Homotopy Limits and Colimits in Nature a Motivation for Derivators Summer School on Derivators — Universit¨Atfreiburg — August 2014 Preliminary Version 21.10.14
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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. -
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. -
Giraud's Theorem
Giraud’s Theorem 25 February and 4 March 2019 The Giraud’s axioms allow us to determine when a category is a topos. Lurie’s version indicates the equivalence between the axioms, the left exact localizations, and Grothendieck topologies, i.e. Proposition 1. [2] Let X be a category. The following conditions are equivalent: 1. The category X is equivalent to the category of sheaves of sets of some Grothendieck site. 2. The category X is equivalent to a left exact localization of the category of presheaves of sets on some small category C. 3. Giraud’s axiom are satisfied: (a) The category X is presentable. (b) Colimits in X are universal. (c) Coproducts in X are disjoint. (d) Equivalence relations in X are effective. 1 Sheaf and Grothendieck topologies This section is based in [4]. The continuity of a reald-valued function on topological space can be determined locally. Let (X;t) be a topological space and U an open subset of X. If U is S covered by open subsets Ui, i.e. U = i2I Ui and fi : Ui ! R are continuos functions then there exist a continuos function f : U ! R if and only if the fi match on all the overlaps Ui \Uj and f jUi = fi. The previuos paragraph is so technical, let us see a more enjoyable example. Example 1. Let (X;t) be a topological space such that X = U1 [U2 then X can be seen as the following diagram: X s O e U1 tU1\U2 U2 o U1 O O U2 o U1 \U2 If we form the category OX (the objects are the open sets and the morphisms are given by the op inclusion), the previous paragraph defined a functor between OX and Set i.e it sends an open set U to the set of all continuos functions of domain U and codomain the real numbers, F : op OX ! Set where F(U) = f f j f : U ! Rg, so the initial diagram can be seen as the following: / (?) FX / FU1 ×F(U1\U2) FU2 / F(U1 \U2) then the condition of the paragraph states that the map e : FX ! FU1 ×F(U1\U2) FU2 given by f 7! f jUi is the equalizer of the previous diagram. -
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. -
Left Kan Extensions Preserving Finite Products
Left Kan Extensions Preserving Finite Products Panagis Karazeris, Department of Mathematics, University of Patras, Patras, Greece Grigoris Protsonis, Department of Mathematics, University of Patras, Patras, Greece 1 Introduction In many situations in mathematics we want to know whether the left Kan extension LanjF of a functor F : C!E, along a functor j : C!D, where C is a small category and E is locally small and cocomplete, preserves the limits of various types Φ of diagrams. The answer to this general question is reducible to whether the left Kan extension LanyF of F , along the Yoneda embedding y : C! [Cop; Set] preserves those limits (see [12] x3). Such questions arise in the classical context of comparing the homotopy of simplicial sets to that of spaces but are also vital in comparing, more generally, homotopical notions on various combinatorial models (e.g simplicial, bisimplicial, cubical, globular, etc, sets) on the one hand, and various \realizations" of them as spaces, categories, higher categories, simplicial categories, relative categories etc, on the other (see [8], [4], [15], [2]). In the case where E = Set the answer is classical and well-known for the question of preservation of all finite limits (see [5], Chapter 6), the question of finite products (see [1]) as well as for the case of various types of finite connected limits (see [11]). The answer to such questions is also well-known in the case E is a Grothendieck topos, for the class of all finite limits or all finite connected limits (see [14], chapter 7 x 8 and [9]). -
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. -
Free Models of T-Algebraic Theories Computed As Kan Extensions Paul-André Melliès, Nicolas Tabareau
Free models of T-algebraic theories computed as Kan extensions Paul-André Melliès, Nicolas Tabareau To cite this version: Paul-André Melliès, Nicolas Tabareau. Free models of T-algebraic theories computed as Kan exten- sions. 2008. hal-00339331 HAL Id: hal-00339331 https://hal.archives-ouvertes.fr/hal-00339331 Preprint submitted on 17 Nov 2008 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Free models of T -algebraic theories computed as Kan extensions Paul-André Melliès Nicolas Tabareau ∗ Abstract One fundamental aspect of Lawvere’s categorical semantics is that every algebraic theory (eg. of monoid, of Lie algebra) induces a free construction (eg. of free monoid, of free Lie algebra) computed as a Kan extension. Unfortunately, the principle fails when one shifts to linear variants of algebraic theories, like Adams and Mac Lane’s PROPs, and similar PROs and PROBs. Here, we introduce the notion of T -algebraic theory for a pseudomonad T — a mild generalization of equational doctrine — in order to describe these various kinds of “algebraic theories”. Then, we formulate two conditions (the first one combinatorial, the second one algebraic) which ensure that the free model of a T -algebraic theory exists and is computed as an Kan extension. -
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. -
Quasicategories 1.1 Simplicial Sets
Quasicategories 12 November 2018 1.1 Simplicial sets We denote by ∆ the category whose objects are the sets [n] = f0; 1; : : : ; ng for n ≥ 0 and whose morphisms are order-preserving functions [n] ! [m]. A simplicial set is a functor X : ∆op ! Set, where Set denotes the category of sets. A simplicial map f : X ! Y between simplicial sets is a natural transforma- op tion. The category of simplicial sets with simplicial maps is denoted by Set∆ or, more concisely, as sSet. For a simplicial set X, we normally write Xn instead of X[n], and call it the n set of n-simplices of X. There are injections δi :[n − 1] ! [n] forgetting i and n surjections σi :[n + 1] ! [n] repeating i for 0 ≤ i ≤ n that give rise to functions n n di : Xn −! Xn−1; si : Xn+1 −! Xn; called faces and degeneracies respectively. Since every order-preserving function [n] ! [m] is a composite of a surjection followed by an injection, the sets fXngn≥0 k ` together with the faces di and degeneracies sj determine uniquely a simplicial set X. Faces and degeneracies satisfy the simplicial identities: n−1 n n−1 n di ◦ dj = dj−1 ◦ di if i < j; 8 sn−1 ◦ dn if i < j; > j−1 i n+1 n < di ◦ sj = idXn if i = j or i = j + 1; :> n−1 n sj ◦ di−1 if i > j + 1; n+1 n n+1 n si ◦ sj = sj+1 ◦ si if i ≤ j: For n ≥ 0, the standard n-simplex is the simplicial set ∆[n] = ∆(−; [n]), that is, ∆[n]m = ∆([m]; [n]) for all m ≥ 0. -
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∗. -
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: