infin-Categories for the Working Mathematician
Emily Riehl and Dominic Verity
Department of Mathematics Johns Hopkins University Baltimore MD 21218 USAEmail address eriehlmathjhuedu
Centre of Australian Category Theory Macquarie University NSW 2109 AustraliaEmail address dominicveritymqeduau
This text is a rapidly-evolving work in progress mdash use at your own risk The most recent version canalways be found here
wwwmathjhuedusimeriehlICWMpdfWe would be delighted to hear about any comments corrections or confusions readers might havePlease send to
eriehlmathjhuedu
Commenced on January 14 2018
Last modified on June 15 2018
Contents
Preface vii
Part I Basicinfin-category theory 1
Chapter 1 infin-Cosmoi and their homotopy 2-categories 311 Quasi-categories 312 infin-Cosmoi 1213 Cosmological functors 2214 The homotopy 2-category 24
Chapter 2 Adjunctions limits and colimits I 3121 Adjunctions and equivalences 3122 Initial and terminal elements 3723 Limits and colimits 3924 Preservation of limits and colimits 45
Chapter 3 Weak 2-limits in the homotopy 2-category 4931 Smothering functors 5032 infin-categories of arrows 5333 The comma construction 5834 Representable commainfin-categories 6435 Sliced homotopy 2-categories and fibered equivalences 73
Chapter 4 Adjunctions limits and colimits II 8141 The universal property of adjunctions 8142 infin-categories of cones 8443 The universal property of limits and colimits 8744 Loops and suspension in pointedinfin-categories 97
Chapter 5 Fibrations and Yonedarsquos lemma 10351 The 2-category theory of cartesian fibrations 10352 Cocartesian fibrations and bifibrations 12353 The quasi-category theory of cartesian fibrations 12554 Discrete cartesian fibrations 12955 The external Yoneda lemma 135
Part II Homotopy coherent category theory 145
Chapter 6 Simplicial computads and homotopy coherence 14761 Simplicial computads 148
iii
62 Free resolutions and homotopy coherent simplices 15363 Homotopy coherent realization and the homotopy coherent nerve 157
Chapter 7 Weighted limits ininfin-cosmoi 16571 Weighted limits and colimits 16572 Flexible weighted limits and the collage construction 17173 Homotopical properties of flexible weighed limits 17674 Moreinfin-cosmoi 18375 Weak 2-limits revisited 193
Chapter 8 Homotopy coherent adjunctions and monads 19781 The free homotopy coherent adjunction 19882 Homotopy coherent adjunction data 20983 Building homotopy coherent adjunctions 21684 Homotopical uniqueness of homotopy coherent adjunctions 219
Chapter 9 The formal theory of homotopy coherent monads 22791 Homotopy coherent monads 22792 Homotopy coherent algebras and the monadic adjunction 23093 Limits and colimits in theinfin-category of algebras 23694 The monadicity theorem 24095 Monadic descent 24996 Homotopy coherent monad maps 256
Part III The calculus of modules 263
Chapter 10 Two-sided fibrations 265101 Four equivalent definitions of two-sided fibrations 265102 Theinfin-cosmos of two-sided fibrations 272103 Representable two-sided fibrations and the Yoneda lemma 275104 Modules as discrete two-sided fibrations 280
Chapter 11 The calculus of modules 285111 The double category of two-sided fibrations 286112 The virtual equipment of modules 293113 Composition of modules 297114 Representable modules 303
Chapter 12 Formal category theory in a virtual equipment 311121 Exact squares 311122 Pointwise right and left extensions 316123 Formal category theory in a virtual equipment 321124 Limits and colimits in cartesian closedinfin-cosmoi 323
Part IV Change of model and model independence 329
Chapter 13 Cosmological biequivalences 331131 Cosmological functors 331
iv
132 Biequivalences ofinfin-cosmoi as change-of-model functors 336133 Properties of change-of-model functors 340134 Ad hoc model invariance 344
Chapter 14 Proof of model independence 347141 A biequivalence of virtual equipments 350
Appendix of Abstract Nonsense 353
Appendix A Basic concepts of enriched category theory 355A1 Cartesian closed categories 356A2 Enriched categories 359A3 Enriched natural transformations and the enriched Yoneda lemma 362A4 Tensors and cotensors 367A5 Conical limits 369A6 Change of base 372
Appendix B Introduction to 2-category theory 377B1 2-categories and the calculus of pasting diagrams 377B2 The 3-category of 2-categories 382B3 Adjunctions and mates 383B4 Right adjoint right inverse adjunctions 384B5 A bestiary of 2-categorical lemmas 384
Appendix C Abstract homotopy theory 385C1 Lifting properties weak factorization systems and Leibniz closure 385
Appendix of Semantic Considerations 387
Appendix D The combinatorics of simplicial sets 389D1 Simplicial sets and markings 389
Appendix E Examples ofinfin-cosmoi 391
Appendix F Compatibility with the analytic theory of quasi-categories 393
Bibliography 395
v
Preface
In this text we develop the theory of infin-categories from first principles in a model-independentfashion using a common axiomatic framework that is satisfied by a variety of models In contrast withprior ldquoanalyticrdquo treatments of the theory of infin-categories mdash in which the central categorical notionsare defined in reference to the combinatorics of a particular model mdash our approach is ldquosyntheticrdquoproceeding from definitions that can be interpreted simultaneously in many models to which ourproofs then apply While synthetic our work is not schematic or hand-wavy with the details of howto make things fully precise left to ldquothe expertsrdquo and turtles all the way downsup1 Rather we prove ourtheorems starting from a short list of clearly-enumerated axioms and our conclusions are valid in anymodel ofinfin-categories satisfying these axioms
The synthetic theory is developed in any infin-cosmos which axiomatizes the universe in whichinfin-categories live as objects So that our theorem statements suggest their natural interpretation werecast infin-category as a technical term to mean an object in some (typically fixed) infin-cosmos Severalmodels of (infin 1)-categoriessup2 areinfin-categories in this sense but ourinfin-categories also include certainmodels of (infin 119899)-categoriessup3 as well as fibered versions of all of the above This usage is meant to in-terpolate between the classical one which refers to any variety of weak infinite-dimensional categoryand the common one which is often taken to mean quasi-categories or complete Segal spaces
Much of the development of the theory ofinfin-categories takes place not in the fullinfin-cosmos butin a quotient that we call the homotopy 2-category the name chosen because an infin-cosmos is some-thing like a category of fibrant objects in an enriched model category and the homotopy 2-categoryis then a categorification of its homotopy category The homotopy 2-category is a strict 2-category mdashlike the 2-category of categories functors and natural transformations mdash and in this way the foun-dational proofs in the theory of infin-categories closely resemble the classical foundations of ordinarycategory theory except that the universal properties that characterize eg when a functor betweeninfin-categories defines a cartesian fibration are slightly weaker than in the familiar case In Part Iwe define and develop the notions of equivalence and adjunction between infin-categories limits andcolimits in infin-categories cartesian and cocartesian fibrations and their discrete variants and provean external version of the Yoneda lemma all from the comfort of the homotopy 2-category In PartII we turn our attention to homotopy coherent structures present in the full infin-cosmos to define
sup1A less rigorous ldquomodel-independentrdquo presentation ofinfin-category theory might confront a problem of infinite regresssince infinite-dimensional categories are themselves the objects of an ambient infinite-dimensional category and in de-veloping the theory of the former one is tempted to use the theory of the latter We avoid this problem by using a veryconcrete model for the ambient (infin 2)-category of infin-categories that arises frequently in practice and is designed to fa-cilitate relatively simple proofs While the theory of (infin 2)-categories remains in its infancy we are content to cut theGordian knot in this way
sup2 (naturally marked quasi-categories) all define theinfin-categories in aninfin-cosmossup3Θ119899-spaces iterated complete Segal spaces and 119899-complicial sets also define theinfin-categories in aninfin-cosmos as do
(nee weak) complicial sets a model for (infininfin)-categories We hope to add other models of (infin 119899)-categories to this list
vii
and study homotopy coherent adjunctions and monads borne by infin-categories as a mechanism foruniversal algebra
Whatrsquos missing from this basic account of the category theory of infin-categories is a satisfactorytreatment of the ldquohomrdquo bifunctor associated to an infin-category which is the prototypical example ofwhat we call a module In Part III we develop the calculus of modules betweeninfin-categories and applythis to define and study pointwise Kan extensions This will give us an opportunity to repackageuniversal properties proven in Part I as parts of the ldquoformal category theoryrdquo ofinfin-categories
This work is all ldquomodel-agnosticrdquo in the sense of being blind to details about the specificationsof any particular infin-cosmos In Part IV we prove that the category theory of infin-categories is alsoldquomodel-independentrdquo in a precise sense all categorical notions are preserved reflected and createdby any ldquochange-of-modelrdquo functor that defines what we call a biequivalence This model-independencetheorem is stronger than our axiomatic framework might initially suggest in that it also allows usto transfer theorems proven using ldquoanalyticrdquo techniques to all biequivalent infin-cosmoi For instancethe four infin-cosmoi whose objects model (infin 1)-categories are all biequivalent It follows that theanalytically-proven theorems about quasi-categories⁴ from [36] transfer to complete Segal spaces andvice versa
⁴Importantly the synthetic theory developed in theinfin-cosmos of quasi-categories is fully compatible with the analytictheory developed by Joyal Lurie and many others This is the subject of Appendix F
viii
Part I
Basicinfin-category theory
CHAPTER 1
infin-Cosmoi and their homotopy 2-categories
11 Quasi-categories
Before introducing an axiomatic framework that will allow us to develop infin-category theory ingeneral we first consider one model in particular namely quasi-categories which were first analyzedby Joyal in [24] and [25] and in several unpublished draft book manuscripts
111 Notation (the simplex category) Let 120491 denote the simplex category of finite non-empty ordi-nals [119899] = 0 lt 1 lt ⋯ lt 119899 and order-preserving maps These include in particular the
elementary face operators [119899 minus 1] [119899] 0 le 119894 le 119899 and the
elementary degeneracy operators [119899 + 1] [119899] 0 le 119894 le 119899
120575119894
120590119894
whose images respectively omit and double up on the element 119894 isin [119899] Every morphism in 120491 fac-tors uniquely as an epimorphism followed by a monomorphism these epimorphisms the degeneracyoperators decompose as composites of elementary degeneracy operators while the monomorphismsthe face operators decompose as composites of elementary face operators
The category of simplicial sets is the category 119982119982119890119905 ≔ 119982119890119905120491op
of presheaves on the simplex cat-egory We write Δ[119899] for the standard 119899-simplex the simplicial set represented by [119899] isin 120491 andΛ119896[119899] sub 120597Δ[119899] sub Δ[119899] for its 119896-horn and boundary sphere respectively
Given a simplicial set 119883 it is conventional to write 119883119899 for the set of 119899-simplices defined byevaluating at [119899] isin 120491 By the Yoneda lemma each119899-simplex 119909 isin 119883119899 corresponds to amap of simplicialsets 119909∶ Δ[119899] rarr 119883 Accordingly we write 119909 sdot 120575119894 for the 119894th face of the 119899-simplex an (119899 minus 1)-simplexclassified by the composite map
Δ[119899 minus 1] Δ[119899] 119883120575119894 119909
Geometrically 119909 sdot 120575119894 is the ldquoface opposite the vertex 119894rdquo in the 119899-simplex 119909Since the morphisms of120491 are generated by the elementary face and degeneracy operators the data
of a simplicial setsup1 119883 is often presented by a diagram
⋯ 1198833 1198832 1198831 1198830
1205750
1205751
1205752
1205753
1205901
1205900
1205902
1205751
1205752
12057501205900
1205901 1205751
12057501205900
identifying the set of 119899-simplices for each [119899] isin 120491 as well as the (contravariant) actions of the elemen-tary operators conventionally denoted using subscripts
sup1This presentation is also used for more general simplicial objects valued in any category
3
112 Definition (quasi-category) A quasi-category is a simplicial set119860 in which any inner horn canbe extended to a simplex solving the displayed lifting problem
Λ119896[119899] 119860 119899 ge 2 0 lt 119896 lt 119899
Δ[119899]
(113)
Quasi-categories were first introduced by Boardman and Vogt [8] under the name ldquoweak Kancomplexesrdquo a Kan complex being a simplicial set admitting extensions as in (113) along all horninclusions 119899 ge 1 0 le 119896 le 119899 Since any topological space can be encoded as a Kan complexsup2 in thisway spaces provide examples of quasi-categories
Categories also provide examples of quasi-categories via the nerve construction
114 Definition (nerve) The category 119966119886119905 of 1-categories embeds fully faithfully into the categoryof simplicial sets via the nerve functor An 119899-simplex in the nerve of a 1-category 119862 is a sequence of 119899composable arrows in 119862 or equally a functor [119899] rarr 119862 from the ordinal category 120159 + 1 ≔ [119899] withobjects 0hellip 119899 and a unique arrow 119894 rarr 119895 just when 119894 le 119895
115 Remark The nerve of a category 119862 is 2-coskeletal as a simplicial set meaning that every sphere120597Δ[119899] rarr 119862 with 119899 ge 3 is filled uniquely by an 119899-simplex in 119862 (see Definition D11) This is becausethe simplices in dimension 3 and above witness the associativity of the composition of the path ofcomposable arrows found along their spine the 1-skeletal simplicial subset formed by the edges con-necting adjacent vertices In fact as suggested by the proof of the following proposition any simplicialset in which inner horns admit unique fillers is isomorphic to the nerve of a 1-category see Exercise11iii
We decline to introduce explicit notation for the nerve functor preferring instead to identify1-categories with their nerves As we shall discover the theory of 1-categories extends toinfin-categoriesmodeled as quasi-categories in such a way that the restriction of each infin-categorical concept alongthe nerve embedding recovers the corresponding 1-categorical concept For instance the standardsimplex Δ[119899] is the nerve of the ordinal category 120159 + 1 and we frequently adopt the latter notationmdash writing 120793 ≔ Δ[0] 120794 ≔ Δ[1] 120795 ≔ Δ[2] and so on mdash to suggest the correct categorical intuition
To begin down this path we must first verify the assertion that has implicitly just been made
116 Proposition (nerves are quasi-categories) Nerves of categories are quasi-categories
Proof Via the isomorphism 119862 cong cosk2 119862 and the adjunction sk2 ⊣ cosk2 of D11 the requiredlifting problem displayed below-left transposes to the one displayed below-right
Λ119896[119899] 119862 cong cosk2 119862 sk2Λ119896[119899] 119862
Δ[119899] sk2 Δ[119899]
sup2The total singular complex construction defines a functor from topological spaces to simplicial sets that is an equiv-alence on their respective homotopy categories mdash weak homotopy types of spaces correspond to homotopy equivalenceclasses of Kan complexes
4
For 119899 ge 4 the inclusion sk2Λ119896[119899] sk2 Δ[119899] is an isomorphism in which case the lifting problemson the right admit (unique) solutions So it remains only to solve the lifting problems on the left inthe cases 119899 = 2 and 119899 = 3
To that end consider
Λ1[2] 119862 Λ1[3] 119862 Λ2[3] 119862
Δ[2] Δ[3] Δ[3]
An inner hornΛ1[2] rarr 119862 defines a composable pair of arrows in119862 an extension to a 2-simplex existsprecisely because any composable pair of arrows admits a (unique) composite
An inner hornΛ1[3] rarr 119862 specifies the data of three composable arrows in 119862 as displayed in thediagram below together with the composites 119892119891 ℎ119892 and (ℎ119892)119891
1198881
1198880 1198883
1198882
ℎ119892119891
119892119891
(ℎ119892)119891
ℎ119892
Because composition is associative the arrow (ℎ119892)119891 is also the composite of 119892119891 followed by ℎ whichproves that the 2-simplex opposite the vertex 1198881 is present in119862 by 2-coskeletality the 3-simplex fillingthis boundary sphere is also present in119862 The filler for a hornΛ2[3] rarr 119862 is constructed similarly
117 Definition (homotopy relation on 1-simplices) A parallel pair of 1-simplices 119891 119892 in a simplicialset 119883 are homotopic if there exists a 2-simplex of either of the following forms
1199091 1199090
1199090 1199091 1199090 1199091
119891119891
119892 119892
(118)
or if 119891 and 119892 are in the same equivalence class generated by this relation
In a quasi-category the relation witnessed by any of the types of 2-simplex on display in (118) isan equivalence relation and these equivalence relations coincide
119 Lemma (homotopic 1-simplices in a quasi-category) Parallel 1-simplices 119891 and 119892 in a quasi-categoryare homotopic if and only if there exists a 2-simplex of any or equivalently all of the forms displayed in (118)
Proof Exercise 11i
1110 Definition (the homotopy category) By 1-truncating any simplicial set 119883 has an underlyingreflexive directed graph
1198831 11988301205751
1205750
1205900
the 0-simplices of 119883 defining the ldquoobjectsrdquo and the 1-simplices defining the ldquoarrowsrdquo by conventionpointing from their 0th vertex (the face opposite 1) to their 1st vertex (the face opposite 0) The free
5
category on this reflexive directed graph has1198830 as its object set degenerate 1-simplices serving as iden-tity morphisms and non-identity morphisms defined to be finite directed paths of non-degenerate1-simplices The homotopy category h119883 of 119883 is the quotient of the free category on its underlyingreflexive directed graph by the congruencesup3 generated by imposing a composition relation ℎ = 119892 ∘ 119891witnessed by 2-simplices
1199091
1199090 1199092
119892119891
ℎThis implies in particular that homotopic 1-simplices represent the same arrow in the homotopy cat-egory
1111 Proposition The nerve embedding admits a left adjoint namely the functor which sends a simplicialset to its homotopy category
119966119886119905 119982119982119890119905perph
Proof Using the description of h119883 as a quotient of the free category on the underlying reflexivedirected graph of 119883 we argue that the data of a functor h119883 rarr 119862 can be extended uniquely to asimplicial map 119883 rarr 119862 Presented as a quotient in this way the functor h119883 rarr 119862 defines a mapfrom the 1-skeleton of119883 into119862 and since every 2-simplex in119883 witnesses a composite in h119883 this mapextends to the 2-skeleton Now119862 is 2-coskeletal so via the adjunction sk2 ⊣ cosk2 of Definition D11this map from the 2-truncation of 119883 into 119862 extends uniquely to a simplicial map 119883 rarr 119862
The homotopy category of a quasi-category admits a simplified description
1112 Lemma (the homotopy category of a quasi-category) If 119860 is a quasi-category then its homotopycategory h119860 hasbull the set of 0-simplices 1198600 as its objectsbull the set of homotopy classes of 1-simplices 1198601 as its arrowsbull the identity arrow at 119886 isin 1198600 represented by the degenerate 1-simplex 119886 sdot 1205900 isin 1198601bull a composition relation ℎ = 119892 ∘ 119891 in h119860 if and only if for any choices of 1-simplices representing these
arrows there exists a 2-simplex with boundary
1198861
1198860 1198862
119892119891
ℎ
Proof Exercise 11ii
1113 Definition (isomorphisms in a quasi-category) A 1-simplex in a quasi-category is an isomor-phism just when it represents an isomorphism in the homotopy category By Lemma 1112 this meansthat 119891∶ 119886 rarr 119887 is an isomorphism if and only if there exist a 1-simplex 119891minus1 ∶ 119887 rarr 119886 together with apair of 2-simplices
119887 119886
119886 119886 119887 119887
119891minus1 119891119891 119891minus1
sup3A relation on parallel pairs of arrows of a 1-category is a congruence if it is an equivalence relation that is closedunder pre- and post-composition if 119891 sim 119892 then ℎ119891119896 sim ℎ119892119896
6
The properties of the isomorphisms in a quasi-category are most easily proved by arguing in aslightly different category where simplicial sets have the additional structure of a ldquomarkingrdquo on a spec-ified subset of the 1-simplices subject to the condition that all degenerate 1-simplices are markedmaps of these so-called marked simplicial sets must then preserve the markings Because these objectswill seldom appear outside of the proofs of certain combinatorial lemmas about the isomorphisms inquasi-categories we save the details for Appendix D
Let us now motivate the first of several results proven using marked techniques Quasi-categoriesare defined to have extensions along all inner horns But if in an outer hornΛ0[2] rarr 119860 orΛ2[2] rarr 119860the initial or final edges respectively are isomorphisms then intuitively a filler should exist
1198861 1198861
1198860 1198862 1198860 1198862
ℎ119891minus1
sim
119892sim119891
ℎ
119892minus1ℎ
ℎ
and similarly for the higher-dimensional outer horns
1114 Proposition (special outer horn lifting)(i) Let 119860 be a quasi-category Then any outer horns
Λ0[119899] 119860 Λ119899[119899] 119860
Δ[119899] Δ[119899]
119892 ℎ
in which the edges 119892|01 and ℎ|119899minus1119899 are isomorphisms admit fillers(ii) Let 119860 and 119861 be quasi-categories and 119891∶ 119860 rarr 119861 a map that lifts against the inner horn inclusions
Then any outer horns
Λ0[119899] 119860 Λ119899[119899] 119860
Δ[119899] 119861 Δ[119899] 119861
119892
119891
ℎ
119891
in which the edges 119892|01 and ℎ|119899minus1119899 are isomorphisms admit fillers
The proof of Proposition 1114 requires clever combinatorics due to Joyal and is deferred toAppendix D⁴ Here we enjoy its myriad consequences Immediately
1115 Corollary A quasi-category is a Kan complex if and only if its homotopy category is a groupoid
Proof If the homotopy category of a quasi-category is a groupoid then all of its 1-simplices areisomorphisms and Proposition 1114 then implies that all inner and outer horns have fillers Thusthe quasi-category is a Kan complex Conversely in a Kan complex all outer horns can be filled andin particular fillers for the horns Λ0[2] and Λ2[2] can be used to construct left and right inverses forany 1-simplex of the form displayed in Definition 1113⁵
⁴The second statement subsumes the first but the first is typically used to prove the second⁵In a quasi-category any left and right inverses to a common 1-simplex are homotopic but as Corollary 1116 proves
any isomorphism in fact has a single two-sided inverse
7
A quasi-category contains a canonical maximal sub Kan complex the simplicial subset spannedby those 1-simplices that are isomorphisms Just as the arrows in a quasi-category 119860 are representedby simplicial maps 120794 rarr 119860 whose domain is the nerve of the free-living arrow the isomorphisms in aquasi-category are represented by diagrams 120128 rarr 119860 whose domain is the free-living isomorphism
1116 Corollary An arrow 119891 in a quasi-category119860 is an isomorphism if and only if it extends to a homo-topy coherent isomorphism
120794 119860
120128
119891
Proof If 119891 is an isomorphism the map 119891∶ 120794 rarr 119860 lands in the maximal sub Kan complex con-tained in 119860 The postulated extension also lands in this maximal sub Kan complex because the inclu-sion 120794 120128 can be expressed as a sequential composite of outer horn inclusions see Exercise 11iv
The category of simplicial sets like any category of presheaves is cartesian closed By the Yonedalemma and the defining adjunction an 119899-simplex in the exponential 119884119883 corresponds to a simplicialmap 119883 times Δ[119899] rarr 119884 and its faces and degeneracies are computed by precomposing in the simplexvariable Our aim is now to show that the quasi-categories define an exponential ideal in the simpli-cially enriched category of simplicial sets if 119883 is a simplicial set and119860 is a quasi-category then119860119883 isa quasi-category We will deduce this as a corollary of the ldquorelativerdquo version of this result involving aclass of maps called isofibrations that we now introduce
1117 Definition (isofibrations between quasi-categories) A simplicial map 119891∶ 119860 rarr 119861 is a isofibra-tion if it lifts against the inner horn inclusions as displayed below left and also against the inclusionof either vertex into the free-standing isomorphism 120128
Λ119896[119899] 119860 120793 119860
Δ[119899] 119861 120128 119861
119891 119891
To notationally distinguish the isofibrations we depict them as arrows ldquo↠rdquo with two heads
1118 Observation(i) For any simplicial set 119883 the unique map 119883 rarr lowast whose codomain is the terminal simplicial
set is an isofibration if and only if 119883 is a quasi-category(ii) Any class of maps characterized by a right lifting property is automatically closed under com-
position product pullback retract and limits of towers see Lemma C11(iii) Combining (i) and (ii) if 119860 ↠ 119861 is an isofibration and 119861 is a quasi-category then so is 119860(iv) The isofibrations generalize the eponymous categorical notion The nerve of any functor
119891∶ 119860 rarr 119861 between categories defines a map of simplicial sets that lifts against the innerhorn inclusions This map then defines an isofibration if and only if given any isomorphism in119861 and specified object in119860 lifting either its domain or codomain there exists an isomorphismin 119860 with that domain or codomain lifting the isomorphism in 119861
We typically only deploy the term ldquoisofibrationrdquo for a map between quasi-categories because our usageof this class of maps intentionally parallels the classical categorical case
8
Much harder to establish is the stability of the class of isofibrations under forming ldquoLeibniz ex-ponentialsrdquo as displayed in (1120) The proof of this result is given in Proposition in AppendixD
1119 Proposition If 119894 ∶ 119883 119884 is a monomorphism and 119891∶ 119860 ↠ 119861 is an isofibration then the inducedLeibniz exponential map
119860119884
bull 119860119883
119861119884 119861119883
1198941114023⋔119891
119860119894
119891119884
119891119883
119861119894
(1120)
is again an isofibration⁶
1121 Corollary If119883 is a simplicial set and119860 is a quasi-category then119860119883 is a quasi-category Moreovera 1-simplex in 119860119883 is an isomorphism if and only if its components at each vertex of 119883 are isomorphisms in 119860
Proof The first statement is a special case of Proposition 1119 see Exercise 11vi The secondstatement is proven similarly by arguing with marked simplicial sets See Lemma
1122 Definition (equivalences of quasi-categories) A map 119891∶ 119860 rarr 119861 between quasi-categories isan equivalence if it extends to the data of a ldquohomotopy equivalencerdquo with the free-living isomorphism120128 serving as the interval that is if there exist maps 119892∶ 119861 rarr 119860 and
119860 119861
119860 119860120128 119861 119861120128
119860 119861119892119891
120572
ev0
ev1
120573
119891119892 ev0
ev1
We write ldquo⥲rdquo to decorate equivalences and119860 ≃ 119861 to indicate the presence of an equivalence119860 ⥲ 119861
1123 Definition A map 119891∶ 119883 rarr 119884 between simplicial sets is a trivial fibration if it admits liftsagainst the boundary inclusions for all simplices
120597Δ[119899] 119883 119899 ge 0
Δ[119899] 119884
119891 (1124)
We write ldquo⥲rarrrdquo to decorate trivial fibrations
1125 Remark The simplex boundary inclusions 120597Δ[119899] Δ[119899] ldquocellularly generaterdquo the monomor-phisms of simplicial sets mdash see Definition C12 and Lemma D12 Hence the dual of Lemma C11
⁶Degenerate cases of this result taking 119883 = empty or 119861 = 1 imply that the other six maps in this diagram are alsoisofibrations see Exercise 11vi
9
implies that trivial fibrations lift against any monomorphism between simplicial sets In particularapplying this to the map empty rarr 119884 it follows that any trivial fibration 119883 ⥲rarr 119884 is a split epimorphism
The notation ldquo⥲rarrrdquo is suggestive the trivial fibrations between quasi-categories are exactly thosemaps that are both isofibrations and equivalences This can be proven by a relatively standard althoughrather technical argument in simplicial homotopy theory given as Proposition in Appendix D
1126 Proposition For a map 119891∶ 119860 rarr 119861 between quasi-categories the following are equivalent(i) 119891 is at trivial fibration(ii) 119891 is both an isofibration and an equivalence(iii) 119891 is a split fiber homotopy equivalence an isofibration admitting a section 119904 that is also an equivalence
inverse via a homotopy from 119904119891 to 1119860 that composes with 119891 to the constant homotopy from 119891 to 119891
As a class characterized by a right lifting property the trivial fibrations are also closed undercomposition product pullback limits of towers and contain the isomorphisms The stability ofthese maps under Leibniz exponentiation will be verified along with Proposition 1119 in Proposition
1127 Proposition If 119894 ∶ 119883 rarr 119884 is a monomorphism and 119891∶ 119860 rarr 119861 is an isofibration then if either 119891 isa trivial fibration or if 119894 is in the class cellularly generated⁷ by the inner horn inclusions and the map 120793 120128then the induced Leibniz exponential map
119860119884 119861119884 times119861119883 1198601198831198941114023⋔119891
a trivial fibration
1128 Digression (the Joyal model structure) The category of simplicial set bears a model structure(see Appendix D) whose fibrant objects are exactly the quasi-categories all objects are cofibrant Thefibrations weak equivalences and trivial fibrations between fibrant objects are precisely the classesof isofibrations equivalences and trivial fibrations defined above Proposition 1126 proves that thetrivial fibrations are the intersection of the classes of fibrations and weak equivalences Propositions1119 and 1127 reflect the fact that the Joyal model structure is a closed monoidal model category withrespect to the cartesian closed structure on the category of simplicial sets
We have declined to elaborate on the Joyal model structure for quasi-categories alluded to in Di-gression 1128 because the only aspects of it that we will need are those described above The resultsproven here suffice to show that the category of quasi-categories defines an infin-cosmos a concept towhich we now turn
Exercises
11i Exercise Consider the set of 1-simplices in a quasi-category with initial vertex 1198860 and finalvertex 1198861
(i) Prove that the relation defined by 119891 sim 119892 if and only if there exists a 2-simplex with boundary1198861
1198860 1198861
119891
119892
is an equivalence relation
⁷See Definition C12
10
(ii) Prove that the relation defined by 119891 sim 119892 if and only if there exists a 2-simplex with boundary1198860
1198860 1198861
119891
119892
is an equivalence relation
(iii) Prove that the equivalence relations defined by (i) and (ii) are the sameThis proves Lemma 119
11ii Exercise Consider the free category on the reflexive directed graph
1198601 11986001205751
1205750
1205900
underlying a quasi-category 119860(i) Consider the relation that identifies a pair of sequences of composable 1-simplices with com-
mon source and common target whenever there exists a simplex of119860 in which the sequences of1-simplices define two paths from its initial vertex to its final vertex Prove that this relation isstable under pre- and post-composition with 1-simplices and conclude that its transitive clo-sure is a congruence an equivalence relation that is closed under pre- and post-composition⁸
(ii) Consider the congruence relation generated by imposing a composition relation ℎ = 119892 ∘ 119891witnessed by 2-simplices
1198861
1198860 1198862
119892119891
ℎand prove that this coincides with the relation considered in (i)
(iii) In the congruence relations of (i) and (ii) prove that every sequence of composable 1-simplicesin 119860 is equivalent to a single 1-simplex Conclude that every morphism in the quotient of thefree category by this congruence relation is represented by a 1-simplex in 119860
(iv) Prove that for any triple of 1-simplices 119891 119892 ℎ in 119860 ℎ = 119892 ∘ 119891 in the quotient category if andonly if there exists a 2-simplex with boundary
1198861
1198860 1198862
119892119891
ℎ
This proves Lemma 1112
11iii Exercise Show that any quasi-category inwhich inner horns admit unique fillers is isomorphicto the nerve of its homotopy category
11iv Exercise(i) Prove that 120128 contains exactly two non-degenerate simplices in each dimension
⁸Given a congruence relation on the hom-sets of a 1-category the quotient category can be formed by quotientingeach hom-set see [39 sectII8]
11
(ii) Inductively build 120128 from 120794 by expressing the inclusion 120794 120128 as a sequential composite ofpushouts of outer horn inclusions⁹ Λ0[119899] Δ[119899] one in each dimension starting with 119899 =2sup1⁰
11v Exercise Prove the relative version of Corollary 1116 for any isofibration 119901∶ 119860 ↠ 119861 betweenquasi-categories and any isomorphism119891∶ 120794 rarr 119860 any homotopy coherent isomorphism in119861 extending119901119891 lifts to a homotopy coherent isomorphism in 119860 extending 119891
120794 119860
120128 119861
119891
119901
11vi Exercise Specialize Proposition 1119 to prove the following(i) If 119860 is a quasi-category and 119883 is a simplicial set then 119860119883 is a quasi-category(ii) If 119860 is a quasi-category and 119883 119884 is a monomorphism then 119860119884 ↠ 119860119883 is an isofibration(iii) If 119860 ↠ 119861 is an isofibration and 119883 is a simplicial set then 119860119883 ↠ 119861119883 is an isofibration
11vii Exercise(i) Prove that the equivalences defined in Definition 1122 are closed under retracts(ii) Prove that the equivalences defined in Definition 1122 satisfy the 2-of-3 property
12 infin-Cosmoi
In sect11 we presented ldquoanalyticrdquo proofs of a few of the basic facts about quasi-categories Thecategory theory of quasi-categories can be developed in a similar style but we aim instead to developthe ldquosyntheticrdquo theory of infinite-dimensional categories so that our results will apply to many modelsat once To achieve this our strategy is not to axiomatize what these infinite-dimensional categoriesare but rather axiomatize the ldquouniverserdquo in which they live
The following definition abstracts the properties of the quasi-categories and the classes of isofibra-tions equivalences and trivial fibrations introduced in sect11 Firstly the category of quasi-categoriesand simplicial maps is enriched over the category of simplicial sets mdash the set of morphisms from 119860to 119861 coincides with the set of vertices of the simplicial set 119861119860 mdash and moreover these hom-spaces areall quasi-categories Secondly a number of limit constructions that can be defined in the underlying1-category of quasi-categories and simplicial maps satisfy universal properties relative to this simpli-cial enrichment And finally the classes of isofibrations equivalences and trivial fibrations satisfyproperties that are familiar from abstract homotopy theory In particular the use of isofibrations indiagrams guarantees that their strict limits are equivalence invariant so we can take advantage of up-to-isomorphism universal properties and strict functoriality of these constructions while still workingldquohomotopicallyrdquo
As will be explained in Digression 129 there are a variety of models of infinite-dimensional cat-egories for which the category of ldquoinfin-categoriesrdquo as we will call them and ldquofunctorsrdquo between them is
⁹By duality mdash the opposite of a simplicial set 119883 is the simplicial set obtained by reindexing along the involution(minus)op ∶ 120491 rarr 120491 that reverses the ordering in each ordinal mdash the outer horn inclusions Λ119899[119899] Δ[119899] can be used instead
sup1⁰This decomposition of the inclusion 120794 120128 reveals which data can always be extended to a homotopy coherentisomorphism for instance the 1- and 2-simplices of Definition 1113 together with a single 3-simplex that has these as itsouter faces with its inner faces degenerate
12
enriched over quasi-categories and admits classes of isofibrations equivalences and trivial fibrationssatisfying analogous properties This motivates the following axiomatization
121 Definition (infin-cosmoi) Aninfin-cosmos119974 is a category enriched over quasi-categoriessup1sup1 mean-ing that it hasbull objects 119860119861 that we callinfin-categories andbull its morphisms define the vertices of functor-spaces Fun(119860 119861) which are quasi-categories
that is also equipped with a specified class of maps that we call isofibrations and denote by ldquo↠rdquoFrom these classes we define a map 119891∶ 119860 rarr 119861 to be an equivalence if and only the induced map119891lowast ∶ Fun(119883119860) ⥲ Fun(119883 119861) on functor-spaces is an equivalence of quasi-categories for all 119883 isin 119974and we define 119891 to be a trivial fibration just when 119891 is both an isofibration and an equivalence theseclasses are denoted by ldquo⥲rdquo and ldquo⥲rarrrdquo respectively These classes must satisfy the following two axioms
(i) (completeness) The simplicial category 119974 possesses a terminal object small products pull-backs of isofibrations limits of countable towers of isofibrations and cotensors with all sim-plicial sets each of these limit notions satisfying a universal property that is enriched oversimplicial setssup1sup2
(ii) (isofibrations) The class of isofibrations contains all isomorphisms and any map whose codo-main is the terminal object is closed under composition product pullback forming inverselimits of towers and Leibniz cotensors with monomorphisms of simplicial sets and has theproperty that if 119891∶ 119860 ↠ 119861 is an isofibration and119883 is any object then Fun(119883119860) ↠ Fun(119883 119861)is an isofibration of quasi-categories
122 Digression (simplicial categories) A simplicial category 119964 is given by categories 119964119899 with acommon set of objects and whose arrows are called 119899-arrows that assemble into a diagram120491op rarr 119966119886119905of identity-on-objects functors
⋯ 1199643 1199642 1199641 1199640 ≕ 119964
1205750
1205751
1205752
1205753
1205901
1205900
1205902
1205751
1205752
12057501205900
1205901 1205751
12057501205900 (123)
The data of a simplicial category can equivalently be encoded by a simplicially enriched categorywith a set of objects and a simplicial set 119964(119909 119910) of morphisms between each ordered pair of ob-jects an 119899-arrow in119964119899 from 119909 to 119910 corresponds to an 119899-simplex in119964(119909 119910) (see Exercise 12i) Eachendo-hom-space contains a distinguished identity 0-arrow (the degenerate images of which define thecorresponding identity 119899-arrows) and composition is required to define a simplicial map
119964(119910 119911) times 119964(119909 119910) 119964(119909 119911)∘
the single map encoding the compositions in each of the categories 119964119899 and also the functoriality ofthe diagram (123) The composition is required to be associative and unital in a sense expressed by
sup1sup1This is to say119974 is a simplicially enriched category whose hom-spaces are quasi-categories this will be unpacked in122
sup1sup2This will be elaborated upon in 124
13
the commutative diagrams
119964(119910 119911) times 119964(119909 119910) times 119964(119908 119909) 119964(119909 119911) times 119964(119908 119909)
119964(119910 119911) times 119964(119908 119910) 119964(119908 119911)1times∘
∘times1
∘
∘
119964(119909 119910) 119964(119910 119910) times 119964(119909 119910)
119964(119909 119910) times 119964(119909 119909) 119964(119909 119910)
id119910 times1
1timesid119909 ∘
∘
the latter making use of natural isomorphisms 119964(119909 119910) times 1 cong 119964(119909 119910) cong 1 times 119964(119909 119910) in the domainvertex
On account of the equivalence between these two presentations the terms ldquosimplicial categoryrdquoand ldquosimplicially-enriched categoryrdquo are generally taken to be synonymssup1sup3 The category1199640 of 0-arrowsis the underlying category of the simplicial category119964 which forgets the higher dimensional simpli-cial structure
In particular the underlying category of an infin-cosmos 119974 is the category whose objects are theinfin-categories in119974 and whose morphisms are the 0-arrows in the functor spaces In all of the examplesto appear below this recovers the expected category ofinfin-categories in a particularmodel and functorsbetween them
124 Digression (simplicially enriched limits) Let 119964 be a simplicial category The cotensor of anobject 119860 isin 119964 by a simplicial set 119880 is characterized by an isomorphism of simplicial sets
119964(119883119860119880) cong 119964(119883119860)119880 (125)
natural in 119883 isin 119964 Assuming such objects exist the simplicial cotensor defines a bifunctor
119982119982119890119905op times119964 119964
(119880119860) 119860119880
