^{D})_{vf} ⊂ M^{D}

the full subcategories of M^{D} spanned by these virtually cofibrant or fibrant diagrams.

Now we can finally give a

23.4. Proof (of the colimit half) of 20.5. To prove 20.5(i) one notes that in view of 23.2, for every left deformation (r, s) of M^{Δ D} into (M^{Δ D})_{c} (8.3, 10.2 and 22.3), the pair consisting of the functor

colim^{p_{t}} rp_{t}^{∗}: M^{D} ⟶ M^{D}

and the natural transformation

colim^{p_{t}} rp_{t}^{∗} ⟶ 1_{M}^{D}

which associates with every object Y ∈ M^{D} the adjunct of the weak equivalence

sp_{t}^{∗}Y : rp_{t}^{∗}Y ⟶ p_{t}^{∗}Y

is a left deformation of M^{D} into (M^{D})_{vc}, while 20.5(ii) readily follows from 22.4 and the observation that the functor f^{Δ D}: M^{Δ D} → N^{Δ D} sends restricted diagrams (23.1) to restricted diagrams.

It thus remains to prove 20.5(iii), i.e. to verify that, for every two Reedy cofibrant diagrams X_{1}, X_{2} ∈ (M^{Δ A})_{res} and objectwise weak equivalences

f: colim^{p_{t}} X_{1} ⟶ colim^{p_{t}} X_{2} ∈ M^{A}

the induced map

colim^{u} f: colim^{u} colim^{p_{t}} X_{1} ⟶ colim^{u} colim^{p_{t}} X_{2} ∈ M^{B}

is an objectwise weak equivalence between objectwise cofibrant diagrams. To do this one constructs a commutative diagram in M^{Δ A} of the form

[figure: X_{1}; a; ∼; ∼; η_{t}X_{1}; X_{3}; b; ∼; X_{2}; c; η_{t}X_{2}; ∼; p_{t}^{∗} colim^{p_{t}} X_{1}; p_{t}^{∗}f; ∼; p_{t}^{∗} colim^{p_{t}} X_{2}]

in which

(i) the vertical maps are the unit maps, i.e. the adjuncts of the identity maps of colim^{p_{t}} X_{1} and colim^{p_{t}} X_{2}, which, in view of 23.2 are weak equivalences (ii) maps a and b are (9.1) a factorization of the weak equivalence (p_{t}^{∗}f)(η_{t}X_{1}) into a trivial Reedy cofibration a followed by a trivial Reedy fibration b, and

(iii) the map c is obtained by lifting η_{t}X_{2} along b.

Adjunction then yields the commutative diagram in M^{A}

\xymatrix{colim^{p_{t}} X_{1} \ar[r]^{colim^{p_{t}} a} \ar[d]_{1} & colim^{p_{t}} X_{3} \ar[dr] & colim^{p_{t}} X_{2} \ar[l]_{colim^{p_{t}} c} \ar[d]^{1} \\ colim^{p_{t}} X_{1} \ar[rr]^{f} &  & colim^{p_{t}} X_{2}}

and application of the functor colim^{u} to this diagram yields a commutative diagram in M^{B} of the form

\xymatrix{colim^{u} colim^{p_{t}} X_{1} \ar[r] \ar[d] & colim^{u} colim^{p_{t}} X_{3} \ar[dr] & colim^{u} colim^{p_{t}} X_{2} \ar[l] \ar[d] \\ colim^{u} colim^{p_{t}} X_{1} \ar[rr]^{colim^{u} f} &  & colim^{u} colim^{p_{t}} X_{2}}

in which, in view of (ii), (iii), 19.6(iii) and 22.12, the two top maps are objectwise weak equivalences between objectwise cofibrant objects, which readily implies that so is the bottom map.

It thus remains to give the promised

23.5. Proof (of the colimit half) of 23.2. For every object D ∈ D, let p_{t}^{−1}D ⊂ Δ D denote the subcategory which has as objects the functors E: [n] → D (n ≥ 0) such that En = D and as maps (E_{1} : [n_{1}] → D) → (E_{2} : [n_{2}] → D) the functors e: [n_{1}] → [n_{2}] such that E_{1} = E_{2}e and en_{1} = n_{2}. The one readily verifies that

(i) the category p_{t}^{−1}D has as an initial object the functor E_{D}: [0] → D such that E_{D}0 = D, and is a Reedy category for which

\overrightarrow{p_{t}^{−1}D} = \overrightarrow{Δ D} ∩ p_{t}^{−1}D  and  \overleftarrow{p_{t}^{−1}D} = \overleftarrow{Δ D} ∩ p_{t}^{−1}D

with the property that \overleftarrow{p_{t}^{−1}D} is a disjoint union of categories with a terminal object,

and that

(ii) for every Reedy cofibrant diagram X ∈ M^{Δ D}, the diagram i^{∗}X ∈ M^{p_{t}^{−1}D} induced by the inclusion functor i: p_{t}^{−1}D → Δ D is also Reedy cofibrant, in view of the fact that, if d^{n}: [n−1] → [n] denotes the functor such that d^{n}i = i for 0 ≤ i ≤ n, then for every object E: [n] → D ∈ p_{t}^{−1}D with n > 0, the commutative diagram (22.6)

[figure: LX(Ed^{n}); L(i^{∗}X)E; (i^{∗}X)E; 1; X(Ed^{n}); LXE; XE]

has the property that the square on the left is a pushout square and the map LX(Ed^{n}) → X(Ed^{n}) and XLE → XE are cofibrations and that therefore the map L(i^{∗}X)E → (i^{∗}X)E is also a cofibration.

One can therefore, for every object D ∈ D, diagram Y ∈ M^{D}, Reedy cofibrant diagram X ∈ (M^{Δ D})_{res} and map X → p_{t}^{∗}Y ∈ (M^{Δ D})_{res} consider the induced commutative diagram

\xymatrix{(colim^{p_{t}} X)D \ar[r]^{a_{1}} \ar[d] & colim^{(p_{t}↓D)} j^{∗}X \ar[d] & colim^{p_{t}^{−1}D} i^{∗}X \ar[l]_{b_{1}} \ar[d] & XE_{D} \ar[l]_{c_{1}} \ar[d] \\ (colim^{p_{t}} p_{t}^{∗}Y)D \ar[r]^{a_{2}} & colim^{(p_{t}↓D)} j^{∗}p_{t}^{∗}Y & colim^{p_{t}^{−1}D} i^{∗}p_{t}^{∗}Y \ar[l]_{b_{2}} & p_{t}^{∗}Y E_{D} \ar[l]|{=}_{c_{2}} & Y D}

in which (iii) j: (p_{t} ↓ D) = Δ(D ↓ D) → Δ D is the forgetful functor and a_{1} and a_{2}

are the isomorphism implied by 19.7(ii), (iv) b_{1} and b_{2} are induced by the functor k: p_{t}^{−1}D → (p_{t} ↓ D) which sends an

object E ∈ p_{t}^{−1}D to the object (E, 1_{D}) ∈ (p_{t} ↓ D), and are isomorphisms because the functor k is right cofinal (i.e. for every object E ∈ (p_{t} ↓ D), the category (E ↓ k) is non-empty and every two of its objects can be connected by a zigzag of maps), and

(v) c_{1} and c_{2} are the restrictions to E_{D} of the respective colimiting cocones and hence ((i) and 22.9) c_{1} is a weak equivalence while c_{2} is actually an isomorphism.

A somewhat technical but essentially straightforward calculation then yields that

(vi) the composition c_{2}^{−1}b_{2}^{−1}a_{2} : (colim^{p_{t}} p_{t}^{∗}Y)D → Y D is the restriction to D

of the counit of the adjunction colim^{p_{t}} : (M^{Δ D})_{res} ↔ M^{D} :p_{t}^{∗}

and that therefore

(vii) the composition of this map with the map (colim^{p_{t}} X)D → (colim^{p_{t}} p_{t}^{∗}Y)D

is the restriction to D of the adjunct of the given map X → p_{t}^{∗}Y

and from this one now readily deduces 23.2.

24. Homotopical comments

We end with noting that the main results of this chapter, i.e. those obtained in §20 and §21, are special cases of, and proven in essentially the same manner, as more general results for cocomplete and complete homotopical categories which we will obtain in chapter VIII.

In more detail:

24.1. Homotopy colimit and limit functors (47.1). Given a homotopical category X which is cocomplete and complete (19.7) and a map u: A → B ∈ cat,

(i) a homotopy u-colimit (resp. u-limit) functor on X will be a left (resp. a right) approximation (18.3) of the functor (19.6)

colim^{u}: X^{A} ⟶ X^{B} (resp. lim^{u}: X^{A} ⟶ X^{B}).

Thus (13.3)

(ii) such u-colimit (resp. u-limit) functors on X, if they exist, are homotopi-

cally unique (13.3)

and essentially the same arguments that (in 20.6) were used to prove 20.2, 20.3 and 20.4 yield that (47.2)

(iii) the result of 20.2 remains valid if one replaces everywhere the model

category M by a homotopical category X for which (iii)′ there exist u-colimit (resp. u-limit) functors on X, and (iii)′′ the functor colim^{u} (resp. lim^{u}): X^{A} → X^{B} is left (resp. right) de-

formable (18.1),

that (47.4)

(iv) the result of 20.3 remains valid if one replaces everywhere the model

category M by a homotopical category X for which (iv)′ there exist u-colimit (resp. u-limit) and v-colimit (resp. v-limit) functors

on X, and

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(iv)′′ the pair (colim^{u}, colim^{v}) (resp. (lim^{u}, lim^{v})) is left (resp. right) deformable (18.4), which is in particular the case if X is saturated (8.4) and this pair is locally left (resp. right) deformable (18.4)

and that (47.6)

(v) the result of 20.4 remains valid if one replaces everywhere the Quillen adjunction f : M ↔ N : g by a deformable adjunction f : X ↔ Y : g (18.2) for which

(v)′ there exist u-colimit (resp. u-limit) functors on both X and Y , and (v)′′ the pairs

(colim^{u}: X^{A} → X^{B}, f^{B})  and  (f^{A}, colim^{u}: Y^{A} → Y^{B})
(resp. (lim^{u}: Y^{A} → Y^{B}, g^{B})  and  (g^{A}, lim^{u}: X^{A} → X^{B}) )

are left (resp. right) deformable (18.4), which is in particular the case if X and Y are saturated (8.4) and these pairs are locally left (resp. right) deformable (18.4).

It thus remains to discuss

24.2. Homotopical cocompleteness and completeness (49.2). Given a homotopical category X which is cocomplete and complete one

(i) defines left and right cat-systems and homotopy colimit and limit systems on X exactly as in 21.1 and 21.3 (i.e. by replacing everywhere in 21.1 and 21.3 the model category M by the homotopical category X), and

(ii) calls X homotopically cocomplete or complete if there exist homotopy colimit or limit systems on X.

The arguments used in the proof of 21.4 then yield that (49.3) (iii) the result of 21.4 remains valid if one replaces everywhere the model

category M by a homotopical category X such that (iii)′ X is cocomplete, complete and saturated (8.4), and (iii)′′ the colimit system colim^{(cat)} (resp. limit system lim^{(cat)}) on X (19.7) is

locally left (resp. right) deformable in the sense, for every object D ∈ cat, there exists a left (resp. a right) deformation retract (X^{D})_{0} ⊂ X^{D} (8.3) such that, for every object E ∈ cat and map x: D → E ∈ cat, the functor colim^{x} (resp. lim^{x}) is homotopical on (X^{D})_{0}.

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Part II

Homotopical Categories

Draft: May 14, 2004

CHAPTER V Summary of part II

25. Introduction

25.1. Motivation. In [Qui67] and [Qui69] Quillen introduced the notion of a model category, i.e. a category with three distinguished classes of maps (weak equivalences, cofibrations and fibrations) satisfying a few simple axioms which enable one to “do homotopy theory”. A closer look at this notion however reveals that the weak equivalences already determine the “homotopy theory”, while the cofibrations and the fibrations provide additional structure which enables one to “do” homotopy theory, in the sense that, while the homotopy notions involved in doing homotopy theory can be defined in terms of the weak equivalences, the verification of their properties (e.g. their existence) requires the cofibrations and/or the fibrations. Consequently many model category arguments are a mix of arguments which only involve weak equivalences and arguments which also involve cofibrations and/or fibrations and as these two kinds of arguments have different flavors, the resulting mix often looks rather mysterious. In this part II we therefore try to isolate the key model category arguments which involve only weak equivalences by developing a kind of “relative category theory” or theory of homotopical categories, which are categories with a single distinguished class of maps called weak equivalences which

(i) includes all the identity maps, and

(ii) has the two out of six property that, for every three maps r, s and t for which the two maps sr and ts exist and are weak equivalences, the four maps r, s, t and tsr are also weak equivalences,

which property is slightly stronger than Quillen’s “two out of three” property.

25.2. Organization of part II. In developing the beginnings of a relative or homotopical category theory we are guided by the desire to be able to

(i) define homotopy or more generally homotopical colimit and limit functors which, if they exist, are in an appropriate sense homotopically unique and the associated notions of homotopical cocompleteness and completeness, and

(ii) describe useful sufficient conditions for the existence of such homotopical colimit and limit functors and for such homotopical cocompleteness and completeness.

In the final chapter VIII we indeed obtain these objectives. However most of the work is done in the two preceding chapters. In the first of these, chapter VI, we introduce homotopical categories and homotopical (i.e. weak equivalence preserving) functors between them, discuss the homotopy category of such a homotopical category and explain what exactly we will mean by homotopical uniqueness.

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In the other chapter, chapter VII, we investigate what we will call left and right deformable functors, which are a frequently occurring kind of functors between homotopical categories which are not necessarily homotopical, but which still have “homotopical meaning”, because they have homotopically unique left or right approximations, i.e. homotopical functors which, in a homotopical sense, are closest to them from the left or from the right.

In order to help the reader navigate through all this we devote this first chapter of part II to a preview of some of the key notions and results of chapters VI, VII and VIII.

25.3. Organization of the present chapter. There are five more sections, of which the first three deal with chapter VI and the other two with the other two chapters.

In §26 we explain why we will not assume that a category necessarily has small hom-sets, introduce homotopical categories and homotopical functors and discuss a few immediate consequences of their definitions. In §27 we then discuss the homotopy category of such a homotopical category and consider in particular the not infrequently occurring case that the homotopical category admits a so-called 3-arrow calculus and in §28 we deal with the notion of homotopical uniqueness.

In §29 we introduce deformable functors and their approximations and describe sufficient conditions on a composable pair of deformable functors in order that their composition is also deformable and all compositions of their approximations are approximations of their composition.

And in §30 we then finally discuss homotopy and homotopical colimit functors and the associated notion of homotopical cocompleteness.

26. Homotopical categories

Before introducing homotopical categories we explain what exactly we will mean by

26.1. Categories, locally small categories and small categories. In order to avoid set theoretical difficulties one often defines categories in terms of a universe, i.e.

(i) one assumes that every set is an element of some universe, where one defines a universe as a set U of sets (called U-sets) satisfying a few simple axioms which imply that U is closed under the usual operations of set theory and that every U-set is a subset of U, but that the set U itself and many of its subsets are not U-sets, and

(ii) one then chooses an arbitrary but fixed universe U and defines a Ucategory as a category of which the hom-sets are U-sets and the set of objects is a subset of U and calls such a U-category small if the set of its objects is actually a U-set.

It turns out that the notion of a small U-category is indeed a convenient one, in the sense that any “reasonable” operation, when applied to small U-categories yields again a small U-category, but that the notion of a U-category is not, as such operations as the formation of the category of functors between two U-categories or the localization of a U-category with respect to a subcategory is not necessarily again a U-category. Still one cannot avoid the notion of a U-category as many of

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the categories one is interested in are not small. However a way of getting around this problem is by not insisting on working in only one universe and noting that

(iii) there exists a unique successor universe U^{+} of our chosen (ii) universe

U, i.e. a unique smallest universe U^{+} such that the set U is a U^{+}-set.

This implies that every U-category is a small U^{+}-category so that any “reasonable” operation when applied to U-categories yields a small U^{+}-category.

In view of all this we choose an arbitrary but fixed universe U and use the term

small category  for small U-category, and
locally small category  for U-category

(instead of the customary use of the term category for U-category) and reserve, unless the context clearly indicates otherwise, the term

category  for small U^{+}-category

Similarly we use the term

small set  for U-set, and
set  for U^{+}-set.

26.2. Homotopical categories and homotopical functors. A homotopical category will be a category C (26.1) with a distinguished set W (26.1) of maps (called weak equivalences) such that

(i) W contains all the identity maps of C, and (ii) W has the two out of six property that, for every three maps r, s

and t ∈ C for which the two compositions sr and ts exist and are in W, the four maps r, s, t and tsr are also in W

which one readily verifies (by assuming that both sr and ts are identity maps or that at least one of r, s and t is an identity map) is equivalent to requiring that

(i)′ W contains all the identity maps of C, (ii)′ W has the weak invertibility property that every map s ∈ C for

which there exist maps r and t ∈ C such that the compositions sr and ts exist and are in W, is itself in W (which, together with (i)′, implies that all isomorphisms of C are in W), and

(iii)′ W has the two out of three property that, for every two maps f and g ∈ C for which gf exists and two of f, g and gf are in W, so is the third (which implies that W is actually a subcategory of C).

This definition suggests, given two homotopical categories C and D

(v) calling a functor f: C → D a homotopical functor whenever f preserves weak equivalences, and

(vi) considering two kinds of homotopical functor or diagram categories, the first, denoted by

Fun(C, D)  or  D^{C}

being the usual functor category (which has as objects the (ordinary) functors C → D and as maps the natural transformations between them) in which the weak equivalences will be the natural weak equivalences, i.e. those natural transformations which send the objects of C to weak equivalences in D, while the other kind of homotopical functor category will be the full homotopical subcategory of the first spanned by the homotopical functors C → D and will be denoted by

Fun_{w}(C, D)  or  (D^{C})_{w} .

It is also convenient to consider the following notions.

26.3. Homotopical equivalences of homotopical categories. A homotopical functor f: C → C′ will be called a homotopical equivalence of homotopical categories if there exists a homotopical functor f′: C′ → C (called homotopical inverse of f) such that the compositions f′f and ff′ are naturally weakly equivalent, i.e. can be connected by a zigzag of natural weak equivalences, to the identity functors 1_{C} and 1_{C}′ respectively.

26.4. Minimal and maximal homotopical categories. Homotopical categories can be seen as a generalization of categories by considering an (ordinary) category as either a minimal or a maximal homotopical category, i.e. a homotopical category in which either only the isomorphism or all the maps are weak equivalences. The first of these approaches seems to be the more useful one and we therefore

(i) usually consider ordinary categories as minimal homotopical categories, and

(ii) when we talk of a homotopical version of some categorical notion or result we usually mean (as for instance in §28 below) a notion or result which on minimal homotopical categories reduces to the given categorical one.

26.5. The homotopy category of a homotopical category. With a homotopical category C one can, in a functorial manner, associate not only its underlying category and its category of weak equivalences, but also its homotopy category Ho C, i.e. the category obtained from C by “formally inverting” the weak equivalences. In view of the assumptions made in 26.1, such homotopy category always exists (although the homotopy category of a locally small homotopical category need not again be locally small).

These homotopy categories and the induced functors between them have the properties that

(i) if two homotopical functors f, g: C → D between homotopical categories are naturally weakly equivalent, then the induced functors

Ho f, Ho g: Ho C ⟶ Ho D

are naturally isomorphic, and hence (ii) if f: C → D is a homotopical equivalence of homotopical categories, then the induced functor Ho f: Ho C → Ho D is an equivalence of categories.

27. The hom-sets of the homotopy categories

27.1. An initial description of the hom-sets. Given a homotopical category C, its homotopy category Ho C (26.5) is obtained from C by “formally inverting” the weak equivalences, i.e. (see 33.8) the category which has the same objects as C and in which, for every two objects X, Y ∈ C, a map X → Y ∈ Ho C is an appropriately defined equivalence class of restricted zigzags in C, i.e. sequences of maps in C

X = C_{0}  C_{1}  ⋯  C_{n} = Y  n ≥ 0

in which some maps are forward (i.e. go to to the right), while the others are backward (i.e. go to the left) and are weak equivalences. It comes with a localization functor γ: C → Ho C which is the identity on objects and which sends each map of C to the class of zigzags containing the zigzag which consists of only this map.

This description in which each equivalence class contains zigzags of countably many different types (i.e. numbers of forward and backward maps and/or the order in which they appear) can be used to obtain a sometimes more useful

27.2. Colimit description of the hom-sets of the homotopy category. This is a description of each hom-set of Ho C as a colimit of a diagram of sets, indexed by a category of types, in which each set consists of equivalence classes of zigzags in C of only one type.

Occasionally one can do even much better than this and describe the hom-sets themselves as equivalences classes of zigzags of only one type. In particular this happens when the homotopical category C admits a

27.3. 3-arrow calculus. By this we mean that the category of weak equivalences W of C has subcategories U and V (like the categories of the trivial cofibrations and the trivial fibrations in a model category (§11)) such that, in a functorial manner,

(i) for every zigzag A′ ⟵^{u} A ⟶^{f} B in C with u ∈ U, there exists a zigzag

A′ ⟶^{f′} B′ ⟵^{u′} B in C with u′ ∈ U such that u′f = f′u and in which u′ is an isomorphism whenever u is,

(ii) for every zigzag X ⟶^{g} Y ⟵^{v} Y′ in C which v ∈ V , there exists a zigzag

X ⟵^{v′} X′ ⟶^{g′} Y′ in C with v′ ∈ V such that gv′ = vg′ and in which v′ is an isomorphism whenever v is, and

(iii) every map w ∈ W admits a factorization w = vu with u ∈ U and v ∈ V .

Then we can formulate

27.4. A 3-arrow description of the hom-sets. If a homotopical category C admits a 3-arrow calculus, then the maps of Ho C be be described as equivalence classes of restricted zigzags (27.1) in C of the form

\xymatrix{· & · \ar[l] \ar[r] & · & · \ar[l]}

in which two such zigzags are in the same equivalence class iff they are the top row and the bottom row in a commutative diagram in C of the form

\xymatrix{· & · \ar[l] \ar[r] & · & · \ar[l] \\ · \ar[u]^{1} \ar[d]_{1} & · \ar[r] \ar[d] \ar[l] \ar[u] & · \ar[d] \ar[u] & · \ar[l] \ar[u]_{1} \ar[d]^{1} \\ · & · \ar[l] \ar[r] & · & · \ar[l] \\ · \ar[u]^{1} & · \ar[l] \ar[u] \ar[r] & · \ar[u] & · \ar[l] \ar[u]_{1}}

in which the middle rows are also restricted zigzags.

Moreover, using for the first time the full strength of the two out of six property (26.2) and not merely the two out of three property, we deduce from the 3-arrow description the following somewhat surprising

27.5. Saturation result. Let C be a homotopical category which admits a 3-arrow calculus (27.3). Then C is saturated in the sense that a map in C is a weak equivalence iff its image in Ho C is an isomorphism.

This last result is not only pleasing, but also quite useful as it allows one, in the many situations (like those which arise from model categories) where one has a 3-arrow calculus, to use the labor saving result mentioned in 29.6 below.

28. Homotopical uniqueness

When working with homotopical categories one often constructs objects in a homotopical category which are homotopically unique, in the sense that they satisfy a homotopical version (26.4) of the categorical notion of uniqueness up to a (unique) isomorphism, which one encounters when certain objects in a category Y (for instance because they have a common universal property) have the property that, for every two of them Y_{1} and Y_{2}, there is exactly one map Y_{1} → Y_{2} ∈ Y and this unique map is moreover an isomorphism. We will refer to this categorical notion as categorical uniqueness and reformulate as follows in a fashion which readily points to a corresponding homotopical notion.

28.1. Categorical uniqueness. Given a nonempty set of objects in a category C, the objects in this set are categorically unique or canonically isomorphic if

(i) the full subcategory of C spanned by these objects, or equivalently

(i)′ the categorically full subcategory of C spanned by these objects, i.e. the full subcategory of C spanned by these objects and all isomorphic ones,

is categorically contractible in the sense that

(ii) this category is a non-empty groupoid in which there is exactly one iso-

morphism between any two objects

or equivalently

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(ii)′ the unique functor from this category to the category [0] which consists of only a single object 0 and its identity map, is an equivalence of categories.

This suggests the following notion of

28.2. Homotopical uniqueness. Given a non-empty set of objects in a homotopical category C, the objects in this set will be called homotopically unique or canonically weakly equivalent if the homotopically full subcategory G ⊂ C spanned by these objects, i.e. the full subcategory of C spanned by these objects and all weakly equivalent ones is homotopically contractible in the sense that the unique functor G → [0] (28.1) is a homotopical equivalence of homotopical categories (26.3).

28.3. Homotopically initial and terminal objects. We also obtain a homotopical version (26.4) of the fact that,

(i) as the full subcategory of a category spanned by its initial or terminal objects is empty or categorically contractible (28.1), one can often verify categorical uniqueness (28.1) by noting that the objects in question are either initial or terminal

and we do this by

(ii) defining homotopically initial or terminal objects in a homotopical category with the property that the full subcategory of a homotopical category spanned by its homotopically initial or terminal objects is a homotopically full (28.2) subcategory which is empty or homotopically contractible (28.2) (the proof of which, for the second time, uses the full strength of the two out of six property (26.2)).

29. Deformable functors

In dealing with homotopical categories one often runs into functors between homotopical categories which are not necessarily homotopical, but which still have “homotopical meaning” because they are homotopical on a so-called left (or right) deformation retract of the domain category and which we will therefore refer to as left (or right) deformable functors.

More precisely

29.1. Left and right deformation retracts. Given a homotopical category X, a left (resp. a right) deformation retract of X will be a full subcategory X_{0} ⊂ X for which there exists a left (resp. a right) deformation of X into X_{0}, i.e. a pair (r, s) consisting of

(i) a homotopical functor f: X → X which sends all of X into X_{0}, and (ii) a natural weak equivalence

s: r → 1_{X} (resp. s: 1_{X} → r)

(which clearly implies that the inclusion X_{0} → X is a homotopical equivalence of homotopical categories (26.3)).

Draft: May 14, 2004

29.2. Left and right deformable functors. A (not necessarily homotopical) functor f: X → Y between homotopical categories will be called left (resp. right) deformable if f is homotopical on a left (resp. a right) deformation retract of X (29.1).

The usefulness of this notion is due to the fact that such left and right deformable functors have homotopically unique (28.2)

29.3. Left and right approximations. Given a (not necessarily homotopical) functor f: X → Y between homotopical categories, a left (resp. a right) approximation of f will be a pair (k, a) consisting of

(i) a homotopical functor k: X → Y , and

(ii) a natural transformation

a: k → f (resp. a: f → k)

which is a homotopically terminal (resp. initial) (28.3) object in “the category of homotopical functors X → Y over (resp. under) f”. Thus (28.3(ii))

(iii) such a left (resp. right) approximation of f, if it exists, is homotopically unique (28.2).

Moreover a sufficient condition for the existence of such an approximation is that f be left (resp. right) deformable, as we show that

(iv) for every left (resp. right) deformation retract X_{0} ⊂ X on which f is homotopical and every left (resp. right) deformation (r, s) of X into X_{0} (29.1), the pair (fr, fs) is a left (resp. a right) approximation of f.

29.4. Compositions of deformable functors. Given two composable left (or right) deformable functors f_{1} : X → Y and f_{2} : Y → Z (29.2), the question arises whether

(i) their composition f_{2}f_{1} : X → Z is also left (or right) deformable, and (ii) the compositions of their left (or right) approximations (29.3) are left (or

right) approximations of their composition,

and we show that a sufficient condition for (i) and (ii) to happen is that the pair (f_{1}, f_{2}) is left (or right) deformable in the following sense.

29.5. Left (or right) deformable composable pairs of functors. Given two composable functors f_{1} : X → Y and f_{2} : Y → Z between homotopical categories, the pair (f_{1}, f_{2}) will be called locally left (resp. right) deformable if there exist left (resp. right) deformation retracts X_{0} ⊂ X and Y_{0} ⊂ Y (29.1) such that

(i) f_{1} is homotopical on X_{0} (and hence f_{1} is left (resp. right) deformable), (ii) f_{2} is homotopical on Y_{0} (and hence f_{2} is left (resp. right) deformable),

and (iii) f_{2}f_{1} is also homotopical on X_{0} (and hence f_{2}f_{1} is also left (resp. right)

deformable),

and the pair (f_{2}, f_{1}) will be called left (resp. right) deformable if in addition

(iv) f_{1} sends all of X_{0} into Y_{0} (which together with (i) and (ii) implies (iii)).

29.6. A special case. Occasionally, to prove the deformability of a composable pair of functors, i.e. the existence of deformation retracts satisfying 29.5(i)– (iv), it suffices to verify the local deformability of the pair, i.e. the existence of deformation retracts satisfying only 29.5(i)–(iii), because

(i) given two composable adjunctions

f_{1}: X ⟷ Y :f′_{1}  and  f_{2}: Y ⟷ Z :f′_{2}

of functors between homotopical categories such that the pair (f_{1}, f_{2}) is locally left deformable (29.5) and the pair (f′_{2}, f′_{1}) is locally right deformable,

(ii) if the category Z is saturated (27.5) and the pair (f′_{2}, f′_{1}) is right de-

formable, then the pair (f_{1}, f_{2}) is left deformable, and dually (iii) if the category X is saturated and the pair (f_{1}, f_{2}) is left deformable, then

the pair (f′_{2}, f′_{1}) is right deformable.

30. Homotopy colimit and limit functors and homotopical ones

As an application of the above results we

(i) generalize the homotopy colimit and limit functors on model categories (see chapter IV) encountered in homotopical algebra, by similarly defining homotopy colimit and limit functors on a homotopical category X as respectively the left and right approximations (29.1) of arbitrary but fixed colimit and limit functors on X, and give sufficient conditions for their existence and composability,

(ii) describe associated notions of homotopical cocompleteness and completeness which are considerably stronger than the requirement that there exist small colimit and limit functors and give sufficient conditions for such homotopical cocompleteness and completeness, and

(iii) note that, while the above homotopy colimit and limit functors are only defined when there exist colimit and limit functors, there is a more general notion of what we will call homotopical colimit and limit functors which is not subject to such a restriction.

In more detail (in the colimit case):

30.1. Homotopy colimit functors. Given a homotopical category X and a category D, we

(i) define a D-colimit functor on X as a left adjoint of the constant diagram functor c^{∗}: X → X^{D} and, if such a D-colimit functor exists, denote by colim^{D}: X^{D} → X an arbitrary but fixed such D-colimit functor, and

(i)′ define a homotopy D-colimit functor on X as a left approximation (29.1)

of the functor colim^{D},

and more generally, given a functor u: A → B, we

(ii) define a u-colimit functor on X as a left adjoint of the induced diagram functor u^{∗}: X^{B} → X^{A} and, if such a u-colimit functor exists, denote by colim^{u}: X^{A} → X^{B} an arbitrary but fixed such u-colimit functor, and

(ii)′ define a homotopy u-colimit functor on X as a left approximation of the functor colim^{u}.

Draft: May 14, 2004

Then

(iii) such homotopy u-colimit functors on X, if they exist, are homotopically

unique (28.2), and (iv) a sufficient condition for their existence is that there exists a u-colimit

functor on X which is left deformable (29.2),

and, given two composable functors u: A → B and v: B → D

(v) a sufficient condition in order that every composition of a homotopy u-colimit functor on X with a homotopy v-colimit functor on X is a homotopy vu-colimit functor on X

is that

(v)′ the pair (colim^{u}, colim^{v}) exists and is left deformable (29.5) which in particular is the case if (29.6)

(v)′′ X is saturated (27.5) and the pair (colim^{u}, colim^{v}) exists and is locally left deformable (29.5).

30.2. Homotopical cocompleteness. While (categorical) cocompleteness, i.e. the existence of D-colimit functors on a homotopical category X for every object D ∈ cat, where cat denotes the category of small categories (26.1), implies

(i) the existence, for every map u ∈ cat, of u-colimit functors on X (29.1(ii)), and

(i)′ the existence, for every composable pair of maps u, v ∈ cat, u-colimit functor f on X, v-colimit functor g on X and vu-colimit functor h on X, of a canonical natural isomorphism gf → h, and hence

(i)′′ the existence of what we will call a colimit system on X, i.e. a function F which assigns to every map u ∈ cat a u-colimit functor F u and to every composable pair of maps u, v ∈ cat, this canonical natural isomorphism (F v)(F u) → F (vu) which is associative in the obvious sense that every three composable maps u, v, w ∈ cat give rise to a commutative diagram of the form

\xymatrix{(F w)(F v)(F u) \ar[r] \ar[d] & F (wv)(F u) \ar[d] \\ (F w)(F vu) \ar[r] & F (wvu) ,}

and in particular of the colimit system colim^{(cat)} which assigns to every map u ∈ cat the functor colim^{u} (30.1)

the existence, for every object D ∈ cat, of a homotopy D-colimit functor on X has in general no such implications.

Therefore we will call a homotopical category X homotopically cocomplete if there exists what we will call a homotopy colimit system on X, which we define as a kind of left approximation of the colimit system colim^{(cat)} (i)′′ and which essentially is a function K which assigns to every map u ∈ cat a homotopy u-colimit functor on X and to every composable pair of maps u, v ∈ cat a natural weak equivalence which is associative in the sense of (i)′′.

