Emily Riehl and Dominic Verity

Transcription

1 Elements o -Category Theory Emily Riehl and Dominic Verity Department o Mathematics, Johns Hopkins University, altimore, MD 21218, US address: eriehl@math.jhu.edu Centre o ustralian Category Theory, Macquarie University, NSW 2109, ustralia address: dominic.verity@mq.edu.au

2 This text is a rapidly-evolving work in progress use at your own risk. The most recent version can always be ound here: eriehl/icwm.pd We would be delighted to hear about any comments, corrections, or conusions readers might have. Please send to: eriehl@math.jhu.edu Commenced on January 14, Last modiied on ugust 14, 2018.

3 Contents Preace ims o this text cknowledgments vii viii ix Part I. asic -category theory 1 Chapter 1. -Cosmoi and their homotopy 2-categories Quasi-categories Cosmoi Cosmological unctors The homotopy 2-category 26 Chapter 2. djunctions, limits, and colimits I djunctions and equivalences Initial and terminal elements Limits and colimits Preservation o limits and colimits 47 Chapter 3. Weak 2-limits in the homotopy 2-category Smothering unctors categories o arrows Pullbacks and limits o towers The comma construction Representable comma -categories Sliced homotopy 2-categories and ibered equivalences 78 Chapter 4. djunctions, limits, and colimits II The universal property o adjunctions categories o cones The universal property o limits and colimits Loops and suspension in pointed -categories 101 Chapter 5. Fibrations and Yoneda s lemma The 2-category theory o cartesian ibrations Cocartesian ibrations and biibrations The quasi-category theory o cartesian ibrations Discrete cartesian ibrations The external Yoneda lemma 139 iii

4 Part II. Homotopy coherent category theory 149 Chapter 6. Simplicial computads and homotopy coherence Simplicial computads Free resolutions and homotopy coherent simplices Homotopy coherent realization and the homotopy coherent nerve 161 Chapter 7. Weighted limits in -cosmoi Weighted limits and colimits Flexible weighted limits and the collage construction Homotopical properties o lexible weighed limits More -cosmoi Weak 2-limits revisited 197 Chapter 8. Homotopy coherent adjunctions and monads The ree homotopy coherent adjunction Homotopy coherent adjunction data uilding homotopy coherent adjunctions Homotopical uniqueness o homotopy coherent adjunctions 223 Chapter 9. The ormal theory o homotopy coherent monads Homotopy coherent monads Homotopy coherent algebras and the monadic adjunction Limits and colimits in the -category o algebras The monadicity theorem Monadic descent Homotopy coherent monad maps 260 Part III. The calculus o modules 267 Chapter 10. Two-sided ibrations Four equivalent deinitions o two-sided ibrations The -cosmos o two-sided ibrations Representable two-sided ibrations and the Yoneda lemma Modules as discrete two-sided ibrations 284 Chapter 11. The calculus o modules The double category o two-sided ibrations The virtual equipment o modules Composition o modules Representable modules 307 Chapter 12. Formal category theory in a virtual equipment Litings and extensions o modules Exact squares Pointwise right and let extensions Formal category theory in a virtual equipment Limits and colimits in cartesian closed -cosmoi 331 iv

5 Part IV. Change o model and model independence 337 Chapter 13. Cosmological biequivalences Cosmological unctors iequivalences o -cosmoi as change-o-model unctors Properties o change-o-model unctors 348 Chapter 14. Proo o model independence d hoc model invariance The context or the model independence theorem biequivalence o virtual equipments 357 ppendix o bstract Nonsense 361 ppendix. asic concepts o enriched category theory Cartesian closed categories Enriched categories Enriched natural transormations and the enriched Yoneda lemma Tensors and cotensors Conical limits Change o base 380 ppendix. n introduction to 2-category theory categories and the calculus o pasting diagrams The 3-category o 2-categories djunctions and mates Right adjoint right inverse adjunctions bestiary o 2-categorical lemmas Representable characterizations o 2-categorical notions 402 ppendix C. bstract homotopy theory 407 C.1. bstract homotopy theory in a category o ibrant objects 407 C.2. Liting properties, weak actorization systems, and Leibniz closure 415 C.3. Model categories and Quillen unctors 422 C.4. Reedy categories and canonical presentations 427 C.5. The Reedy model structure 433 ppendix o Semantic Considerations 441 ppendix D. The combinatorics o simplicial sets 443 D.1. Simplicial sets and markings 443 ppendix E. -cosmoi ound in nature 445 E.1. Quasi-categorically enriched model categories 445 E.2. -cosmoi o (, 1)-categories 448 E.3. -cosmoi o (, n)-categories 455 E.4. Other examples 462 E.5. -cosmoi with non-coibrant objects 462 v

6 ppendix F. Compatibility with the analytic theory o quasi-categories 463 F.1. Initial and terminal elements 463 F.2. Limits and colimits 466 F.3. Right adjoint right inverse adjunctions 467 F.4. Cartesian and cocartesian ibrations 469 F.5. djunctions 475 ibliography 481 vi

7 Preace Mathematical objects o a certain sophistication are requently accompanied by higher homotopical structures in the sense that the maps between them might be connected by homotopies, which might then be connected by higher homotopies, which might then be connected by even higher homotopies ad ininitum. In such contexts, the natural habitat or these mathematical objects is not an ordinary 1-category but instead an -category or, more precisely, an (, 1)-category, with the index 1 reerring to the act that its morphisms above the lowest dimension 1, the homotopies just discussed, are invertible. Here the homotopies deining the higher morphisms o an -category are to be regarded as data rather than mere witnesses to an equivalence relation borne by the 1-dimensional morphisms, which has the consequence that all o the categorical structures in an -category are weak. Even at the level o 1-morphisms, composition is not necessarily uniquely deined but instead witnessed by a 2-morphism and associative up to a 3-morphism whose boundary data involves speciied 2-morphism witnesses. Thus, diagrams valued in an -category cannot be said to commute on the nose but are instead interpreted as homotopy coherent. undamental challenge in deining -categories has to do with giving a precise mathematical meaning o this notion o a weak composition law, not just or the 1-morphisms but also or the morphisms in higher dimensions. This is achieved through a variety o models o (, 1)-categories, which are ourbaki-style mathematical structures that encode ininite-dimensional categories with all morphisms above dimension 1 invertible that satisy a weak composition law. In order o appearance, these include simplicial categories, quasi-categories (nee. weak Kan complexes), relative categories, Segal categories, complete Segal spaces, and 1-complicial sets (nee. saturated 1-trivial weak complicial sets), each o which comes with an associated array o naturally-occurring examples. The prolieration o models o (, 1)-categories begs the question o how they might be compared. In the irst decades o the 21st century, ergner, Joyal Tierney, Verity, Lurie, and arwick Kan built various bridges that prove that each o the models listed above has the same homotopy theory in the sense o deining the objects Quillen equivalent model categories.¹ In parallel with the development o models o (, 1)-categories and the construction o comparisons between them, Joyal pioneered and Lurie and many others extended a wildly successul project to extend basic category theory rom ordinary 1-categories to (, 1)-categories modeled as quasicategories in such a way that the new quasi-categorical notions restrict along the standard embedding Cat QCat to the standard 1-categorical notions. natural question is then: does this work extend to other models o (, 1)-categories? nd to what extent are basic -categorical notions invariant under change o model? For practical, aesthetic, and moral reasons, the ultimate desire o practitioners is to work model independently, meaning that theorems proven with any o the models o (, 1)-categories would ¹ recent book by ergner surveys all but the last o these models and their interrelationships [10]. For a more whirlwind tour, see [19]. vii

8 apply to them all, with the technical details inherent to any particular model never entering the discussion. Since all models o (, 1)-categories have the same homotopy theory the general consensus is that the choice o model should not matter greatly, but one obstacle to proving results o this kind is that, to a large extent, precise versions o the categorical deinitions that have been established or quasi-categories had not been given or the other models. In cases where comparable deinitions do exist in dierent models, an ad-hoc heuristic proo o model-invariance o the categorical notion in question can typically be supplied, with details to be illed in by experts luent in the combinatorics o each model, but it would be more reassuring to have a systematic method o comparing the category theory o (, 1)-categories in dierent models via arguments that are somewhat closer to the ground. ims o this text In this text we develop the theory o -categories rom irst principles in a model-independent ashion using a common axiomatic ramework that is satisied by a variety o models. In contrast with prior analytic treatments o the theory o -categories in which the central categorical notions are deined in reerence to the combinatorics o a particular model our approach is synthetic, proceeding rom deinitions that can be interpreted simultaneously in many models to which our proos then apply. While synthetic, our work is not schematic or hand-wavy, with the details o how to make things ully precise let to the experts and turtles all the way down.² Rather, we prove our theorems starting rom a short list o clearly-enumerated axioms, and our conclusions are valid in any model o -categories satisying these axioms. The synthetic theory is developed in any -cosmos, which axiomatizes the universe in which -categories live as objects. So that our theorem statements suggest their natural interpretation, we recast -category as a technical term, to mean an object in some (typically ixed) -cosmos. Several models o (, 1)-categories³ are -categories in this sense, but our -categories also include certain models o (, n)-categories⁴ as well as ibered versions o all o the above. This usage is meant to interpolate between the classical one, which reers to any variety o weak ininite-dimensional categories, and the common one, which is oten taken to mean quasi-categories or complete Segal spaces. Much o the development o the theory o -categories takes place not in the ull -cosmos but in a quotient that we call the homotopy 2-category, the name chosen because an -cosmos is something like a category o ibrant objects in an enriched model category and the homotopy 2-category is then a categoriication o its homotopy category. The homotopy 2-category is a strict 2-category like the 2-category o categories, unctors, and natural transormations⁵ and in this way the oundational proos in the theory o -categories closely resemble the classical oundations o ordinary category ² less rigorous model-independent presentation o -category theory might conront a problem o ininite regress, since ininite-dimensional categories are themselves the objects o an ambient ininite-dimensional category, and in developing the theory o the ormer one is tempted to use the theory o the latter. We avoid this problem by using a very concrete model or the ambient (, 2)-category o -categories that arises requently in practice and is designed to acilitate relatively simple proos. While the theory o (, 2)-categories remains in its inancy, we are content to cut the Gordian knot in this way. ³Quasi-categories, complete Segal spaces, Segal categories, and 1-complicial sets (naturally marked quasi-categories) all deine the -categories in an -cosmos. ⁴Θ n -spaces, iterated complete Segal spaces, and n-complicial sets also deine the -categories in an -cosmos, as do (nee. weak) complicial sets, a model or (, )-categories. We hope to add other models o (, n)-categories to this list. ⁵In act this is another special case: there is an -cosmos whose objects are ordinary categories and its homotopy 2-category is the usual category o categories, unctors, and natural transormations. viii