in a unique way making the isomorphism (125) natural in 119880 and 119860 as wellThe other simplicial limit notions postulated by axiom 121(i) are conical which is the term used
for ordinary 1-categorical limit shapes that satisfy an enriched analog of the usual universal propertysee Definition 7114 When these limits exist they correspond to the usual limits in the underlying cat-egory but the usual universal property is strengthened Applying the covariant representable functor119964(119883 minus) ∶ 1199640 rarr 119982119982119890119905 to a limit cone (lim119895isin119869119860119895 rarr 119860119895)119895isin119869 in1199640 there is natural comparison map
119964(119883 lim119895isin119869119860119895) rarr lim
119895isin119869119964(119883119860119895) (126)
and we say that lim119895isin119869119860119895 defines a simplicially enriched limit if and only if (126) is an isomorphismof simplicial sets for all 119883 isin 119964
Considerably more details on the general theory of enriched categories can be found in [31] andin Appendix A Enriched limits are the subject of sectA
127 Remark (flexible weighted limits ininfin-cosmoi) The axiom 121(i) implies that anyinfin-cosmos119974 admits all flexible limits (see Corollary 733) a much larger class of simplicially enriched ldquoweightedrdquolimits that will be introduced in sect72
sup1sup3The phrase ldquosimplicial object in119966119886119905rdquo is reserved for themore general yet less commonnotion of a diagram120491op rarr 119966119886119905that is not necessarily comprised of identity-on-objects functors
14
Using the results of Joyal discussed in sect11 we can easily verify
128 Proposition The full subcategory 119980119966119886119905 sub 119982119982119890119905 of quasi-categories defines aninfin-cosmos with theisofibrations equivalences and trivial fibrations of Definitions 1117 1122 and 1123
Proof The subcategory 119980119966119886119905 sub 119982119982119890119905 inherits its simplicial enrichment from the cartesianclosed category of simplicial sets note that for quasi-categories 119860 and 119861 Fun(119860 119861) ≔ 119861119860 is again aquasi-category
The limits postulated in 121(i) exist in the ambient category of simplicial setssup1⁴ The defininguniversal property of the simplicial cotensor is satisfied by the exponentials of simplicial sets We nowargue that the full subcategory of quasi-categories inherits all these limit notions
Since the quasi-categories are characterized by a right lifting property it is clear that they areclosed under small products Similarly since the class of isofibrations is characterized by a right liftingproperty Lemma C11 implies that the fibrations are closed under all of the limit constructions of121(ii) except for the last two Leibniz closure and closure under exponentiation (minus)119883 These lastclosure properties are established in Proposition 1119 This completes the proof of 121(i) and 121(ii)
Remark 1125 proves that trivial fibrations split once we verify that the classes of trivial fibrationsand of equivalences coincide with those defined by 1123 and 1122 By Proposition 1126 the formercoincidence follows from the latter so it remains only to show that the equivalences of 1122 coincidewith the representably-defined equivalences those maps of quasi-categories 119891∶ 119860 rarr 119861 for which119860119883 rarr 119861119883 is an equivalence of quasi-categories in the sense of 1122 Taking 119883 = Δ[0] we seeimmediately that representably-defined equivalences are equivalences and the converse holds sincethe exponential (minus)119883 preserves the data defining a simplicial homotopy
We mention a common source of infin-cosmoi found in nature at the outside to help ground theintuition for readers familiar with Quillenrsquos model categories a popular framework for ldquoabstract ho-motopy theoryrdquo but reassure others that model categories are not needed outside of Appendix E
129 Digression (a source of infin-cosmoi in nature) As explained in Appendix E certain easily de-scribed properties of a model category imply that the full subcategory of fibrant objects defines aninfin-cosmos whose isofibrations equivalences and trivial fibrations are the fibrations weak equiva-lences and trivial fibrations between fibrant objects Namely any model category that is enriched assuch over the Joyal model structure on simplicial sets and with the property that all fibrant objects arecofibrant has this property This compatible enrichment in the Joyal model structure can be definedwhen the model category is cartesian closed and equipped with a right Quillen adjoint to the Joyalmodel structure on simplicial sets whose left adjoint preserves finite products In this case the rightadjoint becomes the underlying quasi-category functor (see Proposition 133(ii)) and the infin-cosmoiso-produced will then be cartesian closed (see Definition 1218) The infin-cosmoi listed in Example1219 all arise in this way
The following results are consequences of the axioms of Definition 121 The first of these resultstells us that the trivial fibrations enjoy all of the same stability properties satisfied by the isofibrations
1210 Lemma (stability of trivial fibrations) The trivial fibrations in aninfin-cosmos define a subcategorycontaining the isomorphisms are stable under product pullback forming inverse limits of towers the Leibnizcotensors of any trivial fibration with a monomorphism of simplicial sets is a trivial fibration as is the Leibniz
sup1⁴Any category of presheaves is cartesian closed complete and cocomplete mdash a ldquocosmosrdquo in the sense of Beacutenabou Ourinfin-cosmoi are more similar to the fibrational cosmoi due to Street [57]
15
cotensor of an isofibration with a map in the class cellularly generated by the inner horn inclusions and the map120793 120128 and if 119864 ⥲rarr 119861 is a trivial fibration then so is Fun(119883 119864) ⥲rarr Fun(119883 119861)
Proof We prove these statements in the reverse order By axiom 121(ii) and the definition of thetrivial fibrations in aninfin-cosmos we know that if 119864 ⥲rarr 119861 is a trivial fibration then is Fun(119883 119864) ⥲rarrFun(119883 119861) is both an isofibration and an equivalence and hence by Proposition 1126 a trivial fibra-tion For stability under the remaining constructions we know in each case that the maps in ques-tion are isofibrations in the infin-cosmos it remains to show only that the maps are also equivalencesThe equivalences in an infin-cosmos are defined to be the maps that Fun(119883 minus) carries to equivalencesof quasi-categories so it suffices to verify that trivial fibrations of quasi-categories satisfy the corre-sponding stability properties This is established in Proposition 1127 and the fact that that class ischaracterized by a right lifting property
Additionally every trivial fibration is ldquosplitrdquo by a section
1211 Lemma (trivial fibrations split) Every trivial fibration admits a section
119864
119861 119861
sim
Proof If 119901∶ 119864 ⥲rarr 119861 is a trivial fibration then by the final stability property of Lemma 1210 sois 119901lowast ∶ Fun(119861 119864) ↠ 119865119906119899(119861 119861) By Definition 1123 we may solve the lifting problem
empty = 120597Δ[0] Fun(119861 119864)
Δ[0] Fun(119861 119861)
sim 119901lowast119904
id119861
to find a map 119904 ∶ 119861 rarr 119864 so that 119901119904 = id119861
A classical construction in abstract homotopy theory proves the following
1212 Lemma (Brown factorization lemma) Any functor 119891∶ 119860 rarr 119861 in an infin-cosmos may be factoredas an equivalence followed by an isofibration where this equivalence is constructed as a section of a trivialfibration
119875119891
119860 119861
119901
sim119902
119891
sim119904
(1213)
16
Proof The displayed factorization is constructed by the pullback of an isofibration formed bythe simplicial cotensor of the inclusion 120793 + 120793 120128 into theinfin-category 119861
119860120128
119860 119875119891 119861120128
119860 times 119861 119861 times 119861
119891120128
sim119904
Δ
(119860119891)(119902119901)
(ev0ev1)
119891times119861
Note the map 119902 is a pullback of the trivial fibration ev0 ∶ 119861120128 ⥲rarr 119861 and is hence a trivial fibration Itssection 119904 constructed by applying the universal property of the pullback to the displayed cone withsummit 119860 is thus an equivalence
1214 Remark (equivalences satisfy the 2-of-6 property) Exercise 11vii proves that the equivalencesbetween quasi-categories mdash and hence also the equivalences in any infin-cosmos mdash are closed underretracts and have the 2-of-3 property Using this latter fact we see that in the case where 119891∶ 119860 rarr 119861 isan equivalence the map 119901 of (1213) is also a trivial fibration and in particular has a section by Lemma1211 Combining these facts a result of Blumberg and Mandell [7 64] reproduced in Appendix Capplies to prove that the equivalences in any infin-cosmos satisfy the stronger 2-of-6 property for anycomposable triple of morphisms
119861
119860 119863
119862
simℎ119892119891
sim119892119891
ℎ119892119891
119892ℎ
if 119892119891 and ℎ119892 are equivalences then 119891 119892 ℎ and ℎ119892119891 are too
By a Yoneda-style argument the ldquohomotopy equivalencerdquo characterization of the equivalences intheinfin-cosmos of quasi-categories extends to an analogous characterization of the equivalences in anyinfin-cosmos
1215 Lemma (equivalences are homotopy equivalences) A map 119891∶ 119860 rarr 119861 between infin-categories inan infin-cosmos 119974 is an equivalence if and only if it extends to the data of a ldquohomotopy equivalencerdquo with thefree-living isomorphism 120128 serving as the interval that is if there exist maps 119892∶ 119861 rarr 119860 and
119860 119861
119860 119860120128 119861 119861120128
119860 119861119892119891
120572
ev0
ev1
120573
119891119892 ev0
ev1
(1216)
in theinfin-cosmos
17
Proof By hypothesis if 119891∶ 119860 rarr 119861 defines an equivalence in theinfin-cosmos119974 then the inducedmap on post-composition 119891lowast ∶ Fun(119861119860) ⥲ Fun(119861 119861) is an equivalence of quasi-categories Evalu-ating the equivalence inverse ∶ Fun(119861 119861) ⥲ Fun(119861119860) and homotopy ∶ Fun(119861 119861) rarr Fun(119861 119861)120128at the 0-arrow 1119861 isin Fun(119861 119861) we obtain a 0-arrow 119892∶ 119861 rarr 119860 together with an isomorphism120128 rarr Fun(119861 119861) from the composite 119891119892 to 1119861 By the defining universal property of the cotensorthis isomorphism internalizes to define the map 120573∶ 119861 rarr 119861120128 in119974 displayed on the right of (1216)
Now the hypothesis that 119891 is an equivalence also provides an equivalence of quasi-categories119891lowast ∶ Fun(119860119860) ⥲ Fun(119860 119861) and the map 120573119891∶ 119860 rarr 119861120128 represents an isomorphism in Fun(119860 119861)from 119891119892119891 to 119891 Since 119891lowast is an equivalence we can conclude that 1119860 and 119892119891 are isomorphic in thequasi-category Fun(119860119860) such an isomorphism may be defined by applying the inverse equivalenceℎ ∶ Fun(119860 119861) rarr Fun(119860119860) and composing with the components at 1119860 119892119891 isin Fun(119860119860) of the iso-morphism ∶ Fun(119860119860) rarr Fun(119860119860)120128 from 1Fun(119860119860) to ℎ119891lowast Now by Corollary 1116 this isomor-phism is represented by a map 120128 rarr Fun(119860119860) from 1119860 to 119892119891 which internalizes to a map120572∶ 119860 rarr 119860120128in119974 displayed on the left of (1216)
The converse is easy the simplicial cotensor construction commutes with Fun(119883 minus) so homotopyequivalences are preserved and by Definition 1122 homotopy equivalences of quasi-categories defineequivalences of quasi-categories
One of the key advantages of the infin-cosmological approaches to abstract category theory is thatthere are a myriad varieties of ldquofiberedrdquo infin-cosmoi that can be built from a given infin-cosmos whichmeans that any theorem proven in this axiomatic framework specializes and generalizes to those con-texts The most basic of these derived infin-cosmos is the infin-cosmos of isofibrations over a fixed basewhich we introduce now Other examples of infin-cosmoi will be introduced in sect74 once we have de-veloped a greater facility with the simplicial limits of axiom 121(i)
1217 Proposition (slicedinfin-cosmoi) For anyinfin-cosmos119974 and any object 119861 isin 119974 there is aninfin-cosmos119974119861 of isofibrations over 119861 whose
(i) objects are isofibrations 119901∶ 119864 ↠ 119861 with codomain 119861(ii) functor-spaces say from 119901∶ 119864 ↠ 119861 to 119902 ∶ 119865 ↠ 119861 are defined by pullback
Fun119861(119901 ∶ 119864 ↠ 119861 119902 ∶ 119865 ↠ 119861) Fun(119864 119865)
120793 Fun(119864 119861)
119902lowast
119901
and abbreviated to Fun119861(119864 119865) when the specified isofibrations are clear from context(iii) isofibrations are commutative triangles of isofibrations over 119861
119864 119865
119861
119903
119901 119902
18
(iv) terminal object is 1∶ 119861 ↠ 119861 and products are defined by the pullback along the diagonal
times119861119894 119864119894 prod119894 119864119894
119861 prod119894 119861
Δ
(v) pullbacks and limits of towers of isofibrations are created by the forgetful functor119974119861 rarr119974(vi) simplicial cotensors 119901∶ 119864 ↠ 119861 by 119880 isin 119982119982119890119905 are denoted 119880 ⋔119861 119901 and constructed by the pullback
119880 ⋔119861 119901 119864119880
119861 119861119880
119901119880
Δ
(vii) and in which a map
119864 119865
119861
119891
119901 119902
over 119861 is an equivalence in theinfin-cosmos119974119861 if and only if 119891 is an equivalence in119974
Proof Note first that the functor spaces are quasi-categories since axiom 121(ii) asserts that forany isofibration 119902 ∶ 119865 ↠ 119861 in 119974 the map 119902lowast ∶ Fun(119864 119865) ↠ Fun(119864 119861) is an isofibration of quasi-categories Other parts of this axiom imply that that each of the limit constructions define isofibra-tions over 119861 The closure properties of the isofibrations in 119974119861 follow from the corresponding onesin 119974 The most complicated of these is the Leibniz cotensor stability of the isofibrations in 119974119861which follows from the corresponding property in 119974 since for a monomorphism of simplicial sets119894 ∶ 119883 119884 and an isofibration 119903 over 119861 as above the map 119894 1114024⋔119861 119903 is constructed by pulling back 119894 1114023⋔ 119903along Δ∶ 119861 rarr 119861119884
The fact that the above constructions define simplicially enriched limits in a simplicially enrichedslice category are standard from enriched category theory It remains only to verify that the equiva-lences in theinfin-cosmos of isofibrations are created by the forgetful functor119974119861 rarr119974 First note thatif 119891∶ 119864 rarr 119865 defines an equivalence in119974 then for any isofibration 119904 ∶ 119860 ↠ 119861 the induced equivalenceon functor-spaces in119974 pulls back to define an equivalence on corresponding functor spaces in119974119861
Fun119861(119860 119864) Fun(119860 119864)
Fun119861(119860 119865) Fun(119860 119865)
120793 Fun(119860 119861)
sim
119891lowast119901lowast sim
119891lowast
119902lowast
119903
19
This can be verified either by appealing to Lemmas 1210 and 1212 and using standard techniquesfrom simplicial homotopy theorysup1⁵ or by appealing to Lemma 1215 and using the fact that pullbackalong 119903 defines a simplicial functor
For the converse implication we appeal to Lemma 1215 If 119891∶ 119864 rarr 119865 is an equivalence in 119974119861then it admits a homotopy inverse in119974119861 The inverse equivalence 119892∶ 119865 rarr 119864 also defines an inverseequivalence in119974 and the required simplicial homotopies in119974 are defined by composing
119864 120128 ⋔119861 119864 119864120128 119865 120128 ⋔119861 119865 rarr 119865120128120572 120573
with the top horizontal leg of the pullback defining the cotensor in119974119861
1218 Definition (cartesian closed infin-cosmoi) An infin-cosmos 119974 is cartesian closed if the productbifunctor minus times minus∶ 119974 times119974 rarr 119974 extends to a simplicially enriched two-variable adjunction
Fun(119860 times 119861119862) cong Fun(119860 119862119861) cong Fun(119861 119862119860)in which the right adjoint (minus)119860 preserve the class of isofibrations
For instance theinfin-cosmos of quasi-categories is cartesian closed with the exponentials definedas (special cases of) simplicial cotensors This is one of the reasons that we use the same notation forcotensor and for exponentialsup1⁶
1219 Example (infin-cosmoi of (infin 1)-categories) The following models of (infin 1)-categories definecartesian closedinfin-cosmoi
(i) Rezkrsquos complete Segal spaces define the objects of aninfin-cosmos 119966119982119982 in which the isofibra-tions equivalences and trivial fibrations are the corresponding classes of the model structureof [45]sup1⁷
(ii) The Segal categories defined byDwyer Kan and Smith [18] and developed byHirschowitz andSimpson [23] define the objects of aninfin-cosmos119982119890119892119886119897 in which the isofibrations equivalencesand trivial fibrations are the corresponding classes of the model structure of [43] and [3]sup1⁸
(iii) The 1-complicial sets of [61] equivalently the ldquonaturally marked quasi-categoriesrdquo of [36] de-fine the objects of an infin-cosmos 1-119966119900119898119901 in which the isofibrations equivalences and trivialfibrations are the corresponding classes of the model structure from either of these sources
Proofs of these facts can be found in Appendix E
Appendix E also proves that certain models of (infin 119899)-categories or even (infininfin)-categories defineinfin-cosmoi
sup1⁵In more detail any functor between the 1-categories underlying infin-cosmoi that preserves trivial fibrations alsopreserves equivalences
sup1⁶Another reason for this convenient notational conflation will be explained in sect23sup1⁷Warning the model category of complete Segal spaces is enriched over simplicial sets in two distinct ldquodirectionsrdquo
mdash one enrichment makes the simplicial set of maps between two complete Segal spaces into a Kan complex that probesthe ldquospacialrdquo structure while another enrichment makes the simplicial set of maps into a quasi-category that probes theldquocategoricalrdquo structure [26] It is this latter enrichment that we want
sup1⁸Here we reserve the term ldquoSegal categoryrdquo for those simplicial objects with a discrete set of objects that are Reedyfibrant and satisfy the Segal condition The traditional definition does not include the Reedy fibrancy condition becauseit is not satisfied by the simplicial object defined as the nerve of a Kan complex enriched category Since Kan complexenriched categories are not among our preferred models of (infin 1)-categories this does not bother us
20
1220 Example (119966119886119905 as an infin-cosmos) The category 119966119886119905 of 1-categories defines a cartesian closedinfin-cosmos inheriting its structure as a full subcategory 119966119886119905 119980119966119886119905 of the infin-cosmos of quasi-categories via the nerve embedding which preserves all limits and also exponentials the nerve of thefunctor category 119861119860 is the exponential of the nerves
In theinfin-cosmos of categories the isofibrations are the isofibrations functors satisfying the dis-played right lifting property
120793 119860
120128 119861
119891
The equivalences are the equivalences of categories and the trivial fibrations are surjective equiva-lences equivalences of categories that are also surjective on objects
1221 Definition (discrete infin-categories) An object 119864 in an infin-cosmos 119974 is discrete just when forall 119883 isin 119974 the functor-space Fun(119883 119864) is a Kan complex
In theinfin-cosmos of quasi-categories the discrete objects are exactly the Kan complexes by Propo-sition the Kan complexes also define an exponential ideal in the category of simplicial sets Similarlyin theinfin-cosmoi of Example 1219 whoseinfin-categories are (infin 1)-categories in some model the dis-crete objects are theinfin-groupoids
1222 Proposition (infin-cosmos of discrete objects) The full subcategory 119967119894119904119888(119974) 119974 spanned bythe discrete objects in anyinfin-cosmos form aninfin-cosmos
Proof We first establish this result for the infin-cosmos of quasi-categories By Proposition 1114an isofibration between Kan complexes is a Kan fibration a map with the right lifting property withrespect to all horn inclusions Conversely all Kan fibrations define isofibrations Since Kan com-plexes are closed under simplicial cotensor (which coincides with exponentiation) It follows that thefull subcategory 119974119886119899 119980119966119886119905 is closed under all of the limit constructions of axiom 121(i) Theremaining axiom 121(ii) is inherited from the analogous properties established for quasi-categoriesin Proposition 128
In a generic infin-cosmos 119974 we need only show that the discrete objects are closed in 119974 underthe limit constructions of 121(i) The definition natural isomorphism (126) characterizing thesesimplicial limits expresses the functor-space Fun(119883 lim119895isin119869119860119895) as an analogous limit of functor spaceFun(119883119860119895) If each 119860119895 is discrete then these objects are Kan complexes and the previous paragraphthen establishes that the limit is a Kan complex as well This holds for all objects 119883 isin 119974 so it followsthat lim119895isin119869119860119895 is discrete as required
Exercises
12i Exercise Prove that the following are equivalent(i) a simplicial category as in 122(ii) a category enriched over simplicial sets
12ii Exercise Elaborate on the proof of Proposition 128 by proving that the simplicially enrichedcategory 119980119966119886119905 admits conical products satisfying the universal property of Digression 124 That is
21
(i) For quasi-categories 119860119861119883 form the cartesian product 119860times 119861 and prove that the projectionmaps 120587119860 ∶ 119860 times 119861 rarr 119860 and 120587119861 ∶ 119860 times 119861 rarr 119861 induce an isomorphism of quasi-categories
(119860 times 119861)119883 119860119883 times 119861119883cong
(ii) Explain how this relates to the universal property of Digression 124(iii) Express the usual 1-categorical universal property of the product119860times119861 as the ldquo0-dimensional
aspectrdquo of the universal property of (i)
12iii Exercise Prove that any object in aninfin-cosmos has a path object
119861120128
119861 119861 times 119861
(ev0ev1)
sim
sim
Δ
constructed by cotensoring with the free-living isomorphism
12iv Exercise(i) Use Exercise 11iv and results from Appendix D to prove that a quasi-category 119876 is a Kan
complex if and only if the map119876120128 ↠ 119876120794 induced by the inclusion 120794 120128 is a trivial fibration(ii) Conclude that aninfin-category 119860 is discrete if and only if 119860120128 ⥲rarr 119860120794 is a trivial fibration
13 Cosmological functors
Certain ldquoright adjoint typerdquo constructions define maps between infin-cosmoi that preserve all ofthe structures axiomatized in Definition 121 The simple observation that such constructions definecosmological functors betweeninfin-cosmoi will streamline many proofs
131 Definition (cosmological functor) A cosmological functor is a simplicial functor 119865∶ 119974 rarr ℒthat preserves the specified classes of isofibrations and all of the simplicial limits enumerated in 121(i)
132 Lemma Any cosmological functor also preserves the equivalences and the trivial fibrations
Proof By Lemma 1215 the equivalences in an infin-cosmos coincide with the homotopy equiva-lences defined relative to cotensoring with the free-living isomorphism Since a cosmological functorpreserves simplicial cotensors it preserves the data displayed in (1216) and hence carries equivalencesto equivalences The statement about trivial fibrations follows
In general cosmological functors preserve any infin-categorical notion that can be characterizedinternally to theinfin-cosmos mdash for instance as a property of a certain map mdash as opposed to externallymdash for instance in a statement that involves a universal quantifier From Definition 1221 it is notclear whether cosmological functors preserve discrete objects but using the internal characterizationof Exercise 12iv mdash aninfin-category 119860 is discrete if and only if 119860120128 ⥲rarr 119860120794 is at trivial fibration mdash thisfollows easily cosmological functors preserve simplicial cotensors and trivial fibrations
133 Proposition(i) For any object 119860 in aninfin-cosmos119974 Fun(119860 minus) ∶ 119974 rarr 119980119966119886119905 defines a cosmological functor(ii) Specializing eachinfin-cosmos has an underlying quasi-category functor
(minus)0 ≔ Fun(1 minus) ∶ 119974 rarr 119980119966119886119905
22
(iii) For anyinfin-cosmos 119974 and any simplicial set 119880 the simplicial cotensor defines a cosmological functor(minus)119880 ∶ 119974 rarr 119974
(iv) For any object 119860 in a cartesian closed infin-cosmos 119974 exponentiation defines a cosmological functor(minus)119860 ∶ 119974 rarr 119974
(v) For any map 119891∶ 119860 rarr 119861 in aninfin-cosmos119974 pullback defines a cosmological functor 119891lowast ∶ 119974119861 rarr119974119860(vi) For any cosmological functor119865∶ 119974 rarr ℒ and any119860 isin 119974 the inducedmap on slices119865∶ 119974119860 rarr ℒ119865119860
defines a cosmological functor
Proof The first four of these statements are nearly immediate the preservation of isofibrationsbeing asserted explicitly as a hypothesis in each case and the preservation of limits following fromstandard categorical arguments
For (v) pullback in aninfin-cosmos119974 is a simplicially enriched limit construction one consequenceof this is that 119891lowast ∶ 119974119861 rarr119974119860 defines a simplicial functor The action of the functor 119891lowast on a 0-arrow119892 in119974119861 is also defined by a pullback square since the front and back squares in the displayed diagramare pullbacks the top square is as well
119891lowast119864 119864
119891lowast119865 119865
119860 119861
119891lowast(119892)
119901
119892
119902
119891
Since isofibrations are stable under pullback it follows that 119891lowast ∶ 119974119861 rarr 119974119860 preserves isofibrationsIt remains to prove that this functor preserves the simplicial limits constructed in Proposition 1217In the case of connected limits which are created by the forgetful functors to 119974 this is clear Forproducts and simplicial cotensors this follows from the commutative cubes
times119860119894 119891lowast119864119894 prod119894 119891
lowast119864119894 119880 ⋔119860 119891lowast(119901) (119891lowast119864)119880
times119861119894 119864119894 prod119894 119864119894 119880 ⋔119861 119901 119864119880
119860 prod119894119860 119860 119860119880
119861 prod119894 119861 119861 119861119880
(119891lowast119901)119880
Δ
119891 prod119894 119891
Δ
119891 119891119880
ΔΔ
119901119880
Since the front back and right faces are pullbacks the left is as well which is what we wanted toshow
The final statement (vi) is left as Exercise 13i
134 Non-Example The forgetful functor119974119861 rarr119974 is simplicial and preserves the class of isofibra-tions but does not define a cosmological functor failing to preserve cotensors and products Howeverby Proposition 133(v) minus times 119861∶ 119974 rarr 119974119861 does define a cosmological functor
135 Definition (biequivalences) A cosmological functor defines a biequivalence 119865∶ 119974 ⥲ ℒ if ad-ditionally it
23
(i) is essentially surjective on objects up to equivalence for all 119862 isin ℒ there exists119860 isin 119974 so that119865119860 ≃ 119862 and
(ii) it defines a local equivalence for all 119860119861 isin 119974 the action of 119865 on functor quasi-categoriesdefines an equivalence
Fun(119860 119861) Fun(119865119860 119865119861)sim
136 Remark Cosmological biequivalences will be studied more systematically in Chapter 13 wherewe think of them as ldquochange-of-modelrdquo functors A basic fact is that any biequivalence of infin-cosmoinot only preserves equivalences but also creates them a pair of objects in aninfin-cosmos are equivalentif and only if their images in any biequivalentinfin-cosmos are equivalent (Exercise 13ii) It follows thatthe cosmological biequivalences satisfy the 2-of-3 property
137 Example (biequivalences betweeninfin-cosmoi of (infin 1)-categories)(i) The underlying quasi-category functors defined on the infin-cosmoi of complete Segal spaces
Segal categories and 1-complicial sets
119966119982119982 119980119966119886119905 119982119890119892119886119897 119980119966119886119905 1-119966119900119898119901 119980119966119886119905sim(minus)0 sim(minus)0 sim(minus)0
are all biequivalences In the first two cases these are defined by ldquoevaluating at the 0th rowrdquoand in the last case this is defined by ldquoforgetting the markingsrdquo
(ii) There is also a cosmological biequivalence 119980119966119886119905 ⥲ 119966119982119982 defined by Joyal and Tierney [26](iii) The functor 119966119982119982 ⥲ 119982119890119892119886119897 defined by Bergner [5] that ldquodiscretizesrdquo a complete Segal spaces
also defines a cosmological biequivalence(iv) There is a further cosmological biequivalence (minus) ∶ 119980119966119886119905 rarr 1-119966119900119898119901 that gives each quasi-
category its ldquonatural markingrdquo with all invertible 1-simplices and all simplices in dimensiongreater than 1 marked
Proofs of these facts can be found in Appendix E
Exercises
13i Exercise Prove that for any cosmological functor 119865∶ 119974 rarr ℒ and any 119860 isin 119974 the inducedmap 119865∶ 119974119860 rarr ℒ119865119860 defines a cosmological functor
13ii Exercise Let 119865∶ 119974 ⥲ ℒ be a cosmological biequivalence and let 119860119861 isin 119974 Sketch a proofthat if 119865119860 ≃ 119865119861 inℒ then 119860 ≃ 119861 in119974 (and see Exercise 14i)
14 The homotopy 2-category
Small 1-categories define the objects of a strict 2-categorysup1⁹119966119886119905 of categories functors and naturaltransformations Many basic categorical notions mdash those defined in terms of categories functorsand natural transformations and their various composition operations mdash can be defined internallyto the 2-category 119966119886119905 This suggests a natural avenue for generalization reinterpreting these same
sup1⁹A comprehensive introduction to strict 2-categories appears as Appendix B Succinctly in parallel with Digression122 2-categories can be understood equally asbull ldquotwo-dimensionalrdquo categories with objects 0-arrows (typically called 1-cells) and 1-arrows (typically called 2-cells)bull or as categories enriched over 119966119886119905
24
definitions in a generic 2-category using its objects in place of small categories its 1-cells in place offunctors and its 2-cells in place of natural transformations
In Chapter 2 we will develop a non-trivial portion of the theory of infin-categories in any fixedinfin-cosmos following exactly this outline working internally to a strict 2-category that we refer to asthe homotopy 2-category that we associate to anyinfin-cosmos The homotopy 2-category of aninfin-cosmosis a quotient of the fullinfin-cosmos replacing each quasi-categorical functor-space by its homotopy cat-egory Surprisingly this rather destructive quotienting operation preserves quite a lot of informationIndeed essentially all of the work in Part I will take place in the homotopy 2-category of aninfin-cosmosThis said we caution the reader against becoming overly seduced by homotopy 2-categories for thatstructure is more of a technical convenience for reducing the complexity of our arguments than afundamental notion ofinfin-category theory
The homotopy 2-category for the infin-cosmos of quasi-categories was first introduced by Joyal inhis work on the foundations of quasi-category theory
141 Definition (homotopy 2-category) Let 119974 be an infin-cosmos Its homotopy 2-category is thestrict 2-category 120101119974 whosebull objects are theinfin-categories ie the objects 119860119861 of119974bull 1-cells 119891∶ 119860 rarr 119861 are the 0-arrows in the functor space Fun(119860 119861) of119974 ie theinfin-functors and
bull 2-cells 119860 119861119891
119892dArr120572 are homotopy classes of 1-simplices in Fun(119860 119861) which we call infin-natural
transformationsPut another way 120101119974 is the 2-category with the same objects as119974 and with hom-categories defined by
hFun(119860 119861) ≔ h(Fun(119860 119861))that is as the homotopy category of the quasi-category Fun(119860 119861)
The underlying category of a 2-category is defined by simply forgetting its 2-cells Note that aninfin-cosmos 119974 and its homotopy 2-category 120101119974 share the same underlying category of infin-categoriesandinfin-functors in119974
142 Digression The homotopy category functor h ∶ 119982119982119890119905 rarr 119966119886119905 preserves finite products as ofcourse does its right adjoint It follows that the adjunction of Proposition 1111 induces a change-of-base adjunction
2-119966119886119905 119982119982119890119905-119966119886119905perp
hlowast
whose left and right adjoints change the enrichment by applying the homotopy category functor orthe nerve functor to the hom objects of the enriched category Here 2-119966119886119905 and 119982119982119890119905-119966119886119905 can each beunderstood as 2-categories mdash of enriched categories enriched functors and enriched natural trans-formations mdash and both change of base constructions define 2-functors [10 643]
143 Observation (functors representing (invertible) 2-cells) By definition 2-cells in the homotopycategory 120101119974 are represented by maps 120794 rarr Fun(119860 119861) valued in the appropriate functor space andtwo such maps represent the same 2-cell if and only if their images are homotopic as 1-simplices inFun(119860 119861) in the sense defined by Lemma 119
25
Now a 2-cell in a 2-category is invertible if and only if it defines an isomorphism in the appropriatehom-category hFun(119860 119861) By Definition 1113 and Corollary 1116 it follows that each invertible2-cell in 120101119974 is represented by a map 120128 rarr Fun(119860 119861)
144 Lemma Any simplicial functor 119865∶ 119974 rarr ℒ between infin-cosmoi induces a 2-functor 119865∶ 120101119974 rarr 120101ℒbetween their homotopy 2-categories
Proof This follows immediately from the remarks on change of base in Digression 142 but wecan also argue directly The action of the induced 2-functor 119865∶ 120101119974 rarr 120101ℒ on objects and 1-cells isgiven by the corresponding action of 119865∶ 119974 rarr ℒ recall an infin-cosmos and its homotopy 2-categoryhave the same underlying 1-category Each 2-cell in 120101119974 is represented by a 1-simplex in Fun(119860 119861)which is mapped via
Fun(119860 119861) Fun(119865119860 119865119861)119865
119860 119861 119865119860 119865119861119891
119892dArr120572
119865119891
119865119892
dArr119865120572
to a 1-simplex representing a 2-cell in 120101ℒ Since the action 119865∶ Fun(119860 119861) rarr Fun(119865119860 119865119861) on functorspaces defines a morphism of simplicial sets it preserves faces and degeneracies In particular homo-topic 1-simplices in Fun(119860 119861) are carried to homotopic 1-simplices in Fun(119865119860 119865119861) so the action on2-cells just described is well-defined The 2-functoriality of these mappings follows from the simplicialfunctoriality of the original mapping
We now begin to relate the simplicially enriched structures of an infin-cosmos to the 2-categoricalstructures in its homotopy 2-category The first result proves that homotopy 2-categories inheritproducts from their infin-cosmoi which satisfy a 2-categorical universal property To illustrate recallthat the terminal infin-category 1 isin 119974 has the universal property Fun(119883 1) cong 120793 for all 119883 isin 119974Applying the homotopy category functor we see that 1 isin 120101119974 has the universal property hFun(119883 1) cong120793 for all119883 isin 120101119974 This 2-categorical universal property has both a 1-dimensional and a 2-dimensionalaspect Since hFun(119883 1) cong 120793 is a category with a single object there exists a unique morphism119883 rarr 1in 119974 And since hFun(119883 1) cong 120793 has only a single identity morphism we see that the only 2-cells in120101119974 with codomain 1 are identities
145 Proposition (cartesian (closure))(i) The homotopy 2-category of anyinfin-cosmos has 2-categorical products(ii) The homotopy 2-category of a cartesian closedinfin-cosmos is cartesian closed as a 2-category
Proof While the functor h ∶ 119982119982119890119905 rarr 119966119886119905 only preserves finite products the restricted functorh ∶ 119980119966119886119905 rarr 119966119886119905 preserves all products on account of the simplified description of the homotopycategory of a quasi-category given in Lemma 1112 Thus for any set 119868 and family of infin-categories(119860119894)119894isin119868 in 119974 the homotopy category functor carries the isomorphism of quasi-categories displayedbelow left to an isomorphisms of hom-categories displayed below right
Fun(119883prod119894isin119868119860119894) prod119894isin119868 Fun(119883119860119894) hFun(119883prod119894isin119868119860119894) prod
119894isin119868 hFun(119883119860119894)cong h cong
This proves that the homotopy 2-category 120101119974 has products whose universal properties have both a 1-and 2-dimensional component as described for terminal objects above
26
If 119974 is a cartesian closed infin-cosmos then for any triple of infin-categories 119860119861 119862 isin 119974 there existexponential objects 119862119860 119862119861 isin 119974 characterized by natural isomorphisms
Fun(119860 times 119861119862) cong Fun(119860 119862119861) cong Fun(119861 119862119860)Passing to homotopy categories we have natural isomorphisms
hFun(119860 times 119861119862) cong hFun(119860 119862119861) cong hFun(119861 119862119860)which demonstrates that 120101119974 is cartesian closed as a 1-category functors 119860 times 119861 rarr 119862 transpose todefine functors 119860 rarr 119862119861 and 119861 rarr 119862119860 and 2-cells transpose similarly
There is a standard definition of isomorphism between two objects in any 1-category Similarlythere is a standard definition of equivalence between two objects in any 2-category
146 Definition (equivalence) An equivalence in a 2-category is given bybull a pair of objects 119860 and 119861bull a pair of 1-cells 119891∶ 119860 rarr 119861 and 119892∶ 119861 rarr 119860bull a pair of invertible 2-cells
119860 119860 119861 119861119892119891
congdArr120572
119891119892
congdArr120573
When119860 and119861 are equivalent we write119860 ≃ 119861 and refer to the 1-cells 119891 and 119892 as equivalences denotedby ldquo⥲rdquo
In the case of the homotopy 2-category of aninfin-cosmos we have a competing definition of equiva-lence from 121 namely a 1-cell 119891∶ 119860 ⥲ 119861 that induces an equivalence 119891lowast ∶ Fun(119883119860) ⥲ Fun(119883 119861)on functor-spaces mdash or equivalently by Lemma 1215 a homotopy equivalence defined relative to theinterval 120128 Crucially these two notions of equivalence coincide
147 Theorem (equivalences are equivalences) A functor 119891∶ 119860 rarr 119861 between infin-categories defines anequivalence in theinfin-cosmos119974 if and only if it defines an equivalence in its homotopy 2-category 120101119974
Proof Given an equivalence 119891∶ 119860 ⥲ 119861 in the infin-cosmos 119974 Lemma 1215 provides an inverseequivalence 119892∶ 119861 ⥲ 119860 and homotopies 120572∶ 119860 rarr 119860120128 and 120573∶ 119861 rarr 119861120128 in119974 By Observation 143 thisdata represents an equivalence in the homotopy 2-category 120101119974
For the converse first note that if parallel 1-cells ℎ 119896 ∶ 119860 119861 in the homotopy 2-category areconnected by an invertible 2-cell as displayed below left then ℎ is an equivalence in theinfin-cosmos119974if and only if 119896 is
119860 119861ℎ
119896
congdArr120574119860
119861 119861120128 119861
120574ℎ 119896
simev0
simev1
By Observation 143 the invertible 2-cell can be represented by a map 120574∶ 119860 rarr 119861120128 in119974 as displayedabove right now apply the 2-of-3 property for the equivalences in119974
Now it follows immediately from the 2-of-6 property of the equivalences in anyinfin-cosmos estab-lished in Remark 1214 and the fact that the equivalences contain the identities that any 2-categorical
27
equivalence defines an equivalence in the infin-cosmos since 119892119891 and 119891119892 are isomorphic to identitiesthey must be equivalences in119974 and hence so must 119891 and 119892
148 Digression (on the importance of Theorem 147) It is hard to overstate the importance ofTheorem 147 to the work that follows The categorical constructions that we will introduce forinfin-categories infin-functors and infin-natural transformations are invariant under 2-categorical equiva-lence in the homotopy 2-category and the universal properties we develop similarly characterize a2-categorical equivalence class of infin-categories Theorem 147 then asserts that such constructionsare ldquohomotopically correctrdquo both invariant under equivalence in theinfin-cosmos and precisely identi-fying equivalence classes of objects
The equivalence invariance of the functor space in the codomain variable is axiomatic but equiv-alence invariance in the domain variable is notsup2⁰ But using 2-categorical techniques there is now ashort proof
149 Corollary Equivalences of infin-categories 119860prime ⥲ 119860 and 119861 ⥲ 119861prime induce an equivalence of functorspaces Fun(119860 119861) ⥲ Fun(119860prime 119861prime)
