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Sheaves and Homotopy Theory
SHEAVES AND HOMOTOPY THEORY DANIEL DUGGER The purpose of this note is to describe the homotopy-theoretic version of sheaf theory developed in the work of Thomason [14] and Jardine [7, 8, 9]; a few enhancements are provided here and there, but the bulk of the material should be credited to them. Their work is the foundation from which Morel and Voevodsky build their homotopy theory for schemes [12], and it is our hope that this exposition will be useful to those striving to understand that material. Our motivating examples will center on these applications to algebraic geometry. Some history: The machinery in question was invented by Thomason as the main tool in his proof of the Lichtenbaum-Quillen conjecture for Bott-periodic algebraic K-theory. He termed his constructions `hypercohomology spectra', and a detailed examination of their basic properties can be found in the first section of [14]. Jardine later showed how these ideas can be elegantly rephrased in terms of model categories (cf. [8], [9]). In this setting the hypercohomology construction is just a certain fibrant replacement functor. His papers convincingly demonstrate how many questions concerning algebraic K-theory or ´etale homotopy theory can be most naturally understood using the model category language. In this paper we set ourselves the specific task of developing some kind of homotopy theory for schemes. The hope is to demonstrate how Thomason's and Jardine's machinery can be built, step-by-step, so that it is precisely what is needed to solve the problems we encounter. The papers mentioned above all assume a familiarity with Grothendieck topologies and sheaf theory, and proceed to develop the homotopy-theoretic situation as a generalization of the classical case. -
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. -
Simplicial Sets, Nerves of Categories, Kan Complexes, Etc
SIMPLICIAL SETS, NERVES OF CATEGORIES, KAN COMPLEXES, ETC FOLING ZOU These notes are taken from Peter May's classes in REU 2018. Some notations may be changed to the note taker's preference and some detailed definitions may be skipped and can be found in other good notes such as [2] or [3]. The note taker is responsible for any mistakes. 1. simplicial approach to defining homology Defnition 1. A simplical set/group/object K is a sequence of sets/groups/objects Kn for each n ≥ 0 with face maps: di : Kn ! Kn−1; 0 ≤ i ≤ n and degeneracy maps: si : Kn ! Kn+1; 0 ≤ i ≤ n satisfying certain commutation equalities. Images of degeneracy maps are said to be degenerate. We can define a functor: ordered abstract simplicial complex ! sSet; K 7! Ks; where s Kn = fv0 ≤ · · · ≤ vnjfv0; ··· ; vng (may have repetition) is a simplex in Kg: s s Face maps: di : Kn ! Kn−1; 0 ≤ i ≤ n is by deleting vi; s s Degeneracy maps: si : Kn ! Kn+1; 0 ≤ i ≤ n is by repeating vi: In this way it is very straightforward to remember the equalities that face maps and degeneracy maps have to satisfy. The simplical viewpoint is helpful in establishing invariants and comparing different categories. For example, we are going to define the integral homology of a simplicial set, which will agree with the simplicial homology on a simplical complex, but have the virtue of avoiding the barycentric subdivision in showing functoriality and homotopy invariance of homology. This is an observation made by Samuel Eilenberg. To start, we construct functors: F C sSet sAb ChZ: The functor F is the free abelian group functor applied levelwise to a simplical set. -
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. -
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. -
Homotopy Coherent Structures
HOMOTOPY COHERENT STRUCTURES EMILY RIEHL Abstract. Naturally occurring diagrams in algebraic topology are commuta- tive up to homotopy, but not on the nose. It was quickly realized that very little can be done with this information. Homotopy coherent category theory arose out of a desire to catalog the higher homotopical information required to restore constructibility (or more precisely, functoriality) in such “up to homo- topy” settings. The first lecture will survey the classical theory of homotopy coherent diagrams of topological spaces. The second lecture will revisit the free resolutions used to define homotopy coherent diagrams and prove that they can also be understood as homotopy coherent realizations. This explains why diagrams valued in homotopy coherent nerves or more general 1-categories are automatically homotopy coherent. The final lecture will venture into ho- motopy coherent algebra, connecting the newly discovered notion of homotopy coherent adjunction to the classical cobar and bar resolutions for homotopy coherent algebras. Contents Part I. Homotopy coherent diagrams 2 I.1. Historical motivation 2 I.2. The shape of a homotopy coherent diagram 3 I.3. Homotopy coherent diagrams and homotopy coherent natural transformations 7 Part II. Homotopy coherent realization and the homotopy coherent nerve 10 II.1. Free resolutions are simplicial computads 11 II.2. Homotopy coherent realization and the homotopy coherent nerve 13 II.3. Further applications 17 Part III. Homotopy coherent algebra 17 III.1. From coherent homotopy theory to coherent category theory 17 III.2. Monads in category theory 19 III.3. Homotopy coherent monads 19 III.4. Homotopy coherent adjunctions 21 Date: July 12-14, 2017. -
