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NOTES on QUASI-CATEGORIES Contents Introduction 2 1 NOTES ON QUASI-CATEGORIES ANDRE´ JOYAL To the memory of Jon Beck Contents Introduction 2 1. Elementary aspects 8 2. The model structure for quasi-categories 12 3. Equivalence with simplicial categories 16 4. Equivalence with Rezk categories 17 5. Equivalence with Segal categories 19 6. Minimal quasi-categories 20 7. Discrete fibrations and coverings 22 8. Left and right fibrations 24 9. Join and slice 27 10. Initial and terminal objects 33 11. Homotopy factorisation systems 34 12. The contravariant model structure 39 13. Distributors, cylinders, correspondances and mediators 42 14. Base change 47 15. Spans 50 16. Yoneda lemma 57 17. Morita equivalence 62 18. Adjoint maps 64 19. Homotopy localisations 66 20. Barycentric localisations 69 21. Limits and colimits 69 22. Grothendieck fibrations 78 23. Proper and smooth maps 81 24. Kan extensions 83 25. The quasi-category U 88 26. Duality 91 27. Factorisation systems in quasi-categories 97 28. n-objects 102 29. Truncated quasi-categories 103 30. Universal algebra 105 31. Limit sketches 108 32. Locally presentable quasi-categories 118 33. Varieties of homotopy algebras 131 34. Para-varieties 138 Date: June 2 2008. 1 2 ANDRE´ JOYAL 35. Stabilisation 139 36. Descent theory 141 37. Exact quasi-categories 146 38. Meta-stable quasi-categories 151 39. Fiber sequences 153 40. Additive quasi-categories 155 41. Stable quasi-categories 164 42. Homotopoi (∞-topos) 169 43. Higher Categories 171 44. Higher Monoidal Categories 174 45. Disks and duality 175 46. Higher quasi-categories 184 47. Appendix on category theory 187 48. Appendix on factorisation systems 191 49. Appendix on weak factorisation systems 199 50. Appendix on simplicial sets 200 51. Appendix on model categories 203 52. Appendix on Cisinski theory 210 References 211 Index of terminology 215 Index of notation 222 Introduction The following notes are a collection of assertions about quasi-categories, many of which have not yet been formally proved. The unproven statements can be regarded as open problems or conjectures, unless they are labelled as theorems or pre-theorems. Our goal is to show that category theory can be extended to quasi- categories. The extended theory has applications to homotopy theory, homotopical algebra, higher category theory and higher topos theory. Parts of the material are taken from a book under preparation [J2], other parts from the talks I gave on the subject during the last decade, and the rest is taken from the literature. A first draft of the notes was written in 2004 in view of its publication in the Proceedings of the Conference on higher categories held at the IMA in Minneapolis. An expanded version was used in a course given at the Fields Institute in January 2007. The latest version was used in a course given at the CRM in Barcelona in February 2008. 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 basic examples. A quasi-category is sometime called a weak Kan complex in the literature [KP]. The purpose of our terminology is to stress the analogy with categories. Quasi-categories abound. The coherent nerve of a category enriched over Kan complexes is a quasi-category. The left (and right) barycentric localisations of a model category is a quasi-category. A quasi-category can be large. For example, the coherent nerve of the category of Kan complexes is QUASI-CATEGORIES 3 a large quasi-category U. The fact that category theory can be extended to quasi- categories is not obvious a priori, but can discovered by working on the subject. It is a kind of experimental fact. Quasi-categories 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). Simplicial categories were introduced by Dwyer and Kan in their work on simplicial localisation. Segal categories were introduced by Schwnzel and Vogt under the name of ∆-categories [ScVo] but they were rediscovered by Hirschowitz and Simpson in their work on higher stacks. Com- plete Segal spaces were introduced by Rezk in his work on homotopy theories. To each of these examples is associated a model category and the four model cate- gories are Quillen equivalent. The equivalence between simplicial categories, Segal categories and complete Segal spaces was established by Bergner [B2]. The equiv- alence between quasi-categories, Segal categories and complete Segal spaces was established by Tierney and the author [JT2]. The equivalence between simplicial categories and quasi-categories was established by Lurie [Lu1] and independantly by the author [J4]. Many aspects of category theory were extended to Segal categories by Hirschowitz, Simpson, Toen and Vezzosi. For examples, a notion of Segal topos was introduced by Simpson and a notion of homotopy topos by Toen and Vezzosi along ideas of Rezk. The development of derived algebraic geometry by Toen and