Proof The simplicial functors Fun(119860 minus) ∶ 119974 rarr 119980119966119886119905 and Fun(minus 119861) ∶ 119974op rarr 119980119966119886119905 induce2-functors Fun(119860 minus) ∶ 120101119974 rarr 120101119980119966119886119905 and Fun(minus 119861) ∶ 120101119974op rarr 120101119980119966119886119905 which preserve the 2-categor-ical equivalences of Definition 146 By Theorem 147 this is what we wanted to show
Similarly there is a standard 2-categorical notion of an isofibration defined in the statementof Proposition 1410 and any isofibration in an infin-cosmos defines an isofibration in its homotopy2-categorysup2sup1
1410 Proposition (isofibrations define isofibrations) Any isofibration 119901∶ 119864 ↠ 119861 in an infin-cosmos119974 also defines an isofibration in the homotopy 2-category 120101119974 given any invertible 2-cell as displayed belowleft abutting to 119861 with a specified lift of one of its boundary 1-cells through 119901 there exists an invertible 2-cellabutting to 119864 with this boundary 1-cell as displayed below right that whiskers with 119901 to the original 2-cell
119883 119864 119883 119864
119861 119861
119890
119887
119901congdArr120573 =
119890
congdArr120574
119901
Proof Put another way the universal property of the statement says that the functor
119901lowast ∶ hFun(119883 119864) ↠ hFun(119883 119861)is an isofibration of categories in the sense defined in Example 1220 By axiom 121(ii) since 119901∶ 119864 ↠119861 is an isofibration in 119974 the induced map 119901lowast ∶ Fun(119883 119864) ↠ Fun(119883 119861) is an isofibration of quasi-categories So it suffices to show that the functor h ∶ 119980119966119886119905 rarr 119966119886119905 carries isofibrations of quasi-categories to isofibrations of categoriessup2sup2
sup2⁰Lemma 132 does not apply since Fun(minus 119861) is not cosmologicalsup2sup1In this case the converse does not hold nor is it the case that a representably-defined isofibration of quasi-categories
is necessarily an isofibration in theinfin-cosmos consider the case of slicedinfin-cosmoi for instancesup2sup2Alternately argue directly using Observation 143
28
So let us now consider an isofibration 119901∶ 119864 ↠ 119861 between quasi-categories By Corollary 1116 ev-ery isomorphism 120573 in the homotopy category h119861 of the quasi-category 119861 is represented by a simplicialmap 120573∶ 120128 rarr 119861 By Definition 1117 the lifting problem
120793 119864
120128 119861
119890
119901120574
120573
can be solved and the map 120574∶ 120128 rarr 119864 so-produced represents a lift of the isomorphism from h119861 to anisomorphism in h119864 with domain 119890
1411 Convention (on ldquoisofibrationsrdquo in homotopy 2-categories) Since the converse to Proposition1410 does not hold there is a potential ambiguity when using the term ldquoisofibrationrdquo to refer to a mapin the homotopy 2-category of an infin-cosmos We adopt the convention that when we declare that amap in 120101119974 is an isofibration we always mean this is the stronger sense of defining an isofibration in119974 This stronger condition gives us access to the 2-categorical lifting property of Proposition 1410but also to the many homotopical properties axiomatized in Definition 121 which guarantee that thestrictly defined limits of 121(i) are automatically equivalence invariant constructions
The 1- and 2-cells in the homotopy 2-category from the terminal infin-category 1 isin 119974 to a genericinfin-category 119860 isin 119974 define the objects and morphisms in the homotopy category of 119860
1412 Definition (homotopy category of aninfin-category) The homotopy category of aninfin-category119860 in aninfin-cosmos119974 is defined to be the homotopy category of its underlying quasi-category that is
h119860 ≔ hFun(1 119860) ≔ h(119865119906119899(1119860))
As we shall discover homotopy categories generally bear ldquoderivedrdquo analogues of structures presentat the level ofinfin-categories See the remark after the statement Proposition 217 for an early exampleof this
Exercises
14i Exercise Let 119865∶ 119974 ⥲ ℒ be a cosmological biequivalence and let 119860119861 isin 119974 Prove that if119865119860 ≃ 119865119861 in ℒ then 119860 ≃ 119861 in 119974 and ruminate on why this exercise is considerably easier thanExercise 13ii)
14ii Exercise(i) What is the homotopy 2-category of theinfin-cosmos 119966119886119905 of 1-categories(ii) Prove that the nerve defines a 2-functor 119966119886119905 120101119980119966119886119905 that is locally fully faithful
14iii Exercise Let 119861 be an infin-category in the infin-cosmos 119974 and let 120101119974119861 denote the 2-categorywhosebull objects are isofibrations 119864 ↠ 119861 in119974 with codomain 119861bull 1-cells are 1-cells in 120101119974 over 119861
119864 119865
119861
29
bull 2-cells are 2-cells in 120101119974 over 119861
119864 119865
119861119901
119891
119892
dArr120572
119902
in the sense that 119902120572 = id119901Argue that the homotopy 2-category 120101(119974119861) of the slicedinfin-cosmos has the same underlying 1-categorybut different 2-cells How do these compare with the 2-cells of 120101119974119861sup2sup3
sup2sup3A more systematic comparison will be given in Proposition 353
30
CHAPTER 2
Adjunctions limits and colimits I
Heuristicallyinfin-categories generalize ordinary 1-categories by adding in higher dimensional mor-phisms and weakening the composition law The dream is that proofs establishing the theory of1-categories similarly generalize to give proofs for infin-categories just by adding a prefix ldquoinfin-rdquo every-where In this chapter we make this dream a reality mdash at least for a library of basic propositionsconcerning equivalences adjunctions limits and colimits and the relationships between these no-tions
Recall that categories functors and natural transformations assemble into a 2-category119966119886119905 Sim-ilarly theinfin-categoriesinfin-functors andinfin-natural transformations in anyinfin-cosmos assemble intoa 2-category namely the homotopy 2-category of the infin-cosmos introduced in sect14 By Exercise 14ii119966119886119905 can be regarded as a special case of a homotopy 2-category In this chapter we will use strict2-categorical techniques to define adjunctions betweeninfin-categories and limits and colimits of diagramsvalued in aninfin-category and prove that these notions interact in the expected ways In the homotopy2-category of categories these recover the classical results from 1-category theory As these proofs areequally valid in any homotopy 2-category our arguments also establish the desired generalizations bysimply appending the prefix ldquoinfin-rdquo
21 Adjunctions and equivalences
In sect14 we encountered the definition of an equivalence between a pair of objects in a 2-categoryIn the case where the ambient 2-category is the homotopy 2-category of aninfin-cosmos we observed inTheorem 147 that the 2-categorical notion of equivalence precisely recaptures the notion of equiva-lence introduced in Definition 121 betweeninfin-categories in the fullinfin-cosmos In each of the exam-ples ofinfin-cosmoi we have considered the representably-defined equivalences in theinfin-cosmos coin-cide with the standard notion of equivalences between infin-categories as presented in that particularmodelsup1 Thus the 2-categorical notion of equivalence is the ldquocorrectrdquo notion of equivalence betweeninfin-categories
Similarly there is a standard definition of an adjunction between a pair of objects in a 2-categorywhich when interpreted in the homotopy 2-category ofinfin-categories functors and natural transfor-mations in aninfin-cosmos will define the correct notion of adjunction betweeninfin-categories
211 Definition (adjunction) An adjunction betweeninfin-categories is comprised ofbull a pair ofinfin-categories 119860 and 119861bull a pair of functors 119906∶ 119860 rarr 119861 and 119891∶ 119861 rarr 119860bull and a pair of natural transformations 120578∶ 1119861 rArr 119906119891 and 120598 ∶ 119891119906 rArr 1119860 called the unit and counit
respectively
sup1For instance as outlined in Digression 129 the equivalences in theinfin-cosmoi of Example 1219 recapture the weakequivalences between fibrant-cofibrant objects in the usual model structure
31
so that the triangle equalities holdsup2
119861 119861 119861 119861 119861 119861
119860 119860 119860 119860 119860 119860dArr120598 119891 dArr120578 = =
119891dArr120578 dArr120598
119891 = = 119891119891119906119906
119906 119906 119906
The functor 119891 is called the left adjoint and 119906 is called the right adjoint a relationship that is denotedsymbolically in text by writing 119891 ⊣ 119906 or in a displayed diagram such assup3
119860 119861119906perp119891
212 Digression (why this is the right definition) For readers who find Definition 211 implausiblemdash perhaps too simple to be trusted mdash we offer a few words of justification Firstly the correct notionof adjunction between quasi-categories is well established though the definition appearing in [36 sect52]takes a quite different form In Appendix F we prove that in the infin-cosmos of quasi-categories ourdefinition of adjunction precisely recovers Luriersquos As explained in Part IV each of the models of(infin 1)-categories described in Example 1219 ldquohas the same category theoryrdquo so Definition 211 agreeswith the community consensus notion of adjunction between (infin 1)-categories
But what about those infin-cosmoi whose objects model (infin 119899)- or (infininfin)-categories For in-stance in the infin-cosmos of complicial sets the adjunctions defined in the homotopy 2-category arethe ldquopseudo-stylerdquo adjunctions While these are not the most general adjunctions that might be con-sidered mdash for instance one could have (op)lax units and counits mdash they are an important class ofadjunctions One reason for the relevance of Definition 211 in allinfin-cosmoi is its formal propertiesvis-a-vis the related notion of equivalence which Theorem 147 has established is morally ldquocorrectrdquoand with the notions of limits and colimits to be introduced
Finally a reasonable objection is that Definition 211 appears too ldquolow dimensionalrdquo comprisedof data found entirely in the homotopy 2-category and ignoring the higher dimensional morphismsin aninfin-cosmos This deficiency will be addressed in Chapter 8 when we prove that any adjunctionbetween infin-categories extends to a homotopy coherent adjunction and moreover such extensions arehomotopically unique
The definition of an adjunction given in Definition 211 is ldquoequationalrdquo in character stated interms of the objects 1-cells and 2-cells of a 2-category and their composites Immediately
213 Lemma Adjunctions in a 2-category are preserved by any 2-functor
Lemma 213 provides an easy source of examples of adjunctions between quasi-categories The2-functors underlying the cosmological functors of Example 137 then transfer adjunctions defined inone model of (infin 1)-categories to adjunctions defined in each of the other models
214 Example (adjunctions between 1-categories) Via the nerve embedding 119966119886119905 120101119980119966119886119905 any ad-junction between 1-categories induces an adjunction between their nerves regarded as quasi-categories
sup2The left-hand equality of pasting diagrams asserts the composition relation 119906120598 sdot 120578119906 = id119906 in the hom-categoryhFun(119860 119861) while the right-hand equality asserts that 120598119891 sdot 119891120578 = id119891 in hFun(119861119860)
sup3Some authors contort adjunction diagrams so that the left adjoint is always on the left we instead use the turnstilesymbol ldquoperprdquo to indicate which adjoint is the left adjoint
32
215 Example (adjunctions between topological categories) The homotopy coherent nerve defines a2-functor 120081∶ 119974119886119899-119966119886119905 rarr 120101119980119966119886119905 from the 2-category of Kan complex enriched categories simpli-cially enriched functors and simplicial natural transformations to the homotopy 2-category 120101119980119966119886119905In this way topologically enriched adjunctions define adjunctions between quasi-categories
216 Remark Topologically enriched adjunctions are relatively rare More prevalent are ldquoup-to-homotopyrdquo topologically enriched adjunctions such as those given by Quillen adjunctions betweensimplicial model categories These also define adjunctions between quasi-categories though the proofwill have to wait until Part II
The preservation of adjunctions by 2-functors proves
217 Proposition Given any adjunction 119860 119861119906perp119891
betweeninfin-categories then
(i) for anyinfin-category 119883
Fun(119883119860) Fun(119883 119861)
119906lowast
perp
119891lowast
defines an adjunction between quasi-categories(ii) for anyinfin-category 119883
hFun(119883119860) hFun(119883 119861)
119906lowast
perp
119891lowast
defines an adjunction between categories(iii) for any simplicial set 119880
119860119880 119861119880
119906119880
perp119891119880
defines an adjunction betweeninfin-categories(iv) and if the ambientinfin-cosmos is cartesian closed then for anyinfin-category 119862
119860119862 119861119862
119906119862
perp119891119862
defines an adjunction betweeninfin-categories
For instance taking 119883 = 1 in (ii) yields a ldquoderivedrdquo adjunction between the homotopy categoriesof theinfin-categories 119860 and 119861
Proof Any adjunction 119891 ⊣ 119906 in the homotopy 2-category 120101119974 is preserved by the 2-functorsFun(119883 minus) ∶ 120101119974 rarr 120101119980119966119886119905 hFun(119883 minus) ∶ 120101119974 rarr 119966119886119905 (minus)119880 ∶ 120101119974 rarr 120101119974 and (minus)119862 ∶ 120101119974 rarr 120101119974
33
218 Remark There are contravariant versions of each of the adjunction-preservation results ofProposition 217 the first of which we explain in detail Fixing the codomain variable of the functor-space at anyinfin-category 119862 isin 119974 defines a 2-functor
Fun(minus 119862) ∶ 120101119974op rarr 120101119980119966119886119905that is contravariant on 1-cells and covariant on 2-cells⁴ Similarly the cotensor or exponential 119862(minus) iscontravariant on 1-cells and covariant on 2-cells⁵ Such 2-functors preserve adjunctions but exchangeleft and right adjoints for instance given 119891 ⊣ 119906 in119974 we obtain an adjunction
Fun(119860 119862) Fun(119861 119862)
119891lowast
perp
119906lowast
between the functor- spaces
219 Proposition Adjunctions compose given adjoint functors
119862 119861 119860 119862 119860119891prime
perp119891
perp119906prime 119906
119891119891prime
perp119906prime119906
the composite functors are adjoint
Proof Writing 120578∶ id119861 rArr 119906119891 120598 ∶ 119891119906 rArr id119860 120578prime ∶ idc rArr 119906prime119891prime and 120598prime ∶ 119891prime119906prime rArr id119861 for therespective units and counits the pasting diagrams
119862 119862 119862
119861 119861 119861 119861
119860 119860 119860
119891prime dArr120578prime dArr120598prime119891prime
119891dArr120578
119906prime119906prime
dArr120598119891
119906119906
define the unit and counit of 119891119891prime ⊣ 119906prime119906 so that the triangle equalities
119862 119862 119862 119862 119862 119862
119861 119861 119861 119861 119861 119861
119860 119860 119860 119860 119860 119860
119891prime dArr120578prime dArr120598prime119891prime
119891119891prime119891119891prime
=dArr120598prime 119891
primedArr120578prime
119891dArr120578
119906prime
dArr120598119891
=
119906prime
dArr120598 119891 dArr120578
119906prime
=119906 119906
119906
119906prime119906 119906prime119906=
hold
⁴On a strict 2-category the superscript ldquooprdquo is used to signal that the 1-cells should be reversed but not the 2-cells thesuperscript ldquocordquo is used to signal that the 2-cells should be reversed but not the 1-cells and the superscript ldquocooprdquo is usedto signal that both the 1- and 2-cells should be reversed see Chapter B
⁵In the case of the simplicial cotensor the domain can safely be restricted to the homotopy 2-category of quasi-categories or can be regarded as an analogously-defined homotopy 2-category of simplicial sets
34
An adjoint to a given functor is unique up to natural isomorphism
2110 Proposition (uniqueness of adjoints)(i) If 119891 ⊣ 119906 and 119891prime ⊣ 119906 then 119891 cong 119891prime(ii) Conversely if 119891 ⊣ 119906 and 119891 cong 119891prime then 119891prime ⊣ 119906
Proof Writing 120578∶ id119861 rArr 119906119891 120598 ∶ 119891119906 rArr id119860 120578prime ∶ idc rArr 119906119891prime and 120598prime ∶ 119891prime119906 rArr id119861 for therespective units and counits the pasting diagrams
119861 119861 119861 119861
119860 119860 119860 119860119891prime
dArr120578prime119891
119891dArr120578
119891prime119906 dArr120598 119906 dArr120598prime
define 2-cells 119891 rArr 119891prime and 119891prime rArr 119891 The composites 119891 rArr 119891prime rArr 119891 and 119891prime rArr 119891 rArr 119891 are computed bypasting these diagrams together horizontally on one side or the other Applying the triangle equalitiesfor the adjunctions 119891 ⊣ 119906 and 119891prime ⊣ 119906 both composites are easily seen to be identities Hence 119891 cong 119891primeas functors from 119861 to 119860
Part (ii) is left as Exercise 21i
We will make repeated use of the following standard 2-categorical result which says that anyequivalence in a 2-category can be promoted to an equivalence that also defines an adjunction
2111 Proposition (adjoint equivalences) Any equivalence can be promoted to an adjoint equivalence bymodifying one of the 2-cells That is the invertible 2-cells in an equivalence can be chosen so as to satisfy thetriangle equalities Hence if 119891 and 119892 are inverse equivalences then 119891 ⊣ 119892 and 119892 ⊣ 119891
Proof Consider an equivalence comprised of functors 119891∶ 119860 rarr 119861 and 119892∶ 119861 rarr 119860 and invertible2-cells
119860 119860 119861 119861119892119891
congdArr120572
119891119892
congdArr120573
We will construct an adjunction 119891 ⊣ 119892 with unit 120578 ≔ 120572 by modifying 120573 The ldquotriangle identitycompositerdquo
120601 ≔ 119891 119891119892119891 119891119891120572 120573119891
is an isomorphism though likely not an identity Define
120598 ≔ 119891119892 119891119892 id119861 ≔ 119891119892 119891119892119891119892 119891119892 id119861120601minus1119892 120573 120573minus1119891119892 119891120572minus1119892 120573
This ldquocorrectsrdquo the counit so that now the composite 120598119891 sdot 119891120578 displayed on the top of the diagram
119891119892119891
119891 119891119892119891 119891
119891
120601minus1119892119891120598119891119891120572
120601minus1
120573119891
119891120572120601
35
which agrees with the bottom composite by ldquonaturality of whiskeringrdquo is the identity id119891Now by another diagram chase the other triangle composite 119892120598 sdot 120578119892 is an idempotent
119892 119892119891119892 119892
119892119891119892 119892119891119892119891119892 119892119891119892
119892119891119892 119892
120578119892
120578119892 120578119892119891119892
119892120598
120578119892119892119891120578119892 119892119891119892120598
119892120598119891119892 119892120598
119892120598
By cancelation any idempotent isomorphism is the identity proving that 119892120598 sdot 120578119892 = id119892
One use of Proposition 2110 is to show that adjunctions are equivalence invariant
2112 Proposition (equivalence-invariance of adjunctions) A functor119906∶ 119860 rarr 119861 betweeninfin-categoriesadmits a left adjoint if and only if for any pair of equivalentinfin-categories119860prime ≃ 119860 and 119861prime ≃ 119861 the equivalentfunctor 119906∶ 119860prime rarr 119861prime admits a right adjoint
Proof Exercise 21ii
As we will discover all ofinfin-category theory is equivalence invariant in this way
2113 Lemma For anyinfin-category 119860 the ldquocompositionrdquo functor
119860120794 times119860119860120794 119860120794∘perpperp
(minusiddom(minus))
(idcod(minus)minus)
(2114)
admits left and right adjoints which respectively ldquoextend an arrow into a composable pairrdquo by pairing it withthe identities at its domain or its codomain
Proof There is a dual adjunction in119966119886119905whose functors we describe using notation for simplicialoperators introduced in 111 the full subcategory of 119966119886119905 spanned by the finite non-empty ordinals isisomorphic to 120491
120795 120794 119860120795 1198601207941199040⊤
1199041⊤1198891 1198601198891
1198601199040
perp
1198601199041perp
Any infin-category 119860 in an infin-cosmos 119974 defines a 2-functor 119860(minus) ∶ 119966119886119905op rarr 120101119974 carrying the adjointtriple displayed above-left to the one displayed above-right
Now we claim there is a trivial fibration 119860120795 ⥲rarr 119860120794 times119860 119860120794 constructed as follows The pushoutdiagram of simplicial sets displayed below-left is carried by the simplicial cotensor119860(minus) ∶ 119982119982119890119905op rarr119974to a pullback diagram displayed below-right since the legs of the pushout square are monomorphisms
36
the legs of the pullback square are isofibrations
Λ1[2] 120794 119860Λ1[2] 119860120794
120794 120793 119860120794 119860
ev0
1198890
1198891
ev1
Lemma 1210 tells us that the cotensor of the inner horn inclusion Λ1[2] 120795 with the infin-category119860 defines a trivial fibration119860120795 ⥲rarr 119860Λ1[2] and the pullback square above left recognizes its codomainas the desiredinfin-category of ldquocomposable pairsrdquo Any section 119904 to 119902 ∶ 119860120795 ⥲rarr 119860120794 times119860 119860120794 can be madeinto an equivalence inverse By Proposition 2111 these functors are both left and right adjointsComposing the adjunction 119902 ⊣ 119904 ⊣ 119902 with the adjunction constructed above defines the desiredadjunction
Note that the adjoint functors of (2114) commute with the ldquoendpoint evaluationrdquo functors to119860 times 119860 In fact the units and counits can similarly be fibered over 119860 times 119860 see Example 3512
Exercises
21i Exercise Prove Proposition 2110(ii)
21ii Exercise Prove Proposition 2112 given an adjunction 119860 119861119906perp119891
and equivalences119860 ≃ 119860prime
and 119861 ≃ 119861prime construct an adjunction between 119860prime and 119861prime
22 Initial and terminal elements
Employing the tactic used to define the homotopy category of 119860 in Definition 1412 we use theterminal infin-category 1 to probe inside the infin-category 119860 The objects 119886 isin h119860 of the homotopycategory of 119860 were defined to be maps of infin-categories 119886 ∶ 1 rarr 119860 but to avoid the proliferation ofthe term ldquoobjectsrdquo we refer to maps 119886 ∶ 1 rarr 119860 as elements of theinfin-category 119860 instead
Before introducing limits and colimits of general diagram shapes we warm up by defining initialand terminal elements in aninfin-category 119860
221 Definition (initialterminal element) An initial element in an infin-category 119860 is a left adjointto the unique functor ∶ 119860 rarr 1 as displayed below left while a terminal element in an infin-category119860 is a right adjoint as displayed below right
1 119860 1 119860119894
perp 119905
perp
Let us unpack the definition of an initial element dual remarks apply to terminal elements
222 Lemma (the minimal data required to present an initial element) To define an initial element in119860 it suffices to specifybull an element 119894 ∶ 1 rarr 119860 and
37
bull a natural transformation1
119860 119860119894
dArr120598
so that the component 120598119894 ∶ 119894 rArr 119894 is the identity in h119860
Proof Proposition 145 demonstrates that the infin-category 1 isin 119974 is terminal in the homotopy2-category 120101119974 The 1-dimensional aspect of this universal property implies that 119894 defines a section ofthe unique map 119860 rarr 1 and from the 2-dimensional aspects we see that there exist no non-identity2-cells with codomain 1 In particular the unit of the adjunction 119894 ⊣ is necessarily an identity andone of the triangle equalities comes for free The data enumerated above is what remains of Definition211 in this setting
Put more concisely an initial element 119894 defines a left adjoint right inverse to the functor ∶ 119860 rarr 1Such adjunctions are studied more systematically in sectB4 In fact it suffices to assume that the counitcomponent 120598119894 is an isomorphism not necessarily the identity see Lemma B41
223 Remark Applying the 2-functor Fun(119883 minus) ∶ 120101119974 rarr 120101119980119966119886119905 to an initial or terminal elementof aninfin-category 119860 isin 119974 yields adjunctions
120793 cong Fun(119883 1) Fun(119883119860) 120793 cong Fun(119883 1) Fun(119883119860)
119894lowast
perp
119905lowast
perp
Via the isomorphisms Fun(119883 1) cong 120793 that express the universal property of the terminalinfin-category1 we see that initial or terminal elements of119860 define initial or terminal elements of the functor-spaceFun(119883119860) namely the composite functors
119883 1 119860 or 119883 1 119860 119894 119905
In particular initial or terminal elements are representably initial or terminal at the level of theinfin-cosmos
This representable universal property is also captured at the level of the homotopy 2-category Thenext lemma shows that the initial element 119894 ∶ 1 rarr 119860 is initial among all generalized elements 119891∶ 119883 rarr 119860in the following precise sense
224 Lemma An element 119894 ∶ 1 rarr 119860 is initial if and only if for all 119891∶ 119883 rarr 119860 there exists a unique 2-cellwith boundary
1
119883 119860119894
dArrexist
119891
38
Proof If 119894 ∶ 1 rarr 119860 is initial then the adjunction of Definition 221 is preserved by the 2-functorhFun(119883 minus) ∶ 120101119974 rarr 119966119886119905 defining an adjunction
120793 cong hFun(119883 1) hFun(119883119860)
119894lowast
perp
Via the isomorphism hFun(119883 1) cong 120793 this adjunction proves that the element 119894 ∶ 119883 rarr 119860 is initial inhFun(119883119860) and thus has the universal property of the statement
Conversely if 119894 ∶ 1 rarr 119860 satisfies the universal property of the statement applying this to thegeneric element of 119860 (the identity map id119860 ∶ 119860 rarr 119860) easily produces the data of Lemma 222
225 Remark Lemma 224 says that initial elements are representably initial in the homotopy 2-cat-egory Specializing the generalized elements to ordinary elements we see that initial and terminalelements in 119860 respectively define initial and terminal elements in the homotopy category h119860
226 Lemma If119860 has an initial element and119860 ≃ 119860prime then119860prime has an initial element and these elements arepreserved up to isomorphism by the equivalences
Proof By Proposition 2111 the equivalence119860 ≃ 119860prime can be promoted to an adjoint equivalencewhich can immediately be composed with the adjunction characterizing an initial element 119894 of 119860
1 119860 119860prime119894
perp
simperpsim
The composite adjunction provided by Proposition 219 proves that the image of 119894 defines an initialelement of 119860prime which by construction is preserved by the equivalence 119860 ⥲ 119860prime
To see that the equivalence 119860prime ⥲ 119860 also preserves initial elements we can use the invertible2-cells of the equivalence to see that 119894 is isomorphic to the image of the image of 119894 in 119860prime In case theinitial objects in mind are not the ones being considered here we can appeal to the uniqueness ofinitial elements proven in Exercise 22ii
Exercises
22i Exercise Prove that initial elements are preserved by left adjoints and terminal elements arepreserved by right adjoints
22ii Exercise Prove that any two initial elements in aninfin-category 119860 are isomorphic in h119860
23 Limits and colimits
Our aim is now to introduce limits and colimits of diagram valued inside aninfin-category119860 in someinfin-cosmos We will consider two varieties of diagramsbull In a genericinfin-cosmos119974 we shall consider diagrams indexed by a simplicial set 119869 and valued in
aninfin-category 119860
39
bull In a cartesian closedinfin-cosmos119974 we shall also consider diagram indexed by aninfin-category 119869 andvalued in aninfin-category 119860⁶
231 Definition (diagraminfin-categories) For a simplicial set 119869mdash or possibly in the case of a cartesianclosed infin-cosmos an infin-category 119869 mdash and an infin-category 119860 we refer to 119860119869 as the infin-category of119869-shaped diagrams in 119860 Both constructions define bifunctors
119982119982119890119905op times119974 119974 119974op times119974 119974
(119869 119860) 119860119869 (119869 119860) 119860119869
In either indexing context there is a terminal object 1 with the property that 1198601 cong 119860 for anyinfin-category 119860 Restriction along the unique map ∶ 119869 rarr 1 induces the constant diagram functorΔ∶ 119860 rarr 119860119869
We are deliberately conflating the notation for infin-categories of diagrams indexed by a simplicialset or by another infin-category because all of the results we will prove in Part I about the former casewill also apply to the latter For economy of language we refer only to simplicial set indexed diagramsfor the remainder of this section
232 Definition An infin-category 119860 admits all colimits of shape 119869 if the constant diagram functorΔ∶ 119860 rarr 119860119869 admits a left adjoint while119860 admits all limits of shape 119869 if the constant diagram functoradmits a right adjoint
119860119869 119860
colimperp
limperpΔ
233 Warning Limits or colimits of set-indexed diagrams mdash the case where the indexing shape is acoproduct of the terminal object 1 indexed by a set 119869mdash are called products or coproducts respectivelyIn this case theinfin-category of diagrams itself decomposes as a product 119860119869 cong prod119869119860 As the functor
120101119974 119966119886119905
119860 h119860
hFun(1minus)
that carries aninfin-category to its homotopy category preserves products when 119869 is a set there is a chainof isomorphisms
h(119860119869) cong h(prod119869119860) cong prod119869 h119860 cong (h119860)119869
Thus in this special case the adjunctions of Definition 232 that define products or coproducts in aninfin-category descend to the adjunctions that define products or coproducts in its homotopy category
However this argument does not extend to more general limit or colimit notions and suchinfin-cat-egorical limits or colimits are generally not limits or colimits in the homotopy category⁷ In sect32 we
⁶In Proposition 1334 proven in Part IV we shall discover that in the case of the infin-cosmoi of (infin 1)-categoriesthere is no essential difference between these notions in 119980119966119886119905 they are tautologically the same and in all biequivalentinfin-cosmsoi theinfin-category of diagrams indexed by aninfin-category119860 is equivalent to theinfin-category of diagrams indexedby its underlying quasi-category regarded as a simplicial set
⁷This sort of behavior is expected in abstract homotopy theory homotopy limits and colimits are not generally limitsor colimits in the homotopy category
40
shall see that the homotopy category construction fails to preserve more complicated cotensors evenin the relatively simple case of 119869 = 120794
The problem with Definition 232 is that it is insufficiently general manyinfin-categories will havecertain but not all limits of diagrams of a particular indexing shape So it would be desirable tore-express Definition 232 in a form that allows us to define the limit of a single diagram 119889∶ 1 rarr 119860119869or of a family of diagrams To achieve this we make use of the following 2-categorical notion thatop-dualizes the more familiar absolute extension diagrams
234 Definition (absolute lifting diagrams) Given a cospan 119862 119860 119861119892 119891
in a 2-categoryan absolute left lifting of 119892 through 119891 is given by a 1-cell and 2-cell as displayed below-left
119861 119883 119861 119883 119861
119862 119860 119862 119860 119862 119860uArr120582
119891 uArr120594
119887
119888 119891 =existuArr120577
uArr120582
119887
119888 119891
119892
ℓ
119892
ℓ
119892
so that any 2-cell as displayed above-center factors uniquely through (ℓ 120582) as displayed above-rightDually an absolute right lifting of 119892 through 119891 is given by a 1-cell and 2-cell as displayed below-left
119861 119883 119861 119883 119861
119862 119860 119862 119860 119862 119860dArr120588
119891 dArr120594
119887
119888 119891 =existdArr120577
dArr120588
119887
119888 119891
119892
119903
119892
119903
119892
so that any 2-cell as displayed above-center factors uniquely through (119903 120588) as displayed above-right
The adjectives ldquoleftrdquo and ldquorightrdquo refer to the handedness of the adjointness of these construc-tions left and right liftings respectively define left and right adjoints to the composition functor119891lowast ∶ hFun(119862 119861) rarr hFun(119862119860) with the 2-cells defining the components of the unit and counit ofthese adjunctions respectively at the object 119892 The adjective ldquoabsoluterdquo refers to the following stabilityproperty
235 Lemma Absolute left or right lifting diagrams are stable under restriction of their domain object if(ℓ 120582) defines an absolute left lifting of 119892 through 119891 then for any 119888 ∶ 119883 rarr 119862 the restricted diagram (ℓ119888 120582119888)defines an absolute left lifting of 119892119888 through 119891
119861
119883 119862 119860uArr120582
119891
119888 119892
ℓ
Proof Exercise 23i
Units and counits of adjunctions provide important examples of absolute left and right liftingdiagrams respectively
41
236 Lemma A 2-cell 120578∶ id119861 rArr 119906119891 defines the unit of an adjunction 119891 ⊣ 119906 if and only if (119891 120578) definesan absolute left lifting diagram displayed below-left
119860 119861
119861 119861 119860 119860uArr120578
119906dArr120598
119891119891 119906
Dually a 2-cell 120598 ∶ 119891119906 rArr id119860 defines the counit of an adjunction if and only if (119906 120598) defines an absolute rightlifting diagram displayed above-right
Proof We prove the universal property of the counit Given a 2-cell 120572∶ 119891119887 rArr 119886 as displayedbelow left
119883 119861 119883 119861
119860 119860 119860 119860
119887
119886 dArr120572 119891 =
119887
119886dArr120573
dArr120598119891119906
there exists a unique transpose 120573∶ 119887 rArr 119906119886 as displayed above-right across the induced adjunction
hFun(119883 119861) hFun(119883119860)
119891lowast
perp
119906lowast
between the hom-categories of the homotopy 2-category see Proposition 217(ii) From right to lefttransposes are composed by pasting with the counit hence the left-hand side above equals the right-hand side The converse is left as Exercise 23ii
In particular the unit of the adjunction colim ⊣ Δ of Definition 232 defines an absolute leftlifting diagram
119860
119860119869 119860119869uArr120578
Δcolim
By Lemma 235 this universal property is retained upon restricting to any subobject of theinfin-categoryof diagrams This motivates the following definitions
237 Definition A colimit of a family of diagrams 119889∶ 119863 rarr 119860119869 indexed by 119869 in aninfin-category 119860 isgiven by an absolute left lifting diagram
119860
119863 119860119869uArr120578
Δcolim
119889
comprised of a colimit functor colim ∶ 119863 rarr 119860 and a colimit cone 120578∶ 119889 rArr Δ colim
42
Dually a limit of a family of diagrams 119889∶ 119863 rarr 119860119869 indexed by 119869 in aninfin-category119860 is given by anabsolute right lifting diagram
119860
119863 119860119869dArr120598
Δlim
119889comprised of a limit functor lim ∶ 119863 rarr 119860 and a limit cone 120598 ∶ Δ limrArr 119889
238 Remark If 119860 has all limits of shape 119869 then Lemma 235 implies that any family of diagrams119889∶ 119863 rarr 119860119869 has a limit defined by evaluating the limit functor lim ∶ 119860119869 rarr 119860 at 119889 ie by restrictinglim along 119889 In certain infin-cosmoi such as 119980119966119886119905 if every diagram 119889∶ 1 rarr 119860119869 has a limit then 119860has all 119869-indexed limits because the quasi-category 1 generates theinfin-cosmos of quasi-categories in asuitable sense but this result is not true for allinfin-cosmoi
For example a 2-categorical lemma enables general proof of a classical result from homotopy the-ory that computes geometric realizations of ldquosplitrdquo simplicial objects Before proving this we introducethe indexing shapes involved
239 Definition (split augmented (co)simplicial objects) Recall 120491 is the simplex category of finitenon-empty ordinals and order-preserving maps introduced in 111 It defines a full subcategory of acategory 120491+ which freely appends the empty ordinal ldquo[minus1]rdquo as an initial object This in turn definesa wide subcategory of a category 120491perp which adds an ldquoextrardquo degeneracy 120590minus1 ∶ [119899 + 1] ↠ [119899] betweeneach pair of consecutive ordinals including 120590minus1 ∶ [0] ↠ [minus1]
Diagrams indexed by 120491 sub 120491+ sub 120491perp are respectively called cosimplicial objects coaugmentedcosimplicial objects and split coaugmented cosimplicial objects if they are covariant and simplicialobjects augmented simplicial objects and split augmented simplicial objects if they are contravariant
A simplicial object 119889∶ 1 rarr 119860120491opin an infin-category 119860 admits an augmentation or admits a split-
ting if it lifts along the restriction functors
119860120491perpop
119860120491op+
1 119860120491op
119889
The family of simplicial objects admitting an augmentation and splitting is then represented by thegeneric element 119860120491perp
op↠ 119860120491op
The following proposition proves that for any simplicial objectadmitting a splitting the augmentation defines the colimit cone dual results apply to colimits of splitcosimplicial objects The limit and colimit cones are defined by cotensoring with the unique naturaltransformation
120491 120491+
120793 [minus1]
uArr120584 (2310)
that exists because [minus1] ∶ 120793 rarr 120491+ is initial see Lemma 22443
2311 Proposition (geometric realizations) Let 119860 be any infin-category For every cosimplicial object in119860 that admits a coaugmentation and a splitting the coaugmentation defines a limit cone Dually for everysimplicial object in 119860 that admits an augmentation and a splitting the augmentation defines a colimit coneThat is there exist absolute right and left lifting diagrams
119860 119860
119860120491perp 119860120491+ 119860120491 119860120491perpop
119860120491op+ 119860120491op
dArr119860120584Δ
uArr119860120584op Δ
res
ev[minus1]
res res res
ev[minus1]
in which the 2-cells are obtained as restrictions of the cotensor of the 2-cell (2310) with 119860 Moreover suchlimits and colimits are absolute preserved by any functor 119891∶ 119860 rarr 119861 ofinfin-categories
Proof By Example B52 the inclusion 120491 120491perp admits a right adjoint which can automaticallybe regarded as an adjunction ldquooverrdquo 120793 since 120793 is 2-terminal The initial element [minus1] isin 120491+ sub 120491perpdefines a left adjoint to the constant functor
120491 120491+ 120491perp
120793
⊤
[minus1] perp
with the counit of this adjunction (2310) defining the colimit cone under the constant functor atthe initial element These adjunctions are preserved by the 2-functor 119860(minus) ∶ 119966119886119905op rarr 120101119974 yielding adiagram
119860
119860120491perp 119860120491+ 119860120491dArr119860120584
Δ
perp
res
ev[minus1]
res⊤
By Lemma B51 these adjunctions witness the fact that evaluation at [minus1] and the 2-cell from (2310)define an absolute right lifting of the canonical restriction functor 119860120491perp rarr 119860120491 through the constantdiagram functor as claimed The colimit case is proven similarly by applying the composite 2-functor
119966119886119905coop 119966119886119905op 120101119974(minus)op 119860(minus)
Finally by the 2-functoriality of the simplicial cotensor any 119891∶ 119860 rarr 119861 commutes with the 2-cellsdefined by cotensoring with 120584 or its opposite
119860 119861 119861
119860120491perp 119860120491+ 119860120491 119861120491 119860120491perp 119861120491perp 119861120491+ 119861120491dArr119860120584
Δ
119891
Δ =dArr119860120584
Δ
res
ev[minus1]
res 119891120491 119891120491perp res
ev[minus1]
res
Since the right-hand composite is an absolute right lifting diagram so is the left-hand compositewhich says that 119891∶ 119860 rarr 119861 preserves the totalization of any split coaugmented cosimplicial object in119860
44
Exercises
23i Exercise Prove Lemma 235
23ii Exercise Re-prove the forwards implication of Lemma 236 by following your nose through apasting diagram calculation and prove the converse similarly
24 Preservation of limits and colimits
Famously right adjoint functors preserve limits and left adjoints preserve colimits Our aim inthis section is to prove this in the infin-categorical context and exhibit the first examples of initial andfinal functors in the sense introduced in Definition 246 below
The commutativity of right adjoints and limits is very easily established in the case where theinfin-categories in question admit all limits of a given shape under these hypotheses the limit functoris right adjoint to the constant diagram functor which commutes with all functors between the baseinfin-categories Since the left adjoints commute the uniqueness of adjoints (Proposition 2110) impliesthat the right adjoints do as well This outline gives a hint for Exercise 24i
A slightly more delicate argument is needed in the general case involving say the preservation ofa single limit diagram without a priori assuming that any other limits exist This follows easily froma general lemma about composition and cancelation of absolute lifting diagrams
241 Lemma (composition and cancelation of absolute lifting diagrams) Suppose (119903 120588) defines an ab-solute right lifting of ℎ through 119891
