DOCSLIB.ORG

Homotopy Colimits - Comparison Lemmas for Combinatorial Applications

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Homotopy Colimits - Comparison Lemmas for Combinatorial Applications J. reine angew. Math. 50 9 (1999), 117-149 Journal für die reine und angewandte Mathematik © W a l t e r d e G r u y t e r Berlin · New York 1999 Homotopy colimits - comparison lemmas for combinatorial applications B y Volkmar Welker1) and Günter M. Ziegler2) at Berlin, and Rade T. Ž i v a l j e v i ć 3 ) at Belgrade A b s t r a c t .W e provide a “toolkit” of basic lemmas f o r the comparison o f homotopy types o f homotopy colimits o f diagrams of spaces over small categories. W e show how this toolkit can b e u s e d i n quite d i f f e r e n t fields o f applications. W e demonstrate this with respect to 1 . Björner’s “Generalized Homotopy Complementation Formula” [5], 2. the topology of toric varieties, 3. the study o f homotopy types o f arrangements o f subspaces, 4. the analysis o f homotopy types o f subgroup complexes. 1. Introduction The aim of this paper i s to advertise homotopy colimit considerations f o r topological investigations i n combinatorics. For this, w e provide a toolkit and demonstrate its u s e f u l n e s s b y a number of applications. A diagram of spaces is a functor from a small category to the category of topological spaces. In various topological, geometric, algebraic and combinatorial situations one has to deal with structures that can profitably be interpreted as (co)limits or homotopy (co)limits of diagrams over small categories, specially over (finite) posets. In fact, if a space is written as a finite union of (simpler) pieces, then it is the colimit of a corresponding diagram of spaces. While (co)limits do not have good functorial properties in homotopy theory, they can usually be replaced by homotopy (co)limits (Puppe [32] may have been the first to exploit this). Homotopy (co)limits have much better functorial properties. Thus there is 1) Supported by Deutsche Forschungsgemeinschaft (DFG). 2) Supported by a “Gerhard-Hess-Förderpreis” of the DFG, which also provided support for the authors’ joint work at the Konrad-Zuse-Zentrum für Informationstechnik Berlin (ZIB) in 1994. 3) Partially supported by the Serbian Science Foundation, Grant 04 M03. Bereitgestellt von | Technische Universität Berlin Angemeldet Heruntergeladen am | 09.11.18 17:08 118 Welker, Ziegler and Živaljević, Homotopy colimits a wide variety of techniques to manipulate diagrams of spaces in such a way that the homotopy (co)limit is preserved (up to homotopy type). Basic work on homotopy (co)limits has been done b y S e g a l [37], B o u s f i e l d and Kan [7], tom Dieck [42], V o g t [44], and Dwyer and Kan [11], [12]. S e e Hollender and V o g t [21] f o r a recent survey. T w o k e y results i n this setting are the “Projection Lemma” [3], [7], XII.3.1(iv), [15], [37], [49] which sometimes allows one to replace colimits b y homotopy colimits, and the “Homotopy Lemma” [42], [7], XII. 4.2, [44] which compares the homotopy types o f diagrams over the same small category. These tools have found striking applications, f o r example, i n the study o f subspace arrangements [49], [41], [36]. The choice o f contents f o r our “toolkit f o r the manipulation o f homotopy colimits” i s partially motivated b y the usefulness o f corresponding lemmas i n the special case o f order complexes (the discrete case, when the spaces o f the diagram are points). In this case, there i s a solid amount o f theory available, which has proved to be extremely powerful and u s e f u l i n quite diverse situations. The k e y result i s the Quillen Fiber Theorem (Quillen's “Theorem A” [33], [34], s e e below). A l l other basic tools o f the “homotopy theory o f posets”, such as the crosscut theorem, order homotopy theorem, complementation f o r ¬ mulas, etc., can be derived f r o m it. W e r e f e r to Björner [4] f o r an excellent account o f the theory [4], Sect. 1 0 , f o r an extensive survey o f applications [4], Part I, and f o r further references. Homotopy limits have not found immediate applications i