We then show that a sufficient condition for such homotopical cocompleteness, i.e. for the existence of such a homotopy colimit system on X is that (ii) X is cocomplete and the colimit system colim^{(cat)} (i)′′ is left deformable

in the sense that there exists a function F_{0} which assigns to every object D ∈ cat, a left deformation retract F_{0}D ∈ X^{D} (29.1) such that, for every map u: A → B ∈ cat

(ii)′ the functor colim^{u}: X^{A} → X^{B} is homotopical on F_{0}A, and (ii)′′ the functor colim^{u}: X^{A} → X^{B} sends all of F_{0}A into F_{0}B,

and note that in particular this is the case if

(iii) X is cocomplete and is saturated (27.5) and the colimit system colim^{(cat)} (i)′′ is locally left deformable in the sense that there exists a function F_{0} as in (ii) which satisfies (ii)′ but not necessarily (ii)′′

as in that case the saturation ensures the existence of a (possibly different) such function which satisfies both (ii)′ and (ii)′′.

It thus remains to discuss

30.3. Homotopical colimit functor. To get around the fact that (30.1), given a functor u: A → B, the homotopy u-colimit functors on a category X were only defined when there existed u-colimit functors on X, we note

(i) that the u-colimit functors on X, i.e. the left adjoints of the induced diagram functor u^{∗}: X^{B} → X^{A}, are also exactly the Kan extension of the identity functor 1_{X}^{B} along the functor u^{∗}, and

(ii) that there is a notion of homotopical Kan extension of 1_{X}^{B} along u^{∗} with the property that, if there exist Kan extensions of 1_{X}^{B} along u^{∗}, then these homotopical Kan extensions of 1_{X}^{B} along u^{∗} are essentially the same as the left approximations of an arbitrary but fixed such Kan extension.

It follows that

(iii) if there exist u-colimit functors on X, then these homotopical Kan extensions of 1_{X}^{B} along u^{∗} are essentially the same as the homotopy u-colimit functors

and we therefore define a homotopical u-colimit functor on X as a homotopical Kan extension of 1_{X}^{B} along u^{∗}.

We also note that in a similar fashion the notion of homotopy colimit systems can be generalized to that of homotopical colimit systems.

Draft: May 14, 2004

CHAPTER VI Homotopical Categories and Homotopical Functors

31. Introduction

31.1. Summary. This introductory chapter on homotopical categories and homotopical functors between them consists essentially of three parts.

(i) After attempting to explain why we will not assume that a category necessarily has small hom-sets, we introduce homotopical categories as such categories with a distinguished class of maps (called weak equivalences) satisfying two simple axioms, and discuss a few immediate consequences of their definition.

(ii) Next we investigate the homotopy category Ho C of such a homotopical category C, i.e. the category obtained from C by “formally inverting” the weak equivalences (which homotopy category, in view of the above (i) mentioned assumption, always “exists”), and in particular

(ii)′ show that if C admits a so-called 3-arrow calculus, then the maps in Ho C can be described as equivalence classes of zigzags in C of the form

\xymatrix{· & · \ar[l] \ar[r] & · & · \ar[l]}

in which the two backward maps are weak equivalences, from which in turn we deduce that

(ii)′′ the presence of such a 3-arrow calculus ensures that the homotopical category C has the useful property of being saturated in the sense that a map in C is a weak equivalence iff its image in Ho C is an isomorphism.

(iii) And finally we end the chapter with a discussion of homotopical analogs of such basic categorical notions as uniqueness up to a unique isomorphism (which we will call categorical uniqueness), universal properties, initial and terminal objects and Kan extensions.

In more detail:

31.2. Categories, locally small categories and small categories. In order to avoid set theoretical difficulties one often defines categories in terms of a universe, i.e.

(i) one assumes that every set is an element of some universe, where one defines a universe as a set U of sets (called U-sets) satisfying a few simple axioms which imply that U is closed under the usual operations of set theory and that every U-set is a subset of U, but that the set U itself and many of its subsets are not U-sets, and (ii) one then defines a U-category as a category of which the hom-sets are U-sets and the set of objects is a subset of U and calls such a U-category small if the set of its objects is actually a U-set.

It turns out that the notion of a small U-category is indeed a convenient one, in the sense that any “reasonable” operation, when applied to small U-categories yields again a small U-category, but that the notion of a U-category is not, as such operations as the formation of the category of functors between two U-categories (33.2) or the localization of a U-category with respect to a subcategory (33.9) do not necessarily yield again a U-category. Still one cannot avoid the notion of a U-category as many of the categories one is interested in are not small. However a way of getting around this problem is by not insisting on working in only one universe and noting that

(iii) for every universe U there exists a unique successor universe U^{+}, i.e. a unique smallest universe U^{+} such that the set U is a U^{+}-set.

This implies that every U-category is a small U^{+}-category so that any “reasonable” operation when applied to U-categories yields a small U^{+}-category.

In view of all this we choose an arbitrary but fixed universe U and use the term

small category  for small U-category
locally small category  for U-category, and
category  for small U^{+}-category

(instead of the customary use of the term “category” for U-category) and similarly use the term

small set  for U-set, and
set  for U^{+}-set

31.3. Homotopical categories. A homotopical category will be a category C (31.2) with a distinguished set (31.2) W of maps (called weak equivalences) such that

(i) W contains all the identity maps of C, and (ii) W has the two out of six property that, for every three maps r, s and t ∈ C for which the two compositions sr and ts exist and are in W, the four maps r, s, t and tsr are also in W,

which implies that (iii) W contains all the isomorphisms of C, and (iv) W has the two out of three property (26.2(iii)) and hence is a subcategory

of C.

Furthermore, given two homotopical categories C and D,

(v) a functor C → D will be called homotopical if it preserves weak equivalences,

(vi) a natural transformation between two functors C → D will be called a natural weak equivalence if it sends the objects of C to weak equivalences in D,

(vii) two functors C → D will be called naturally weakly equivalent if they can be connected by a zigzag of natural weak equivalences, and

Draft: May 14, 2004

(viii) a homotopical functor f: C → D will be called a homotopical equivalence of homotopical categories if there exists a homotopical functor f′: D → C such that the compositions f′f and ff′ are naturally weakly equivalent to the identity functors 1_{C} and 1_{D} respectively.

31.4. The homotopy category. Given a homotopical category C (31.3), its homotopy category will be the category (31.2) obtained from C by “formally inverting” the weak equivalences, i.e., the category Ho C with the same objects as C in which, for every pair of objects X, Y ∈ C, the hom-set (31.2) Ho C(X, Y) consists of equivalence classes of zigzags in C from X to Y in which the backward maps are weak equivalences, by the weakest equivalence relation which puts two such zigzags in the same class

(i) when one can be obtained from the other by omitting identity maps, (ii) when one can be obtained from the other by replacing adjacent maps

which go in the same direction by their composition, and (iii) when one can be obtained from the other by omitting two adjacent maps

which are the same but go in opposite directions.

From this description of Ho C(X, Y) in which each equivalence class of zigzag contains zigzags of countably many types one can deduce sometimes more useful descriptions of Ho C(X, Y) as a colimit of sets in which each set consists of equivalence classes of only one type. Moreover for homotopical categories which admit a so-called 3-arrow calculus one can even do much better and describe the hom-sets themselves as equivalence classes of only one type as follows.

31.5. 3-arrow calculi. A homotopical category C will be said to admit a 3-arrow calculus if the category of weak equivalences W of C has subcategories U and V (like the categories of the trivial cofibrations and the trivial fibrations in a model category (§11)) such that, in a functorial manner,

(i) for every zigzag A′ ⟵^{u} A ⟶^{f} B in C with u ∈ U, there exists a zigzag

A′ ⟶^{f′} B′ ⟵^{u′} B in C with u′ ∈ U such that u′f = f′u and in which u′ is an isomorphism whenever u is,

(ii) for every zigzag X ⟶^{g} Y ⟵^{v} Y′ in C which v ∈ V , there exists a zigzag

X ⟵^{v′} X′ ⟶^{g′} Y′ in C with v′ ∈ V such that gv′ = vg′ and in which v′ is an isomorphism whenever v is, and

(iii) every map w ∈ W admits a factorization w = vu with u ∈ U and v ∈ V .

Such a homotopical category C which admits a 3-arrow calculus then has the property that, for every pair of objects X, Y ∈ C, the hom-set Ho C(X, Y) can be described as the set of the equivalence classes of zigzags in C of the form

\xymatrix{X & · \ar[l] \ar[r] & · & Y \ar[l]}

in which the backward maps are weak equivalences, where two such zigzags are in the same class iff they are the top row and the bottom row in a commutative diagram in C of the form

\xymatrix{X & · \ar[l] \ar[r] & · & Y \ar[l] \\ X \ar[u]^{1} \ar[d]_{1} & · \ar[l] \ar[u] \ar[r] \ar[d] & · \ar[u] \ar[d] & Y \ar[u]_{1} \ar[d]^{1} \ar[l] \\ X & · \ar[l] \ar[r] & · & Y \ar[l] \\ X \ar[u]^{1} & · \ar[l] \ar[u] \ar[r] & · \ar[u] & Y \ar[l] \ar[u]_{1}}

in which the backward maps and hence also the vertical maps are weak equivalences, from which in turn we deduce the rather useful

31.6. Saturation result. Using the above 3-arrow description of the homsets we show that if C is a homotopical category which admits a 3-arrow calculus (31.5), then C is saturated in the sense that a map in C is a weak equivalence iff its image in Ho C is an isomorphism.

31.7. Categorical and homotopical uniqueness. We reformulate the usual notion of uniqueness up to a unique isomorphism of certain objects in a category Y , i.e. the fact that for any two of these objects Y_{1} and Y_{2} there is exactly one map Y_{1} → Y_{2} ∈ Y and that this unique map is an isomorphism, by saying that, given a category Y and a non-empty set I (31.2), certain objects Y_{i} (i ∈ I) are categorically unique or canonically isomorphic if the full subcategory of Y spanned by these objects (or equivalently the categorically full subcategory spanned by these objects, i.e. the full subcategory spanned by these objects and all isomorphic ones) is categorically contractible in the sense that it is a non-empty groupoid in which there is exactly one isomorphism between any two objects, or equivalently, that the unique functor from this category to the category [0] (where [0] denotes the category which consists of only one object 0 and its identity map) is an equivalence of categories and if Y is a homotopical category we therefore say similarly that objects Y_{i} (i ∈ I) are homotopically unique or canonically weakly equivalent if the homotopically full subcategory spanned by these objects, i.e. the full subcategory spanned by these objects and all weakly equivalent ones, is homotopically contractible in the sense that the unique functor G → [0] is a homotopical equivalence of homotopical categories (31.3).

As

∗ the full subcategory of a category spanned by its initial or terminal objects

is either empty or categorically contractible (31.7),

one can often verify the categorical uniqueness (31.7) of a non-empty set of objects in a category Y by noting that these objects are initial or terminal objects of Y and similarly one can often verify the homotopical uniqueness (31.7) of a nonempty set of objects in a homotopical category Y by noting that these objects are homotopically initial or homotopically terminal objects of Y in the following sense.

31.8. Homotopically initial and terminal objects. If, given a category Y and an object Y ∈ Y , one denotes by 1_{Y} : Y → Y the identity functor of Y and by cst_{Y} : Y → Y the constant functor which sends all maps of Y to the identity

Draft: May 14, 2004

map of the object Y ∈ Y , then (see 38.1) an object Y ∈ Y is initial (resp. terminal) iff there exists a natural transformation

f: cst_{Y} → 1_{Y} (resp. f: 1_{Y} → cst_{Y})

such that the map fY : Y → Y ∈ Y is an isomorphism.

We therefore call an object Y of a homotopical category Y homotopically initial (or terminal) if there exist homotopical functors F_{0}, F_{1} : Y → Y and a natural transformation

f: F_{0} → F_{1} (resp. f: F_{1} → F_{0})

such that the map fY ∈ Y is a weak equivalence and the functors F_{0} and F_{1} are naturally weakly equivalent (31.3) to cst_{Y} and 1_{Y} respectively, and note that this definition indeed implies that

∗ the full subcategory of Y spanned by the homotopically initial (resp. terminal) objects is a homotopically full subcategory (31.7) which is either empty or homotopically contractible (31.7).

31.9. Homotopical Kan extensions. Just as one defines (ordinary) Kan extensions as terminal or initial objects in an appropriate category, one can define homotopical Kan extensions as homotopically terminal or initial objects in an appropriate homotopical category. Except for a brief mention in 39.3 and 41.1, these Kan extensions will only be used in the second half of chapter VIII.

We end with

31.10. Organization of the chapter. After some categorical preliminaries (in §32) we introduce homotopical categories and define their homotopy categories (in §33). Next (in §34) we give a more explicit description of this homotopy category and investigate (in §36) the consequences of the presence of a 3-arrow calculus. And finally we discuss homotopical uniqueness and homotopically universal properties (in §37) and the related notions of homotopically terminal and initial objects and homotopical Kan extensions (in §38).

32. Universes and categories

The aim of this section is to explain what exactly we mean by the terms category, locally small category, and small category.

In order to avoid the usual set theoretical problems we will work in the setting of

32.1. Universes. ([AGV72], [Mac71], [Sch72]). When one works in the setting of universes one makes the basic assumption that

(∗) every set is an element of some universe where a universe is, roughly speaking, a set of “sufficiently small sets” satisfying a few simple axioms which ensure that the usual operations of set theory, when applied to these sufficiently small sets, produce again such sets. More precisely, a universe is a set U of sets (called U-sets) such that

(i) if x ∈ U and y ∈ x, then y ∈ U, (ii) if x ∈ U and y ∈ U, then {x, y} ∈ U, where {x, y} denotes the 2-element

set with x and y as its elements, (iii) if a ∈ U and, for every i ∈ a, x_{i} ∈ U, then the union ⋃_{i∈a} x_{i} ∈ U, (iv) if x ∈ U, then P(x) ∈ U, where P(x) denotes the power set of x, i.e. the

set of the subsets of x, and

(v) the set of the finite ordinals is an element of U, from which one can deduce such other closure properties as

(vi) if x ∈ U and y ⊆ x, then y ∈ U,

(vii) if a ∈ U and, for every i ∈ a, x_{i} ∈ U, then the product ∏_{a∈a} x_{i} ∈ U, (viii) if x ∈ U and y ∈ U, then x^{y} ∈ U, where x^{y} denotes the set of the functions

y → x, (ix) the intersection of any set of universes is again a universe, and

(x) if x ∈ U, then x ⊆ U. However the opposite of (x) does not hold, i.e. not every subset of U is an element of U. For instance, U itself cannot be an element of U as this would imply (iv) that P(U) would also be an element of U and hence (x) a subset of U, which is impossible as P(U) has greater cardinality than U.

A consequence of (∗) is not only the existence of universes, but also the fact that

(xi) for every universe U, there exists a higher universe, i.e. a universe V such that U ∈ V, and in fact (ix) a unique smallest such higher universe which will be called the successor universe of U and denoted by U^{+}.

In such a setting of universes one then defines as follows

32.2. U-categories and small U-categories. Given a universe U (32.1), a U-category will be a category C such that

(i) for every pair of object X, Y ∈ C, the hom-set C(X, Y), i.e. the set of the maps X → Y ∈ C, is an element of U, and

(ii) the set of the objects of C is a subset of U, and a small U-category will be a U-category such that

(ii)′ the set of the objects of C is actually an element of U, which implies

(32.1(iii)) that so is the set of all its maps. These definitions readily imply that (iii) the small U-categories and the functors between them form a U-category.

However this conclusion does not hold for the (not necessarily small) U-categories as the set of all U-categories has greater cardinality than U. But

(iv) every U-category is a small U^{+}-category, where U^{+} denotes the successor universe of U (32.1). Thus U-categories and the functors between them form a U^{+}-category and a rather straightforward calculation yields that in fact

(v) the U-categories and the functors between them form a small U^{+}-category. Now we can explain what we will mean by

32.3. Categories, locally small categories and small categories. We choose an arbitrary but fixed universe U and use the term

small category  for small U-category, and
locally small category  for U-category

instead of the customary use of the term category for U-categories, and unless the context clearly indicates otherwise we reserve the term

category  for small U^{+}-category.

Moreover we use the term

small set  for U-set, and
set  for U^{+}-set.

In a similar fashion we denote by

cat  the U-category of the small U-categories, and
CAT  the U^{+}-category of the small U^{+}-categories.

Our justification for the above terminology is the following:

The notion of a small U-category is a quite convenient one as any “reasonable” operation, when applied to small U-categories, yields again a small U-category. For instance, in view of the arguments given in 33.2 and 33.8 (with U instead of U^{+}),

(i) the category of functors between two small U-categories (33.2), and (ii) the localization of a small U-category with respect to a subcategory (33.9)

are again small U-categories. However many interesting U-categories, and in particular the model categories of part I to which we want to apply the results of part II, are not small and, while many operations, when applied to U-categories yield again a U-category,

(i)′ the colimit or limit of a diagram of U-categories indexed by a U-category, (ii)′ the category of functors between two U-categories, and (iii)′ the localization of a U-category with respect to a subcategory

are (in view of 32.2 (iv)) always a small U^{+}-category, but need not be a U-category. As a result, in dealing with U-categories it is difficult to avoid the use of small U^{+}- categories and one gets a cleaner exposition if one works most of the time with small U^{+}-categories and, only in the few instances when this is really necessary, notes sufficient conditions in order that some operation on U-categories yields a small U^{+}-category which is actually a U-category.

33. Homotopical categories

We now introduce homotopical categories and homotopical functors between them and discuss several functorial ways in which one can associate with each homotopical category an (ordinary) category and conversely with each category a homotopical category.

33.1. Homotopical categories and homotopical functors. A homotopical category will be a category C (32.3) with a distinguished set W (32.3) of maps (called weak equivalences) such that

(i) W contains all the identity maps of C, and

(ii) W has the two out of six property that, for every three maps r, s and t ∈ C for which the two compositions sr and ts exist and are in W, the four maps r, s, t and tsr are also in W

which one readily verifies (by assuming that both sr and ts are identity maps or that at least one of r, s and t is an identity map) is equivalent to requiring that

(iii) W contains all the identity maps of C,

Draft: May 14, 2004

(iv) W has the weak invertibility property that every map s ∈ C for which there exist maps r and t ∈ C such that the compositions sr and ts exist and are in W, is itself in W (which together with (iii) implies that all isomorphisms of C are in W), and

(v) W has the two out of three property that, for every two maps f and g ∈ C for which gf exists and two of f, g and gf are in W, the third is also in W (which implies that W is a subcategory of C).

Such a homotopical category will be called small or locally small whenever the category C is (32.3).

Two objects of C will be called weakly equivalent if they can be connected by a zigzag of weak equivalences. The category C will be referred to as the underlying category and its subcategory W as the homotopical structure or category of weak equivalences and a homotopical subcategory of C will be a subcategory C_{0} ⊂ C with C_{0} ∩ W as its homotopical structure.

Furthermore, given two homotopical categories C and C′

(v) a functor C → C′ will be called homotopical if it preserves weak equivalences, i.e. sends the weak equivalences in C to weak equivalences in C′,

(vi) a natural transformation f → g between two functors f, g: C → C′ will be called a natural weak equivalence if it sends the objects of C to weak equivalences in C′, and

(vii) a homotopical functor f: C → C′ will be called a homotopical equivalence of homotopical categories if there exists a homotopical functor f′: C′ → C (called a homotopical inverse of f) such that the compositions f′f and ff′ are naturally weakly equivalent, i.e. can be connected by a zigzag of natural weak equivalences, to the identity functors 1_{C} and 1_{C}′ respectively.

We also note that functors between homotopical categories and the natural transformations between them give rise to two kinds of

33.2. Homotopical diagram or functor categories. Given two categories C and D (32.3), a C-diagram in D is just a functor C → D and one sometimes refers to C as the indexing category of the diagram. Such diagrams and the natural transformations between them form a category in the sense of 32.3 (as these natural transformations can be considered as functions from the U^{+}-set of the objects of C to the U^{+}-set of the maps of D and, in view of 32.1(viii), the set of all such functions is also a U^{+}-set). This category is called the functor category or, especially when the category C plays a rather different role than the category D, diagram category and will be denoted by

Fun(C, D)  or  D^{C} .

If C and D are homotopical categories, then one can similarly form two kinds of homotopical diagram or functor categories. The first of these is the homotopical category of (ordinary) functors C → D which has

(i) as objects the (ordinary) functors C → D, (ii) as maps the natural transformations between them, and (iii) as weak equivalences the natural weak equivalences,

Draft: May 14, 2004

for which we will use the same notation

Fun(C, D)  or  D^{C}

as for the (ordinary) functor category, while the second will be the homotopical category of homotopical functors C → D, i.e. its full homotopical subcategory (33.1)

Fun_{w}(C, D)  or  (D^{C})_{w}

spanned by the homotopical functors C → D.

Next we consider

33.3. The category of homotopical categories and homotopical functors. The homotopical categories and the homotopical functors between them (33.1) form a U^{+}-category (32.1 and 32.2) which we will denote by CAT_{w}, and which is related to the U^{+}-category CAT of (ordinary) categories and functors (32.3) by a sequence of five functors (two in one direction and three in the other) which form four pairs of adjoint functors as indicated in the diagram

Ho ⟷ min ⟷ j ⟷ max ⟷ j_{w} ,

and we devote the remainder of this section to a brief discussion of these five functors.

We start with

33.4. The forgetful functors j, j_{w} : CAT_{w} → CAT. These are the functors which send a homotopical category respectively to its underlying category and its category of weak equivalences ((33.1)).

Closely related to both these functors is

33.5. The maximal structure functor max: CAT → CAT_{w}. This is the functor which sends a category C to the maximal homotopical category with C as its underlying category as well as its category of weak equivalences (33.1). It obviously is (33.4)

(i) a left adjoint of the forgetful functor j_{w} : CAT_{w} → CAT, and (ii) a right adjoint of the forgetful functor j: CAT_{w} → CAT.

Of course there is also the other extreme of

33.6. The minimal structure functor min: CAT → CAT_{w}. This is the functor which sends a category C to the minimal homotopical category which has C as its underlying category, but in which the weak equivalences are only the isomorphisms of C. It clearly is

∗ a left adjoint of the forgetful functor j: CAT_{w} → CAT (33.4).

These last two results (33.5 and 33.6) suggest two ways of

33.7. Embedding CAT in CAT_{w}. Homotopical categories can be seen as a generalization of categories by considering an (ordinary) category as either a minimal or a maximal homotopical category, i.e. (33.6 and 33.5) a homotopical category in which either only the isomorphisms or all the maps are weak equivalences. The first of these approaches seems to be the more useful one and we therefore usually will

(i) consider CAT as a full subcategory of CAT_{w} by considering each (ordinary) category as a minimal homotopical one, and

(ii) mean, when we talk of a homotopical version of some categorical notion or result, a notion or result which (for instance as in §37 and §38 below) on minimal homotopical categories reduces to the given categorical one.

It thus remains to discuss the functor Ho: CAT_{w} → CAT which sends homotopical categories to their

33.8. Homotopy categories. Given a homotopical category C, its homotopy category will be the category Ho C obtained from C by “formally inverting” the weak equivalences, i.e. the category defined as follows.

Given two objects X, Y ∈ C and an integer n ≥ 0, a zigzag in C from X to Y of length n will be a sequence

[figure: X = C_{0}; f_{1}; C_{1}; ⋯; f_{n}; C_{n} = Y]

of maps in C, each of which is either forward (i.e. points to the right) or backward (i.e. points to the left) and such a zigzag will be called restricted if all the backward maps are weak equivalences. The homotopy category of C then will be the category Ho C which has the same objects as C, in which, for every two objects X, Y ∈ C, the hom-set Ho C(X, Y) is the set of the equivalence classes of the restricted zigzags in C from X to Y, where two such zigzags are in the same class iff one can be transformed into the other by a finite sequence of operations of the following three types and their inverses:

(i) omitting an identity map,

(ii) replacing two adjacent maps which go in the same direction by their

composition, and (iii) omitting two adjacent maps when they are the same, but go in opposite

directions

and in which the compositions are induced by the compositions of the zigzags involved.

It thus remains to verify that

(iv) the homotopy category Ho C so defined is indeed a category, i.e. (32.3) a

small U^{+}-category

but this follows readily from the observation that the set of the maps of Ho C is a quotient of a subset of the finite ordered subsets of the set of the maps of C and that, as (32.2(ii)′) the set of the maps of C is a U^{+}-set, so is, in view of 32.1(viii), the set of its finite ordered subsets.

We also note that the same argument with U instead of U^{+} yields that (v) the homotopy category of a small homotopical category (32.3) is also

small.

However the homotopy category of a locally small homotopical category (32.3) need not be locally small.

33.9. The localization functor. Given a homotopical category C with category of weak equivalences W, its homotopy category Ho C (33.8) comes with a functor γ: C → Ho C which is the identity on the objects and which sends a map f: X → Y ∈ C to the map γf: X → Y ∈ Ho C which contains the restricted zigzag in C from X to Y which consists of only the map f. We will refer to it as the localization functor of C as the definitions involved imply that

(i) the pair (Ho C, γ) is a localization of C with respect to W in the sense that this pair has the universal property that, for every category B and functor b: C → B which sends the weak equivalences in C to isomorphisms in B, there is a unique functor b_{γ}: Ho C → B such that b_{γ}γ = b.

This universal property implies that

(ii) given two homotopical categories C and D with localization functors γ: C → Ho C and γ′: D → Ho D there is, for every homotopical functor f: C → D a unique functor Ho f: Ho C → Ho D such that (Ho f)γ = γ′f and, for every natural transformation (resp. natural weak equivalence) d: f_{1} → f_{2} between two homotopical functors f_{1}, f_{2} : C → D, a unique natural transformation (resp. natural isomorphism) Ho d: Ho f_{1} → Ho f_{2} such that (Ho d)γ = γ′d,

(iii) the function Ho so defined is a functor Ho: CAT_{w} → CAT which is left

adjoint to the functor min: CAT → CAT_{w} (33.6), and (iv) this functor Ho sends naturally weakly equivalent homotopical functors

C → D (33.1(vii)) to naturally isomorphic functors Ho C → Ho D and homotopical equivalences of homotopical categories C → D (33.1(vii)) to equivalences of categories Ho C → Ho D.

Closely related to the localization functor is the rather useful (see §§42 and 47–49) notion of a saturated homotopical category, i.e. a homotopical category W with localization functor γ: C → Ho C with the property that a map c ∈ C is a weak equivalence iff the map γc ∈ Ho C is an isomorphism. This notion is hereditary in the sense that

(v) if C is a saturated homotopical category, then so are, for every homotopical category D, the homotopical functor categories (33.2)

C^{D}  and  (C^{D})_{w}

which follows readily from the observation that every object D ∈ D gives rise to a commutative diagram of the form

\xymatrix{ & Ho(C^{D}) \ar[dr] \\ _{C_{D}} \ar[ur]^{γ′} \ar[rr]_{γ^{D}} \ar[d] &  & (Ho C)^{D} \ar[d] \\ C \ar[rr]_{γ} &  & _{Ho C}}

in which the vertical maps are “restrictions to D”, γ and γ′ are the localization functors of C and C^{D} and the map in the upper right hand corner is the unique map (i) such that the triangle commutes.

We end this discussion of homotopy categories with an often convenient

33.10. Alternate description of the hom-sets. Given a homotopical category C and objects X, Y ∈ C, the hom-set Ho C(X, Y) (33.8) is the set of the equivalence classes of the restricted zigzags in C from X to Y, where two such zigzags are in the same class if one can be transformed into the other by a finite sequence of operations of the following three types and their inverses:

(i) omitting an identity map, (ii) replacing two adjacent maps which go in the same direction by their com-

position, and (iii) in a commutative diagram in C of the form

[figure: X; f_{1}; 1=a_{0}; ·; a_{1}; ···; ·; a_{n−1}; f_{n}; Y; a_{n}=1; X; g_{1}; ·; ···; ·; g_{n}; Y] n ≥ 0

in which, for every integer i with 1 ≤ i ≤ n, the maps f_{i} and g_{i} go in the same direction, replacing the top row by the bottom row.

Proof. As every weak equivalence c: C_{0} → C_{1} and every composable pair of maps c_{1} : C_{1} → C_{2} and c_{2} : C_{2} → C_{3} ∈ C give rise to commutative diagrams

\xymatrix{C_{0} \ar[r]^{c} & C_{1} & C_{0} \ar[l]_{c} \\ C_{0} \ar[u]^{1} \ar[r]_{1} & C_{0} \ar[u]_{c} & C_{0} \ar[l]^{1} \ar[u]_{1}}, \xymatrix{C_{1} \ar[d]_{1} & C_{0} \ar[l]_{c} \ar[r]^{c} \ar[d]^{c} & C_{1} \ar[d]^{1} \\ C_{1} & C_{1} \ar[l]^{1} \ar[r]_{1} & C_{1}} and \xymatrix{C_{1} \ar[r]^{c_{1}} \ar[d]_{1} & C_{2} \ar[r]^{c_{2}} \ar[d]^{c_{2}} & C_{3} \ar[d]^{1} \\ C_{1} \ar[r] & C_{3} \ar[r]_{1} & C_{3}}

every two restricted zigzags in C which are in the same class in the sense of 33.8 are also in the same class in the sense of 33.10.

To show the converse one has to prove that, for every commutative diagram as in 33.10(iii) and every integer i with 1 ≤ i ≤ n, the restricted zigzag

[figure: X; f_{1}; ·; ···; ·; f_{i−1}; ·; f_{i}; ·; a_{i}; ·; g_{i+1}; ·; ···; ·; g_{n}; Y]

Draft: May 14, 2004

34. A COLIMIT DESCRIPTION OF THE HOM-SETS OF THE HOMOTOPY CATEGORY 99

and

[figure: X; f_{1}; ·; ···; ·; f_{i−1}; ·; a_{i−1}; ·; g_{i}; ·; g_{i+1}; ·; ···; ·; g_{n}; Y]

are in the same class in the sense of 33.8 and one does this by noting that, in the case that f_{i} and g_{i} are forward maps, this follows from 33.8(ii) and the fact that a_{i}f_{i} = g_{i}a_{i−1}, and that, in the case that f_{i} and g_{i} are backward maps, this follows from the observation that, in view of 33.8(ii), these two zigzags are respectively in the same class, in the sense of 33.8, as the zigzags

[figure: X; ···; ·; f_{i−1}; ·; ·; f_{i}; f_{i}; ·; a_{i−1}; ·; g_{i}; ·; g_{i+1}; ·; ···; Y]

and

[figure: X; ···; ·; f_{i−1}; ·; f_{i}; ·; a_{i}; ·; g_{i}; ·; ·; g_{i}; g_{i+1}; ·; ···; Y]

and that, in view of 33.8(ii) and the fact that a_{i−1}f_{i} = g_{i}a_{i}, these last two zigzags are in the same class in the sense of 33.8.

34. A colimit description of the hom-sets of the homotopy category

In this section we deduce from the description of the hom-sets of the homotopy category Ho C of a homotopical category C in which, for every two objects X, Y ∈ C, the hom-set Ho C(X, Y) is described as a set of equivalence classes of restricted zigzags in C from X to Y, where each such class contains zigzags of countably many different types (i.e. numbers of forward and backward maps and the order in which they appear), a description of Ho C(X, Y) as a colimit of a diagram of sets, indexed by the category T of these types, in which each set consists of equivalence classes of zigzags of only one type.

To do this we start with giving a precise definition of

34.1. Types of Zigzags. The type of a zigzag in a category C from an object X to an object Y (33.8)

X = C_{0} ^{f_{1}} C_{1}  ⋯ ^{f_{n}} C_{n} = Y  (n ≥ 0)

will be the pair T = (T_{+}, T_{−}) of complementary subsets of the set of integers {1, …, n} such that i ∈ T_{+} whenever f_{i} is forward and i ∈ T_{−} otherwise.

These types can be considered as objects of a category of types T which has, for every two types (T_{+}, T_{−}) and (T′_{+}, T′_{−}) of length n and n′ respectively, as maps t: (T_{+}, T_{−}) → (T′_{+}, T′_{−}) the weakly monotone functions t: {1, …, n} → {1, …, n′} such that

t(T_{+}) ⊂ T′_{+}  and  t(T_{−}) ⊂ T′_{−}.

With these types of zigzags one can associate

34.2. Arrow categories. Given a homotopical category C with category of weak equivalences W, a pair of objects X, Y ∈ C and a type T of length n (34.1), the associated n-arrow category will be the category which has as objects the restricted zigzags in C from X to Y (i.e. zigzags in which the backward maps are in W) of type T and has as maps the commutative diagrams in C of the form

[figure: X; 1; ·; ···; ·; Y; 1; X; ·; ···; ·; Y]

in which the top row and the bottom row are restricted zigzags in C of type T and the vertical maps are weak equivalences. This n-arrow category will be denoted by

(C, W^{−1})^{T} (X, Y)

Next we note that from these arrow categories one can form

34.3. T -diagrams of arrow categories. Given a homotopical category C with category of weak equivalences W and a pair of objects X, Y ∈ C, one can form a T -diagram of arrow categories

(C, W^{−1})^{T} (X, Y)

which assigns to every object T ∈ T (34.1) the category (34.2)

(C, W^{−1})^{T} (X, Y)

and to every map t: T → T′ ∈ T the functor

t_{∗}: (C, W^{−1})^{T} (X, Y) → (C, W^{−1})^{T′}(X, Y)

which sends a zigzag of type T

X = C_{0} ^{f_{1}} C_{1}  · · · · · ^{f_{n}} C_{n} = Y

to the zigzag of type T′

X = C′_{0} ^{f′_{1}} C′_{1}  · · · · · ^{f′_{n}} C′_{n′} = Y

in which each f′_{j} (1 ≤ j ≤ n′) is the composition of the f_{i} with ti = j or, if no such i exists, the appropriate identity map.