119862
119861
119863 119860
dArr120590119892
dArr120588119891
ℎ
119903
119904
Then (119904 120590) defines an absolute right lifting of 119903 through 119892 if and only if (119904 120588 sdot 119891120590) defines an absolute rightlifting of ℎ through 119891119892
Proof Exercise 24ii
242 Theorem (RAPLLAPC) Right adjoints preserve limits and left adjoints preserve colimits
The usual argument that right adjoints preserve limits proceeds like this a cone over a 119869-shapeddiagram in the image of 119906 transposes across the adjunction 119891119869 ⊣ 119906119869 to a cone over the original diagramwhich factors through the designated limit cone This factorization transposes across the adjunction119891 ⊣ 119906 to define the sought-for unique factorization through the image of the limit cone The use ofabsolute lifting diagrams to express the universal properties of limits and colimits (Definition 237)and adjoint transposition (Lemma 236) allows us to economize on the usual proof by suppressingconsideration of a generic test cone that must be shown to uniquely factor through the limit cone
Proof We prove that right adjoints preserve limits By taking ldquocordquo duals the same argumentdemonstrates that left adjoints preserve colimits
45
Suppose 119906∶ 119860 rarr 119861 admits a left adjoint 119891∶ 119861 rarr 119860 with unit 120578∶ id119861 rArr 119906119891 and counit 120598 ∶ 119891119906 rArrid119860 Our aim is to show that any absolute right lifting diagram as displayed below-left is carried toan absolute right lifting diagram as displayed below-right
119860 119860 119861
119863 119860119869 119863 119860119869 119861119869dArr120588
ΔdArr120588
Δ
119906
Δlim
119889
lim
119889 119906119869
(243)
The cotensor (minus)119869 ∶ 120101119974 rarr 120101119974 carries the adjunction 119891 ⊣ 119906 to an adjunction 119891119869 ⊣ 119906119869 with unit 120578119869and counit 120598119869 In particular by Lemma 236 (119906119869 120598119869) defines an absolute right lifting of the identitythrough 119891119869 which is then preserved by restriction along the functor 119889 Thus by Lemma 241 thediagram on the right of (243) is an absolute right lifting diagram if and only if the pasted compositedisplayed below-left
119860 119861 119861 119861
119863 119860119869 119861119869 119860 119860 119860
119860119869 119863 119860119869 119860119869 119863 119860119869
dArr120588Δ
119906
ΔdArr120598
119891dArr120598 lim
119891lim
119889119906119869
119891119869dArr120598119869=
dArr120588
119906
Δ Δ
=
dArr120588Δ
119889
lim
119889
lim
119906 lim
defines an absolute right lifting diagram Pasting the 2-cell on the right of (243) with the counit inthis way amounts to transposing the cone under 119906 lim across the adjunction 119891119869 ⊣ 119906119869
Wersquoll now observe that this transposed cone factors through the limit cone (lim 120588) in a canonicalway From the 2-functoriality of the simplicial cotensor in its exponent variable 119891119869Δ = Δ119891 and120598119869Δ = Δ120598 Hence the pasting diagram displayed above-left equals the one displayed above-centerand hence also by naturality of whiskering the diagram above-right⁸ This latter diagram is a pastedcomposite of two absolute right lifting diagrams and is hence an absolute right lifting diagram in itsown right this universal property says that any cone over 119889 whose summit factors through 119891 factorsuniquely through the limit cone (lim 120588) through a map that then transposes along the adjunction119891 ⊣ 119906 Hence all of the diagrams in the statement are absolute right lifting diagrams including inparticular the one on the right-hand side of (243)
By combining Theorem 242 with Proposition 2111 we have immediately that
244 Corollary Equivalences preserve limits and colimits
We can also prove a more refined result
245 Proposition If 119860 ≃ 119861 then any family of diagrams in 119860 admitting a limit or colimit in 119861 alsoadmits a limit or colimit in 119860 that is preserved by the equivalence
Proof By Proposition 2111 the equivalence 119861 ⥲ 119860 is both left and right adjoint to its equiva-lence inverse preserving both limits and colimits of the composite family of diagrams119863 rarr 119860119869 ⥲ 119861119869Via the invertible 2-cells of the equivalence 119860119869 ≃ 119861119869 constructed by applying (minus)119869 ∶ 120101119974 rarr 120101119974 to theequivalence 119860 ≃ 119861 the preserved diagram 119863 rarr 119860119869 ⥲ 119861119869 ⥲ 119860119869 is isomorphic to the original family
⁸By naturality of whiskering 120598119869119889 sdot 119891119869119906119869120588 = 120588 sdot 120598119869Δ lim and since 120598119869Δ = Δ120598 this composite equals 120588 sdot Δ120598 lim
46
of diagrams 119863 rarr 119860119869 Thus we conclude that a family of diagrams in 119860 has a limit or colimit if andonly if its image in an equivalentinfin-category 119861 does and such limits and colimits are preserved by theequivalence
The following definition makes sense between small quasi-categories or equally between arbitraryinfin-categories in a cartesian closedinfin-cosmos
246 Definition (initial and final functor) A functor 119896 ∶ 119868 rarr 119869 is final if 119869-indexed colimits existif and only if and in such cases coincide with the restricted 119868-indexed colimits That is 119896 ∶ 119868 rarr 119869 isfinal if and only if for anyinfin-category 119860 the square
119860 119860
119860119869 119860119868Δ Δ
119860119896
preserves and reflects all absolute left lifting diagramsDually a functor 119896 ∶ 119868 rarr 119869 is initial if this square preserves and reflects all absolute right lifting
diagrams or informally if a generalized element defines a limit of a 119869-indexed diagram if and only ifit defines a limit of the restricted 119868-indexed diagrams
Historically final functors were called ldquocofinalrdquo with no obvious name for the dual notion Ourpreferred terminology hinges on the following mnemonic the inclusion of an initial element definesan initial functor while the inclusion of a terminal (aka final) element defines a final functor Theseresults are special cases of a more general result we now establish using exactly the same tactics astaken to prove Theorem 242
247 Proposition Left adjoints define initial functors and right adjoints define final functors
Proof If 119896 ⊣ 119903 with unit 120578∶ id119868 rArr 119903119896 and counit 120598 ∶ 119896119903 rArr id119869 then cotensoring into 119860 yieldsan adjunction
119860119869 119860119868
119860119896perp119860119903
with unit 119860120578 ∶ id119860119868 rArr 119860119896119860119903 and counit 119860120598 ∶ 119860119903119860119896 rArr id119860119869 To prove that 119896 is initial we must show that for any (119889 lim 120588) as displayed below-left
119860 119860 119860
119863 119860119869 119863 119860119869 119860119868dArr120588
ΔdArr120588
Δ Δlim
119889
lim
119889 119860119896
the left-hand diagram is an absolute right lifting diagram if and only if the right-hand diagram is anabsolute right lifting diagram
47
By Lemmas 236 and 241 the right-hand diagram is an absolute right lifting diagram if and onlyif the pasted composite displayed below-left
119860 119860 119860
119863 119860119869 119860119868 119863 119860119869
119860119869
dArr120588Δ Δ
dArr120588Δlim
119889119860119896
119860119903dArr119860120598
=
lim
119889
is also an absolute right lifting diagram On noting that 119860119903Δ = Δ and 119860120598Δ = idΔ the left-hand sidereduces to the right-hand side which proves the claim
Exercises
24i Exercise Show that any left adjoint 119891∶ 119861 rarr 119860 between infin-categories admitting all 119869-shapedcolimits preserves them in the sense that the square of functors
119861119869 119860119869
119861 119860colim
119891119869
colimcong
119891
commutes up to isomorphism
24ii Exercise Prove Lemma 241
24iii Exercise Given a proof of Theorem 242 that does not appeal to Lemma 241 by directlyverifying that the diagram on the right of (243) is an absolute right lifting diagram
24iv Exercise Use Lemma 241 to give a new proof of Proposition 219
48
CHAPTER 3
Weak 2-limits in the homotopy 2-category
In Chapter 2 we introduced adjunctions between infin-categories and limits and colimits of dia-grams valued within aninfin-category through definitions that are particularly expedient for establishingthe expected interrelationships But neither 2-categorical definition clearly articulates the universalproperties of these notions Definition 237 does not obviously express the expected universal prop-erty of the limit cone namely that the limit cone over a diagram 119889 defines the terminal element oftheinfin-category of cones over 119889 yet-to-be-defined Nor have we understood how an adjunction 119891 ⊣ 119906induces an equivalence on as-yet-to-be-defined hom-spaces Hom119860(119891119887 119886) ≃ Hom119861(119887 119906119886) for a pair ofgeneralized elementssup1 In this section we make use of the completeness axiom in the definition of aninfin-cosmos to exhibit a general construction that will specialize to give a definition of thisinfin-categoryof cones and also specialize to define these hom-spaces This construction will also permit us to rep-resent a functor betweeninfin-categories as aninfin-category in dual ldquoleftrdquo or ldquorightrdquo fashions Using thiswe can redefine an adjunction to consist of a pair of functors 119891∶ 119861 rarr 119860 and 119906∶ 119860 rarr 119861 so that theleft representation of 119891 is equivalent to the right representation of 119906 over 119860 times 119861
Our vehicle for all of these new definitions is the commainfin-category associated to a cospan
119862 119860 119861119892 119891
Hom119860(119891 119892)
119862 times 119861
(11990111199010)
Our aim in this chapter is to develop the general theory of comma constructions from the point ofview of the homotopy 2-category of aninfin-cosmos Our first payoff for this work will occur in Chapter4 where we study the universal properties of adjunctions limits and colimits in the sense of the ideasjust outlined The comma construction will also provide the essential vehicle for establishing themodel-independence of the categorical notions we will introduce throughout this text
There is a standard definition of a ldquocomma objectrdquo that can be stated in any strict 2-categorydefined as a particular weighted limit (see Example 7117) Comma infin-categories do not satisfy thisuniversal property in the homotopy 2-category however Instead they satisfy a somewhat peculiarldquoweakrdquo variant of the usual 2-categorical universal property that to our knowledge has not been dis-covered elsewhere in the categorical or homotopical literature expressed in terms of something wecall a smothering functor To introduce these universal properties in a concrete rather than abstractframework we start in sect31 by considering smothering functors involving homotopy categories ofquasi-categories The intrepid and impatient reader may skip the entirety of sect31 if they wish to in-stead first encounter these notions in their full generality
sup1A 2-categorical version of this result mdash exhibiting a bijection between sets of 2-cells mdash appears as Lemma 236 butin aninfin-category wersquod hope for a similar equivalence of hom-spaces
49
31 Smothering functors
Let 119876 be a quasi-category Recall from Lemma 1112 that its homotopy category h119876 hasbull the elements 1 rarr 119876 of 119876 as its objectsbull the set of homotopy classes of 1-simplices of119876 as its arrows where parallel 1-simplices are homo-
topic just when they bound a 2-simplex with the remaining outer edge degenerate andbull a composition relation if and only if any chosen 1-simplices representing the three arrows bound
a 2-simplexFor a 1-category 119869 it is well-known in classical homotopy theory that the homotopy category of dia-grams h(119876119869) is not equivalent to the category (h119876)119869 of diagrams in the homotopy category mdash exceptin very special cases such as when 119869 is a set (see Warning 233) The objects of h(119876119869) are homotopycoherent diagrams of shape 119869 in 119876 while the objects of (h119876)119869 are mere homotopy commutative diagramsThere is however a canonical comparison functor
h(119876119869) rarr (h119876)119869
defined by applying h ∶ 119980119966119886119905 rarr 119966119886119905 to the evaluation functor 119876119869 times 119869 rarr 119876 and then transposing ahomotopy coherent diagram is in particular homotopy commutative
Our first aim in this section is to better understand the relationship between the arrows in thehomotopy category h119876 and what wersquoll refer to as the arrows of 119876 namely the 1-simplices in thequasi-category To study this wersquoll be interested in the quasi-category in which the arrows of119876 live aselements namely 119876120794 where 120794 = Δ[1] is the nerve of the ldquowalkingrdquo arrow Our notation deliberatelyimitates the notation commonly used for the category of arrows if 119862 is a 1-category then 119862120794 is thecategory whose objects are arrows in 119862 and whose morphisms are commutative squares regarded as amorphism from the arrow displayed vertically on the left-hand side to the arrow displayed verticallyon the right-hand side This notational conflation suggests our first motivating question how doesthe homotopy category of 119876120794 relate to the category of arrows in the homotopy category of 119876
311 Lemma The canonical functor h(119876120794) rarr (h119876)120794 isbull surjective on objectsbull full andbull conservative ie reflects invertibility of morphisms
but not injective on objects nor faithful
Proof Surjectivity on objects asserts that every arrow in the homotopy category h119876 is repre-sented by a 1-simplex in 119876 This is the conclusion of Exercise 11ii(iii) which outlines the proof ofLemma 1112
To prove fullness consider a commutative square in h119876 and choose arbitrary 1-simplices repre-senting each morphism and their common composite
bull bull
bull bull119891
ℎ
ℓ 119892
119896
By Lemma 1112 every composition relation in h119876 is witnessed by a 2-simplex in 119876 choosing a pairof such 2-simplices defines a diagram 120794 rarr 119876120794 which represents a morphism from 119891 to 119892 in h(119876120794)proving fullness
50
Surjectivity on objects and fullness of the functor h(119876120794) rarr h(119876)120794 are special properties havingto do with the diagram shape 120794 Conservativity is much more general as a consequence of the secondstatement of Corollary 1121
The properties of the canonical functor h(119876120794) rarr h(119876)120794 will reappear frequently so are worthgiving a name
312 Definition A functor 119891∶ 119860 rarr 119861 between 1-categories is smothering if it is surjective onobjects full and conservative That is a functor is smothering if and only if it has the right liftingproperty with respect to the set of functors
⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩
empty 120793 + 120793 120794
120793 120794 120128
⎫⎪⎪⎪⎪⎬⎪⎪⎪⎪⎭
Some elementary properties of smothering functors are established in Exercise 31i The mostimportant of these is
313 Lemma Each fibre of a smothering functor is a non-empty connected groupoid
Proof Suppose 119891∶ 119860 rarr 119861 is smothering and consider the fiber
119860119887 119860
120793 119861
119891
119887
over an object 119887 of 119861 By surjectivity on objects the fiber is non-empty Its morphisms are definedto be arrows between objects in the fiber of 119887 that map to the identity on 119887 By fullness any twoobjects in the fiber are connected by a morphism indeed by morphisms pointing in both directionsBy conservativity all the morphisms in the fiber are necessarily invertible
The argument used to prove Lemma 311 generalizes to
314 Lemma If 119869 is a 1-category that is free on a reflexive directed graph and119876 is a quasi-category then thecanonical functor h(119876119869) rarr (h119876)119869 is smothering
Proof Exercise 31ii
Cotensors are one of the simplicial limit constructions enumerated in axiom 121(i) Other limitconstructions listed there also give rise to smothering functors
315 Lemma For any pullback diagram of quasi-categories in which 119901 is an isofibration
119864 times119861 119860 119864
119860 119861
119901
119891
the canonical functor h(119864 times119861 119860) rarr h119864 timesh119861 h119860 is smothering
51
Proof As h ∶ 119980119966119886119905 rarr 119966119886119905 does not preserve pullbacks the canonical comparison functor of thestatement is not an isomorphism It is however bijective on objects since the composite functor
119980119966119886119905 119966119886119905 119982119890119905h obj
is given by evaluation on the set of vertices of each quasi-category and this functor does preservepullbacks
For fullness note that a morphism in h119864timesh119861 h119860 is represented by a pair of 1-simplices 120572∶ 119886 rarr 119886primeand 120598 ∶ 119890 rarr 119890prime in 119860 and 119864 whose images in 119861 are homotopic a condition that implies in particularthat 119891(119886) = 119901(119890) and 119891(119886prime) = 119901(119890prime) By Lemma 119 we can arrange this homotopy however we likeand thus we choose a 2-simplex witness 120573 so as to define a lifting problem
Λ1[2] 119864 ni
Δ[2] 119861 ni
119901
120573
119890119890 119890prime
119901(119890)
119891(119886) 119891(119886prime) = 119901(119890prime)
120598
119901(120598)
119891(120572)
Since 119901 is an isofibration a solution exists defining an arrow ∶ 119890 rarr 119890prime in 119864 in the same homotopyclass as 120598 so that 119901() = 119891(120572) The pair (120572 ) now defines the lifted arrow in h(119864 times119861 119860)
Finally consider an arrow 120794 rarr 119864 times119861 119860 whose image in h119864 timesh119861 h119860 is an isomorphism whichis the case just when the projections to 119864 and 119860 define isomorphisms By Corollary 1116 we maychoose a homotopy coherent isomorphism 120128 rarr 119860 extending the given isomorphism 120794 rarr 119860 Thisdata presents us with a lifting problem
120794 119864 times119861 119860 119864
120128 119860 119861
119901
119891
which Exercise 11v tells us we can solve This proves that h(119864 times119861 119860) rarr h119864 timesh119861 h119860 is conservativeand hence also smothering
A similar argument proves
316 Lemma For any tower of isofibrations between quasi-categories
⋯ 119864119899 119864119899minus1 ⋯ 1198642 1198641 1198640the canonical functor h(lim119899 119864119899) rarr lim119899 h119864119899 is smothering
Proof Exercise 31iii
52
317 Lemma For any cospan between quasi-categories 119862 119860 119861119892 119891
consider the quasi-categorydefined by the pullback
Hom119860(119891 119892) 119860120794
119862 times 119861 119860 times 119860
(11990111199010)
(coddom)
119892times119891
The canonical functor hHom119860(119891 119892) rarr Homh119860(h119891 h119892) is smothering
Proof Here the codomain is the category defined by an analogous pullback
Homh119860(h119891 h119892) (h119860)120794
h119862 times h119861 h119860 times h119860
(coddom)
h119892timesh119891
in 119966119886119905 and the canonical functor factors as
hHom119860(119891 119892) rarr h(119860120794) timesh119860timesh119860 (h119862 times h119861) rarr (h119860)120794 timesh119860timesh119860 (h119862 times h119861)By Lemma 315 the first of these functors is smothering By Lemma 311 the second is a pullback of asmothering functor By Exercise 31i(i) it follows that the composite functor is smothering
In the sections that follow we will discover that the smothering functors just constructed expressparticular ldquoweakrdquo universal properties of arrow pullback and comma constructions in the homotopy2-category of anyinfin-cosmos It is to the first of these that we now turn
Exercises
31i Exercise Prove that(i) The class of smothering functors is closed under composition retract product and pullback(ii) The class of smothering functors contains all surjective equivalences of categories(iii) All smothering functors are isofibrations that is maps that have the right lifting property
with respect to 120793 120128(iv) Prove that if 119891 and 119892119891 are smothering functors then 119892 is a smothering functorsup2
31ii Exercise Prove Lemma 314
31iii Exercise Prove Lemma 316
32 infin-categories of arrows
In this section we replicate the discussion from the start of the previous section using an arbitraryinfin-category 119860 in place of the quasi-category 119876 The analysis of the previous section could have beendeveloped natively in this general setting but at the cost of an extra layer of abstraction and moreconfusing notation mdash with a functor space Fun(119883119860) replacing the quasi-category 119876
Recall an element of aninfin-category is defined to be a functor 1 rarr 119860 Tautologically the elementsof 119860 are the vertices of the underlying quasi-category Fun(1 119860) of 119860 In this section we will define
sup2It suffices in fact to merely assume that 119891 is surjective on objects and arrows
53
and study an infin-category 119860120794 whose elements are the 1-simplices in the underlying quasi-category of119860 We refer to 119860120794 as theinfin-category of arrows in 119860 and call its elements simply arrows of 119860
In fact wersquove tacitly introduced this construction already Recall 120794 is our preferred notation forthe quasi-categoryΔ[1] as this coincides with the nerve of the 1-category 120794with a single non-identitymorphism 0 rarr 1321 Definition (arrow infin-category) Let 119860 be an infin-category The infin-category of arrows in 119860 isthe simplicial cotensor 119860120794 together with the canonical endpoint-evaluation isofibration
119860120794 ≔ 119860Δ[1] 119860120597Δ[1] cong 119860 times 119860(11990111199010)
induced by the inclusion 120597Δ[1] Δ[1] For conciseness we write 1199010 ∶ 119860120794 ↠ 119860 for the domain-evaluation induced by the inclusion 0∶ 120793 120794 and write 1199011 ∶ 119860120794 ↠ 119860 for the codomain-evaluationinduced by 1∶ 120793 120794
As an object of the homotopy 2-category 120101119974 the infin-category of arrows comes equipped with acanonical 2-cell that we now construct
322 Lemma For anyinfin-category 119860 theinfin-category of arrows comes equipped with a canonical 2-cell
119860120794 1198601199010
1199011
dArr120581 (323)
that we refer to as the generic arrow with codomain 119860
Proof The simplicial cotensor has a strict universal property described inDigression 124 namely119860120794 is characterized by the natural isomorphism
Fun(119883119860120794) cong Fun(119883119860)120794 (324)
By the Yoneda lemma the data of the natural isomorphism (324) is encoded by its ldquouniversal elementrdquowhich is defined to be the image of the identity at the representing object Here the identity functorid ∶ 119860120794 rarr 119860120794 is mapped to an element of Fun(119860120794 119860)120794 a 1-simplex in Fun(119860120794 119860) which representsa 2-cell in the homotopy 2-category defining (323)
To see that its source and target must be the domain-evaluation and codomain-evaluation mapsnote that the action of the simplicial cotensor 119860(minus) on morphisms of simplicial sets is defined so thatthe isomorphism (324) is natural in the cotensor variable as well Thus by restricting along theendpoint inclusion 120793+120793 120794 we may regard the isomorphism (324) as lying over Fun(119883119860times119860) congFun(119883119860) times Fun(119883119860)
There is a 2-categorical limit notion that is analogous to Definition 321 which constructs forany object 119860 the universal 2-cell with codomain 119860 namely the cotensor with the 1-category 120794 Itsuniversal property is analogous to (324) but with the hom-categories of the 2-category in place of thefunctor spaces In 119966119886119905 this constructs the arrow category associated to a 1-category
In the homotopy 2-category 120101119974 by the Yoneda lemma again the data (323) encodes a naturaltransformation
hFun(119883119860120794) rarr hFun(119883119860)120794
of categories but this is not a natural isomorphism nor even a natural equivalence of categories butdoes express the arrowinfin-category as a ldquoweakrdquo arrow object with a universal property of the followingform
54
325 Proposition (the weak universal property of the arrow infin-category) The generic arrow (323)with codomain 119860 has a weak universal property in the homotopy 2-category given by three operations
(i) 1-cell induction Given a 2-cell over 119860 as below-left
119883 119883
= 119860120794
119860 119860
119904119905120572lArr 119904119905
119886
11990101199011 120581lArr
there exists a 1-cell 119886 ∶ 119883 rarr 119860120794 so that 119904 = 1199010119886 119905 = 1199011119886 and 120572 = 120581119886(ii) 2-cell induction Given a pair of functors 119886 119886prime ∶ 119883 119860120794 and a pair of 2-cells 1205910 and 1205911 so that
119883 119883
119860120794 119860120794 = 119860120794 119860120794
119860 119860
119886119886prime1205911lArr
119886119886prime1205910lArr
1199011
1199011
1199010
120581lArr1199011
1199010120581lArr 1199010
there exists a 2-cell 120591∶ 119886 rArr 119886prime so that119883 119883 119883 119883
119860120794 119860120794 119860120794 119860120794 119860120794 119860120794
119860 119860 119860 119860
119886119886prime
1205911lArr
119886119886prime 120591lArr119886119886prime 120591lArr
119886119886prime
1205910lArr
1199011 11990111199011
= and1199010
=
1199010 1199010
(iii) 2-cell conservativity Any 2-cell119883
119860120794
119886119886prime 120591lArr
with the property that both 1199011120591 and 1199010120591 are isomorphisms is an isomorphism
Proof Let 119876 = Fun(119883119860) and apply Lemma 311 to observe that the natural map of hom-categories
hFun(119883119860120794) hFun(119883119860)120794
hFun(119883119860) times hFun(119883119860)((1199011)lowast(1199010)lowast) (ev1ev0)
over hFun(119883119860times119860) cong hFun(119883119860) times hFun(119883119860) is a smothering functor Surjectivity on objects isexpressed by 1-cell induction fullness by 2-cell induction and conservativity by 2-cell conservativity
55
Note that the functors 119883 rarr 119860120794 that represent a fixed 2-cell with domain 119883 and codomain119860 arenot unique However they are unique up to ldquofiberedrdquo isomorphisms that whisker with (1199011 1199010) ∶ 119860120794 ↠119860times119860 to an identity 2-cell
326 Proposition Whiskering with (323) induces a bijection between 2-cells with domain119883 and codomain119860 as displayed below-left
⎧⎪⎪⎨⎪⎪⎩119883 119860
119904
119905
dArr120572
⎫⎪⎪⎬⎪⎪⎭
⎧⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎩
119883
119860 119860
119860120794
119905 119904
119886
11990101199011
⎫⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎭cong
and fibered isomorphism classes of functors119883 rarr 119860120794 as displayed above-right where the fibered isomorphismsare given by invertible 2-cells
119883
119860 119860
119860120794
119905 119904119886119886prime 120574
cong
11990101199011
so that 1199010120574 = id119904 and 1199011120574 = id119905
Proof Lemma 313 proves that the fibers of the smothering functor of Proposition 325 are con-nected groupoids The objects of these fibers are functors 119883 rarr 119860120794 and the morphisms are invertible2-cells that whisker with (1199011 1199010) ∶ 119860120794 ↠ 119860 times 119860 to an identity 2-cell The action of the smotheringfunctor defines a bijection between the objects of its codomain and their corresponding fibers
Our final task is to observe that the universal property of Proposition 325 is also enjoyed by anyobject (1198901 1198900) ∶ 119864 ↠ 119860 times 119860 that is equivalent to the arrow infin-category (1199011 1199010) ∶ 119860120794 ↠ 119860 times 119860 inthe slice infin-cosmos 119974119860times119860 We have special terminology to allow us to concisely express the type ofequivalence we have in mind
327 Definition (fibered equivalence) A fibered equivalence over aninfin-category 119861 in aninfin-cosmos119974 is an equivalence
119864 119865
119861
sim(328)
in the slicedinfin-cosmos119974119861 We write 119864 ≃119861 119865 to indicate that that specified isofibrations with thesedomains are equivalent over 119861
By Proposition 1217(vii) a fibered equivalence is just a map between a pair of isofibrations over acommon base that defines an equivalence in the underlyinginfin-cosmos the forgetful functor119974119861 rarr119974preserves and reflects equivalences Note however that it does not create them it is possible for twoinfin-categories 119864 and 119865 to be equivalent without there existing any equivalence compatible with a pairof specified isofibration 119864 ↠ 119861 and 119865 ↠ 119861
56
329 Remark At this point there is some ambiguity about the 2-categorical data that presents afibered equivalence related to the question posed in Exercise 14iii But since Proposition 1217(vii)tells us that a mere equivalence in 120101119974 involving a functor of the form (328) is sufficient to guaranteethat this as-yet-unspecified 2-categorical data exists we defer a careful analysis of this issue to sect35
3210 Proposition (uniqueness of arrow infin-categories) For any isofibration (1198901 1198900) ∶ 119864 ↠ 119860 times 119860equipped with a fibered equivalence 119890 ∶ 119864 ⥲ 119860120794 the corresponding 2-cell
119864 1198601198900
1198901
dArr120598
satisfies the weak universal property of Proposition 325 Conversely if (1198891 1198890) ∶ 119863 ↠ 119860times119860 and (1198901 1198900) ∶ 119864 ↠119860 times 119860 are equipped with 2-cells
119863 119860 and 119864 1198601198890
1198891
dArr120575
1198900
1198901
dArr120598
satisfying the weak universal property of Proposition 325 then 119863 and 119864 are fibered equivalent over 119860 times 119860
Proof We prove the first statement By the definition equation of 1-cell induction 120598 = 120581119890 where120581 is the canonical 2-cell of (323) Hence pasting with 120598 induces a functor
hFun(119883 119864) hFun(119883119860120794) hFun(119883119860)120794
hFun(119883119860) times hFun(119883119860)
119890lowast
((1199011)lowast(1199010)lowast) (ev1ev0)
and our task is to prove that this composite functor is smothering We see that the first functordefined by post-composing with the equivalence 119890 ∶ 119864 rarr 119860120794 is an equivalence of categories and thesecond functor is smothering Thus the composite is clearly full and conservative To see that it isalso surjective on objects note first that by 1-cell induction any 2-cell
119883 119860119904
119905
dArr120572
is represented by a functor 119886 ∶ 119883 rarr 119860120794 over119860times119860 Composing with any fibered inverse equivalence119890prime to 119890 yields a functor
119883 119860120794 119864
119860 times 119860
119886
(119905119904)(11990111199010)
sim119890prime
(11989011198900)
whose image after post-composing with 119890 is isomorphic to 119886 over119860times119860 Because this isomorphism isfibered (see Proposition 326) the image of 119886119890prime under the functor hFun(119883 119864) rarr hFun(119883119860)120794 returnsthe 2-cell 120572 This proves that this mapping is surjective on objects and hence defines a smotheringfunctor as claimed
The converse is left to Exercise 32ii and proven in a more general context in Proposition 3311
57
3211 Convention On account of Proposition 3210 we extend the appellation ldquoinfin-category of ar-rowsrdquo from the strict model constructed in Definition 321 to anyinfin-category that is fibered equivalentto it
Via Lemma 314 the discussion of this section extends to establish corresponding weak universalproperties for the cotensors 119860119869 of an infin-category 119860 with a free category 119869 We leave the explorationof this to the reader
Exercises
32i Exercise(i) Prove that a parallel pair of 1-simplices in a quasi-category 119876 are homotopic if and only if
they are isomorphic as elements of 119876120794 via an isomorphism that projects to an identity along(1199011 1199010) ∶ 119876120794 ↠ 119876times119876
(ii) Conclude that a parallel pair of 1-arrows in the functor space Fun(119883119860) between twoinfin-categories119883 and119860 in anyinfin-cosmos represent the same natural transformation if and only if theyare isomorphic as elements of Fun(119883119860)120794 cong Fun(119883119860120794) via an isomorphism whose domainand codomain components are an identity
(iii) Conclude that a parallel pair of 1-arrows in the functor space Fun(119883119860) which may be en-coded as functors 119883 119860120794 represent the same natural transformation if and only if they areconnected by a fibered isomorphism
119883 119860120794
119860 times 119860
cong
(11990111199010)
32ii Exercise Prove the second statement of Proposition 3210
33 The comma construction
The comma infin-category is defined by restricting the domain and codomain of the infin-category ofarrows 119860120794 along specified functors with codomain 119860
331 Definition (comma infin-category) Let 119862 119860 119861119892 119891
be a diagram of infin-categoriesThe commainfin-category is constructed as a pullback of the simplicial cotensor 119860120794 along 119892 times 119891
Hom119860(119891 119892) 119860120794
119862 times 119861 119860 times 119860
(11990111199010)
120601
(11990111199010)
119892times119891
(332)
58
This construction equips the commainfin-categorywith a specified isofibration (1199011 1199010) ∶ Hom119860(119891 119892) ↠119862 times 119861 and a canonical 2-cell
Hom119860(119891 119892)
119862 119861
119860
1199011 1199010
120601lArr
119892 119891
(333)
in the homotopy 2-category called the comma cone
334 Example (arrowinfin-categories as commainfin-categories) The arrowinfin-category arises as a spe-cial case of the comma construction applied to the identity span This provides us with alternatenotation for the generic arrow of (323) which may be regarded as a particular instance of a commacone
Hom119860
119860 119860
119860
1199011 1199010
120601lArr
The following proposition encodes the homotopical properties of the comma construction Theproof is by a standard argument in abstract homotopy theory which can be found in Appendix C Ahint for this proof is given in Exercise 33i
335 Proposition (maps between commas) A commutative diagram
119862 119860 119861
119903
119892
119901 119902
119891
119891
induces a map between the commainfin-categories
Hom119860(119891 119892) Hom( 119891 )
119862 times 119861 times
(11990111199010)
Hom119901(119902119903)
(11990111199010)119903times119902
Moreover if 119901 119902 and 119903 are all(i) equivalences(ii) isofibrations or(iii) trivial fibrations
then the induced map is again an equivalence isofibration or trivial fibration respectively
59
There is a 2-categorical limit notion that is analogous to Definition 331 which constructs theuniversal 2-cell inhabiting a square over a specified cospan In 119966119886119905 the category so-constructed isreferred to as a comma category from when we borrow the name As with the case ofinfin-categories ofarrow commainfin-categories do not satisfy this 2-universal property strictly Instead
336 Proposition (the weak universal property of the comma infin-category) The comma cone (333)has a weak universal property in the homotopy 2-category given by three operations
(i) 1-cell induction Given a 2-cell over 119862 119860 119861119892 119891
as below-left
119883
119862 119861
119860
119888 119887
120572lArr
119892 119891
=
119883
Hom119860(119891 119892)
119862 119861
119860
119888 119887119886
1199011 1199010
120601lArr
119892 119891
there exists a 1-cell 119886 ∶ 119883 rarr Hom119860(119891 119892) so that 119887 = 1199010119886 119888 = 1199011119886 and 120572 = 120601119886(ii) 2-cell induction Given a pair of functors 119886 119886prime ∶ 119883 Hom119860(119891 119892) and a pair of 2-cells 1205910 and 1205911 so
that
119883 119883
Hom119860(119891 119892) Hom119860(119891 119892) = Hom119860(119891 119892) Hom119860(119891 119892)
119862 119861 119862 119861
119860 119860
119886119886prime
1205911lArr
119886119886prime
1205910lArr
11990111199011 1199010
120601lArr
1199011 1199010
120601lArr
1199010
119892 119891 119892 119891
there exists a 2-cell 120591∶ 119886 rArr 119886prime so that119883 119883 119883 119883
Hom119860(119891 119892) Hom119860(119891 119892) Hom119860(119891 119892) Hom119860(119891 119892) Hom119860(119891 119892) Hom119860(119891 119892)
119862 119862 119861 119861
119886119886prime
1205911lArr
119886119886prime 120591lArr119886119886prime 120591lArr
119886119886prime
1205910lArr
1199011 11990111199011
=
and 1199010
=
1199010 1199010
(iii) 2-cell conservativity Any 2-cell119883
Hom119860(119891 119892)
119886119886prime 120591lArr
60
with the property that both 1199011120591 and 1199010120591 are isomorphisms is an isomorphism
Proof The cosmological functor Fun(119883 minus) ∶ 119974 rarr 119980119966119886119905 carries the pullback (332) to a pull-back
Fun(119883Hom119860(119891 119892)) cong HomFun(119883119860)(Fun(119883 119891) Fun(119883 119892)) Fun(119883119860)120794
Fun(119883 119862) times Fun(119883 119861) Fun(119883119860) times Fun(119883119860)
(11990111199010)
120601
(11990111199010)
Fun(119883119892)timesFun(119883119891)
of quasi-categories Now Lemma 317 demonstrates that the canonical 2-cell (333) induces a naturalmap of hom-categories
hFun(119883Hom119860(119891 119892)) HomhFun(119883119860)(hFun(119883 119891) hFun(119883 119892))
hFun(119883 119862) times hFun(119883 119861)((1199011)lowast(1199010)lowast) (ev1ev0)
over hFun(119883 119862times119861) cong hFun(119883 119862)timeshFun(119883 119861) that is a smothering functor The properties of 1-cellinduction 2-cell induction and 2-cell conservativity follow from surjectivity on objects fullness andconservativity of this smothering functor respectively
The 1-cells 119883 rarr Hom119860(119891 119892) that are induced by a fixed 2-cell 120572∶ 119891119887 rArr 119892119888 are unique up tofibered isomorphism over 119862 times 119861
337 Proposition Whiskering with the comma cone (333) induces a bijection between 2-cells as displayedbelow-left ⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩
119883
119862 119861
119860
119888 119887
120572lArr
119892 119891
⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭
⎧⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎩
119883
119862 119861
Hom119860(119891 119892)
119888 119887
119886
11990101199011
⎫⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎭cong
and fibered isomorphism classes of maps of spans from 119862 to 119861 as displayed above-right where the fiberedisomorphisms are given by invertible 2-cells
119883
119862 119861
Hom119860(119891 119892)
119888 119887
119886119886prime 120574cong
11990101199011
so that 1199010120574 = id119887 and 1199011120574 = id119888
Proof Lemma 313 proves that the fibers of the smothering functor of Proposition 336 areconnected groupoids The objects of these fibers are functors 119883 rarr Hom119860(119891 119892) and the morphismsare invertible 2-cells that whisker with
(1199011 1199010) ∶ Hom119860(119891 119892) ↠ 119862 times 11986161
to an identity 2-cell The action of the smothering functor defines a bijection between the objects ofits codomain and their corresponding fibers
The construction of the comma infin-category is also pseudo-functorial in lax maps defined in thehomotopy 2-category
338 Observation By 1-cell induction a diagram
119862 119860 119861
119903
119892
lArr120574 119901 lArr120573 119902
119891
119891
induces a map between commainfin-categories as displayed below-right
Hom119860(119891 119892) Hom119860(119891 119892)
119862 119861 Hom( 119891 )
119860
1199011 1199010
120601lArr
120573darr12057411990211990101199031199011
119892119903 119891 119902 =1199011 1199010
120601lArrlArr120574
119901
lArr120573
119891 119891
that is well-defined and functorial up to fibered isomorphism
One of many uses of comma infin-categories is to define the internal mapping spaces between twoelements of aninfin-category 119860 This is one motivation for our notation ldquoHom119860rdquo
339 Definition For any two elements 119909 119910 ∶ 1 119860 of an infin-category 119860 their mapping space isthe commainfin-category Hom119860(119909 119910) defined by the pullback diagram
Hom119860(119909 119910) 119860120794
1 119860 times 119860
(11990111199010)
120601
(11990111199010)
(119910119909)
The mapping spaces in anyinfin-category are discrete in the sense of Definition 1221
3310 Proposition (internal mapping spaces are discrete) For any pair of elements 119909 119910 ∶ 1 119860 of aninfin-category 119860 the mapping space Hom119860(119909 119910) is discrete
Proof Our task is to prove that for any infin-category 119883 the functor space Fun(119883Hom119860(119909 119910))is a Kan complex This is so just when hFun(119883Hom119860(119909 119910)) is a groupoid ie when any 2-cell withcodomain Hom119860(119909 119910) is invertible By 2-cell conservativity a 2-cell with codomain Hom119860(119909 119910) isinvertible just when its whiskered composite with the isofibration (1199011 1199010) ∶ Hom119860(119909 119910) ↠ 1 times 1 isan invertible 2-cell but in fact this whiskered composite is an identity since 1 is terminal
62
As in our convention forinfin-categories of arrows it will be convenient to weaken the meaning ofldquocommainfin-categoryrdquo to extend this appellation to any object of119974119862times119861 that is fibered equivalent (seeDefinition 327) to the strict model (1199011 1199010) ∶ Hom119860(119891 119892) ↠ 119862 times 119861 defined by 331 This is justifiedbecause such objects satisfy the weak universal property of Proposition 336 and conversely any twoobjects satisfying this weak universal property are equivalent over 119862 times 119861
3311 Proposition (uniqueness of comma infin-categories) For any isofibration (1198901 1198900) ∶ 119864 ↠ 119862 times 119861that is fibered equivalent to Hom119860(119891 119892) ↠ 119862 times 119861 the 2-cell
119864
119862 119861
119860
1198901 1198900