n combinatorics and discrete geometry so far, and this i s the reason w h y w e restrict our attention to the case o f homotopy colimits. Also note that many results about homotopy limits can be derived f r o m the case o f homotopy colimits b y standard duality procedures ( s e e B o u s f i e l d and Kan [7], XII. 4.1, and Hollender and V o g t [21], Sect. 3). W e provide several applications o f our methods to various areas within mathematics. As a first application, i n the field o f topological combinatorics, w e present a n e w proof o f a result b y Björner on the homotopy type o f complexes [5], which generalizes the Homotopy Complementation Formula o f Björner and Walker [6], a tool which has proved to be v e r y powerful i n combinatorics. S i n c e this proof a f f o r d s the application o f many o f the techniques provided i n this paper, w e g i v e a detailed exposition of it here. Then w e present a n e w v i e w o f toric varieties. Namely, w e start with the observation that toric varieties are homeomorphic to homotopy colimits over the f a c e poset o f the f a n d e f i n i n g the variety. This immediately leads to a spectral sequence to compute the homology o f toric varieties isomorphic to one already employed b y Danilov [9]. We derive a new “combinatorial formula” for the homotopy types of quite general arrangements (such as “Grassmannian” arrangements) that are associated to linear sub- space arrangements by suitable functorial constructions. More briefly we cover two appli¬ cations for which details are contained in other papers: We describe a new result on the homotopy type of the order complex of the poset Sp(G) of non-trivial p-subgroups of a finite group G [31], and we review results obtained by homotopy limit methods on the topology of subspace arrangements in [49], and provide the equivalence with the results of Vassiliev [43]. Bereitgestellt von | Technische Universität Berlin Angemeldet Heruntergeladen am | 09.11.18 17:08 Welker, Ziegler and Živaljević, Homotopy colimits 119 2. F u n d a m e n t a l c o n c e p t s a n d c o n s t r u c t i o n s 2.1. Basic definitions and motivating examples of diagrams. In the following, all cate¬ gories are small, so their objects and morphisms form sets. Any partially ordered set can be considered as a category “with morphisms pointing down”, that is, for $ there is a (unique) arrow x → y if and only if $. A diagram of spaces over a small category A is a covariant functor $ into the category Top of topological spaces. We denote for an object $ the image under $ by Fa or by F (a), and for a morphism $ the image $ or b y F(g). I f there i s a unique morphism g : a → b i n A between a and b, then w e write fab for $. A morphism $ of diagrams $ and $ is a functor F:A → B together with a natural transformation α from $ to $. Given a diagram $, the homotopy colimit hocolim $ i s a space associated to $ b y a homotopy mixing construction, s e e Section 2.2. B e f o r e w e proceed with a reasonably detailed outline o f the theory, w e introduce several motivating examples o f diagrams o f spaces. Constant diagram. For a topological space X the constant diagram $ is defined by sending each object of A to the space X and each morphism to the identity id : X → X . Of particular interest is the case when $ is the one-point space. In this case the constant diagram $ leads, via homotopy colimits, to the construction of the classifying space BA of the category A. In the special case when A = P is a partially ordered set—a po s e t for short—the classifying space BP will be seen to coincide with the order complex Δ (P) of P . Group diagram. Given a discrete group G, let A G be th e ca t e g o r y wh i c h co n s i s t s of one single object and a morphism for each element $. Then the classifying space of this category is the K(G,1)-space $ of the group G. Its universal cover EG is constructed in a similar way from the category whose objects are the elements of the group. Here for each pair of elements $ the unique morphism from g to h is given by h g -1. Finally, any AG-diagram $ can be interpreted as a G-space X and it turns out that $.
