DOCSLIB.ORG

SIMPLICIAL SETS, I 1. Definitions Let ∆ Be the Following Category. • Objects

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
SIMPLICIAL SETS, I 1. Definitions Let ∆ Be the Following Category. • Objects LECTURE 8: SIMPLICIAL SETS, I 1. Definitions Let ∆ be the following category. • Objects: totally ordered sets [n] = f0 < 1 < ··· < ng where n ≥ 0 is any non-negative integer. • Morphisms: order preserving maps of sets θ :[n] ! [m]. That is, i < j =) θ(i) < θ(j). Remark 1.1. If we think about the objects of ∆ as categories themselves [n] = f0 ! 1 !···! ng then the morphisms in ∆ are precisely the functors. We refer to ∆ as the finite ordinal category. Definition 1.2. A simplicial set is a functor X : ∆op ! Set: In general, if C is any category, a simplicial object in C is a functor X : ∆op ! C: If X is a simplicial set, we let X[n] denote the image of the object [n] 2 ∆. The category of simplicial sets is the functor category op Set∆ = Fun(∆ ; Set): Unwinding the definition, a morphism of simplicial sets f : X ! Y is equivalent to prescribing a map of sets f[n]: X[n] ! Y [n] for each n satisfying some conditions that we spell out below. Example 1.3. Singular set. Let Top be the category of topological spaces, and let n n+1 X j∆ j = f(t0; : : : ; tn) 2 (R≥0) j ti = 1g i≥0 be the standard n-simplex. For any order preserving map θ :[n] ! [m] there is a natural map of spaces n m θ∗ : j∆ j ! j∆ j: 1 This defines a functor ∆ ! Top; [n] 7! j∆nj. Further, for any topological space Y 2 Top, we have the simplicial set Sing(Y ) : ∆op ! Set n [n] 7! HomTop(j∆ j;Y ) called the singular simplicial set associated to Y . Example 1.4. Standard simplex. For each n let n op ∆ = Hom∆(−; [n]) : ∆ ! Set: This is called the standard n-simplex. Note that this is simply the image of [n] under the Yoneda embedding op ∆ ! Fun(∆ ; Set) = Set∆: In particular n HomSet∆ (∆ ;X) = X[n] for each n. In pictures, we can think about ∆1 as simply the two vertex diagram ∆1 : 0 ! 1 and ∆2 as being prescribed by the following diagram 1 ∆2 : 0 2: For instance, ∆2([1]) = ∆([1]; [2]) = f0 ! 1; 1 ! 2; 1 ! 2g reads off all the 1-simplices inside ∆2. 1.1. A combinatorial description. So far, the definition of a simplicial set might seem a little abstract. One can prescribe the data of a simplicial set more explicitly as a collection of sets fX[n]gn≥0 together with the following data of face and degeneracy maps. The category ∆ is generated by two classes of morphisms: (1) Face maps. For each n ≥ 1 and 1 ≤ j ≤ n define the maps dj :[n − 1] ! [n] ( i ; i < j i 7! : i + 1 ; i ≥ j 2 (2) Degeneracy maps. For each n ≥ 1 and 1 ≤ j ≤ n define the maps sj :[n + 1] ! [n] ( i ; i ≤ j i 7! : i − 1 ; i > j Generated means that any order preserving morphism [n] ! [m] can be written as a composition of these two classes of maps. The category is not, however, freely generated by these maps. There are a number of relations involving sj's and di's such as djdi = didj−1 and sjdi = disj−1 whenever both sides are defined. We won't list all the relations here, but refer to Chapter 1 of [?] for a textbook reference. Consequently we can define a simplicial set by the following data: • a collection of sets X = fX[n]gn≥0, where we refer to X[n] as the n-simplices; ∗ ∗ • a collection of morphisms fdj : X[n] ! X[n−1]g (the face maps) and fsj : X[n] ! X[n + 1]g (the degeneracy maps) satisfying some compatibilities. We will often define a simplicial set in this hands-on way. 2. Geometric Realization We have already constructed a functor Sing(−): Top ! Set∆ given by the singular set. One utility of simplicial sets is that they give us combinatorial rules for gluing