120598lArr
119892 119891
encoded by the equivalence 119864 ⥲ Hom119860(119891 119892) satisfies the weak universal property of Proposition 336 Con-versely if (1198891 1198890) ∶ 119863 ↠ 119862 times 119861 and (1198901 1198900) ∶ 119864 ↠ 119862 times 119861 are equipped with 2-cells
119863 119864
119862 119861 and 119862 119861
119860 119860
1198891 1198890
120575lArr
1198901 1198900
120598lArr
119892 119891 119892 119891
(3312)
satisfying the weak universal property of Proposition 336 then 119863 and 119864 are fibered equivalent over 119862 times 119861
Proof The proof of the first statement proceeds exactly as in the special case of Proposition3210 We prove the converse solving Exercise 32ii
Consider a pair of 2-cells (3312) satisfying the weak universal properties enumerated in Proposi-tion 336 1-cell induction supplies maps of spans
119863
119862 119861
119860
1198891 1198890120575lArr
119892 119891
=
119863
119864
119862 119861
119860
1198891 1198890119889
1198901 1198900120598lArr
119892 119891
and
119864
119862 119861
119860
1198901 1198900120598lArr
119892 119891
=
119864
119863
119862 119861
119860
1198901 1198900119890
1198891 1198890120575lArr
119892 119891
with the property that 120598119889119890 = 120598 and 120575119890119889 = 120575 By Proposition 337 it follows that 119889119890 cong id119864 over 119862 times 119861and 119890119889 cong id119863 over 119862 times 119861 This defines the data of a fibered equivalence 119863 ≃ 119864sup3
3313 Convention On account of Proposition 3311 we extend the appellation ldquocommainfin-categoryrdquofrom the strict model constructed in Definition 331 to any infin-category that is fibered equivalent toit and refer to its accompanying 2-cell as the ldquocomma conerdquo
sup3For the reader uncomfortable with Remark 329 Proposition 353 and Lemma 354 provides a small boost to finishthe proof
63
For example in sect43we define theinfin-category of cones over a fixed diagram as a commainfin-categoryProposition 3311 gives us the flexibility to use multiple models for thisinfin-category which will be use-ful in characterizing the universal properties of limits and colimits
Exercises
33i Exercise Prove Proposition 335 by observing that the map Hom119901(119902 119903) factors as a pullback ofthe Leibniz cotensor of 120597Δ[1] Δ[1] with 119901 followed by a pullback of 119903 times 119902
33ii Exercise Use Proposition 337 to justify the pseudofunctoriality of the comma constructionin lax morphisms described in Observation 338
34 Representable commainfin-categories
Definition 331 constructs a commainfin-category for any cospan Of particular importance are thespecial cases of this construction where one of the legs of the cospan is an identity
341 Definition (left and right representations) Any functor 119891∶ 119860 rarr 119861 admits a left representa-tion and a right representation as a commainfin-category displayed below-left and below-right respec-tively
Hom119861(119891 119861) Hom119861(119861 119891)
119861 119860 119860 119861
119861 119861
1199011 1199010
120601lArr
1199011 1199010
120601lArr
119891 119891
To save space we typically depict the left comma cone over 119901 displayed above-left and the right commacone over 119901 displayed above-right as inhabiting triangles rather than squares
By Proposition 3311 the weak universal property of the comma cone characterizes the commaspan up to fibered equivalence over the product of the codomain objects Thus
342 Definition A commainfin-category Hom119860(119891 119892) ↠ 119862 times 119861 isbull left representable if there exists a functor ℓ ∶ 119861 rarr 119862 so that Hom119860(119891 119892) ≃ Hom119862(ℓ 119862) over119862 times 119861 and
bull right representable if there exists a functor 119903 ∶ 119862 rarr 119861 so that Hom119860(119891 119892) ≃ Hom119861(119861 119903) over119862 times 119861
In this section we prove the first of many representability theorems demonstrating that a functor119892∶ 119862 rarr 119860 admits an absolute right lifting along 119891∶ 119861 rarr 119860 if and only if the comma infin-categoryHom119860(119891 119892) is right representable the representing functor then defining the postulated lifting Weprove this over the course of three theorems each strengthening the previous statement The firsttheorem characterizes 2-cells
119861
119862 119860dArr120588
119891
119892
119903
64
that define absolute right lifting diagrams via an induced equivalence Hom119861(119861 119903) ≃119862times119861 Hom119860(119891 119892)between commainfin-categories The second theorem proves that a functor 119903 defines an absolute rightlifting of 119892 through 119891 just when Hom119860(119891 119892) is right-represented by 119903 the difference is that no 2-cell120588∶ 119891119903 rArr 119892 need be postulated a priori to exist The final theorem gives a general right-representabilitycriterion that can be applied to construct a right representation to Hom119860(119891 119892) without a priori spec-ifying the representing functor 119903
343 Theorem The triangle below-left defines an absolute right lifting diagram if and only if the induced1-cell below-right
119861
119862 119860dArr120588
119891
119892
119903
Hom119861(119861 119903)
119862 119861
119860
1199011 1199010120601lArr
120588lArr
119903
119892 119891
=
Hom119861(119861 119903)
Hom119860(119891 119892)
119862 119861
119860
1199010 1199011119910
1199011 1199010
120601lArr
119892 119891
(344)
defines a fibered equivalence Hom119861(119861 119903) ≃ Hom119860(119891 119892) over 119862 times 119861
Proof Suppose that (119903 120588) defines an absolute right lifting of 119892 through 119891 and consider the cor-responding unique factorization of the comma cone under Hom119860(119891 119892) through 120588 as displayed below-center
Hom119860(119891 119892)
119862 119861
119860
1199011 1199010
120601lArr
119892 119891
=
Hom119860(119891 119892)
119862 119861
119860
1199011 1199010120577lArr
120588lArr
119903
119892 119891
=
Hom119860(119891 119892)
Hom119861(119861 119903)
119862 119861
119860
1199011 1199010119911
1199011 1199010120601lArr
120588lArr
119903
119892 119891
(345)By 1-cell induction the 2-cell 120577 factors through the right comma cone over 119903 as displayed above-right Substituting the right-hand side of (344) into the bottom portion of the above-right diagramwe see that 119910119911 ∶ Hom119860(119891 119892) rarr Hom119860(119891 119892) is a 1-cell that factors the comma cone for Hom119860(119891 119892)through itself Applying the universal property of Proposition 337 it follows that there is a fiberedisomorphism 119910119911 cong idHom119860(119891119892) over 119862 times 119861
To prove that 119911119910 cong idHom119861(119861119903) it suffices to argue similarly that the right comma cone over 119903restricts along 119911119910 to itself Since 120588 is absolute right lifting it suffices to verify the equality 120601119911119910 = 120601
65
after pasting below with 120588 But now reversing the order of the equalities in (345) and (344) we have
Hom119861(119861 119903)
Hom119860(119891 119892)
Hom119861(119861 119903)
119862 119861
119860
1199011 1199010
119910
1199011 1199010
119911
1199011 1199010120601lArr
120588lArr
119903
119892 119891
=
Hom119861(119861 119903)
Hom119860(119891 119892)
119862 119861
119860
1199011 1199010119910
1199011 1199010120577lArr
120588lArr
119903
119892 119891
=
Hom119861(119861 119903)
Hom119860(119891 119892)
119862 119861
119860
1199011 1199010119910
1199011 1199010
120601lArr
119892 119891
=
Hom119861(119861 119903)
119862 119861
119860
1199011 1199010120601lArr
120588lArr
119903
119892 119891
which is exactly what we wanted to show Thus we see that if (119903 120588) is an absolute right lifting of 119892through 119891 then the induced map (344) defines a fibered equivalence Hom119861(119861 119903) ≃ Hom119860(119891 119892)
Now conversely suppose the 1-cell 119910 defined by (344) is a fibered equivalence and let us arguethat (119903 120588) is an absolute right lifting of 119892 through 119891 By Proposition 3311 via this fibered equivalencethe 2-cell displayed on the left-hand side of (344) inherits the weak universal property of a commacone from Hom119860(119891 119892) So Proposition 337 supplies a bijection displayed below-left-center⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩
119883
119862 119861
119860
119888 119887
120572lArr
119892 119891
⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭
⎧⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎩
119883
119862 119861
Hom119861(119861 119903)
119888 119887
119886
11990101199011
⎫⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎭cong
⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩
119883
119862 119861
119888 119887120585lArr
119903
⎫⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎭
between 2-cells over the cospan and fibered isomorphism classes of maps of spans that is implementedfrom center to left by whiskering with the 2-cell 1205881199011 sdot 119891120601∶ 1198911199010 rArr 1198921199011 in the center of (344)Proposition 337 also applies to the right comma cone 120601 over 119903 ∶ 119862 rarr 119861 giving us a second bijectiondisplayed above center-right between the same fibered isomorphism classes of maps of spans and2-cells over 119903 This second bijection is implemented from center to right by pasting with the rightcomma cone 120601∶ 1199010 rArr 1199031199011 Combining these yields a bijection between the 2-cells displayed onthe right and the 2-cells displayed on the left implemented by pasting with 120588 which is precisely theuniversal property that characterizes absolute right lifting diagrams
As a special case of this result we can now present several equivalent characterizations of fullyfaithful functors betweeninfin-categories
346 Corollary The following are equivalent and define what it means for a functor 119891∶ 119860 rarr 119861 betweeninfin-categories to be fully faithful
66
(i) The identity defines an absolute right lifting diagram
119860
119860 119861
119891
119891
(ii) The identity defines an absolute left lifting diagram
119860
119860 119861
119891
119891
(iii) For all 119883 isin 119974 the induced functor
119891lowast ∶ hFun(119883119860) rarr hFun(119883 119861)is a fully faithful functor of 1-categories
(iv) The functor induced by the identity 2-cell id119891 is an equivalence
119964120794
119860 119860
Hom119861(119891 119891)
1199011 1199010
sim id1198911199011 1199010
Proof The statement (iii) is an unpacking of the meaning of both (i) and (ii) Theorem 343specializes to prove (i)hArr(iv) or dually (ii)hArr(iv)
It is not surprising that post-composition with a fully faithful functor of infin-categories shouldinduce a fully-faithful functor of hom-categories in the homotopy 2-category What is surprising isthat this definition is strong enough This result together with the general case of Theorem 343should be provide some retroactive justification for our use of absolute lifting diagrams in Chapter 2
Having proven Theorem 343 our immediate aim is to strengthen it to show that a fibered equiv-alence Hom119861(119861 119903) ≃ Hom119860(119891 119892) over 119862 times 119861 implies that 119903 ∶ 119862 rarr 119861 defines an absolute right liftingof 119892 through 119891 without a previously specified 2-cell 120588∶ 119891119903 rArr 119892
347 Theorem Given a trio of functors 119903 ∶ 119862 rarr 119861 119891∶ 119861 rarr 119860 and 119892∶ 119862 rarr 119860 there is a bijection between2-cells as displayed below-left and fibered isomorphism classes of maps of spans as displayed below-right
⎧⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎩
119861
119862 119860dArr120588
119891
119892
119903
⎫⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎭
⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎩
Hom119861(119861 119903)
119862 119861
Hom119860(119891 119892)
1199011 1199010
119910
11990101199011
⎫⎪⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎪⎭cong
67
that is constructed by pasting the right comma cone over 119903 and then applying 1-cell induction to factor throughthe comma cone for Hom119860(119891 119892)
Hom119861(119861 119903)
119862 119861
119860
1199011 1199010120601lArr
120588lArr
119903
119892 119891
=
Hom119861(119861 119903)
Hom119860(119891 119892)
119862 119861
119860
1199011 1199010119910
1199011 1199010
120601lArr
119892 119891
Moreover a 2-cell 120588∶ 119891119903 rArr 119892 displays 119903 as an absolute right lifting of 119892 through 119891 if and only if the corre-sponding map of spans 119910∶ Hom119861(119861 119903) rarr Hom119860(119891 119892) is an equivalence
The second clause is the statement of Theorem 343 so it remains only to prove the first We showthe claimed construction is a bijection by exhibiting its inverse the construction of which involves arather mysterious lemma the significance of which will gradually reveal itself For instance Lemma348 figures prominently in the proof of the external Yoneda lemma in sect55 and is also the mainingredient in a ldquocheaprdquo version of the Yoneda lemma appearing as Corollary 3410
348 Lemma Let 119891∶ 119860 rarr 119861 be any functor and denote the right comma cone over 119891 by
Hom119861(119861 119891)
119860 119861
1199011 1199010120601lArr
119891
Then the codomain-projection functor 1199011 ∶ Hom119861(119861 119891) ↠ 119860 admits a right adjoint right inverse⁴ inducedfrom the identity 2-cell id119891 defining an adjunction
119860 perp Hom119861(119861 119891)
119860119894 1199011
1199011
over 119860 whose counit is an identity and whose unit 120578∶ idrArr 1198941199011 satisfies the conditions 120578119894 = id119894 1199011120578 = id1199011and 1199010120578 = 120601
⁴A functor admits a right adjoint right inverse just when it admits a right adjoint in an adjunction whose counit isthe identity When the original functor is an isofibration as is the case here it suffices to merely assume that the counit isinvertible see Lemma 359 and Appendix B
68
Proof This adjunctionwill be constructed using theweak universal properties of the right commacone over 119891 The identity 2-cell id119891 induces a 1-cell over the right comma cone over 119891
119860
Hom119861(119861 119891)
119860 119861
119891119894
1199011 1199010120601lArr
119891
=119860
119860 119861
119891=
119891
Note that 1199011119894 = id119860 so we may take the counit to be the identity 2-cell Since 120601119894 = id119891 we have apasting equality
Hom119861(119861 119891)
Hom119861(119861 119891) 119860
Hom119861(119861 119891) Hom119861(119861 119891) = Hom119861(119861 119891)
119860 119861 119860 119861
1199010
1199011
120601lArr
1198941199011
=
119894 119891
11990111199011 1199010120601
lArr1199011 1199010120601
lArr
119891 119891
while allows us to induce a 2-cell 120578∶ idrArr 1198941199011 with defining equations 1199011120578 = id1199011 and 1199010120578 = 120601 Thefirst of these conditions ensures one triangle identity for the other we must verify that 120578119894 = id119894 By2-cell conservativity 120578119894 is an isomorphism since 1199011120578119894 = id119860 and 1199010120578119894 = 120601119894 = id119891 are both invertibleBy naturality of whiskering we have
119894 119894
119894 119894
120578119894
120578119894 120578119894
1198941199011120578119894
and since 1199011120578 = id1199011 the bottom edge is an identity So 120578119894 sdot 120578119894 = 120578119894 and since 120578119894 is an isomorphismcancelation implies that 120578119894 = id119894 as required
One interpretation of Lemma 348 is best revealed though a special case
349 Corollary For any element 119887 ∶ 1 rarr 119861 the identity at 119887 defines a terminal element in Hom119861(119861 119887)
Proof By Lemma 348 the codomain-projection from the right representation of any functor ad-mits a right adjoint right inverse induced from its identity 2-cell In this case the codomain-projectionis the unique functor ∶ Hom119861(119861 119887) rarr 1 so by Definition 221 this right adjoint identifies a terminalelement of Hom119861(119861 119887) corresponding to the identity morphism id119887 in the homotopy category h119861
The general version of Lemma 348 has a similar interpretation in the slicedinfin-cosmos119974119860 theidentity functor at119860 defines the terminal object and Lemma 348 asserts that id119891 induces a terminalelement of Hom119861(119861 119891) ldquoover 119860rdquo
69
Proof of Theorem 347 The inverse to the function that takes a 2-cell 119891119903 rArr 119892 and producesan isomorphism class of maps Hom119861(119861 119903) rarr Hom119860(119891 119892) over 119862 times 119861 is constructed by applyingLemma 348 to the functor 119903 ∶ 119862 rarr 119861 given a map of spans restrict along the right adjoint 119894 ∶ 119862 rarrHom119861(119861 119903) and paste with the comma cone for Hom119860(119891 119892) to define a 2-cell 119891119903 rArr 119892
Starting from a 2-cell 120588∶ 119891119903 rArr 119892 the composite of these two functions constructs the 2-celldisplayed below-left
119862
Hom119861(119861 119903)
Hom119860(119891 119892)
119862 119861
119860
119894
119903
1199011 1199010
119910
1199011 1199010
120601lArr
119892 119891
=
119862
Hom119861(119861 119903)
119862 119861
119860
119894119903
1199011 1199010120601lArr
120588lArr
119903
119892 119891
=
119862
119862 119861
119860
=119903
120588lArr
119903
119892 119891
which equals the above-center pasted composite by the definition of 119910 from 120588 and equals the above-right composite since 120601119894 = id119903 Thus when a 2-cell 120588∶ 119891119903 rArr 119892 is encoded as a map 119910∶ Hom119861(119861 119903) rarrHom119860(119891 119892) over 119862 times 119861 and then re-converted into a 2-cell the original 2-cell 120588 is recovered
For the converse starting with a map 119911 ∶ Hom119861(119861 119903) rarr Hom119860(119891 119892) over 119862 times 119861 the compositeof these two functions constructs an isomorphism class of maps of spans 119908 displayed below-left byapplying 1-cell induction for the comma cone Hom119860(119891 119892) to the composite 2-cell pasted below-center-left
Hom119861(119861 119903)
Hom119860(119891 119892)
119862 119861
119860
1199011 1199010
119908
1199011 1199010
120601lArr
119892 119891
=
Hom119861(119861 119903)
119862
Hom119861(119861 119903)
Hom119860(119891 119892)
119862 119861
119860
1199011
1199010
120601lArr
119894
119903
1199011 1199010
119911
1199011 1199010
120601lArr
119892 119891
=
Hom119861(119861 119903)
119862
Hom119861(119861 119903)
Hom119860(119891 119892)
119862 119861
119860
1199011
120578lArr119894
1199011 1199010
119911
1199011 1199010
120601lArr
119892 119891
=
Hom119861(119861 119903)
Hom119860(119891 119892)
119862 119861
119860
1199011 1199010
119911
1199011 1199010
120601lArr
119892 119891
70
Applying Lemma 348 there exists a 2-cell 120578∶ id rArr 1198941199011 so that 1199010120578 = 120601 mdash this gives the pastingequality above center mdash and 1199011120578 = id mdash which gives the pasting equality above right Proposition337 now implies that 119908 cong 119911 over 119862 times 119861
A dual version of Theorem 347 represents 2-cells 119892 rArr 119891ℓ as fibered isomorphism classes of mapsHom119861(ℓ 119861) rarr Hom119860(119892 119891) over 119861 times 119862 Specializing these results to the case where one of 119891 or 119892 isthe identity we immediately recover a ldquocheaprdquo form of the Yoneda lemma
3410 Corollary Given a parallel pair of functors 119891 119892 ∶ 119860 119861 there are bijections between 2-cells asdisplayed below-center and fibered isomorphism classes of maps between their left and right representations ascommainfin-categories as displayed below-left and below-right respectively⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎩
Hom119861(119892 119861)
119861 119860
Hom119861(119891 119861)
1199011 1199010
119886
11990101199011
⎫⎪⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎪⎭cong
⎧⎪⎪⎪⎨⎪⎪⎪⎩119860 119861
119891
119892dArr120572
⎫⎪⎪⎪⎬⎪⎪⎪⎭
⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎩
Hom119861(119861 119891)
119860 119861
Hom119861(119861 119892)
1199011 1199010
119886
11990101199011
⎫⎪⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎪⎭cong
that are constructed by pasting with the left comma cone over 119892 and right comma cone over 119891 respectively
Hom119861(119892 119861)
119861 119860
1199011 1199010lArr120601
119892
119891uArr120572
=
Hom119861(119892 119861)
Hom119861(119891 119861)
119861 119860
1199011 1199010119886
1199011 1199010120601lArr
119891
Hom119861(119861 119891)
119860 119861
1199011 1199010
119891
119892dArr120572
lArr120601 =
Hom119861(119861 119891)
Hom119861(119861 119892)
119860 119861
1199011 1199010119886
1199011 1199010120601lArr
119892
and then applying 1-cell induction to factor through the left comma cone over 119891 in the former case or the rightcomma cone over 119892 in the latter
Combining the results of this section we prove one final representability theorem that allowsus to recognize when a comma infin-category is right representable in the absence of a predeterminedrepresenting functor This result specializes to give existence theorems for adjoint functors and limitsand colimits in the next chapter
3411 Theorem The commainfin-category Hom119860(119891 119892) associated to a cospan 119862 119860 119861119892 119891
isright representable if and only if its codomain-projection functor admits a right adjoint right inverse
Hom119860(119891 119892)
119862 119861
11990101199011perp119894
in which case the composite 1199010119894 ∶ 119862 rarr 119861 defines the representing functor and the 2-cell represented by thefunctor 119894 ∶ 119862 rarr Hom119860(119891 119892) defines an absolute right lifting of 119892 through 119891
71
Proof If Hom119860(119891 119892) is represented on the right by a functor 119903 ∶ 119862 rarr 119861 then Hom119860(119891 119892) ≃Hom119861(119861 119903) over 119862 times 119861 and the codomain-projection functor is equivalent to 1199011 ∶ Hom119861(119861 119903) ↠ 119862which admits a right adjoint right inverse 119894 by Lemma 348 The proof of Theorem 343 then showsthat 119894 represents an absolute right lifting diagram Thus it remains only to prove the converse
To that end suppose we are given a right adjoint right inverse adjunction 1199011 ⊣ 119894 Unpacking thedefinition this provides an adjunction
119862 perp Hom119860(119891 119892)
119862119894 1199011
1199011
over 119862 whose counit is an identity and whose unit 120578∶ id rArr 1198941199011 satisfies the conditions 120578119894 = id119894 and1199011120578 = id1199011 By Theorem 347 to construct the fibered equivalence Hom119861(119861 119903) ≃119862times119861 Hom119860(119891 119892)with 119903 ≔ 1199010119894 it suffices to demonstrate that the 2-cell defined by restricting the comma cone forHom119860(119891 119892) along 119894
119862
Hom119860(119891 119892)
119862 119861
119860
119903119894
1199011 1199010
120601lArr
119892 119891
defines an absolute right lifting diagramBy 1-cell induction any 2-cell as displayed below-left induces a 1-cell119898 as displayed below-center
119883 119861
119862 119860
119887
119888 dArr120594 119891
119892
=119883 Hom119860(119891 119892) 119861
119862 119860
119887
119898
1198881199011
1199010
dArr120601 119891
119892
=119883 Hom119860(119891 119892) Hom119860(119891 119892) 119861
119862 119862 119860
119887
119898
1198881199011
dArr120578 1199011
1199010
dArr120601 119891119894
119892
Inserting the triangle equality 1199011120578 = id1199011 as displayed above-right constructs the desired factorization1199010120578119898∶ 119887 rArr 119903119888 of 120594 through 120601119894
In fact by 2-cell induction for the comma cone 120601 any 2-cell 1205910 ∶ 119887 rArr 119903119888 defining a factorizationof 120594∶ 119891119887 rArr 119892119888 through 120601119894 must have the form 1205910 = 1199010120591 for some 2-cell 120591∶ 119898 rArr 119894119888 so that 1205871120591 =id119888 The pair (1205910 id119888) satisfies the compatibility condition of Proposition 336(ii) to induce a 2-cell120591∶ 119898 rArr 119894119888 Wersquoll argue that the 2-cell 120591 is unique proving that the factorization 1199010120591∶ 119887 rArr 119903119888 is alsounique
To see this note that the adjunction 1199011 ⊣ 119894 over119862 exhibits the right adjoint as a terminal elementof the object 1199011 ∶ Hom119860(119891 119892) ↠ 119862 in the slice 2-category (120101119974)119862 It follows as in Lemma 224 thatfor any object 119888 ∶ 119883 rarr 119862 and any morphism119898∶ 119883 rarr Hom119860(119891 119892) over119862 there exists a unique 2-cell
72
119898 rArr 119894119888 over 119862 Thus there is a unique 2-cell 120591∶ 119898 rArr 119894119888 with the property that 1199011120591 = id119888 and so thefactorization 1199010120591∶ 119887 rArr 119903119888 of 120594 through 120601119894 must also be unique
More concisely Theorem 3411 shows that a commainfin-category Hom119860(119891 119892) is right representablejust when its codomain-projection functor 1199011 ∶ Hom119860(119891 119892) ↠ 119862 admits a terminal element as anobject of the sliced infin-cosmos 119974119862 dually Hom119860(119891 119892) is left representable just when its domain-projection functor admits an initial element as an object of the sliced infin-cosmos 119974119861 see Corollary3510 There is a small gap between this statement and the version proven in Theorem 3411 havingto do with the discrepancy between the homotopy 2-category of 119974119862 and the slice of the homotopy2-category 120101119974 over 119862 This is the subject to which we now turn
Exercises
34i Exercise How might one encode the existence of an adjunction 119891 ⊣ 119906 between a given oppos-ing pair of functors using commainfin-categories
35 Sliced homotopy 2-categories and fibered equivalences
Theinfin-category 119860120794 of arrows in 119860 together with its domain- and codomain-evaluation functors(1199010 1199011) ∶ 119860120794 ↠ 119860 times 119860 satisfies a weak universal property in the homotopy 2-category that charac-terizes the infin-category up to equivalence over 119860 times 119860 see Proposition 3210 Similarly the commainfin-category is characterized up to fibered equivalence as defined in Definition 327
As commented upon in Remark 329 there is some ambiguity regarding the 2-categorical datarequired to specify a fibered equivalence that we shall now address head-on The issue is that foran infin-category 119861 in an infin-cosmos 119974 the homotopy 2-category 120101(119974119861) of the sliced infin-cosmos ofProposition 1217 is not isomorphic to the 2-category (120101119974)119861 of isofibrations functors and 2-cellsover 119861 in the homotopy 2-category 120101119974 of119974 see Exercise 14iii
However there is a canonical comparison functor relating this pair of 2-categories that satisfies aproperty we now introduce
351 Definition (smothering 2-functor) A 2-functor 119865∶ 119964 rarr ℬ is smothering if it isbull surjective on 0-cellsbull full on 1-cells for any pair of objects 119860119860prime in 119964 and 1-cell 119896 ∶ 119865119860 rarr 119865119860prime in ℬ there exists119891∶ 119860 rarr 119860prime in119964 with 119865119891 = 119896
bull full on 2-cells for any parallel pair 119891 119892 ∶ 119860 119860prime in 119964 and 2-cell 119865119860 119865119860prime119865119891
119865119892
dArr120573 in ℬ there
exists a 2-cell 120572∶ 119891 rArr 119892 in119964 with 119865120572 = 120573 andbull conservative on 2-cells for any 2-cell 120572 in119964 if 119865120572 is invertible in ℬ then 120572 is invertible in119964
352 Remark Note that smothering 2-functors are surjective on objects 2-functors that are ldquolocallysmotheringrdquo meaning that the action on hom-categories is by a smothering functor as defined in312
The prototypical example of a smothering 2-functor solves Exercise 14iii
353 Proposition Let 119861 be aninfin-category in aninfin-cosmos119974 There is a canonical 2-functor
120101(119974119861) rarr (120101119974)11986173
from the homotopy 2-category of the slicedinfin-cosmos119974119861 to the 2-category of isofibrations functors and 2-cellsover 119861 in 120101119974 and this 2-functor is smothering
This follows more-or-less immediately from Lemma 315 but we spell out the details nonetheless
Proof The 2-categories 120101(119974119861) and (120101119974)119861 have the same objects mdash isofibrations with codomain119861mdash and 1-cells mdash functors between the ldquototal spacesrdquo that commute with these isofibrations to 119861mdashso the canonical mapping may be defined to act as the identity on underlying 1-categories
By the definition of the sliced infin-cosmos given in Proposition 1217 a 2-cell between functors119891 119892 ∶ 119864 119865 from 119901∶ 119864 ↠ 119861 to 119902 ∶ 119865 ↠ 119861 is a homotopy class of 1-simplices in the quasi-categorydefined by the pullback of simplicial sets below-left
Fun119861(119864 119865) Fun(119864 119865) (hFun)119861(119864 119865) hFun(119864 119865)
120793 Fun(119864 119861) 120793 hFun(119864 119861)
119902lowast
119902lowast
119901 119901
Unpacking a 2-cell 120572∶ 119891 rArr 119892 is represented by a 1-simplex 120572∶ 119891 rarr 119892 in Fun(119864 119865) that whiskerswith 119902 to the degenerate 1-simplex on the vertex 119901 isin Fun(119864 119861) and two such 1-simplices representthe same 2-cell if and only if they bound a 2-simplex of the form displayed in (118) that also whiskerswith 119902 to the degenerate 2-simplex on 119901
By contrast a 2-cell in (120101119974)119861 is a morphism in the category defined by the pullback of categoriesabove-right Such 2-cells are represented by 1-simplices 120572∶ 119891 rarr 119892 in Fun(119864 119865) that whisker with 119902 to1-simplices in Fun(119864 119861) that are homotopic to the degenerate 1-simplex on 119901 and two such 1-simplicesrepresent the same 2-cell if and only if they are homotopic in Fun(119864 119865)
Applying the homotopy category functor h ∶ 119980119966119886119905 rarr 119966119886119905 to the above-left pullback produces acone over the above-right pullback inducing a canonical map
120101(Fun119861(119864 119865)) rarr (hFun)119861(119864 119865)which is the action on homs of the canonical 2-functor 120101(119974119861) rarr (120101119974)119861
The 2-functor just constructed is bijective on 0- and 1-cells To see that it is full on 2-cells we mustshow that any 1-simplex 120572∶ 119891 rarr 119892 in Fun(119864 119865) for which 119902120572∶ 119901 rarr 119901 is homotopic to 119901 sdot 1205900 ∶ 119901 rarr 119901in Fun(119864 119861) is homotopic in Fun(119864 119865) to a 1-simplex from 119891 to 119892 over 119901 sdot 1205900 By Lemma 119 anysuch 120572 defines a lifting problem
Λ1[2] Fun(119864 119865) ni
Δ[2] Fun(119864 119861) ni
119902lowast
119891
119891 119892
119901
119901 119901
120572
119902120572
A solution exists since 119902lowast ∶ Fun(119864 119865) ↠ Fun(119864 119861) is an isofibration proving that 120101(119974119861) rarr (120101119974)119861is full on 2-cells
Now suppose 120572∶ 119891 rarr 119892 represents a 2-cell in Fun119861(119864 119865) whose image in (hFun)119861(119864 119865) is anisomorphism A map in a 1-category defined by a pullback is invertible if and only if its projectionsalong the legs of the pullback cone are isomorphisms Thus the image of 120572 is invertible if and only if
74
120572∶ 119891 rarr 119892 defines an isomorphism in hFun(119864 119865) which by Definition 1113 is the case if and only if120572∶ 119891 rarr 119892 represents an isomorphism in Fun(119864 119865) Since 120572 is fibered over the degenerate 1-simplexat 119901 this presents us with a lifting problem
120794 Fun119861(119864 119865) Fun(119864 119865)
120128 120793 Fun(119864 119861)
120572
119902lowast
119901
which Exercise 11v tells us we can solve This proves that 120101(119974119861) rarr (120101119974)119861 reflects invertibility of2-cells and hence defines a smothering 2-functor
Smothering 2-functors are not strictly speaking invertible but nevertheless 2-categorical struc-tures from the codomain can be lifted to the domain
354 Lemma Smothering 2-functors reflect equivalences for any smothering 2-functor 119865∶ 119964 rarr ℬ and1-cell 119891∶ 119860 rarr 119861 in119964 if 119865119891∶ 119865119860 ⥲ 119865119861 is an equivalence in ℬ then 119891 is an equivalence in119964
Proof By fullness on 1-cells an equivalence inverse 119892prime ∶ 119865119861 ⥲ 119865119860 to 119865119891 lifts to a 1-cell 119892∶ 119861 rarr119860 in119964 By fullness on 2-cells the isomorphisms id119865119860 cong 119892prime ∘ 119865119891 and 119865119891 ∘ 119892prime cong id119865119861 also lift to119964 andby conservativity on 2-cells these lifted 2-cells are also invertible
Applying Lemma 354 to the smothering 2-functor
120101(119974119861) rarr (120101119974)119861we resolve the ambiguity about the 2-categorical data of a fibered equivalence
355 Proposition(i) Any equivalence in (120101119974)119861 lifts to an equivalence in 120101(119974119861) That is fibered equivalences over 119861 may
be specified by defining an opposing pair of 1-cells 119891∶ 119864 rarr 119865 and 119892∶ 119865 rarr 119864 over 119861 together withinvertible 2-cells id119864 cong 119892119891 and 119891119892 cong id119865 that lie over 119861 in 120101119974
(ii) Moreover if 119891∶ 119864 rarr 119865 is a map between isofibrations over 119861 that admits an not-necessarily fiberedequivalence inverse 119892∶ 119865 rarr 119864 with not-necessarily fibered 2-cells id119864 cong 119892119891 and 119891119892 cong id119865 then thisdata is isomorphic to a genuine fibered equivalence
Proof The first statement is proven by Lemma 354 and Proposition 353 The second statementasserts that the forgetful 2-functor (120101119974)119861 rarr 120101119974 reflects equivalences Exercise 35i shows that forany map between isofibrations over 119861 that admits an equivalence inverse in the underlying 2-categorythe inverse equivalence and invertible 2-cells can be lifted to also lie over 119861
This gives a 2-categorical proof of Proposition 1217(vii) that for anyinfin-category119861 in aninfin-cosmos119974 the forgetful functor119974119861 rarr119974 preserves and reflects equivalences
The smothering 2-functor 120101(119974119861) rarr (120101119974)119861 can also be used to lift adjunctions that are fibered2-categorically over 119861 to adjunctions in the slicedinfin-cosmos119974119861
75
356 Definition (fibered adjunction) A fibered adjunction over aninfin-category 119861 in aninfin-cosmos119974 is an adjunction
119864 perp 119865
119861
119891
119906
in the slicedinfin-cosmos119974119861 We write 119891 ⊣119861 119906 to indicate that specified maps over 119861 are adjoint over119861
357 Lemma (pullback and pushforward of fibered adjunctions)(i) A fibered adjunction over 119861 can be pulled back along any functor 119896 ∶ 119860 rarr 119861 to define a fibered
adjunction over 119860(ii) A fibered adjunction over 119860 can be pushed forward along any isofibration 119901∶ 119860 ↠ 119861 to define a
fibered adjunction over 119861
Proof By Proposition 133(v) pullback defines a cosmological functor 119896lowast ∶ 119974119861 rarr 119974119860 whichdescends to a 2-functor 119896lowast ∶ 120101(119974119861) rarr 120101(119974119860) that carries fibered adjunctions over 119861 to fibered ad-junctions over 119860 This proves (i)
Composition with an isofibration 119901∶ 119860 ↠ 119861 also defines a 2-functor 119901lowast ∶ 120101(119974119860) rarr 120101(119974119861) thereason we ask 119901 to be an isofibration is due to our convention that the objects in the slicedinfin-cosmoiare isofibrations over a fixed base Thus composition with an isofibration carries a fibered adjunctionover 119860 to a fibered adjunction over 119861 proving (ii)
In analogy with Lemma 354 we have
358 Lemma If 119865∶ 119964 rarr ℬ is a smothering 2-functor then any adjunction in ℬ may be lifted to anadjunction in119964
Proof Exercise 35ii
Many of the examples of fibered adjunctions we will encounter are right adjoint right inverses orleft adjoint right inverses to a given isofibration The next result shows that whenever an isofibration119901∶ 119864 ↠ 119861 admits a left adjoint with unit an isomorphism then this left adjoint may be modified soas to define a left adjoint right inverse making the adjunction fibered over 119861 The dual also holds
359 Lemma Let 119901∶ 119864 ↠ 119861 be any isofibration that admits a right adjoint 119903prime ∶ 119861 rarr 119864 with counit120598 ∶ 119901119903prime cong id119861 an isomorphism Then 119903prime is isomorphic to a functor 119903 that lies strictly over 119861 and defines a rightadjoint right inverse to 119901 Thus any such 119901 defines a fibered adjunction
119864 perp 119861
119861119901
119901
119903
in119974119861 whose right adjoint 119903 lies strictly over 119861 whose counit is the identity 2-cell and in which the unit 120578 liesover 119861 in the sense that 119901120578 = id119901
Proof Exercise 35iii
76
Since the identity on 119861 defines the terminal object of the slicedinfin-cosmos119974119861 Lemma 359 canbe summarized more compactly as follows
3510 Corollary An isofibration 119901∶ 119864 ↠ 119861 admits a right adjoint right inverse if and only if it admitsa terminal element as an object of119974119861 Dually 119901∶ 119864 ↠ 119861 admits a left adjoint right inverse if and only if itadmits an initial element as an object of119974119861
3511 Example Lemma 348 constructs an adjunction in the sliced 2-category 120101119974119860 Lemma 358now allows us to lift it to a genuine adjunction
119860 perp Hom119861(119861 119891)
119860119894 1199011
1199011
in the sliced infin-cosmos 119974119860 By Corollary 3510 this situation may be summarized by saying that1199011 ∶ Hom119861(119861 119891) ↠ 119860 admits a terminal element over 119860
By Lemma 357(i) we may pull back the fibered adjunction along any element 119886 ∶ 1 rarr 119860 to obtainan adjunction
1 perp Hom119861(119861 119891119886)
119894119886
that identifies a terminal element in the fiber Hom119861(119861 119891119886) of 1199011 ∶ Hom119861(119861 119891) ↠ 119860 over 119886 Thisgeneralizes the result of Corollary 349
3512 Example (the fibered adjoints to composition) For any infin-category 119860 the adjoints to theldquocompositionrdquo functor ∘ ∶ 119860120794 times
119860119860120794 rarr 119860120794 constructed in Lemma 2113 may be constructed by com-
posing a triple of adjoint functors that are fibered over the endpoint-evaluation functors
119860120795 119860120794
119860 times 119860
1198601198891
(ev2ev0)
1198601199040
perp
1198601199041perp
(ev1ev0)
with an adjoint equivalence involving a functor119860120795 ⥲rarr 119860120794 times119860119860120794 which also lies over119860times119860 Lemma
359 and its dual implies that these adjoint equivalences can be lifted to fibered adjoint equivalencesover 119860 times 119860 and now both adjoint triples and hence also the composite adjunctions
119860120794 times119860119860120794 119860120794
119860 times 119860
∘perpperp
(ev2ev0)
(minusiddom(minus))
(idcod(minus)minus)(ev1ev0)
77
lie in119974119860times119860
This fibered adjunction figures in the proof of a result that will allow us to convert limit andcolimit diagrams into right and left Kan extension diagrams in the next chapter
3513 Proposition A cospan as displayed below-left admits an absolute right lifting if and only if the cospandisplayed below-right admits an absolute right lifting
119861 Hom119860(119891 119860)
119862 119860 119862 119860dArr120588
119891dArr120598
1199011
119892
119903
119892
119894
in which case the the 2-cell 120598 is necessarily an isomorphism and can be chosen to be an identity
Proof By Theorem 3411 a cospan admits an absolute right lifting if and only if the codomain-projection functor from the associated comma infin-category admits a right adjoint right inverse Ourtask is thus to show that this right adjoint right inverse exists for Hom119860(119891 119892) if and only if this rightadjoint right inverse exists for Hom119860(1199011 119892)
From the defining pullback (332) that constructs the comma infin-category Hom119860(1199011 119892) repro-duced below-left we have the below-right pullback square
Hom119860(1199011 119892) Hom119860(119860 119892) 119860120794
119862 times Hom119860(119891 119860) 119862 times 119860 119860 times 119860
Hom119860(119891 119860) 119860
(11990111199010)
(11990111199010)
119862times1199011120587
119892times119860120587
1199011
Hom119860(1199011 119892) Hom119860(119860 119892)
Hom119860(119891 119860) 119860
1199010
1199011
(3514)By Lemma 357 the composition-identity fibered adjunction of Example 3512 pulls back along 119892 times119891∶ 119862 times 119861 rarr 119860 times 119860 to define a fibered adjunction
Hom119860(1199011 119892) cong Hom119860(119860 119892) times119860 Hom119860(119891 119860) Hom119860(119891 119892)
119862 times 119861
∘
(11990111199010)
perp
perp
(11990111199010)
which then pushes forward along the projection 120587∶ 119862 times 119861 ↠ 119862 to a fibered adjunction over 119862
Hom119860(1199011 119892) Hom119860(119891 119892)
119862
∘
1199011
perp
perp
1199011
78
between the codomain-projection for Hom119860(1199011 119892) and the codomain projection for Hom119860(119891 119892)Now by Corollary 3510 1199011 ∶ Hom119860(119891 119892) ↠ 119862 admits a right adjoint right inverse just when theobject on the right admits a terminal element while 1199011 ∶ Hom119860(1199011 119892) ↠ 119862 similarly admits a rightadjoint right inverse just when the object on the left admits a terminal element By Theorem 242 aterminal element on either side is carried by the appropriate right adjoint to a terminal element onthe other side This proves the equivalence of these conditions
It remains only to prove that the 2-cell for the absolute right lifting of 119892 through 1199011 is invertible ByTheorem 3411 this 2-cell is constructed as by restricting the comma cone along the terminal elementso it is given by the composite