9 theory except that the universal properties that characterize, e.g. when a unctor between -categories deines a cartesian ibration, are slightly weaker than in the amiliar case. In Part I, we deine and develop the notions o equivalence and adjunction between -categories, limits and colimits in -categories, cartesian and cocartesian ibrations and their discrete variants, and prove an external version o the Yoneda lemma all rom the comort o the homotopy 2-category. In Part II, we turn our attention to homotopy coherent structures present in the ull -cosmos to deine and study homotopy coherent adjunctions and monads borne by -categories as a mechanism or universal algebra. What s missing rom this basic account o the category theory o -categories is a satisactory treatment o the hom biunctor associated to an -category, which is the prototypical example o what we call a module. In Part III, we develop the calculus o modules between -categories and apply this to deine and study pointwise Kan extensions. This will give us an opportunity to repackage universal properties proven in Part I as parts o the ormal category theory o -categories. This work is all model-agnostic in the sense o being blind to details about the speciications o any particular -cosmos. In Part IV we prove that the category theory o -categories is also model-independent in a precise sense: all categorical notions are preserved, relected, and created by any change-o-model unctor that deines what we call a biequivalence. This model-independence theorem is stronger than our axiomatic ramework might initially suggest in that it also allows us to transer theorems proven using analytic techniques to all biequivalent -cosmoi. For instance, the our -cosmoi whose objects model (, 1)-categories are all biequivalent. It ollows that the analytically-proven theorems about quasi-categories rom [52] transer to complete Segal spaces, and vice versa. The ideal reader might already have some acquaintance with enriched category theory, with 2-category theory, and with abstract homotopy theory so that the constructions and proos with antecedents in these traditions will be amiliar. ecause -categories are o interest to mathematicians with a wide variety o backgrounds, we review all o the material we need on each o these topics in ppendices,, and C respectively. Some basic acts about quasi-categories irst proven by Joyal are needed to establish the corresponding eatures o general -cosmoi in Chapter 1. We state all o these results in 1.1 but deer the proos that require a lengthy combinatorial digression to ppendix D, where we also review amiliar material about the category o simplicial sets. The proos that many examples o -cosmoi appear in the wild can be ound in ppendix E, where we also present general techniques that the reader might use to ind even more examples. The inal appendix addresses a crucial bit o uninished business. Importantly, the synthetic theory developed in the -cosmos o quasi-categories is ully compatible with the analytic theory developed by Joyal, Lurie, and many others. This is the subject o ppendix F. cknowledgments The irst drat o much o this material was written over the course o a semester long topics course taught at Johns Hopkins in the Spring o 2018 and beneitted considerably rom the perspicuous questions asked during lecture by Qingci n, Thomas razelton, Tslil Clingman, Daniel Fuentes- Keuthan, urel Malapani-Scala, Mona Merling, David Myers, purv Nakade, Martina Rovelli, and Xiyuan Wang. Further revisions were made during the 2018 MIT Talbot Workshop on the model-independent theory o -categories, organized by Eva elmont, Calista ernard, Inbar Klang, Morgan Opie, and Sean Pohorence, with Haynes Miller serving as the aculty sponsor. We were inspired by compelling ix

10 lectures given at that workshop by Timothy Campion, Kevin Carlson, Kyle Ferendo, Daniel Fuentes- Keuthan, Joseph Heler, Paul Lessard, Lyne Moser, Emma Phillips, Nima Rasekh, Martina Rovelli, Maru Sarazola, Matthew Weatherley, Jonathan Weinberger, Laura Wells, and Liang Ze Wong as well as by myriad discussions with Elena Dimitriadis ermejo, Luciana asualdo onatto, Olivia orghi, Tai-Danae radley, Tejas Devanur, ras Ergus, Matthew Feller, Sina Hazratpour, Peter James, Zhulin Li, David Myers, Maximilien Péroux, Mitchell Riley, Luis Scoccola, randon Shapiro, Pelle Steens, Raael Stenzel, Paula Verdugo, and Marco Vergura. Gabriel Drummond-Cole encouraged us to do some much needed restructuring o Chapter 5. Yuri Sulyma made a number o useul suggestions during the drating o 9.5, and 9.6 was written in heavy consultation with Dimitri Zaganidis who also suggested a conceptual simpliication to the geometric description o the proo o Proposition The presentation and organization in Chapter 12 was greatly improved by observations o Kevin Carlson and David Myers, who irst stated the result appearing as Proposition lexander Campbell was consulted several times during the writing o ppendix, in particular regarding the subtle interaction between change-o-enrichment unctors and underlying categories, and also alerted us to the -cosmos o n-quasi-categories appearing in Proposition E.3.3 and supplied its proo. Tslil Clingman suggested a better way to typeset proarrows in the virtual equipment, greatly improving the displayed diagrams in Chapter 11. lexander Campbell, Gabriel Drummond-Cole, Emma Phillips, Maru Sarazola, Paula Verdugo, and Jonathan Weinberger wrote to point out typos. Finally, the authors are grateul to inancial support provided by the National Science Foundation or support via grants DMS and DMS and the ustralian Research Council via the grant DP and to the Department o Mathematics at Johns Hopkins University and the Centre o ustralian Category Theory at Macquarie University or hosting our respective visits. x