spaces together in terms of simplices. Geometric realization makes this precise. I'm going to write the abstract definition, then interpret it in a way that hopefully makes clear what is going on. Given any category C and object X 2 C one can consider the overcategory (or slice category) C=X whose objects are maps Y ! X and whose morphisms are commuting triangles Y Y 0 X There is a dual notion of an undercategory. Given a simplicial set X we can then consider the overcategory Set∆=X . Think about this as the category of maps of simplical sets into X. We want to think about a smaller category, namely the category of maps of just the n-simplices into X, that we denote ∆=X . This category fits into a commutative diagram ∆=X Set∆=X ∆(−) ∆ Set∆ 3 The lower horizontal map is the Yoneda embedding sending [n] 7! ∆n. The right vertical map is the map that forgets the map into X defining the object of the overcategory and just remembers the source object. 1 Concretely, the objects ∆=X consists of maps, for each n ∆n ! X: These are precisely the n-simplices X[n] of X. The morphisms are commuting triangles of simplicial sets ∆n ∆m X Note that any simplicial set defines a composition of functors ∆(−) ∆=X ! ∆ −−−! Set∆: The following lemma consists of throwing definitions around Lemma 2.1. There is an isomorphism of simplicial sets ∆(−) =∼ colim ∆=X ! ∆ −−−! Set∆ −! X Roughly, this lemma just says that X is glued together from its n-simplices. Remark 2.2. An important analogous phenomena to keep in mind is the gluing condition for sheaves. Suppose Y is a topological space and F is a sheaf on Y . Let Open(Y ) be the poset of open sets of Y . For an open set U ⊂ Y , we can present the value F(U) as the following limit ∼ op op F F(U) = lim Open(Y )=U ! Open(Y ) −! Set In the limit, the first functor is the forgetful functor and the second functor is defined by the sheaf F. Perhaps even more analogous is the case of a cosheaf G on Y . Then, the cosheaf condition is equivalent to ∼ G G(U) = colim Open(Y )=U ! Open(Y ) −! Set : Using this perspective, the following definition is properly motivated. Definition 2.3. Let X be a simplicial set. Define the geometric realization of X to be the topological space j∆(−)j jXj := colim ∆=X ! ∆ −−−−! Top : Here, j∆(−)j is the functor from Example 1.3 which sends [n] to the geometric n-simplex n n j∆ j ⊂ R . 1This is a pullback diagram in the category of categories. 4 Remark 2.4. We have already seen how colimits can be used to glue together a space. Morally, this is what the colimit definition above is encoding. This becomes more apparent when one uses the following alternative description of the realization jXj. For each n, we consider the topolgical space X[n] × j∆nj. Here, we are viewing X[n] with the discrete topology. We write a generic element in this space as (x; t0; : : : ; tn). We can present the realization as a quotient of the form 0 1 G n jXj = @ X[n] × j∆ jA = ∼ n≥0 where the equivalence relation is of the form (di(x); t0; : : : ; tn) ∼ (x; t0; : : : ; ti−1; 0; ti; : : : ; tn): Almost tautologically, one has the following. Proposition 2.5. The functor j − j is left adjoint to Sing(−): |−| a Set∆ Top: Sing Proof. We use the categorical fact that colimits / limits commute with the Yoneda em- bedding in the following way Hom(colim (−);Y ) = lim Hom(−;Y ): (−) Also, note that the singular space was defined via the formula Sing(Y ) = HomTop(j∆ j;Y ). Using these two facts, for any space Y we obtain isomorphisms (−) HomTop(jXj;Y ) = HomTop(colim ∆ ;Y ) ∼ (−) = lim HomTop(j∆ j;Y ) ∼ (−) = lim HomSet∆ (∆ ; Sing(Y )) ∼ (−) = HomSet∆ (colim ∆ ; Sing(Y )) = HomSet∆ (X; Sing(Y )): 5.