119862 Hom119860(1199011 119892) 119860120794 1198601199010
dArr1205811199011
where the left-hand map is the terminal element just constructed and the middle one comes comesfrom the defining pullback diagram displayed on the left of (3514) As just argued that terminalelement may be chosen to be in the image of the right adjoint Hom119860(119891 119892) rarr Hom119860(119860 119892) times119860Hom119860(119891 119860) cong Hom119860(1199011 119892) whose component on the left factor is the identity Simultaneouslythe pullback defining the map Hom119860(1199011 119892) rarr 119860120794 factors through the projection onto the left factorso we see that the 2-cell in the absolute right lifting diagram is represented by the composite
119862 Hom119860(119891 119892) Hom119860(119860 119892) times119860 Hom119860(119891 119860) Hom119860(119860 119892) 119860120794 119860119894 (idcod(minus)minus) 1205871199010
dArr1205811199011
and hence that this cell is invertible
Exercises
35i Exercise Let 119861 be an object in a 2-category 119966 and consider a map
119864 119865
119861
119891
between isofibrations over 119861 Prove that if 119891 is an equivalence in119966 then 119891 is also an equivalence in theslice 2-category119966119861 of isofibrations over 119861 1-cells that form commutative triangles over 119861 and 2-cellsthat lie over 119861 in the sense that they whisker with the codomain isofibration to the identity 2-cell onthe domain isofibration
35ii Exercise Let 119865∶ 119964 rarr ℬ be a smothering 2-functor Show that any adjunction in ℬ can belifted to an adjunction in 119964 Demonstrate furthermore that if we have previously specified a lift ofthe objects 1-cells and either the unit or counit of the adjunction in ℬ then there is a lift of theremaining 2-cell that combines with the previously specified data to define an adjunction in119964 Thisproves a more precise version of Lemma 358
35iii Exercise Prove Lemma 359
79
CHAPTER 4
Adjunctions limits and colimits II
Comma infin-categories provide a vehicle for encoding the universal properties of categorical con-structions that restrict to define equivalences between the internal mapping spaces introduced in Def-inition 339 between suitable pairs of elements Using the theory developed in Chapter 3 we quicklyprove a variety of results of this type first for adjunctions in sect41 and then for limits and colimits insect43 In an interlude in sect42 we introduce the infin-categories of cones over or under a diagram as acomma infin-category and then give a second model for these infin-categories of cones in the case of dia-grams indexed by simplicial sets built from Joyalrsquos join construction Then we conclude in sect44 withan application constructing the loops ⊢ suspension adjunction for pointed infin-categories containingan element that is both initial and terminal
41 The universal property of adjunctions
Our first result shows that an adjunction between an opposing pair of functors can equally beencoded by a ldquotransposing equivalencerdquo between their left and right representations as commainfin-cat-egories
411 Proposition An opposing pair of functors 119906∶ 119860 rarr 119861 and 119891∶ 119861 rarr 119860 define an adjunction 119891 ⊣ 119906if and only if Hom119860(119891 119860) ≃ Hom119861(119861 119906) over 119860 times 119861
Proof This is a special case of Theorem 347 If 119891 ⊣ 119906 then Lemma 236 tells us that its counit120598 ∶ 119891119906 rArr id119860 defines an absolute right lifting diagram Theorem 347 then tells us that the 1-cellinduced by the left-hand pasted composite
Hom119861(119861 119906)
119860 119861
119860
1199011 1199010120601lArr
120598lArr
119906
119891
=
Hom119861(119861 119906)
Hom119860(119891 119860)
119860 119861
119860
1199011 1199010119910
1199011 1199010
120601lArr
119891
defines a fibered equivalence Hom119861(119861 119906) ⥲ Hom119860(119891 119860) over 119860 times 119861 We interpret this result assaying that in the presence of an adjunction 119891 ⊣ 119906 the right comma cone over 119906 transposes to definethe left comma cone over 119891sup1
sup1If desired an inverse equivalence can be constructed by applying the dual of Theorem 347 to the absolute left liftingdiagram presented by the unit
81
Conversely Theorem 347 tells us that from a fibered equivalence Hom119861(119861 119906) ⥲ Hom119860(119891 119860)over 119860 times 119861 one can extract a 2-cell that defines an absolute right lifting diagram
119861
119860 119860dArr120598
119891119906
Lemma 236 then tells us that this 2-cell defines the counit of an adjunction 119891 ⊣ 119906
412 Observation (the transposing equivalence) To justify referring to the induced functor
Hom119861(119861 119906) ⥲ Hom119860(119891 119860)as a transposing equivalence recall that the transpose of a 2-cell 120594∶ 119887 rArr 119906119886 across the adjunction119891 ⊣ 119906 is computed by the left-hand pasting diagram below
119883
119860 119861
119860
119887119886 120594lArr
120598lArr
119906
119891
=
119883
Hom119861(119861 119906)
119860 119861
119860
119909119887119886
1199011 1199010120601lArr
120598lArr
119906
119891
=
119883
Hom119861(119861 119906)
Hom119860(119891 119860)
119860 119861
119909
119887119886
1199011 1199010
sim
1199011 1199010120601lArr
119891
By the weak universal property of the right comma cone over 119906 the 2-cell 120594 is represented by theinduced functor 119883 rarr Hom119861(119861 119906) which then composes with the transposing equivalence to definea functor119883 rarr Hom119860(119891 119860) that represents the transpose of120594 by the pasting diagram equalities fromright to left
413 Corollary An adjunction 119861 119860119891
perp119906
induces an equivalence Hom119860(119891119887 119886) ≃ Hom119861(119887 119906119886)
over 119883 times 119884 for any pair of generalized elements 119886 ∶ 119883 rarr 119860 and 119887 ∶ 119884 rarr 119861
Proof By the pullback construction of comma infin-categories given in (332) the equivalenceHom119860(119891 119860) ≃ Hom119861(119861 119906) in119974119860times119861 pulls back along 119886times119887∶ 119883times119884 rarr 119860times119861 to define an equivalenceHom119860(119891119887 119886) ≃ Hom119861(119887 119906119886) in119974119883times119884
In particular the equivalence of Proposition 411 pulls back to define an equivalence of internalmapping spaces introduced in 339
414 Proposition (the universal property of units and counits) Consider an adjunction
119861 119860119891
perp119906
with unit 120578∶ id119861 rArr 119906119891 and counit 120598 ∶ 119891119906 rArr id119860
Then for each element 119886 ∶ 1 rarr 119860 the component 120598119886 defines a terminal element of Hom119860(119891 119886) and for eachelement 119887 ∶ 1 rarr 119861 the component 120578119887 defines an initial element of Hom119861(119887 119906)
82
Proof By Corollary 413 the fibered equivalence Hom119860(119891 119860) ≃119860times119861 Hom119861(119861 119906) of Proposition411 pulls back to define equivalences
Hom119860(119891 119886) ≃119861 Hom119861(119861 119906119886) and Hom119860(119891119887 119860) ≃119860 Hom119861(119887 119906)By Corollary 349 id119906119886 induces a terminal element of Hom119861(119861 119906119886) and by Observation 412 its imageacross the equivalence Hom119861(119861 119906119886) ⥲ Hom119860(119891 119886) is again a terminal element which represents thetransposed 2-cell the component of the counit 120598 at the element 119886 The proof that the unit componentdefines a terminal element of Hom119861(119887 119906) is dual
A more sophisticated formulation of the universal property of unit and counit components willappear in Proposition 832 where it will form a key step in the proof that any adjunction extends toa homotopy coherent adjunction
The universal property of unit and counit components captured in Proposition 414 gives the mainidea behind the adjoint functor theorems a functor 119891∶ 119861 rarr 119860 admits a right adjoint just when foreach element 119886 ∶ 1 rarr 119860 the infin-category Hom119860(119891 119886) admits a terminal element The image of thisterminal element under the domain-projection functor 1199010 ∶ Hom119860(119891 119886) ↠ 119861 then defines the element119906119886∶ 1 rarr 119861 and the comma cone defines the component of the counit at 119886 The universal property ofthese unit components is then used to extend the mapping on elements to a functor 119906∶ 119860 rarr 119861
The result just stated is true in the infin-cosmos of quasi-categories and in other infin-cosmoi whereuniversal properties are generated by the terminalinfin-category 1sup2 What is true in allinfin-cosmoi is theversion of the result just stated where the quantifier ldquofor each element 119886 ∶ 1 rarr 119860rdquo is replaced withldquofor each generalized element 119886 ∶ 119883 rarr 119860rdquo in which case the meaning of ldquoterminal elementrdquo should beenhanced to ldquoterminal element over 119883rdquo see the remark after Corollary 349 Since every generalizedelement factors through the universal generalized element namely the identity functor at119860 it sufficesto prove
415 Proposition A functor 119891∶ 119861 rarr 119860 admits a right adjoint if and only if Hom119860(119891 119860) admits aterminal element over 119860 Dually 119891∶ 119861 rarr 119860 admits a left adjoint if and only if Hom119860(119860 119891) admits aninitial element over 119860
Proof By Proposition 411 119891∶ 119861 rarr 119860 admits a right adjoint if and only if the commainfin-categoryHom119860(119891 119860) is right representable Theorem 3411 specializes to tell us that this is the case if and onlyif the codomain-projection functor 1199011 ∶ Hom119860(119891 119860) ↠ 119860 admits a right adjoint right inverse whichby Corollary 3510 is equivalent to postulating a terminal element over 119860
The same suite of results from sect34 specialize to theorems that encode the universal propertiesof limits and colimits Before proving these we first construct the infin-category of cones over a fixeddiagram and also construct alternate models for the infin-categories of cones over varying 119869-indexeddiagrams in the case where 119869 is a simplicial set
Exercises
41i Exercise Prove that the transposing equivalence of Proposition 411 as elaborated upon inObservation 412 is natural with respect to pre-composing with a 2-cell 120573∶ 119887prime rArr 119887 or post-composingwith a 2-cell 120572∶ 119886 rArr 119886prime
sup2We delay the discussion of ldquoanalytically-provenrdquo theorems about quasi-categories until we demonstrate in Part IVthat such results apply also in biequivalentinfin-cosmoi
83
42 infin-categories of cones
421 Definition (theinfin-category of cones) Let 119889∶ 1 rarr 119860119869 be a 119869-shaped diagram in aninfin-category119860 Theinfin-category of cones over 119889 is the commainfin-category Hom119860119869(Δ 119889)with comma cone displayedbelow-left while theinfin-category of cones under 119889 is the commainfin-category Hom119860119869(119889 Δ)with commacone displayed below-right
Hom119860119869(Δ 119889) Hom119860119869(119889 Δ)
1 119860 119860 1
119860119869 119860119869
1199011 1199010
120601lArr
1199011 1199010
120601lArr
119889 Δ Δ 119889
By replacing the ldquo119889rdquo leg of the cospans Definition 421 can be modified to allow 119889∶ 119863 rarr 119860119869 tobe a family of diagrams or to defineinfin-categories of cones over any diagram of shape 119869 an element ofHom119860119869(Δ119860119869) is a cone with any summit over any 119869-indexed diagram
In the case where the indexing shape 119869 is a simplicial set (and not an infin-category in a cartesianclosedinfin-cosmos) there is another model of theinfin-categories of cones over or under a diagram thatmay be constructed using Joyalrsquos join construction The reason for the equivalence is that joins ofsimplicial sets are known to be equivalent to so-called ldquofat joinsrdquo of simplicial sets and a particularinstance of the fat join construction gives the shape of the cones appearing in Definition 421 Wenow introduce these notions
422 Definition (fat join) The fat join of simplicial sets 119868 and 119869 is the simplicial set constructed bythe following pushout
(119868 times 119869) ⊔ (119868 times 119869) 119868 ⊔ 119869
119868 times 120794 times 119869 119868 ⋄ 119869
120587119868⊔120587119869
from which it follows that(119868 ⋄ 119869)119899 ≔ 119868119899 ⊔ ( 1114018
[119899]↠[1]119868119899 times 119869119899) ⊔ 119869119899
Note there is a natural map 119868 ⋄ 119869 ↠ 120794 induced by the projection 120587∶ 119868 times 120794 times 119869 ↠ 120794 so that 119868 is thefiber over 0 and 119869 is the fiber over 1
119868 ⊔ 119869 119868 ⋄ 119869
120793 + 120793 120794
(01)
423 Lemma For any simplicial set 119869 andinfin-category 119860 we have natural isomorphisms
Hom119860119869(Δ119860119869) cong 119860120793⋄119869 and Hom119860119869(119860119869 Δ) cong 119860119869⋄120793
84
Proof The simplicial cotensor119860(minus) ∶ 119982119982119890119905op rarr119974 carries the pushout of Definition 422 to thepullback squares that define the left and right representations ofΔ∶ 119860 rarr 119860119869 as a commainfin-category
119860120793⋄119869 (119860119869)120794 119860119869⋄120793 (119860119869)120794
119860119869 times 119860 119860119869 times 119860119869 119860 times 119860119869 119860119869 times 119860119869
(11990111199010)
(11990111199010)
idtimesΔ Δtimesid
424 Definition (join) The join of simplicial sets 119868 and 119869 is the simplicial set 119868 ⋆ 119869
119868 ⊔ 119869 119868 ⋆ 119869
120793 + 120793 120794
(01)
with(119868 ⋆ 119869)119899 ∶= 119868119899 ⊔ ( 1114018
0le119896lt119899119868119899minus119896minus1 times 119869119896) ⊔ 119869119899
and with the vertices of these 119899-simplices oriented so that there is a canonical map 119868 ⋆ 119869 rarr 120794 so that119868 is the fiber over 0 and 119869 is the fiber over 1 See [24 sect3] for more details
The join functor minus ⋆ 119869∶ 119982119982119890119905 rarr 119982119982119890119905 preserves connected colimits but not the initial object orother coproducts but this issue can be rectified by replacing the codomain by the slice category under119869 Contextualized in this way the join admits a right adjoint defined by Joyalrsquos slice construction
425 Proposition The join functors admit right adjoints
119982119982119890119905 119868119982119982119890119905 119982119982119890119905 119869119982119982119890119905
119868⋆minus
perpminusminus
minus⋆119869
perpminusminus
defined by the natural bijections⎧⎪⎪⎪⎨⎪⎪⎪⎩
119868
119868 ⋆ Δ[119899] 119883ℎ
⎫⎪⎪⎪⎬⎪⎪⎪⎭cong 1114107 Δ[119899] ℎ119883 1114110 and
⎧⎪⎪⎪⎨⎪⎪⎪⎩
119869
Δ[119899] ⋆ 119869 119883119896
⎫⎪⎪⎪⎬⎪⎪⎪⎭cong 1114107 Δ[119899] 119883119896 1114110
Proof As in the statement the simplicial set 119883119896 is defined to have 119899-simplices correspondingto maps Δ[119899] ⋆ 119869 rarr 119883 under 119869 with the right action by the simplicial operators [119898] rarr [119899] givenby pre-composition with Δ[119898] rarr Δ[119899] Since the join functor minus ⋆ 119869∶ 119982119982119890119905 rarr 119869119982119982119890119905 preservescolimits this extends to a bijection between maps 119868 rarr 119883119896 and maps 119868 ⋆ 119869 rarr 119883 under 119869 that isnatural in 119868 and in 119896 ∶ 119869 rarr 119883
426 Notation For any simplicial set 119869 we write
119869◁ ≔ 120793 ⋆ 119869 and 119869▷ ≔ 119869 ⋆ 120793and write ⊤ for the cone vertex of 119869◁ and perp for the cone vertex of 119869▷ These simplicial sets areequipped with canonical inclusions
119869◁ 119869 119869▷
85
427 Proposition (an alternate model) For any simplicial sets 119868 and 119869 and anyinfin-category 119860 there is anatural equivalence
119860119868⋆119869 119860119868⋄119869
119860119868⊔119869
sim
res res
In particular there are comma squares
119860119869◁ 119860119869▷
119860119869 119860 119860 119860119869
119860119869 119860119869
res ev⊤
120601lArr
evperp res
120601lArr
Δ Δ
(428)
Proof There is a canonical map of simplicial sets
(119868 times 119869) ⊔ (119868 times 119869) 119868 ⊔ 119869
119868 times 120794 times 119869 119868 ⋄ 119869 119868 ⋆ 119869
120794
120587119868⊔120587119869
that commutes with the inclusions of the fibers 119868 ⊔ 119869 over the endpoints of 120794 This dashed mapdisplayed above is defined on those 119899-simplices over 119868 ⋄ 119869 that map surjectively onto 120794 to send a triple(120572 ∶ [119899] ↠ [1] 120590 isin 119868119899 120591 isin 119869119899) representing an 119899-simplex of 119868 ⋄ 119869 to the pair (120590|0hellip119896 isin 119868119896 120591|119896+1hellip119899 isin119869119899minus119896minus1) representing an 119899-simplex of 119868 ⋆ 119869 where 119896 isin [119899] is the maximal vertex in 120572minus1(0) Proposition of Appendix C proves that this map induces a natural equivalence119876119868⋆119869 ⥲ 119876119868⋄119869 of quasi-categoriesover119876119869times119876119868 Taking119876 to be the functor space Fun(119883119860) proves the claimed equivalence for generalinfin-categories
Now Proposition 3311 and Lemma 423 implies that this fibered equivalence equips119860119869◁ and119860119869▷
with comma cone squares The 2-cells in (428) are represented by the maps
119869 ⊔ 119869 120793 ⊔ 119869 119869 ⊔ 119869 119869 ⊔ 120793
119869 times 120794 120793 ⋄ 119869 119869◁ 119869 times 120794 119869 ⋄ 120793 119869▷
120794 120794
86
which yield 2-cells
119860119869◁ (119860119869)120794 119860119869 119860119869▷ (119860119869)120794 119860119869
Δ ev⊤
res
1199010
1199011
dArr120581
Δ evperp
res
1199010
1199011
dArr120581
upon cotensoring into 119860
Exercises
42i Exercise Compute Δ[119899] ⋆ Δ[119898] and Δ[119899] ⋄ Δ[119898] and define a section
Δ[119899] ⋆ Δ[119898] rarr Δ[119899] ⋄ Δ[119898]to the map constructed in the proof of Proposition 427
43 The universal property of limits and colimits
We now return to the general context of Definition 231 simultaneously considering diagramsvalued in aninfin-category119860 that are indexed either by a simplicial set or by anotherinfin-category in thecase where the ambientinfin-cosmos is cartesian closed As was the case for Proposition 411 Theorem347 specializes immediately to prove
431 Proposition (colimits represent cones) A family of diagrams 119889∶ 119863 rarr 119860119869 admits a limit if andonly if theinfin-category of cones Hom119860119869(Δ 119889) over 119889 is right representable
Hom119860119869(Δ 119889) ≃119863times119860 Hom119860(119860 ℓ)in which case the representing functor ℓ ∶ 119863 rarr 119860 defines the limit functor Dually 119889∶ 119863 rarr 119860119869 admits acolimit if and only if theinfin-category of cones Hom119860119869(119889 Δ) under 119889 is left representable
Hom119860119869(119889 Δ) ≃119860times119863 Hom119860(119888 119860)in which case the representing functor 119888 ∶ 119863 rarr 119860 defines the colimit functor
Theorem 3411 now specializes to tell us that such representations can be encoded by terminalor initial elements a result which is easiest to interpret in the case of a single diagram rather than afamily of diagrams
432 Proposition (limits as terminal elements) Consider a diagram 119889∶ 1 rarr 119860119869 of shape 119869 in aninfin-category 119860
(i) If 119889 admits a limit then the 1-cell 1 rarr Hom119860119869(Δ 119889) induced by the limit cone 120598 ∶ Δℓ rArr 119889 definesa terminal element of theinfin-category of cones
(ii) Conversely if theinfin-category of cones Hom119860119869(Δ 119889) admits a terminal element then the cone repre-sented by this element defines a limit cone
Dually 119889 admits a colimit if and only if the infin-category Hom119860119869(119889 Δ) of cones under 119889 admits an initialelement in which case the initial element defines the colimit cone
87
Proof By Definition 237 a limit cone defines an absolute right lifting diagram which by Theo-rem 343 induces an equivalence Hom119860(119860 ℓ) ⥲ Hom119860119869(Δ 119889) over119860 By Corollary 349 the identityat ℓ induces a terminal element of Hom119860(119860 ℓ) which the equivalence carries to a terminal element oftheinfin-category of cones this being the element that represents the limit cone 120598 ∶ Δℓ rArr 119889
Conversely if Hom119860119869(Δ 119889) admits a terminal element this defines a right adjoint right inverse tothe codomain-projection functor Hom119860119869(Δ 119889) Theorem 3411 then tells us that the cone representedby this element 1 rarr Hom119860119869(Δ 119889) defines an absolute right lifting of 119889 through Δ
433 Remark The proof of Proposition 432 extends without change to the case of a family of di-agrams 119889∶ 119863 rarr 119860119869 in place of a single diagram since Theorem 3411 applies at this level of gen-erality For a family of diagrams 119889 parametrized by 119863 the infin-category of cones defines an object1199011 ∶ Hom119860119869(Δ 119889) ↠ 119863 of the slicedinfin-cosmos119974119863 and the terminal elements referred to in both (i)and (ii) should be interpreted as terminal elements in119974119863
434 Proposition An infin-category 119860 admits a limit of a family of diagrams 119889∶ 119863 rarr 119860119869 indexed by asimplicial setsup3 119869 if and only if there exists an absolute right lifting of 119889 through the restriction functor
119860119869◁
119863 119860119869dArr120598
res
119889
ran
When these equivalent conditions hold 120598 is necessarily an isomorphism and may be chosen to be the identity
Proof ByDefinition 237 the family of diagrams admits a limit if and only if 119889 admits an absoluteright lifting through Δ∶ 119860 rarr 119860119869 By Proposition 3513 such an absolute lifting diagram exists if andonly if 119889 admits an absolute right lifting through codomain-projection functor 1199011 ∶ Hom119860119869(Δ119860119869) ↠119860119869 in which case the 2-cell of this latter absolute right lifting diagram is invertible By Proposition427 the restriction functor res ∶ 119860119869◁ ↠ 119860119869 is equivalent to this codomain-projection functor soabsolute right liftings of 119889 through 1199011 are equivalent to absolute right liftings of 119889 through res If thisabsolute lifting diagram is inhabited by an invertible 2-cell the isomorphism lifting property of theisofibration proven in Proposition 1410 can be used to replace the functor ran ∶ 119863 rarr 119860119869◁ with anisomorphic functor making the triangle commute strictly
Recall from Lemma 236 that absolute lifting diagrams can be used to encode the existence ofadjoint functors Combining this with Definition 232 Proposition 434 specializes to prove
435 Corollary Aninfin-category119860 admits all limits indexed by a simplicial set 119869 if and only if the restrictionfunctor
119860119869◁ 119860119869res
perpran
sup3We have stated this result for diagrams indexed by simplicial sets because its means is easiest to interpret but weactually prove it with the codomain-projection functor 1199011 ∶ Hom119860119869(Δ119860119869) ↠ 119860119869 in place of the equivalent isofibration119860119869◁ ↠ 119860119869 and this proof applies equally in the case of diagrams indexed byinfin-categories 119869 in cartesian closedinfin-cosmoithat may or may not have a join operation available
88
admits a right adjoint Dually aninfin-category119860 admits all colimits indexed by a simplicial set 119869 if and only ifthe restriction functor
119860119869▷ 119860119869res
perplan
admits a left adjoint
436 Remark Since the restriction functor is an isofibration Lemma 359 applies and the adjunc-tions of Corollary 435 can be defined so as to be fibered over theinfin-category of diagrams 119860119869
The adjunctions of Corollary 435 are particular useful in the case of pullbacks and pushouts
437 Definition (pushouts and pullbacks) A pushout in an infin-category 119860 is a colimit indexed bythe simplicial set
⟔≔ Λ0[2]Dually a pullback in aninfin-category 119860 is a limit indexed by the simplicial set
⟓≔ Λ2[2]Cones over diagrams of shape ⟓ or cones under diagrams of shape ⟔ define commutative squaresdiagrams of shape
⊡ ≔ Δ[1] times Δ[1] cong⟔▷cong⟓◁ A pullback square in an infin-category 119860 is an element of 119860⊡ in the essential image of the functor
ran of Proposition 434 for some diagram of shape ⟓ When 119860 admits all pullbacks these are exactlythose elements of 119860⊡ at which the component of the unit of the adjunction res ⊣ ran of Corollary435 is an isomorphism Dually a pushout square in119860 is an element in the essential image of the dualfunctor lan for some diagram of shape ⟔ ie those elements for which the component of the counitof the adjunction lan ⊣ res is an isomorphism
An 119883-indexed commutative square in 119860 is a diagram 119883 rarr 119860⊡ or equivalently an element ofFun(119883119860)⊡ We label the 0- and 1-simplex components as follows
119889 119887
119888 119886
119906
119907 119908 119891
119892
isin Fun(119883119860)
The diagram also determines a pair of 2-simplices that witness commutativity 119891119906 = 119908 = 119892119907 inhFun(119883119860) but the names of these witnesses wonrsquot matter for this discussion
438 Lemma An119883-indexed commutative square valued in aninfin-category119860 in119974 as below-left is a pullbacksquare if and only if the induced 2-cell below-right is an absolute right lifting diagram in119974119883
119889 119887 Hom119860(119860 119887)
119888 119886 119883 Hom119860(119860 119886)
119883
119906
119907 119908 119891 dArr(id119886119907) 119891lowast
1199011119892 119892
119906
1199011
89
The statement requires some explanation The 1-simplex 119892∶ 119888 rarr 119886 represents a 2-cell 119883 119860119888
119886dArr119892
inducing the map 119892 ∶ 1 rarr Hom119860(119860 119886) The map 119906 ∶ 119883 rarr Hom119860(119860 119887) is defined similarly Themap 119891lowast ∶ Hom119860(119860 119887) rarr Hom119860(119860 119886) is characterized by the pasting diagram
Hom119860(119860 119887)
119883 119860
1199011 1199010120601lArr119887dArr119891119886
=
Hom119860(119860 119887)
Hom119860(119860 119886)
119883 119860
1199011 1199010119891lowast
1199011 1199010120601lArr
119886
By Proposition 337 the composite 119891lowast119906 is isomorphic to 119891119906 By 2-cell induction the 2-cell maybe constructed by specifying its domain and codomain components the former of which we take to
be 119883 119860119889
119888dArr119907 and the latter of which we take to be id119886 Note that the 2-cell just constructed lies
in 120101119974119883 and so can be lifted to 120101(119974119883) by Proposition 353
Proof We prove the result in the case 119883 = 1 and then deduce the result for families of pullbackdiagrams from this case By Proposition 427 the pullback
119860⊡119892or119891 119860⊡
1 119860⟓
res
119892or119891
is equivalent to the infin-category of cones over the cospan diagram 119892 or 119891 By Proposition 432 toshow that the commutative square defines a pullback diagram is to show that (119906 119907 119891 119892) ∶ 1 rarr 119860⊡119892or119891defines a terminal element in the pullback We will show that this pullback119860⊡119892or119891 is also equivalent tothe commainfin-category HomHom119860(119860119886)(119891lowast 119892) By Theorem 3411 the pair (119906 (id119886 119907)) defines anabsolute right lifting if and only if it represents a terminal element in this commainfin-category whichwill prove the claimed equivalence
To see this first consider the diagram which induces a map between the two pullbacks
119860119891 119860120795 119860120794
Hom119860(119860 119886) 119860120794
119883 119860120794 119860
119883 119860
119901119891 11990102
11990112
11990101
1199011
119891
1199011
1199010
119886
1199011
90
Since 119860120795 ≃ 119860Λ1[2] the right-hand back square is equivalent to a pullback Composing the pullbacksquares in the back face of the diagram we obtain an equivalence 119860119891 ⥲ Hom119860(119860 119887) and by in-spection see that the map 119901119891 ∶ 119860119891 ↠ Hom119860(119860 119886) is equivalent to the map 119891lowast ∶ Hom119860(119860 119887) rarrHom119860(119860 119886) over Hom119860(119860 119886)
By applying (minus)120794 to the pullback diagram that defines Hom119860(119860 119886) we obtain a pullback squarethat factors as
Hom119860(119860 119886)120794 119860120794⋄120793 119860120794times120794
1 119860 119860120794
1199011207941
119886 Δ
By the equivalence 119860120794⋄120793 ≃ 119860120794⋆120793 of Proposition 427 the left-hand pullback square shows thatHom119860(119860 119886)120794 is equivalent to the pullback of 1199012 ∶ 119860120795 ↠ 119860 along 119886 ∶ 1 rarr 119860 Modulo this equiv-alence the map 1199010 ∶ Hom119860(119860 119886)120794 ↠ Hom119860(119860 119886) is the pullback of the fibered map
119860120795 119860120794
119860
11990102
1199012 1199011
along 119886 ∶ 1 rarr 119860 and the codomain projection is similarly the pullback of the fibered map 11990112 ∶ 119860120795 ↠119860120794
Putting this together it follows that the pullback
HomHom119860(119860119886)(119891lowastHom119860(119860 119886)) Hom119860(119860 119886)120794
Hom119860(119860 119887) Hom119860(119860 119886)
1199010
119891lowast
is equivalently computed by forming the limit
bull 119860⊡ 119860120795 119860120794
119860120795 119860120794
1 119860120794
11990102
11990112
1199011211990102
119891
The codomain projection 1199011 ∶ HomHom119860(119860119886)(119891lowastHom119860(119860 119886)) ↠ Hom119860(119860 119886) is the pullback of thetop-horizontal composite in the above diagram along Hom119860(119860 119886) rarr 119860120794 So we see that the commainfin-category HomHom119860(119860119886)(119891lowast 119892) is equivalently computed by the limit below-left or equivalently
91
by the limit below right exactly as we claimed
bull bull 1
bull 119860⊡ 119860120795 119860120794
119860120795 119860120794
1 119860120794
119892
11990102
11990112
1199011211990102
119891
119860⊡119892or119891 119860⊡
1 119860⟓
res
119892or119891
The same computation proves the general case for 119883 ne 1 when the comma infin-category is con-structed in 119974119883 see Proposition 1217 for a description of the simplicial limits in sliced infin-cosmoiAlternatively a diagram 119904 ∶ 119883 rarr 119860⊡ in 119974 also defines a 119883-indexed diagram in the infin-cosmos 119974119883valued in the infin-category 120587∶ 119860 times 119883 rarr 119883 This takes the form of a functor (119889 id119883) ∶ 119883 rarr 119860⊡ times 119883over 119883 Itrsquos easy to verify that a diagram valued in 120587∶ 119860 times 119883 ↠ 119883 whose component at 119883 is theidentity has a limit in119974119883 if and only if the 119860-component of the diagram has a limit in119974 Since id119883is the terminal object of119974119883 this object is theinfin-category 1 isin 119974119883 so the proof just give applies toprove the general case of 119883-indexed families of commutative squares
There is an automorphism of the simplicial set 120794times 120794 that swaps the ldquointermediaterdquo vertices (0 1)and (1 0) which induces a ldquotranspositionrdquo automorphism of119860⊡ By symmetry a commutative squarein 119860 is a pullback if and only if its transposed square is a pullback This gives a dual form of Lemma438 with the roles of 119891 and 119892 and of 119906 and 119907 interchanged As a corollary we can easily prove thatpullback squares compose both ldquoverticallyrdquo and ldquohorizontallyrdquo and can be cancelled from the ldquorightrdquoand ldquobottomrdquo
439 Proposition (composition and cancelation of pullback squares) Given a composable pair of119883-indexed commutative squares in 119860 and their composite rectangle defined via the equivalence 119860120795times120794 ≃119860⊡ times
119860120794119860⊡
119901 119889 119887
119890 119888 119886
119909
119910 119911
119906
119907 119908 119891
ℎ 119892
if the right-hand square is a pullback then the left-hand square is a pullback if and only if the composite rectangleis a pullback
Proof By Lemma 438 we are given an absolute right lifting diagram in119974119883
Hom119860(119860 119888)
119883 Hom119860(119860 119886)dArr(id119886119906)
119892lowast
119891
119907
92
By Lemma 241 the composite diagram
Hom119860(119860 119890)
Hom119860(119860 119888)
119883 Hom119860(119860 119886)
dArr(id119888119909)ℎlowast
dArr(id119886119906)119892lowast
119891
119907
119910
is an absolute right lifting diagram in 119974119883 if and only if the top triangle is an absolute right liftingdiagram in119974119883 By Lemma 438 this is exactly what we wanted to show
Terminal elements are special cases of limits where the diagram shape is empty For anyinfin-category119860 theinfin-category of diagrams119860empty cong 1 which tells us that there is a uniqueempty-indexed diagram in119860In this context theinfin-categories of cones over or under the unique diagram constructed in Definition421 are isomorphic to119860 In the case of cones over an empty diagram the domain-evaluation functorcarrying a cone to its summit is the identity on119860 while in the case of cones under the empty diagramthe codomain-evaluation functor carrying a cone to its nadir is the identity on 119860 The followingcharacterization of terminal elements can be deduced as a special case of Proposition 431 though wefind it easier to argue from Proposition 411
4310 Proposition For an element 119905 ∶ 1 rarr 119860 of aninfin-category 119860(i) 119905 defines a terminal element of119860 if and only if the domain-projection functor 1199010 ∶ Hom119860(119860 119905) ↠ 119860
is a trivial fibration(ii) 119905 defines an initial element of119860 if and only if the codomain-projection functor 1199011 ∶ Hom119860(119905 119860) ↠ 119860
is a trivial fibration
Proof Recall from Definition 221 that an element is terminal if and only if it is right adjointto the unique functor
1 119860119905perp
By Proposition 411 ⊣ 119905 if and only if there is an equivalence Hom1( 1) ≃119860 Hom119860(119860 119905) By thedefining pullback (332) for the comma infin-category the left representation of ∶ 119860 rarr 1 is 119860 itselfwith domain-projection functor the identity So the component of the equivalence Hom119860(119860 119905) ⥲ 119860over 119860 must be the domain projection functor 1199010 ∶ Hom119860(119860 119905) ↠ 119860 and we conclude that 119905 is aterminal element if and only if this isofibration is a trivial fibration
4311 Digression (terminal elements of a quasi-category) In the infin-cosmos of quasi-categories theisofibration 1199010 ∶ Hom119860(119860 119905) ↠ 119860 is equivalent over119860 to the slice quasi-category119860119905 defined as a rightadjoint to the join construction of Definition 424 Proposition 4310 proves that 119905 is terminal if andonly if the projection 119860119905 ↠ 119860 is a trivial fibration in the sense of Definition 1123 which transposesto Joyalrsquos original definition of a terminal element of a quasi-category See Appendix F for a full proof
We conclude with two results that could have been proven in Chapter 2 were it not for one smallstep of the argument as we explain A functor 119891∶ 119860 rarr 119861 preserves limits if the image of a limit conein119860 also defines a limit cone in 119861 In the other direction a functor 119891∶ 119860 rarr 119861 reflects limits if a conein 119860 that defines a limit cone in 119861 is also a limit cone in 119860
93
4312 Proposition A fully faithful functor 119891∶ 119860 rarr 119861 reflects any limits or colimits that exist in 119860
Proof The statement for limits asserts that given any family of diagrams 119889∶ 119863 rarr 119860119869 of shape119869 in 119860 any functor ℓ ∶ 119863 rarr 119860 and cone 120588∶ Δℓ rArr 119889 as below-left so that the whiskered compositewith 119891119869 ∶ 119860119869 rarr 119861119869 displayed below is an absolute right lifting diagram
119860 119861
119863 119860119869 119861119869dArr120588
Δ
119891
Δℓ
119889 119891119869
then (ℓ 120588) defines an absolute right lifting of 119889∶ 119863 rarr 119860119869 through Δ∶ 119860 rarr 119860119869 Our proof strategymirrors the results of sect24 By Corollary 346(i) to say that 119891 is fully faithful is to say that id119860 ∶ 119860 rarr 119860defines an absolute right lifting of 119891 through itself So by Lemma 241 and the hypothesis just statedthe composite diagram below-left is an absolute right lifting diagram and by 2-functoriality of thesimplicial cotensor with 119869 the diagram below-left coincides with the diagram below-right
119860 119860
119860 119861 119860119869
119863 119860119869 119861119869 119863 119860119869 119861119869
119891 Δ
dArr120588Δ
119891Δ
=dArr120588
119891119869ℓ
119889 119891119869 119889
ℓ
119891119869
By Corollary 346(iv) to say that 119891 is fully faithful is to say that id119891 ∶ 119860120794 ⥲ Hom119861(119891 119891) is a fiberedequivalence over119860times119860 Applying (minus)119869 ∶ 119974 rarr 119974 this maps to a fibered equivalence id119891119869 ∶ (119860119869)120794 ⥲Hom119861119869(119891119869 119891119869) over 119860119869 times 119860119869 proving that if 119891∶ 119860 rarr 119861 is fully faithful then 119891119869 ∶ 119860119869 rarr 119861119869 is also⁴Hence by Corollary 346(i) id119860119869 ∶ 119860119869 rarr 119860119869 defines an absolute right lifting of 119891119869 through itselfApplying Lemma 241 again we now conclude that (ℓ 120588) is an absolute right lifting of 119889 through Δas required
An alternate approach to proving this result is suggested as Exercise 43iiiOur final result proves that for 119868 and 119869 simplicial sets whenever we are given a 119869-indexed diagram
valued in theinfin-category 119860119868 of 119868-indexed diagrams in 119860 its limit may be computed pointwise in thevertices of 119868 as the limit of the corresponding 119869-indexed diagram in 119860 Our argument requires thefollowing representable characterization of absolute lifting diagrams whose proof again makes use ofthe fact that they are preserved by cosmological functors
4313 Proposition A natural transformation defined in aninfin-cosmos119974 as below-left is an absolute rightlifting diagram if and only if its ldquoexternalizationrdquo displayed below-right defines a right lifting diagram in119980119966119886119905
119861 Fun(119862 119861)
119862 119860 120793 Fun(119862119860)dArr120588
119891 dArr120588119891lowast
119903
119892
119903
119892
⁴This is the statement that we could not yet prove in Chapter 2
94
that is preserved by precomposition with any functor 119888 ∶ 119883 rarr 119862 in 119974 in the sense that the diagram below isalso right lifting
Fun(119862 119861) Fun(119883 119861) Fun(119883 119861)
120793 Fun(119862119860) Fun(119883119860) 120793 Fun(119883 119862)dArr120588
119891lowast
119888lowast
119891lowast = dArr120588119888119891lowast119903
119892 119888lowast
119903119888
119892119888
Moreover the externalized right lifting diagrams in 119980119966119886119905 are in fact absolute
Proof Since Fun(120793 minus) ∶ 119980119966119886119905 rarr 119980119966119886119905 is naturally isomorphic to the identity functor for anyinfin-categories 119883119860 isin 119974 we have Fun(119883119860) cong Fun(120793Fun(119883119860)) and hence
hFun(119883119860) cong hFun(120793Fun(119883119860)) (4314)
This justifies our use of the same name 120588 for the 2-cell in 120101119974 and the 2-cell in 120101119980119966119886119905 in the statementTo say that (119903 120588) defines a right lifting of 119892 through 119891 in 120101119974 asserts a bijection between 2-cells
in hFun(119862 119861) with codomain 119903 and 2-cells in hFun(119862119860) with codomain 119892 and domain factoringthrough 119891 implemented by pasting with 120588 Under the correspondence of (4314) this asserts equallythat 119903 defines a right lifting of 119892 through 119891lowast in 120101119980119966119886119905 To say that (119903 120588) defines an absolute rightlifting of 119892 through 119891 is to assert the analogous right lifting property for the pair (119903119888 120588119888) defined byrestricting along any 119888 ∶ 119883 rarr 119862 This is exactly the first claim of the statement
It remains only to argue that if (119903 120588) is an absolute right lifting diagram in119974 then its externalizedright lifting diagram of quasi-categories is also absolute To see this first note that the cosmologicalfunctor Fun(119862 minus) ∶ 119974 rarr 119980119966119886119905 preserves this absolute right lifting diagram yielding an absoluteright lifting diagram of quasi-categories as below left
Fun(119862 119861) Fun(119862 119861) Fun(119862 119861)
Fun(119862 119862) Fun(119862119860) 120793 Fun(119862119860) 120793 Fun(119862 119862) Fun(119862119860)dArrFun(119862120588)
119891lowast dArr120588119891lowast = dArrFun(119862120588)
119891lowast119903lowast
119892lowast
119903
119892id119862
119903lowast
119892lowast
We obtain the desired absolute right lifting diagram by evaluating at the identity
4315 Proposition Let 119868 and 119869 be simplicial sets and let119860 be aninfin-category Then a diagram as below-leftis an absolute right lifting diagram
119860119868 119860119868 119860
119863 119860119868times119869 119863 119860119868times119869 119860119869dArr120588 Δ119868 dArr120588 Δ119868
ev119894
Δlim
119889
lim
119889 ev119894
if for each vertex 119894 isin 119868 the diagram above right is an absolute right lifting diagram
Note this statement is not a biconditional Even in the case of strict 1-categories there may existcoincidental limits of diagram valued in functor categories that are not defined pointwise [9 21710]
95
Proof By Proposition 4313 we may externalize and instead show that the diagram of quasi-categories displayed below left
Fun(119863119860)119868 Fun(119863119860)119868 Fun(119863119860)
120793 Fun(119863119860)119868times119869 120793 Fun(119863119860)119868times119869 Fun(119863119860)119869dArr120588 Δ119868 dArr120588 Δ119868
ev119894
Δlim
119889
lim
119889 ev119894
