Homotopy Theoretic Models of Type Theory
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FIELDS MEDAL for Mathematical Efforts R
Recognizing the Real and the Potential: FIELDS MEDAL for Mathematical Efforts R Fields Medal recipients since inception Year Winners 1936 Lars Valerian Ahlfors (Harvard University) (April 18, 1907 – October 11, 1996) Jesse Douglas (Massachusetts Institute of Technology) (July 3, 1897 – September 7, 1965) 1950 Atle Selberg (Institute for Advanced Study, Princeton) (June 14, 1917 – August 6, 2007) 1954 Kunihiko Kodaira (Princeton University) (March 16, 1915 – July 26, 1997) 1962 John Willard Milnor (Princeton University) (born February 20, 1931) The Fields Medal 1966 Paul Joseph Cohen (Stanford University) (April 2, 1934 – March 23, 2007) Stephen Smale (University of California, Berkeley) (born July 15, 1930) is awarded 1970 Heisuke Hironaka (Harvard University) (born April 9, 1931) every four years 1974 David Bryant Mumford (Harvard University) (born June 11, 1937) 1978 Charles Louis Fefferman (Princeton University) (born April 18, 1949) on the occasion of the Daniel G. Quillen (Massachusetts Institute of Technology) (June 22, 1940 – April 30, 2011) International Congress 1982 William P. Thurston (Princeton University) (October 30, 1946 – August 21, 2012) Shing-Tung Yau (Institute for Advanced Study, Princeton) (born April 4, 1949) of Mathematicians 1986 Gerd Faltings (Princeton University) (born July 28, 1954) to recognize Michael Freedman (University of California, San Diego) (born April 21, 1951) 1990 Vaughan Jones (University of California, Berkeley) (born December 31, 1952) outstanding Edward Witten (Institute for Advanced Study, -
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]. -
Affine Grassmannians in A1-Algebraic Topology
AFFINE GRASSMANNIANS IN A1-ALGEBRAIC TOPOLOGY TOM BACHMANN Abstract. Let k be a field. Denote by Spc(k)∗ the unstable, pointed motivic homotopy category and A1 1 by R ΩGm : Spc(k)∗ →Spc(k)∗ the (A -derived) Gm-loops functor. For a k-group G, denote by GrG the affine Grassmannian of G. If G is isotropic reductive, we provide a canonical motivic equivalence A1 A1 R ΩGm G ≃ GrG. We use this to compute the motive M(R ΩGm G) ∈ DM(k, Z[1/e]). 1. Introduction This note deals with the subject of A1-algebraic topology. In other words it deals with with the ∞- category Spc(k) of motivic spaces over a base field k, together with the canonical functor Smk → Spc(k) and, importantly, convenient models for Spc(k). Since our results depend crucially on the seminal papers [1, 2], we shall use their definition of Spc(k) (which is of course equivalent to the other definitions in the literature): start with the category Smk of smooth (separated, finite type) k-schemes, form the universal homotopy theory on Smk (i.e. pass to the ∞-category P(Smk) of space-valued presheaves on k), and A1 then impose the relations of Nisnevich descent and contractibility of the affine line k (i.e. localise P(Smk) at an appropriate family of maps). The ∞-category Spc(k) is presentable, so in particular has finite products, and the functor Smk → Spc(k) preserves finite products. Let ∗ ∈ Spc(k) denote the final object (corresponding to the final k-scheme k); then we can form the pointed unstable motivic homotopy category Spc(k)∗ := Spc(k)/∗. -
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. -
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. -
An Intuitive Introduction to Motivic Homotopy Theory Vladimir Voevodsky
What follows is Vladimir Voevodsky's snapshot of his Fields Medal work on motivic homotopy, plus a little philosophy and \from my point of view the main fun of doing mathematics" Voevodsky (2002). Voevodsky uses extremely simple leading ideas to extend topological methods of homotopy into number theory and abstract algebra. The talk takes the simple ideas right up to the edge of some extremely refined applications. The talk is on-line as streaming video at claymath.org/video/. In a nutshell: classical homotopy studies a topological space M by looking at maps to M from the unit interval on the real line: [0; 1] = fx 2 R j 0 ≤ x ≤ 1 g as well as maps to M from the unit square [0; 1]×[0; 1], the unit cube [0; 1]3, and so on. Algebraic geometry cannot copy these naively, even over the real numbers, since of course the interval [0; 1] on the real line is not defined by a polynomial equation. The only polynomial equal to 0 all over [0; 1] is the constant polynomial 0, thus equals 0 over the whole line. The problem is much worse when you look at algebraic geometry over other fields than the real or complex numbers. Voevodsky shows how category theory allows abstract definitions of the basic apparatus of homotopy, where the category of topological spaces is replaced by any category equipped with objects that share certain purely categorical properties with the unit interval, unit square, and so on. For example, the line in algebraic geometry (over any field) does not have \endpoints" in the usual way but it does come with two naturally distinguished points: 0 and 1. -
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 . -
References Is Provisional, and Will Grow (Or Shrink) As the Course Progresses
Note: This list of references is provisional, and will grow (or shrink) as the course progresses. It will never be comprehensive. See also the references (and the book) in the online version of Weibel's K- Book [40] at http://www.math.rutgers.edu/ weibel/Kbook.html. References [1] Armand Borel and Jean-Pierre Serre. Le th´eor`emede Riemann-Roch. Bull. Soc. Math. France, 86:97{136, 1958. [2] A. K. Bousfield. The localization of spaces with respect to homology. Topology, 14:133{150, 1975. [3] A. K. Bousfield and E. M. Friedlander. Homotopy theory of Γ-spaces, spectra, and bisimplicial sets. In Geometric applications of homotopy theory (Proc. Conf., Evanston, Ill., 1977), II, volume 658 of Lecture Notes in Math., pages 80{130. Springer, Berlin, 1978. [4] Kenneth S. Brown and Stephen M. Gersten. Algebraic K-theory as gen- eralized sheaf cohomology. In Algebraic K-theory, I: Higher K-theories (Proc. Conf., Battelle Memorial Inst., Seattle, Wash., 1972), pages 266{ 292. Lecture Notes in Math., Vol. 341. Springer, Berlin, 1973. [5] William Dwyer, Eric Friedlander, Victor Snaith, and Robert Thoma- son. Algebraic K-theory eventually surjects onto topological K-theory. Invent. Math., 66(3):481{491, 1982. [6] Eric M. Friedlander and Daniel R. Grayson, editors. Handbook of K- theory. Vol. 1, 2. Springer-Verlag, Berlin, 2005. [7] Ofer Gabber. K-theory of Henselian local rings and Henselian pairs. In Algebraic K-theory, commutative algebra, and algebraic geometry (Santa Margherita Ligure, 1989), volume 126 of Contemp. Math., pages 59{70. Amer. Math. Soc., Providence, RI, 1992. [8] Henri Gillet and Daniel R. -
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. -
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.