Load more

Recommended publications
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • The Simplicial Parallel of Sheaf Theory Cahiers De Topologie Et Géométrie Différentielle Catégoriques, Tome 10, No 4 (1968), P
    CAHIERS DE TOPOLOGIE ET GÉOMÉTRIE DIFFÉRENTIELLE CATÉGORIQUES YUH-CHING CHEN Costacks - The simplicial parallel of sheaf theory Cahiers de topologie et géométrie différentielle catégoriques, tome 10, no 4 (1968), p. 449-473 <http://www.numdam.org/item?id=CTGDC_1968__10_4_449_0> © Andrée C. Ehresmann et les auteurs, 1968, tous droits réservés. L’accès aux archives de la revue « Cahiers de topologie et géométrie différentielle catégoriques » implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ CAHIERS DE TOPOLOGIE ET GEOMETRIE DIFFERENTIELLE COSTACKS - THE SIMPLICIAL PARALLEL OF SHEAF THEORY by YUH-CHING CHEN I NTRODUCTION Parallel to sheaf theory, costack theory is concerned with the study of the homology theory of simplicial sets with general coefficient systems. A coefficient system on a simplicial set K with values in an abe - . lian category 8 is a functor from K to Q (K is a category of simplexes) ; it is called a precostack; it is a simplicial parallel of the notion of a presheaf on a topological space X . A costack on K is a « normalized» precostack, as a set over a it is realized simplicial K/" it is simplicial « espace 8ta18 ». The theory developed here is functorial; it implies that all &#x3E;&#x3E; homo- logy theories are derived functors. Although the treatment is completely in- dependent of Topology, it is however almost completely parallel to the usual sheaf theory, and many of the same theorems will be found in it though the proofs are usually quite different.
    [Show full text]
  • Fields Lectures: Simplicial Presheaves
    Fields Lectures: Simplicial presheaves J.F. Jardine∗ January 31, 2007 This is a cleaned up and expanded version of the lecture notes for a short course that I gave at the Fields Institute in late January, 2007. I expect the cleanup and expansion processes to continue for a while yet, so the interested reader should check the web site http://www.math.uwo.ca/∼jardine periodi- cally for updates. Contents 1 Simplicial presheaves and sheaves 2 2 Local weak equivalences 4 3 First local model structure 6 4 Other model structures 12 5 Cocycle categories 13 6 Sheaf cohomology 16 7 Descent 22 8 Non-abelian cohomology 25 9 Presheaves of groupoids 28 10 Torsors and stacks 30 11 Simplicial groupoids 35 12 Cubical sets 43 13 Localization 48 ∗This research was supported by NSERC. 1 1 Simplicial presheaves and sheaves In all that follows, C will be a small Grothendieck site. Examples include the site op|X of open subsets and open covers of a topo- logical space X, the site Zar|S of Zariski open subschemes and open covers of a scheme S, or the ´etale site et|S, again of a scheme S. All of these sites have “big” analogues, like the big sites Topop,(Sch|S)Zar and (Sch|S)et of “all” topological spaces with open covers, and “all” S-schemes T → S with Zariski and ´etalecovers, respectivley. Of course, there are many more examples. Warning: All of the “big” sites in question have (infinite) cardinality bounds on the objects which define them so that the sites are small, and we don’t talk about these bounds.