Load more

Recommended publications
  • Op → Sset in Simplicial Sets, Or Equivalently a Functor X : ∆Op × ∆Op → Set
    Contents 21 Bisimplicial sets 1 22 Homotopy colimits and limits (revisited) 10 23 Applications, Quillen’s Theorem B 23 21 Bisimplicial sets A bisimplicial set X is a simplicial object X : Dop ! sSet in simplicial sets, or equivalently a functor X : Dop × Dop ! Set: I write Xm;n = X(m;n) for the set of bisimplices in bidgree (m;n) and Xm = Xm;∗ for the vertical simplicial set in horiz. degree m. Morphisms X ! Y of bisimplicial sets are natural transformations. s2Set is the category of bisimplicial sets. Examples: 1) Dp;q is the contravariant representable functor Dp;q = hom( ;(p;q)) 1 on D × D. p;q G q Dm = D : m!p p;q The maps D ! X classify bisimplices in Xp;q. The bisimplex category (D × D)=X has the bisim- plices of X as objects, with morphisms the inci- dence relations Dp;q ' 7 X Dr;s 2) Suppose K and L are simplicial sets. The bisimplicial set Kט L has bisimplices (Kט L)p;q = Kp × Lq: The object Kט L is the external product of K and L. There is a natural isomorphism Dp;q =∼ Dpט Dq: 3) Suppose I is a small category and that X : I ! sSet is an I-diagram in simplicial sets. Recall (Lecture 04) that there is a bisimplicial set −−−!holim IX (“the” homotopy colimit) with vertical sim- 2 plicial sets G X(i0) i0→···→in in horizontal degrees n. The transformation X ! ∗ induces a bisimplicial set map G G p : X(i0) ! ∗ = BIn; i0→···→in i0→···→in where the set BIn has been identified with the dis- crete simplicial set K(BIn;0) in each horizontal de- gree.
    [Show full text]
  • Appendices Appendix HL: Homotopy Colimits and Fibrations We Give Here
    Appendices Appendix HL: Homotopy colimits and fibrations We give here a briefreview of homotopy colimits and, more importantly, we list some of their crucial properties relating them to fibrations. These properties came to light after the appearance of [B-K]. The most important additional properties relate to the interaction between fibration and homotopy colimit and were first stated clearly by V. Puppe [Pu] in a paper dealing with Segal's characterization of loop spaces. In fact, much of what we need follows formally from the basic results of V. Puppe by induction on skeletons. Another issue we survey is the definition of homotopy colimits via free resolutions. The basic definition of homotopy colimits is given in [B-K] and [S], where the property of homotopy invariance and their associated mapping property is given; here however we mostly use a different approach. This different approach from a 'homotopical algebra' point of view is given in [DF-1] where an 'invariant' definition is given. We briefly recall that second (equivalent) definition via free resolution below. Compare also the brief review in the first three sections of [Dw-1]. Recently, two extensive expositions combining and relating these two approaches appeared: a shorter one by Chacholsky and a more detailed one by Hirschhorn. Basic property, examples Homotopy colimits in general are functors that assign a space to a strictly commutative diagram of spaces. One starts with an arbitrary diagram, namely a functor I ---+ (Spaces) where I is a small category that might be enriched over simplicial sets or topological spaces, namely in a simplicial category the morphism sets are equipped with the structure of a simplicial set and composition respects this structure.