Similarly, for every subcategory T′ ⊂ T the restriction of the above T -diagram to T′ yields a T′-diagram of arrow categories which we will denote by

(C, W^{−1})^{T′}(X, Y)

We also need the notion of

34.4. Connected components of a category. A connected component of a category D is a maximal subcategory D_{0} ⊂ D with the property that every pair of objects in D_{0} can be connected by a zigzag of maps in D_{0}. We will denote by π_{0}D the resulting set of the equivalence classes of the objects of D, where two objects will be in the same class when they are in the same connected component, and denote, for a functor c: D → E, by π_{0}c: π_{0}D → π_{0}E the function induced by c.

Now we can finally formulate the promised

34.5. Colimit description of the hom-sets. Let C be a homotopical category with category of weak equivalences W. Then, for every pair of objects X, Y ∈

C (34.3)
Ho C(X, Y) = colim^{T} π_{0}(C, W^{−1})^{T} (X, Y) .

Proof. This follows from 33.8 and the observation that the functor π_{0} takes care of the relations imposed by 33.8(iii) and that the functors between the arrow categories take exactly care of the relations imposed by 33.8(i) and (ii).

35. A Grothendieck construction

As an application of the above colimit description of the hom-sets of the homotopy category of a homotopical category C, we

(i) obtain, for every pair of objects X, Y ∈ C, from the T -diagram of arrow categories (C, W^{−1})^{T} (X, Y) a single category Gr(C, W^{−1})^{T} (X, Y) (called its Grothendieck construction) which has as its objects all the restricted zigzags in C from X to Y and of which any two objects are in the same connected component iff they represent the same map X → Y ∈ Ho C,

(ii) combine all these Grothendieck constructions into a single category

Gr(C, W^{−1})^{T}

which has the same objects as C and is enriched over CAT (32.3) and which has the property that its “category of connected components” is exactly the category Ho C, and

(iii) note that this enriched category is closely related to the (simplicial) ham-

mock localization of [DK80a]. We thus start with

35.1. Grothendieck construction. We recall from [Tho79] that, given a homotopical category C with category of weak equivalences W and a pair of objects X, Y ∈ C, the Grothendieck construction on the T -diagram of categories (C, W^{−1})^{T} (X, Y) (34.3) is the category

Gr(C, W^{−1})^{T} (X, Y)

which has (34.2 and 34.3)

(i) as objects the restricted zigzags in C from X to Y, i.e. the pairs (T, Z) consisting of the objects

T ∈ T  and  Z ∈ (C, W^{−1})^{T} (X, Y)

and

(ii) for every two such objects (T′, Z′) and (T′′, Z′′), as maps (T′, Z′) → (T′′, Z′′) the pairs (t, z) consisting of maps

t: T′ → T′′ ∈ T  and  z: t_{∗}Z′ → Z′′ ∈ (C, W^{−1})^{T′′}(X, Y)

and note that, by essentially the same argument as was used in the proof of 34.5, this definition implies that

35.2. Proposition. For every homotopical category C and pair of objects X, Y ∈ C, (34.5)

Ho C(X, Y) = colim^{T} π_{0}(C, W^{−1})^{T} (X, Y) = π_{0}Gr(C, W^{−1})^{T} (X, Y) .

Next we note that these Grothendieck constructions give rise to what we will call the

35.3. Grothendieck enrichment. Given a homotopical category C with category of weak equivalences W, its Grothendieck enrichment will be the category Gr(C, W^{−1})^{T} enriched over CAT (32.3) which

(i) has the same objects as C, (ii) has, for every two objects X, Y ∈ C, as its hom-category the category

Gr(C, W^{−1})^{T} (X, Y), and (iii) has, for every three objects X_{1}, X_{2} and X_{3} ∈ C as its composition functor

Gr(C, W^{−1})^{T} (X_{1}, X_{2}) × Gr(C, W^{−1})^{T} (X_{2}, X_{3}) ⟶ Gr(C, W^{−1})^{T} (X_{1}, X_{3})

the functor induced by the compositions of the zigzags involved.

If π_{0}Gr(C, W^{−1})^{T} denotes the category obtained from Gr(C, W^{−1})^{T} by applying the functor π_{0} (34.4) to each of its hom-categories, then 34.5 and 35.2 imply the following

35.4. Grothendieck description of the homotopy category. If C is a homotopical category with category of weak equivalences W, then

Ho C = π_{0}Gr(C, W^{−1})^{T} .

We end with a brief discussion of the connection between these Grothendieck enrichments and simplicial localizations and for this we need a few

35.5. Simplicial preliminaries. We need the following definitions and results.

(i) a simplicial set will be a functor A: Δ^{op} → SET, where SET denotes the U^{+}-category (32.1 and 32.2) of sets (32.3) and Δ^{op} is the category which has as its objects the categories [n] (n ≥ 0) (22.10) and as maps all the functors between them, and the elements of the set A[n] (n ≥ 0) will be referred to as the n-simplices of A,

(ii) every category X then gives, in a functorial manner, rise to a simplicial set NX (called its nerve or classifying space) which is the functor Δ^{op} → SET which sends [n] (n ≥ 0) to the set of the functors [n] → X, and

(iii) the U^{+}-category of simplicial sets SET^{Δ^{op}} admits a model structure (9.1) (the details of which can be found in [GJ99], [Hir03] and [Hov99]) in which

(iii)′ the cofibrations are the objectwise monomorphisms, and (iii)′′ if f: X → Y is a functor which has a right inverse g: Y → X for which

there exists a natural transformation 1_{X} → gf, then the induced map of simplicial sets Nf: NX → NY is a weak equivalence.

Now we can start dealing with

35.6. The connection with simplicial localizations. Given a homotopical category C with category of weak equivalences W, its Grothendieck enrichment (35.3) gives rise to a simplicial category, i.e. a category enriched over simplicial sets (35.5)

NGr(C, W^{−1})^{T}

which has the same objects as C and which, for every pair of objects X, Y ∈ C, has the simplicial set

NGr(C, W^{−1})^{T} (X, Y)

as its simplicial hom-set. This simplicial category is closely related to the (simplicial) hammock localization of C, i.e. the simplicial category L^{H}C which has the same objects as C and in which, for every pair of objects X, Y ∈ C, the simplicial hom-set can be described as [DK80a, 5.5]

L^{H}C(X, Y) = colim^{T} N(C, W^{−1})^{T} (X, Y) .

These two simplicial categories then are related by the following

35.7. Proposition. Let C be a homotopical category with category of weak equivalences W. Then the map of simplicial categories (35.6)

p: NGr(C, W^{−1})^{T} → L^{H}C

which is the identity on the objects and which, for every pair of object X, Y ∈ C, consists of the map

p(X, Y): NGr(C, W^{−1})^{T} (X, Y) ⟶ colim^{T} N(C, W^{−1})^{T} (X, Y)

which sends an n-simplex of NGr(C, W^{−1})^{T} (X, Y), i.e. a sequence of maps in Gr(C, W^{−1})^{T} (X, Y) (35.1)

(T^{0}, Z^{0}) ⟶^{(t^{1},z^{1})} (T^{1}, Z^{1}) ⟶ ⋯ ⟶ (T^{n−1}, Z^{n−1}) ⟶^{(t^{n},z^{n})} (T^{n}, Z^{n})

to the n-simplex of colim^{T} N(C, W^{−1})^{T} (X, Y) which contains the sequence of maps in (C, W^{−1})^{T^{n}(X,Y)}

(t^{n} ⋯ t^{1})_{∗}Z^{0} ⟶^{(t^{n}···t^{2})_{∗}z^{1}} (t^{n} ⋯ t^{2})_{∗}Z^{1} ⟶ ⋯ ⟶ t_{∗}^{n}Z^{n−1} ⟶^{z^{n}} Z^{n}

is a weak equivalence in the sense that, for every pair of objects (X, Y) ∈ C, the map p(X, Y) is a weak equivalence of simplicial sets (35.5).

To prove this we need the following

35.8. Colimit description of the Grothendieck construction. Given a homotopical category C with category of weak equivalences W and two objects X, Y ∈ C, let

(C, W^{−1})^{(−↓T)}(X, Y)

denote the T -diagram of categories which assigns to every object T ∈ T the category

(C, W^{−1})^{(−↓T)}(X, Y)

which has

(i) as objects the pairs (t′, Z′) consisting of a map t′: T′ → T ∈ T and an object Z′ ∈ (C, W^{−1})^{T′}(X, Y), and (i)′ as maps (t′, Z′) → (t′′, Z′′), where

Z′ ∈ (C, W^{−1})^{T′}(X, Y) and Z′′ ∈ (C, W^{−1})^{T′′}(X, Y) the pairs (t, z) consisting of a map t: T′ → T′′ ∈ T such that t′′t = t′ and a map (35.1)

z: t_{∗}Z′ ⟶ Z′′ ∈ (C, W^{−1})^{T′′}(X, Y)

and which assigns to every map t̄: T → \overline{T} ∈ T the functor

(C, W^{−1})^{(−↓T)}(X, Y) ⟶ (C, W^{−1})^{(−↓\overline{T})}(X, Y)

which sends (ii) an object (t′, z′) to the object (t̄t′, Z′), and (ii)′ a map (t, z): (t′, Z′) → (t′′, Z′′) to the map (t, z): (t̄t′, Z′) → (t̄g′′, Z′′),

and let, for every object T ∈ T,

(C, W^{−1})^{(−↓T)}(X, Y) ⟶ Gr(C, W^{−1})^{T} (X, Y)

denote the functor which sends

(iii) an object (t′, Z′), where Z′ ∈ (C, W^{−1})^{T′}(X, Y), to the object (T′, Z′),

and (iii)′ a map (t, z): (t′, Z′) → (t′′, Z′′), where

Z′ ∈ (C, W^{−1})^{T′}(X, Y) and Z′′ ∈ (C, W^{−1})^{T′′}(X, Y) , to the map (t, z): (T′, Z′) → (T′′, Z′′).

Then (iv) these functors are natural in T, and

(v) the induced functor

colim^{T} (C, W^{−1})^{(−↓T)}(X, Y) ⟶ Gr(C, W^{−1})^{T} (X, Y) is an isomorphism

and moreover

(vi) so is the induced (35.6) map of simplicial sets colim^{T} N(C, W^{−1})^{(−↓T)}(X, Y) ⟶ NGr(C, W^{−1})^{T} (X, Y) .

Proof. This is a lengthy but essentially straightforward calculation which will be left to the reader.

It thus remains to give a

35.9. Proof of 35.7. To prove 35.7 one first notes that

(i) every object T ∈ T gives, for every pair of objects X, Y ∈ C, rise to a

functor

(C, W^{−1})^{(−↓T)}(X, Y) ⟶ (C, W^{−1})^{T} (X, Y)

which sends an object (t′, Z′) to the object (T, t′_{∗}Z′) and a map (t, z): (t′, Z′) → (t′′, Z′′) to the map

(T, t′_{∗}Z′) = (T, t′′_{∗}t_{∗}Z′) ⟶^{(1_{T} ,t′′_{∗}z)} (T, t′′_{∗}Z′′)

and that

Draft: May 14, 2004

(ii) the map p(X, Y) admits a factorization

\xymatrix{ & colim^{T} N(C, W^{−1})^{(−↓T)}(X, Y) \ar@{-->}[dr] \\ NGr(C, W^{−1})^{T} (X, Y) \ar@{-->}[ur] \ar[rr]^{p(X,Y)} &  & colim^{T} N(C, W^{−1})^{T} (X, Y)}

in which the first map is the inverse of the isomorphism of 35.8(vi) and the second map is induced by the maps of (i) above.

The desired result now follows from 22.7 and the observations that

(iii) the category T can be considered as a Reedy category with fibrant constants (22.1 and 22.8) in which the required subcategories \overrightarrow{T} and \overleftarrow{T} consist of the maps which are respectively 1-1 and onto,

(iv) the T -diagrams of simplicial sets

N(C, W^{−1})^{(−↓T)}(X, Y)  and  N(C, W^{−1})^{T} (X, Y)

are, in view of 35.5(iii)′ both Reedy cofibrant diagrams (10.2, 22.2 and 22.6),

and the fact that,

(v) as one readily verifies that the functors considered in (i) above have the property mentioned in 35.5(iii)′′, the nerves of these maps are weak equivalences of simplicial sets.

36. 3-arrow calculi

Next we use the description (34.5) of the hom-sets of the homotopy category of a homotopical category as a colimit of a diagram of sets, indexed by the category of types T, in which each set consists of equivalence classes of restricted zigzags of only one type, to show that

(i) if C admits a so-called 3-arrow calculus, then these hom-sets can be described in terms of zigzags in C of type

\xymatrix{· & · \ar[l] \ar[r] & · & · \ar[l]}

only.

Moreover we deduce from this (for the first time using the full strength of the “two out of six” property (33.1) and not merely the “two out of three” property) the somewhat unexpected result that

(ii) if C admits a 3-arrow calculus, then C is saturated, i.e. (33.9) a map in C is a weak equivalence iff its image in Ho C is an isomorphism.

This last result turns out to be of use in chapters VII and VIII where we obtain sufficient conditions for the composability of approximations (in §42) and of homotopy colimit and limit functors (in §47) and for the homotopical cocompleteness and completeness of homotopical categories (in §49), which are considerably simpler, and therefore easier to verify, when the homotopical categories involved are saturated.

We start (motivated by some of the properties of the subcategories of the trivial cofibrations and the trivial fibrations in a model category (see chapter II)) with defining

36.1. 3-arrow calculi. Given a homotopical category C with category of weak equivalences W (33.1), C is said to admit a 3-arrow calculus {U, V } if there exist subcategories U, V ⊂ W such that

(i) for every zigzag A′ ⟵^{u} A ⟶^{f} B in C which u ∈ U, there exists a functorial

zigzag A′ ⟶^{f′} B′ ⟵^{u′} B in C with u′ ∈ U such that

u′f = f′u and

u′ is an isomorphism whenever u is

(e.g. if C is closed under pushouts and every pushout of a map in U is again in U),

(ii) for every zigzag X ⟶^{g} Y ⟵^{v} Y′ in C with v ∈ V , there exists a functorial

zigzag X ⟵^{v′} X ⟶^{g′} Y in C with v′ ∈ V such that

gv′ = vg′ and

v′ is an isomorphism whenever v is

(e.g. if C is closed under pullbacks and every pullback of a map in V is again in V ), and

(iii) every map w ∈ W admits a functorial factorization w = vu with u ∈ U

and v ∈ V . Note that (iii) implies that U and V contain all the objects of W and hence

of C.

Another immediate consequence of this definition is that

(iv) for every homotopical category D, the homotopical diagram categories

(33.2)

C^{D} and (C^{D})_{w}

inherit from C a 3-arrow calculus in which the desired subcategories consist of the natural transformations which send the objects of D to maps in U and V respectively.

Similarly one has

36.2. Proposition. Let C be a homotopical category with category of weak equivalences W which admits a 3-arrow calculus {U, V } (36.1). Then, for every pair of objects X, Y ∈ C, the 3-arrow category (34.2)

(C, W^{−1})^{({2},{1,3})}(X, Y)

and its full subcategory

(C, W^{−1})_{{U,V }}^{({2},{1,3})}(X, Y)

spanned by the zigzags of the form

\xymatrix{X & · \ar[r] \ar[l]_{v} & · & Y \ar[l]_{u}}

in which u ∈ U and v ∈ V , when considered as maximal homotopical categories (33.5), inherit from C a 3-arrow calculus of which (i) the desired subcategories consist of the commutative diagrams in C of the

form

\xymatrix{X \ar[d]_{1} & · \ar[d] \ar[r] \ar[l] & · \ar[d] & Y \ar[d]^{1} \ar[l] \\ X & · \ar[r] \ar[l] & · & Y \ar[l]}

in which the vertical maps are respectively in U and in V , and (ii) the functorial factorization consists of the commutative diagrams in C of

the form

\xymatrix{X \ar[d]_{1} & · \ar[l] \ar[d] \ar[r] & · \ar[d] & Y \ar[l] \ar[d]^{1} \\ X \ar[d]_{1} & · \ar[l]_{p} \ar[d] \ar[r] & · \ar[d] & Y \ar[l]_{q} \ar[d]^{1} \\ X & · \ar[l] \ar[r] & · & Y \ar[l]}

in which the squares in the middle are produced by the given functorial factorization and the maps p and q are the unique ones which make the other squares commutative.

Now we formulate the promised

36.3. 3-arrow description of the hom-sets. Let C be a homotopical category with category of weak equivalences W, which admits a 3-arrow calculus {U, V } (36.1). Then, for every pair of objects X, Y ∈ C,

(i) the map ( 34.1–34.5)

π_{0}(C, W^{−1})^{({2},{1,3})}(X, Y) ⟶^{incl._{∗}} colim^{T} π_{0}(C, W^{−1})^{T} (X, Y) = Ho C(X, Y)

induced by the inclusion ({2}, {1, 3}) ∈ T and the map

π_{0}(C, W^{−1})_{{U,V }}^{({2},{1,3})}(X, Y) ⟶^{incl._{∗}} π_{0}(C, W^{−1})^{({2},{1,3})}(X, Y)

induced by the inclusion of 36.2, are both isomorphisms.

Moreover, in view of (i) and 36.2

(ii) two objects of (CW^{−1})^{({2}.{1,3})}(X, Y) are in the same connected component iff they are the top row and the bottom row in a commutative diagram in C of the form

\xymatrix{X & · \ar[l] \ar[r] & · & Y \ar[l] \\ X \ar[u]^{1} \ar[d]_{1} & · \ar[l]_{v_{3}} \ar[u]_{v_{1}} \ar[r] \ar[d]^{w_{1}} & · \ar[u]_{v_{2}} \ar[d]^{w_{2}} & Y \ar[u]_{1} \ar[l]_{u_{3}} \ar[d]^{1} \\ X & · \ar[l]_{v_{4}} \ar[r] & · & Y \ar[l]_{u_{4}} \\ X \ar[u]^{1} & · \ar[l] \ar[u]_{u_{1}} \ar[r] & · \ar[u]_{u_{2}} & Y \ar[l] \ar[u]_{1}}

in which u_{1} and u_{2} are in U, v_{1} and v_{2} are in V and the w’s are in W, and

(iii) two objects of (C, W^{−1})_{{U,V }}^{({2},{1,3})}(X, Y) are in the same connected component iff they are the top row and the bottom row in such a diagram in which all the u’s are in U, all the v’s are in V and the w’s are in W.

Before proving this result we first use it to obtain the earlier mentioned

36.4. Saturation proposition. Let C be a homotopical category with localization functor γ: C → Ho C (33.9), which admits a 3-arrow calculus (36.1). Then C is saturated, i.e. (33.9) a map f ∈ C is a weak equivalence iff the map γf ∈ Ho C is an isomorphism and moreover ( 33.9(v) or 36.1(iv)) so are, for every homotopical category D, the homotopical functor categories (33.2)

Fun(D, C)  and  Fun_{w}(D, C).

Proof. Let {U, V } be a 3-arrow calculus on C, let f: A → B ∈ C be a map such that γf ∈ Ho C is an isomorphism and let (36.3(i))

\xymatrix{B & B′ \ar[l]_{v} \ar[r]^{g} & A′ & A \ar[l]_{u}}

be a zigzag in C with u ∈ U and v ∈ V which represents (γf)^{−1}. Then one can form the commutative diagram

\xymatrix{A \ar[r]_{f} & B \\ · \ar[u]^{v_{1}} \ar[r]_{f_{1}} & B′ \ar[u]^{v} \ar[r]_{g} & A′ \ar[r]_{f_{2}} & · \\  &  & A \ar[u]^{u} \ar[r]_{f} & B \ar[u]^{u_{2}}}

in which the squares are as in 36.1(ii) and (i). In this diagram the zigzags

[figure: ·; f; ·; ·; v; g; ·; u; and; ·; ·; v; g; ·; ·; u; f; ·]

represent identity maps in Ho C and hence so do the zigzags

\xymatrix{· & · \ar[l]_{v_{1}} \ar[r]^{gf_{1}} & · & · \ar[l]_{u} & and & · & · \ar[l]_{v} \ar[r]^{f_{2}g} & · & · \ar[l]_{u_{2}}}

This implies, in view of 36.3(ii) and the “two out of three” property (33.1) that gf_{1} and f_{2}g are weak equivalences and so are therefore in view of the “two out of six” property (33.1), the maps g, f_{1} and f_{2} and another application of the two out of three property now yields the desired result that f is a weak equivalence.

It thus remains to give a

36.5. Proof of 36.3. That the second map in 36.3(i) is an isomorphism follows from the fact that (36.1) every restricted zigzag in C of the form X ← · → · ← Y can, in a functorial manner, be embedded as the top row in a zigzag in W^{−1}CW^{−1}(X, Y) of the form

\xymatrix{X \ar[d]_{1} & · \ar[l] \ar[d]_{u_{1}} \ar[r] & · \ar[d] & Y \ar[l] \ar[d]^{1} \\ X & · \ar[l]_{v_{1}} \ar[r] & · & Y \ar[l] \\ X \ar[u]^{1} & · \ar[l]_{v_{4}} \ar[u]^{v_{3}} \ar[r] & · \ar[u]_{v_{2}} & Y \ar[l]_{u_{2}} \ar[u]_{1}}

in which the u’s are in U and the v’s are in V and the squares in the middle are as in 36.1(i) and (ii) respectively.

That the first map in 36.3(i) is also an isomorphism, we will show by proving that, in the commutative diagram (34.3)

\xymatrix{π_{0}(C, W^{−1})^{(\{2\},\{1,3\})}(X, Y) \ar[d]_{p} \ar[r]^{^{incl.^{∗}}} & colim^{T} π_{0}(C, W^{−1})^{T} (X, Y) \ar[d]^{r} \ar@{-->}[dl]_{q} \\ colim^{ET} π_{0}(C, W^{−1})^{ET} (X, Y) \ar[r]^{_{incl._{∗}}} & colim^{T} π_{0}(C, W^{−1})^{T} (X, Y)}

in which ET ⊂ T denotes the image of T in T under the functor E: T → T given by the formula

E({i_{1}, ⋯, i_{a}}, {j_{1}, ⋯ j_{b}}) = ({i_{1} + 1, ⋯, i_{a} + 1}, {1, j_{1} + 1, ⋯, j_{b} + 1, a + b + 2})

and q is the map which sends, for every restricted zigzag in C from X to Y, the element containing it to the element containing the zigzag obtained from it by adding a backward identity map on both sides, the maps p and r (and hence the other three maps) are isomorphisms.

That the map r is an isomorphism follows readily from 33.8(i), but in order to be able to prove that the map p is also an isomorphism we first note that p admits a factorization

π_{0}(C, W^{−1})^{({2},{1,3})}(X, Y) ⟶^{p′} colim^{F ET} π_{0}(C, W^{−1})^{F ET} (X, Y)

⟶^{incl._{∗}} colim^{ET} π_{0}(C, W^{−1})^{ET} (X, Y)

in which F ET ⊂ ET is the image of ET in ET under the functor F: ET → ET given by the formula

F ({i_{1}, ⋯, i_{c}}, {1, j_{1}, ⋯, j_{d}, c + d + 2}) = ({2, ⋯, c + 1}, {1, c + 2}) .

That the second of these maps is an isomorphism then follows from the observations that the functor F is the identity on F ET and comes with a natural transformation f: F → 1_{ET} which sends each object (T_{+}, T_{−}) ∈ ET to the unique monomorphism F (T_{+}, T_{−}) → (T_{+}, T_{−}) ∈ ET and that, for every object (T_{+}, T_{−}) ∈ ET the map (34.3)

π_{0}(f(T_{+}, T_{−}))_{∗}: π_{0}(C, W^{−1})^{F (T_{+},T_{−})}(X, Y) ⟶ π_{0}(C, W^{−1})^{(T_{+},T_{−})}(X, Y)

Draft: May 14, 2004

is an isomorphism, which last observation can be verified by iterated application of the fact that, in view of the presence of the 3-arrow calculus {U, V } (36.1), one can, in a functorial manner, embed every restricted zigzag in C of the form

[figure: ·; ·; ···; ·; ·; ···; ·; ·]

in which each ⋯ indicates a possibly empty sequence of forward maps, in a commutative diagram in C of the form

[figure: ·; 1; ·; 1; ···; ·; 1; ·; u; ···; ·; u′; ·; 1; ·; ·; ···; ·; ·; v; ···; ·; ·; ·; 1; ·; v′; ···; ·; v; ·; 1; 1; ···; ·; 1; ·; 1]

in which u, u′ ∈ U, v, v′ ∈ V and u′ and v′ are obtained from u and v by iterated applications of 36.1(i) and 36.1(ii) respectively.

To finally prove that the map p′ is also an isomorphism we note that, for every restricted zigzag of the form

\xymatrix{X & · \ar[l]_{w_{1}} \ar[r]^{z} & · & Y \ar[l]_{w_{2}}}

p′ sends the element containing it to the element containing the zigzag

\xymatrix{X & · \ar[l]_{1} & · \ar[l]_{w_{1}} \ar[r]^{z} & · & · \ar[l]_{w_{2}} & Y \ar[l]_{1}}

which, in view of the commutativity of the diagram

\xymatrix{X & · \ar[l]_{1} & · \ar[l]_{w_{1}} \ar[r]^{z} & · & · \ar[l]_{w_{2}} & Y \ar[l]_{1} \\ X \ar[d]_{1} \ar[u]^{1} & · \ar[l]_{w_{1}} \ar[u]^{w_{1}} \ar[d]_{1} & · \ar[l]_{1} \ar[u]^{1} \ar[r]^{z} \ar[d]_{1} & · \ar[u]_{1} \ar[d]^{1} & · \ar[l]_{w_{2}} \ar[u]_{1} \ar[d]^{w_{2}} & Y \ar[l]_{1} \ar[u]_{1} \ar[d]^{1} \\ X & · \ar[l]^{w_{1}} & · \ar[l]^{1} \ar[r]_{z} & · & · \ar[l]^{1} & Y \ar[l]^{w_{2}}}

also contains its bottom row and hence the original zigzag

\xymatrix{X & · \ar[l]_{w_{1}} \ar[r]^{z} & · & Y \ar[l]_{w_{2}}}

and that therefore the map p′ is exactly the map induced by the inclusion of ({2}, {1, 3}) in F ET, and the desired result now follows immediately from the observation that ({2}, {1, 3}) is the terminal object of F ET.

37. Homotopical uniqueness

In classical category theory one often notes that certain objects in a category are canonically isomorphic or unique up to a unique isomorphism in the sense that, for any two of these objects X and Y, there is exactly one map X → Y in the category and that this unique map is an isomorphism (for instance (see 37.3 and 37.4) because they have a common universal property) and our aim in this section is to describe a homotopical version of this phenomenon.

We start with a slight reformulation of the above mentioned classical results.

37.1. Categorical uniqueness. Given a category Y and a non-empty set I (32.3), we will say that objects Y_{i} ∈ Y (i ∈ I) are canonically isomorphic or categorically unique if

(i) the full subcategory of Y spanned by these objects

or equivalently

(ii) the categorically full subcategory of Y spanned by these objects, i.e. the full subcategory spanned by these objects and all isomorphic ones

is categorically contractible in the following sense.

37.2. Categorically contractible categories. A category G will be called categorically contractible if

a) G is a non-empty groupoid in which there is exactly one isomorphism between any two objects,

or equivalently

b) the unique functor G → [0] (where [0] denotes the category which consists of only one object 0 and its identity map) is an equivalence of categories.

One then readily verifies that this is equivalent to requiring that G be a non-empty category which has one, and hence all, of the following properties:

(i) for one, and hence every, object G ∈ G, the identity functor 1_{G}: G → G is naturally isomorphic to the constant functor cst_{G}: G → G which sends every map of G to the identity map of the object G,

(ii) every object of G is an initial object, and (iii) every object of G is a terminal object.

Also not difficult to show is that

(iv) if G is a categorically contractible category, then so is its opposite G^{op},

(v) if G is a categorically contractible category then so is, for every category D, the diagram category G^{D} (33.2),

and last but not least

(vi) if Y is a category, then the full subcategory of Y spanned by the initial (resp. terminal) objects is either empty or categorically contractible

which property is behind the notion of

37.3. Universal properties. In view of 37.2(vi) one can sometimes verify the categorical uniqueness (37.1) of a non-empty set of objects in a category Y by noting that these objects are initial (or terminal) objects of Y . This is for instance what happens when one makes the somewhat imprecise statement that

(i) some objects in a category X are unique up to a unique isomorphism because they have a certain initial (or terminal) universal property,

as what one really means by this statement is that

(ii) these objects together with some, often not explicitly mentioned, additional structure are initial (or terminal) objects in the category of all objects of X with such additional structure.

37.4. Examples. Let Y be a category and let A, B and C be objects of Y . Then, if they exist,

(i) the coproducts of A and B, i.e. the pairs consisting of an object Q ∈ Y and an (often suppressed) diagram in Y of the form A → Q ← B which is an initial object in the category of all such pairs, are canonically isomorphic,

(ii) dually the products of B and C, i.e. the pairs consisting of an object P ∈ Y and an (often suppressed) diagram in Y of the form B ← P → C which is a terminal object in the category of all such pairs, are canonically isomorphic, and

(iii) somewhat less obviously, the coproducts of A with a product of B and C, i.e. the quadruples consisting of a pair of objects P, R ∈ Y and an (often suppressed) pair of diagrams in Y of the form A → R ← P and B ← P → C such that (iii)′ the pair consisting of P and B ← P → C is a product of B and C

(ii), and (iii)′′ the pair consisting of R and A → R ← P is a coproduct of A and

P, are canonically isomorphic.

In this last case the quadruples satisfying (iii)′ and (iii)′′ are initial objects in the category of the quadruples satisfying (iii)′′, but not in the category of all quadruples.

The obvious homotopical analog of categorical uniqueness (37.1) is the following notion of

37.5. Homotopical uniqueness. Given a homotopical category Y and a non-empty set I, we will say that objects Y_{i} ∈ Y (i ∈ I) are homotopically unique or canonically weakly equivalent if the homotopically full subcategory of Y spanned by these objects, i.e. the full subcategory spanned by these objects and all weakly equivalent ones is homotopically contractible in the following sense.

37.6. Homotopically contractible categories. A homotopical category G (33.1) will be called homotopically contractible if

a) the unique functor G → [0] (37.2) is a homotopical equivalence of homo-

topical categories (33.1)

or equivalently

b) there exists an object G ∈ G with the property that the identity functor 1_{G}: G → G is naturally weakly equivalent (33.1) to the constant functor cst_{G}: G → G (37.2).

This is also equivalent to the requirement that

c) G is a non-empty category with the property that, for every object G ∈ G, the identity functor 1_{G}: G → G is naturally weakly equivalent to the constant functor cst_{G}: G → G

in view of the fact that iterated application of the two out of three property yields that

(i) if G is homotopically contractible then every map in G is a weak equivalence.

Draft: May 14, 2004

One also readily verifies that

(ii) if G is homotopically contractible, then the homotopy category Ho C of

C (33.8) is categorically contractible (37.2), (iii) if every weak equivalence in G is an isomorphism, then G is homotopi-

cally contractible iff it is categorically contractible, and (iv) if G is homotopically contractible, then so are its opposite G^{op} and, for

every homotopical category D, the homotopical diagram categories G^{D} and (G^{D})_{w} (33.2).

Moreover in the next section (in 38.2) we will define notions of homotopically initial and homotopically terminal objects in a homotopical category with the properties that (as we will prove in 38.5)

(v) if G is a homotopically contractible category, then every object of G is

both homotopically initial and homotopically terminal

and, last but not least,

(vi) if Y is a homotopical category, then the full subcategory of Y spanned by the homotopically initial (resp. homotopically terminal) objects is a homotopically full subcategory (37.5) which is either empty or homotopically contractible.

This last property then suggests the following formulation of

37.7. Homotopical universal properties. In view of 37.6(vi) one can sometimes verify the homotopical uniqueness (37.5) of a non-empty set of objects in a homotopical category Y by noting that these objects are homotopically initial (or terminal) objects of Y and, as in the categorical case (37.3), it is therefore sometimes convenient to make the somewhat imprecise statement that

(i) some objects in a homotopical category X are homotopically unique because they have a certain homotopically initial (or terminal) homotopically universal property

as a shorthand for saying that

(ii) these objects together with some, not necessarily explicitly mentioned, additional structure are homotopically initial (or terminal) objects in the category of all objects of X with such additional structure.

38. Homotopically initial and terminal objects

This last section is devoted to the homotopically initial and terminal objects mentioned in 37.6 and 37.7.

As it is not immediately clear how to define homotopically initial and terminal objects we start with some

38.1. Motivation. Our definition of homotopically initial and terminal objects is motivated by the following unorthodox characterization of initial and terminal objects:

(i) Given a category Y , an object Y ∈ Y is initial (resp. terminal) iff there exists a natural transformation (37.2)

f: cst_{Y} → 1_{Y} (resp. f: 1_{Y} → cst_{Y})

such that the map fY : Y → Y ∈ Y is an isomorphism which one verifies by noting that every map m: Y′ → Y ∈ Y gives rise to commutative diagrams

\xymatrix{ & Y \ar[dd]^{fY}_{≈} &  &  & Y \ar[dd]^{m} \\ Y \ar[ur]^{fY}_{≈} \ar[dr]^{≈}_{fY} &  & and & Y \ar[ur]^{fY} \ar[dr]_{fY′} \\  & Y &  &  & Y′}

of which the first implies that fY = 1_{Y}, while the second, together with this equality, yields that m = fY′.