    [Show full text]
  • The 2-Category Theory of Quasi-Categories
    THE 2-CATEGORY THEORY OF QUASI-CATEGORIES EMILY RIEHL AND DOMINIC VERITY Abstract. In this paper we re-develop the foundations of the category theory of quasi- categories (also called 1-categories) using 2-category theory. We show that Joyal’s strict 2-category of quasi-categories admits certain weak 2-limits, among them weak comma objects. We use these comma quasi-categories to encode universal properties relevant to limits, colimits, and adjunctions and prove the expected theorems relating these notions. These universal properties have an alternate form as absolute lifting diagrams in the 2- category, which we show are determined pointwise by the existence of certain initial or terminal vertices, allowing for the easy production of examples. All the quasi-categorical notions introduced here are equivalent to the established ones but our proofs are independent and more “formal”. In particular, these results generalise immediately to model categories enriched over quasi-categories. Contents 1. Introduction 2 1.1. A generalisation 3 1.2. Outline 4 1.3. Acknowledgments 5 2. Background on quasi-categories 6 2.1. Some standard simplicial notation 6 2.2. Quasi-categories 8 2.3. Isomorphisms and marked simplicial sets 10 2.4. Join and slice 14 3. The 2-category of quasi-categories 19 3.1. Relating 2-categories and simplicially enriched categories 19 3.2. The 2-category of quasi-categories 21 3.3. Weak 2-limits 24 arXiv:1306.5144v4 [math.CT] 6 May 2015 3.4. Slices of the category of quasi-categories 33 3.5. A strongly universal characterisation of weak comma objects 37 4.
    [Show full text]
  • 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.
    [Show full text]
  • A Model Structure for Quasi-Categories
    A MODEL STRUCTURE FOR QUASI-CATEGORIES EMILY RIEHL DISCUSSED WITH J. P. MAY 1. Introduction Quasi-categories live at the intersection of homotopy theory with category theory. In particular, they serve as a model for (1; 1)-categories, that is, weak higher categories with n-cells for each natural number n that are invertible when n > 1. Alternatively, an (1; 1)-category is a category enriched in 1-groupoids, e.g., a topological space with points as 0-cells, paths as 1-cells, homotopies of paths as 2-cells, and homotopies of homotopies as 3-cells, and so forth. The basic data for a quasi-category is a simplicial set. A precise definition is given below. For now, a simplicial set X is given by a diagram in Set o o / X o X / X o ··· 0 o / 1 o / 2 o / o o / with certain relations on the arrows. Elements of Xn are called n-simplices, and the arrows di : Xn ! Xn−1 and si : Xn ! Xn+1 are called face and degeneracy maps, respectively. Intuition is provided by simplical complexes from topology. There is a functor τ1 from the category of simplicial sets to Cat that takes a simplicial set X to its fundamental category τ1X. The objects of τ1X are the elements of X0. Morphisms are generated by elements of X1 with the face maps defining the source and target and s0 : X0 ! X1 picking out the identities. Composition is freely generated by elements of X1 subject to relations given by elements of X2. More specifically, if x 2 X2, then we impose the relation that d1x = d0x ◦ d2x.
    [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]
  • Exposé I – Elements of Parametrized Higher Category Theory
    PARAMETRIZED HIGHER CATEGORY THEORY AND HIGHER ALGEBRA: EXPOSÉ I – ELEMENTS OF PARAMETRIZED HIGHER CATEGORY THEORY CLARK BARWICK, EMANUELE DOTTO, SAUL GLASMAN, DENIS NARDIN, AND JAY SHAH Abstract. We introduce the basic elements of the theory of parametrized ∞-categories and functors between them. These notions are defined as suitable fibrations of ∞-categories and functors between them. We give as many examples as we are able at this stage. Simple operations, such as the formation of opposites and the formation of functor ∞-categories, become slightly more involved in the parametrized setting, but we explain precisely how to perform these constructions. All of these constructions can be performed explicitly, without resorting to such acts of desperation as straightening. The key results of this Exposé are: (1) a universal characterization of the 푇-∞-category of 푇-objects in any ∞-category, (2) the existence of an internal Hom for 푇-∞-categories, and (3) a parametrized Yoneda lemma. Contents 1. Parametrized ∞-categories 1 2. Examples of parametrized ∞-categories 3 3. Parametrized opposites 7 4. Parametrized subcategories 7 5. Constructing 푇-∞-categories via pairings 8 6. A technical result: the strong pushforward 9 7. 푇-objects in ∞-categories 11 8. Parametrized fibrations 14 9. Parametrized functor categories 15 10. The parametrized Yoneda embedding 19 Appendix A. Notational glossary 20 References 21 1. Parametrized ∞-categories Suppose 퐺 a finite group. At a minimum, a 퐺-∞-category should consist of an ∞- category 퐶 along with a weak action 휌 of 퐺. In particular, for every element 푔 ∈ 퐺, one should have an equivalence 휌(푔)∶ 퐶 ∼ 퐶, and for every 푔, ℎ ∈ 퐺, one should have a natu- ral equivalence 휌(푔ℎ) ≃ 휌(푔) ∘ 휌(ℎ), and these natural equivalences should then in turn be constrained by an infinite family of homotopies that express the higher associativity of 휌.