11 Part I asic -category theory

12

13 CHPTER 1 -Cosmoi and their homotopy 2-categories 1.1. Quasi-categories eore introducing an axiomatic ramework that will allow us to develop -category theory in general, we irst consider one model in particular: namely, quasi-categories, which were irst analyzed by Joyal in [40] and [41] and in several unpublished drat book manuscripts Notation (the simplex category). Let Δ denote the simplex category o inite non-empty ordinals [n] = {0 < 1 < < n} and order-preserving maps. These include in particular the elementary ace operators [n 1] [n] 0 i n and the elementary degeneracy operators [n + 1] [n] 0 i n whose images respectively omit and double up on the element i [n]. Every morphism in Δ actors uniquely as an epimorphism ollowed by a monomorphism; these epimorphisms, the degeneracy operators, decompose as composites o elementary degeneracy operators, while the monomorphisms, the ace operators, decompose as composites o elementary ace operators. The category o simplicial sets is the category SSet Set Δop o presheaves on the simplex category. We write Δ[n] or the standard n-simplex, the simplicial set represented by [n] Δ, and Λ k [n] Δ[n] Δ[n] or its k-horn and boundary sphere respectively. Given a simplicial set X, it is conventional to write X n or the set o n-simplices, deined by evaluating at [n] Δ. y the Yoneda lemma, each n-simplex x X n corresponds to a map o simplicial sets x Δ[n] X. ccordingly, we write x δ i or the ith ace o the n-simplex, an (n 1)-simplex classiied by the composite map δ Δ[n 1] i x Δ[n] X. Geometrically, x δ i is the ace opposite the vertex i in the n-simplex x. Since the morphisms o Δ are generated by the elementary ace and degeneracy operators, the data o a simplicial set¹ X is oten presented by a diagram δ 3 σ 2 δ 2 X σ 3 1 X 2 δ 1 X σ 1 0 X 0, δ 1 σ 0 δ 0 δ i σ i δ 2 σ 1 δ 1 identiying the set o n-simplices or each [n] Δ as well as the (contravariant) actions o the elementary operators, conventionally denoted using subscripts. ¹This presentation is also used or more general simplicial objects valued in any category. 3 σ 0 δ 0 δ 0

14 Deinition (quasi-category). quasi-category is a simplicial set in which any inner horn can be extended to a simplex, solving the displayed liting problem: Λ k [n] n 2, 0 < k < n. (1.1.3) Δ[n] Quasi-categories were irst introduced by oardman and Vogt [14] under the name weak Kan complexes, a Kan complex being a simplicial set admitting extensions as in (1.1.3) along all horn inclusions n 1, 0 k n. Since any topological space can be encoded as a Kan complex,² in this way spaces provide examples o quasi-categories. Categories also provide examples o quasi-categories via the nerve construction Deinition (nerve). The category Cat o 1-categories embeds ully aithully into the category o simplicial sets via the nerve unctor. n n-simplex in the nerve o a 1-category C is a sequence o n composable arrows in C, or equally a unctor [n] C rom the ordinal category n + 1 [n] with objects 0,, n and a unique arrow i j just when i j Remark. The nerve o a category C is 2-coskeletal as a simplicial set, meaning that every sphere Δ[n] C with n 3 is illed uniquely by an n-simplex in C (see Deinition D.1.1). This is because the simplices in dimension 3 and above witness the associativity o the composition o the path o composable arrows ound along their spine, the 1-skeletal simplicial subset ormed by the edges connecting adjacent vertices. In act, as suggested by the proo o the ollowing proposition, any simplicial set in which inner horns admit unique illers is isomorphic to the nerve o a 1-category; see Exercise 1.1.iii. We decline to introduce explicit notation or the nerve unctor, preerring instead to identiy 1-categories with their nerves. s we shall discover the theory o 1-categories extends to -categories modeled as quasi-categories in such a way that the restriction o each -categorical concept along the nerve embedding recovers the corresponding 1-categorical concept. For instance, the standard simplex Δ[n] is the nerve o the ordinal category n + 1, and we requently adopt the latter notation writing 1 Δ[0], 2 Δ[1], 3 Δ[2], and so on to suggest the correct categorical intuition. To begin down this path, we must irst veriy the assertion that has implicitly just been made: Proposition (nerves are quasi-categories). Nerves o categories are quasi-categories. Proo. Via the isomorphism C cosk 2 C and the adjunction sk 2 cosk 2 o D.1.1, the required liting problem displayed below-let transposes to the one displayed below-right Λ k [n] C cosk 2 C sk 2 Λ k [n] C Δ[n] sk 2 Δ[n] ²The total singular complex construction deines a unctor rom topological spaces to simplicial sets that is an equivalence on their respective homotopy categories weak homotopy types o spaces correspond to homotopy equivalence classes o Kan complexes. 4

15 For n 4, the inclusion sk 2 Λ k [n] sk 2 Δ[n] is an isomorphism, in which case the liting problems on the right admit (unique) solutions. So it remains only to solve the liting problems on the let in the cases n = 2 and n = 3. To that end consider Λ 1 [2] C Λ 1 [3] C Λ 2 [3] C Δ[2] Δ[3] Δ[3] n inner horn Λ 1 [2] C deines a composable pair o arrows in C; an extension to a 2-simplex exists precisely because any composable pair o arrows admits a (unique) composite. n inner horn Λ 1 [3] C speciies the data o three composable arrows in C, as displayed in the diagram below, together with the composites g, hg, and (hg). c 1 hg c 0 c 3 g g c 2 ecause composition is associative, the arrow (hg) is also the composite o g ollowed by h, which proves that the 2-simplex opposite the vertex c 1 is present in C; by 2-coskeletality, the 3-simplex illing this boundary sphere is also present in C. The iller or a horn Λ 2 [3] C is constructed similarly Deinition (homotopy relation on 1-simplices). parallel pair o 1-simplices, g in a simplicial set X are homotopic i there exists a 2-simplex o either o the ollowing orms (hg) x 1 x 0 x 0 x 1 x 0 x 1 or i and g are in the same equivalence class generated by this relation. g h g (1.1.8) In a quasi-category, the relation witnessed by any o the types o 2-simplex on display in (1.1.8) is an equivalence relation and these equivalence relations coincide: Lemma (homotopic 1-simplices in a quasi-category). Parallel 1-simplices and g in a quasi-category are homotopic i and only i there exists a 2-simplex o any or equivalently all o the orms displayed in (1.1.8). Proo. Exercise 1.1.i Deinition (the homotopy category). y 1-truncating, any simplicial set X has an underlying relexive directed graph δ 1 X σ 1 0 X 0, δ 0 the 0-simplices o X deining the objects and the 1-simplices deining the arrows, by convention pointing rom their 0th vertex (the ace opposite 1) to their 1st vertex (the ace opposite 0). The ree 5

