Volume II The Theory of Quasi-Categories and its Applications Andr´eJoyal Contents Introduction 153 Perspective 157 I Twelve lectures 207 1 Elementary aspects 209 2 Three classes of fibrations 221 3 Join and slices 241 4 Quasi-categories and Kan complexes 259 5 Pseudo-fibrations and function spaces 273 6 The model structure for quasi-categories 293 7 The model structure for cylinders 309 8 The contravariant model structure 323 9 Minimal fibrations 339 10 Base changes 353 11 Proper and smooth maps 363 12 Higher quasi-categories 369 151 152 Contents II Appendices 377 A Accessible categories 379 B Simplicial sets 381 C Factorisation systems 393 D Weak factorisation systems 403 E Model categories 427 F Homotopy factorisation systems 447 G Reedy theory 463 H Open boxes and prisms 467 Bibliography 487 Indices 489 Introduction The notion of quasi-category was introduced by Boardman and Vogt in their work on homotopy invariant algebraic structures [BV]. A Kan complex and the nerve of a category are examples. The goal of our work is to extend category theory to quasi-categories and to develop applications to homotopy theory, higher category theory and (higher) topos theory. Quasi-category are examples of (∞, 1)-categories in the sense of Baez and Dolan. Other examples are simplicial categories, Segal categories and complete Segal spaces (here called Rezk categories). To each example is associated a model category and the model categories are connected by a network of Quillen equiva- lences. Simplicial categories were introduced by Dwyer and Kan in their work on simplicial localisation. Segal categories were introduced by Hirschowitz and Simp- son in their work on higher stacks in algebraic geometry. Many aspects of category theory were extended to Segal categories. A notion of Segal topos was introduced by Toen and Vezzosi, and a notion of stable Segal category by Hirschowitz, Simp- son and Toen. A notion of higher Segal category was studied by Tamsamani, and a notion of enriched Segal category by Pellisier. The theory of Segal categories is a source of inspiration for the theory of quasi-categories. Jacob Lurie has recently formulated his work on higher topoi in the language of quasi-categories [Lu1]. In doing so, he has extended a considerable amount of category theory to quasi-categories. He also developed a theory of stable quasi- categories [Lu2] and applications to geometry in [Lu3], [Lu4] and [Lu5]. Our lec- tures may serve as an introduction to his work. The present notes were prepared for a course on quasi-categories given at the CRM in Barcelona in February 2008. The material is taken from two manuscripts under preparation. The first is a book in two volumes called the ”Theory of Quasi- categories” which I hope to finish before I leave this world if God permits. The second is a paper called ”Notes on Quasi-categories” to appear in the Proceedings of an IMA Conference in Minneapolis in 2004. The two manuscripts have somewhat different goals. The aim of the book is to teach the subject at a technical level by giving all the relevant details while the aim of the paper is to brush the subject 153 154 Introduction in perspective. Our goal in the course is to bring the participants at the cutting edge of the subject. The perspective presented in the course is very sketchy and a more complete one will be found in the IMA Conference Proceedings. To the eight lectures that were originally planned for the course we added four complementary lectures for a total of twelve. We included support material organised in eight appendices. The last appendix called ”Open boxes and prismes” was originally a chapter of the book. But it is so technical that we putted it as an appendix. The results presented here are the fruits of a long term research project which began around thirty years ago. We suspect that some of our results could be given a simpler proofs. The extension by Cisinski [Ci2] of the homotopy theory of Grothendieck [Mal2] appears to be the natural framework for future develope- ments. We briefly describe this theory in the perspective and we use some of the results. The fact that category theory can be extended to quasi-categories is not obvious a priori but it can discovered by working on the subject. The theory of quasi-categories depends strongly on homotopical algebra. Quasi-categories are the fibrant objects of a Quillen model structure on the category of simplicial sets. Many results of homotopical algebra become more conceptual and simpler when reformulated in the language of quasi-categories. We hope that this reformula- tion will help to shorten the proofs. In mathematics, many details of a proof are omitted because they are considered obvious. But what is ”obvious” in a given subject evolves through times. It is the result of an implicit agreement between the reseachers based on their knowledge and experience. A mathematical theory is a social construction. The theory of quasi-categories is presently in its infancy. The theory of quasi-categories can analyse phenomena which belong properly to homotopy theory. The notion of stable quasi-category is an example. The notion of meta-stable quasi-category introduced in the notes is another. We give a proof that the quasi-category of parametrized spectra is an utopos (joint work with Georg Biedermann). All the machinery of universal algebra can be transfered to homotopy theory. We introduce the notion of para-variety (after a suggestion by Mathieu Anel). In the last chapters we venture a few steps in the theory of (∞, n)-categories. We introduce a notion of n-disk and of n-cellular sets. If n = 1, a n-disk is an interval and a n-cellular set is a simplicial set. A n-quasi-category is defined to be a fibrant n-cellular set for a certain model structure on n-cellular sets. In the course, we shall formulate a conjecture of Cisinski about this model structure. A few words on terminology. A quasi-category is sometime called a weak Kan complex in the literature [KP]. The name Boardman complex was recently proposed by Vogt. The purpose of our terminology is to stress the analogy with categories. The theory of quasi-categories is very closely apparented to category theory. We are calling utopos (upper topos) a “higher topos”; alternatives are “homotopy topos” Introduction 155 or “homotopos”. We are calling pseudo-fibration a fibration in the model structure for quasi-categories; alternatives are “iso-fibration”, “categorical fibration” and “quasi-fibration”. We are calling isomorphism a morphism which is invertible in a quasi-category; alternatives are “quasi-isomorphism”,“equimorphism” and “equiv- alence”. I warmly thank Carles Casacuberta and Joachim Kock for the organisation of the Advanced Course and their support. The 2007-2008 CRM research program on Homotopy Theory and Higher Categories is the fruit of their initiative. I thank the director of the CRM Joaquim Bruna and the former director Manuel Castellet for their support. The CRM is a great place for mathematical research and Barcelona a marvelous cultural center. Long live to Catalunya! 156 Introduction Perspective The notion of quasi-category Recall that a simplicial set X is called a Kan complex if it satisfies the Kan condition: every horn Λk[n] → X can be filled by a simplex ∆[n] → X, ∀ Λk[n] / X _ = {{ {{ {{∃ {{ ∆[n]. The notion of quasi-category is a slight modification of this notion. A simplicial set X is called a quasi-category if it satisfies the Boardman condition: every horn Λk[n] → X with 0 < k < n can be filled by a simplex ∆[n] → X. A quasi- category is sometime called a weak Kan complex in the literature [KP]. The name Boardman complex was recently proposed by Vogt. A Kan complex and the nerve of a category are examples of quasi-categories. The purpose of our terminology is to stress the analogy with categories. The theory of quasi-categories is very closely apparented to category theory. We often say that a vertex of a quasi-category is an object of this quasi-category, and that an arrow is a morphism.A map of quasi- categories f : X → Y is a map of simplicial sets. We denote the category of (small) categories by Cat and the category of (small) quasi-categories by QCat. If X is a quasi-category, then so is the simplicial set XA for any simplicial sets A. Hence the category QCat is cartesian closed. The notion of quasi-category has many equivalent descriptions. For n > 0, the n-chain I[n] ⊆ ∆[n] is defined to be the union of the edges (i − 1, i) ⊆ ∆[n] for 1 ≤ i ≤ n. We shall put I[0] = 1. A simplicial set X is a quasi-category iff the projection X∆[2] → XI[2] defined by the inclusion I[2] ⊂ ∆[2] is a trivial fibration. 157 158 Perspective The nerve functor The nerve functor N : Cat → S, from the category of small categories to the category of simplicial sets is fully faithful. It can be regarded as an inclusion by adopting the same notation for a small category and its nerve. If I denotes the category generated by one arrow 0 → 1 and J the groupoid generated by one isomorphism 0 → 1, then we have ∆[1] = I ⊂ J. The nerve functor has a left adjoint τ1 : S → Cat which associates to a simplicial set X its fundamental category τ1X. The fundamental category of a quasi-category X is isomorphic to the homotopy category hoX constructed by Boardman and Vogt. If X is a simplicial and a, b ∈ X0, let us denote by X(a, b) the fiber at (a, b) of the projection (s, t): XI → X{0,1} = X × X defined by the inclusion {0, 1} ⊂ I.