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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. -
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]). -
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. -
Kan Extensions Along Promonoidal Functors
Theory and Applications of Categories, Vol. 1, No. 4, 1995, pp. 72{77. KAN EXTENSIONS ALONG PROMONOIDAL FUNCTORS BRIAN DAY AND ROSS STREET Transmitted by R. J. Wood ABSTRACT. Strong promonoidal functors are de¯ned. Left Kan extension (also called \existential quanti¯cation") along a strong promonoidal functor is shown to be a strong monoidal functor. A construction for the free monoidal category on a promonoidal category is provided. A Fourier-like transform of presheaves is de¯ned and shown to take convolution product to cartesian product. Let V be a complete, cocomplete, symmetric, closed, monoidal category. We intend that all categorical concepts throughout this paper should be V-enriched unless explicitly declared to be \ordinary". A reference for enriched category theory is [10], however, the reader unfamiliar with that theory can read this paper as written with V the category of sets and for V as cartesian product; another special case is obtained by taking all categories and functors to be additive and V to be the category of abelian groups. The reader will need to be familiar with the notion of promonoidal category (used in [2], [6], [3], and [1]): such a category A is equipped with functors P : AopAopA¡!V, J : A¡!V, together with appropriate associativity and unit constraints subject to some axioms. Let C be a cocomplete monoidal category whose tensor product preserves colimits in each variable. If A is a small promonoidal category then the functor category [A, C] has the convolution monoidal structure given by Z A;A0 F ¤G = P (A; A0; ¡)(FAGA0) (see [7], Example 2.4). -
Quantum Kan Extensions and Applications
Quantum Kan Extensions and their Applications IARPA QCS PI Meeting z 16–17 July 2012 0 | i F ϕ ψ A B | i ǫ =⇒ X Lan (X ) F θ y Sets x 1 | i BakeÖ ÅÓÙÒØaiÒ Science Technology Service Dr.RalphL.Wojtowicz Dr.NosonS.Yanofsky Baker Mountain Research Corporation Department of Computer Yellow Spring, WV and Information Science Brooklyn College Contract: D11PC20232 Background Right Kan Extensions Left Kan Extensions Theory Plans Project Overview Goals: Implement and analyze classical Kan extensions algorithms Research and implement quantum algorithms for Kan extensions Research Kan liftings and homotopy Kan extensions and their applications to quantum computing Performance period: 26 September 2011 – 25 September 2012 Progress: Implemented Carmody-Walters classical Kan extensions algorithm Surveyed complexity of Todd-Coxeter coset enumeration algorithm Proved that hidden subgroups are examples of Kan extensions Found quantum algorithms for: (1) products, (2) pullbacks and (3) equalizers [(1) and (3) give all right Kan extensions] Found quantum algorithm for (1) coproducts and made progress on (2) coequalizers [(1) and (2) give all left Kan extensions] Implemented quantum algorithm for coproducts Report on Kan liftings in progress ÅÓÙÒØaiÒ BakeÖ Quantum Kan Extensions — 17 July 2012 1/34 Background Right Kan Extensions Left Kan Extensions Theory Plans Outline 1 Background Definitions Examples and Applications The Carmody-Walters Kan Extension Algorithm 2 Right Kan Extensions Products Pullbacks Equalizers 3 Left Kan Extensions Coproducts Coequalizers -
UC Berkeley UC Berkeley Electronic Theses and Dissertations
UC Berkeley UC Berkeley Electronic Theses and Dissertations Title Smooth Field Theories and Homotopy Field Theories Permalink https://escholarship.org/uc/item/8049k3bs Author Wilder, Alan Cameron Publication Date 2011 Peer reviewed|Thesis/dissertation eScholarship.org Powered by the California Digital Library University of California Smooth Field Theories and Homotopy Field Theories by Alan Cameron Wilder A dissertation submitted in partial satisfaction of the requirements for the degree of Doctor of Philosophy in Mathematics in the Graduate Division of the University of California, Berkeley Committee in charge: Professor Peter Teichner, Chair Associate Professor Ian Agol Associate Professor Michael Hutchings Professor Mary K. Gaillard Fall 2011 Smooth Field Theories and Homotopy Field Theories Copyright 2011 by Alan Cameron Wilder 1 Abstract Smooth Field Theories and Homotopy Field Theories by Alan Cameron Wilder Doctor of Philosophy in Mathematics University of California, Berkeley Professor Peter Teichner, Chair In this thesis we assemble machinery to create a map from the field theories of Stolz and Teichner (see [ST]), which we call smooth field theories, to the field theories of Lurie (see [Lur1]), which we term homotopy field theories. Finally, we upgrade this map to work on inner-homs. That is, we provide a map from the fibred category of smooth field theories to the Segal space of homotopy field theories. In particular, along the way we present a definition of symmetric monoidal Segal space, and use this notion to complete the sketch of the defintion of homotopy bordism category employed in [Lur1] to prove the cobordism hypothesis. i To Kyra, Dashiell, and Dexter for their support and motivation. -
Ends and Coends
THIS IS THE (CO)END, MY ONLY (CO)FRIEND FOSCO LOREGIAN† Abstract. The present note is a recollection of the most striking and use- ful applications of co/end calculus. We put a considerable effort in making arguments and constructions rather explicit: after having given a series of preliminary definitions, we characterize co/ends as particular co/limits; then we derive a number of results directly from this characterization. The last sections discuss the most interesting examples where co/end calculus serves as a powerful abstract way to do explicit computations in diverse fields like Algebra, Algebraic Topology and Category Theory. The appendices serve to sketch a number of results in theories heavily relying on co/end calculus; the reader who dares to arrive at this point, being completely introduced to the mysteries of co/end fu, can regard basically every statement as a guided exercise. Contents Introduction. 1 1. Dinaturality, extranaturality, co/wedges. 3 2. Yoneda reduction, Kan extensions. 13 3. The nerve and realization paradigm. 16 4. Weighted limits 21 5. Profunctors. 27 6. Operads. 33 Appendix A. Promonoidal categories 39 Appendix B. Fourier transforms via coends. 40 References 41 Introduction. The purpose of this survey is to familiarize the reader with the so-called co/end calculus, gathering a series of examples of its application; the author would like to stress clearly, from the very beginning, that the material presented here makes arXiv:1501.02503v2 [math.CT] 9 Feb 2015 no claim of originality: indeed, we put a special care in acknowledging carefully, where possible, each of the many authors whose work was an indispensable source in compiling this note. -