16 category on this relexive directed graph has X 0 as its object set, degenerate 1-simplices serving as identity morphisms, and non-identity morphisms deined to be inite directed paths o non-degenerate 1-simplices. The homotopy category hx o X is the quotient o the ree category on its underlying relexive directed graph by the congruence³ generated by imposing a composition relation h = g witnessed by 2-simplices x 1 x 0 x h 2 This implies in particular that homotopic 1-simplices represent the same arrow in the homotopy category Proposition. The nerve embedding admits a let adjoint, namely the unctor which sends a simplicial set to its homotopy category: Cat h g SSet Proo. Using the description o hx as a quotient o the ree category on the underlying relexive directed graph o X, we argue that the data o a unctor hx C can be extended uniquely to a simplicial map X C. Presented as a quotient in this way, the unctor hx C deines a map rom the 1-skeleton o X into C, and since every 2-simplex in X witnesses a composite in hx, this map extends to the 2-skeleton. Now C is 2-coskeletal, so via the adjunction sk 2 cosk 2 o Deinition D.1.1, this map rom the 2-truncation o X into C extends uniquely to a simplicial map X C. The homotopy category o a quasi-category admits a simpliied description Lemma (the homotopy category o a quasi-category). I is a quasi-category then its homotopy category h has the set o 0-simplices 0 as its objects the set o homotopy classes o 1-simplices 1 as its arrows the identity arrow at a 0 represented by the degenerate 1-simplex a σ 0 1 a composition relation h = g in h i and only i, or any choices o 1-simplices representing these arrows, there exists a 2-simplex with boundary a 1 g Proo. Exercise 1.1.ii. a 0 a 2 h Deinition (isomorphisms in a quasi-category). 1-simplex in a quasi-category is an isomorphism just when it represents an isomorphism in the homotopy category. y Lemma this means that a b is an isomorphism i and only i there exist a 1-simplex 1 b a together with a pair o 2-simplices b a 1 a a b b ³ relation on parallel pairs o arrows o a 1-category is a congruence i it is an equivalence relation that is closed under pre- and post-composition: i g then hk hgk. 6 1

17 The properties o the isomorphisms in a quasi-category are most easily proved by arguing in a slightly dierent category where simplicial sets have the additional structure o a marking on a speciied subset o the 1-simplices subject to the condition that all degenerate 1-simplices are marked; maps o these so-called marked simplicial sets must then preserve the markings. ecause these objects will seldom appear outside o the proos o certain combinatorial lemmas about the isomorphisms in quasi-categories, we save the details or ppendix D. Let us now motivate the irst o several results proven using marked techniques. Quasi-categories are deined to have extensions along all inner horns. ut i in an outer horn Λ 0 [2] or Λ 2 [2], the initial or inal edges, respectively, are isomorphisms, then intuitively a iller should exist a 1 a 1 h 1 a 0 a 2 a 0 a 2 and similarly or the higher-dimensional outer horns Proposition (special outer horn liting). (i) Let be a quasi-category. Then any outer horns g h g 1 h Λ 0 [n] Λ n [n] h g h Δ[n] Δ[n] in which the edges g {0,1} and h {n 1,n} are isomorphisms admit illers. (ii) Let and be quasi-categories and a map that lits against the inner horn inclusions. Then any outer horns g Λ 0 [n] Λ n [n] Δ[n] Δ[n] in which the edges g {0,1} and h {n 1,n} are isomorphisms admit illers. The proo o Proposition requires clever combinatorics, due to Joyal, and is deerred to ppendix D.⁴ Here, we enjoy its myriad consequences. Immediately: Corollary. quasi-category is a Kan complex i and only i its homotopy category is a groupoid. Proo. I the homotopy category o a quasi-category is a groupoid, then all o its 1-simplices are isomorphisms, and Proposition then implies that all inner and outer horns have illers. Thus, the quasi-category is a Kan complex. Conversely, in a Kan complex, all outer horns can be illed and in particular illers or the horns Λ 0 [2] and Λ 2 [2] can be used to construct let and right inverses or any 1-simplex o the orm displayed in Deinition ⁵ ⁴The second statement subsumes the irst, but the irst is typically used to prove the second. ⁵In a quasi-category, any let and right inverses to a common 1-simplex are homotopic, but as Corollary proves, any isomorphism in act has a single two-sided inverse. h 7

18 quasi-category contains a canonical maximal sub Kan complex, the simplicial subset spanned by those 1-simplices that are isomorphisms. Just as the arrows in a quasi-category are represented by simplicial maps 2 whose domain is the nerve o the ree-living arrow, the isomorphisms in a quasi-category are represented by diagrams I whose domain is the ree-living isomorphism: Corollary. n arrow in a quasi-category is an isomorphism i and only i it extends to a homotopy coherent isomorphism 2 I Proo. I is an isomorphism, the map 2 lands in the maximal sub Kan complex contained in. The postulated extension also lands in this maximal sub Kan complex because the inclusion 2 I can be expressed as a sequential composite o outer horn inclusions; see Exercise 1.1.iv. The category o simplicial sets, like any category o presheaves, is cartesian closed. y the Yoneda lemma and the deining adjunction, an n-simplex in the exponential Y X corresponds to a simplicial map X Δ[n] Y, and its aces and degeneracies are computed by precomposing in the simplex variable. Our aim is now to show that the quasi-categories deine an exponential ideal in the simplicially enriched category o simplicial sets: i X is a simplicial set and is a quasi-category, then X is a quasi-category. We will deduce this as a corollary o the relative version o this result involving a class o maps called isoibrations that we now introduce Deinition (isoibrations between quasi-categories). simplicial map is a isoibration i it lits against the inner horn inclusions, as displayed below let, and also against the inclusion o either vertex into the ree-standing isomorphism I. Λ k [n] 1 Δ[n] I To notationally distinguish the isoibrations, we depict them as arrows with two heads Observation. (i) For any simplicial set X, the unique map X whose codomain is the terminal simplicial set is an isoibration i and only i X is a quasi-category. (ii) ny class o maps characterized by a right liting property is automatically closed under composition, product, pullback, retract, and limits o towers; see Lemma C.2.3. (iii) Combining (i) and (ii), i is an isoibration, and is a quasi-category, then so is. (iv) The isoibrations generalize the eponymous categorical notion. The nerve o any unctor between categories deines a map o simplicial sets that lits against the inner horn inclusions. This map then deines an isoibration i and only i given any isomorphism in and speciied object in liting either its domain or codomain, there exists an isomorphism in with that domain or codomain liting the isomorphism in. We typically only deploy the term isoibration or a map between quasi-categories because our usage o this class o maps intentionally parallels the classical categorical case. 8