is absolute right lifting lifting if the diagrams above-right are for each vertex 119894 isin 119868 By naturality ofour methods our proof will show that the absolute right lifting diagrams on the left are preserved bypostcomposition with the restriction functors induced by 119889∶ 119883 rarr 119863 if the same is true on the right
We simplify our notation and write119876 for the quasi-category Fun(119863119860) and assume that for each119894 isin 119868 the diagram
119876119868 119876
120793 119876119868times119869 119876119869dArr120588 Δ119868
ev119894
Δlim
119889 ev119894
is absolute right lifting By 2-adjunction (minus times 119868) ⊣ (minus)119868 a diagram as below-left is absolute rightlifting if and only if the transposed diagram below-right is a right lifting diagram and this remains thecase upon restricting along functors of the form 120587∶ 119883 times 119868 rarr 119868⁵
119876119868 119876
120793 119876119868times119869 119868 119876119869dArr120588 Δ119868 dArr120588
Δlim
119889 119889
lim (4316)
Wersquoll argue that this right-hand diagram is in fact absolute right lifting which implies that the left-hand diagram is absolute right lifting as well as desired
To see this we appeal to Corollary a consequence of a general theorem proven in Appendix Fthat universal properties in 119980119966119886119905 are detected pointwise Specifically this tells us that the triangleabove right is an absolute right lifting diagram if and only if the restricted diagram is absolute rightlifting for each vertex 119894 isin 119868
119876
120793 119868 119876119869dArr120588
Δ
119894119889
lim
and this is exactly what we have assumed in (4316)
⁵If the reader is concerned that 119868 is not a quasi-category there are two ways to proceed One is to replace 119868 by aquasi-category 119868 prime by inductively attaching fillers for inner horns note that 119868 and 119868 prime will have the same sets of verticesBy Proposition 1127 the diagram quasi-categories 119876119868 prime and 119876119868 are equivalent The other option is to observe that itdoesnrsquot matter if 119868 is a quasi-category or not because we may define hFun(119868 119876) ≔ h(119876119868) and by Corollary 1121 119876119868 is aquasi-category regardless of whether 119868 is
96
Exercises
43i Exercise Prove that if 119860 has a terminal element 119905 then for any element 119886 the mapping spaceHom119860(119886 119905) is contractible ie is equivalent to the terminalinfin-category 1
43ii Exercise Prove that a square in 119860 is a pullback if and only if its ldquotransposedrdquo square definedby composing with the involution119860⊡ cong 119860⊡ induced from the automorphism of 120794times120794 that swaps theldquooff-diagonalrdquo elements is a pulllback square
43iii Exercise ([60 37]) Use Theorem 343 and Corollary 346(iv) to prove that a fully faithfulfunctor 119891∶ 119860 rarr 119861 reflects all limits or colimits that exist in 119860 Why does this argument not alsoshow that 119891∶ 119860 rarr 119861 preserves them
44 Loops and suspension in pointedinfin-categories
441 Definition (pointed infin-categories) An infin-category 119860 is pointed if it admits a zero elementan element lowast ∶ 1 rarr 119860 that is both initial and terminal
The counit of the adjunction lowast ⊣ that witnesses the initiality of the zero element defines a naturaltransformation 120588∶ lowast rArr id119860 that we refer to as the family of points of 119860 Dually the unit of theadjunction ⊣ lowast that witnesses the terminality of the zero element defines a natural transformation120585∶ id119860 rArr lowast that we refer to as the family of copoints
Cospans and spans in aninfin-category 119860 may be defined by gluing together a pair of arrows alongtheir codomains or domains respectively
119860⟓ 119860120794 119860⟔ 119860120794
119860120794 119860 119860120794 119860
1199011
1199010
1199011 1199010
For instance the family of points in a pointedinfin-category 119860 is represented by a functor 120588∶ 119860 rarr 119860120794whose domain-component is constant at lowast and whose codomain component is id119860 Gluing two copiesof this map along their codomain defines a diagram ∶ 119860 rarr 119860⟓ Dually there is a diagram ∶ 119860 rarr119860⟔ defined by gluing the functor 120585∶ 119860 rarr 119860120794 that represents the family of copoints to itself alongtheir domains
442 Definition (loops and suspension) A pointed infin-category 119860 admits loops if it admits a limitof the family of diagrams
119860
119860 119860⟓dArr
Δ
Ω
in which case the limit functorΩ∶ 119860 rarr 119860 is called the loops functor Dually a pointedinfin-category119860 admits suspensions if it admits a colimit of the family of diagrams
119860
119860 119860⟔uArr
Δ
Σ
97
in which case the colimit functor Σ∶ 119860 rarr 119860 is called the suspension functor
Importantly if119860 admits loops and suspensions then the loops and suspension functors are adjoint
443 Proposition (the loops-suspension adjunction) If 119860 is a pointed infin-category that admits loopsand suspensions then the loops functor is right adjoint to the suspension functor
119860 119860Ω
perpΣ
The main idea of the proof is easy to describe If 119860 admits all pullbacks and all pushouts thenCorollary 435 supplies adjunctions
119860⟓ 119860⊡ 119860⟔
ran
perp
res
res
perp
lan
that are fibered over 119860 times 119860 upon evaluating at the intermediate vertices of the commutative squareBy pulling back along (lowast lowast) ∶ 1 rarr 119860 times 119860 we can pin these vertices at the zero element Since thezero element is initial and terminal theinfin-categories of pullback and pushout diagrams of this formare both equivalent to 119860 and the pulled-back adjoints now coincide with the loops and suspensionfunctors
The only subtlety in the proof that follows is that we have assumed weaker hypotheses that 119860admits only loops and suspensions but perhaps not all pullbacks and pushouts
Proof The diagram lands in a subobject119860⟓lowast of119860⟓ defined below-left that is comprised of thosepullback diagrams whose source elements are pinned at the zero element lowast of 119860
119860 119860
119860⟓lowast 119860⟓ 119860⟓lowast Hom119860(lowast 119860)
1 119860 times 119860 Hom119860(lowast 119860) 119860
120588or120588
lowast
120588
lowast
120588sim
sim
sim 1199011
(lowastlowast)sim1199011
From a second construction of 119860⟓lowast displayed above-right and the characterization of initiality givenin Proposition 4310 we may apply the 2-of-3 property of equivalences to conclude first that 120588∶ 119860 ⥲Hom119860(lowast 119860) and then that the induced diagram lowast ∶ 119860 ⥲ 119860⟓lowast are equivalences Dually the diagram ∶ 119860 rarr 119860⟔ defines an equivalence lowast ∶ 119860 ⥲ 119860⟔lowast when its codomain is restricted to the subobject ofpushout diagrams whose target elements are pinned at the zero element lowast
By Proposition 434 a pointed infin-category 119860 admits loops or admits suspensions if and only ifthere exist absolute lifting diagrams as below-left and below-right respectively
119860⊡ 119860⊡
119860 119860⟓ 119860 119860⟔congdArr
resconguArr
res
ran
lan
98
By Theorem 343 the absolute right lifting diagram defines a right representation Hom119860⊡(119860⊡ ran) ≃Hom119860⟓(res ) this being a fibered equivalence over119860times119860⊡ The represented commainfin-category maybe pulled back along the inclusion of the subobject 119860⊡lowast 119860⊡ of commutative squares in 119860 whoseintermediate vertices are pinned at the zero object
Hom119860⟓(res ) (119860⟓)120794
Hom119860⟓lowast (res lowast) (119860⟓lowast )120794
119860⊡ times 119860 119860⟓ times 119860⟓
119860⊡lowast times 119860 119860⟓lowast times 119860⟓lowast
restimes
restimeslowast
The right-hand face of this commutative cube is not strictly a pullback but the universal property ofthe zero element implies that the induced map from (119860⟓lowast )120794 to the pullback is an equivalence It followsthat Hom119860⟓lowast (res lowast) is equivalent over 119860⊡lowast times 119860 to the pullback of Hom119860⟓(res ) along this inclusionand so the fibered equivalence pulls back to define a right representation for Hom119860⟓lowast (res lowast) Duallythe left representation for Hom119860⟔( res) pulls back to a left representation for Hom119860⟔lowast (lowast res) ByTheorem 347 these unpack to define absolute lifting diagrams
119860⊡lowast 119860⊡lowast 119860⊡lowast 119860⊡lowast
119860 119860⟓lowast 119860⟓lowast 119860⟓lowast 119860 119860⟔lowast 119860⟔lowast 119860⟔lowastcongdArr
res congdArr
resconguArr
res conguArr
res
lowast
ran ran
lowast
lan lan
Restricting along the inverse equivalences 119860⟓lowast ⥲ 119860 and 119860⟔lowast ⥲ 119860 and pasting with the invertible2-cell we obtain absolute lifting diagrams whose bottom edge is the identity
By Lemma 236 these lifting diagrams define adjunctions
119860 ≃ 119860⟓lowast 119860⊡lowast 119860⟔lowast ≃ 119860
ran
perp
res
res
perp
lan
which compose to the desired adjunction Σ ⊣ Ω
444 Definition An arrow 119891∶ 1 rarr 119860120794 from 119909 to 119910 in a pointedinfin-category 119860 admits a fiber if 119860admits a pullback of the diagram defined by gluing 119891 to the component 120588119910 of the family of points ByProposition 434 such pullbacks give rise to a pullback square
119860⊡
1 119860⟓congdArr
resran
120588119910or119891
that is referred to as the fiber sequence for 119891 Dually 119891 admits a cofiber if 119860 admits a pushout of thediagram defined by gluing 119891 to the component 120585119909 of the family of copoints in which case the pushout
99
square119860⊡
1 119860⟔conguArr
reslan
120585119909or119891
defines the cofiber sequence for 119891
Fiber and cofiber sequences define commutative squares in 119860 whose lower-left vertex is the zeroelement lowast The data of such squares is given by a commutative triangle in 119860 mdash an element of 119860120795mdash together with a nullhomotopy of the diagonal edge a witness that this edge factors through thezero element in h119860 Borrowing a classical term from homological algebra a commutative square in119860whose lower-left vertex is the zero element is referred to as a triangle in 119860
445 Definition (stableinfin-category) A stableinfin-category is a pointedinfin-category 119860 in which(i) every morphism admits a fiber and a cofiber that is there exist absolute lifting diagrams
119860⊡ 119860⊡
119860120794 119860⟓ 119860120794 119860⟔congdArr
resconguArr
resran
120588orid
lan
120585orid
(ii) and a triangle in 119860 defines a fiber sequence if and only if it also defines a cofiber sequenceSuch triangles are called exact
A stableinfin-category admits loops and admits suspensions formed by taking fibers of the arrowsin the family of points and cofibers of arrows in the family of copoints respectively
446 Proposition If 119860 is a stableinfin-category then Σ ⊣ Ω are inverse equivalences
Proof In the proof of Proposition 443 the adjunction Σ ⊣ Ω is constructed as a composite ofadjunctions
119860 ≃ 119860⟓lowast 119860⊡lowast 119860⟔lowast ≃ 119860
ran
perp
res
res
perp
lan
that construct fiber and cofiber sequences By Proposition 219 the unit and counit of this compositeadjunction are given by
119860 119860 119860
119860⊡lowast 119860⊡lowast 119860⊡lowast 119860⊡lowast
119860 119860 119860
lan
Σ
dArrcong ΣdArr120598 lan
res dArr120578
res res
dArrcong resran Ω
Ω
ran
By Definition 437 the unit of res ⊣ ran restricts to an isomorphism on the subobject of pushoutsquares In a stable infin-category the cofiber sequences in the image of lan ∶ 119860 rarr 119860⊡lowast are pullbacksquares so this tells us that 120578 lan is an isomorphism Dually the fiber sequences in the image of
100
ran ∶ 119860 rarr 119860⊡lowast are pushout squares which tells us that 120598 ran is an isomorphism Hence the unit andcounit of Σ ⊣ Ω are invertible so these functors define an adjoint equivalence
447 Proposition (finite limits and colimits in stable infin-categories) A stable infin-category admits allpushouts and all pullbacks and moreover a square is pushout if and only if it is a pullback
Proof Given a cospan 119892 or 119891∶ 119883 rarr 119860⟓ in 119860 form the cofiber of 119891 followed by the fiber of thecomposite map 119888 rarr 119886 rarr coker 119891
ker(119902119892) 119887 lowast
119888 119886 coker(119891)
119906
119907
119891
119892 119902
By Definition 445(ii) the cofiber sequence 119887 rarr 119886 rarr ker 119891 is also a fiber sequence By the pullbackcancelation result of Proposition 439 we conclude that ker(119902119892) computes the pullback of the cospan119892 or 119891
To see that this pullback square is also a pushout form the fiber of the map 119907
ker(119907) ker(119902119892) 119887
lowast 119888 119886
119906
119907
119891
119892
By the pullback composition result of Proposition 439 ker(119907) is also the fiber of the map 119891 By Def-inition 445(ii) the fiber sequences ker(119907) rarr ker(119902119892) rarr 119888 and ker(119907) rarr 119887 rarr 119886 are also cofibersequences Now by the pushout cancelation result of Proposition 439 we see that the right-handpullback square is also a pushout square A dual argument proves that pushouts coincide with pull-backs
Exercises
44i Exercise Arguing in the homotopy category show that if aninfin-category 119860bull admits an initial element 119894bull admits a terminal element 119905 andbull there exists an arrow 119905 rarr 119894
then 119860 is a pointedinfin-category
101
CHAPTER 5
Fibrations and Yonedarsquos lemma
The fibers 119864119887 of an isofibration 119901∶ 119864 ↠ 119861 over an element 119887 ∶ 1 rarr 119861 are necessarilyinfin-categoriesso it is natural to ask where the isofibration also encodes the data of functors between the fibers in thisfamily of infin-categories Roughly speaking an isofibration defines a cartesian fibration just when thearrows of 119861 act contravariantly functorially on the fibers and a cocartesian fibration when the arrowsof 119861 act covariantly functorially on the fibers This action is not strict but rather pseudofunctorial orhomotopy coherent in a sense that will be made precise when we study the comprehension construc-tion
One of the properties that characterizes cocartesian fibrations is an axiom that says that for any2-cell with codomain 119861 and specified lift of its source 1-cell there is a lifted 2-cell with codomain 119864with that one cell as its source In particular this lifting property can be applied in the case where the2-cell in question is a whiskered composite of an arrow in the homotopy category of 119861 as below-leftand the lift of the source 1-cell is the canonical inclusion of its fiber
119864119886 119864 119864119886 119864
119864119887
1 119861 1 119861
ℓ119886
=119901120573lowast
120573lowast(ℓ119886)dArr120594120573
ℓ119886
119901ℓ119887119886
119887
dArr120573
119887
(501)
In this case the codomain 120573lowast(ℓ119886) of the lifted cell 120594120573 displayed above right lies strictly above thecodomain of the original 2-cell and thus factors through the pullback defining its fiber This de-fines a functor 120573lowast ∶ 119864119886 rarr 119864 the ldquoactionrdquo of the arrow 120573 on the fibers of 119901 The pseudofunctoriality ofthese action maps arises from a universal property required of the specified lifted 2-cells namely thatthey are cartesian in a sense we now define
51 The 2-category theory of cartesian fibrations
There is a standard notion of cartesian fibration in a 2-category developed by Street [55] thatrecovers the Grothendieck fibrations when specialized to the 2-category 119966119886119905 This is not the correctnotion of cartesian fibration betweeninfin-categories as the universal property the usual notion demandsof lifted 2-cells is too strict Instead of referring to the notions defined here as ldquoweakrdquo we would preferto refer to the classical notion of cartesian fibration in a 2-category as ldquostrictrdquo were we to refer to itagain which we largely will not
To remind the reader of the interpretation of the data in the homotopy 2-category of aninfin-cosmoswe refer to 1- and 2-cells as ldquofunctorsrdquo and ldquonatural transformationsrdquo Before defining the notion of
103
cartesian fibration we describe the weak universal property enjoyed by a certain class of ldquoupstairsrdquonatural transformations
511 Definition (119901-cartesian transformations) Let 119901∶ 119864 ↠ 119861 be an isofibration A natural trans-
formation 119883 119864119890prime
119890dArr120594 with codomain 119864 is 119901-cartesian if
(i) induction Given any natural transformations 119883 119864119890Prime
119890dArr120591 and 119883 119861
119901119890Prime
119901119890primedArr120574 so that 119901120591 =
119901120594 sdot 120574 there exists a lift 119883 119864119890Prime
119890primedArr of 120574 so that 120591 = 120594 sdot
119890Prime 119890
119890prime
119901119890Prime 119901119890
119901119890prime
120591
120594
119901120591
120574 119901120594
isin hFun(119883 119864)
isin hFun(119883 119861)
119901lowast
(ii) conservativity Any fibered endomorphism of 120594 is invertible if 119883 119864119890prime
119890primedArr120577 is any natural
transformation so that 120594 sdot 120577 = 120594 and 119901120577 = id119901119890prime then 120577 is invertible
512 Remark (why ldquocartesianrdquo) The induction property for a 119901-cartesian natural transformation120594∶ 119890prime rArr 119890 says that for any 119890Prime ∶ 119883 rarr 119864 there is a surjective function from the set hFun(119883 119864)(119890Prime 119890prime)of natural transformations from 119890Prime to 119890prime to the pullback induced in the commutative square
hFun(119883 119864)(119890Prime 119890prime) hFun(119883 119864)(119890Prime 119890)
hFun(119883 119861)(119901119890Prime 119901119890prime) hFun(119883 119861)(119901119890Prime 119901119890)
120594∘minus
119901 119901
119901120594∘minus
Were the induction property strict and not weak this square would be a pullback ie a ldquocartesianrdquosquare
It follows that 119901-cartesian lifts of a given 2-cell with specified codomain are unique up to fiberedisomorphism
513 Lemma (uniqueness of cartesian lifts) If 119883 119864119890prime
119890dArr120594 and 119883 119864
119890Prime
119890dArr120594prime are 119901-cartesian lifts of
a common 2-cell 119901120594 then there exists an invertible 2-cell 119883 119864119890Prime
119890primecongdArr120577 so that 120594prime = 120594 sdot 120577 and 119901120577 = id
104
Proof By induction there exists 2-cells 120577∶ 119890Prime rArr 119890prime and 120577prime ∶ 119890prime rArr 119890Prime so that 120594prime = 120594 sdot 120577 and120594 = 120594prime sdot 120577prime with 119901120577 = 119901120577prime = id The composites 120577 sdot 120577prime and 120577prime sdot 120577 are then fibered automorphisms of 120594and 120594prime and thus invertible by conservativity Now the 2-of-6 property for isomorphisms implies that120577 and 120577prime are also isomorphisms (though perhaps not inverses)
We frequently make use of the isomorphism stability of the 119901-cartesian transformations given bythe following suite of observations
514 Lemma Let 119901∶ 119864 ↠ 119861 be an isofibration(i) Isomorphisms define 119901-cartesian transformations(ii) Any 119901-cartesian lift of an identity is a natural isomorphism(iii) The class of 119901-cartesian transformations is closed under pre- and post-composition with natural isomor-
phisms
Proof Exercise 51i
Furthermore
515 Lemma (more conservativity) If 119883 119864119890prime
119890dArr120594 119883 119864
119890Prime
119890dArr120594prime and 119883 119864
119890Prime
119890primedArr120577 are 2-cells so
that 120594 and 120594prime are 119901-cartesian 120594prime = 120594 sdot 120577 and 119901120577 is invertible then 120577 is invertible
Proof Given the data in the statement we can use the induction property for 120594prime applied to thepair (120594 119901120577minus1) to induce a candidate inverse for 120577 and then apply the conservativity property toconclude that 120577 sdot and sdot 120577 are both isomorphisms By the 2-of-6 property 120577 is an isomorphism asdesired
We now introduce the class of cartesian fibrations
516 Definition (cartesian fibration) An isofibration 119901∶ 119864 ↠ 119861 is a cartesian fibration if(i) Any natural transformation 120573∶ 119887 rArr 119901119890 as below-left admits a 119901-cartesian liftsup1120594120573 ∶ 120573lowast119890 rArr 119890 as
below-right
119883 119864 119883 119864
119861 119861
119890
119887
uArr120573 119901 =
119890
120573lowast119890
uArr120594120573
119901
(ii) The class of 119901-cartesian transformations is closed under restriction that is if 119883 119864119890prime
119890dArr120594 is
119901-cartesian and 119891∶ 119884 rarr 119883 is any functor then 119884 119864119890prime119891
119890119891
dArr120594119891 is 119901-cartesian
The lifting property (i) implies that a 119901-cartesian transformation 120594∶ 119890prime rArr 119890 is the ldquouniversalnatural transformation over 119901120594 with codomain 119890rdquo in the following weak sense any transformation
sup1To ask that 120594120573 ∶ 120573lowast119890 rArr 119890 is a lift of 120573∶ 119887 rArr 119901119890 asserts that 119901120594120573 = 120573 and hence 119901120573lowast119890 = 119887
105
120595∶ 119890prime rArr 119890 factors through a 119901-cartesian lift 120594120595 ∶ (119901120595)lowast119890 rArr 119890 of 119901120595 via a 2-cell 120574∶ 119890prime rArr (119901120595)lowast119890 overan identity and moreover 120595 is 119901-cartesian if and only if this factorization 120574 is invertible
The reason for condition (ii) will become clearer in sect53 For now note that since all 119901-cartesianlifts of a given 2-cell 120573 are isomorphic and the class of 119901-cartesian cells is stable under isomorphismto verify the condition (ii) it suffices to show that for any functor 119891∶ 119884 rarr 119883 there is some 119901-cartesianlift of 120573 that restricts along 119891 to another 119901-cartesian transformation
517 Lemma (composites of cartesian fibrations) If 119901∶ 119864 ↠ 119861 and 119902 ∶ 119861 ↠ 119860 are cartesian fibrations
then so is 119902119901 ∶ 119864 ↠ 119860 Moreover a natural transformation 119883 119864119890prime
119890dArr120594 is 119902119901-cartesian if and only if 120594 is
119901-cartesian and 119901120594 is 119902-cartesian
Proof The first claim follows immediately from the second for the lifts required by Definition516(i) can be constructed by first taking a 119902-cartesian lift 120594120573 and then taking a 119901-cartesian lift 120594120594120573 ofthis lifted cell
119883 119864 119883 119864 119883 119864
119861 = 119861 = 119861
119860 119860 119860
119890
119886
uArr120573 119901
119890
119887
uArr120594120573 119901
119890
120573lowast119890
uArr120594120594120573119901
119902 119902 119902
and the stability condition 516(ii) is then inherited from the stability of 119901- and 119902-cartesian transfor-mations
To prove the second claim first consider a natural transformation 119883 119864119890prime
119890dArr120594 that is 119901-cartesian
and so that 119901120594 is 119902-cartesian Given any natural transformations 119883 119864119890Prime
119890dArr120591 and 119883 119860
119902119901119890Prime
119902119901119890primedArr120574 so
that 119902119901120591 = 119902119901120594 sdot 120574 119902-cartesianness of 119901120594 induces a lift 119883 119861119890Prime
119890primedArr of 120574 so that 119901120591 = 119901120594 sdot
Now 119901-cartesianness of 120594 induces a further lift 119883 119864119890Prime
119890primedArr of so that 120591 = 120594 sdot Moreover if
119883 119864119890prime
119890primedArr120577 is any natural transformation so that 120594 sdot 120577 = 120594 and 119902119901120577 = id then 119901120594 sdot 119901120577 = 119901120594 and by
conservativity for 119901120594 119901120577 is invertible Applying Lemma 515 we conclude that 120577 is invertible Thisproves that 120594 is 119902119901-cartesian
Conversely if 120594 is 119902119901-cartesian then Lemma 513 implies it is isomorphic to all other 119902119901-cartesianlifts of 119902119901120594 The construction given above produces a 119902119901-cartesian lift of any 2-cell that is 119901-cartesian
106
and whose image under 119901 is 119902-cartesian By the isomomorphism stability of 119901- and 119902-cartesian trans-formations of Lemma 514 120594 must also have these properties
The following lemma proves that cartesian fibrations come equipped with a ldquogeneric 119901-cartesiantransformationrdquo
518 Lemma An isofibration 119901∶ 119864 ↠ 119861 is a cartesian fibration if and only if the right comma cone over 119901displayed below-left admits a lift 120594 as displayed below-right
Hom119861(119861 119901) 119864 Hom119861(119861 119901) 119864
119861 119861
1199011
1199010
uArr120601 119901 =
1199011
119903
uArr120594
119901 (519)
with the property that the restriction of 120594 along any 119883 rarr Hom119861(119861 119901) is a 119901-cartesian transformation
Proof If 119901∶ 119864 ↠ 119861 is cartesian then the right comma cone120601 admits a 119901-cartesian lift120594∶ 119903 rArr 1199011by 516(i) which by 516(ii) has the property that the restriction of this 119901-cartesian transformationalong any 119883 rarr Hom119861(119861 119901) is also 119901-cartesian
For the converse suppose we are given the generic 119901-cartesian transformation 120594∶ 119903 rArr 1199011 of thestatement and consider a 2-cell 120573∶ 119887 rArr 119901119890 as below-left
119883 119864 119883 Hom119861(119861 119901) 119864 119883 Hom119861(119861 119901) 119864
119861 119861 119861
119890
119887
dArr120573 119901 =
119890
119887
120573 1199011
1199010uArr120601 119901 =
119890
119887
1205731199011
119903uArr120594
119901
By 1-cell induction 120573 = 120601120573 for some functor 120573 ∶ 119883 rarr Hom119861(119861 119901) as above- center By substitut-ing the equation (519) as above-right we see that 120594120573 is a lift of 120573 whose codomain is 1199011120573 = 119890as required The hypothesis that restrictions of 120594 are 119901-cartesian implies that this lift is a 119901-cartesiantransformation
Now Lemma 513 implies that any 119901-cartesian natural transformation is isomorphic to a restric-tion of 120594 Thus restrictions of 119901-cartesian transformations are isomorphic to restrictions of 120594 and itfollows from Lemma 514 that the class of 119901-cartesian transformation is closed under restriction
The first major result of this section is an internal characterization of cartesian fibrations inspiredby a similar result of Street [55 58 59] see also [63] Before stating this result recall from Lemma 348that from a functor 119901∶ 119864 ↠ 119861 we can build a fibered adjunction 1199011 ⊣119864 119894 where the right adjoint isinduced from the identity 2-cell id119901
119864
Hom119861(119861 119901)
119864 119861
119901119894
1199011 1199010120601lArr
119901
=119864
119864 119861
119901=
119901
119864 perp Hom119861(119861 119901)
119864119894 1199011
1199011
107
Similarly 1-cell induction for the right comma cone over 119901 applied to the generic arrow for 119864 inducesa functor
119864120794
119864
119861
11990101199011 120581lArr
119901
=
119864120794
Hom119861(119861 119901)
119864 119861
1199011 1199011199010119896
1199011 1199010120601lArr
119901
(5110)
5111 Theorem (an internal characterization of cartesian fibrations) For an isofibration 119901∶ 119864 ↠ 119861the following are equivalent
(i) 119901∶ 119864 ↠ 119861 defines a cartesian fibration(ii) The functor 119894 ∶ 119864 rarr Hom119861(119861 119901) admits a right adjoint over 119861
119864 perp Hom119861(119861 119901)
119861
119894
119901 1199010119903
(iii) The functor 119896 ∶ 119864120794 rarr Hom119861(119861 119901) admits a right adjoint with invertible counit
119864120794 Hom119861(119861 119901)
119896
perp
When these equivalent conditions hold then for a natural transformation 119883 119864119890prime
119890dArr120595 the following are
equivalent(iv) 120595 is 119901-cartesian(v) 120595 factors through a restriction of the 2-cell 1199011120598 where 120598 is the counit of the adjunction 119894 ⊣ 119903 via a
natural isomorphism so that 119901 = id
119883 119864 = 119883 Hom119861(119861 119901) Hom119861(119861 119901) 119864
119864
119890
119890primeuArr120595 119901120595
119890prime
119890
119903
uArr120598 1199011
conguArr 119894
(vi) The component
119883 119864120794 119864120794120595
dArr
119896of the unit for 119896 ⊣ is invertible that is 120595 is in the essential image of the right adjoint
108
The right adjoint of (ii) is the domain-component of the generic cartesian lift of (519) that carte-sian transformation is then recovered as 1199011120598 where 120598 is the counit of the fibered adjunction 119894 ⊣ 119903This explains the statement of (v) By 1-cell induction the generic cartesian lift 120594 can be representedby a functor ∶ Hom119861(119861 119901) rarr 119864120794 and this defines the right adjoint of (iii) and explains the statementof (vi)
Before proving Theorem 5111 we make two further remarks on these postulated adjunctions
5112 Remark By Lemma 348 the functor 119894 ∶ 119864 rarr Hom119861(119861 119901) is itself a right adjoint over 119861 to thecodomain-projection functor Since the counit of the adjunction 1199011 ⊣ 119894 is an isomorphism it followsformally that the unit of the adjunction 119894 ⊣ 119903must also be an isomorphism whenever the adjunctionpostulated in (ii) exists see Lemma B32
5113 Remark In the case where the infin-categories 119864120794 and Hom119861(119861 119901) are defined by the strictsimplicial limits of Definitions 321 and 331 the 1-cell 119896 induced in (5110) can be modeled by anisofibration
119864120794
Hom119861(119861 119901) 119861120794
119864 119861
119901120794
1199011
119896
1199011
1199011
119901
namely the Leibniz cotensor of the codomain inclusion 1∶ 120793 120794 and the isofibration 119901∶ 119864 ↠ 119861Now Lemma 359 can be used to rectify the adjunction 119896 ⊣ 119903 of (iii) to a right adjoint right inverseadjunction that is then fibered over Hom119861(119861 119901) So when Theorem 5111(iii) holds we may modelthe postulated adjunction by a right adjoint right inverse to the isofibration 119896
Proof Wersquoll prove (i)rArr(iii)rArr(ii)rArr(i) and demonstrate the equivalences (iv)hArr(vi) and (iv)hArr(v)in parallel
(i)rArr(iii) If 119901∶ 119864 ↠ 119861 is cartesian then the right comma cone over 119901 admits a cartesian lift along119901
Hom119861(119861 119901) 119864 Hom119861(119861 119901) 119864
119861 119861
1199011
1199010
uArr120601 119901 =
1199011
119903
uArr120594
119901
defining a functor 119903 ∶ Hom119861(119861 119901) rarr 119864 over 119861 together with a 119901-cartesian transformation 120594∶ 119903 rArr 1199011By 1-cell induction this generic cartesian transformation is represented by a functor
Hom119861(119861 119901) 119864 = Hom119861(119861 119901) 119864120794 119864119903
1199011
dArr120594 1199010
1199011
dArr120581
which we take as our definition of the putative right adjoint By the definition (5110) of 119896 120601119896 =119901120581 = 119901120594 = 120601 so Proposition 337 supplies a fibered isomorphism ∶ 119896 cong id with 1199010 = id1199010 and1199011 = id1199011
109
To prove that 119896 ⊣ it remains to define the unit 2-cell which we do by 2-cell induction from apair given by an identity 2-cell 1199011 = id1199011 and a 2-cell 1199010 that remains to be specified The requiredcompatibility condition of Proposition 336(ii) asserts that this 1199010must define a factorization of thegeneric arrow
1199010 1199011
1199010119896 = 119903119896 1199011119896
120581
1199010 1199011=id
120594119896
(5114)
through120581119896 = 120594119896 Note 119901120581 = 120601119896 has120594119896 as its 119901-cartesian lift so we define 1199010 by the induction prop-erty for the cartesian transformation 120594119896 applied to the generic arrow 120581∶ 1199010 rArr 1199011 By construction1199011199010 = id
By Lemma B41 once we verify that 119896 and are invertible then this data this together with ∶ 119896 cong id defines an adjunction with invertible counit We prove 119896 is invertible by 2-cell conserva-tivity 1199011119896 = 1199011 = id and 1199010119896 = 1199011199010 = id
Similarly by 2-cell conservativity to conclude that is invertible it suffices to prove that 1199010is an isomorphism Restricting (5114) along we see that 1199010 defines a fibered isomorphism of119901-cartesian transformations
119903 1199011
1199010119896 = 119903119896
120594
1199010 120594119896
so this follows from Lemma 515Finally note that a transformation 120595∶ 119890prime rArr 119890 is 119901-cartesian if and only if its factorization through
the generic 119901-cartesian lift of 119901120595 is invertible This factorization may be constructed by restrictingthe 2-cells of (5114) along 120595 ∶ 119883 rarr 119864120794 since 120581120595 = 120595 so we see that 120595 is 119901-cartesian if and onlyif 1199010120595 is invertible By 2-cell conservativity 120595 is invertible if and only if its domain componentis invertible This proves that (iv)hArr(vi)
(iii)rArr(ii) By Remark 5113 we can model the left adjoint of (iii) by an isofibration 119896 ∶ 119864120794 ↠Hom119861(119861 119901) and use Lemma 359 to rectify the adjunction 119896 ⊣ with invertible counit into a rightadjoint right inverse adjunction that is then fibered over Hom119861(119861 119901) Composing with the projection1199010 ∶ Hom119861(119861 119901) ↠ 119861 Lemma 357(ii) then gives us a fibered adjunction over 119861
119864120794 perp Hom119861(119861 119901)
119861
119896
1199011199010 1199010
By the dual of Lemma 348 the 1-cell 119895 ∶ 119864 rarr 119864120794 induced by the identity defines a left adjoint rightinverse to the domain projection
119864 119864120794119895
1199010perp
1199011perp
(5115)
110
supplying a fibered adjunction 119895 ⊣ 1199010 over 119864 that we push forward along 119901∶ 119864 ↠ 119861 to a fiberedadjunction over 119861
119864 perp 119864120794
119861119901
119895
1199010 1199011199010
This pair of fibered adjunctions composes to define a fibered adjunction over 119861 with left adjoint 119896119895and right adjoint 119903 ≔ 1199010 Proposition 337 supplies a fibered isomorphism 119894 cong 119896119895 since both 119894 and 119896119895define functors 119864 Hom119861(119861 119901) are induced by the identity id119901 over the right comma cone over 119901Composing with this fibered isomorphism we can replace the left adjoint of the composite adjunctionby 119894
119864 119864120794 Hom119861(119861 119901) = 119864 perp Hom119861(119861 119901)
119861 119861119901
119895
119894
cong
perp1199010 1199011199010
perp
119896
1199010
119894
119901 1199010119903
proving (ii)(ii)rArr(i) Now suppose given a fibered adjunction
119864 perp Hom119861(119861 119901)
119861
119894
119901 1199010119903
We will show that the codomain component of the counit
Hom119861(119861 119901) Hom119861(119861 119901) 119864
119864119903
uArr1205981199011
119894
satisfies the conditions of the generic 119901-cartesian transformation described in Lemma 518 This willthen also demonstrate the equivalence (iv)hArr(v)
The first thing to check is that 1199011120598 defines a lift of the right comma cone along 119901 To see thisconsider the horizontal composite
119864
Hom119861(119861 119901) Hom119861(119861 119901)
119864 119861
119901
119903
uArr120598
1199011
1199010
uArr120601
119894
111
Naturality of whiskering provides a commutative square
1199010119894119903 1199010
1199011199011119894119903 1199011199011
1199010120598
120601119894119903 120601
1199011199011120598
in which 1199010120598 = id as a fibered counit and 120601119894119903 = id since 120601119894 = id119901 Thus 1199011199011120598 = 120601 and we see that1199011120598 is a lift of 120601 along 119901
It remains only to verify that the restriction of 1199011120598 along any functor defines a 119901-cartesian trans-formation To that end consider 120573 ∶ 119883 rarr Hom119861(119861 119901) representing a 2-cell 120573∶ 119887 rArr 119901119890 our taskis to verify that 1199011120598120573 is 119901-cartesian Note that 1199011199011120598120573 = 120601120573 = 120573 so to prove the inductionproperty consider a 2-cell 120591∶ 119890Prime rArr 119890 and a factorization 119901120591 = 120573 sdot 120574 for some 120574∶ 119901119890Prime rArr 119887 Our taskis to define a 2-cell ∶ 119890Prime rArr 119903120573 so that the pasted composite
119883 Hom119861(119861 119901) Hom119861(119861 119901) 119864
119864120573
119890Prime
119890
119903
uArr120598 1199011
uArr 119894
is 120591 Transposing across the adjunction 119894 ⊣ 119903 it suffices instead to define a 2-cell ∶ 119894119890Prime rArr 120573 so that1199011 = 120591 We define by 2-cell induction from this condition and 1199010 = 120574 a pair which satisfies the2-cell induction compatibility condition
119901119890Prime 119901119890Prime
119887 119901119890120574=1199010 1199011199011=119901120591
120573
precisely on account of the postulated factorization 119901120591 = 120573 sdot 120574 This verifies the induction conditionof Definition 511(i)
Now consider an endomorphism 120577∶ 119903 rArr 119903 so that 1199011120598120573 sdot120577 = 1199011120598120573 and 119901120577 = id119887 Write ≔1199011120598 ∶ Hom119861(119861 119901) rarr 119864120794 for the functor induced by 1-cell induction from the generic 119901-cartesiantransformation Now the conditions defining 120577 allow us to induce a 2-cell ∶ 120573 rArr 120573 satisfying1199010 = 120577 and 1199011 = id To prove that 120577 is invertible will make use of the naturality of whiskeringsquare for the horizontal composite
Hom119861(119861 119901)
119883 119864120794 119864
Hom119861(119861 119901) Hom119861(119861 119901)
120573
120573dArr
1199010
119896dArr
119903
1199010119903120573 119903119896120573
1199010120573 119903119896120573
120573cong
120577=1199010 119903119896cong
120573cong
where is a special case of the 2-cell just given this name to be described momentarily for whichwe will demonstrate that is an isomorphism The composite 119896 is a 2-cell induced from 1199010119896 =1199011199010 = 119901120577 = id and 1199011119896 = 1199011 = id so by 2-cell conservativity this is an isomorphism Now 120577 is acomposite of three isomorphisms and hence is invertible
112
To complete the proof we must define and prove that is invertible Specializing the construc-tion just given we define to be the induced 2-cell satisfying 119901 = id and 120581 = 1199011120598119896 sdot Transposingacross the fibered adjunction 119894 ⊣ 119903 it suffices to define the transposed 2-cell ∶ 1198941199010 rArr 119896 so that1199011 = 120581 and 1199010 = id And we may transpose once more along an adjunction 119895 ⊣ 1199010 of (5115) con-structed by the dual of Lemma 348 The counit 120584∶ 1198951199010 rArr id of this adjunction satisfied the definingconditions that 1199010120584 = id and 1199011120584 = 120581 so to construct satisfying the conditions just described itsuffices to define instead a 2-cell 120585∶ 119894 rArr 119896119895 satisfying the conditions 1199011120585 = id and 1199010120585 = id Theseconditions are satisfied by the fibered isomorphism 120585∶ 119894 cong 119896119895 that arises by Proposition 337 sinceboth 1-cells 119864 Hom119861(119861 119901) are induced by the identity id119901 Unpacking these transpositions isdefined to be the composite
≔ 1199010 1199031198941199010 1199031198961198951199010 119903119896120578cong
120585cong
119903119896120584
To verify that is invertible it thus suffices to demonstrate that 119903119896120584 is invertible To see thiswe consider another pasting diagram
119864 119864
Hom119861(119861 119901) Hom119861(119861 119901) 119864120794 119864120794 Hom119861(119861 119901) 119864
119894dArr120598
119895dArr120584
119903
congdArr
1199031199010
119896 119903
By the definition (5110) of 119896 120601119896 = 119901120581 = 119901120594 = 120601 so Proposition 337 supplies a fibered isomor-phism ∶ 119896 cong id with 1199010 = id1199010 and 1199011 = id1199011
Now naturality of whiskering supplies a commutative diagram of 2-cells
1199031198961198951199010119894119903 119903119896119894119903 119903119894119903
1199031198961198951199010 119903119896 119903
119903119896119895119903120598 cong
119903119896120584119894119903
119903119896120598
119903119894119903cong
119903120598cong
119903119896120584cong119903
Since 120598 is the counit of an adjunction 119894 ⊣ 119903 with invertible unit 119903120598 is an isomorphism so we see that119903119896120584 is an isomorphism if and only if 119903119896120584119894119903 is And this is the case since 119896120584119894 is an isomorphism by2-cell conservativity 1199010119896120584119894 = id119901 while 1199011119896120584119894 = 1199011120598119894 which is an isomorphism again because 120598 isthe counit of an adjunction 119894 ⊣ 119903 with invertible unit
One of the myriad applications of Theorem 5111 is
5116 Corollary Cosmological functors preserve cartesian fibrations and cartesian natural transforma-tions
Proof By Theorem 5111(i)hArr(iii) an isofibration 119901∶ 119864 ↠ 119861 in an infin-cosmos 119974 is cartesian ifand only if the isofibration 119896 ∶ 119864120794 ↠ Hom119861(119861 119901) defined in Remark 5113 admits a right adjointright inverse A cosmological functor 119865∶ 119974 rarr ℒ preserves the class of isofibrations and the simpli-cial limits that define the domain and codomain of this 119896 Moreover cosmological functors preserveadjunctions and natural isomorphisms so if this adjoint exists in 119974 it also does in ℒ Similarly theinternal characterization of 119901-cartesian natural transformations given by Theorem 5111(iv)hArr(vi) isalso preserved by cosmological functors
113
Another application of Theorem 5111 is that it allows us to conclude that cartesianness is anequivalence-invariant property of isofibrations
5117 Corollary Consider a commutative square between isofibrations whose horizontals are equivalences
119865 119864
119860 119861
119902
sim
119892
119901
sim119891