    [Show full text]
  • Homotopy Colimits in Model Categories
    Homotopy colimits in model categories Marc Stephan July 13, 2009 1 Introduction In [1], Dwyer and Spalinski construct the so-called homotopy pushout func- tor, motivated by the following observation. In the category Top of topo- logical spaces, one can construct the n-dimensional sphere Sn by glueing together two n-disks Dn along their boundaries Sn−1, i.e. by the pushout of i i Dn o Sn−1 / Dn ; where i denotes the inclusion. Let ∗ be the one point space. Observe, that one has a commutative diagram i i Dn o Sn−1 / Dn ; idSn−1 ∗ o Sn−1 / ∗ where all vertical maps are homotopy equivalences, but the pushout of the bottom row is the one-point space ∗ and therefore not homotopy equivalent to Sn. One probably prefers the homotopy type of Sn. Having this idea of calculating the prefered homotopy type in mind, they equip the functor category CD, where C is a model category and D = a b ! c the category consisting out of three objects a, b, c and two non-identity morphisms as indicated, with a suitable model category structure. This enables them to construct out of the pushout functor colim : CD ! C its so-called total left derived functor Lcolim : Ho(CD) ! Ho(C) between the corresponding homotopy categories, which defines the homotopy pushout functor. Dwyer and Spalinski further indicate how to generalize this construction to define the so-called homotopy colimit functor as the total left derived functor for the colimit functor colim : CD ! C in the case, where D is a so-called very small category.
    [Show full text]
  • Exercises on Homotopy Colimits
    EXERCISES ON HOMOTOPY COLIMITS SAMUEL BARUCH ISAACSON 1. Categorical homotopy theory Let Cat denote the category of small categories and S the category Fun(∆op, Set) of simplicial sets. Let’s recall some useful notation. We write [n] for the poset [n] = {0 < 1 < . < n}. We may regard [n] as a category with objects 0, 1, . , n and a unique morphism a → b iff a ≤ b. The category ∆ of ordered nonempty finite sets can then be realized as the full subcategory of Cat with objects [n], n ≥ 0. The nerve of a category C is the simplicial set N C with n-simplices the functors [n] → C . Write ∆[n] for the representable functor ∆(−, [n]). Since ∆[0] is the terminal simplicial set, we’ll sometimes write it as ∗. Exercise 1.1. Show that the nerve functor N : Cat → S is fully faithful. Exercise 1.2. Show that the natural map N(C × D) → N C × N D is an isomorphism. (Here, × denotes the categorical product in Cat and S, respectively.) Exercise 1.3. Suppose C and D are small categories. (1) Show that a natural transformation H between functors F, G : C → D is the same as a functor H filling in the diagram C NNN NNN F id ×d1 NNN NNN H NN C × [1] '/ D O pp7 ppp id ×d0 ppp ppp G ppp C (2) Suppose that F and G are functors C → D and that H : F → G is a natural transformation. Show that N F and N G induce homotopic maps N C → N D. Exercise 1.4.
    [Show full text]
  • Derivators: Doing Homotopy Theory with 2-Category Theory
    DERIVATORS: DOING HOMOTOPY THEORY WITH 2-CATEGORY THEORY MIKE SHULMAN (Notes for talks given at the Workshop on Applications of Category Theory held at Macquarie University, Sydney, from 2{5 July 2013.) Overview: 1. From 2-category theory to derivators. The goal here is to motivate the defini- tion of derivators starting from 2-category theory and homotopy theory. Some homotopy theory will have to be swept under the rug in terms of constructing examples; the goal is for the definition to seem natural, or at least not unnatural. 2. The calculus of homotopy Kan extensions. The basic tools we use to work with limits and colimits in derivators. I'm hoping to get through this by the end of the morning, but we'll see. 3. Applications: why homotopy limits can be better than ordinary ones. Stable derivators and descent. References: • http://ncatlab.org/nlab/show/derivator | has lots of links, including to the original work of Grothendieck, Heller, and Franke. • http://arxiv.org/abs/1112.3840 (Moritz Groth) and http://arxiv.org/ abs/1306.2072 (Moritz Groth, Kate Ponto, and Mike Shulman) | these more or less match the approach I will take. 1. Homotopy theory and homotopy categories One of the characteristics of homotopy theory is that we are interested in cate- gories where we consider objects to be \the same" even if they are not isomorphic. Usually this notion of sameness is generated by some non-isomorphisms that exhibit their domain and codomain as \the same". For example: (i) Topological spaces and homotopy equivalences (ii) Topological spaces and weak homotopy equivalence (iii) Chain complexes and chain homotopy equivalences (iv) Chain complexes and quasi-isomorphisms (v) Categories and equivalence functors Generally, we call morphisms like this weak equivalences.