19 Much harder to establish is the stability o the class o isoibrations under orming Leibniz exponentials as displayed in (1.1.20). The proo o this result is given in Proposition?? in ppendix D Proposition. I i X Y is a monomorphism and is an isoibration, then the induced Leibniz exponential map Y i i Y X X (1.1.20) is again an isoibration.⁶ Y Corollary. I X is a simplicial set and is a quasi-category, then X is a quasi-category. Moreover, a 1-simplex in X is an isomorphism i and only i its components at each vertex o X are isomorphisms in. Proo. The irst statement is a special case o Proposition ; see Exercise 1.1.vi. The second statement is proven similarly by arguing with marked simplicial sets. See Lemma?? Deinition (equivalences o quasi-categories). map between quasi-categories is an equivalence i it extends to the data o a homotopy equivalence with the ree-living isomorphism I serving as the interval: that is, i there exist maps g and i X I I g α ev 0 ev 1 We write to decorate equivalences and to indicate the presence o an equivalence Remark. I is an equivalence o quasi-categories, then the unctor h h h is an equivalence o categories, with equivalence inverse hg h h and natural isomorphisms encoded by the composite unctors hα h h( I ) (h) I hβ h h( I ) (h) I Deinition. map X Y between simplicial sets is a trivial ibration i it admits lits against the boundary inclusions or all simplices g Δ[n] X n 0 β ev 0 ev 1 (1.1.25) Δ[n] Y ⁶Degenerate cases o this result, taking X = or = 1, imply that the other six maps in this diagram are also isoibrations; see Exercise 1.1.vi. 9

20 We write to decorate trivial ibrations Remark. The simplex boundary inclusions Δ[n] Δ[n] cellularly generate the monomorphisms o simplicial sets see Deinition C.2.4 and Lemma D.1.2. Hence the dual o Lemma C.2.3 implies that trivial ibrations lit against any monomorphism between simplicial sets. In particular, applying this to the map Y, it ollows that any trivial ibration X Y is a split epimorphism. The notation is suggestive: the trivial ibrations between quasi-categories are exactly those maps that are both isoibrations and equivalences. This can be proven by a relatively standard although rather technical argument in simplicial homotopy theory, given as Proposition?? in ppendix D Proposition. For a map between quasi-categories the ollowing are equivalent: (i) is at trivial ibration (ii) is both an isoibration and an equivalence (iii) is a split iber homotopy equivalence: an isoibration admitting a section s that is also an equivalence inverse via a homotopy rom s to 1 that composes with to the constant homotopy rom to. s a class characterized by a right liting property, the trivial ibrations are also closed under composition, product, pullback, limits o towers, and contain the isomorphisms. The stability o these maps under Leibniz exponentiation will be veriied along with Proposition in Proposition?? Proposition. I i X Y is a monomorphism and is an isoibration, then i either is a trivial ibration or i i is in the class cellularly generated⁷ by the inner horn inclusions and the map 1 I then the induced Leibniz exponential map a trivial ibration. Y i Y X X Digression (the Joyal model structure). The category o simplicial set bears a model structure (see ppendix D) whose ibrant objects are exactly the quasi-categories; all objects are coibrant. The ibrations, weak equivalences, and trivial ibrations between ibrant objects are precisely the classes o isoibrations, equivalences, and trivial ibrations deined above. Proposition proves that the trivial ibrations are the intersection o the classes o ibrations and weak equivalences. Propositions and relect the act that the Joyal model structure is a closed monoidal model category with respect to the cartesian closed structure on the category o simplicial sets. We have declined to elaborate on the Joyal model structure or quasi-categories alluded to in Digression because the only aspects o it that we will need are those described above. The results proven here suice to show that the category o quasi-categories deines an -cosmos, a concept to which we now turn. Exercises. 1.1.i. Exercise. Consider the set o 1-simplices in a quasi-category with initial vertex a 0 and inal vertex a 1. ⁷See Deinition C