This suggests that, given a homotopical category Y (33.1), one considers objects Y ∈ Y for which there exists a zigzag of natural transformations between homotopical functors

from cst_{Y} to 1_{Y}  (resp. from 1_{Y} to cst_{Y})

such that (ii) the backward natural transformations are natural weak equivalences, and (iii) each of the forward natural transformations sends Y to a weak equiva-

lence in Y .

Such objects clearly have the property mentioned in 37.6(v). However, in order to be able to prove that they also have the property mentioned in 37.6(vi), we had to require in addition to (ii) and (iii) that

(iv) all but (at most) one of the forward natural transformations are also

natural weak equivalences

unless Y admits a 3-arrow calculus (36.1) in which case this would be a consequence of 36.3.

We therefore define as follows

38.2. Homotopically initial and terminal objects. Given a homotopical category Y (33.1), an object Y ∈ Y will be called homotopically initial (resp. terminal) if there exists a zigzag of natural transformations

cst_{Y} ⋯ F_{0} ⟶^{f} F_{1} ⋯ 1_{Y} (resp. 1_{Y} ⋯ F_{1} ⟶^{f} F_{0} ⋯ cst_{Y})

in which

(i) the ⋯ ’s denote (possibly empty) zigzags of natural weak equivalences, and

(ii) the map fY ∈ Y is a weak equivalence. This definition then readily implies

38.3. Proposition. Let Y be a homotopical category and let Y ∈ Y be an object. Then

(i) if Y is homotopically initial (resp. terminal), then so is every object of Y which is weakly equivalent (33.2) to Y,

(ii) if Y is homotopically initial (resp. terminal), then the object γY ∈ Ho Y

(33.9) is initial (resp. terminal), (iii) if Y is initial (resp. terminal), then Y is homotopically initial (resp.

terminal), and (iv) if every weak equivalence in Y is an isomorphism and Y is homotopically

initial (resp. terminal), then Y is initial (resp. terminal).

One also has

38.4. Proposition. Given a homotopical category Y , two subcategories Y_{1} and Y_{2} ⊂ Y of which Y_{1} (resp. Y_{2}) is categorically contractible (37.2) and two objects Y_{1} ∈ Y_{1} and Y_{2} ∈ Y_{2},

(i) a map Y_{1} → Y_{2} ∈ Y is a homotopically initial (resp. terminal) object (38.2) of the comma category (Y_{1} ↓ Y_{2}) iff

(ii) it is a homotopically initial (resp. terminal) object of its subcategory (Y_{1} ↓ Y_{2}) (resp. (Y_{1} ↓ Y_{2})).

Proof (of the initial case). This follows readily from the observation that, for every object Y′_{1} → Y_{2} ∈ (Y_{1} ↓ Y_{2}) there is a unique map Y_{1} → Y_{2} ∈ (Y_{1} ↓ Y_{2}) and a unique isomorphism Y′_{1} ≈ Y_{1} such that the following diagram commutes

\xymatrix{Y′_{1} \ar[dr] \ar[dd]_{≈} \\  & Y_{2} \\ Y_{1} \ar[ur]}  .

It thus remains to give a

38.5. Proof of 37.6(v) and (vi). Proposition 37.6(v) follows readily from 37.6(i) and in order to prove (the initial half of) 37.6(vi) it suffices, in view of 38.3(i), to show that, for every two homotopically initial objects Y, Y′ ∈ Y and natural transformations

f: F_{0} → F_{1}  and  f′: F′_{0} → F′_{1}

as in 38.2, the map fY′ ∈ Y is a weak equivalence. To do this consider the commutative diagram in Y

\xymatrix{F_{0}F′_{0}F_{0}Y′ \ar[r]^{fF′_{0}F_{0}Y′} \ar[d]_{F_{0}f′F_{0}Y′}^{∼} & F_{1}F′_{0}F_{0}Y′ \ar[r]^{F_{1}F′_{0}fY′}_{∼} \ar[d]_{F_{1}f′F_{0}Y′} & F_{1}F′_{0}F_{1}Y′ \ar[d]_{F_{1}f′F_{1}Y′}^{∼} \\ F_{0}F′_{1}F_{0}Y′ \ar[r]^{fF′_{1}F_{0}Y′}_{∼} & F_{1}F′_{1}F_{0}Y′ \ar[r]^{F_{1}F′_{1}fY′} & F_{1}F′_{1}F_{1}Y′}

A straightforward calculation which repeatedly uses the two out of three property (33.1) then yields that

(i) as F_{0} is naturally weakly equivalent to cst_{Y} it sends every map and in particular the map f′F_{0}Y′ to a weak equivalence

(ii) as F_{0} is naturally weakly equivalent to cst_{Y} and F′_{1} is naturally weakly equivalent to 1_{Y}, F′_{1}F_{0}Y′ is weakly equivalent to Y and hence fF′_{1}F_{0}Y′ is a weak equivalence

(iii) as F′_{0} is naturally weakly equivalent to cst_{Y′}, F′_{0}fY′ and therefore F_{1}F′_{0}fY′

are weak equivalences, (iv) as F_{1} is naturally weakly equivalent to 1_{Y}, F_{1}Y′ is weakly equivalent to Y′ and hence f′F_{1}Y′ and therefore F_{1}f′F_{1}Y′ are weak equivalences.

Thus the maps indicated ∼ are weak equivalences and the desired result now follows from the observations that (v) in view of the two out of six property (33.1), the map F_{1}F′_{1}fY′ is a weak

equivalence

and that

(vi) as F_{1} and F′_{1} are both naturally weakly equivalent to 1_{Y}, the map fY′ is a weak equivalence iff the map F_{1}F′_{1}fY′ is so.

As an application of homotopically initial and terminal objects, which, except for a brief mention in 39.3, 41.1 and 41.5, we will only need in the second half of chapter VIII, we end this section with a definition of

38.6. Homotopical Kan extensions. Given two (not necessarily homotopical) functors

p: R ⟶ P and q: R ⟶ Q

between homotopical categories, denote by

(−p ↓ q) (resp. (q ↓ −p))

the homotopical category which has as objects the pairs (r, e) consisting of a (not necessarily homotopical) functor r: P → Q and a natural transformation

e: rp ⟶ q (resp. e: q ⟶ rp)

and which has as maps and weak equivalences t: (r_{1}, e_{1}) → (r_{2}, e_{2}) the natural transformations and natural weak equivalences t: r_{1} → r_{2} such that e_{2}(tp) = e_{1} (resp. (tp)e_{1} = e_{2}), and let

(−p ↓ q)_{w} ⊂ (−p ↓ q) (resp. (q ↓ −p)_{w} ⊂ (q ↓ −p))

denote the full subcategory spanned by the objects (r, e) for which r is a homotopical functor. The Kan extensions of q along p then are defined as

(i) the terminal objects of (−p ↓ q), and (ii) the initial objects of (q ↓ −p).

The first of these, the terminal objects of (−p ↓ q) are the objects of (−p ↓ q) which are “as far as possible to the right” and some authors therefore refer to these Kan extensions as right Kan extensions (with the awkward consequence that left derived functors and left adjoints become right Kan extensions), while other authors, noting that these terminal objects of (−p ↓ q) are “closest to q from the left” call them left Kan extensions. To avoid this confusion we will therefore use neither terminology, but will refer to a Kan extension (r, e) as in (i) as a terminal Kan extension of q along p with e as its counit and to a Kan extension (r, e) as in (ii) as an initial Kan extension of q along p with e as its unit.

Of course the same terminological problem comes up when dealing with homotopical Kan extensions and we therefore define

(i)′ a homotopically terminal Kan extension of q along p as a homo-

topically terminal object (38.2) of (−p ↓ q)_{w}, and (ii)′ a homotopically initial Kan extension of q along p as a homotopically

initial object of (q ↓ −p)_{w}.

and note that

(iii) such homotopically terminal (resp. initial) Kan extensions of q along p, if they exist, are homotopically unique (37.5).

CHAPTER VII Deformable Functors and Their Approximations

39. Introduction

39.1. Summary. In dealing with homotopical categories one often runs into functors between such categories which are not homotopical, but which still have “homotopical meaning” because they are homotopical on a so-called left (or right) deformation retract of the domain category. Moreover such functors, which we will call left (or right) deformable, frequently are part of deformable adjunctions i.e. adjunctions of which the left adjoint is left deformable and the right adjoint is right deformable. Our main aim in this chapter then is

(i) to show that such a left (or right) deformable functor f: X → Y has left (or right) approximations, i.e. pairs (k, a) consisting of a homotopical functor k: X → Y and a natural transformation

a: k ⟶ f (or a: f ⟶ k)

which are homotopically terminal (or initial) objects (38.2) in the homotopical category of all such pairs,

(ii) to show that, for such a deformable adjunction f : X ↔ Y : f′, its adjunction induces, for every left approximation (k, a) of f and right approximation (k′, a′) of f′, a derived adjunction (33.9)

Ho k: Ho X ⟷ Ho Y :Ho k′

and (iii) investigate the behavior of these approximations and derived adjunctions

under composition. In more detail:

39.2. Deformable functors and deformable adjunctions. Given a homotopical category X,

(i) a left (or right) deformation of X will be a pair (r, s) which consists of a homotopical functor r: X → X and a natural weak equivalence

s: r → 1_{X} (or s: 1_{X} → r)

and a left (or right) deformation retract of X will be a full subcategory X_{0} ⊂ X for which there exists a left (or right) deformation (r, s) such that X_{0} contains the image of r.

We then

(ii) call a not necessarily homotopical functor f: X → Y between homotopical categories left (or right) deformable if f is homotopical on a left (or right) deformation retract of X, and

117 Draft: May 14, 2004

(iii) call an adjunction f : X ↔ Y : f′ between homotopical categories a deformable adjunction if the left adjoint f is left deformable and the right adjoint f′ is right deformable.

A useful property of deformable functors is that they have

39.3. Approximations. A left (or right) approximation of a not necessarily homotopical functor f: X → Y between homotopical categories will be a homotopically terminal (or initial) Kan extension of f along 1_{X} (38.6), i.e. a pair (k, a) consisting of a homotopical functor k: X → Y and a natural transformation

a: k → f (or a: f → k)

which is a homotopically terminal (or initial) object (38.2) in the homotopical category of all such pairs. Thus (37.6(vi))

(i) all such left (or right) approximations of f, they they exist, are canoni-

cally weakly equivalent (37.5).

Moreover

(ii) if (r, s) is a left (or right) deformation of X such that f is homotopical on the full subcategory spanned by the image of r, then the pair (fr, fs) is a left (or right) approximation of f, and hence

(iii) a sufficient condition for the existence of left (or right) approximations

of f is that f be left (or right) deformable (39.2). To next deal with the question when, for two composable deformable functors,

their composition is also deformable and the compositions of their approximations are approximations of their composition we introduce the notions of

39.4. Deformable and locally deformable composable pairs of functors. Given two functors f_{1} : X → Y and f_{2} : Y → Z between homotopical categories, we call the pair (f_{1}, f_{2}) locally left (resp. right) deformable if there exist left (or right) deformation retracts X_{0} ⊂ X and Y_{0} ⊂ Y (39.2) such that

(i) f_{1} is homotopical on X_{0}, (ii) f_{2} is homotopical on Y_{0}, and (iii) f_{2}f_{1} is homotopical on X_{0}

and call the pair (f_{1}, f_{2}) left (or right) deformable if there exist such deformation retracts for which in addition

(iv) f_{1} sends all of X_{0} into Y_{0} (which last condition together with (i) and (ii) implies (iii)).

Now we can formulate the desired results on

39.5. Compositions of deformable functors. Given a composable pair of left (or right) deformable functors f_{1} : X → Y and f_{2} : Y → Z, a sufficient condition in order that

(i) their composition f_{2}f_{1} : X → Z is also left (or right) deformable, and (ii) all compositions of a left (or right) approximation of f_{1} with a similar

approximation of f_{2} is such an approximation of f_{2}f_{1}

is that (iii) the pair (f_{1}, f_{2}) be left (or right) deformable (39.4).

However not infrequently it suffices to verify only that the pair (f_{1}, f_{2}) is deformable (39.4), because the categories and functors involved have additional properties, which allow one to apply the following result on

39.6. Compositions of deformable adjunctions. Given two composable deformable adjunctions (39.2)

f_{1}: X ⟷ Y :f′_{1}  and  f_{2}: X ⟷ Y :f′_{2}

for which the pairs (f_{1}f_{2}) and (f′_{2}, f′_{1}) are locally left and right deformable (39.4) respectively, a sufficient condition in order that

(i) the pairs (f_{1}, f_{2}) and (f′_{2}, f′_{1}) are left and right deformable (39.4) respec-

tively

is that either

(ii) the pair (f′_{2}, f′_{1}) is right deformable and the homotopical category Z is

saturated (33.9)

or dually that

(iii) the pair (f_{1}, f_{2}) is left deformable and the homotopical category X is

saturated. The proof of this involves

39.7. Derived adjunctions. Given a deformable adjunctions f : X ↔ Y : f′ (39.2),

(i) its adjunction induces, for every left approximation (k, a) of f and right approximation (k′, a′) of f′ (39.3), a derived adjunction (33.9)

Ho k: Ho X ⟷ Ho Y :Ho k′

which is natural with respect to these approximations,

and given two composable deformable adjunctions

f_{1}: X ⟷ Y :f′_{1}  and  f_{2}: Y ⟷ Z :f′_{2}

for which the pairs (f_{1}, f_{2}) and (f′_{2}, f′_{1}) are respectively locally left and right deformable (39.4)

(ii) the compositions of their derived adjunctions and the derived adjunctions of their composition are connected by a conjugate pair of natural transformations.

The second of these results then is used to prove 39.6, while we use the first in a brief discussion of

39.8. A Quillen condition. We end the chapter by showing that, given a deformable adjunction f: X ↔ Y :f′ (39.2),

(i) a sufficient condition in order that the left approximations of f and the right approximations of f′ (39.3) are homotopically inverse homotopical equivalences of homotopical categories (33.1)

is that

(ii) the adjunction satisfies the Quillen condition that there exist a left deformation retract X_{0} ⊂ X on which f is homotopical (39.2) and a right deformation retract Y_{0} ⊂ Y on which f′ is homotopical such that,

Draft: May 14, 2004

for every pair of objects X_{0} ∈ X_{0} and Y_{0} ∈ Y_{0}, a map fX_{0} → Y_{0} ∈ Y is a weak equivalence iff its adjunct X_{0} → f′Y_{0} ∈ X is so

and that

(iii) if the homotopical categories X and Y are saturated (33.9), then this condition is also necessary.

39.9. Organization of the chapter. After recalling, in the remainder of this section, some results on adjoint functors which we will need in the last three sections of this chapter as well as in the next chapter, we introduce deformable functors (in§40) and their approximations (in §41) and discuss their compositions (in §42). The next two sections (§43 and §44) are devoted to the derived adjunctions which we mention in 39.7 and the last section deals with the Quillen condition (39.8).

It thus remains to review some results involving

39.10. Adjunctions [Mac71, Ch. IV]. The unit and counit of an adjunction f: X ↔ Y :f′ are the natural transformations

η: 1_{X} ⟶ f′f  and  ε: ff′ ⟶ 1_{Y}

which assign to an object X ∈ X and Y ∈ Y the adjuncts of the identity maps of the objects fX ∈ Y and f′Y ∈ X respectively. Some of their properties are:

(i) the compositions

f ⟶^{fη} ff′f ⟶^{εf} f  and  f′ ⟶^{ηf′} f′ff′ ⟶^{f′ε} f′

are identity natural transformations of f and f′ respectively, (ii) η and ε are both natural isomorphisms iff one (and hence both) of the

functors f and f′ are equivalences of categories, (iii) given two adjunctions

f_{1}: X ⟷ Y :f′_{1}  and  f_{2}: Y ⟷ Z :f′_{2}

with units η_{1} and η_{2} and counits ε_{1} and ε_{2}, the unit and the counit of their composition f_{2}f_{1}: X ↔ Z :f′_{1}f′_{2} are the compositions

1_{X} ⟶^{η_{1}} f′_{1}f_{1} ⟶^{η_{2}} f′_{1}f′_{2}f_{2}f_{1}  and  f_{2}f_{1}f′_{1}f′_{2} ⟶^{ε_{1}} f_{2}f′_{2} ⟶^{ε_{2}} 1_{Z} ,

and (iv) given two functors

f: X ⟶ Y  and  f′: Y ⟶ X

and a natural transformation

e: 1_{X} ⟶ f′f (resp. e: ff′ ⟶ 1_{Y}),

there is at most one adjunction f: X ↔ Y :f′ with e as its unit (resp. counit).

Given two adjunctions

f: X ⟷ Y :f′  and  g: X ⟷ Y :g′

two natural transformations

h: f ⟶ g  and  h′: g′ ⟶ f′

are called conjugate (with respect to the given adjunctions) if, for every pair of objects X ∈ X and Y ∈ Y , the following diagram

[figure: Y (gX, Y); ≈; h^{∗}; X(X, g′Y); h′_{∗}; Y (fX, Y); ≈; X(X, f′Y)]

in which the horizontal isomorphisms are the adjunction isomorphisms, commutes, which is the case iff any one of the following four diagrams, in which ε_{1} and ε_{2} and η_{1} and η_{2} denote the counits and the units of these adjunctions, commutes

\xymatrix{f \ar[r]^{h} \ar[d]_{η_{2}} & g & g′ \ar[r]^{h′} \ar[d]_{η_{1}} & f′ \\ fg′g \ar[r]^{h′} & ff′g \ar[u]_{ε_{1}} & f′fg′ \ar[r]^{h} & f′gg′ \ar[u]_{ε_{2}}}

\xymatrix{fg′ \ar[r]^{h′} \ar[d]_{h} & ff′ \ar[d]^{ε_{1}} & 1_{X} \ar[r]^{η_{1}} \ar[d]_{η_{2}} & f′f \ar[d]^{h} \\ gg′ \ar[r]^{ε_{2}} & 1_{Y} & g′g \ar[r]^{h′} & f′g}

Some of their properties are:

(v) given two adjunctions

f: X ⟷ Y :f′  and  g: X ⟷ Y :g′

every natural transformation f → g (resp. g′ → f′) has a unique conjugate and the conjugate of a natural isomorphism is also a natural isomorphism, and

(vi) given an adjunction f: X ↔ Y :f′, a pair of functors

g: X ⟶ Y  and  g′: Y ⟶ X

and a pair of natural isomorphisms

h: f ⟶ g  and  h′: g′ ⟶ f′ ,

there is a unique adjunction g: X ↔ Y : g′ such that h and h′ are conjugate with respect to the adjunctions

f: X ⟷ Y :f′  and  g: X ⟷ Y :g′ .

40. Deformable functors

In this section we introduce left and right deformable functors and discuss some immediate consequences of their definition.

We start with the auxiliary notions of

40.1. Left and right deformations and deformation retracts. A left (resp. a right) deformation of a homotopical category X (33.1) will be a pair (r, s) consisting of

(i) a homotopical functor r: X → X, and

(ii) a natural weak equivalence

s: r → 1_{X} (resp. s: 1_{X} → r)

and a left (resp. a right) deformation retract of X will be a full subcategory X_{0} ⊂ X for which there exists a left (resp. a right) deformation (r, s) of X into X_{0}, i.e. such that fX ∈ X_{0} for every object X ∈ X.

An immediate consequence of this definition is that

(iii) if X_{0} is a left or right deformation retract of X, then so is every full

subcategory of X which contains X_{0}. Now we can define

40.2. Left and right deformable functors. A (not necessarily homotopical) functor f: X → Y between homotopical categories will be called left (resp. right) deformable if

(i) there exists a left (resp. a right) f-deformation retract, i.e. a left (resp. a right) deformation retract X_{0} ⊂ X (40.1) on which f is homotopical.

It is sometimes convenient to have a somewhat more explicit description of such deformability and we therefore also consider the notion of a left (resp. a right) fdeformation, i.e. a left (resp. a right) deformation (r, s) of X (40.1) such that

(ii) f is homotopical on the full subcategory spanned by the image of the

functor r

or equivalently

(ii)′ the functor fr is a homotopical functor and the natural transformation

fsr is a natural weak equivalence,

so that

(iii) a functor f: X → Y is left (resp. right) deformable iff there exists a left

(resp. a right) f-deformation. Moreover clearly (iv) every homotopical functor is both left and right deformable, and

(v) if f_{1}, f_{2} : X → Y are functors between homotopical categories which are naturally weakly equivalent (33.1), and one of them is left or right deformable, then so is the other.

Next we note that following “f-deformation retract version” of 40.1(iii).

40.3. Proposition. Let f: X → Y be a left (resp. a right) deformable functor, let X_{0} ⊂ X be a left (resp. a right) f-deformation retract (40.2) and let X_{1} ⊂ X be a full subcategory containing X_{0}. Then X_{1} is a left (resp. a right) fdeformation retract iff, for every object X_{p} ∈ X_{1}, there exists an object X_{p,0} ∈ X_{0} and a weak equivalence

b_{p}: X_{p,0} → X_{p} (resp. b_{p}: X_{p} → X_{p,0})

such that the map fb_{p} ∈ Y is also a weak equivalence.

Draft: May 14, 2004

Proof (of the left half). The “only if” part of the proposition is obvious and to prove the “if” part one has to show that, for every pair of objects X_{p}, X_{q} ∈ X_{1} and weak equivalences a: X_{p} → X_{q} ∈ X, the map fa ∈ Y is also a weak equivalence. To do this one chooses a left deformation (r, s) of X into X_{0} and notes that all the other outside maps in the commutative diagram

\xymatrix{frX_{p,0} \ar[r]^{frb_{p}} \ar[d]_{fsX_{p,0}} & frX_{p} \ar[r]^{fra} \ar[d]_{fsX_{p}} & frX_{q} \ar[d]_{fsX_{q}} & frX_{q,0} \ar[l]_{frb_{q}} \ar[d]_{fsX_{q,0}} \\ fX_{p,0} \ar[r]_{fb_{p}} & fX_{p} \ar[r]_{fa} & fX_{q} & fX_{q,0} \ar[l]^{fb_{q}}}

are weak equivalences and applies the two out of three property (33.1).

Somewhat less obvious is the existence of a unique maximal left or right deformation retract, i.e.

40.4. Proposition. Let f: X → Y be a left (resp. a right) deformable functor. Then

(i) the full subcategory of X spanned by the left (resp. the right) f-deformation retracts is itself a left (resp. a right) f-deformation retract, and (ii) if X_{0} ⊂ X is any left (resp. right) f-deformation retract then this unique maximal left (resp. right) f-deformation retract (see (i)) is the full subcategory of X spanned by the objects X_{p} ∈ X for which there exists an object X_{p,0} ∈ X_{0} and a weak equivalence

b_{p}: X_{p,0} → X_{p} (resp. b_{p}: X_{p} → X_{p,0})

such that the map fb_{p} ∈ Y is also a weak equivalence.

Proof (of the left half). Part (ii) is an immediate consequence of 40.3 and part (i) and to prove (i) it suffices to show that, for every two left f-deformation retracts X_{1} and X_{2} and every weak equivalence w: X_{1} → X_{2} ∈ X with X_{1} ∈ X_{1} and X_{2} ∈ X_{2}, the map fw: fX_{1} → fX_{2} ∈ Y is also a weak equivalence. To do this one chooses left f-deformations (r_{1}, s_{1}) and (r_{2}, s_{2}) of X into X_{1} and X_{2} respectively (40.1) and notes that in the commutative diagram in Y

\xymatrix{fr_{1}r_{2}X_{1} \ar[r]^{fs_{1}r_{2}X_{1}} \ar[d]_{fr_{1}s_{2}X_{1}}^{∼} & fr_{2}X_{1} \ar[r]^{fr_{2}w}_{∼} \ar[d]_{fs_{2}X_{1}} & fr_{2}X_{2} \ar[d]_{fs_{2}X_{2}}^{∼} \\ fr_{1}X_{1} \ar[r]^{fs_{1}X_{1}}_{∼} & fX_{1} \ar[r]^{fw} & fX_{2}}

the maps indicated ∼ are weak equivalences. The desired result then is an immediate consequence of the two out of six property (33.1).

A somewhat similar result for f-deformations is

Draft: May 14, 2004

40.5. Proposition. Let f: X → Y be a left (resp. a right) deformable functor and let X_{0} ⊂ X be a left (resp. a right) f-deformation retract (40.2). Then the homotopical category which has

(i) as objects the left (resp. the right) deformations of X into X_{0}, and (ii) for every two such deformations (r_{1}, s_{1}) and (r_{2}, s_{2}), as maps and weak equivalences (r_{1}, s_{1}) → (r_{2}, s_{2}) the natural transformations and natural weak equivalences t: r_{1} → r_{2} such that s_{2}t = s_{1} (resp. ts_{1} = s_{2})

is homotopically contractible (37.6).

Proof (of the left half). This follows readily from the observation that every two left deformations (r_{1}, s_{1}) and (r_{2}, s_{2}) give rise to a zigzag

\xymatrix{(r_{1}, s_{1}) & (r_{1}r_{2}, s_{1}s_{2}) \ar[l]_{r_{1}s_{2}} \ar[r]^{s_{1}r_{2}} & (r_{2}, s_{2})}

in which s_{1}s_{2} denotes the diagonal of the commutative diagram

\xymatrix{r_{1}r_{2} \ar[r]^{s_{1}} \ar[d]_{s_{2}} & r_{2} \ar[d]^{s_{2}} \\ r_{1} \ar[r]^{s_{1}} & 1}

which is natural in both (r_{1}, s_{1}) and (r_{2}, s_{2}).

40.6. Corollary. Let f: X → Y be a left (resp. a right) deformable functor. Then the category of all left (resp. right) f-deformations ( 40.4 and 40.5) is homotopically contractible (37.6).

We end with a brief mention of the notion of

40.7. Left and right deformable natural transformations. Given two homotopical categories X and Y , a natural transformation h: f → g between (not necessarily homotopical) functors X → Y will be called left (resp. right) deformable if there exists a left (resp. a right) h-deformation retract, i.e. a left (resp. a right) deformation retract X_{0} ⊂ X on which both f and g are homotopical, and we similarly will call a left (resp. a right) deformation (r, s) of X a left (resp. a right) h-deformation if (r, s) is both a left (resp. a right) f-deformation and g-deformation.

Note that, if h is a natural weak equivalence, then the following three statements are equivalent:

(i) (r, s) is a left (resp. a right) f-deformation, (ii) (r, s) is a left (resp. a right) g-deformation and (iii) (r, s) is a left (resp. a right) h-deformation.

41. Approximations

Next we consider, for a (not necessarily homotopical) functor f: X → Y between homotopical categories, its left and right approximations, i.e. homotopical functors X → Y which, in a homotopical sense, are closest to f from the left or from the right, and note in particular that

(i) if they exist, then such left or right approximations of f are homotopically unique (37.5), and (ii) a sufficient condition for their existence is that the functor f: X → Y

be left or right deformable (40.2), and (iii) if f is left or right deformable, then its left or right approximations are a kind of “not in the homotopy category” version of Quillen’s total left and right derived functors.

We thus start with defining

41.1. Left and right approximations. Let f: X → Y be a (not necessarily homotopical) functor between homotopical categories, i.e. an object of the homotopical functor category Fun(X, Y ) (33.2). A left (resp. a right) approximation of f then will be a homotopically terminal (resp. initial) Kan extension of f along 1_{X}, i.e. (38.6) a homotopically terminal (resp. initial) (38.2) object of the “homotopical category of homotopical functors X → Y over (resp. under) f”, i.e. the homotopical category

(Fun_{w}(X, Y ) ↓ f)  (resp. (f ↓ Fun_{w}(X, Y )))

which has

(i) as objects the pairs (k, a) consisting of a homotopical functor k: X → Y (i.e. an object of the homotopical functor category Fun_{w}(X, Y ) (33.2)) and a natural transformation

a: k → f (resp. a: f → k)

and

(ii) for every pair of objects (k_{1}, a_{1}) and (k_{2}, a_{2}) as maps and weak equivalences (k_{1}, a_{1}) → (k_{2}, a_{2}) the natural transformations and natural weak equivalences t: k_{1} → k_{2} such that a_{2}t = a_{1} (resp. ta_{1} = a_{2}).

Thus (37.5 and 37.6(vi))

(iii) such a left (resp. right) approximation of f, if it exists, is homotopically unique (37.5).

We now prove the promised existence of

41.2. Approximations of deformable functors. Let f: X → Y be a left (resp. a right) deformable functor (40.2). Then

(i) the functor f has left (resp. right) approximations. In fact, if (40.2) (r, s) is a left (resp. a right) f-deformation of X then

(ii) the pair (fr, fs) is a left (resp. a right) approximation of f.

Proof (of the left half). Every object (k, a) ∈ (Fun_{w}(X, Y ) ↓ f) gives rise to a zigzag

\xymatrix{(k, a) & (kr, as) \ar[l]_{ks} \ar[r]^{ar} & (fr, fs)}

in which ar denotes the diagonal of the commutative diagram

\xymatrix{kr \ar[r]^{ar} \ar[d]_{ks} & fr \ar[d]^{fs} \\ k \ar[r]^{a} & f}

which is natural in (k, a) and in which

Draft: May 14, 2004

(i) the map ks is a weak equivalence, and

(ii) the map ar is so whenever (k, a) = (fr, fs), in which case ar = fsr and (40.2(ii)′) this readily implies (ii).

A special case is provided by

41.3. Approximations of homotopical functors. If f: X → Y is a homotopical functor, then clearly

(i) the pair (f, 1_{f}) consisting of f and its identity natural transformation is

both a left and a right approximation of f,

and more generally (37.6)

(ii) a pair (k, a) consisting of a homotopical functor k: X → Y and a natural transformation a: k → f (resp. a: f → k) is a left (resp. a right) approximation of f iff the natural transformation a is a natural weak equivalence.

In particular

(iii) for every homotopical category X, the left (resp. the right) approximations of the identity functor 1_{X}: X → X are exactly the left (resp. the right) deformations (40.1) of X.

Another immediate consequence of 41.2 is

41.4. Proposition. Let f: X → Y be a left (resp. a right) deformable functor (40.2), let γ′: Y → Ho Y denote the localization functor of Y (33.9) and consider Ho Y as a minimal homotopical category (33.6). Then

(i) the composition γ′f: X → Ho Y is left (resp. right) deformable. In fact,

(ii) every left (resp. right) f-deformation (40.2) is also a left (resp. a right)

γ′f-deformation and hence ( 37.6 and 41.2) (iii) for every left (resp. right) approximation (k, a) of f (41.1), the pair

(γ′k, γ′a) is a left (resp. a right) approximation of γ′f.

And from this in turn we deduce the following

41.5. Relation to Quillen’s total derived functors. Let f: X → Y be a left (resp. a right) deformable functor (40.2), let γ: X → Ho X and γ′: Y → Ho Y be the localization functors of X and Y (33.9) and let (k, a) be a left (resp. a right) approximation of f (41.1). Then the pair (Ho k, γ′a) consisting of the functor Ho k: Ho X → Ho Y and the natural transformation

γ′a: (Ho k)γ = γ′k ⟶ γ′f (resp. γ′a: γ′f ⟶ γ′k = (Ho k)γ)

is a total left (resp. right) derived functor of f in the sense of Quillen [Qui67] (or, in the terminology of 38.6 , a terminal (resp. an initial) Kan extension of γ′f along γ), i.e. the pair ((Ho k)γ, γ′a) is a terminal (resp. an initial) object of the category

(Fun(Ho X, Ho Y ) ↓ γ′f)  (resp. (γ′f ↓ Fun(Ho X, Ho Y )))

which has as objects the pairs (n, c) which consist of a functor n: Ho X → Ho Y and a natural transformation

c: nγ ⟶ γ′f (resp. c: γ′f ⟶ nγ)

and as maps (n_{1}, c_{1}) → (n_{2}, c_{2}) the natural transformations t: n_{1} → n_{2} such that c_{2}(tγ) = c_{1} (resp. (tγ)c_{1} = c_{2}).

Proofs of (the left half of) 41.5.

This follows from 41.4 and the observation that

(i) (Fun_{w}(X, Ho Y ) ↓ γ′f) is a minimal homotopical category (33.6) and hence (38.3) its homotopically terminal objects are its terminal objects,

and that (33.9(ii))

(ii) composition with γ induces an isomorphism

(Fun(Ho X, Ho Y ) ↓ γ′f) ≈ (Fun_{w}(X, Ho Y ) ↓ γ′f).

We end with a mention of

41.6. Left and right approximations of natural transformations. Given two homotopical categories X and Y and a natural transformation h: f → g between (not necessarily homotopical) functors X → Y , let

(Nat_{w}(X, Y ) ↓ h)  (resp. (h ↓ Nat_{w}(X, Y )))

denote “the homotopical category of the natural transformations between homotopical functors X → Y over (resp. under) h”, i.e. the homotopical category which has

(i) as objects the commutative diagrams of the form

\xymatrix{k_{1} \ar[r]^{a_{1}} \ar[d]_{k} & f \ar[d]^{h} \\ k_{2} \ar[r]^{a_{2}} & g}  (resp. \xymatrix{f \ar[r]^{a_{1}} \ar[d]_{h} & k_{1} \ar[d]^{k} \\ g \ar[r]^{a_{2}} & k_{2}})

in which k_{1} and k_{2} are homotopical functors X → Y (and which we will denote by k: (k_{1}, a_{1}) → (k_{2}, a_{2})),

(ii) as maps (k: (k_{1}, a_{1}) → (k_{2}, a_{2})) → (k̄: (k̄_{1}, ā_{1}) → (k̄_{2}, ā_{2})) the commutative diagrams of the form

\xymatrix{k_{1} \ar[rr]^{t_{1}} \ar[dr]^{a_{1}} \ar[ddd]_{k} &  & k̄_{1} \ar[dl]_{ā_{1}} \ar[ddd]^{k̄} \\  & f \ar[d]^{h} \\  & g \\ k_{2} \ar[rr]_{t_{2}} \ar[ur]_{a_{2}} &  & k̄_{2} \ar@{-->}[ul]^{ā_{2}}}  (resp. \xymatrix{k_{1} \ar[rr]^{t_{1}} \ar[ddd]_{k} &  & k̄_{1} \ar[ddd]^{k̄} \\  & f \ar[ul]_{a_{1}} \ar[ur]^{ā_{1}} \ar[d]^{h} \\  & g \ar[dl]^{a_{2}} \ar[dr]_{ā_{2}} \\ k_{2} \ar[rr]_{t_{2}} &  & k̄_{2}})

and

(iii) as weak equivalences the diagrams as in (ii) in which t_{1} and t_{2} are natural weak equivalences.