    [Show full text]
  • Double Homotopy (Co) Limits for Relative Categories
    Double Homotopy (Co)Limits for Relative Categories Kay Werndli Abstract. We answer the question to what extent homotopy (co)limits in categories with weak equivalences allow for a Fubini-type interchange law. The main obstacle is that we do not assume our categories with weak equivalences to come equipped with a calculus for homotopy (co)limits, such as a derivator. 0. Introduction In modern, categorical homotopy theory, there are a few different frameworks that formalise what exactly a “homotopy theory” should be. Maybe the two most widely used ones nowadays are Quillen’s model categories (see e.g. [8] or [13]) and (∞, 1)-categories. The latter again comes in different guises, the most popular one being quasicategories (see e.g. [16] or [17]), originally due to Boardmann and Vogt [3]. Both of these contexts provide enough structure to perform homotopy invariant analogues of the usual categorical limit and colimit constructions and more generally Kan extensions. An even more stripped-down notion of a “homotopy theory” (that arguably lies at the heart of model categories) is given by just working with categories that come equipped with a class of weak equivalences. These are sometimes called relative categories [2], though depending on the context, this might imply some restrictions on the class of weak equivalences. Even in this context, we can still say what a homotopy (co)limit is and do have tools to construct them, such as model approximations due to Chach´olski and Scherer [5] or more generally, left and right arXiv:1711.01995v1 [math.AT] 6 Nov 2017 deformation retracts as introduced in [9] and generalised in section 3 below.
    [Show full text]
  • Homotopical Algebra and Homotopy Colimits
    Homotopical algebra and homotopy colimits Birgit Richter New interactions between homotopical algebra and quantum field theory Oberwolfach, 19th of December 2016 Homotopical algebra { What for? Often, we are interested in (co)homology groups. 2 1 H (X ; Z) = [X ; CP ] = [X ; BU(1)] classifies line bundles on a space X . Many geometric invariants of a manifold M can be understood via its de Rham cohomology groups. In order to calculate or understand such (co)homology groups, we often have to perform constructions on the level of (co)chain complexes: quotients, direct sums,... For these constructions one needs models. Homotopical algebra: Study of homological/homotopical questions via model categories. Definition given by Quillen in 1967 [Q]. Flexible framework, can be used for chain complexes, topological spaces, algebras over operads, and many more { allows us to do homotopy theory. Chain complexes, I Let R be an associative ring and let ChR denote the category of non-negatively graded chain complexes of R-modules. The objects are families of R-modules Cn; n ≥ 0, together with R-linear maps, the differentials, d = dn : Cn ! Cn−1 for all n ≥ 1 such that dn−1 ◦ dn = 0 for all n. Morphisms are chain maps f∗ : C∗ ! D∗. These are families of R-linear maps fn : Cn ! Dn such that dn ◦ fn = fn−1 ◦ dn for all n. The nth homology group of a chain complex C∗ is Hn(C∗) = ker(dn : Cn ! Cn−1)=im(dn+1 : Cn+1 ! Cn): ker(dn : Cn ! Cn−1) are the n-cycles of C∗, ZnC∗, and im(dn+1 : Cn+1 ! Cn) are the n-boundaries of C∗, BnC∗.