21 (i) Prove that the relation deined by g i and only i there exists a 2-simplex with boundary a 1 is an equivalence relation. a 0 g a 1 (ii) Prove that the relation deined by g i and only i there exists a 2-simplex with boundary a 0 is an equivalence relation. a 0 g a 1 (iii) Prove that the equivalence relations deined by (i) and (ii) are the same. This proves Lemma ii. Exercise. Consider the ree category on the relexive directed graph δ 1 σ 1 0 0, δ 0 underlying a quasi-category. (i) Consider the relation that identiies a pair o sequences o composable 1-simplices with common source and common target whenever there exists a simplex o in which the sequences o 1-simplices deine two paths rom its initial vertex to its inal vertex. Prove that this relation is stable under pre- and post-composition with 1-simplices and conclude that its transitive closure is a congruence: an equivalence relation that is closed under pre- and post-composition.⁸ (ii) Consider the congruence relation generated by imposing a composition relation h = g witnessed by 2-simplices a 1 a 0 a h 2 and prove that this coincides with the relation considered in (i). (iii) In the congruence relations o (i) and (ii), prove that every sequence o composable 1-simplices in is equivalent to a single 1-simplex. Conclude that every morphism in the quotient o the ree category by this congruence relation is represented by a 1-simplex in. (iv) Prove that or any triple o 1-simplices, g, h in, h = g in the quotient category i and only i there exists a 2-simplex with boundary g a 1 g This proves Lemma a 0 a 2 h 1.1.iii. Exercise. Show that any quasi-category in which inner horns admit unique illers is isomorphic to the nerve o its homotopy category. 1.1.iv. Exercise. ⁸Given a congruence relation on the hom-sets o a 1-category, the quotient category can be ormed by quotienting each hom-set; see [55, II.8]. 11

22 (i) Prove that I contains exactly two non-degenerate simplices in each dimension. (ii) Inductively build I rom 2 by expressing the inclusion 2 I as a sequential composite o pushouts o outer horn inclusions⁹ Λ 0 [n] Δ[n], one in each dimension starting with n = 2.¹⁰ 1.1.v. Exercise. Prove the relative version o Corollary : or any isoibration p between quasi-categories and any isomorphism 2 any homotopy coherent isomorphism in extending p lits to a homotopy coherent isomorphism in extending. 2 p I 1.1.vi. Exercise. Specialize Proposition to prove the ollowing: (i) I is a quasi-category and X is a simplicial set then X is a quasi-category. (ii) I is a quasi-category and X Y is a monomorphism then Y X is an isoibration. (iii) I is an isoibration and X is a simplicial set then X X is an isoibration. 1.1.vii. Exercise. nticipating Lemma : (i) Prove that the equivalences deined in Deinition are closed under retracts. (ii) Prove that the equivalences deined in Deinition satisy the 2-o-3 property. 1.1.viii. Exercise. Prove that i X Y is a trivial ibration between quasi-categories then the unctor h hx hy is a surjective equivalence o categories Cosmoi In 1.1, we presented analytic proos o a ew o the basic acts about quasi-categories. The category theory o quasi-categories can be developed in a similar style, but we aim instead to develop the synthetic theory o ininite-dimensional categories, so that our results will apply to many models at once. To achieve this, our strategy is not to axiomatize what these ininite-dimensional categories are, but rather axiomatize the universe in which they live. The ollowing deinition abstracts the properties o the quasi-categories and the classes o isoibrations, equivalences, and trivial ibrations introduced in 1.1. Firstly, the category o quasi-categories and simplicial maps is enriched over the category o simplicial sets the set o morphisms rom to coincides with the set o vertices o the simplicial set and moreover these hom-spaces are all quasi-categories. Secondly, a number o limit constructions that can be deined in the underlying 1-category o quasi-categories and simplicial maps satisy universal properties relative to this simplicial enrichment, with the usual isomorphism o sets extending to an isomorphism o simplicial sets. nd inally, the classes o isoibrations, equivalences, and trivial ibrations satisy properties that are ⁹y duality the opposite o a simplicial set X is the simplicial set obtained by reindexing along the involution ( ) op Δ Δ that reverses the ordering in each ordinal the outer horn inclusions Λ n [n] Δ[n] can be used instead. ¹⁰This decomposition o the inclusion 2 I reveals which data can always be extended to a homotopy coherent isomorphism: or instance, the 1- and 2-simplices o Deinition together with a single 3-simplex that has these as its outer aces with its inner aces degenerate. 12

23 amiliar rom abstract homotopy theory. In particular, the use o isoibrations in diagrams guarantees that their strict limits are equivalence invariant, so we can take advantage o up-to-isomorphism universal properties and strict unctoriality o these constructions while still working homotopically. s will be explained in Digression , there are a variety o models o ininite-dimensional categories or which the category o -categories, as we will call them, and -unctors between them is enriched over quasi-categories and admits classes o isoibrations, equivalences, and trivial ibrations satisying analogous properties. This motivates the ollowing axiomatization: Deinition ( -cosmoi). n -cosmos K is a category enriched whose objects, we call -categories and whose morphisms we call -unctors that is enriched over quasicategories,¹¹ meaning in particular that its morphisms deine the vertices o unctor-spaces Fun(, ), which are quasicategories, that is also equipped with a speciied class o maps that we call isoibrations and denote by and satisies the ollowing two axioms: (i) (completeness) The quasi-categorically enriched category K possesses a terminal object, small products, pullbacks o isoibrations, limits o countable towers o isoibrations, and cotensors with all simplicial sets, each o these limit notions satisying a universal property that is enriched over simplicial sets.¹² (ii) (isoibrations) The class o isoibrations contains all isomorphisms and any map whose codomain is the terminal object; is closed under composition, product, pullback, orming inverse limits o towers, and Leibniz cotensors with monomorphisms o simplicial sets; and has the property that i is an isoibration and X is any object then Fun(X, ) Fun(X, ) is an isoibration o quasi-categories Deinition. In an -cosmos K, we deine a morphism to be an equivalence i and only i the induced map Fun(X, ) Fun(X, ) on unctor-spaces is an equivalence o quasi-categories or all X K, and a trivial ibration just when is both an isoibration and an equivalence. These classes are denoted by and respectively. Put more concisely, one might say that an -cosmos is a quasi-categorically enriched category o ibrant objects. See Deinition C.1.1 and Lemma C Digression (simplicial categories). simplicial category is given by categories n, with a common set o objects and whose arrows are called n-arrows, that assemble into a diagram Δ op Cat o identity-on-objects unctors δ 3 σ 2 δ 2 σ δ 1 σ 1 0 0, (1.2.4) δ 1 σ 0 δ 0 δ 2 σ 1 δ 1 σ 0 δ 0 ¹¹This is to say K is a simplicially enriched category whose hom-spaces are all quasi-categories; this will be unpacked in ¹²This will be elaborated upon in δ 0 13