Draft: May 14, 2004

A left (resp. a right) approximation of h then will be an object k: (k_{1}, a_{1}) → (k_{2}, a_{2}) of

(Nat_{w}(X, Y ) ↓ h)  (resp. (h ↓ Nat_{w}(X, Y )))

(iv) which is homotopically terminal (resp. initial) and for which

(v) the pairs (k_{1}, a_{1}) and (k_{2}, a_{2}) are left (resp. right) approximations of f

and g (41.1) respectively. One then readily verifies that (vi) such a left (resp. right) approximation of h, if it exists, is homotopically

unique (37.5).

Moreover

(vii) a sufficient condition for its existence is that h be left (resp. right) de-

formable (40.7)

as the argument used in the proof of 41.2 yields that (viii) for every left (resp. right) h-deformation (r, s) (40.7)

hr: (fr, fs) ⟶ (gr, gs)

is a left (resp. a right) approximation of h.

42. Compositions

Our aim in this section is to find sufficient conditions on a composable pair of left (or right) deformable functors (40.2) in order that

(i) their composition is also left (or right) deformable, and (ii) every composition of their left (or right) approximation is a left (or right)

approximation of their composition. To do this we start with defining

42.1. Compositions of approximations. Let f_{1} : X → Y and f_{2} : Y → Z be (not necessarily homotopical) functors between homotopical categories. The functors and natural transformations involved in the definition of approximations then give rise to homotopical composition functors (41.1)

(Fun_{w}(X, Y ) ↓ f_{1}) × (Fun_{w}(Y , Z) ↓ f_{2}) ⟶ (Fun_{w}(X, Z) ↓ f_{2}f_{1})

and

(f_{1} ↓ Fun_{w}(X, Y )) × (f_{2} ↓ Fun_{w}(Y , Z)) ⟶ (f_{2}f_{1} ↓ Fun_{w}(X, Z))

which sends an object ((k_{1}, a_{1}), (k_{2}, a_{2})) to the object (k_{2}k_{1}, a_{2}a_{1}) in which a_{2}a_{1} denotes the diagonals of

\xymatrix{k_{2}k_{1} \ar[r]^{a_{2}} \ar[dd]_{a_{1}} & f_{2}k_{1} \ar[dd]^{a_{1}} &  & f_{2}f_{1} \ar[r]^{a_{2}} \ar[dd]_{a_{1}} & k_{2}f_{1} \ar[dd]^{a_{1}} \\  &  & and \\ k_{2}f_{1} \ar[r]^{a_{2}} & f_{2}f_{1} &  & f_{2}k_{1} \ar[r]^{a_{2}} & k_{2}k_{1}}

respectively.

The application of these functors to a pair of approximations does, in general, not produce an approximation. However one has the following

Draft: May 14, 2004

42.2. All or none proposition. If, given a composable pair f_{1} : X → Y and f_{2} : Y → Z of (not necessarily homotopical) functors between homotopical categories, there exist a left (resp. a right) approximation of f_{1} and a left (resp. a right) approximation of f_{2} whose composition is a left (resp. a right) approximation of f_{2}f_{1}, then every composition of a left (resp. a right) approximation of f_{1} with a similar approximation of f_{2} is such an approximation of f_{2}f_{1}.

Proof. This follows readily from the definition of approximations (41.1) and the fact that, in view of 37.6(i) and (vi) and 38.3(i), every homotopical functor C → D sends either all or none of the homotopically terminal or initial objects of C to similar objects of D.

The desired sufficient conditions involve the following notions of

42.3. Deformable and locally deformable composable pairs of functors. Given a composable pair f_{1} : X → Y and f_{2} : Y → Z of (not necessarily homotopical) functors between homotopical categories, merely assuming that the functors f_{1}, f_{2} and f_{2}f_{1} are all three left (resp. right) deformable does not seem to be very useful and we therefore call the pair (f_{1}, f_{2}) locally left (resp. right) deformable if it has the slightly stronger property that there exist left (resp. right) deformation retract X_{0} ⊂ X and Y_{0} ⊂ Y (40.1) such that

(i) f_{1} is homotopical on X_{0} (and hence f_{1} is left (resp. right) deformable (40.2)),

(ii) f_{2} is homotopical on Y_{0} (and hence f_{2} is left (resp. right) deformable),

and (iii) f_{2}f_{1} is homotopical on X_{0} (and hence f_{2}f_{1} is left (resp. right) de-

formable)

and call the pair (f_{1}, f_{2}) left (resp. right) deformable if it has the even stronger property that in addition

(iv) f_{1} sends all of X_{0} into Y_{0},

which together with (i) and (ii) implies (iii).

One then readily verifies that such left (resp. right) deformability is equivalent to the requirement that

(v) there exists a left (resp. a right) f_{1}-deformation (r_{1}, s_{1}) (40.2) and a left (resp. a right) f_{2}-deformation (r_{2}, s_{2}) such that the natural transformation

f_{2}s_{2}f_{1}r_{1} : f_{2}r_{2}f_{1}r_{1} ⟶ f_{2}f_{1}r_{1} (resp. f_{2}s_{2}f_{1}r_{1} : f_{2}f_{1}r_{1} ⟶ f_{2}r_{2}f_{1}r_{1})

is a natural weak equivalence,

as in that case, in view of 40.3, f_{2} is homotopical on the full subcategory of Y spanned by the images of r_{2} and f_{1}r_{1}.

In view of 42.2 this last observation (42.3(v)) together with 41.2(ii) implies the desired

42.4. Sufficient condition. Let f_{1} : X → Y and f_{2} : Y → Z be (not necessarily homotopical) functors between homotopical categories such that the pair (f_{1}, f_{2}) is left (resp. right) deformable (42.3). Then

(i) the composition f_{2}f_{1} : X → Z is left (resp. right) deformable, and (ii) every composition (42.1) of a left (resp. a right) approximation of f_{1} with a similar approximation of f_{2} is such an approximation of f_{2}f_{1}.

Furthermore it follows from 37.6(i) that

42.5. Proposition. Let f_{1} : X → Y and f_{2} : Y → Z be (not necessarily homotopical) functors between homotopical categories such that the pair (f_{1}, f_{2}) is locally left (resp. right) deformable (42.3). Then

(i) the pair (f_{1}, f_{2}) is left (resp. right) deformable (42.3) iff

(ii) every composition (42.1) of a left (resp. a right) approximation of f_{1} with a similar approximation of f_{2} is such an approximation of f_{2}f_{1}.

We end with noting that, if one wants to prove the deformability of a composable pair of deformable functors (i.e. the existence of deformation retracts satisfying 42.3(i)–(iv)) and these functors have the property that

(i) they are each one of an adjoint pair of deformable functors of which the left adjoint is left deformable and the right adjoint is right deformable, (ii) the pair consisting of their adjoints are known to be (or easily verified to

be) deformable, and (iii) the codomain of their composition is saturated (33.9),

then the following proposition implies that it suffices to merely verify the local deformability of the pair (i.e. the existence of deformation retracts satisfying 42.3(i)– (iii)).

42.6. Proposition. Given two composable adjunctions

f_{1}: X ⟷ Y :f′_{1}  and  f_{2}: Y ⟷ Z :f′_{2}

of functors between homotopical categories such that the pair (f_{1}, f_{2}) is locally left deformable (42.3) and the pair (f′_{2}, f′_{1}) is locally right deformable,

(i) if the category Z is saturated (33.9) and the pair (f′_{2}, f′_{1}) is right deformable (42.3), then the pair (f_{1}, f_{2}) is left deformable,

and dually

(ii) if the category X is saturated and the pair (f_{1}, f_{2}) is left deformable, then the pair (f′_{2}, f′_{1}) is right deformable

and hence

(iii) if the categories X and Z are both saturated then the pair (f_{1}, f_{2}) is left deformable iff the pair (f′_{2}, f′_{1}) is right deformable.

Proof. In view of 33.9, 42.3(v) and 42.5 this follows readily from the following technical result.

42.7. Proposition. Let

f_{1}: X ⟷ Y :f′_{1}  and  f_{2}: Y ⟷ Z :f′_{2}

be adjoint pairs of functors for which the pairs (f_{1}, f_{2}) and (f′_{2}, f′_{1}) are locally left and right deformable (42.3) respectively and let

(r_{1}, s_{1}), (r_{2}, s_{2}), (r′_{1}, s′_{1}) and (r′_{2}, s′_{2})

respectively be a left f_{1}- as well as a left f_{2}f_{1}-deformation, a left f_{2}-deformation, a right f′_{1}-deformation and a right f′_{2}- as well as a right f′_{1}f′_{2}-deformation (40.2). Then the natural transformations

Ho(f_{2}s_{2}f_{1}r_{1}): Ho(f_{2}r_{2}f_{1}r_{1}) ⟶ Ho(f_{2}f_{1}r_{1}) and

Ho(f′_{1}s′_{1}f′_{2}r′_{2}): Ho(f′_{1}f′_{2}r′_{2}) ⟶ Ho(f′_{1}r′_{1}f′_{2}r′_{2})

have the property that, if one of them is a natural isomorphism, then so is the other.

Proof. This follows from 39.10(v) and the special case of 44.4 below considered in its proof.

43. Induced partial adjunctions

In preparation for the next section, where in 44.4 we complete the proof of 42.7, we first

(i) observe that, for every adjoint pair of deformable functors between homotopical categories (of which the left adjoint is left deformable and the right adjoint is right deformable), its adjunction induces a “partial” adjunction between the homotopy categories, and

(ii) discuss the behavior of these partial adjunctions under composition. In more detail:

43.1. Deformable adjunctions. An adjunction f : X ↔ Y : f′ between homotopical categories will be called a deformable adjunction if the left adjoint f is left deformable (40.2) and the right adjoint f′ is right deformable.

Similarly (40.7) given two deformable adjunctions f: X ↔ Y :f′ and g: X ↔ Y : g′, a conjugate pair of natural transformations h: f → g and h′: g′ → f′ between them (39.10) will be called a conjugate pair of deformable natural transformations if the natural transformation h: f → g between the left adjoints is left deformable (40.7) and the natural transformation h′: g′ → f′ between the right adjoints is right deformable.

The following result then implicitly defines

43.2. Induced partial adjunctions. Let X and Y be homotopical categories with localization functors γ: X → Ho X and γ′: Y → Ho Y , and let f: X ↔ Y : f′ be a deformable adjunction (43.1). Then

(i) for every left f- and right f′-deformation retract X_{0} ⊂ X and Y_{0} ⊂ Y (40.2),there exist, for every pair of objects X ∈ X and Y_{0} ∈ Y_{0} and every pair of objects X_{0} ∈ X_{0} and Y ∈ Y , unique partial adjunction functions

Ho Y (fX, Y_{0}) ⟶^{ϕ} Ho X(X, f′Y_{0})  and  Ho X(X_{0}, f′Y) ⟶^{ψ} Ho Y (fX_{0}, Y)

Draft: May 14, 2004

which are natural in both variables and which are compatible with the given adjunction in the sense that the diagrams

\xymatrix{Y (fX, Y_{0}) \ar[r]^{≈} \ar[dd]_{γ′} & X(X, f′Y_{0}) \ar[dd]^{γ} &  & X(X_{0}, f′Y) \ar[r]^{≈} \ar[dd]^{γ} & Y (fX_{0}, Y) \ar[dd]^{γ′} \\  &  & and \\ Ho Y (fX, Y_{0}) \ar[r]^{ϕ} & Ho X(X, f′Y_{0}) &  & Ho X(X_{0}, f′Y) \ar[r]^{ψ} & Ho Y (fX_{0}, Y)}

in which the top maps are the adjunction isomorphisms of the given adjunction, commute,

This terminology is justified by the fact that

(ii) for every pair of objects X_{0} ∈ X_{0} and Y_{0} ∈ Y_{0}, the functions

Ho Y (fX_{0}, Y_{0}) ⟶^{ϕ} Ho X(X_{0}, f′Y_{0})  and  Ho X(X_{0}, f′Y_{0}) ⟶^{ψ} Ho Y (fX_{0}, Y_{0})

are inverse partial adjunction isomorphisms

while the uniqueness of these partial adjunction functions associated with X_{0} and Y_{0} implies that

(iii) these functions are the restriction to X × Y_{0} and X_{0} × Y of the partial adjunction functions associated with the unique (40.4) maximal left fand right f′-deformation retracts X_{f} and Y_{f′}.

Moreover these partial adjunctions are also natural in the given adjunction in the sense that

(iii) if f : X ↔ Y : f′ and g: X ↔ Y : g′ are deformable adjunctions, h: f → g and h′: g′ → f′ are a conjugate pair of deformable natural transformations between them (43.1) and X_{h} ⊂ X and Y_{h′} ⊂ Y denote be left h and right h′-deformation retracts ( 40.7), then, for every pair of objects X ∈ X and Y_{0} ∈ Y_{h′} and every pair of objects X_{0} ∈ X_{h} and Y ∈ Y , the diagrams

\xymatrix{Ho Y (gX, Y_{0}) \ar[r]^{ϕ} \ar[dd]_{h^{∗}} & Ho X(X, g′Y_{0}) \ar[dd]_{h′_{∗}} &  & Ho X(X_{0}, g′Y) \ar[r]^{ψ} \ar[dd]^{h′_{∗}} & Ho Y (gX_{0}, Y) \ar[dd]^{h^{∗}} \\  &  & and \\ Ho Y (fX, Y_{0}) \ar[r]^{ϕ} & Ho X(X, f′Y_{0}) &  & Ho X(X_{0}, f′Y) \ar[r]^{ψ} & Ho Y (fX_{0}, Y)}

commute.

We also consider their

43.3. Compositions. Let

f_{1}: X ⟷ Y :f′_{1}  and  f_{2}: Y ⟷ Z :f′_{2}

be two composable deformable adjunctions (43.1) such that the pairs (f_{1}, f_{2}) and (f′_{2}, f′_{1}) are left and right deformable (42.3) respectively and let X_{0} ⊂ X and Z_{0} ⊂ Z be left and right deformation retracts on which respectively f_{1} and f_{2}f_{1} and f′_{2} and f′_{1}f′_{2} are homotopical. Then, for every pair of objects X_{0} ∈ X_{0} and Z_{0} ∈ Z_{0},

Draft: May 14, 2004

the following diagrams commute

\xymatrix{Ho Z(f_{2}f_{1}X_{0}, Z_{0}) \ar[rr]^{ϕ}_{≈} \ar[dr]_{ϕ} &  & Ho X(X_{0}, f′_{1}f′_{2}Z_{0}) \ar[dl]^{ψ} \\  & Ho Y (f_{1}X_{0}, f′_{2}Z_{0})}  and

\xymatrix{Ho X(X_{0}, f′_{1}f′_{2}Z_{0}) \ar[rr]^{ψ}_{≈} \ar[dr]_{ψ} &  & Ho Z(f_{2}f_{1}X_{0}, Z_{0}) \ar[dl]^{ϕ} \\  & Ho Y (f_{1}X_{0}, f′_{2}Z_{0})}

The remainder of this section will be devoted to a proof of these two propositions, starting with a

43.4. Proof of 43.2. Let η and ε denote the unit and the counit (39.10) of the given adjunction and let (r, s) and (r′, s′) respectively be a left deformation of X into X_{0} and a right deformation of Y into Y_{0} (40.1). Then it follows readily from 33.10 and 39.10 that the functions which, for every pair of objects X ∈ X and Y_{0} ∈ Y_{0}, and every pair of objects X_{0} ∈ X_{0} and Y ∈ Y , assign to restricted zigzags (34.2) of the form

fX ⋯ Y_{0} in Y  and  X_{0} ⋯ f′Y in X

the restricted zigzags

[figure: X; s′η; f′r′fX; ⋯; f′r′Y_{0}; f′Y_{0}; s′; in X,; and; fX_{0}; frX_{0}; s; ⋯; frf′Y; εs; Y; in Y ,]

induce functions

ϕ: Ho Y (fX, Y_{0}) ⟶ Ho X(X, f′Y_{0}) and ψ: Ho X(X_{0}, f′Y) ⟶ Ho Y (fX_{0}, Y)

which do not depend on the choice of the deformations (r, s) and (r′, s′) and which are

(i) natural in both variables, and (ii) compatible with the given adjunction, in the sense of 43.2(i).

To prove 43.2(i) it thus remains to prove the uniqueness of ϕ (and hence by duality of ψ), i.e. we have to show that

(iii) if ϕ′ is a function which, to every pair of objects X ∈ X and Y_{0} ∈ Y_{0}, assigns a function ϕ′: Ho Y (fX, Y_{0}) → Ho X(X, f′Y_{0}) satisfying (i) and (ii) above, then ϕ′ = ϕ.

To prove (iii) we first show that if, for a homotopical category Z and restricted zigzag Z_{0} ⋯ Z_{n} in Z (n ≥ 0), we denote by [ Z_{0} ⋯ Z_{n} ] the element of Ho Z(Z_{0}, Z_{n}) “containing” it (33.10), then

(iv) for every restricted zigzag in Y of the form [figure: fX; Y_{1}; ⋯; Y_{n}] (n ≥ 1)

with Y_{1}, ⋯, Y_{n} ∈ Y_{0}, the assumptions of (iii) imply that
ϕ′[ [figure: fX; Y_{1}; ⋯; Y_{n}] ] = [ [figure: X; f′Y_{1}; ⋯; f′Y_{n}] ]

where the map X → f′Y_{1} is the adjunct of the map fX → Y_{1} and the zigzag f′Y_{1} ⋯ f′Y_{n} is obtained by applying the functor f′ to the zigzag Y_{1} ⋯ Y_{n}

by noting that

(v) in view of (ii) above, (iv) holds for n = 1, and (vi) if, for some integer k > 0, (iv) holds for all integers n ≤ k, then (i)

above implies that, for every such zigzag [figure: fX; Y_{1}; ⋯; Y_{k}] and map Y_{k} → Y_{k+1} ∈ Y_{0},

ϕ′[ [figure: fX; Y_{1}; ⋯; Y_{k}; Y_{k+1}] ] = [ [figure: X; f′Y_{1}; ⋯; f′Y_{k}; f′Y_{k+1}] ]

and that, for every such zigzag [figure: fX; Y_{1}; ⋯; Y_{k}] and weak equivalence w: Y_{k+1} → Y_{k} ∈ Y_{f′}, the map

[ [figure: X; f′Y_{1}; ⋯; f′Y_{k}] ] = ϕ′[ [figure: fX; Y_{1}; ⋯; Y_{k}] ]
= ϕ′[ [figure: fX; Y_{1}; ⋯; Y_{k}; Y_{k+1}; w; w; Y_{k}] ]

is the composition of ϕ[ [figure: fX; Y_{1}; ⋯; Y_{k}; Y_{k+1}; w] ] with the isomorphism Ho f′w which implies that

ϕ′[ [figure: fX; Y_{1}; ⋯; Y_{k}; Y_{k+1}; w] ] = [ [figure: X; f′Y_{1}; ⋯; f′Y_{k}; f′Y_{k+1}; f′w] ] .

The desired result, i.e. (iii), now readily follows from the observation that, for every restricted zigzag of the form fX ⋯ Y_{0} with Y_{0} ∈ Y_{0}, the existence of the associated commutative diagram

[figure: fX; s′; ⋯; Y_{0}; s′; r′fX; ⋯; r′Y_{0}]

implies that

[ fX ⋯ Y_{0} ] = [ [figure: fX; s′; r′fX; ⋯; r′Y_{0}; Y_{0}; s′] ] .

Next we prove 43.2(ii) by noting that, for every pair of objects X_{0} ∈ X_{0} and Y_{0} ∈ Y_{0}, the function ψϕ sends, for every restricted zigzag in Y of the form fX_{0} ⋯ Y_{0} , the corresponding element of Ho Y (fX_{0}, Y_{0}) to the element corresponding to the top row, and hence the bottom row, of the following commutative

Draft: May 14, 2004

diagram (39.10) in which all three rows are restricted zigzags

[figure: fX_{0}; f; frX_{0}; s; η; s; frf′fX_{0}; s′; s; frf′r′fX_{0}; εs; ⋯; frf′r′Y_{0}; εs; frf′Y_{0}; εs; s′; εs; Y_{0}; 1; fX_{0}; η; 1; ff′fX_{0}; ε; fX_{0}; s′; r′fX_{0}; ⋯; r′Y_{0}; Y_{0}; s′; fX_{0}; 1; s′; ⋯; Y_{0}; s′; 1]

and that a dual assertion holds for the function ϕψ.

It thus remains to prove 43.2(iv). But this now follows from the observation that, in view of the commutative diagram which we mentioned just before 39.10(v), every reduced zigzag in Y of the form gX ⋯ Y_{0} gives rise to a commutative diagram in X of the form

[figure: X; η_{2}; 1; g′gX; s′; h′; g′r′gX; h′; ⋯; g′r′Y_{0}; h′; g′Y_{0}; s′; h′; h′; f′Y_{0}; 1; f′gX; h; s′; X; η_{1}; f′fX; s′; f′r′fX; h; f′r′gX; ⋯; f′r′Y_{0}; f′Y_{0}; s′]

in which η_{1} and η_{2} denote the units of the two adjunctions.

43.5. Proof of 43.3. The commutativity of the first triangle readily follows from the fact that (39.10) every restricted zigzag in Z of the form f_{2}f_{1}X_{0} ⋯ Z_{0} gives rise to a commutative diagram in Y of the form

[figure: f_{1}X_{0}; 1; f_{1}r_{1}X_{0}; s_{1}; η_{1}; s_{1}; f_{1}r_{1}f′_{1}f_{1}X_{0}; η_{2}; s_{1}; f_{1}r_{1}f′_{1}f′_{2}f_{2}f_{1}X_{0}; s′_{2}; s_{1}; f_{1}r_{1}f′_{1}f′_{2}r′_{2}f_{2}f_{1}X_{0}; s_{1}; ⋯; f_{1}X_{0}; η_{1}; 1; f_{1}f′_{1}f_{1}X_{0}; η_{2}; ε_{1}; f_{1}f′_{1}f′_{2}f_{2}f_{1}X_{0}; s′_{2}; ε_{1}; f_{1}f′_{1}f′_{2}r′_{2}f_{2}f_{1}X_{0}; ε_{1}; ⋯; f_{1}X_{0}; η_{2}; f′_{2}f_{2}f_{1}X_{0}; s′_{2}; f′_{2}r′_{2}f_{2}f_{1}X_{0}; ⋯]

[figure: ⋯; f_{1}r_{1}f′_{1}f′_{2}r′_{2}Z_{0}; s_{1}; f_{1}r_{1}f′_{1}f′_{2}Z_{0}; s′_{2}; s_{1}; s_{1}; f_{1}f′_{1}f′_{2}Z_{0}; ε_{1}; 1; f′_{2}Z; 1; ⋯; f_{1}f′_{1}f′_{2}r′_{2}Z_{0}; ε_{1}; f_{1}f′_{1}f′_{2}Z_{0}; s′_{2}; ε_{1}; ⋯; f′_{2}r′_{2}Z_{0}; f′_{2}Z_{0}; s′_{2}]

in which η_{1} and ε_{1} denote the unit and the counit of the first adjunction and η_{2} denotes the unit of the second.

The commutativity of the second triangle follows immediately from the commutativity of the first triangle and 43.2(ii).

44. Derived adjunctions

We now use the results of the preceding section on partial adjunctions to show that

(i) for every deformable adjunction f : X ↔ Y : f′ (43.1), its adjunction induces, for every left approximation (k, a) of f and every right approximation (k′, a′) of f′, a derived adjunction

Ho k: Ho X ⟷ Ho Y :Ho k′

which is natural with respect to these approximations, (ii) for every two deformable adjunctions

f: X ⟷ Y :f′  and  g: X ⟷ Y :g′,

every conjugate pair of deformable natural transformations between them (43.1) induces conjugate pairs of natural transformations between their derived adjunctions, and

(iii) for every two composable deformable adjunctions

f_{1}: X ⟷ Y :f′_{2}  and  f_{2}: Y ⟷ Z :f′_{2}

for which the pairs (f_{1}, f_{2}) and (f′_{2}, f′_{1}) are respectively locally left and right deformable (42.3), the compositions of their derived adjunctions and the derived adjunctions of their composition are connected by conjugate pairs of natural transformations.

We start with noting the existence of certain auxiliary

44.1. Derived natural transformations. Let f: X → Y be a left (resp. a right) deformable functor (40.2). Then there exists a unique function u (resp. u′) which assigns to every pair ((m, b), (k, a)) of objects in

(Fun_{w}(X, Y ) ↓ f)  (resp. (f ↓ Fun_{w}(X, Y )))

of which (k, a) is a left (resp. a right) approximation of f (41.1), a derived natural transformation

u(m, k): Ho m ⟶ Ho k (resp. u′(k, m): Ho k ⟶ Ho m)

such that

(i) if (m, b) is also a left (resp. a right) approximation of f, then u(m, k) (resp. u′(k, m)) is a natural isomorphism,

(ii) the function u (resp. u′) is natural in both variables, and (iii) for every map

t: (m, b) ⟶ (k, a) ∈ (Fun_{w}(X, Y ) ↓ f)

(resp. t′: (k, a) ⟶ (m, b) ∈ (f ↓ Fun_{w}(X, Y )))

one has

u(m, k) = Ho t (resp. u′(k, m) = Ho t′).

Proof (of the left half). The composition

\xymatrix{(Fun_{w}(X, Y ) ↓ f) \ar[r]^{j} & Fun_{w}(X, Y ) \ar[r]^{Ho} & Fun(Ho X, Ho Y )}

in which j denotes the forgetful functor and Ho is as in 33.9, sends weak equivalences to isomorphisms and hence (33.9) admits a unique factorization

\xymatrix{(Fun_{w}(X, Y ) ↓ f) \ar[r]^{γ′} & Ho (Fun_{w}(X, Y ) ↓ f) \ar[r] & Fun(Ho X, Ho Y )}

in which γ′ denotes the localization functor and one now readily verifies that the function which sends a pair ((m, b), (k, a)) to the image in Fun(Ho X, Ho Y ) of the (in view of 38.3(ii)) unique map

γ′(m, b) ⟶ γ′(k, a) ∈ Ho (Fun_{w}(X, Y ) ↓ f)

has the desired properties.

Using these derived natural transformations we now implicitly define

44.2. Derived adjunctions. Given a deformable adjunction (43.1) f: X ↔ Y :f′, there is a unique function which assigns to every pair consisting of a left approximation (k, a) of f (41.1) and a right approximation (k′, a′) of f′, a derived adjunction

Ho k: Ho X ⟷ Ho Y :Ho k′

such that

Draft: May 14, 2004

(i) for every two left approximations (k_{1}, a_{1}) and (k_{2}, a_{2}) of f and right approximations (k′_{1}, a′_{1}) and (k′_{2}, a′_{2}) of f′, the derived natural isomorphisms (44.1)

u(k_{1}, k_{2}): Ho k_{1} ⟶ Ho k_{2}  and  u′(k′_{2}, k′_{1}): Ho k′_{2} ⟶ Ho k′_{1}

are conjugate (39.10) with respect to the derived adjunctions

Ho k_{1}: Ho X ⟷ Ho Y :Ho k′_{1}  and  Ho k_{2}: Ho X ⟷ Ho Y :Ho k′_{2}

and,

(ii) for every left f-deformation (r, s) (40.2) and right f′-deformation (r′, s′), the adjunction isomorphism of the derived adjunction

Ho(fr): Ho X ⟷ Ho Y :Ho(f′r′)

associated with the pair ((fr, fs), (f′r′, f′s′)) (41.2) assigns to every pair of objects X ∈ X and Y ∈ Y the composition

Ho Y (frX, Y) ≈ Ho Y (frX, r′Y) ≈^{ϕ} Ho X(rX, f′r′Y) ≈ Ho X(X, f′r′Y)

in which ϕ is the partial adjunction isomorphism (43.2) and the other isomorphisms are induced by s′ and s.

Proof. In view of 39.10(vi) it suffices to show that, for every two left f-deformations (r_{1}, s_{1}) and (r_{2}, s_{2}) and right f-deformations (r′_{1}, s′_{1}) and (f′_{2}, s′_{2}), the unique natural isomorphisms (44.1)

u(fr_{1}, fr_{2}): Ho(fr_{1}) ⟶ Ho(fr_{2})  and  u′(f′r′_{2}, f′r′_{1}): Ho(f′r′_{2}) ⟶ Ho(f′r′_{1})

are conjugate with respect to the adjunctions

Ho(fr_{1}): Ho X ⟷ Ho Y :Ho(f′r′_{1}) and Ho(fr_{2}): Ho X ⟷ Ho Y :Ho(f′r′_{2})

defined in (ii), but as (40.2, 41.3(iii) and 42.1) the compositions

(r_{2}r_{1}, s_{2}s_{1})  and  (r′_{2}r′_{1}, s′_{2}s′_{1})

of respectively (r_{1}, s_{1}) and (r_{2}, s_{2}) and r′_{1}, s′_{1} and (r′_{2}, s′_{2}) (in which s_{2}s_{1} and s′_{2}s′_{1} denote the diagonals of the commutative diagrams

\xymatrix{r_{2}r_{1} \ar[r]^{s_{2}r_{1}} \ar[dd]_{r_{2}s_{1}} & r_{1} \ar[dd]^{s_{1}} &  & 1 \ar[r]^{s′_{2}} \ar[dd]_{s′_{1}} & r′_{2} \ar[dd]^{r′_{2}s′_{1}} \\  &  & and &  &  & ) \\ r_{2} \ar[r]^{s_{2}} & 1 &  & r′_{1} \ar[r]_{s′_{2}r′_{1}} & r′_{2}r′_{1}}

are a left f-deformation and a right f′-deformation, this follows readily from the commutativity of the diagram

[figure: Ho Y (fr_{2}X, Y); ≈; (fr_{2}s_{1})^{∗}; ⋯; ≈; ϕ; ⋯; ≈; Ho X(X, f′r′_{2}Y); (f′r′_{2}s′_{1})_{∗}; Ho Y (fr_{2}r_{1}X, Y); ≈; ⋯; ≈; ϕ; ⋯; ≈; Ho X(X, f′r′_{2}r′_{1}Y); Ho Y (fr_{1}X, Y); ≈; (fs_{2}r_{1})^{∗}; ⋯; ≈; ϕ; ⋯; ≈; Ho X(X, f′r′_{1}Y); (f′s′_{2}r′_{1})_{∗}]

in which the horizontal sequences are as in (ii), and the observation that (44.1)

Ho(fr_{2}s_{1}) Ho(fs_{2}r_{1})^{−1} = u(fr_{1}, fr_{2})

and

Ho(f′s′_{2}r′_{1})^{−1} Ho(f′r′_{2}s′_{1}) = u′(f′r′_{2}, f′r′_{1}) .

Next we consider

44.3. Conjugations between derived adjunctions. Let

f: X ⟷ Y :f′  and  g: X ⟷ Y :g′

be deformable adjunctions, let

h: f ⟶ g  and  h′: g′ ⟶ f′

be a conjugate pair of deformable natural transformations between them (43.1), and let

k: (k_{1}, a_{1}) ⟶ (k_{2}, a_{2})  and  k′: (k′_{2}, a′_{2}) ⟶ (k′_{2}, a′_{1})

respectively be a left approximation of h and a right approximation of h′ (41.6). Then the natural transformations

Ho k: Ho k_{1} ⟶ Ho k_{2}  and  Ho k′: Ho k′_{2} ⟶ Ho k′_{1}

are conjugate with respect to the derived adjunction (44.2)

Ho k_{1}: Ho X ⟷ Ho Y :Ho k′_{1}  and  Ho k_{2}: Ho X ⟷ Ho Y :Ho k′_{2} .

Proof. This follows from 39.10(vi), 44.2(i) and the observation that, in view of 43.2(iii), for every left h-deformation (r, s) and right h′-deformation (r′, s′) (40.7) and every pair of objects X ∈ X and Y ∈ Y , the diagram

[figure: Ho Y (grX, Y); ≈; h^{∗}; Ho Y (grX, r′Y); ϕ; ≈; h^{∗}; Ho X(rX, g′r′Y); ≈; h′_{∗}; Ho X(X, g′r′Y); h′_{∗}; Ho Y (frX, Y); ≈; Ho Y (frX, r′Y); ϕ; ≈; Ho X(rX, f′r′Y); ≈; Ho X(X, f′r′Y)]

in which the horizontal sequences are as in 44.2(ii), commutes.

