Mapping Spaces in Quasi-Categories
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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]. -
SHEAFIFIABLE HOMOTOPY MODEL CATEGORIES, PART II Introduction
SHEAFIFIABLE HOMOTOPY MODEL CATEGORIES, PART II TIBOR BEKE Abstract. If a Quillen model category is defined via a suitable right adjoint over a sheafi- fiable homotopy model category (in the sense of part I of this paper), it is sheafifiable as well; that is, it gives rise to a functor from the category of topoi and geometric morphisms to Quillen model categories and Quillen adjunctions. This is chiefly useful in dealing with homotopy theories of algebraic structures defined over diagrams of fixed shape, and unifies a large number of examples. Introduction The motivation for this research was the following question of M. Hopkins: does the forgetful (i.e. underlying \set") functor from sheaves of simplicial abelian groups to simplicial sheaves create a Quillen model structure on sheaves of simplicial abelian groups? (Creates means here that the weak equivalences and fibrations are preserved and reflected by the forgetful functor.) The answer is yes, even if the site does not have enough points. This Quillen model structure can be thought of, to some extent, as a replacement for the one on chain complexes in an abelian category with enough projectives where fibrations are the epis. (Cf. Quillen [39]. Note that the category of abelian group objects in a topos may fail to have non-trivial projectives.) Of course, (bounded or unbounded) chain complexes in any Grothendieck abelian category possess many Quillen model structures | see Hovey [28] for an extensive discussion | but this paper is concerned with an argument that extends to arbitrary universal algebras (more precisely, finite limit definable structures) besides abelian groups. -
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. -
Sizes and Filtrations in Accessible Categories
SIZES AND FILTRATIONS IN ACCESSIBLE CATEGORIES MICHAEL LIEBERMAN, JIRˇ´I ROSICKY,´ AND SEBASTIEN VASEY Abstract. Accessible categories admit a purely category-theoretic replace- ment for cardinality: the internal size. Generalizing results and methods from [LRV19b], we examine set-theoretic problems related to internal sizes and prove several L¨owenheim-Skolem theorems for accessible categories. For example, assuming the singular cardinal hypothesis, we show that a large ac- cessible category has an object in all internal sizes of high-enough cofinality. We also prove that accessible categories with directed colimits have filtrations: any object of sufficiently high internal size is (the retract of) a colimit of a chain of strictly smaller objects. Contents 1. Introduction 1 2. Preliminaries 4 3. Directed systems and cofinal posets 9 4. Presentation theorem and axiomatizability 12 5. On successor presentability ranks 14 6. The existence spectrum of a µ-AEC 16 7. The existence spectrum of an accessible category 19 8. Filtrations 22 References 26 1. Introduction Recent years have seen a burst of research activity connecting accessible categories with abstract model theory. Abstract model theory, which has always had the aim of generalizing|in a uniform way|fragments of the rich classification theory of first order logic to encompass the broader nonelementary classes of structures that Date: June 5, 2019 AMS 2010 Subject Classification: Primary 18C35. Secondary: 03C45, 03C48, 03C52, 03C55, 03C75, 03E05. Key words and phrases. internal size, presentability rank, existence spectrum, accessibility spectrum, filtrations, singular cardinal hypothesis. The second author is supported by the Grant agency of the Czech republic under the grant 19-00902S. -
Arxiv:0708.2185V1 [Math.CT] 16 Aug 2007 Ro.Oragmn Sbsdo H Atta Oooyequivalen Homotopy That Fact the on Based Is Argument Our Proof
ON COMBINATORIAL MODEL CATEGORIES J. ROSICKY´ ∗ Abstract. Combinatorial model categories were introduced by J. H. Smith as model categories which are locally presentable and cofibrantly generated. He has not published his results yet but proofs of some of them were presented by T. Beke or D. Dugger. We are contributing to this endeavour by proving that weak equiv- alences in a combinatorial model category form an accessible cat- egory. We also present some new results about weak equivalences and cofibrations in combinatorial model categories. 