    [Show full text]
  • Lecture Notes on Simplicial Homotopy Theory
    Lectures on Homotopy Theory The links below are to pdf files, which comprise my lecture notes for a first course on Homotopy Theory. I last gave this course at the University of Western Ontario during the Winter term of 2018. The course material is widely applicable, in fields including Topology, Geometry, Number Theory, Mathematical Pysics, and some forms of data analysis. This collection of files is the basic source material for the course, and this page is an outline of the course contents. In practice, some of this is elective - I usually don't get much beyond proving the Hurewicz Theorem in classroom lectures. Also, despite the titles, each of the files covers much more material than one can usually present in a single lecture. More detail on topics covered here can be found in the Goerss-Jardine book Simplicial Homotopy Theory, which appears in the References. It would be quite helpful for a student to have a background in basic Algebraic Topology and/or Homological Algebra prior to working through this course. J.F. Jardine Office: Middlesex College 118 Phone: 519-661-2111 x86512 E-mail: [email protected] Homotopy theories Lecture 01: Homological algebra Section 1: Chain complexes Section 2: Ordinary chain complexes Section 3: Closed model categories Lecture 02: Spaces Section 4: Spaces and homotopy groups Section 5: Serre fibrations and a model structure for spaces Lecture 03: Homotopical algebra Section 6: Example: Chain homotopy Section 7: Homotopical algebra Section 8: The homotopy category Lecture 04: Simplicial sets Section 9:
    [Show full text]
  • A Primer on Homotopy Colimits
    A PRIMER ON HOMOTOPY COLIMITS DANIEL DUGGER Contents 1. Introduction2 Part 1. Getting started 4 2. First examples4 3. Simplicial spaces9 4. Construction of homotopy colimits 16 5. Homotopy limits and some useful adjunctions 21 6. Changing the indexing category 25 7. A few examples 29 Part 2. A closer look 30 8. Brief review of model categories 31 9. The derived functor perspective 34 10. More on changing the indexing category 40 11. The two-sided bar construction 44 12. Function spaces and the two-sided cobar construction 49 Part 3. The homotopy theory of diagrams 52 13. Model structures on diagram categories 53 14. Cofibrant diagrams 60 15. Diagrams in the homotopy category 66 16. Homotopy coherent diagrams 69 Part 4. Other useful tools 76 17. Homology and cohomology of categories 77 18. Spectral sequences for holims and hocolims 85 19. Homotopy limits and colimits in other model categories 90 20. Various results concerning simplicial objects 94 Part 5. Examples 96 21. Homotopy initial and terminal functors 96 22. Homotopical decompositions of spaces 103 23. A survey of other applications 108 Appendix A. The simplicial cone construction 108 References 108 1 2 DANIEL DUGGER 1. Introduction This is an expository paper on homotopy colimits and homotopy limits. These are constructions which should arguably be in the toolkit of every modern algebraic topologist, yet there does not seem to be a place in the literature where a graduate student can easily read about them. Certainly there are many fine sources: [BK], [DwS], [H], [HV], [V1], [V2], [CS], [S], among others.
    [Show full text]
  • Arxiv:Math/0509655V2 [Math.CT] 26 May 2006 Nt Rdcs Aigacategory a Having Products
    ON HOMOTOPY VARIETIES J. ROSICKY´ ∗ Abstract. Given an algebraic theory T , a homotopy T -algebra is a simplicial set where all equations from T hold up to homotopy. All homotopy T -algebras form a homotopy variety. We will give a characterization of homotopy varieties analogous to the character- ization of varieties. 1. Introduction Algebraic theories were introduced by F. W. Lawvere (see [31] and also [32]) in order to provide a convenient approach to study algebras in general categories. An algebraic theory is a small category T with finite products. Having a category K with finite products, a T -algebra in K is a finite product preserving functor T →K. Algebras in the category Set of sets are usual many-sorted algebras. Algebras in the category SSet of simplicial sets are called simplicial algebras and they can be also viewed as simplicial objects in the category of algebras in Set. In homotopy theory, one often needs to consider algebras up to homotopy – a homotopy T -algebra is a functor A : T → SSet such that the canonical morphism A(X1 ×···× Xn) → A(X1) ×···× A(Xn) is a weak equivalence for each finite product X1 ×···× Xn in T . These homotopy algebras have been considered in recent papers [6], [7] and [9] but the subject is much older (see, e.g., [16], [8], [35] or [38]). arXiv:math/0509655v2 [math.CT] 26 May 2006 A category is called a variety if it is equivalent to the category Alg(T ) of all T -algebras in Set for some algebraic theory T . There is a char- acterization of varieties proved by F.