Finally we discuss

44.4. Compositions of derived adjunctions. Let

f_{1}: X ⟷ Y :f′_{1}  and  f_{2}: Y ⟷ Z :f′_{2}

be deformable adjunctions (43.1) for which the pairs (f_{1}, f_{2}) and (f′_{2}, f′_{1}) are respectively locally left and right deformable (42.3). Then

(i) their composition

f_{2}f_{1}: X ⟷ Z :f′_{1}f′_{2}

is also a deformable adjunction.

Draft: May 14, 2004

Moreover, if

(k_{1}, a_{1}), (k_{2}, a_{2}) and (k_{21}, a_{21})

are left approximations of f_{1}, f_{2} and f_{2}f_{1} and

(k′_{1}, a′_{1}), (k′_{2}, a′_{2}) and (k′_{12}, a′_{12})

are right approximations of f′_{1}, f′_{2} and f′_{1}f′_{2} respectively (41.1), then

(ii) the derived natural transformations (44.1)

u(k_{2}k_{1}, k_{21}): Ho(k_{2}k_{1}) ⟶ Ho k_{21}  and  u′(k′_{12}, k′_{1}k′_{2}): Ho k′_{12} ⟶ Ho(k′_{1}k′_{2})

associated with the composition (42.1) (k_{2}k_{1,a_{2}a_{1}}) of (k_{1}, a_{1}) and (k_{2}, a_{2}) and the left approximation (k_{21}, a_{21}) and with the composition (k′_{1}k′_{2}, a′_{1}a′_{2}) of (k′_{2}, a′_{2}) and (k′_{1}, a′_{1}) and the right approximation (k′_{12}, a′_{12}), are conjugate with respect to the derived adjunction (44.2)

Ho(k_{2}k_{1}): Ho X ⟷ Ho Z :Ho(k′_{1}k′_{2})

and the composition of the derived adjunctions

Ho k_{1}: Ho X ⟷ Ho Y :Ho k′_{1}  and  Ho k_{2}: Ho Y ⟷ Ho Z :Ho k′_{2} .

44.5. Corollary. Let

f_{1}: X ⟷ Y :f′_{1}  and  f_{2}: Y ⟷ Z :f′_{2}

be deformable adjunctions for which the pairs (f_{1}, f_{2}) and (f′_{2}, f′_{1}) are respectively left and right deformable (42.3). Then every composition of a derived adjunction of the first adjunction with a derived adjunction of the second adjunction is a derived adjunction of their composition.

Proof. In view of 39.10(vi), 44.1 and 44.2 it suffices to prove that if

(r_{1}, s_{1}), (r_{2}, s_{2}), (r′_{1}, s′_{1}) and (r′_{2}, s′_{2})

respectively are a left f_{1}- as well as a left f_{2}f_{1}-deformation, a left f_{2}-deformation, a right f′_{1}-deformation and a right f′_{2}- as well as a right f′_{1}f′_{2}-deformation, then the natural transformations

Ho(f_{2}s_{2}f_{1}r_{1}): Ho(f_{2}r_{2}f_{1}r_{1}) ⟶ Ho(f_{2}f_{1}r_{1}) and

Ho(f′_{1}s′_{1}f′_{2}r′_{2}): Ho(f′_{1}f′_{2}r′_{2}) ⟶ Ho(f′_{1}r′_{1}f′_{2}r′_{2})

are conjugate with respect to the derived adjunction

Ho(f_{2}f_{1}r): Ho X ⟷ Ho Z :Ho(f′_{1}f′_{2}r′_{2})

and the composition of the derived adjunctions

Ho(f_{1}r_{1}): Ho X ⟷ Ho Y :Ho(f′_{1}r′_{1}) and Ho(f_{2}r_{2}): Ho Y ⟷ Ho Z :Ho(f′_{2}r′_{2}) .

But this follows readily from the observation that, in view of 43.3 and the naturality of the partial adjunction functions ϕ and ψ (43.2(i)) and of the maps induced by s_{1}, s_{2}, s′_{1} and s′_{2}, for every pair of objects X ∈ X and Z ∈ Z, the following diagram commutes

\xymatrix{Ho Z(f_{2}r_{2}f_{1}r_{1}X, Z) \ar[r]_{≈}^{(s′_{2})_{∗}} & Ho Z(f_{2}r_{2}f_{1}r_{1}X, r′_{2}Z) \ar@{-->}[dr]^{ϕ}_{≈} \\  &  & Ho Y (r_{2}f_{1}r_{1}X, f′_{2}r′_{2}Z) \\ Ho Z(f_{2}f_{1}r_{1}X, Z) \ar[r]_{≈}^{(s′_{2})_{∗}} \ar[uu]^{s_{2}^{∗}} & Ho Z(f_{2}f_{1}r_{1}X, r′_{2}Z) \ar[uu]^{s_{2}^{∗}} \ar@{-->}[dr]^{ϕ} \ar[dd]^{≈}_{ϕ} \\  &  & Ho Y (f_{1}r_{1}X, f′_{2}r′_{2}Z) \ar[uu]^{≈}_{s_{2}^{∗}} \ar[dd]_{≈}^{(s′_{1})_{∗}} \\ Ho X(X, f′_{1}f′_{2}r′_{2}Z) \ar[r]^{≈}_{s_{1}^{∗}} \ar[dd]_{(s′_{1})_{∗}} & Ho X(r_{1}X, f′_{1}f′_{2}r′_{2}Z) \ar@{-->}[ur]_{ψ} \ar[dd]_{(s′_{1})_{∗}} \\  &  & Ho Y (f_{1}r_{1}X, r′_{1}f′_{2}r′_{2}Z) \\ Ho X(X, f′_{1}r′_{1}f′_{2}r′_{2}Z) \ar[r]^{≈}_{s_{1}^{∗}} & Ho X(r_{1}X, f′_{1}r′_{1}f′_{2}r′_{2}Z) \ar@{-->}[ur]^{≈}_{ψ}}

45. The Quillen condition

As another application of the results of 44.2 we show in this last section of the chapter

(i) that a sufficient condition on a deformable adjunction (43.1) in order that the left approximations of the left adjoints (41.1) and the right approximations of the right adjoint are homotopically inverse homotopical equivalences of homotopical categories (33.1) is that the adjoint pair satisfies a so-called Quillen condition, and

(ii) that, if the homotopical categories in question are saturated (33.9), then this condition is also necessary.

In more detail

45.1. The Quillen condition. A deformable adjunction (43.1) f: X ↔ Y : f′ will be said to satisfy the Quillen condition if

(i) for some left f-deformation retract X_{0} ⊂ X (40.2) and right f′-deforma-tion retract Y_{0} ⊂ Y and every pair of objects X_{0} ∈ X_{0} and Y_{0} ∈ Y_{0}, a map fX_{0} → Y_{0} ∈ Y is a weak equivalence iff its adjoint X_{0} → f′Y_{0} ∈ X is so.

which one readily verifies (using 40.3) is equivalent to the seemingly stronger requirement that

(ii) for every left f-deformation retract X_{0} ⊂ X and right f′-deformation retract Y_{0} ⊂ Y and every pair of objects X_{0} ∈ X_{0} and Y_{0} ∈ Y_{0}, a map fX_{0} → Y_{0} ∈ Y is a weak equivalence iff its adjunct X_{0} → f′Y_{0} ∈ X is so.

The usefulness of this condition is due to

45.2. Proposition. Given a deformable adjunctions (43.1) f: X ↔ Y :f′, a left approximation (k, a) of f and a right approximation (k′, a′) of f′ (41.1), consider the following statements:

(i) the adjunction satisfies the Quillen condition (45.1),

(ii) the functors k and k′ are homotopically inverse homotopical equivalences

of homotopical categories (33.1), (ii)′ the functor k is a homotopical equivalence of homotopical categories (33.1), (ii)′′ the functor k′ is a homotopical equivalence of homotopical categories, (iii) the functors Ho k and Ho k′ are inverse equivalences of categories, (iii)′ the functor Ho k is a equivalence of categories, and (iii)′′ the functor Ho k′ is an equivalence of categories.

Then (iv) the following implications hold

[figure: (ii)′; (iii)′; (i); (ii); (iii); (ii)′′; (iii)′′]

Moreover

(v) if X and Y are saturated (33.8), then each of (iii), (iii)′ and (iii)′′ implies (i), and hence each of the seven statements (i)–(iii)′′ implies all the others.

Proof. In view of the homotopical uniqueness of approximations (41.1), this result follows readily from 44.2 and

45.3. Proposition. Given a deformable adjunction (43.1) f : X ↔ Y : f′, a left f-deformation (r, s) (40.2) and a right f′-deformation (r′, s′), consider the following three statements:

(i) for every pair of objects X_{0} ∈ X and Y_{0} ∈ Y which are in the images of r and r′ respectively, a map X_{0} → f′Y_{0} ∈ X is a weak equivalence if (resp. only if) its adjunct fX_{0} → Y_{0} ∈ Y is so,

(ii) the zigzag of natural transformations

\xymatrix{1_{X} & r \ar[l]_{s} \ar[r] & f′r′fr}  (resp. \xymatrix{frf′r′ \ar[r] & r′ & 1_{Y} \ar[l]_{s′}} )

in which the unnamed map is the adjunct of the natural weak equivalence

\xymatrix{fr \ar[r]^{s′fr} & r′fr}  (resp. \xymatrix{rf′r′ \ar[r]^{sf′r′} & f′r′} )

is a zigzag of natural weak equivalences, and (iii) the unit (resp. the counit) (39.10)

1_{Ho X} ⟶ Ho(f′r′) Ho(fr) (resp. Ho(fr) Ho(f′r′) ⟶ 1_{Ho Y})

of the derived adjunction Ho(fr) : Ho X ↔ Ho Y : Ho(f′r′) (44.2) is a natural isomorphism.

Then (i) implies (ii) and (ii) implies (iii). Moreover, if the homotopical category X (resp. Y ) is saturated (33.9), then (iii) implies (i) and hence (i), (ii) and (iii) are equivalent statements.

Draft: May 14, 2004

Proof (of the first half). Clearly (i) implies (ii) and a straightforward calculation using 43.2(i) and 44.2 yields that the image in Ho X of the zigzag 1_{X} ⟵^{s} r → f′r′fr is exactly the unit 1_{Ho X} → Ho(f′r) Ho(fr), so that (ii) implies (iii).

It thus remains to show that, if X is saturated, then (iii) implies (i) and we do this by successively noting that if p: X_{0} → f′Y_{0} ∈ X is a map such that its adjunct q: fX_{0} → Y_{0} ∈ Y is a weak equivalence, then

(i) the map q(fsX_{0}): frX_{0} → Y_{0} ∈ Y is also a weak equivalence and hence its image γ′(q(fsX_{0})) ∈ Ho Y under the localization functor γ′: Y → Ho Y is an isomorphism,

(ii) in view of 43.2(i) and 44.2 the adjunct of γ′(q(fsX_{0})) is the image γ((f′s′Y_{0})p) ∈ Ho X of the map (f′s′Y_{0})p: X_{0} → f′r′Y_{0} ∈ X under the localization functor γ: X → Ho X,

(iii) this adjunct γ((f′s′Y_{0})p) is the composition of the unit (which is assumed to be an isomorphism) with the image of γ′(q(fsX_{0})) under the functor Ho(f′r′) and hence is also an isomorphism, and

(iv) in view of the saturation of X, the map (f′s′Y_{0})p is thus a weak equivalence and so is therefore, because of the two out of three property, the original map p: X_{0} → fY_{0} ∈ X.

Draft: May 14, 2004

CHAPTER VIII Homotopy Colimit and Limit Functors and Homotopical Ones

46. Introduction

46.1. Summary. This last chapter consists of two parts.

In the first part (i.e. §§47–49), motivated by the homotopy colimit and limit functors on model categories (see chapter IV), we define homotopy colimit and limit functors on arbitrary cocomplete and complete homotopical categories and obtain sufficient conditions for their existence and composability as well as sufficient conditions for the homotopical cocompleteness and completeness of such homotopical categories, in a sense which is much stronger than one would expect.

In the second part (i.e. §§50 and 51) we define, on not necessarily cocomplete and complete homotopical categories, homotopical colimit and limit functors which, in the cocomplete and complete case, turn out to be essentially the same as the above homotopy colimit and limit functors, and note that there are corresponding notions of homotopical cocompleteness and completeness which, in the cocomplete and complete case, reduce to the ones mentioned above.

46.2. Some of the tools. To deal with homotopy colimit and limit functors we need the results of chapter VII on the approximations of (not necessarily homotopical) functors between homotopical categories and of composable pairs of such functors, but in order to be able to deal with the associated notions of homotopical cocompleteness and completeness we first have to generalize these results on composable pairs of such functors to diagram-like collections of homotopical categories and such functors between them, indexed by a category C, which we will call left and right C-systems and which are functions F which assign

(i) to every object D ∈ C a homotopical category F D, (ii) to every map u: A → B ∈ C a (not necessarily homotopical) functor

F u: F A → F B, and (iii) to every composable pair of maps u: A → B and v: B → D ∈ C, a

natural weak equivalence

F (v, u): (F v)(F u) ⟶ F (vu)  or  F (v, u): F (vu) ⟶ (F v)(F u)

which is associative in the obvious sense.

The treatment of homotopical colimit and limit functors and the associated notions of homotopical cocompleteness and completeness requires in addition a discussion of Kan extensions and homotopical Kan extensions of the identity functor along a (not necessarily homotopical) functor between homotopical categories and of the more general Kan extensions and homotopical Kan extensions along a system.

145 Draft: May 14, 2004

146 VIII. HOMOTOPY COLIMIT AND LIMIT FUNCTORS AND HOMOTOPICAL ONES

46.3. Organization of the chapter. After a warning on the use of the term adjoint (in 46.4), we discuss homotopy colimit and limit functors (in §47), introduce left and right systems (in §48) and use these to describe the associated notions of homotopical cocompleteness and completeness (in §49). Next we investigate Kan extensions and homotopical Kan extensions of identity functors and use these to deal with homotopical colimit and limit functors (in §50). And in the last section we introduce the Kan extensions and homotopical Kan extensions along systems which we need to deal with the more general notions of homotopical cocompleteness and completeness.

We thus end with a

46.4. Warning on the term adjoint. Given a functor f: X → Y , we will call a functor g: Y → X a left (or a right) adjoint of f if there is given an adjunction

g: Y ⟷ X :f (or f: X ⟷ Y :g)

and not, as is done by some authors, if there only exists such an adjunction.

47. Homotopy colimit and limit functors

In this section, assuming the existence of colimit and limit functors, we

(i) define homotopy colimit and limit functors on a homotopical category respectively as left and right approximations of an arbitrary but fixed such colimit or limit functor, and

(ii) describe sufficient conditions for their existence, composability and ho-

motopical compatibility with deformable adjoints. We thus start with defining

47.1. Homotopy colimit and limit functors. Given a homotopical category X and a category D, we

(i) define a D-colimit (resp. D-limit) functor on X as a left (resp. a right) adjoint (46.4) X^{D} → X of the constant diagram functor c^{∗}: X → X^{D} (which sends every object of X to the corresponding constant Ddiagram), and

(ii) if such a D-colimit (resp. D-limit) functor on X exists, denote by

colim^{D}  (resp. lim^{D})

an arbitrary but fixed such D-colimit (resp. D-limit) functor, and define a homotopy D-colimit (resp. D-limit) functor on X as a left (resp. a right) approximation (41.1) of colim^{D} (resp. lim^{D}),

and more generally, given a functor u: A → B, we

(i)′ define a u-colimit (resp. u-limit) functor on X as a left (resp. a right) adjoint X^{A} → X^{B} of the induced diagram functor

X^{u} = u^{∗}: X^{B} ⟶ X^{A}

(which sends every functor B → X to its composition with u), and (ii)′ if such a u-colimit (resp. u-limit) functor on X exists, denote by

colim^{u}  (resp. lim^{u})

Draft: May 14, 2004

an arbitrary but fixed such u-colimit (resp. u-limit) functor, and define a homotopy u-colimit (resp. u-limit) functor on X as a left (resp. a right) approximation of colim^{u} (resp. lim^{u}).

Then

47.2. Proposition. For every homotopical category X and functor u: A → B

(i) such homotopy u-colimit (resp. u-limit) functors on X, if they exist, are homotopically unique (37.5), and

(ii) a sufficient condition for their existence is that the functor colim^{u} (resp.

lim^{u}) exist (47.1) and be left (resp. right) deformable (40.2).

Moreover in that case

(iii) every homotopy u-colimit (resp. u-limit) functor on X, i.e. left (resp. right) approximation (k, a) of colim^{u} (resp. lim^{u}), comes with a derived adjunction (44.2)

Ho k: Ho X^{A} ⟷ Ho X^{B} :Ho u^{∗} (resp. Ho u^{∗}: Ho X^{B} ⟷ Ho X^{A} :Ho k)

associated with the approximations

((k, a), (u^{∗}, 1_{u_{∗}})) (resp. ((u^{∗}, 1_{u^{∗}}), (k, a)))

which has as its counit (resp. unit) the natural transformation

Ho(e(au^{∗})): (Ho k)(Ho u^{∗}) ⟶ 1_{Ho}_{X}^{B} (resp. Ho((au^{∗})e): 1_{Ho}_{X}^{B} ⟶ (Ho k)(Ho u^{∗}))

where

e: colim^{u} u^{∗} ⟶ 1_{X}^{B} (resp. e: 1_{X}^{B} ⟶ lim^{u} u^{∗})

denotes the counit (resp. the unit) of the adjunction

colim^{u}: X^{A} ⟷ X^{B} :u^{∗} (resp. u^{∗}: X^{B} ⟷ X^{A} :lim^{u}) .

Proof. Parts (i) and (ii) follow from 41.1(iii) and 41.2(ii), while the colimit half of part (iii) follows from 44.2 and the observation that if (r, s) is a left colim^{u}- deformation (40.2), then the resulting commutative diagram

\xymatrix{(kr)u^{∗} \ar[r]^{(ks)u^{∗}} \ar[d]_{(ar)u^{∗}} & ku^{∗} \ar[dr]^{e(au^{∗})} \ar[d]_{au^{∗}} \\ (colim^{u} r)u^{∗} \ar[r]^{(colim^{u} s)u^{∗}} & colim^{u} u^{∗} \ar[r]^{e} & 1_{X}^{B}}

gives rise to the commutative diagram

\xymatrix{(Ho kr)(Ho u^{∗}) \ar[r] \ar[d] & (Ho k)(Ho u^{∗}) \ar[d]^{Ho(e(au^{∗}))} \\ (Ho colim^{u} r)(Ho u^{∗}) \ar[r] & 1_{Ho}_{X}^{B}}

and the first part of the proposition now follows from the observation that, in view of 44.1 and 44.2, the natural transformation (Ho k)(Ho u^{∗}) → 1_{Ho}_{X}^{B} obtained by going counter clockwise around this diagram is exactly the desired counit.

Draft: May 14, 2004

47.3. Compositions. Given a homotopical category X and two composable functors u: A → B and v: B → D for which there exist u- and v-colimit (resp. u- and v-limit) functors on X, the composition of a homotopy u-colimit (resp. u-limit) functor on X with a homotopy v-colimit (resp. v-limit) functor will be a slight modification (by a natural isomorphism) of the composition of approximations which we considered in §42. More precisely, for every homotopy u-colimit (resp. u-limit) functor (k_{u}, a_{u}) and homotopy v-colimit (resp. v-limit) functor (k_{v}, a_{v}) on X, their “composition” will be the pair

(k_{v}k_{u}, colim^{(v,u)} a_{v}a_{u}) ∈ (Fun_{w}(X^{A}, X^{D}) ↓ colim^{vu})
(resp. (k_{v}k_{u}, lim^{(v,u)} a_{v}a_{u}) ∈ (lim^{vu} ↓ Fun_{w}(X^{A}, X^{D})) )

where a_{v}a_{u} denotes the diagonal of the commutative diagram

\xymatrix{k_{v}k_{u} \ar[r]^{a_{v}} \ar[dd]_{a_{u}} & colim^{v} k_{u} \ar[dd]^{a_{u}} &  & lim^{v} lim^{u} \ar[r]^{a_{v}} \ar[dd]_{a_{u}} & k_{v} lim^{u} \ar[dd]^{a_{u}} \\  &  & (resp. &  &  & ) \\ k_{u} colim^{v} \ar[r]_{a_{v}} & _{colim_{v} colim_{u}} &  & lim^{v} k_{u} \ar[r]_{a_{v}} & k_{v}k_{u}}

and

colim^{(v,u)} : colim^{v} colim^{u} ⟶ colim^{vu} (resp. lim^{(v,u)} : lim^{vu} ⟶ lim^{v} lim^{u})

is the natural isomorphism which is the conjugate of the identity natural transformation of the functor

(vu)^{∗} = u^{∗}v^{∗}: X^{D} ⟶ X^{A} .

One then has the following

47.4. Sufficient conditions for composability. A sufficient condition in order that, given a homotopical category X and two composable functors u: A → B and v: B → D for which there exist u- and v-colimit (resp. u- and v-limit) functors on X, for every homotopy u-colimit (resp. u-limit) functor (k_{u}, a_{u}) and v-colimit (resp. v-limit) functor (k_{v}, a_{v}) on X, their (47.3) “composition” is a homotopy vu-colimit (resp. vu-limit) functor on X is that

(i) the pair

(colim^{u}, colim^{v}) (resp. (lim^{u}, lim^{v}))

exists and is left (resp. right) deformable (42.3)

which in particular is the case if

(ii) X is saturated (33.9) and the pair

(colim^{u}, colim^{v}) (resp. (lim^{u}, lim^{v}))

exists and is locally left (resp. right) deformable (42.3).

Moreover, if (i) holds, then

(iii) the composition of the derived adjunctions (47.2) of (k_{u}, a_{u}) and (k_{v}, a_{v}) is exactly the derived adjunction of their “composition” (47.3).

Proof. Part (i) follows readily from 42.4 and the fact that colim^{(v,u)} (resp. lim^{v,u}) is a natural isomorphism, while (ii) follows from 33.9(v) and 42.6 and the observation that, as u^{∗} and v^{∗} are both homotopical functors, the pair (v^{∗}, u^{∗}) is right (resp. left) deformable (42.3).

To prove the colimit half of (iii) one notes that in the commutative diagram

\xymatrix{k_{v}k_{u}u^{∗}v^{∗} \ar[r]^{a_{u}} \ar[dd]_{1} & k_{v} colim^{u} u^{∗}v^{∗} \ar[r]^{e_{u}} \ar[d]_{a_{v}} & k_{v}v^{∗} \ar[r]^{a_{v}} \ar[d]_{a_{v}} & colim^{v} v^{∗} \ar[r]^{e_{v}} & 1_{M}^{B} \ar[d]^{1} \\  & colim^{v} colim^{u} u^{∗}v^{∗} \ar[r]^{e_{u}} \ar[d]_{_{colim_{(v,u)}}} & colim^{v} v^{∗} \ar[rr]^{e_{v}} &  & 1_{M}^{B} \ar[d]^{1} \\ k_{v}k_{u}(vu)^{∗} \ar[r] & colim^{vu}(vu)^{∗} \ar[rrr]^{e_{vu}} &  &  & 1_{M}^{B}}

in which e_{u}, e_{v} and e_{vu} are (induced by) the relevant counits, the top row, in view of 39.10(iii) and 47.2, is the counit of the composition of the derived adjunction of (k_{u}, a_{u}) and (k_{v}, a_{v}), while (47.2) the bottom row is the counit of the derived adjunction of their “composition”.

It thus remains to discuss

47.5. Homotopical compatibility with deformable adjoints. This will be a homotopical version of the following categorical notion.

Given an adjunction f: X ↔ Y :g and a functor u: A → B, we will say that f (resp. g) is compatible with u-colimit (resp. u-limit) functors if

(i) there exist u-colimit (resp. u-limit) functors (47.1) on both X and Y

as in that case

(ii) for every pair of u-colimit (resp. u-limit) functors s on X and t on Y , the compositions

f^{B}s and tf^{A}: X^{A} ⟶ Y^{B} (resp. g^{B}t and sg^{A}: Y^{A} ⟶ X^{B})

are left (resp. right) adjoints of the composition

X^{u}g^{B} = g^{A}Y^{u}: Y^{B} ⟶ X^{A} (resp. Y^{u}f^{B} = f^{A}X^{u}: X^{B} ⟶ Y^{A})

and hence canonically naturally isomorphic (37.1).

Similarly, given a deformable adjunction (43.1) and a functor u: A → B, we will therefore say that f (resp. g) is homotopically compatible with homotopy u-colimit (resp. u-limit) functors if

(i)′ there exist u-colimit (resp. u-limit) and homotopy u-colimit (resp. u-

limit) functors on X and Y , and (ii)′ the compositions (42.1) of a homotopy u-colimit (resp. u-limit) functor on X (resp. Y ) with a left (resp. a right) approximation of f^{B} (resp. g^{B}) and of a left (resp. a right) approximation of f^{A} (resp. g^{A}) with a homotopy u-colimit (resp. u-limit) functor on Y (resp. X) are left (resp. right) approximations of the canonically (naturally) isomorphic

Draft: May 14, 2004

(ii) compositions

X^{A} ⟶^{colim^{u}} X^{B} ⟶^{f^{B}} Y^{B}  and  X^{A} ⟶^{f^{A}} Y^{A} ⟶^{colim^{u}} Y^{B}
(resp. Y^{A} ⟶^{lim^{u}} Y^{B} ⟶^{g^{B}} X^{B}  and  Y^{A} ⟶^{g^{A}} X^{A} ⟶^{lim^{u}} X^{B} )

respectively, and hence (38.4) are canonically weakly equivalent (37.5). Then one has the following

47.6. Sufficient conditions for homotopical compatibility. Given a deformable adjunction f: X ↔ Y :g (43.1) and a functor u: A → B, a sufficient condition in order that f (resp. g) is homotopically compatible (47.5) with homotopy u-colimit (resp. u-limit) functors is that there exist u-colimit (resp. u-limit) functors on X and Y , and

(i) the pairs

(colim^{u}: X^{A} → X^{B}, f^{B})  and  (f^{A}, colim^{u}: Y^{A} → Y^{B})
(resp. the pairs (lim^{u}: Y^{A} → Y^{B}, g^{B})  and  (g^{A}, lim^{u}: X^{A} → X^{B}) )

are left (resp. right) deformable (42.3)

which in particular is the case if

(ii) X and Y are saturated (33.9) and the above pairs are locally left (resp. right) deformable (42.3).

Proof. Part (i) is a consequence of 42.4, while part (ii) follows from 33.9(v) and 42.6 and the observation that it readily follows from the fact that the functors X^{u} and Y^{u} are homotopical and that g (resp. f) is right (resp. left) deformable (40.2), that the pairs

(X^{u}, g^{B}) and (g^{A}, Y^{u}) (resp. (Y^{u}, f^{B}) and (f^{A}, X^{u}))

are right (resp. left) deformable (42.3).

48. Left and right systems

In order to be able to deal with the notions of homotopical cocompleteness and completeness (in §49), we first have to generalize the results of §42 and §44 on composable pairs of (not necessarily homotopical) functors between homotopical categories to diagram-like collections of such categories and functors which we will call left and right systems.

We thus start with defining these

48.1. Left and right systems. Given a category C, a left (resp. a right) C-system will be a function F which assigns

(i) to every object D ∈ C a homotopical category F D (ii) to every map u: A → B ∈ C a (not necessarily homotopical) functor

F u: F A → F B, and (iii) to every composable pair of maps u: A → B and v: B → D ∈ C, a

natural weak equivalence (called composer)

F (v, u): (F v)(F u) → F (vu) (resp. F (v, u): F (vu) → (F v)(F u))

which is associative in the sense that, for every three composable maps u: A → B, v: B → D and x: D → E ∈ C, the following diagram commutes

[figure: (F x)(F v)(F u); (F x)F (v,u); F (x,v)(F u); (F x)F (vu); F (x,vu); F (xv)(F u); F (xv,u); F (xvu); (F x)F (vu); (F x)F (v,u); (F x)(F v)(F u); F (xvu); F (x,vu); (resp.; F (xv,u); F (xv)(F u); F (x,v)(F u); )]

and such a C-system will be called

(iv) homotopical if, for every map u: A → B ∈ C, the functor F u is homo-

topical, and

(v) saturated if, for every object D ∈ C, F D is saturated, i.e. (33.9) a map in F D is a weak equivalence iff its image in Ho F D is an isomorphism.

Similarly there are

48.2. Maps and weak equivalences between systems. We only consider maps and weak equivalences between two left or right systems F and G when F D = GD for every object D ∈ C and we then define a map F → G between two such left (resp. right) C-systems F and G as a function h which assigns to every map u: A → B ∈ C a natural transformation hu: F u → Gu, which is compatible with the composers in the sense that, for every composable pair of maps u: A → B and v: B → D ∈ C, the following diagram commutes

[figure: (F v)(F u); F (v,u); (hv)(hu); F (vu); h(vu); (Gv)(Gu); G(v,u); G(vu); F (vu); F (v,u); (resp.; h(vu); (F v)(F u); (hv)(hu); ); G(vu); G(v,u); (Gv)(Gu)]

and call such a map a weak equivalence whenever, for every map u ∈ C, the natural transformation hu is a natural weak equivalence.

We denote the resulting homotopical category of left (resp. right) C-systems by

C_{L}-syst  (resp. C_{R}-syst)

and its full subcategory spanned by the homotopical C-systems (48.1) by

C_{L}-syst_{w} (resp. C_{R}-syst_{w}).

48.3. Example. If H is a C-diagram of homotopical categories and (not necessarily homotopical) functors between them, then H gives rise to a left C-system H_{L} and a right C-system H_{R} which assigns to the objects and the maps of C the same homotopical categories and functors as H and in which, for every composable pair of maps u: A → B and v: B → D ∈ C, the composers (48.1) H_{L}(v, u) and H_{R}(v, u) are the identity natural transformations of the functor

H(vu) = (Hv)(Hu): HA → HD.

48.4. Example. For a homotopical left (resp. right) C-system K (48.1) one can consider its homotopy system, i.e. the function Ho K which assigns (33.9)

(i) to every object D ∈ C, the homotopy category Ho KD, considered as a minimal homotopical category (33.6)

(ii) to every map u: A → B ∈ C, the induced functor Ho Ku: Ho KA →

Ho KB, and (iii) to every composable pair of maps u: A → B and v: B → D ∈ C, the

induced natural isomorphism Ho K(v, u),

and note that (33.9)

(iv) Ho K is a left (resp. a right) C-system.

Next we define

48.5. Left and right approximations. Given a category C and a left or right C-system F (48.1), left or right approximations of F will, roughly speaking, be homotopical C-systems which

(i) consist of left or right approximations of the functors of F, and (ii) in a homotopical sense, are closest to F from the left or the right.

More precisely, a left (resp. a right) approximation of a left (resp. a right) C-system F will be a pair (K, a) consisting of an object (48.2)

K ∈ C_{L}-syst_{w} (resp. K ∈ C_{R}-syst_{w})

and a map (48.2)

a: K → F ∈ C_{L}-syst (resp. a: F → K ∈ C_{R}-syst)

such that

(iii) for every map u ∈ C, the pair (Ku, au) is a left (resp. a right) approxi-

mation of F u (41.1), and (iv) (K, a) is a homotopically terminal (resp. initial) object of “the homo-

topical category of homotopical left (resp. right) C-systems over (resp. under) F ”, i.e. the homotopical category

(C_{L}-syst_{w} ↓ F) (resp. (F ↓ C_{R}-syst_{w}))

which has as objects the above pairs (K, a) and, for every two such pairs (K_{1}, a_{1}) and (K_{2}, a_{2}) as maps and weak equivalences (K_{1}, a_{1}) → (K_{2}, a_{2}) the maps and the weak equivalences t: K_{1} → K_{2} such that a_{2}t = a_{1} (resp. ta_{1} = a_{2}).

It then follows readily from 37.6(vi), 38.3(i) and 48.2 that

(v) if one object of (C_{L}-syst_{w} ↓ F) (resp. (F ↓ C_{R}-syst_{w})) satisfies (iii) and (iv), then every object that satisfies one of (iii) and (iv) also satisfies the other

so that (37.6(vi))

(vi) the left (resp. right) approximations of F, if they exist, are homotopically

unique (37.5). In order to obtain sufficient conditions for the existence of such left and right

approximations we generalize the notions of deformable and locally deformable composable pairs of functors of 42.3 to those of

Draft: May 14, 2004

48.6. Deformable and locally deformable systems. Given a category C, we call a left (resp. a right) C-system F locally left (resp. right) deformable if

(i) there exists a function F_{0} which associates with every object D ∈ C a left (resp. a right) deformation retract F_{0}D ⊂ F D (40.1) such that, for every map u: A → B ∈ C, the functor F u: F A → F B is homotopical on F_{0}A

which (40.2)(ii)′ is equivalent to requiring that

(i)′ there exists a local left (resp. a right) F -deformation, i.e. a function (r, s) which assigns to every object D ∈ C a left (resp. a right) deformation (r_{S}, s_{D}) of F D (40.1) such that, for every map u: A → B ∈ C, the functor (F u)r_{A} is homotopical and the natural transformation (F u)s_{A}r_{A} is a natural weak equivalence,

and we call F left (resp. right) deformable if

(ii) there exists a function F_{0} as in (i) with the additional property that, for every map u: A → B ∈ C, the functor F u: F A → F B sends all of F_{0}A into F_{0}B,

which (as in 42.3(v)) is equivalent to requiring that

(ii)′ there exists a left (resp. right) F -deformation, i.e. a local left (resp. right) F -deformation (r, s) (i)′ such that, for every composable pair of maps u: A → B and v: B → D ∈ C, the natural transformation (F v)s_{B}(F u)r_{A} is a natural weak equivalence

as for such a left (resp. right) F -deformation (r, s), the function F_{0} which assigns to every object D ∈ C, the full subcategory F_{0}D ⊂ F D spanned by the union, taken over all objects A ∈ C and finite sequences of maps

A ⟶^{u_{1}} ⋯ ⟶^{u_{n}} D in C (n ≥ 0)

of the images of the functors

(F u_{n}) ⋯ (F u_{1})r_{A}: F A ⟶ F D (n ≥ 0),

has the property that, for every map v: D → E ∈ C, the functor F v: F D → F E sends all of F_{0}D into F_{0}E and, in view of 40.3 and the observation that the iterated composer (48.1)

(F u_{n}) ⋯ (F u_{1}) ⟶ ⋯ ⟶ F (u_{n} ⋯ u_{k+1})(F u_{k}) ⋯ (F u_{1}) ⟶ ⋯ ⟶ F (u_{n} ⋯ u_{1})

is a natural weak equivalence, is homotopical on F_{0}D.