Quasi-Categories Vs Simplicial Categories
Quasi-categories vs Simplicial categories Andr´eJoyal January 07 2007 Abstract We show that the coherent nerve functor from simplicial categories to simplicial sets is the right adjoint in a Quillen equivalence between the model category for simplicial categories and the model category for quasi-categories. Introduction A quasi-category is a simplicial set which satisfies a set of conditions introduced by Boardman and Vogt in their work on homotopy invariant algebraic structures [BV]. A quasi-category is often called a weak Kan complex in the literature. The category of simplicial sets S admits a Quillen model structure in which the cofibrations are the monomorphisms and the fibrant objects are the quasi- categories [J2]. We call it the model structure for quasi-categories. The resulting model category is Quillen equivalent to the model category for complete Segal spaces and also to the model category for Segal categories [JT2]. The goal of this paper is to show that it is also Quillen equivalent to the model category for simplicial categories via the coherent nerve functor of Cordier. We recall that a simplicial category is a category enriched over the category of simplicial sets S. To every simplicial category X we can associate a category X0 enriched over the homotopy category of simplicial sets Ho(S). A simplicial functor f : X → Y is called a Dwyer-Kan equivalence if the functor f 0 : X0 → Y 0 is an equivalence of Ho(S)-categories. It was proved by Bergner, that the category of (small) simplicial categories SCat admits a Quillen model structure in which the weak equivalences are the Dwyer-Kan equivalences [B1]. -
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. -
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. -
Comparison of Waldhausen Constructions
COMPARISON OF WALDHAUSEN CONSTRUCTIONS JULIA E. BERGNER, ANGELICA´ M. OSORNO, VIKTORIYA OZORNOVA, MARTINA ROVELLI, AND CLAUDIA I. SCHEIMBAUER Dedicated to the memory of Thomas Poguntke Abstract. In previous work, we develop a generalized Waldhausen S•- construction whose input is an augmented stable double Segal space and whose output is a 2-Segal space. Here, we prove that this construc- tion recovers the previously known S•-constructions for exact categories and for stable and exact (∞, 1)-categories, as well as the relative S•- construction for exact functors. Contents Introduction 2 1. Augmented stable double Segal objects 3 2. The S•-construction for exact categories 7 3. The S•-construction for stable (∞, 1)-categories 17 4. The S•-construction for proto-exact (∞, 1)-categories 23 5. The relative S•-construction 26 Appendix: Some results on quasi-categories 33 References 38 Date: May 29, 2020. 2010 Mathematics Subject Classification. 55U10, 55U35, 55U40, 18D05, 18G55, 18G30, arXiv:1901.03606v2 [math.AT] 28 May 2020 19D10. Key words and phrases. 2-Segal spaces, Waldhausen S•-construction, double Segal spaces, model categories. The first-named author was partially supported by NSF CAREER award DMS- 1659931. The second-named author was partially supported by a grant from the Simons Foundation (#359449, Ang´elica Osorno), the Woodrow Wilson Career Enhancement Fel- lowship and NSF grant DMS-1709302. The fourth-named and fifth-named authors were partially funded by the Swiss National Science Foundation, grants P2ELP2 172086 and P300P2 164652. The fifth-named author was partially funded by the Bergen Research Foundation and would like to thank the University of Oxford for its hospitality. -
Homotopical Categories: from Model Categories to ( ,)-Categories ∞
HOMOTOPICAL CATEGORIES: FROM MODEL CATEGORIES TO ( ;1)-CATEGORIES 1 EMILY RIEHL Abstract. This chapter, written for Stable categories and structured ring spectra, edited by Andrew J. Blumberg, Teena Gerhardt, and Michael A. Hill, surveys the history of homotopical categories, from Gabriel and Zisman’s categories of frac- tions to Quillen’s model categories, through Dwyer and Kan’s simplicial localiza- tions and culminating in ( ;1)-categories, first introduced through concrete mod- 1 els and later re-conceptualized in a model-independent framework. This reader is not presumed to have prior acquaintance with any of these concepts. Suggested exercises are included to fertilize intuitions and copious references point to exter- nal sources with more details. A running theme of homotopy limits and colimits is included to explain the kinds of problems homotopical categories are designed to solve as well as technical approaches to these problems. Contents 1. The history of homotopical categories 2 2. Categories of fractions and localization 5 2.1. The Gabriel–Zisman category of fractions 5 3. Model category presentations of homotopical categories 7 3.1. Model category structures via weak factorization systems 8 3.2. On functoriality of factorizations 12 3.3. The homotopy relation on arrows 13 3.4. The homotopy category of a model category 17 3.5. Quillen’s model structure on simplicial sets 19 4. Derived functors between model categories 20 4.1. Derived functors and equivalence of homotopy theories 21 4.2. Quillen functors 24 4.3. Derived composites and derived adjunctions 25 4.4. Monoidal and enriched model categories 27 4.5. -
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.