Category Theory and Diagrammatic Reasoning 3 Universal Properties, Limits and Colimits
Category theory and diagrammatic reasoning 13th February 2019 Last updated: 7th February 2019 3 Universal properties, limits and colimits A division problem is a question of the following form: Given a and b, does there exist x such that a composed with x is equal to b? If it exists, is it unique? Such questions are ubiquitious in mathematics, from the solvability of systems of linear equations, to the existence of sections of fibre bundles. To make them precise, one needs additional information: • What types of objects are a and b? • Where can I look for x? • How do I compose a and x? Since category theory is, largely, a theory of composition, it also offers a unifying frame- work for the statement and classification of division problems. A fundamental notion in category theory is that of a universal property: roughly, a universal property of a states that for all b of a suitable form, certain division problems with a and b as parameters have a (possibly unique) solution. Let us start from universal properties of morphisms in a category. Consider the following division problem. Problem 1. Let F : Y ! X be a functor, x an object of X. Given a pair of morphisms F (y0) f 0 F (y) x , f does there exist a morphism g : y ! y0 in Y such that F (y0) F (g) f 0 F (y) x ? f If it exists, is it unique? 1 This has the form of a division problem where a and b are arbitrary morphisms in X (which need to have the same target), x is constrained to be in the image of a functor F , and composition is composition of morphisms. -
Kan Extensions and the Calculus of Modules for $\Infty $-Categories
Kan extensions and the calculus of modules for ∞-categories EMILY RIEHL DOMINIC VERITY Various models of (∞, 1)-categories, including quasi-categories, complete Segal spaces, Segal categories, and naturally marked simplicial sets can be considered as the objects of an ∞-cosmos. In a generic ∞-cosmos, whose objects we call ∞-categories, we introduce modules (also called profunctors or correspondences) between ∞-categories, incarnated as as spans of suitably-defined fibrations with groupoidal fibers. As the name suggests, a module from A to B is an ∞-category equipped with a left action of A and a right action of B, in a suitable sense. Applying the fibrational form of the Yoneda lemma, we develop a general calculus of modules,provingthattheynaturallyassemble intoa multicategory-like structure called a virtual equipment, which is known to be a robust setting in which to develop formal category theory. Using the calculus of modules, it is straightforward to define andstudy pointwise Kan extensions,which we relate, in thecaseofcartesian closed ∞-cosmoi, to limits and colimits of diagrams valued in an ∞-category, as introduced in previous work. 18G55, 55U35, 55U40; 18A05, 18D20, 18G30, 55U10 arXiv:1507.01460v3 [math.CT] 13 Jun 2016 2 Riehl and Verity 1 Introduction Previous work [12, 15, 13, 14] shows that the basic theory of (∞, 1)-categories — categories that are weakly enriched over ∞-groupoids, i.e., topological spaces — can be developed “model independently,” at least if one is content to work with one of the better-behaved models: namely, quasi-categories, complete Segal spaces, Segal categories, or naturally marked simplicial sets. More specifically, we show that a large portion of the category theory of quasi-categories—one model of (∞, 1)-categories that has been studied extensively by Joyal, Lurie, and others—can be re-developed from the abstract perspective of the homotopy 2-category of the ∞-cosmos of quasi- categories. -
Three Studies in Higher Category Theory: Fibrations, Skew-Monoidal Structures and Excision of Extremals
Bull. Aust. Math. Soc. 94 (2016), 337–338 doi:10.1017/S0004972716000356 THREE STUDIES IN HIGHER CATEGORY THEORY: FIBRATIONS, SKEW-MONOIDAL STRUCTURES AND EXCISION OF EXTREMALS MITCHELL BUCKLEY (Received 4 November 2015; first published online 21 July 2016) 2010 Mathematics subject classification: primary 18D10; secondary 18D05, 18D30. Keywords and phrases: fibration, bicategory, fibred category, fibred bicategory, Catalan simplicial set, monoidal category, skew-monoidal category, parity complex, excision of extremals. Many fundamental constructions from ordinary category theory can be generalised to higher categories. Obvious examples include adjunctions, monads, algebras, limits and colimits. This thesis explores three cases where a construction from category theory is extended to higher categories. We first consider the theory of (Grothendieck) fibrations and the Grothendieck construction. Our goal is to generalise this theory to the contexts of 2-categories and bicategories. Previous authors had made significant steps in this direction [1,6] but attempts at a full description were incomplete or contained errors. The ordinary Grothendieck construction describes a correspondence between fibrations over a fixed category B and contravariant pseudo-functors from B into the 2-category of categories. By examining contravariant 2-functors from a 2-category B into the 3-category of 2-categories, we are able to define a very strict kind of fibration for 2-categories. Following this, we examine contravariant trihomomorphisms from a bicategory B into the