Now we can state

48.7. Proposition. Given a category C, a sufficient condition in order that a left (resp. a right) C-system F (48.1) has left (resp. right) approximations (48.5) is that F is left (resp. right) deformable ( 48.6(ii)).

Proof (of the left half). Given a left C-system F and a left F -deformation (r, s) (48.6(ii)′) denote, for every left C-system K such that KD = F D for every object D ∈ C, by Kr the function which assigns

(i) to every object D ∈ C, the homotopical category (Kr)D = KD, (ii) to every map u: A → B ∈ C, the functor (Kr)u = (Ku)r_{A}, and (iii) to every composable pair of maps u: A → B and v: B → D ∈ C, the

composition (Kr)(v, u) of the natural transformation

(Kr)v(Kr)u = (Kv)r_{B}(Ku)r_{A} ⟶^{(Kv)s_{B}(Ku)r_{A}} (Kv)(Ku)r_{A}

with the natural weak equivalence

(Kv)(Ku)r_{A} ⟶^{K(v,u)r_{A}} K(vu)r_{A} = (Kr)(vu)

and denote by Ks the function which assigns

(iv) to every map u: A → B ∈ C, the natural transformation

(Ks)u = (Ku)s_{A}.

A sufficient condition in order that Kr so defined is a left C-system is that, for every composable pair of maps u: A → B and v: B → D ∈ C, the natural transformation (Kv)s_{B}(Ku)r_{A} is a natural weak equivalence and it follows that

(v) if K is a homotopical left C-system ( 48.1(iv)), then so is Kr and the function Ks is a weak equivalence Kr → K ∈ C_{L}-syst_{w} (48.5), and (vi) F r is a homotopical left C-system and the function F s is a map F r →

F ∈ C_{L}-syst (48.5),

and that consequently, in view of the naturality in K of the functions Kr and Ks,

(vii) every object (K, a) ∈ (C_{L}-syst_{w} ↓ F) (48.5) gives rise to a zigzag in (C_{L}-syst_{w} ↓ F)

(K, a) ⟵^{Ks} (Kr, as) ⟶^{ar} (F r, F s)

in which as denotes the diagonal of the commutative diagram

\xymatrix{Kr \ar[r]^{ar} \ar[d]_{Ks} & F r \ar[d]^{F s} \\ K \ar[r]^{a} & F}

which is natural in (K, a) and in which ar: Kr → F r is the map given by (ar)u = (au)r_{A} for every map u: A → B ∈ C,

The desired result now follows from 38.2 and 48.6(i)′ and the observation that, as for every map u: A → B ∈ C, the natural transformation

((F s)r)u = (F s)ur_{A} = (F u)s_{A}r_{A}

is a natural weak equivalence, the map (F s)r: (F r)r → F r ∈ C_{L}-syst_{w} is a weak equivalence.

It remains to generalize to systems the rather useful sufficient condition (42.6) in order that a composable pair of functors between saturated homotopical categories be deformable (42.3), namely that the pair is locally deformable (42.3) and has a deformable pair of adjoints.

To do this we thus first have to define

Draft: May 14, 2004

48.8. Adjunctions between left and right systems. Given a category C, a left C-system F and a right C^{op}-system F′, F is said to be a left adjoint of F′ and F′ is said to be a right adjoint of F, if F D = F′D for every object D ∈ C and there is given an adjunction F ↔ F′, i.e. there are given adjunctions

F u: F A ⟷ F′B :F′u^{op},

one for every map u: A → B ∈ C, such that, for every composable pair of maps u: A → B and v: B → D ∈ C, the composers F (v, u) and F′(u^{op}, v^{op}) are conjugate (39.10) with respect to the adjunction

F (vu): F A ⟷ F′D :F′(u^{op}v^{op})

and the composition of the adjunctions

F u: F A ⟷ F′B :F′u^{op}  and  F v: F B ⟷ F′D :F′v^{op} .

The functions ε and η which assign to every map u: A → B ∈ C the counit and the unit

εu: (F u)(F′u^{op}) ⟶ 1_{F B}  and  ηu^{op} : 1_{F A} ⟶ (F′u^{op})(F u)

of the adjunction F u: F A ↔ F B :F′u^{op}, will be called the counit and the unit of the adjunction F ↔ F′.

One then readily verifies that (39.10)

(i) if a left (resp. a right) C-system has a right (resp. a left) adjoint, then all such adjoints are canonically isomorphic (37.1),

(ii) if F is a left (resp. a right) C-system of which the composers are natural isomorphisms, then F has a right (resp. a left) adjoint iff, for every map u: A → B ∈ C, the functor F u: F A → F B has a right (resp. a left) adjoint.

We also define

48.9. Deformable and locally deformable adjunctions of systems. Given a category C, an adjunction F ↔ F′ (48.8) between a left C-system F and a right C^{op}-system F′ will be called

(i) locally deformable, if F and F′ are locally left and right deformable (48.6(i)) respectively, and

(ii) deformable if F and F′ are left and right deformable (48.6(ii)).

Now we can state

48.10. Proposition. Let C be a category and F ↔ F′ a locally deformable adjunction ( 48.8 and 48.9) between a saturated left C-system F ( 48.1(v)) and a saturated right C^{op}-system F′. Then F is left deformable ( 48.8(ii)) iff F′ is right deformable.

Proof (of the “if” part). If (r, s) and (r′, s′) are a local left F -deformation (48.6(i)′) and a right F′-deformation (48.6(ii)′) then, for every composable pair of maps u: A → B and v: G → D ∈ C, the natural transformation (F′u^{op})s′_{B}(F′v^{op})r′_{D} is a natural weak equivalence and hence, in view of the saturation of F D and 42.7, so is the natural transformation (F v)s_{B}(F u)r_{A}, which implies that (r, s) is actually a left F -deformation.

Draft: May 14, 2004

We end with noting that deformable adjunctions of systems have

48.11. Derived adjunctions. Let C be a category and let F ↔ F′ be a deformable adjunction ( 48.8 and 48.9) between a left C-system F ( 48.1) and a right C-system F′ Then the adjunction F ↔ F′ induces, for every left approximation (K, a) of F and every right approximations (K′, a′) of F′ (48.5) a derived adjunction (48.4)

Ho K ⟷ Ho K′

which ( 48.8(ii)), for every map u: A → B ∈ C, consists of the derived adjunction (44.2)

Ho Ku: Ho F A ⟷ Ho F′B :Ho K′u^{op}

associated with the left approximation (Ku, au) of F u and the right approximation (K′u^{op}, a′u^{op}) of F′u^{op}.

Proof. It follows from 44.4 that, for every composable pair of maps u: A → B and v: B → D ∈ C, the composition

Ho(Kv)(Ku): Ho X^{A} ⟷ Ho X^{D} :Ho(K′u^{op})(K′v^{op})

of the derived adjunctions (44.2)

Ho Ku: Ho X^{A} ⟷ Ho X^{B} :Ho K′u^{op}

and Ho Kv: Ho X^{B} ⟷ Ho X^{D} :Ho K′v^{op}

associated with the approximations

Ku ⟶^{au} F u, F′u^{op} ⟶^{a′u^{op}} K′u^{op}, Kv ⟶^{av} F v and F′v^{op} ⟶^{a′v^{op}} K′v^{op}

is the derived adjunction associated with the approximations

(Kv)(Ku) ⟶^{(av)(au)} (F v)(F u) and (F′u^{op})(F′v^{op}) ⟶^{(a′u^{op})(a′v^{op})} (K′u^{op})(K′v^{op}) .

We then have to show that the natural isomorphisms

Ho K(v, u)  and  Ho K′(u^{op}, v^{op})

are conjugate with respect to this composite derived adjunction and the derived adjunction

Ho K(vu): Ho X^{A} ⟷ Ho X^{D} :Ho K′(u^{op}v^{op})

associated with the approximations

K(vu) ⟶^{a(vu)} F (vu)  and  F′(u^{op}v^{op}) ⟶^{a′(u^{op}v^{op})} K′(u^{op}v^{op})

49. HOMOTOPICAL COCOMPLETENESS AND COMPLETENESS (SPECIAL CASE) 157

and do this by noting that this follows from 44.3 and the observation that the commutative diagrams

\xymatrix{(Kv)(Ku) \ar[r]^{(av)(au)} \ar[d]_{K(v,u)} & (F v)(F u) \ar[d]^{F (vu)} \\ K(vu) \ar[r]^{a(vu)} & F (vu)}

and  \xymatrix{(F′u^{op})(F′v^{op}) \ar[r]^{(a′u^{op})(a′v^{op})} & (K′u^{op})(K′v^{op}) \\ F′(u^{op}v^{op}) \ar[r]^{a′(u^{op}v^{op})} \ar[u]^{F′(u^{op},v^{op})} & K′(u^{op}v^{op}) \ar[u]_{K′(u^{op},v^{op})}}

are (41.6) left and right approximations respectively of the conjugate natural transformations (43.1)

F (v, u)  and  F (u^{op}, v^{op}) .

49. Homotopical cocompleteness and completeness (special case)

We now use the results of the preceding two sections to obtain, for homotopical categories which are cocomplete and complete, notions of homotopical cocompleteness and completeness which are considerably stronger than requiring the existence of homotopy D-colimit or D-limit functors for every small category D, or even of homotopy u-colimit or u-limit functors for every functor u: A → B between small categories.

We start with a brief discussion of the categorical notions of

49.1. Cocompleteness and completeness. Let cat denote the category of the small categories (32.3) and for every homotopical category X, let X^{(cat)} denote the cat^{op}-diagram which associates with every object D ∈ cat the diagram category X^{D} and with every map u: A → B ∈ cat the induced diagram functor X^{u}: X^{B} → X^{A} (47.1(i)′). Then X is called cocomplete (resp. complete) if, for every object D ∈ cat, there exists a D-colimit (resp. D-limit) functor on X (47.1(i)) and such cocompleteness (resp. completeness) implies that

(i) for every map u: A → B ∈ cat there exist u-colimit (resp. u-limit) functors on X (for instance the functor which sends an object T ∈ X^{A} to the functor B → X which associates with each object B ∈ B the object ( 47.1(ii)′)

colim^{(u↓B)} j^{∗}T (resp. lim^{(B↓u)} j^{∗}T)

where

j: (u ↓ B) ⟶ A (resp. j: (B ↓ u) ⟶ A)

denotes the forgetful functor), and (ii) for every composable pair of maps u: A → B and v: B → D ∈ cat, every composition of a u-colimit (resp. u-limit) functor on X with a vcolimit (resp. v-limit) functor is a vu-colimit (resp. vu-limit) functor,

Draft: May 14, 2004

and even though in general it is not possible to choose, for every map u ∈ cat a u-colimit (resp. u-limit) functor on X such that these functors form a diagram indexed by cat,

(iii) there exists what we will call a colimit(resp. a limit) system on X, i.e. a left (resp. a right) adjoint of the right (resp. left) cat^{op}-system X_{R}^{(cat)} (resp. X_{L}^{(cat)}) (48.3 and 48.8).

Clearly (48.8(i) and (ii))

(iv) such a colimit (resp. limit) system on a category X, if it exists, is cate-

gorically unique (37.1), and

(v) such a colimit (resp. limit) system on a category X exists iff X is co-

complete (resp. complete).

Moreover (vi) if X is cocomplete (resp. complete), then the function

colim^{(cat)}  (resp. lim^{(cat)})

given by the formulas ( 47.1 and 47.3)

colim^{(cat)} u = colim^{u} (resp. lim^{(cat)} u = lim^{u})

and

colim^{(cat)}(v, u) = colim^{(v,u)} (resp. lim^{(cat)}(v, u) = lim^{(v,u)})

is such a colimit (resp. limit) system on X. Now we turn to

49.2. Homotopical cocompleteness and completeness. If X is a homotopical category, then

(i) the existence of homotopy D-colimit (or D-limit) functors on X for every

small category D, i.e. object D ∈ cat (32.3)

need not imply

(ii) the existence of homotopy u-colimit (or a u-limit) functors on X for every

map u ∈ cat,

nor need (ii) imply that

(iii) for every composable pair of maps u: A → B and v: B → D ∈ cat, the compositions of homotopy u-colimit (or u-limit) functors on X with homotopy v-colimit (or v-limit) functors on X are homotopy vu-colimit (or vu-limit) functors on X.

In view of 49.1(v) we therefore will call a homotopical category X homotopically cocomplete (resp. complete) if

(iv) X is cocomplete (resp. complete) and

(v) there exists a homotopy colimit (resp. limit) system on X, which we will define as a left (resp. a right) approximation (48.5) of the colimit (resp. the limit) system colim^{(cat)} (resp. lim^{(cat)}).

Clearly (48.5(vi)) (vi) such homotopy colimit (resp. limit) systems on X, if they exist, are ho-

motopically unique (37.5).

Moreover one has, in view of 48.6, 48.7 and 48.10, the following sufficient conditions for their existence, i.e.

49.3. Sufficient conditions for homotopical cocompleteness and completeness. Sufficient conditions for the homotopical cocompleteness (resp. completeness) of a homotopical category X are that

(i) X is cocomplete (resp. complete) and the colimit (resp. the limit) system colim^{(cat)} (resp. lim^{(cat)}) on X is left (resp. right) deformable, i.e. there exists a function F_{0} which assigns to every object D ∈ cat a left (resp. a right) deformation retract F_{0}D ⊂ X^{D} (40.1) such that, for every map u: A → B ∈ cat, the functor colim^{u}: X^{A} → X^{B} (resp. lim^{u}: X^{A} →

X^{B}

(i)′ is homotopical on F_{0}A, and (i)′′ sends all of F_{0}A into F_{0}B

which, in view of the right (resp. the left) deformability of X_{R}^{(cat)} (resp. X_{L}^{(cat)}) ( 48.3 and 49.1), is in particular the case if

(ii) X is cocomplete (resp. complete) and saturated ( 33.9 and 48.1(v)) and the colimit (resp. the limit) system colim^{(cat)} (resp. lim^{(cat)}) on X is locally left (resp. right) deformable, i.e. there exists a function F_{0} which assigns to every object D ∈ cat a left (resp. a right) deformation retract F_{0}D ⊂ X^{D} such that, for every map u: A → B ∈ cat, the functor colim^{u}: X^{A} → X^{B} (resp. lim^{u}: X^{A} → X^{B}) satisfies (i)′ but not necessarily (i)′′ (in which case there exists a possibly different such function which satisfies both (i)′) and (i)′′.

We end with noting that 48.11 implies that, under the assumptions made in 49.3, homotopy colimit and limit systems come with

49.4. Derived adjunctions. Let X be a homotopical category which satisfies 49.3(i). Then every homotopy colimit (resp. limit) system (K, a) on X comes with the derived adjunction

Ho K ⟷ Ho X_{R}^{(cat)} (resp. Ho X_{L}^{(cat)} ⟷ Ho K)

associated with (K, a) and the pair consisting of X_{R}^{(cat)} (resp. X_{L}^{(cat)}) and its identity map.

50. Homotopical colimit and limit functors

The homotopy colimit and limit functors which we considered in the preceding sections assume the existence of colimit or limit functors and choice of an arbitrary but fixed such functor, and even though it seems, for the moment at least, that this is the only situation in which one is really interested, we will now discuss a notion of what we will call homotopical colimit and limit functors which in the following sense “generalize” the homotopy colimit and limit functors:

Given a homotopical category X and a functor u: A → B, we define a homotopical u-colimit (or u-limit) functor on X as a homotopically terminal (or initial) Kan extension (38.6) of the identity functor along the induced diagram functor X^{u}: X^{B} → X^{A}, and note that as [Mac71, Ch. X, §7]

Draft: May 14, 2004

(i) the left (or right) adjoints of X^{u} (i.e. the u-colimit (or u-limit) functors on X) are exactly the same as the terminal (or initial) Kan extensions of the identity functor along X^{u},

and as

(ii) for every terminal (or initial) Kan extension f of the identity functor of X^{B} along X^{u}, the left (or right) approximations of f (41.1) are the f-presentations of the homotopically terminal (or initial) Kan extensions of the identity functor along X^{u}, in the sense that they are the approximations associated with these homotopical Kan extensions under a 1-1 correspondence which is induced by the counit (or unit) of the Kan extension f,

it follows that

(iii) the homotopy u-colimit (or u-limit) functors on X are exactly the colim^{u}- (or lim^{u}-)presentations of the homotopical u-colimit (or u-limit) functors on X.

We start with discussing

50.1. Kan extensions of the identity. Given a functor g: Y → X, we recall from 38.6 that a terminal (resp. an initial) Kan extension of the identity (i.e. 1_{Y}) along g is a functor f: X → Y , together with an often suppressed natural transformation

e: fg ⟶ 1_{Y} (resp. e: 1_{Y} ⟶ fg),

called counit (resp. unit) such that the pair (f, e) is a terminal (resp. an initial) object of the category (38.6)

(−g ↓ 1_{Y}) (resp. (1_{Y} ↓ −g)) .

Thus (37.2(vi))

(i) such Kan extensions of the identity, if they exist, are categorically unique

(37.1).

Moreover [Mac71, Ch. X, §7]

(ii) for every functor g: Y → X which has left (resp. right) adjoints (46.4), the pairs (f, e) consisting of a terminal (resp. an initial) Kan extension of the identity along g and its counit (resp. unit) are exactly the same as the pairs (f, e) consisting of a left (resp. a right) adjoint of g and the counit (resp. the unit) of its adjunction

and hence

(iii) given a functor g: Y → X, a sufficient condition for the existence of a terminal (resp. an initial) Kan extension of the identity along g is the existence of a left (resp. a right) adjoint of g.

Next we consider

50.2. Homotopical Kan extensions of the identity. Given a (not necessarily homotopical) functor g: Y → X between homotopical categories, we recall from 38.6 that a homotopically terminal (resp. initial) Kan extension of the identity (i.e. 1_{Y}) along g is a homotopical functor k: X → Y , together with an often suppressed natural transformation

d: kg ⟶ 1_{Y} (resp. d: 1_{Y} ⟶ kg)

Draft: May 14, 2004

such that the pair (k, d) is a homotopically terminal (resp. initial) object (38.2) of the category (38.6)

(−g ↓ 1_{Y})_{w} (resp. (1_{Y} ↓ −g)_{w}) .

Thus (37.6(vi))

(i) such homotopical Kan extensions of the identity, if they exist, are homo-

topically unique (37.5).

Moreover one readily verifies that

(ii) for every (not necessarily homotopical) functor g: Y → X between homotopical categories, every terminal (resp. initial) Kan extension f of the identity along g gives rise to a 1-1 correspondence between the left (resp. the right) approximations of f (41.1) and the homotopically terminal (resp. initial) Kan extensions of the identity along g, which is induced by the isomorphism (41.1)

(Fun_{w}(X, Y ) ↓ f) ≈ (−g ↓ 1_{Y})_{w} (resp. (f ↓ Fun_{w}(X, Y )) ≈ (1_{Y} ↓ − g)_{w})

which sends an object (k, a) to the pair (k, d) in which, if e denotes the counit (resp. the unit) of f (50.1), the natural transformation d is the composition

kg ⟶^{ag} fg ⟶^{e} 1_{Y} (resp. 1_{Y} ⟶^{e} fg ⟶^{ag} kg).

and this, together with 41.2 implies that

(iii) given a functor g: Y → X, a sufficient condition for the existence of a homotopically terminal (resp. initial) Kan extension of the identity along g is the existence of a left (resp. right) deformable (40.2) terminal (resp. initial) Kan extension of the identity along g.

We also note that, in view of (ii), it is convenient (iv) given a functor g: Y → X, a terminal (resp. initial) Kan extension f of

the identity along g, and a pair consisting of a left (resp. a right) approximation (k, a) of f (41.1) and the corresponding (as in (ii)) homotopically terminal (resp. initial) Kan extension (k, d) of the identity along g, to say that (k, a) is the f-presentation of (k, d).

50.3. Compositions. For every composable pair of (not necessarily homotopical) functors

g_{2} : Z ⟶ Y and g_{1} : Y ⟶ X

between homotopical categories one can consider the homotopical composition functors

(−g_{1} ↓ 1_{Y}) × (−g_{2} ↓ 1_{Z}) ⟶ (−g_{1}g_{2} ↓ 1_{Z})

(resp. (1_{Y} ↓ −g_{1}) × (1_{Z} ↓ −g_{2}) ⟶ (1_{Z} ↓ −g_{1}g_{2}))

and

(−g_{1} ↓ 1_{Y})_{w} × (−g_{2} ↓ 1_{Z})_{w} ⟶ (−g_{1}g_{2} ↓ 1_{Z})_{w}

(resp. (1_{Y} ↓ −g_{1})_{w} × (1_{Z} ↓ −g_{2})_{w} ⟶ (1_{Z} ↓ −g_{1}g_{2})_{w})

which send an object ((f_{1}, e_{1}), (f_{2}, e_{2})) to the pair (f_{2}f_{1}, e_{2}e_{1}) (resp. (f_{2}f_{1}, e_{1}e_{2})) where e_{2}e_{1} (resp. e_{1}e_{2}) denotes the composite natural transformation

f_{2}f_{1}g_{1}g_{2} ⟶^{e_{1}} f_{2}g_{2} ⟶^{e_{2}} 1_{Z} (resp. 1_{Z} ⟶^{e_{2}} f_{2}g_{2} ⟶^{e_{1}} f_{2}f_{1}g_{1}g_{2}).

Draft: May 14, 2004

It then follows from 39.10(iii), 46.4 and 50.1(ii) that

(i) a sufficient condition in order that, for two terminal (resp. initial) Kan extensions of the identity f_{1} and f_{2} along g_{1} and g_{2} respectively, their composition f_{2}f_{1} is similar Kan extension along g_{1}g_{2}, is that g_{1} and g_{2} both have left (resp. right) adjoints, in which case ( 50.1(ii)) f_{1} and f_{2} are such adjoints and their composition as Kan extensions coincides with their composition as adjoints ( 39.10(iii)).

Furthermore, it follows from 50.2(ii) that

(ii) for every two terminal (resp. initial) Kan extensions of the identity f_{1} and f_{2} along g_{1} and g_{2} respectively for which the composition f_{2}f_{1} is a similar Kan extension along g_{1}g_{2}, the diagram

\xymatrix{(Fun_{w}(X, Y ) ↓ f_{1}) × (Fun_{w}(Y , Z) ↓ f_{2}) \ar[r] \ar[d]_{≈} & (Fun_{w}(X, Z) ↓ f_{2}f_{1}) \ar[d]^{≈} \\ (−g_{1} ↓ 1_{Y})_{w} × (−g_{2} ↓ 1_{Z})_{w} \ar[r] & (−g_{1}g_{2} ↓ 1_{Z})_{w}}

(resp. \xymatrix{(f_{1} ↓ Fun_{w}(X, Y )) × (f_{2} ↓ Fun_{w}(Y , Z)) \ar[r] \ar[d]_{≈} & (f_{2}f_{1} ↓ Fun_{w}(X, Z)) \ar[d]^{≈} \\ (1_{Y} ↓ −g_{1})_{w} × (1_{Z} ↓ −g_{2})_{w} \ar[r] & (1_{Z} ↓ −g_{1}g_{2})_{w}} )

in which the horizontal maps are the composition functors and the vertical ones are as in 50.2(i), commutes.

and this, together with 42.4, implies that

(iii) if f_{1} and f_{2} are terminal (resp. initial) Kan extensions of the identity along g_{1} and g_{2} respectively for which the composition f_{2}f_{1} is a similar Kan extension along g_{1}g_{2}, then a sufficient condition in order that every composition of a homotopically terminal (resp. initial) Kan extension of the identity along g_{1} with one along g_{2} is one along g_{2}g_{2} is that

(iii)′ the pair (f_{1}, f_{2}) is left (resp. right) deformable (42.3)

Now we are ready to deal with

50.4. Homotopical colimit and limit functors. Given a homotopical category X and a functor u: A → B, a homotopical u-colimit (resp. u-limit) functor on X will be a homotopically terminal (resp. initial) Kan extension of the identity (50.2) along the induced diagram functor X^{u}: X^{B} → X^{A} (47.1). It then follows from 50.2(i), 50.2(iii) and 50.3(iii) that

(i) such homotopical colimit and limit functors, if they exist, are homotopically unique (37.5),

(ii) a sufficient condition for the existence of such a homotopical u-colimit (resp. u-limit) functor on X is the existence of a left (resp. a right) deformable (40.2) terminal (resp. initial) Kan extension of the identity along the induced diagram functor X^{u}, and

(iii) given two composable functors u: A → B and v: B → D, a sufficient condition in order that every composition of a homotopical u-colimit

Draft: May 14, 2004

(resp. u-limit) functor on X with a homotopical v-colimit (resp. v-limit) functor on X is a homotopical vu-colimit (resp. vu-limit) functor on X is that

(iii)′ there exist terminal (resp. initial) Kan extensions of the identity f_{1} and f_{2} along X^{u} and X^{v} respectively for which the composition f_{2}f_{1} is a similar Kan extension along X^{vu}, and

(iii)′′ the pair (f_{1}, f_{2}) is left (resp. right) deformable (42.3).

We end with two propositions which formalize the statement that, in the case where there exist colimit and limit functors, the above homotopical colimit and limit functors are “essentially the same” as the homotopy colimit and limit functors which we considered in §47.

50.5. Proposition. Let X be a homotopical category and let u: A → B be a functor for which there exist u-colimit (resp. u-limit) functors on X. Then

(i) there exist homotopical u-colimit (resp. u-limit) functors on X (50.4)

iff

(ii) there exist homotopy u-colimit (resp. u-limit) functors on X (47.1)

and

(iii) if (i) and (ii) hold, then the homotopy u-colimit (resp. u-limit) functors on X are exactly the colim^{u}- (resp. lim^{u}-) presentations ( 50.2(iv)) of the homotopical u-colimit (resp. u-limit) functors on X.

Proof. This follows from 50.1(iii) and 50.2(ii).

50.6. Proposition. Let X be a homotopical category and let u: A → B and v: B → D be functors for which there exist u- and v-colimit (resp. u- and v-limit) functors on X. Then

(i) every composition of a homotopical u-colimit (resp. u-limit) functor on X with a homotopical v-colimit (resp. v-limit) functor on X is a homotopical vu-colimit (resp. vu-limit) functor on X

iff

(ii) every “composition” (47.3) of a homotopy u-colimit (resp. u-limit) functor on X (47.1) with a homotopy v-colimit (resp. v-limit) functor on X is a homotopy vu-colimit (resp. vu-limit) functor on X

and

(iii) if (i) and (ii) hold, then, for every pair consisting of a homotopical ucolimit (resp. u-limit) functor (k_{u}, d_{u}) on X with (k_{u}, a_{u}) as its colim^{u}- (resp. lim^{u}-) presentation (50.5) and homotopical v-colimit (resp. v-limit) functor (k_{v}, d_{v}) on X with (k_{v}, a_{v}) as its colim^{v}- (resp. lim^{v}-) presentation, the “composition” (47.3) of (k_{u}, a_{u}) with (k_{v}, a_{v}) is the colim^{vu}- (resp. lim^{vu}-) presentation ( 50.2(iv)) of the composition of (k_{u}, d_{u}) and (k_{v}, d_{v}) (50.3).

Proof (of the colimit half). This follows from 50.3(ii) and the commutativity of the diagram

\xymatrix{(Fun_{w}(X^{A}, X^{D}) ↓ colim^{v} colim^{u}) \ar[rr]^{^{colim{(v,u) \atop ∗}}}_{≈} \ar@{-->}[dr]_{≈} &  & (Fun_{w}(X^{A}, X^{D}) ↓ colim^{vu}) \ar[dl]^{≈} \\  & (−X^{u}X^{v} ↓ 1_{X}^{D})_{w}}

in which the slanted maps are as in 50.2(ii).

51. Homotopical cocompleteness and completeness (general case)

In this last section we describe, for arbitrary homotopical categories, notions of homotopical cocompleteness and completeness which, for categories which are cocomplete or complete, reduce to the one we considered in §49, by calling a homotopical category X homotopically cocomplete if there exist what we will call homotopical colimit systems on X which we define as “homotopically terminal Kan extensions along the right cat^{op}-system X_{R}^{(cat)} (49.1)”.

To do this we have to generalize the results of §50 on Kan extensions and homotopical Kan extensions of the identity to systems (§48) and as systems consist not only of categories and functors, but also natural transformations we first have to consider

51.1. A notion of conjugation for Kan extensions. Given two functors g_{1}, g_{2} : Y → X, a natural transformation v: g_{2} → g_{1} and terminal (resp. initial) Kan extensions of the identity f_{1} and f_{2} along g_{1} and g_{2} respectively we

(i) will refer to the unique natural transformation u: f_{1} → f_{2} for which the diagram

\xymatrix{f_{1}g_{2} \ar[r]^{v} \ar[d]_{u} & f_{1}g_{1} \ar[d]^{e_{1}} \\ f_{2}g_{2} \ar[r]_{e_{2}} & 1_{Y}}  (resp. \xymatrix{1_{Y} \ar[r]^{e_{1}} \ar[d]_{e_{2}} & f_{1}g_{1} \ar[d]^{u} \\ f_{2}g_{2} \ar[r]_{v} & f_{2}g_{1}} )

commutes as the conjugate of v with respect to these Kan extensions

and note that it follows from the diagram preceding 39.10(v) and 50.1(ii) that

(ii) if g_{1} and g_{2} both have left (resp. right) adjoints, this conjugate of v with respect to these Kan extensions is also the conjugate of v with respect o the associated ( 50.1(ii)) adjunctions.

Next we also note that the counit and the unit of an adjunction F ↔ F′ between a left C-system F and a right C^{op}-system F′ (48.8) are compatible with the composers of F and F′ in the sense that they are what we will call

51.2. Dinatural transformations. Given a category C, a left C-system F and a right C^{op}-system F′ (48.1) such that F D = F′D for every object D ∈ C, a dinatural transformation

e: F F′ ⟶ 1 (resp. e: 1 ⟶ F′F)

51. HOMOTOPICAL COCOMPLETENESS AND COMPLETENESS (GENERAL CASE) 165

will be a function e which assigns to every map u: A → B ∈ C a natural transformation

eu: (F u)(F′u^{op}) ⟶ 1_{F B} (resp. eu^{op} : 1_{F A} ⟶ (F′u^{op})(F u))

which is compatible with the composers of F and F′ in the sense that for every composable pair of maps u: A → B and v: B → D ∈ C, the diagram

\xymatrix{(F v)(F u)F′(u^{op}v^{op}) \ar[r]^{F (v,u)} \ar[d]^{F′(u^{op},v^{op})} & F (vu)F′(vu)^{op} \ar[d]^{e(vu)} \\ (F v)(F u)(F′u^{op})(F′v^{op}) \ar[r]^{(ev)(eu)} & 1_{F D}}

(resp. \xymatrix{1_{F A} \ar[r]^{(ev^{op})(eu^{op})} \ar[d]^{e(vu)^{op}} & (F′u^{op})(F′v^{op})(F v)(F u) \ar[d]^{F (v,u)} \\ F′(vu)^{op}F (vu) \ar[r]^{F′(u^{op},v^{op})} & (F′u^{op})(F′v^{op})F (vu)} )

commutes. Using 39.10(iii) and the diagrams preceding 39.10(v) one then can show that

(i) for every adjunction F ↔ F′ between a left C-system F and a right C^{op}-system F′ (48.8) its counit and unit are dinatural transformations

F F′ ⟶ 1  and  1 ⟶ F′F

and (ii) for every left C-system F, right C^{op}-system F′ and dinatural transfor-

mation

e: F F′ ⟶ 1 (resp. e: 1 ⟶ F′F)

there is at most one adjunction F ↔ F′ with e as its counit (resp. unit).

Now we are ready to deal with

51.3. Kan extensions along systems. Given a category C and a right (resp. a left) C^{op}-system G, let

(−G ↓ 1) (resp. (1 ↓ −G))

denote the homotopical category which has as objects the pairs (F, e) consisting of a left (resp. a right) C-system F and a dinatural transformation (51.2)

e: F G ⟶ 1 (resp. e: 1 ⟶ F G)

and as maps and weak equivalences (F_{1}, e_{1}) → (F_{2}, e_{2}) the maps and weak equivalences t: F_{1} → F_{2} (48.2) such that, for every map u: A → B ∈ C, the diagram

\xymatrix{(F_{1}u)(Gu^{op}) \ar@{-->}[dr]^{e_{1}u} \ar[dd]_{tu} \\  & 1 \\ (F_{2}u)(Gu^{op}) \ar@{-->}[ur]_{e_{2}u}}  (resp. \xymatrix{ & (F_{1}u)(Gu^{op}) \ar[dd]^{tu} \\ 1 \ar[ur]^{e_{1}u} \ar[dr]_{e_{2}u} \\  & (F_{2}u)(Gu^{op})} )

commutes, and let

(−G ↓ 1)_{w} ⊂ (−G ↓ 1) (resp. (1 ↓ −G)_{w} ⊂ (1 ↓ −G))

denote the full subcategory spanned by the pairs (F, e) for which F is homotopical (48.1).

