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Notes on Quasi-Categories

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Notes on Quasi-Categories NOTES ON QUASI-CATEGORIES ANDRE´ JOYAL To the memory of Jon Beck Contents Introduction 2 1. Elementary aspects 8 2. The model structure for quategories 12 3. Equivalence with simplicial categories 15 4. Equivalence with Rezk categories 16 5. Equivalence with Segal categories 18 6. Minimal quategories 19 7. Discrete fibrations and covering maps 21 8. Left and right fibrations 23 9. Join and slice 26 10. Initial and terminal objects 31 11. Homotopy factorisation systems 34 12. The covariant and contravariant model structures 39 13. Base changes 42 14. Cylinders, correspondances, distributors and spans 46 15. Yoneda lemmas 61 16. Morita equivalences 65 17. Adjoint maps 68 18. Quasi-localisations 70 19. Limits and colimits 72 20. Grothendieck fibrations 80 21. Proper and smooth maps 84 22. Kan extensions 85 23. The quategory K 92 24. Factorisation systems in quategories 96 25. n-objects 101 26. Truncated quategories 102 27. Accessible quategories and directed colimits 104 28. Limit sketches and arenas 109 29. Duality for prestacks and null-pointed prestacks 118 30. Cartesian theories 120 31. Sifted colimits 131 32. Algebraic theories and theaters 135 33. Fiber sequences 150 34. Additive quategories 153 Date: June 22 2008. 1 2 ANDRE´ JOYAL 35. Dold-Kan correspondance and finite differences calculus 162 36. Stabilisation 164 37. Perfect quategories and descent 168 38. Stable quategories 173 39. Para-varieties 178 40. Homotopoi (∞-topoi) 180 41. Meta-stable quasi-categories 183 42. Higher categories 184 43. Higher monoidal categories 186 44. Disks and duality 188 45. Higher quasi-categories 197 46. Appendix on category theory 200 47. Appendix on factorisation systems 206 48. Appendix on weak factorisation systems 213 49. Appendix on simplicial sets 215 50. Appendix on model categories 217 51. Appendix on simplicial categories 225 52. Appendix on Cisinski theory 229 References 230 Index of terminology 235 Index of notation 244 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 basic examples. The following notes are a collection of assertions on quasi-categories, many of which have not yet been formally proved. Our goal is to show that category theory has a natural extension to quasi-categories, The extended theory has applications to homotopy theory, homotopical algebra, higher category theory and higher topos theory. 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 at the CRM in Barcelona in February 2008. Remarks on terminology: a quasi-category is sometime called a weak Kan com- plex in the literature [KP]. The term ”quasi-category” was introduced to suggest a similarity with categories. We shall use the term quategory as an abreviation. Quategories abound. The coherent nerve of a category enriched over Kan com- plexes is a quategory. The quasi-localisation of a model category is a quategory. A quategory can be large. For example, the coherent nerve of the category of Kan complexes is a large quategory K. The coherent nerve of the category of (small) quategories is a large quategory Q1. QUASI-CATEGORIES 3 Quategories 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 by Schwnzel and Vogt under the name of ∆-categories [ScVo] and rediscovered by Hirschowitz and Simpson in their work on higher stacks. Complete Segal spaces (Rezk categories) were introduced by Rezk in his work on homotopy theories. To each of these examples is associated a model category and the four model categories are Quillen equivalent. The equivalence between simplicial categories, Segal categories and Rezk categories was established by Bergner [B2]. The equivalence between Rezk categories and quategories was established by Tierney and the author [JT2]. The equivalence between simplicial categories and quategories was established by Lurie [Lu1] and independantly by the author [J4]. Many aspects of category theory were extended to simplicial categories by Bousfield, Dwyer and Kan. The theory of homotopical categories of Dwyer, Hirschhorn, Kan and Smith is closely related to that of quategories [DHKS]. Many aspects of category theory were extended to Segal categories by Hirschowitz, Simpson, Toen and Vezzosi. Jacob Lurie has recently formulated his work on homotopoi in the language of quategories. In doing so, he has developped a formidable amount of quategory theory and our notes may serve as an introduction to his work. Many notions introduced here are due to Charles Rezk. The notion of homotopoi is an example. The notion of reduced category object is another. 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 whole subject. A theory of A∞-categories was developped by Batanin [Bat1]. A theory of homotopy coherent diagrams was developped by Cordier and Porter[CP2]. The theory of quategories depends on homotopical algebra. A basic result states that the category of simplicial sets S admits a Quillen model structure in which the fibrant objects are the quategories (and the cofibration are the monomorphisms). This defines the model structure for quategories. The classical model structure on the category S is a Bousfield localisation of this model structure. Many aspects of category theory can be formulated in the language of homo- topical algebra. The category of small categories Cat admits a model structure in which the weak equivalences are the equivalence of categories; it is the natu- ral model structure on Cat. Homotopy limits in the natural model structure are closely related to the pseudo-limits introduced by category theorists. Many aspects of homotopical algebra can be formulated in the language of quat- egories. This is true for example of the theory of homotopy limits and colimits. Many results of homotopical algebra becomes simpler when formulated in the lan- guage of quategories. We hope a similar simplification of the proofs. But this is not be entirely clear at present, since the theory of quategories 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. 4 ANDRE´ JOYAL The quategory K has many properties in common with the category of sets. It is the archetype of a homotopos.A prestack on a simplicial set A is defined to be a map Ao → K. A general homotopos is a left exact reflection of a quategory of prestacks. Homotopoi can be described abstractly by a system of axioms similar to the those of Giraud for a Grothendieck topos [Lu1]. They also admit an elegant characterization (due to Lurie) in terms of a strong descent property discovered by Rezk. All the machinery of universal algebra can be extended to quategories. An al- gebraic theory is defined to be a small quategory with finite products T , and a model of T to be a map T → K which preserves finite products. The models of T form a large quategory Model(T ) which is complete an cocomplete. A variety of homotopy algebras, or an homotopy variety is defined to be a quategory equivalent to a quategory Mod(T ) for some algebraic theory T . 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 the essentially algebraic structures defined by a limit sketch. For example, the notions of groupoid object and of category object in a category are essentially algebraic. The classical theory of limit sketches and of essentially algebraic structures is easily extended to quategories. A category object in a quategory X is defined to be a simplicial object C : ∆o → X satisfying the Segal condition. The theory of limit sketches is a natu- ral framework for studying homotopy coherent algebraic structures in general and higher weak categories in particular. The quategory of models of a limit sketch is locally presentable and conversely, every locally presentable quategory is equivalent to the quategory of models of a limit sketch. The theory of accessible categories and of locally presentable categories was extended to quategories by Lurie. A para-variety is defined to be a left exact reflection of a variety of homotopy algebras. For example, a homotopos is a para-variety. The quategories of spectra and of ring spectra are also examples. Para-varieties can be characterized by a system of axioms closely related to those of Vitale [Vi]. Factorisation systems are playing an important role in the theory of quategories. We introduce a general notion of homotopy factorisation system in a model category with examples in category theory, in classical homotopy theory and in the theory of quategories. A basic example is provided by the theory of Dwyer-Kan localisations. This is true also of the theory of prestacks. The theory of quategories can analyse phenomena which belong properly to homotopy theory. The notion of stable quategory is an example. The notion of meta-stable quategory introduced in the notes is another. We give a proof that the quategory of parametrized spectra is a homotopos (joint work with Georg Biedermann).
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