24 The data o a simplicial category can equivalently be encoded by a simplicially enriched category with a set o objects and a simplicial set (x, y) o morphisms between each ordered pair o objects: an n-arrow in n rom x to y corresponds to an n-simplex in (x, y) (see Exercise 1.2.i). Each endo-hom-space contains a distinguished identity 0-arrow (the degenerate images o which deine the corresponding identity n-arrows) and composition is required to deine a simplicial map (y, z) (x, y) (x, z) the single map encoding the compositions in each o the categories n and also the unctoriality o the diagram (1.2.4). The composition is required to be associative and unital, in a sense expressed by the commutative diagrams (y, z) (x, y) (w, x) 1 (y, z) (w, y) 1 (x, z) (w, x) (w, z) (x, y) 1 id x (x, y) (x, x) id y 1 (y, y) (x, y) (x, y) the latter making use o natural isomorphisms (x, y) 1 (x, y) 1 (x, y) in the domain vertex. On account o the equivalence between these two presentations, the terms simplicial category and simplicially-enriched category are generally taken to be synonyms.¹³ The category 0 o 0-arrows is the underlying category o the simplicial category, which orgets the higher dimensional simplicial structure. In particular, the underlying category o an -cosmos K is the category whose objects are the -categories in K and whose morphisms are the 0-arrows in the unctor spaces. In all o the examples to appear below, this recovers the expected category o -categories in a particular model and unctors between them Digression (simplicially enriched limits). Let be a simplicial category. The cotensor o an object by a simplicial set U is characterized by an isomorphism o simplicial sets (X, U ) (X, ) U (1.2.6) natural in X. ssuming such objects exist, the simplicial cotensor deines a biunctor SSet op (U, ) in a unique way making the isomorphism (1.2.6) natural in U and as well. The other simplicial limit notions postulated by axiom 1.2.1(i) are conical, which is the term used or ordinary 1-categorical limit shapes that satisy an enriched analog o the usual universal property; see Deinition When these limits exist they correspond to the usual limits in the underlying category, but the usual universal property is strengthened. pplying the covariant representable unctor (X, ) 0 SSet to a limit cone (lim j J j j ) j J in 0, there is natural comparison map U (X, lim j J j ) lim j J (X, j ) (1.2.7) ¹³The phrase simplicial object in Cat is reserved or the more general yet less common notion o a diagram Δ op Cat that is not necessarily comprised o identity-on-objects unctors. 14

25 and we say that lim j J j deines a simplicially enriched limit i and only i (1.2.7) is an isomorphism o simplicial sets or all X. Considerably more details on the general theory o enriched categories can be ound in [47] and in ppendix. Enriched limits are the subjects o.4 and Remark (lexible weighted limits in -cosmoi). The axiom 1.2.1(i) implies that any -cosmos K admits all lexible limits (see Corollary 7.3.3), a much larger class o simplicially enriched weighted limits that will be introduced in 7.2. Using the results o Joyal discussed in 1.1, we can easily veriy: Proposition. The ull subcategory QCat SSet o quasi-categories deines an -cosmos with the isoibrations, equivalences, and trivial ibrations o Deinitions , , and Proo. The subcategory QCat SSet inherits its simplicial enrichment rom the cartesian closed category o simplicial sets: note that or quasi-categories and, Fun(, ) is again a quasi-category. The limits postulated in 1.2.1(i) exist in the ambient category o simplicial sets.¹⁴ The deining universal property o the simplicial cotensor is satisied by the exponentials o simplicial sets. We now argue that the ull subcategory o quasi-categories inherits all these limit notions. Since the quasi-categories are characterized by a right liting property, it is clear that they are closed under small products. Similarly, since the class o isoibrations is characterized by a right liting property, Lemma C.2.3 implies that the isoibrations are closed under all o the limit constructions o 1.2.1(ii) except or the last two: Leibniz closure and closure under exponentiation ( ) X. These last closure properties are established in Proposition This completes the proo o 1.2.1(i) and 1.2.1(ii). It remains to veriy that the classes o trivial ibrations and o equivalences coincide with those deined by and y Proposition the ormer coincidence ollows rom the latter, so it remains only to show that the equivalences o coincide with the representably-deined equivalences: those maps o quasi-categories or which X X is an equivalence o quasi-categories in the sense o Taking X = Δ[0], we see immediately that representablydeined equivalences are equivalences, and the converse holds since the exponential ( ) X preserves the data deining a simplicial homotopy. We mention a common source o -cosmoi ound in nature at the outside to help ground the intuition or readers amiliar with Quillen s model categories, a popular ramework or abstract homotopy theory, but reassure others that model categories are not needed outside o ppendix E Digression (a source o -cosmoi in nature). s explained in ppendix E, certain easily described properties o a model category imply that the ull subcategory o ibrant objects deines an -cosmos whose isoibrations, equivalences, and trivial ibrations are the ibrations, weak equivalences, and trivial ibrations between ibrant objects. Namely, any model category that is enriched as such over the Joyal model structure on simplicial sets and with the property that all ibrant objects are coibrant has this property. This compatible enrichment in the Joyal model structure can be deined when the model category is cartesian closed and equipped with a right Quillen adjoint to the Joyal model structure on simplicial sets whose let adjoint preserves inite products. In this case, the right ¹⁴ny category o presheaves is cartesian closed, complete, and cocomplete a cosmos in the sense o énabou. Our -cosmoi are more similar to the ibrational cosmoi due to Street [78]. 15