We then define

(i) a terminal (resp. an initial) Kan extension along G as a terminal (resp. an initial) object

(F, e) ∈ (−G ↓ 1) (resp. (F, e) ∈ (1 ↓ −G))

such that, for every map u: A → B ∈ C, the pair (F u, eu) is a terminal (resp. an initial) Kan extension of the identity along Gu^{op} (47.1), and

(ii) a homotopically terminal (resp. initial) Kan extension along G as a homotopically terminal (resp. initial) object

(K, d) ∈ (−G ↓ 1)_{w} (resp. (K, d) ∈ (1 ↓ −G)_{w})

such that, for every map u: A → B ∈ C, the pair (Ku, du) is a homotopically terminal (resp. initial) Kan extension of the identity along Gu^{op} (50.2)

and note that (37.2(vi) and 37.6(vi))

(iii) such Kan extensions and homotopical Kan extensions, if they exist, are

respectively categorically unique (37.1) and homotopically unique (37.5).

Moreover it follows from 50.1(ii) and 51.1(ii) that

(iv) for every right (resp. left) C^{op}-system G which has left (resp. right) adjoints (48.8), a pair (F, e) is a terminal (resp. an initial) Kan extension along G iff there exists an adjunction F ↔ G (resp. G ↔ F) with e as its counit (resp. unit),

so that

(iv)′ a sufficient condition for the existence of a terminal (resp. an initial) Kan extension along G is the existence of a left (resp. a right) adjoint of G.

Similarly it follows from 50.2(ii) that

(v) for every right (resp. left) C^{op}-system G, every terminal (resp. initial) Kan extension (F, e) along G gives rise to a 1-1 correspondence between the left (resp. the right) approximations of F (48.5) and the homotopically terminal (resp. initial) Kan extensions along G, which is induced by the isomorphism (48.2)

(C_{L}-syst_{w} ↓ F) ≈ (−G ↓ 1)_{w} (resp. (F ↓ C_{R}-syst_{w}) ≈ (1 ↓ −G)_{w})

which sends an object (K, a) to the pair (K, d) in which d denotes the dinatural transformation (51.2)

(Ku)(Gu^{op}) ⟶^{au} (F u)(Gu^{op}) ⟶^{eu} 1_{GB} (resp. 1_{GB} ⟶^{eu} (F u)(Gu^{op}) ⟶^{au} (Ku)(Gu^{op}))

which together with (iv)′ and 48.7 implies that

(vi) a sufficient condition for the existence of a homotopically terminal (resp.

initial) Kan extension along G

is that

(vi)′ there exists a terminal (resp. initial) Kan extension along G which is left

(resp. right) deformable (48.6)

We will also, as in 50.2(iv)

Draft: May 14, 2004

(vii) given a a right (resp. a left) C^{op}-system G, a terminal (resp. an initial) Kan extension (F, e) along G and a pair consisting of a left (resp. a right) approximation (K, a) of F and the corresponding (see (v) above) homotopically terminal (resp. initial) Kan extension (K, d) along G, say that (K, a) is the F -presentation of (K, d).

Finally we are ready to discuss

51.4. Homotopical cocompleteness and completeness. Given a homotopical category X, we will say that X is homotopically cocomplete (resp. complete) if there exists a homotopical colimit (resp. limit) system on X, which we define as a homotopically terminal (resp. initial) Kan extension (51.3(ii)) along the right (resp. the left) cat^{op}-system X_{R}^{(cat)} (resp. X_{L}^{(cat)}) (49.1). Then (51.3(iii))

(i) such homotopical colimit (resp. limit) systems on X, if they exist, are

homotopically unique (37.5)

while 49.1 and 51.3(vi) imply that

(ii) a sufficient condition for the homotopical cocompleteness (resp. completeness) of X, i.e. the existence of homotopical colimit (resp. limit) systems on X,

is that

(ii)′ there exists a terminal (resp. initial) Kan extension along X_{R}^{(cat)} (resp.

X_{L}^{(cat)}) which is left (resp. right) deformable (48.6)

Moreover it follows from 49.1 and 51.3(iv) and (vii) that, if X is cocomplete (resp. complete), then

(iii) there exist homotopical colimit (resp. limit) systems on X iff

(iii)′ there exist homotopy colimit (resp. limit) systems on X (49.2) and

(iii)′′ if (iii) and (iii)′ hold then the homotopy colimit (resp. limit) systems on X are exactly the colim^{(cat)}- (resp. lim^{(cat)}-) presentations ( 51.3(vii)) of the homotopical colimit (resp. limit) systems on X

so that

(iv) for cocomplete (resp. complete) homotopical categories the above notion of homotopical cocompleteness (resp. completeness) coincides with the one we considered in §49.

Draft: May 14, 2004

Index

Page numbers in bold point to the definitions of the indexed terms.

adjunctions, 120–121 counits of, 120, 120–121, 133–136, 155 deformable, 14–15, 49, 49–52, 117–120,

131 derived, 34, 44, 51, 42–52, 119, 137, 136–

141 of systems, 155 partial, 42, 51, 131, 131–136 Quillen, 38 units of, 120, 120–121, 133–136, 155

adjunctions of systems and dinatural transformations, 165 and Kan extensions along systems, 166

all or none proposition for compositions, 129 approximations, 33–34, 40–42, 49, 118–120,

125, 124–128 of Quillen functors, 40

arrow categories, 99, 99–110

T-diagrams of, 100 axiom

factorization, 24 lifting, 24 limit, 23 retract, 17–18, 24 two out of six, 23 two out of three, 17–18

canonically isomorphic objects, 36, 111 canonically weakly equivalent objects, 37,

112 CAT, 93, 95–97 cat, 93 CAT_{w}, 95, 95–97 cat-systems

left, 60, 60–63, 71 right, 60, 60–63, 71

categorical uniqueness, 36, 36–37, 90–91, 111,

111–112 of colimit systems, 158 of initial objects, 36, 111 of Kan extensions along systems, 166 of left adjoints of left systems, 155 of limit systems, 158 of right adjoints of right systems, 155

of terminal objects, 36, 111 categorically contractible categories, 36, 111 categorically full subcategories, 36, 111 categorically unique objects, 36, 111 categories, 4, 21, 20–21, 76–77, 87–88, 93,

92–93 arrow, 99, 99–110 categorically contractible, 36, 111 cocomplete, 23, 56, 56, 157 complete, 23, 56, 56, 157 connected components of, 100 diagram, 22, 94 functor, 22, 94 homotopical, 11–12, 21, 18–23, 75, 77–78,

88–89, 93, 93–99 homotopically cocomplete, 71, 158, 167 homotopically complete, 71, 158, 167 homotopically contractible, 37, 112 homotopy, 5–8 indexing, 94 locally small, 21, 92 model, 3–4, 17–18, 23, 23–27 n-arrow, 99 of simplices, 65, 65–70 of weak equivalences, 94 Reedy, 63 Reedy model, 63–70 small, 21, 92 small U-, 20, 92, 92–93 U-, 20, 91, 92, 92–93 underlying, 94 we-, 4–5 with cofibrant constants, 64, 64–66 with fibrant constants, 64, 64–66

category of types, 99 classical homotopy categories, 29 classifying space, 102 closed model categories, 24–26 closure properties of model categories, 25–26 cocomplete categories, 23, 56, 56, 157

homotopically, 71, 158, 167 cocompleteness, 56, 56, 157–158

and colimit systems, 56, 158 homotopical, 54, 60–63, 71, 84–85, 157–

159, 167 cofibrant fibrant objects, 28 cofibrant objects, 27 cofibrations, 23

characterization of, 25–26 Reedy, 63 trivial, 24

colim^{D}, 146 colim^{u}, 146 colimit description of Grothendieck construc-

tions, 103 colimit description of the homotopy cate-

gory, 101 colimit functors, 55, 55–56

compatibility with left adjoints, 149 D-, 55, 146 homotopical, 83–85, 162–163 homotopy, 8–11, 15–16, 53–54, 57, 57–60,

70, 83–85, 146–150 u-, 55, 146

colimit systems, 56, 158 and cocompleteness, 56, 158 categorical uniqueness of, 158 homotopical, 167 homotopy, 61, 61–63, 71, 158

combined natural transformations, 164 compatibility of

left adjoints with colimit functors, 149 right adjoints with limit functors, 149

complete categories, 23, 56, 56, 157

homotopically, 71, 158, 167 completeness, 56, 56, 157–158

and limit systems, 56, 158 homotopical, 60–63, 71, 157–159, 167

components

connected, 100 composers, 150

of cat-systems, 60 composition functors, 41, 128, 161 compositions

all or none proposition, 129 of deformable functors, 50–51, 82–83 of derived adjunctions, 139–140 of derived adjunctions of Quillen adjunc-

tions, 44 of homotopy colimit functors, 148 of homotopy limit functors, 148 of left approximations, 118–119, 128–131 of left approximations of left Quillen func-

tors, 42 of left deformable functors, 129–131 of partial adjunctions, 132 of right approximations, 118–119, 128–131 of right approximations of right Quillen

functors, 42 of right deformable functors, 129–131 conjugate deformable natural transformations,

131 conjugate natural transformations, 121, 121,

164 connected components, 100 constant diagram functors, 55, 146 contractible categories categorically, 36, 111 homotopically, 37, 112

counits of adjunctions, 120, 120–121, 133–

136, 155 and dinatural transformations, 165 counits of terminal Kan extension, 116 counits of terminal Kan extensions, 160

D-colimit functors, 55, 146 homotopy, 57, 146 D-colimit functors homotopy, 57

D-limit functors, 55, 146

homotopy, 146 deformability result, 58 deformable adjunctions, 14–15, 49, 49–52,

117–120, 131 Quillen condition for, 52 deformable adjunctions of systems derived adjunctions of, 156

deformable functors, 13–15, 49, 49–52, 81–

83, 122, 117–124 compositions of, 82–83 of compositions, 50–51

deformable systems left, 153 locally left, 153 locally right, 153 right, 153

deformation retracts left, 22, 122 right, 22, 122

deformations left, 22, 122 right, 22, 122 degree function, 63 derived adjunctions, 34, 44, 51, 42–52, 119,

137, 136–141 compositions of, 139–140 conjugations between, 139 of deformable adjunctions of systems, 156 of homotopy colimit functors, 57 of homotopy colimit systems, 159 of homotopy limit functors, 57 of homotopy limit systems, 159 of homotopy u-colimit functors, 147 of homotopy u-limit functors, 147

derived functors total left, 126 total right, 126

derived natural isomorphisms, 43 derived natural transformations, 137, 137–

140 diagram categories, 22, 94

homotopical, 94 diagrams, 94

restricted, 67, 67–70 virtually cofibrant, 67, 66–70 virtually fibrant, 67, 66–70

dinatural transformations, 164, 164–166 and adjunctions of systems, 165 and counits of adjunctions, 165 and units of adjunctions, 165

embedding CAT in CAT_{w}, 96 enrichment

Grothendieck, 102 existence of

left approximation of left Quillen functor,

40 right approximations of right Quillen func-

tor, 40

F -deformation left, 153 local left, 153 local right, 153 right, 153

f-deformation left, 122 right, 122

f-deformation retracts left, 122 right, 122

F -presentations, 167 f-presentations, 161 factorization axiom, 24 fibrant objects, 27 fibrations, 23

characterization of, 25–26 Reedy, 63 trivial, 24

forgetful functors CAT_{w} → CAT, 95 Fun, 22, 95 Fun_{w}, 22, 95 functor categories, 22, 94

homotopical, 94 functors

colimit, 55, 55–56 composition, 41, 128, 161 constant diagram, 55, 146 D-colimit, 55, 146 D-limit, 55, 146 deformable, 13–15, 49, 49–52, 81–83, 122,

117–124 homotopical, 21, 21–23, 77–78, 94 homotopical categories of, 94 homotopical colimit, 83–85, 162–163 homotopical limit, 162–163 homotopical u-colimit, 162 homotopical u-limit, 162 homotopy colimit, 8–11, 15–16, 53–54, 57,

57–60, 70, 83–85, 146–150 homotopy D-colimit, 57, 146 homotopy D-limit, 57, 146 homotopy limit, 15–16, 57, 57–60, 70, 146–

150 homotopy u-colimit, 57, 70, 146 homotopy u-limit, 57, 70, 146 induced diagram, 55, 146 initial projection, 66, 66–70 left deformable, 13–15, 49, 49–52, 122,

117–124 left deformable pairs, 50, 129 left Quillen, 33, 38, 38–52 limit, 55, 55–56 localization, 22, 27 locally left deformable pairs, 50, 129 locally right deformable pairs, 50, 129 naturally weakly equivalent, 22, 94 projection, 66, 66–70 Quillen, 33, 38, 38–52 right deformable, 13–15, 49, 49–52, 122,

117–124 right deformable pairs, 50, 129 right Quillen, 33, 38, 38–52 terminal projection, 66, 66–70 total left derived, 126 total right derived, 126 u-colimit, 55, 146 u-limit, 55, 146

γ, 22, 97 Gr, 101, 101–105 Grothendieck construction, 101, 101–105

and simplicial localizations, 103 colimit description of, 103

Grothendieck description of homotopy cate-

gories, 102 Grothendieck enrichment, 102

h-deformation left, 124 right, 124

h-deformation retracts left, 124 right, 124

higher universes, 92 Ho, 22, 96, 96–102 homotopic maps, 28 homotopical cat-systems, 60 homotopical categories, 11–12, 21, 18–23,

75, 77–78, 88–89, 93, 93–99 homotopical equivalences of, 22, 94 homotopy categories of, 12–13, 22, 22–23,

78–80, 89–90, 96, 96–105 locally small, 94 maximal, 95 minimal, 95 of functors, 94 of homotopical functors, 95 of homotopical functors over/under a func-

tor, 40 saturated, 23 small, 94 weak equivalences in, 21, 93

homotopical category

saturated, 97 homotopical cocompleteness, 54, 60–63, 71,

84–85, 157–159, 167 of model categories, 61 sufficient conditions for, 159, 167

homotopical colimit functors, 83–85, 162–

163 and homotopy colimit functors, 163 homotopical uniqueness of, 162 sufficient conditions for composability, 162 sufficient conditions for existence, 162

homotopical colimit systems, 167 and homotopy colimit systems, 167 homotopical uniqueness of, 167

homotopical compatibility

of left deformable left adjoints with homo-

topy colimit functors, 149–150 of right deformable right adjoints with ho-

motopy limit functors, 149–150 homotopical compatibility of

left Quillen functors with homotopy co-

limit functors, 54, 58 right Quillen functors with homotopy limit

functors, 58 homotopical completeness, 60–63, 71, 157–

159, 167 of model categories, 61 sufficient conditions for, 159, 167

homotopical diagram categories, 94 3-arrow calculi on, 106 and saturation, 23 saturation of, 97

homotopical equivalences of homotopical cat-

egories, 22, 94 homotopical functor categories, 94

3-arrow calculi on, 106 and saturation, 23 saturation of, 97

homotopical functors, 21, 21–23, 77–78, 94 homotopical categories of, 95 left approximations of, 126 right approximations of, 126

homotopical inverses, 22, 94 homotopical Kan extensions, 116, 160

along systems, 166 homotopical uniqueness of, 116, 161 presentations of, 161 sufficient conditions for composability, 162

homotopical Kan extensions along systems homotopical uniqueness of, 166

presentations of, 167 homotopical left systems, 151 homotopy systems of, 152 homotopical limit functors, 162–163 and homotopy limit functors, 163 homotopical uniqueness of, 162 sufficient conditions for composability, 162 sufficient conditions for existence, 162

homotopical limit systems, 167 and homotopy limit systems, 167 homotopical uniqueness of, 167 homotopical right systems, 151 homotopy systems of, 152 homotopical structures, 21, 94 homotopical subcategories, 94 homotopical u-colimit functors, 162 homotopical u-limit functors, 162 homotopical uniqueness, 9–10, 37, 37–38,

80–81, 90–91, 112, 110–116 of homotopical colimit functors, 162 of homotopical colimit systems, 167 of homotopical Kan extensions, 116, 161 of homotopical Kan extensions along sys-

tems, 166 of homotopical limit functors, 162 of homotopical limit systems, 167 of homotopically initial objects, 38, 113 of homotopically terminal objects, 38, 113 of homotopy colimit functors, 57 of homotopy colimit systems, 61, 158 of homotopy limit functors, 57 of homotopy limit systems, 61, 158 of homotopy u-colimit functors, 70, 147 of homotopy u-limit functors, 70, 147 of left approximations, 40, 49, 125, 152 of left f-deformations, 124 of right approximations, 40, 49, 125, 152 of right f-deformations, 124

homotopical version, 96

homotopically cocomplete categories, 71, 158,

167 homotopically complete categories, 71, 158,

167 homotopically contractible categories, 37, 112 homotopically full subcategories, 37, 112 homotopically initial

homotopically universal properties, 38, 113 homotopically initial Kan extensions, 116,

160 along left systems, 166 and right approximations of initial Kan

extensions, 161 sufficient conditions for existence, 161 homotopically initial Kan extensions along

systems and right approximations of initial Kan

extensions along systems, 166 sufficient conditions for existence, 166 homotopically initial objects, 13, 37, 37–38,

114, 113–116 homotopical uniqueness of, 38, 113 motivation, 113

homotopically terminal

homotopically universal properties, 38, 113 homotopically terminal Kan extensions, 116,

160 along right systems, 166 and left approximations of terminal Kan

extensions, 161 sufficient conditions for existence, 161

homotopically terminal Kan extensions along

systems and left approximations of terminal Kan

extensions along systems, 166 sufficient conditions for existence, 166

homotopically terminal objects, 13, 37, 37–

38, 114, 113–116 homotopical uniqueness of, 38, 113 motivation, 113

homotopically unique objects, 37, 112 homotopically universal properties, 38

homotopically initial, 38, 113 homotopically terminal, 38, 113

homotopically universal property, 113 homotopy categories, 5–8

3-arrow description of, 107 classical, 29 colimit description of, 101 description of, 96–110 explicit description of, 133–136 Grothendieck description of, 102 of homotopical categories, 12–13, 22, 22–

23, 78–80, 89–90, 96, 96–105 of model categories, 27, 27–30 3-arrow description of, 29

homotopy colimit functors, 8–11, 15–16, 53–

54, 57, 57–60, 70, 83–85, 146–150 and homotopical colimit functors, 163 compositions of, 148 derived adjunctions of, 57 homotopical compatibility with left deformable

left adjoints, 149–150 homotopical compatibility with left Quillen

functors, 54, 58 homotopical uniqueness, 57 sufficient conditions for composability, 148

homotopy colimit systems, 61, 61–63, 71,

158 and homotopical colimit systems, 167 derived adjunctions of, 159 homotopical uniqueness of, 61, 158

homotopy D-colimit functors, 57, 146 homotopy D-limit functors, 57, 146 homotopy equivalence, 28 homotopy inverses, 28 homotopy limit functors, 15–16, 57, 57–60,

70, 146–150 and homotopical limit functors, 163 compositions of, 148 derived adjunctions of, 57 homotopical compatibility with right de-

formable right adjoints, 149–150 homotopical compatibility with right Quillen

functors, 58 homotopical uniqueness of, 57 sufficient conditions for composability, 148

homotopy limit systems, 61, 61–63, 71, 158 and homotopical limit systems, 167 derived adjunctions of, 159 homotopical uniqueness of, 61, 158

homotopy relations, 28–30 homotopy systems

of homotopical left systems, 152 of homotopical right systems, 152

homotopy u-colimit functors, 57, 70, 146 derived adjunctions of, 147 homotopical uniqueness of, 70, 147 sufficient conditions for existence, 147

homotopy u-limit functors, 57, 70, 146 derived adjunctions of, 147 homotopical uniqueness of, 70, 147 sufficient conditions for existence, 147

indexing categories, 94 induced diagram functors, 55, 146 initial Kan extensions, 116, 126, 160

along left systems, 166 homotopically, 160 units of, 116, 160

initial Kan extensions along left systems

sufficient conditions for existence, 166 initial objects

categorical uniqueness of, 36, 111 homotopically, 13, 37, 37–38, 114, 113–

116 initial projection functors, 66, 66–70 initial universal properties, 36, 111 inverses

homotopical, 22, 94 homotopy, 28

invertibility property weak, 21, 94

Kan extensions, 116, 126, 160 along systems, 166 homotopical, 116, 160 homotopically initial, 116, 160 homotopically terminal, 116, 160 initial, 116, 160 terminal, 116, 160

Kan extensions along systems and adjunctions of systems, 166 categorical uniqueness of, 166

Ken Brown’s lemma, 39 latching objects, 64 left adjoints

compatibility with colimit functors, 149 left adjoints of left systems, 155

categorical uniqueness of, 155 left approximations, 33–34, 40–42, 49, 118–

120, 125, 124–128 and total left derived functors, 126 compositions of, 118–119, 128–131 homotopical uniqueness of, 40, 49, 125,

152 of homotopical functors, 126 of left deformable functors, 49, 125 of left Quillen functors, 40 of left systems, 152 of natural transformations, 127, 127–128 sufficient conditions for composability, 50,

130 left approximations of terminal Kan exten-

sions and homotopically terminal Kan exten-

sions, 161 left approximations of terminal Kan exten-

sions along systems and homotopically terminal Kan exten-

sions along systems, 166 left cat-systems, 60, 60–63, 71

composers of, 60 homotopical, 60 maps between, 60 weak equivalences between, 60

left deformable functors, 13–15, 49, 49–52,

122, 117–124 compositions of, 50–51, 129–131 left approximations of, 49, 125

left deformable left adjoints

homotopical compatibility with homotopy

colimit functors, 149–150 left deformable natural transformations, 124 left deformable pairs of functors, 50, 129 left deformable systems, 153 left deformation retracts, 22, 122

and model categories, 28 left deformations, 22, 122 left F -deformation, 153 left f-deformation, 122 left f-deformation retracts, 122

maximal, 123 left f-deformations

homotopical uniqueness of, 124 left h-deformation, 124 left h-deformation retracts, 124 left homotopic maps, 28 left lifting property, 24 left Quillen equivalences, 34–35, 47, 46–48 left Quillen functors, 33, 38, 38–52

compositions of left approximations, 42 existence of left approximations, 40 homotopical compatibility with homotopy

colimit functors, 54, 58 left approximations of, 40

left systems, 150 homotopical, 151 homotopically initial Kan extensions along,

166 initial Kan extensions along, 166 left adjoints of, 155 left approximations of, 152 maps between, 151 saturated, 151 sufficient conditions for existence of left

approximations, 153 sufficient conditions for left deformability,

155 weak equivalences between, 151

length

of a zigzag, 96 lifting axiom, 24 lim^{D}, 146 lim^{u}, 146 limit axiom, 23 limit functors, 55, 55–56

compatibility with right adjoints, 149 D-, 55, 146 homotopical, 162–163 homotopy, 15–16, 57, 57–60, 70, 146–150 u-, 55, 146

limit systems, 56, 158 and completeness, 56, 158 categorical uniqueness of, 158 homotopical, 167 homotopy, 61, 61–63, 71, 158

local left F -deformation, 153 local right F -deformation, 153 localization

simplicial, 103 localization functor, 97 localization functors, 22, 27 locally left deformable pairs of functors, 50,

129 locally left deformable systems, 153 locally right deformable pairs of functors,

50, 129 locally right deformable systems, 153 locally small categories, 21, 92 locally small homotopical categories, 94

maps between cat-systems, 60 between left systems, 151 between right systems, 151 homotopic, 28 left homotopic, 28 right homotopic, 28

matching objects, 64 maximal homotopical categories, 95 maximal model structures, 26 maximal structure functors, 95 minimal homotopical categories, 95 minimal model structures, 26 minimal structure functor, 95 model categories, 3–4, 17–18, 23, 23–27

and left deformation retracts, 28 and right deformation retracts, 28 closed, 24–26 closure properties, 25–26 homotopical cocompleteness of, 61 homotopical completeness of, 61 homotopy categories of, 27, 27–30 Ken Brown’s lemma, 39 Reedy, 63–70 saturation of, 29 3-arrow calculi of, 32 weak equivalences, 23 weak equivalences in, 18

model structures, 23 maximal, 26 minimal, 26 Reedy, 63, 63–66

n-arrow categories, 99 natural transformations

combined, 164 conjugate, 121, 121, 164 conjugate deformable, 131 derived, 137, 137–140 di-, 164, 164–166 left approximations of, 127, 127–128 left deformable, 124 right approximations of, 127, 127–128 right deformable, 124

natural weak equivalences, 21, 94 naturally weakly equivalent functors, 22, 94 nerve, 102

objects canonically isomorphic, 36, 111 canonically weakly equivalent, 37, 112 categorically unique, 36, 111 cofibrant, 27 cofibrant fibrant, 28 fibrant, 27 homotopically initial, 13, 37, 37–38, 114,

113–116 homotopically terminal, 13, 37, 37–38, 114,

113–116 homotopically unique, 37, 112 latching, 64 matching, 64 weakly equivalent, 21, 94

pairs of functors left deformable, 50, 129 locally left deformable, 50, 129 locally right deformable, 50, 129 right deformable, 50, 129 sufficient conditions for left deformability,

50, 130 sufficient conditions for right deformabil-

ity, 50, 130 partial adjunction functors, 131 partial adjunction isomorphisms, 42, 51, 132,

138–141 partial adjunctions, 42, 51, 131, 131–136

compositions of, 132 naturality of, 132

presentation f-, 161 presentations F -, 167 of homotopical Kan extensions, 161 of homotopical Kan extensions along sys-

tems, 167 projection functors, 66, 66–70

initial, 66, 66–70 terminal, 66, 66–70

Quillen adjunctions, 38 and Reedy model structures, 64 compositions of derived adjunctions, 44 Quillen condition for, 47

Quillen condition, 119, 141, 141–143 consequences of, 142 equivalent statements, 142 for deformable adjunctions, 52 for Quillen adjunctions, 35, 47 Quillen equivalences, 34–35, 47, 46–48

left, 34–35, 47, 46–48 right, 34–35, 47, 46–48 Quillen functors, 33, 38, 38–52 approximations of, 40 compositions of approximations, 42 existence of approximations, 40 key property of, 39 left, 33, 38, 38–52 right, 33, 38, 38–52

Reedy categories, 63 Reedy cofibrations, 63 Reedy fibrations, 63 Reedy model categories, 63–70 Reedy model structures, 63, 63–66 and Quillen adjunctions, 64 explicit description of, 64 Reedy weak equivalences, 63 restricted diagrams, 67, 67–70 restricted zigzag, 96 restricted zigzags, 100 retract axiom, 17–18, 24 right adjoints

compatibility with limit functors, 149 right adjoints of right systems, 155 categorical uniqueness of, 155 right approximations, 33–34, 40–42, 49, 118–

120, 125, 124–128 and total right derived functors, 126 compositions of, 118–119, 128–131 homotopical uniqueness of, 40, 49, 125,

152 of homotopical functors, 126 of natural transformations, 127, 127–128 of right deformable functors, 49, 125 of right Quillen functors, 40 of right systems, 152 sufficient conditions for composability, 50,

130 right approximations of initial Kan exten-

sions and homotopically initial Kan extensions,

161 right approximations of initial Kan exten-

sions along systems and homotopically initial Kan extensions

along systems, 166 right cat-systems, 60, 60–63, 71

composers of, 60 homotopical, 60 maps between, 60 weak equivalences between, 60

right deformable functors, 13–15, 49, 49–52,

122, 117–124 compositions of, 50–51, 129–131 right approximations of, 49, 125

right deformable natural transformations, 124 right deformable pairs of functors, 50, 129 right deformable right adjoints

homotopical compatibility with homotopy

limit functors, 149–150 right deformable systems, 153 right deformation retracts, 22, 122

and model categories, 28 right deformations, 22, 122 right F -deformation, 153 right f-deformation, 122 right f-deformation retracts, 122

maximal, 123 right f-deformations

homotopical uniqueness of, 124 right h-deformation, 124 right h-deformation retracts, 124 right homotopic maps, 28 right lifting property, 24 right Quillen equivalences, 34–35, 47, 46–48 right Quillen functors, 33, 38, 38–52

compositions of right approximations, 42 existence of right approximations, 40 homotopical compatibility with homotopy

limit functors, 58 right approximations of, 40

right systems, 150 homotopical, 151 homotopically terminal Kan extensions along,

166 maps between, 151 right adjoints of, 155 right approximations of, 152 saturated, 151 sufficient conditions for existence of right

approximations, 153 sufficient conditions for right deformabil-

ity, 155 terminal Kan extensions along, 166 weak equivalences between, 151

saturated homotopical categories, 23, 97 saturated left systems, 151 saturated right systems, 151 saturation, 23, 97

and homotopical diagram categories, 23 and homotopical functor categories, 23 and 3-arrow calculi, 31, 108 of homotopical diagram categories, 97 of homotopical functor categories, 97 of model categories, 29

sets, 21, 93 simplicial, 102 small, 21, 93 U-, 20, 91

simplicial localizations, 103

and Grothendieck construction, 103 simplicial sets, 102 small categories, 21, 92

locally, 21 small homotopical categories, 94 small sets, 21, 93 small U-categories, 20, 92, 92–93 structures

homotopical, 21, 94 model, 23 Reedy model, 63, 63–66

subcategories categorically full, 36, 111 homotopical, 94 homotopically full, 37, 112

successor universes, 20, 92 sufficient conditions for

homotopical cocompleteness, 159, 167 homotopical completeness, 159, 167 left deformability of left systems, 155 left deformability of pairs of functors, 50 right deformability of pairs of functors, 50 right deformability of right systems, 155

sufficient conditions for composability of homotopical colimit functors, 162 homotopical Kan extensions, 162 homotopical limit functors, 162 homotopy colimit functors, 148 homotopy limit functors, 148 left approximations, 50, 130

right approximations, 50, 130 sufficient conditions for existence of

homotopical colimit functors, 162 homotopical limit functors, 162 homotopically initial Kan extensions, 161 homotopically initial Kan extensions along

systems, 166 homotopically terminal Kan extensions, 161 homotopically terminal Kan extensions along

systems, 166 homotopy u-colimit functors, 147 homotopy u-limit functors, 147 initial Kan extensions along left systems,

166 left approximations of left systems, 153 right approximations of right systems, 153 terminal Kan extensions along right sys-

tems, 166 systems, 150

adjunctions of, 155 colimit, 56, 158 homotopical colimit, 167 homotopical Kan extensions along, 166 homotopical limit, 167 homotopy colimit, 61, 61–63, 71, 158 homotopy limit, 61, 61–63, 71, 158 Kan extensions along, 166 left, 150 left cat-, 60, 60–63, 71 left deformable, 153 limit, 56, 158 locally left deformable, 153 locally right deformable, 153 right, 150 right cat-, 60, 60–63, 71 right deformable, 153

T, 99 T-diagrams of arrow categories, 100 terminal Kan extensions, 116, 126, 160

along right systems, 166 counits of, 116, 160 homotopically, 160

terminal Kan extensions along right systems

sufficient conditions for existence, 166 terminal objects

categorical uniqueness of, 36, 111 homotopically, 13, 37, 37–38, 114, 113–

116 terminal projection functors, 66, 66–70 terminal universal properties, 36, 111 3-arrow calculi, 30, 106, 105–110

and saturation, 31, 108 and 3-arrow description of homotopy cat-

egories, 31 of model categories, 32 on homotopical diagram categories, 106 on homotopical functor categories, 106 3-arrow description of homotopy categories,

29, 107 and 3-arrow calculi, 31 total left derived functors, 126 and left approximations, 126 total right derived functors, 126 and right approximations, 126

trivial cofibrations, 24

characterization of, 25–26 trivial fibrations, 24

characterization of, 25–26 two out of six axiom, 23 two out of six property, 10–11, 21, 93, 108,

115, 123 two out of three axiom, 17–18 two out of three property, 21, 94 types

category of, 99 types of zigzags, 99

U-categories, 20, 91, 92, 92–93

small, 20, 92, 92–93 u-colimit functors, 55, 146 homotopical, 162 homotopy, 57, 70, 146 u-limit functors, 55, 146 homotopical, 162 homotopy, 57, 70, 146

U-sets, 20, 91

underlying categories, 94

uniqueness

categorical, 36, 36–37, 90–91, 111, 111–

112 homotopical, 9–10, 37, 37–38, 80–81, 90–

91, 112, 110–116 units of adjunctions, 120, 120–121, 133–136,

155 and dinatural transformations, 165 units of initial Kan extensions, 116, 160 universal properties, 36, 111 homotopically, 38, 113 initial, 36, 111 terminal, 36, 111

universes, 20, 20–21, 76–77, 87–88, 91, 91–

93 basic assumption, 91 higher, 92 successor, 20, 92

virtually cofibrant diagrams, 67, 66–70 virtually fibrant diagrams, 67, 66–70

we-categories, 4–5 weak equivalences

between cat-systems, 60 between left systems, 151 between right systems, 151 categories of, 94 in homotopical categories, 21, 93 in model categories, 18, 23 natural, 21, 94 Reedy, 63

weak invertibility property, 21, 94 weakly equivalent objects, 21, 94

canonically, 37, 112

zigzag, 96 length, 96 restricted, 96

zigzags, 99–110 restricted, 100 type of, 99

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