Vezzosi [TV1] and independantly by Lurie [Lu1] is one of the main incentive for developing the theory of quasi-categories. A notion of stable Segal category was introduced by Hirschowitz, Simpson and Toen. Lurie has recently formulated his work on ∞-topoi (here called homotopoi) in the language of quasi-categories [Lu1]. In doing so, he has extended a considerable amount of category theory to quasi- categories. Our notes may serve as an introduction to his work [Lu2][Lu3] [Lu4] [Lu5]. Many ideas introduced in the notes are due to Charles Rezk. The notion of reduced category is inspired by his notion of complete Segal space, and the notion of homotopos is a reformulation of his notion of homotopy topos. The observation that every diagram in a homotopos is a descent diagram is due to him. Remark: the list (∞, 1)-categories given above is not exhaustive and our account of the history of the subject is incomplete. The notion of A∞-space introduced by Stasheff is a seminal idea in the subject. A theory of A∞-categories was developped by Batanin [Bat1]. Many aspects of category theory were extended to simpicial categories by Bousfield, Dwyer and Kan. See also Cordier and Porter [CP2]. The theory of homotopical categories by Dwyer, Hirschhorn, Kan and Smith is closely related to that of quasi-categories [DHKS]. The theory of quasi-categories depends on homotopical algebra for its formula- tion. A basic result states that the category of simplicial sets S admits a Quillen model structure in which the fibrant objects are the quasi-categories (and the cofibration are the monomorphisms). This defines the model structure for quasi- categories. The classical model structure on S is a Bousfield localisation of this model structure. Many aspects of category theory can be formulated in the lan- guage of homotopical algebra. For example, the category of small categories Cat admits a natural model structure in which the weak equivalences are the equiva- lence of categories. The corresponding notion of homotopy limit is closely related to the notion of pseudo-limit defined by category theorists. 4 ANDRE´ JOYAL Many aspects of homotopical algebra become simpler and more conceptual when formulated in the language of quasi-categories. This is true for example of the theory of homotopy limits and colimits which is equivalent to the theory of limits and colimits in a quasi-category. We hope that this reformulation of homotopical algebra will greatly simplify and clarify the proofs in this subject. This may not be entirely clear at present because the theory of quasi-categories is presently in its infancy. A mathematical theory is a kind of social construction, and the complexity of a proof depends on the degree of maturity of the subject. What is considered to be ”obvious” is the result of an implicit agreement between the experts based on their knowledge and experience. The quasi-category U has many properties in common with the category of sets. For example, it is cocomplete and freely generated by one object (its terminal object). It is the archetype of a homotopos,A prestack on a simplicial set A is defined to be a map Ao → U. A general homotopos is a left exact reflection of a quasi-category of prestacks. Homotopoi can be characterized abstractly by a system of axioms similar to those of Giraud for a Grothendieck topos. They also admit an elegant characterization due to Lurie in terms of a descent property discovered by Rezk. All the machinery of universal algebra can be extended to homotopy theory via the theory of quasi-categories. An algebraic theory is defined to be a small quasi-category with finite products T , and a model of T to be a map T → U which preserves finite products. The models T → U form a large quasi-category Mod(T ) which is complete an cocomplete. A variety of homotopy algebras is a quasi-category equivalent to a quasi-category Mod(T ) for some algebraic theory T . The homotopy varieties can be characterized by system of axioms closely related to those of Rosicky [Ros]. The notion of algebraic structure was extended by Ehresman to include essentially algebraic structures defined by a limit sketch. The classical theory of limit sketches and of essentially algebraic structures are easily extended to quasi-categories. For example, the notions of groupoid and of category are essentially algebraic. A category object in a finitely complete quasi-category X is defined to be a simplicial object C : ∆o → X satisfying the Segal condition. The theory of limit sketches is a natural framework for studying homotopy coherent algebraic structures in general and higher weak categories in particular. The quasi- category of models of a limit sketch is locally presentable and conversely, every locally presentable quasi-category is equivalent to the quasi-category of models of a limit sketch.
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