1. Introduction Model categories were introduced by Quillen [24] as a foundation of homotopy theory. Their modern theory can found in [20] or [19]. Combinatorial model categories were introduced by J. H. Smith as model categories which are locally presentable and cofibrantly gener- ated. The latter means that both cofibrations and trivial cofibrations are cofibrantly generated by a set of morphisms. He has not published his results yet but some of them can be found in [7] or [14]. In partic- ular, [7] contains the proof of the theorem characterizing when a class W of weak equivalences makes a locally presentable category K to be a combinatorial model category with a given cofibrantly generated class C of cofibrations. The characterization combines closure properties of W together with a smallness condition saying that W satisfies the solution set condition at the generating set X of cofibrations. We will show that arXiv:0708.2185v1 [math.CT] 16 Aug 2007 these conditions are also necessary. This is based or another result of J. H. Smith saying that, in a combinatorial model category, W is always accessible and accessibly embedded in the category K→ of morphisms of K (he informed me about this result in 2002 without indicating a proof). -
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. -
Homotopy Theoretic Models of Type Theory
Homotopy Theoretic Models of Type Theory Peter Arndt1 and Krzysztof Kapulkin2 1 University of Oslo, Oslo, Norway [email protected] 2 University of Pittsburgh, Pittsburgh, PA, USA [email protected] Abstract. We introduce the notion of a logical model category which is a Quillen model category satisfying some additional conditions. Those conditions provide enough expressive power so that we can soundly inter- pret dependent products and sums in it while also have purely intensional iterpretation of the identity types. On the other hand, those conditions are easy to check and provide a wide class of models that are examined in the paper. 1 Introduction Starting with the Hofmann-Streicher groupoid model [HS98] it has become clear that there exist deep connections between Intensional Martin-L¨ofType Theory ([ML72], [NPS90]) and homotopy theory. These connections have recently been studied very intensively. We start by summarizing this work | a more complete survey can be found in [Awo10]. It is well-known that Identity Types (or Equality Types) play an important role in type theory since they provide a relaxed and coarser notion of equality between terms of a given type. For example, assuming standard rules for type Nat one cannot prove that n: Nat ` n + 0 = n: Nat but there is still an inhabitant n: Nat ` p: IdNat(n + 0; n): Identity types can be axiomatized in a usual way (as inductive types) by form, intro, elim, and comp rules (see eg. [NPS90]). A type theory where we do not impose any further rules on the identity types is called intensional. -
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. -
All (,1)-Toposes Have Strict Univalent Universes
All (1; 1)-toposes have strict univalent universes Mike Shulman University of San Diego HoTT 2019 Carnegie Mellon University August 13, 2019 One model is not enough A (Grothendieck{Rezk{Lurie)( 1; 1)-topos is: • The category of objects obtained by \homotopically gluing together" copies of some collection of \model objects" in specified ways. • The free cocompletion of a small (1; 1)-category preserving certain well-behaved colimits. • An accessible left exact localization of an (1; 1)-category of presheaves. They are a powerful tool for studying all kinds of \geometry" (topological, algebraic, differential, cohesive, etc.). It has long been expected that (1; 1)-toposes are models of HoTT, but coherence problems have proven difficult to overcome. Main Theorem Theorem (S.) Every (1; 1)-topos can be given the structure of a model of \Book" HoTT with strict univalent universes, closed under Σs, Πs, coproducts, and identity types. Caveats for experts: 1 Classical metatheory: ZFC with inaccessible cardinals. 2 We assume the initiality principle. 3 Only an interpretation, not an equivalence. 4 HITs also exist, but remains to show universes are closed under them. Towards killer apps Example 1 Hou{Finster{Licata{Lumsdaine formalized a proof of the Blakers{Massey theorem in HoTT. 2 Later, Rezk and Anel{Biedermann{Finster{Joyal unwound this manually into a new( 1; 1)-topos-theoretic proof, with a generalization applicable to Goodwillie calculus. 3 We can now say that the HFLL proof already implies the (1; 1)-topos-theoretic result, without manual translation. (Modulo closure under HITs.) Outline 1 Type-theoretic model toposes 2 Left exact localizations 3 Injective model structures 4 Remarks Review of model-categorical semantics We can interpret type theory in a well-behaved model category E : Type theory Model category Type Γ ` A Fibration ΓA Γ Term Γ ` a : A Section Γ ! ΓA over Γ Id-type Path object . -
Takeaways from Talbot 2018: the Model Independent Theory of ∞-Categories