    [Show full text]
  • The Log Product Formula and Deformations of Modules
    The Log Product Formula and Deformations of Modules by L. S. Herr B.S., Stony Brook University, 2014 A thesis submitted to the Faculty of the Graduate School of the University of Colorado in partial fulfillment of the requirements for the degree of Doctor of Philosophy Department of Mathematics 2019 This thesis entitled: The Log Product Formula and Deformations of Modules written by L. S. Herr has been approved for the Department of Mathematics Prof. Jonathan Wise Prof. Sebastian Casalaina-Martin Date The final copy of this thesis has been examined by the signatories, and we find that both the content and the form meet acceptable presentation standards of scholarly work in the above mentioned discipline. Herr, L. S. (Ph.D., Mathematics) The Log Product Formula and Deformations of Modules Thesis directed by Prof. Prof. Jonathan Wise The material of this thesis is drawn from two distinct papers. One part concerns deformations of modules. We pose an extension problem, the possibility and number of solutions of which are encoded in a banded gerbe on a topos of modules. This object represents a class Def 2 2 H (A-mod=M; hK ) in cohomology. 2 Work of L. Illusie [29] produced a class f ^ ! 2 ExtA(M; K) with similar properties, and questioned whether an approach along the lines of the present work was possible. We show that 2 2 the two groups H (A-mod=M; hK ) and ExtA(M; K) are isomorphic in such a way that our class is mapped to the inverse of Illusie's. Our topology is an analogue for modules of that found in [71], [70].
    [Show full text]
  • A Simplicial Set Is a Functor X : a Op → Set, Ie. a Contravariant Set-Valued
    Contents 9 Simplicial sets 1 10 The simplex category and realization 10 11 Model structure for simplicial sets 15 9 Simplicial sets A simplicial set is a functor X : Dop ! Set; ie. a contravariant set-valued functor defined on the ordinal number category D. One usually writes n 7! Xn. Xn is the set of n-simplices of X. A simplicial map f : X ! Y is a natural transfor- mation of such functors. The simplicial sets and simplicial maps form the category of simplicial sets, denoted by sSet — one also sees the notation S for this category. If A is some category, then a simplicial object in A is a functor A : Dop ! A : Maps between simplicial objects are natural trans- formations. 1 The simplicial objects in A and their morphisms form a category sA . Examples: 1) sGr = simplicial groups. 2) sAb = simplicial abelian groups. 3) s(R − Mod) = simplicial R-modules. 4) s(sSet) = s2Set is the category of bisimplicial sets. Simplicial objects are everywhere. Examples of simplicial sets: 1) We’ve already met the singular set S(X) for a topological space X, in Section 4. S(X) is defined by the cosimplicial space (covari- ant functor) n 7! jDnj, by n S(X)n = hom(jD j;X): q : m ! n defines a function ∗ n q m S(X)n = hom(jD j;X) −! hom(jD j;X) = S(X)m by precomposition with the map q : jDmj ! jDmj. The assignment X 7! S(X) defines a covariant func- tor S : CGWH ! sSet; called the singular functor.
    [Show full text]