    [Show full text]
  • On Colimits in Various Categories of Manifolds
    ON COLIMITS IN VARIOUS CATEGORIES OF MANIFOLDS DAVID WHITE 1. Introduction Homotopy theorists have traditionally studied topological spaces and used methods which rely heavily on computation, construction, and induction. Proofs often go by constructing some horrendously complicated object (usually via a tower of increasingly complicated objects) and then proving inductively that we can understand what’s going on at each step and that in the limit these steps do what is required. For this reason, a homotopy theorist need the freedom to make any construction he wishes, e.g. products, subobjects, quotients, gluing towers of spaces together, dividing out by group actions, moving to fixed points of group actions, etc. Over the past 50 years tools have been developed which allow the study of many different categories. The key tool for this is the notion of a model category, which is the most general place one can do homotopy theory. Model categories were introduced by the Fields Medalist Dan Quillen, and the best known reference for model categories is the book Model Categories by Mark Hovey. Model categories are defined by a list of axioms which are all analogous to properties that the category Top of topological spaces satisfies. The most important property of a model category is that it has three distinguished classes of morphisms–called weak equivalences, cofibrations, and fibrations–which follow rules similar to the classes of maps with those names in the category Top (so the homotopy theory comes into play by inverting the weak equivalences to make them isomorphisms). The fact that homotopy theorists need to be able to do lots of constructions is contained in the axiom that M must be complete and cocomplete.
    [Show full text]
  • Arxiv:Math/0110316V1 [Math.AT] 30 Oct 2001 Ne ..Bx288,Nwhvn Onciu 62-23 U 06520-8283, Connecticut Haven, New 208283, Box P.O
    Homotopy theory of diagrams Wojciech Chach´olski J´erˆome Scherer Author address: Yale University, Department of Mathematics, 10 Hillhouse Av- enue, P.O. Box 208283, New Haven, Connecticut 06520-8283, USA E-mail address: [email protected] Universite´ de Lausanne, Institut de Mathematiques,´ CH-1015 Lau- sanne, Switzerland E-mail address: [email protected] arXiv:math/0110316v1 [math.AT] 30 Oct 2001 Contents Introduction 1 Chapter I. Model approximations and bounded diagrams 5 1. Notation 5 2. Model categories 6 3. Left derived functors 12 4. Left derived functors of colimits and left Kan extensions 14 5. Model approximations 16 6. Spaces and small categories 19 7. Thepull-backprocessandlocalproperties 22 8. Colimits of diagrams indexed by spaces 22 9. Left Kan extensions 24 10. Bounded diagrams 26 ChapterII. Homotopytheoryofdiagrams 29 11. Statements of the main results 29 12. Cofibrations 30 13. Funb(K, M) as a model category 32 14. Ocolimit of bounded diagrams 35 15. Bousfield-Kan approximation of Fun(I, C) 36 16. Homotopy colimits and homotopy left Kan extensions 37 17. Relative boundedness 38 18. Reduction process 40 19. Relative cofibrations 44 20. Cofibrations and colimits 46 b 21. Funf (L, M) as a model category 48 22. Cones 49 23. Diagrams indexed by cones I 51 ChapterIII. Propertiesofhomotopycolimits 54 24. Fubini theorem 54 25. Bounded diagrams indexed by Grothendieck constructions 57 26. Thomason’s theorem 59 27. Etale´ spaces 63 28. Diagrams indexed by cones II 67 29. Homotopy colimits as ´etale spaces 69 30. Cofinality 70 31. Homotopy limits 71 Appendix A.
    [Show full text]