TAKEAWAYS FROM TALBOT 2018: THE MODEL INDEPENDENT THEORY OF ∞-CATEGORIES EMILY RIEHL AND DOMINIC VERITY Monday, May 28 What is an ∞-category? It depends on who you ask: • A category theorist: “It refers to some sort of weak infinite-dimensional category, perhaps an (∞, 1)-category, or an (∞, 푛)-category, or an (∞, ∞)-category — or perhaps an internal or fibered version of one of these.” • Jacob Lurie: “It’s a good nickname for an (∞, 1)-category.” • A homotopy theorist: “I don’t want to give a definition but if you push me I’ll say it’s a quasi- category (or maybe a complete Segal space).” ⋆ For us (interpolating between all of the usages above): “An ∞-category is a technical term mean- ing an object in some ∞-cosmos (and hence also an adjunction in its homotopy 2-category), ex- amples of which include quasi-categories or complete Segal spaces; other models of (∞, 1)-cat- egories; certain models of (∞, 푛)-categories; at least one model of (∞, ∞)-categories; fibered and internal versions of all of the above; and other things besides. The reason for this terminology is so that our theorem statements suggest their natural interpreta- tion. For instance an adjunction between ∞-categories is defined to be an adjunction in the homo- topy 2-category: this consists of a pair of ∞-categories 퐴 and 퐵, a pair of ∞-functors 푓 ∶ 퐵 → 퐴 and 푢 ∶ 퐴 → 퐵, and a pair of ∞-natural transformations 휂 ∶ id퐵 ⇒ 푢푓 and 휖 ∶ 푓푢 ⇒ id퐴 satisfying the triangle equalities. Now any theorem about adjunctions in 2-categories specializes to a theorem about adjunctions between ∞-categories: e.g. -
Accessible Categories and Model Theory: Hits and Misses Accessible Categories Workshop, Leeds
singular compactness calculus of λ-equivalences problems Accessible categories and model theory: hits and misses Accessible Categories Workshop, Leeds Tibor Beke University of Massachusetts tibor [email protected] July 17, 2018 Tibor Beke Accessible categories and model theory: hits and misses Wilfred Hodges in [Hodges 2008] singular compactness calculus of λ-equivalences problems epithet \ It would still be very welcome to be told that the whole scheme coincides with, say, something known to category theorists in another context. " Tibor Beke Accessible categories and model theory: hits and misses singular compactness calculus of λ-equivalences problems epithet \ It would still be very welcome to be told that the whole scheme coincides with, say, something known to category theorists in another context. " Wilfred Hodges in [Hodges 2008] Tibor Beke Accessible categories and model theory: hits and misses singular compactness cellular maps calculus of λ-equivalences cellular singular compactness problems beyond cellular prehistory Theorem (Schreier, 1926): Every subgroup of a free group is free. Also true for I abelian groups I abelian p-groups I over a field k: I Lie algebras (Witt) I commutative (non-associative) algebras (Shirshov) I magmas (Kurosh) Remark: there is still no complete characterization of these varieties known. Tibor Beke Accessible categories and model theory: hits and misses singular compactness cellular maps calculus of λ-equivalences cellular singular compactness problems beyond cellular conversely: Does this characterize free groups? Question Suppose G is an infinite group such that all subgroups of G of cardinality less than G, are free. (We will call such a group G almost free.) Is G free? Answer No. -
The 2-Category Theory of Quasi-Categories
Advances in Mathematics 280 (2015) 549–642 Contents lists available at ScienceDirect Advances in Mathematics www.elsevier.com/locate/aim The 2-category theory of quasi-categories Emily Riehl a,∗, Dominic Verity b a Department of Mathematics, Harvard University, Cambridge, MA 02138, USA b Centre of Australian Category Theory, Macquarie University, NSW 2109, Australia a r t i c l e i n f o a b s t r a c t Article history: In this paper we re-develop the foundations of the category Received 11 December 2013 theory of quasi-categories (also called ∞-categories) using Received in revised form 13 April 2-category theory. We show that Joyal’s strict 2-category of 2015 quasi-categories admits certain weak 2-limits, among them Accepted 22 April 2015 weak comma objects. We use these comma quasi-categories Available online xxxx Communicated by the Managing to encode universal properties relevant to limits, colimits, Editors of AIM and adjunctions and prove the expected theorems relating these notions. These universal properties have an alternate MSC: form as absolute lifting diagrams in the 2-category, which primary 18G55, 55U35, 55U40 we show are determined pointwise by the existence of certain secondary 18A05, 18D20, 18G30, initial or terminal vertices, allowing for the easy production 55U10 of examples. All the quasi-categorical notions introduced here are equiv- Keywords: alent to the established ones but our proofs are independent Quasi-categories and more “formal”. In particular, these results generalise 2-Category theory Formal category theory immediately to model categories enriched over quasi-cate- gories. © 2015 Elsevier Inc.