Then 119901 is a cartesian fibration if and only if 119902 is a cartesian fibration in which case 119892 preserves and reflectscartesian transformations 120594 is 119902-cartesian if and only if 119892120594 is 119901-cartesian
Proof The commutative square of the statement induces a commutative square up to isomor-phism whose horizontals are equivalences
119865120794 119864120794
Hom119860(119860 119902) Hom119861(119861 119901)
sim
119892120794
119896 cong 119896
simHom119891(119891119892)
By the equivalence-invariance of adjunctions the left-hand vertical admits a right adjoint with invert-ible counit if and only if the right-hand vertical does these adjunctions being defined in such a waythat the mate of the given square is an isomorphism built by composing with the natural isomorphismsof the horizontal equivalences (see Proposition B31) By Theorem 5111(i)hArr(iii) it follows that 119901 iscartesian if and only if 119902 is cartesian
Supposing the postulated adjoints exist via their construction the whiskered composites 119892120794120578 and120578119892120794 of the units of the respective adjunctions are isomorphic Hence the component at an element120594 ∶ 119883 rarr 119865120794 of 119892120794120578 is invertible if and only if the component of 120578119892120794 is invertible since 119892120794 isan equivalence the former is the case if and only if the component of 120578 is invertible By Theorem5111(iv)hArr(vi) this proves that 120594 is 119901-cartesian if and only if 119892120594 is 119902-cartesian
In terminology we now introduce the square defined by the equivalences and also the squaredefined by their inversessup2 defines a cartesian functor from 119902 to 119901
5118 Definition (cartesian functor) Let 119901∶ 119864 ↠ 119861 and 119902 ∶ 119865 ↠ 119860 be cartesian fibrations Acommutative square
119865 119864
119860 119861
119902
119892
119901
119891
defines a cartesian functor if 119892 preserves cartesian transformations if 120594 is 119902-cartesian then 119892120594 is119901-cartesian
sup2For any inverse equivalences 119892prime and 119891prime to 119892 and 119891 there is a natural isomorphism 119902119892prime cong 119891prime119891119902119892prime = 119891prime119901119892119892prime cong 119891prime119901Using the isofibration property of 119902 of Proposition 1410 119892prime may be replaced by an isomorphic functor 119892Prime which alsodefines an inverse equivalence to 119892 and for which the square 119902119892Prime = 119891prime119901 commutes strictly
114
The internal characterization of cartesian functorsmakes use of amap between right representablecommainfin-categories induced by the commutative square 119891119902 = 119901119892 defined by Proposition 335
5119 Theorem (an internal characterization of cartesian functors) For a commutative square
119865 119864
119860 119861
119902
119892
119901
119891
between cartesian fibrations the following are equivalent(i) The square (119892 119891) defines a cartesian functor from 119902 to 119901(ii) The mate of the canonical isomorphism
119865 119864 119865 119864
Hom119860(119860 119902) Hom119861(119861 119901) Hom119860(119860 119902) Hom119861(119861 119901)
119892
119894 cong 119894
119892
cong
Hom119891(119891119892) Hom119891(119891119892)
119903 119903
in the diagram of functors over 119891∶ 119860 rarr 119861 is an isomorphism(iii) The mate of the canonical isomorphism
119865120794 119864120794 119865120794 119864120794
Hom119860(119860 119902) Hom119861(119861 119901) Hom119860(119860 119902) Hom119861(119861 119901)
119892120794
119896 cong 119896
119892120794
cong
Hom119891(119891119892) Hom119891(119891119892)
in the diagram of functors is an isomorphism
Proof We will prove (i)hArr(ii) and (i)hArr(iii) The idea in each case is similar Conditions (ii) and(iii) imply that 119892 preserves the explicitly chosen cartesian lifts up to isomorphism which by Lemma513 implies that 119892 preserves all cartesian lifts Conversely assuming (i) we need to show that awhiskered copy of the counit of 119894 ⊣ 119903 and of the unit of 119896 ⊣ are isomorphisms The counit of 119894 ⊣ 119903and unit of 119896 ⊣ each encode the data of the factorizations of a natural transformation through thecartesian lift of its projection It will follow from (i) that the cells in question are cartesian and thefactorizations live over identities so Lemma 515 will imply that these natural transformations areinvertible
(i)hArr(iii) For convenience we take the functors 119896 to be the isofibrations of Remark 5113 sothe square on the left hand side of (iii) commutes strictly and its mate is the 2-cell 119892120794 By Theorem5111(iv)hArr(vi) this component of is invertible if and only if 119892120594 is 119901-cartesian where 120594 is the generic119902-cartesian lift of Lemma 518 for the cartesian fibration 119902 ∶ 119865 ↠ 119860 recall = 120594 By that lemmaagain 119892120594 is 119901-cartesian if and only if 119892 preserves cartesian transformations since the other canonical119902-cartesian lifts are constructed as restrictions of 120594
(i)hArr(ii) Let us write for Hom119891(119891 119892) to save space Since the unit of 119894 ⊣ 119903 is an isomorphismby Remark 5112 the mate of the isomorphism on the left hand side of (ii) is isomorphic to 119903120598 soour task is to show that this natural transformation is invertible if and only if 119892 defines a cartesianfunctor Recall from the proof of Theorem 5111(ii)rArr(i) that 1199011120598 defines the generic 119902-cartesian lift of
115
Lemma 518 for the cartesian fibration 119902 ∶ 119865 ↠ 119860 whiskering with ≔ Hom119891(119891 119892) ∶ Hom119860(119860 119902) rarrHom119861(119861 119901) carries this to a 2-cell whose projection with 1199010 is an identity since 1199010120598 = id and whoseprojection along 1199011 is 1198921199011120598 by the commutativity of the left-hand portion of the diagram below
119865 119864 119864
Hom119860(119860 119902) Hom119860(119860 119902) Hom119861(119861 119901) Hom119861(119861 119901)
119865 119864 119864
119892
119894 119894 119894119903dArr120598
1199011
119903
1199011
dArr120598
1199011
119892
Now naturality of whiskering provides a commutative square of natural transformations
1199011119894119903119894119892119903 119903
1199011119894119903 1199011
119903120598
1199011120598119894119892119903 cong 1199011120598
1199011120598
where wersquove simplified some of the names since 1199011119894 = id Since 120598 is the counit of an adjunction 119894 ⊣ 119903with invertible unit 120598119894 is an isomorphism Note 119901119903120598 = 1199010120598 = id and the right-hand vertical 1199011120598 isa 119901-cartesian lift of the restriction of 120601 this 120601 being the right comma cone over 119901 which equals thewhiskered right comma cone 119891120601 this120601 being the right comma cone over 119902 by the definition of Thebottom horizontal 1199011120598 is similarly a lift of 119891120601 = 120601 So if 119892 is a cartesian functor the right-handvertical and bottom horizontal are both 119901-cartesian lifts of a common 2-cell and the conservativityproperty implies that 119903120598 is invertible Conversely if 119903120598 is invertible the 1199011120598 = 1198921199011120598 is isomorphicto a 119901-cartesian transformation and is consequently 119901-cartesian Since Lemma 518 constructs theother canonical 119902-cartesian lifts as restrictions of 1199011120598 this is the case if and only if 119892 is a cartesianfunctor
5120 Proposition (pullback stability) If
119865 119864
119860 119861
119902
119892
119901
119891
is a pullback square and 119901 is a cartesian fibration then 119902 is a cartesian fibration Moreover a natural transfor-mation 120594 with codomain 119865 is 119902-cartesian if and only if 119892120594 is 119901-cartesian and in particular the pullback squaredefines a cartesian functor
116
Proof We apply Theorem 5111(i)rArr(iii) in the form described in Remark 5113 to the cartesianfibration 119901 which yields a fibered adjunction
119864120794 perp Hom119861(119861 119901)
Hom119861(119861 119901)119896
119896
(5121)
We will now argue that this functor 119896 pulls back to the corresponding functor for 119902 To that endfirst note that the top face of the following cube is a pullback since the front back and bottom facesare
119865120794
119864120794
Hom119860(119860 119902) 119860120794
Hom119861(119861 119901) 119861120794
119865 119860
119864 119861
119892120794
119896
119902120794
119896Hom119891(119891119892)
1199011 119891120794 1199011
1199011
119901120794
119892119902
119891119901
1199011
The right adjoint (minus)120794 preserves pullbacks so 119865120794 is the pullback of 119901120794 along 119891120794 and since this pullbacksquare factors through the top face of the cube along the square inducing the maps 119896 we conclude thatthis last square is a pullback as claimed
Now pullback defines a cosmological functor Hom119891(119891 119892)lowast ∶ 119974Hom119861(119861119901) rarr 119974Hom119860(119860119902) that car-ries the fibered adjunction (5121) to a fibered adjunction
119865120794 perp Hom119860(119860 119902)
Hom119860(119860 119902)119896
119896
which by Theorem 5111(iii)rArr(i) proves that 119902 is a cartesian fibration Moreover by construction ofthe adjunction 119896 ⊣ as a pullback of the adjunction 119896 ⊣ both of the mates in Theorem 5119(iii)are identities proving that (119892 119891) defines a cartesian functor
To see that (119892 119891) creates cartesian natural transformations note that a natural transformation 120594with codomain119865 is represented by an element 120594 ∶ 119883 rarr 119865120794 and 119892120594 is represented by the image of thiselement under the functor 119892120794 ∶ 119865120794 rarr 119864120794 By Theorem 5111(iv)hArr(vi) 120594 is 119902-cartesian just when thecomponent 120594 of the unit of 119896 ⊣ This unit component 120578120594 is the pullback of the corresponding
117
unit component 120578119892120794120594 indexed by 119892120594 and by conservativity of the smothering functor
hFun(119883 119865120794) rarr hFun(119883Hom119860(119860 119902)) timeshFun(119883Hom119861(119861119901))
hFun(119883 119864120794)
if 120578119892120794120594 is invertible then so is 120578120594
Pullback squares provide a key instance of cartesian functors Another is given by the followinglemma which can be proven using Theorem 5119
5122 Lemma If
119865 119864
119861
119892
119902 119901
is a functor between cartesian fibrations that admits a left adjoint over 119861 then 119892 defines a cartesian functor
Proof If ℓ ⊣119861 119892 then the cosmological functor 119901lowast1 ∶ 119974119861 rarr119974119861120794 carries this fibered adjunctionto a fibered adjunction
Hom119861(119861 119902) Hom119861(119861 119901)
Homid119861(id119861119892)
perp
Homid119861(id119861ℓ)
Now both horizontal functors in the commutative square
119865120794 119864120794
Hom119860(119860 119902) Hom119861(119861 119901)
119892120794
119896 cong 119896
Hom119891(119891119892)
admit left adjoints and a standard result from the calculus of mates tells us that the mate with respectto the vertical adjunctions 119896 ⊣ is an isomorphism if and only if the mate with respect to the horizon-tal adjunctions is an isomorphism the latter natural transformation between left adjoints being thetranspose of the former natural transformation between their right adjoints This is the case becausethe mate with respect to the left adjoints lies is the fiber of the smothering functor of Proposition 325for 119865120794
Examples of cartesian fibrations are overdue
5123 Proposition (domain projection) For anyinfin-category119860 the domain-projection functor 1199010 ∶ 119860120794 ↠119860 defines a cartesian fibration Moreover a natural transformation 120594 with codomain119860120794 is 1199010-cartesian if andonly if 1199011120594 is invertible
Before giving the proof we explain the idea A 2-cell
119883 119860120794
119860
120573
119886
1199010uArr120572
118
defines a composable pair of 2-cells 120572∶ 119886 rArr 119909 and 120573∶ 119909 rArr 119910 in hFun(119883119860) Composing these we
induce a 2-cell 119883 119860120794120573∘120572
120573
dArr120594 representing the commutative square
119886 119909
119910 119910
120572
120573∘120572 120573
so that 1199010120594 = 120572 as required and 1199011120594 = idsup3
Proof We use Theorem 5111(i)hArr(iii) and prove that 1199010 is cartesian by constructing an appro-priate adjoint to the functor
(119860120794)120794 120794 times 120794
Hom119860(119860 1199010) 119860120794 120795 120794
119860120794 119860 120794 120793
1199011207940
1199011
119896
1199011
1199011
0times120794
1199010
120794times1
0
1
defined by cotensoring with the 1-categories displayed above right⁴To construct a right adjoint with invertible counit to the map 119896 it suffices to construct a left
adjoint left inverse to the inclusion of 1-categories 120795 120794 times 120794 with image (0 0) rarr (0 1) rarr (1 1)The left adjoint ℓ ∶ 120794 times 120794 rarr 120795 is a left inverse on the image of 120795 and sends (1 0) to the terminalelement of 120795
120794 times 120794 ni
⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩
(0 0) (0 1)
(1 0) (1 1)
⎫⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎭
↦
⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩
0 1
2 2
⎫⎪⎪⎪⎪⎬⎪⎪⎪⎪⎭
isin 120795
Now
(119860120794)120794 119860120795 ≃ Hom119860(119860 1199010)
119896
perp
119860ℓ
defines the desired right adjoint with invertible counit
sup3While the weak universal property of the arrowinfin-category can be used to induce 2-cells with codomain119860120794 it cannotbe used to prove equations between the induced 2-cells as required to demonstrate that induction condition of Definition511 Thus some sort ofinfin-cosmos-level argument is necessary to establish this result
⁴The cotensor 119860(minus) carries pushouts of simplicial sets to pullbacks of infin-categories and the pushout of 120794 cup120793 120794 ofsimplicial sets is Λ1[2] not 120795 = Δ[2] However on account of the equivalence of infin-categories 119860120795 ≃ 119860Λ1[2] no harmcomes from making the indicated substitution
119
The characterization of 1199010-cartesian transformations now follows from Theorem 5111(iv)hArr(vi)A cell 120594 with codomain 119860120794 is 1199010-cartesian if and only if it its representing element of (119860120794)120794 is in theessential image of the right adjoint 119860ℓ ∶ 119860120795 rarr 119860120794times120794 Clearly if 120594 is in the essential image then itscodomain component must be invertible
Conversely suppose the codomain component
119883 119860120794times120794 119860120794120594 ev(1minus)
represents a natural isomorphism Applying Lemma 1112 to Fun(119883119860) one can build a diagram inFun(119883119860)120794times120794 cong Fun(119883119860120794times120794) whose top edge is the given isomorphism whose vertical edges areisomorphisms and whose bottom edge is an identity This glues onto 120594 to define a diagram 119883 rarr119860Λ1[2]times120794 and now the composite
119883 119860Λ1[2]times120794 119860120795times120794 119860120794times120794times120794sim 119860119888
where 119888 ∶ 120794times120794 rarr 120795 is the surjective functor that sends (0 0) and (0 1) to 0 witnesses an isomorphismbetween 120594 and a diagram in the image of 119860ℓ whose codomain component is the identity
The same argument proves that for any 119892∶ 119862 rarr 119860 the domain-projection 1199010 ∶ Hom119860(119860 119892) ↠119860 is a cartesian fibration For 119891∶ 119861 rarr 119860 the domain-projections 1199010 ∶ Hom119860(119891 119860) ↠ 119861 and1199010 ∶ Hom119860(119891 119892) ↠ 119861 are obtained by pullback via Proposition 5120 These cartesian fibrationsfigures in a final important lemma about the class of 119901-cartesian transformations
5124 Lemma Let 119901∶ 119864 ↠ 119861 be a cartesian fibration and consider a composable pair of natural transfor-
mations 119883 119864119890prime
119890dArr120595 and 119883 119864
119890Prime
119890primedArr120595prime with codomain 119864
(i) If 120595 and 120595prime are 119901-cartesian then so is 120595 sdot 120595prime(ii) If 120595 and 120595 sdot 120595prime are 119901-cartesian then so is 120595prime
Proof For (i) recall from the proof of Lemma 518 that a 119901-cartesian lift of 119901120595 is given by thecomposite
119883 Hom119861(119861 119901) 119864119901120595
119903
1199011
dArr120594
with the natural transformation 120594∶ 119903 rArr 1199011 of (519) whose domain 119903 is the right adjoint of Theorem5111(ii) Since we are assuming that120595 is 119901-cartesian 120595 is isomorphic to this whiskered natural trans-formation so we may redefine 120595 to equal it and redefine 120595prime to absorb the isomorphism By Lemma514 this new 120595prime is still 119901-cartesian since wersquove assumed 120595prime is This modification does not change thecomposition transformation 120595 sdot 120595prime that we desire to show is 119901-cartesian
Now by 2-cell induction the diagram as below-left defines a 2-cell as below-right
119901119890Prime 119901119890prime
119901119890 119901119890
119901120595prime
119901(120595sdot120595prime) 119901120595 119883 Hom119861(119861 119901)
119901(120595sdot120595prime)
119901120595
dArr120574
120
By the generalization of Proposition 5123 120574 is a 1199010-cartesian cell By Lemma 5122 the fibered rightadjoint 119903
Hom119861(119861 119901) perp 119864
1198611199010 119903
119894
119901
carries 120574 to a 119901-cartesian transformation 119903120574Now the horizontal composite
119883 Hom119861(119861 119901) 119864
119901(120595sdot120595prime)
119901120595
dArr120574
119903
1199011
dArr120594
provides a commutative diagram of natural transformations
119890prime
119890 119890
120594119901(120595sdot120595prime)
119903120574
120595
In particular the composite 120595 sdot 119903120574 is a 119901-cartesian natural transformation Since 119901119903120574 = 1199010120574 = 119901120595prime119903120574 is a 119901-cartesian lift of the cartesian transformation 120595prime so 120595prime and 119903120574 are isomorphic But now120595 sdot 120595prime is isomorphic to the 119901-cartesian transformation 120594119901(120595sdot120595prime) and Lemma 514 proves that 120595 sdot 120595prime is119901-cartesian
Now for (ii) suppose 120595 and 120595 sdot 120595prime are 119901-cartesian and let 120594prime ∶ rArr 119890prime denote a 119901-cartesian lift of119901120595prime Consider the factorization 120595prime = 120594prime sdot 120579 of 120595prime through its 119901-cartesian lift with 119901120579 = id Nowpart (i) implies that 120595 sdot 120594prime is 119901-cartesian and 120595 sdot 120595prime = 120595 sdot 120594prime sdot 120579 so Lemma 515 implies that 120579 isan isomorphism Now 120595prime is isomorphic to a 119901-cartesian cell so Lemma 514 implies that 120595prime must be119901-cartesian
For infin-categories admitting pullbacks the codomain-projection functor also defines a cartesianfibration
5125 Proposition (codomain projection) Let 119860 be an infin-category that admits pullbacks in the senseof Definition 437 Then the codomain-projection functor 1199011 ∶ 119860120794 ↠ 119860 is a cartesian fibration and the1199011-cartesian arrows are just those 2-cells that represent pullback squares
121
Proof Via Theorem 5111(i)hArr(iii) we desire a right adjoint right inverse to the functor 119896 definedbelow-left applying 119860minus to the diagram of simplicial sets appearing below-right
(119860120794)120794 120794 times 120794
Hom119860(119860 1199011) 119860120794 ⟓ 120794
119860120794 119860 120794 120793
1199011207941
1199011
119896
1199011
1199011
1times120794
1199010
120794times1
1
1
This is done in Corollary 435
119860⊡ 119860⟓(119860120794)120794 congres
perpran
cong Hom119860(119860 1199011)
Exercises
51i Exercise Prove Lemma 514
51ii Exercise Attempt to prove directly from Definition 516 that cosmological functors preservecartesian fibrations and explain what goes wrong
51iii Exercise Categorify the intuition that cartesian fibrations 119901∶ 119864 ↠ 119861 and 119902 ∶ 119865 ↠ 119861 defineldquocontravariant 119861-indexed functors valued ininfin-categoriesrdquo by proving that a cartesian functor
119864 119865
119861
119892
119901 119901
defines a ldquonatural transformationrdquo show that there exists a natural isomorphism in the square of fibers
119864119887 119865119887
119864119886 119865119886
120573lowast
119892
existcong 120573lowast
119892
where the action of an arrow 120573 in the homotopy category of 119861 on the fibers is defined by factoring thedomain of a 119901- or 119902-cartesian lift of 120573
119864119887 119864 119864119887 119864
119864119886
1 119861 1 119861
ℓ119887
=119901 120573lowast 120573lowast(ℓ119887)
uArr120594120573
ℓ119887
119901ℓ119886119887
119886
uArr120573
119886
51iv Exercise Use Proposition 5120 to prove that cartesian functors pull back
122
52 Cocartesian fibrations and bifibrations
By the dual of Proposition 5123 the codomain-projection functor is also a cocartesian fibrationa notion we now introduce
521 Definition (119901-cocartesian transformations) Let 119901∶ 119864 ↠ 119861 be an isofibration A natural trans-
formation 119883 119864119890
119890primedArr120594 with codomain 119864 is 119901-cocartesian if
(i) induction Given any natural transformations 119883 119864119890
119890PrimedArr120591 and 119883 119861
119901119890prime
119901119890PrimedArr120574 so that 119901120591 =
120574 sdot 119901120594 there exists a lift 119883 119864119890prime
119890PrimedArr of 120574 so that 120591 = sdot 120594
119890 119890Prime
119890prime
119901119890 119901119890Prime
119901119890prime
120591
120594
119901120591
119901120594 120574
isin hFun(119883 119864)
isin hFun(119883 119861)
119901lowast
(ii) conservativity Any fibered endomorphism of 120594 is invertible if 119883 119864119890prime
119890primedArr120577 is any natural
transformation so that 120577 sdot 120594 = 120594 and 119901120577 = id119901119890prime then 120577 is invertible
522 Definition (cocartesian fibration) An isofibration 119901∶ 119864 ↠ 119861 is a cocartesian fibration if(i) Any natural transformation 120573∶ 119901119890 rArr 119887 as below-left admits a lift 120594120573 ∶ 119890 rArr 120573lowast119890 as below-right
119883 119864 119883 119864
119861 119861
119890
119887
dArr120573 119901 =
119890
120573lowast119890
dArr120594120573
119901
that is a 119901-cartesian transformation so that 119901120594 = 120573(ii) The class of 119901-cocartesian transformations is closed under restriction along any functor that
is if 119883 119864119890
119890primedArr120594 is 119901-cocartesian and 119891∶ 119884 rarr 119883 is any functor then 119884 119864
119890119891
119890prime119891
dArr120594119891 is 119901-co-
cartesian
The dual to Theorem 5111 asks for left adjoints to 119894 ∶ 119864 rarr Hom119861(119901 119861) and 119896 ∶ 119864120794 rarr Hom119861(119901 119861)in place of right adjoints See Exercise 52i This result can be deduced immediately by consideringthe isofibration 119901∶ 119864 ↠ 119861 as a map in the dualinfin-cosmos
123
523 Observation (dual infin-cosmoi) There is an identity-on-objects functor (minus)∘ ∶ 120491 rarr 120491 thatreverses the ordering of the elements in each ordinal [119899] isin 120491 The functor (minus)∘ sends a face map120575119894 ∶ [119899minus1] ↣ [119899] to the face map 120575119899minus119894 ∶ [119899minus1] ↣ [119899] and sends the degeneracy map 120590119894 ∶ [119899+1] ↠ [119899]to the degeneracy map 120590119899minus119894 ∶ [119899 + 1] ↠ [119899] Precomposition with this involutive automorphisminduces an involution (minus)op ∶ 119982119982119890119905 rarr 119982119982119890119905 that sends a simplicial set 119883 to its opposite simplicialset 119883op with the orientation of the vertices in each simplex reversed This construction preserves allconical limits and colimits and induces an isomorphism (119884119883)op cong (119884op)119883op
on exponentialsFor anyinfin-cosmos119974 there is a dualinfin-cosmos119974co with the same objects but with functor spaces
defined byFun119974co(119860 119861) ≔ Fun119974(119860 119861)op
The isofibrations equivalences and trivial fibrations in119974co coincide with those of119974Conical limits in 119974co coincide with those in 119974 while the cotensor of 119860 isin 119974 with 119880 isin 119982119982119890119905 is
defined to be 119860119880op In particular the cotensor of aninfin-category with 120794 is defined to be 119860120794op
whichexchanges the domain and codomain projections from arrow and commainfin-categories
524 Definition An isofibration 119901∶ 119864 ↠ 119861 defines a bifibration if 119901 is both a cartesian fibrationand a cocartesian fibration
Projections give trivial examples of bifibrations
525 Example For anyinfin-categories119860 and 119861 the projection functor 120587∶ 119860times119861 ↠ 119861 is a bifibrationin which a 2-cell with codomain119860times119861 is120587-cocartesian or120587-cartesian if and only if its composite withthe projection 120587∶ 119860 times 119861 ↠ 119860 is an isomorphism
526 Proposition Let 119901∶ 119864 ↠ 119861 be a bifibration Then any arrow 119883 119861119886
119887
dArr120573 induces a fibered
adjunction
119864119886 perp 119864119887 119864119886 119864 119864119887
119883 119883 119861 119883
120573lowast
119901119886 120573lowast 119901119887
ℓ119886
119901119886
119901
ℓ119887
119901119887
119886 119887between the fibers of 119901 over 119886 and 119887
As will be remarked upon following the proof of this result the left adjoint 120573lowast is the covariantlypseudofunctorial action of the arrow 120573 on the fibers of 119901 defined in (501) while the right adjoint 120573lowastis the dual contravariantly pseudofunctorial action
Proof Write 120573 ∶ 119883 rarr 119861120794 for the functor induced by 120573 Note that the pullbacks defining thefibers over its domain edge 119886 and codomain edge 119887 factor as
119864119886 Hom119861(119901 119861) 119864 Hom119861(119861 119901) 119864119887
119883 119861120794 119861 119861120794 119883
ℓ119886
119901119886
119902
1199010119901 119902
1199011
ℓ119887
119901119887
119886
120573 1199010 1199011 120573
119887
124
Now via Remark 5113 Theorem 5111(i)rArr(iii) and its dual provide a right adjoint right inverseto 119896 ∶ 119864120794 rarr Hom119861(119861 119901) and a left adjoint right inverse to 119896 ∶ 119864120794 rarr Hom119861(119901 119861) Composingthe former fibered adjunction with 119902 ∶ Hom119861(119861 119901) ↠ 119861120794 and the latter fibered adjunction with119902 ∶ Hom119861(119901 119861) ↠ 119861120794 we obtain a composable pair of adjunctions
Hom119861(119901 119861) 119864120794 Hom119861(119861 119901)
119861120794
ℓ
perp
119902119896 119901120794
119896
perp
119902
fibered over 119861120794 note in both cases that 119902119896 = 119901120794 Pulling back the composite adjunction along120573 ∶ 119883 rarr 119861120794 yields the fibered adjunction of the statement
527 Remark (action of arrows on the fibers of a cocartesian fibration) If 119901∶ 119864 ↠ 119861 is a cocartesianfibration but not a cartesian fibration the construction in the proof of Proposition 526 still definesthe functor 120573lowast ∶ 119864119886 rarr 119864119887 Examining the details of this construction we see that it produces thefunctor given this name in (501) The functor just constructed as the pullback over 120573 of 119896ℓ isinduced by the composite commutative square
119864119886 Hom119861(119901 119861) 119864
119883 119861120794 119861
119901119886
119902
1199011ℓ
119901
120573 1199011
which defines a cone over the pullback defining 119864119887 The top-horizontal functor takes the codomain-component of the 119901-cocartesian lift of 120573 with domain ℓ119886 This recovers the description given at thestart of this chapter
Exercises
52i Exercise Formulate the dual to Theorem 5111 providing an internal characterization of co-cartesian fibrations
52ii Exercise Prove that for any infin-category 119860 the codomain-projection functor 1199011 ∶ 119860120794 ↠ 119860defines a cocartesian fibration
53 The quasi-category theory of cartesian fibrations
In this section we reinterpret the notion of cartesian fibration and cartesian natural transforma-tion from the point of view of the infin-cosmos 119974 rather than its quotient homotopy 2-category 120101119974In doing so we recall that the functors 119890 ∶ 119883 rarr 119864 between infin-categories are precisely the verticesor 0-arrows in the quasi-categorical functor space Fun(119883 119864) The 1-simplices or 1-arrows 120594∶ 119890prime rarr 119890
of Fun(119883 119864) represent natural transformations 119883 119864119890prime
119890dArr120594 Every natural transformation from 119890prime
to 119890 is represented by a 1-arrow 119890prime rarr 119890 and a parallel pair of 1-arrows represent the same naturaltransformation if and only if they are homotopic bounding a 2-arrow in Fun(119883 119864) whose 0th or 2ndedge is degenerate
125
Before analyzing the 0- and 1-arrows in the functor spaces in particular we consider the collectionof functor spaces of an infin-cosmos globally and prove another important corollary of the internalcharacterization of cartesian fibrations The notion of cartesian fibrations are representably definedin the following sense
531 Proposition Let 119901∶ 119864 ↠ 119861 be an isofibration betweeninfin-categories in aninfin-cosmos119974 Then 119901 isa cartesian fibration if and only if
(i) For all 119883 isin 119974 the isofibration 119901lowast ∶ Fun(119883 119864) ↠ Fun(119883 119861) is a cartesian fibration between quasi-categories
(ii) For all 119891∶ 119884 rarr 119883 isin 119974 the square
Fun(119883 119864) Fun(119884 119864)
Fun(119883 119861) Fun(119884 119861)
119901lowast
119891lowast
119901lowast
119891lowast
is a cartesian functor
Proof If 119901∶ 119864 ↠ 119861 is a cartesian fibration then Theorem 5111(i)rArr(iii) constructs a right ad-joint right inverse to 119896 ∶ 119864120794 ↠ Hom119861(119861 119901) The simplicial bifunctor Fun(minus minus) ∶ 119974op times 119974 rarr 119980119966119886119905defines a 2-functor Fun(minus minus) ∶ 120101119974op times 120101119974 rarr 120101119980119966119886119905 which transposes to a Yoneda-type embeddingFun(minus minus) ∶ 120101119974 rarr 120101119980119966119886119905120101119974
opfrom the homotopy 2-category of 119974 to the 2-category of 2-functors
2-natural transformations and modifications This 2-functor carries the adjunction 119896 ⊣ to an ad-junction in the 2-category 120101119980119966119886119905120101119974
op This latter adjunction defines for each 119883 isin 119974 a right adjoint
right inverse adjunction
Fun(119883 119864)120794 Fun(119883 119864120794) perp Fun(119883Hom119861(119861 119901)) HomFun(119883119861)(Fun(119883 119861) 119901lowast)cong119896lowast
conglowast
and for each 119891∶ 119884 rarr 119883 in119974 a strict adjunction morphism⁵ commuting strictly with the left adjointsand with the right adjoints
Fun(119883 119864)120794 cong Fun(119883 119864120794) Fun(119884 119864120794) cong Fun(119884 119864)120794
Hom(Fun(119883 119861) 119901lowast) cong Fun(119883Hom119861(119861 119901)) Fun(119884Hom119861(119861 119901)) cong Hom(Fun(119884 119861) 119901lowast)
119891lowast
119896lowast⊢ 119896lowast ⊣
119891lowast
lowast lowast
(532)By Theorems 5111(iii)rArr(i) and 5119(iii)rArr(i) this demonstrates the two conditions of the statement
Conversely supposing 119901∶ 119864 ↠ 119861 satisfies conditions (i) and (ii) by Theorems 5111(i)rArr(iii) and5119(i)rArr(iii) there is a commutative square 119896lowast119891lowast = 119891lowast119896lowast where both verticals 119896lowast admit right adjointright inverses 119896lowast ⊣ and the mate of the identity 119896lowast119891lowast = 119891lowast119896lowast defines an isomorphism 119891lowast cong 119891lowastBy Proposition in Appendix B this data suffices to internalize the right adjoints to the repre-sentable functors 119896lowast ∶ Fun(119883 119864120794) rarr Fun(119883Hom119861(119861 119901)) to a functor lowast ∶ Fun(119883Hom119861(119861 119901)) rarr
⁵A strict adjunctionmorphism is given by a pair of functors the horizontals of (532) that define strictly commutativesquares with both the left and with the right adjoints and so that the units of each adjunction whisker along these functorsto each other and the counits of each adjunction whisker along these functors to each other See Appendix B for more
126
Fun(119883 119864120794) arising from post-composition with some ∶ Hom119861(119861 119901) rarr 119864120794 This right adjoint isextracted as the image of the identity element
Fun(Hom119861(119861 119901)Hom119861(119861 119901)) Fun(Hom119861(119861 119901) 119864120794)
id
and the unit and counit are internalized similarly the condition on mates is used to verify the triangleequalities that demonstrate that 119896 ⊣ Now Theorem 5111(iii)rArr(i) proves that 119901∶ 119864 ↠ 119861 is acartesian fibration
An easier argument along the same lines demonstrates
533 Corollary A commutative square between cartesian fibrations as displayed below-left
119865 119864 Fun(119883 119865) Fun(119883 119864)
119860 119861 Fun(119883119860) Fun(119883 119861)
119902
119892
119901 119902lowast
119892lowast
119901lowast
119891 119891lowast
defines a cartesian functor in aninfin-cosmos 119974 if and only if for all 119883 isin 119974 the square displayed above rightdefines a cartesian functor between cartesian fibrations of quasi-categories
Proof Exercise 53i
Our aim is now to characterize those 1-arrows in Fun(119883 119864) that represent 119901-cartesian naturaltransformation for some cartesian fibration 119901∶ 119864 ↠ 119861 Recall that the 1-arrows in Fun(119883 119864) are inbijection with the 0-arrows of Fun(119883 119864)120794 cong Fun(119883 119864120794)
534 Definition (119901-cartesian 1-arrow) Let 119901∶ 119864 ↠ 119861 be a cartesian fibration and consider a 1-arrow120594 in Fun(119883 119864) defining an element 120594 isin Fun(119883 119864)120794 cong Fun(119883 119864120794) Say 120594 is a 119901-cartesian 1-arrow ifit is isomorphic to some object in the image of the right adjoint right inverse functor
Fun(119883Hom119861(119861 119901)) Fun(119883 119864120794)lowast
of Theorem 5111(iii)
The new notion of 119901-cartesian 1-arrow coincides exactly with the previous notion of 119901-cartesiannatural transformation
535 Lemma Consider a cartesian fibration 119901∶ 119864 ↠ 119861 betweeninfin-categories For a 1-arrow 120594∶ 119890prime rarr 119890 inFun(119883 119864) the following are equivalent
(i) 120594 is a 119901-cartesian 1-arrow
(ii) 120594 represents a 119901-cartesian natural transformation 119883 119864119890prime
119890dArr120594
(iii) 120594 represents a natural transformation
120793 Fun(119883 119864)119890prime
119890
dArr120594
127
that is cartesian for 119901lowast ∶ Fun(119883 119864) ↠ Fun(119883 119861)
Conversely a natural transformation 119883 119864119890prime
119890dArr120594 is 119901-cartesian if and only if any representing 1-arrow
120594 ∶ 119883 rarr 119864120794 satisfies all of these equivalent conditions
Proof We start with the final clause Note from Exercise 32i that homotopic 1-arrows are iso-morphic as objects of Fun(119883 119864120794) so if some 1-arrow representing a 119901-cartesian transformation is a119901-cartesian 1-arrow then all representatives of that natural transformation are
Now the equivalence (i)hArr(ii) follows fromTheorem 5111(iv)hArr(vi) oncewe establish that a 1-arrowof Fun(119883 119864) when encoded as a functor 120594∶ 119883 rarr 119864120794 is in the essential image of lowast if and only if thecomponent 120594 of the unit of 119896 ⊣ is invertible
If 120594∶ 119890prime rarr 119890 is a 119901-cartesian 1-arrow then by definition there exists some 120573∶ 119883 rarr Hom119861(119861 119901)and an invertible 2-cell
119883 119864120794
Hom119861(119861 119901)
120594
120573 dArrcong
The unit of the adjunction 119896 ⊣ of Theorem 5111(iii) has the property that is invertible so thecomponent 120573 is invertible and so 120594 is also invertible By Theorem 5111(vi)rArr(iv) this implies that120594 represents a 119901-cartesian natural transformation
Conversely if 120594 defines a 119901-cartesian natural transformation then Theorem 5111(iv)rArr(vi) tellsus that for any representing 1-arrow 120594 ∶ 119883 rarr 119864120794 the component 120594 is an isomorphism Inparticular 120594 is isomorphic to 119896120594 which proves that 120594 is in the essential image of lowast and thusdefines a 119901-cartesian 1-arrow
Finally via the characterization of cartesian transformations given in Theorem (vi) and the factthat the adjunction 119896 ⊣ in 120101119974 induces an adjunction 119896lowast ⊣ lowast in 120101119980119966119886119905 the conditions (ii) and(iii) are tautologically equivalent The former refers to the adjunction between categories hFun(119883 119864)and hFun(119883 119861) while the latter refers to the adjunction between categories hFun(120793Fun(119883 119864)) andhFun(120793Fun(119883 119861)) and these are isomorphic
Combining Lemma 535with Proposition 531 we arrive at a new equivalent definition of cartesianfibrations
536 Corollary An isofibration 119901∶ 119864 ↠ 119861 is a cartesian fibration if and only if(i) Any 1-arrow with codomain 119861 admits a 119901-cartesian lift with specified codomain 0-arrow(ii) 119901-cartesian 1-arrows are stable under precomposition with 0-arrows
537 Lemma Let 119901∶ 119864 ↠ 119861 be a cartesian fibration and consider a 2-arrow
119890prime
119890Prime 119890
120595120595prime
120595Prime
in Fun(119883 119864)(i) If 120595 and 120595prime are 119901-cartesian 1-arrows so is 120595Prime(ii) If 120595 and 120595Prime are 119901-cartesian 1-arrows so is 120595prime
128
(iii) If 120595 and 120595Prime are 119901-cartesian 1-arrows and 119901120595prime is invertible then 120595prime is invertible
Proof Via Lemma 535 (i) and (ii) are Lemmas 5124(i) and (ii) while (iii) is Lemma 515
538 Lemma A 2-cell as below left
119876 Fun(119883 119864) 120793 119876 Fun(119883 119864) 119883 119864119890prime
119890
dArr120594119902
119890prime
119890
dArr120594
119890prime119902
119890119902dArr120594119902
is 119901lowast-cartesian if and only if each of its components 120594119902 is 119901-cartesian
Proof If 120594 is 119901lowast-cartesian then so is the restriction 120594119902 along any element 119902 ∶ 120793 rarr 119876 By Lemma535 this tells us that 120594119902 defines a 119901-cartesian transformation
Conversely if120594119902 is a 119901-cartesian transformation then Lemma 535 tells us that120594119902 is a 119901lowast-cartesiantransformation Now consider the factorization120594 = 120594120594sdot120579 through 119901lowast-cartesian lift120594120594 of 119901lowast120594 Becausethe components 120594119902 of 120594 are 119901lowast-cartesian the components 120579119902 of 120579 are isomorphisms By Lemma an arrow in an exponetial Fun(119883 119864)119876 is an isomorphism if and only if it is a pointwise isomorphismso this implies that 120594 By isomorphism stability of cartesian transformations we thus conclude that120594 is 119901lowast-cartesian
Exercises
53i Exercise Prove Corollary 533
54 Discrete cartesian fibrations
Recall from Definition 1221 that an object 119864 in an infin-cosmos 119974 is discrete if for all 119883 isin 119974the functor-space Fun(119883 119864) is a Kan complex Since a quasi-category is a Kan complex just when itshomotopy category is a groupoid (see Corollary 1115) equivalently 119864 is discrete if and only if everynatural transformation with codomain 119864 is invertible
From this definition it follows that an isofibration 119901∶ 119864 ↠ 119861 considered as an object of 119974119861 isdiscrete if and only if any 2-cell with codomain 119864 that whiskers with 119901 to an identity is invertible Infact the discrete objects are exactly those isofibrations that define conservative functors in 120101119974
541 Lemma An isofibration 119901∶ 119864 ↠ 119861 is a discrete object of119974119861 if and only if 119901∶ 119864 ↠ 119861 is a conservative
functor meaning any 119883 119864119886
119887
dArr120574 for which 119901120574 is an isomorphism is invertible
Proof Exercise 54i
Our aim in this section is to study a special class of cartesian fibrations and cocartesian fibrations
542 Definition An isofibration 119901∶ 119864 ↠ 119861 is a discrete cartesian fibration if it is a cartesian fibra-tion and if it is discrete as an object of119974119861 Dually an isofibration 119901∶ 119864 ↠ 119861 is a discrete cocartesianfibration if it is a cocartesian fibration and if it is discrete as an object of119974119861
The fibers of a discrete object 119901∶ 119864 ↠ 119861 in 119974119861 are discrete infin-categories in infin-cosmoi whoseobjects model (infin 1)-categories the discrete cartesian fibrations and discrete cocartesian fibrationsare ldquoinfin-groupoid-valued pseudofunctorsrdquo
129
There is also a direct 2-categorical characterization of the discrete cartesian fibrations whichreveals that unlike the case for cartesian and cocartesian fibrations for their discrete analogues thereare no special classes of 119901-cartesian or 119901-cocartesian cells
543 Proposition(i) If 119901∶ 119864 ↠ 119861 is a discrete cartesian fibration every natural transformation with codomain 119864 is119901-cartesian
(ii) An isofibration 119901∶ 119864 ↠ 119861 is a discrete cartesian fibration if and only if every 2-cell 120573∶ 119887 rArr 119901119890 hasan essentially unique lift given 120594∶ 119890prime rArr 119890 and 120595∶ 119890Prime rArr 119890 so that 119901120594 = 119901120595 = 120573 then there existsan isomorphism 120574∶ 119890Prime rArr 119890prime with 120594 sdot 120574 = 120595 and 119901120574 = id
Note that (i) implies immediate that any commutative square
119865 119864
119860 119861
119902
119892
119901
119891
from a cartesian fibration 119902 to a discrete cartesian fibration 119901 defines a cartesian functor
Proof By the definition of cartesian fibration any 2-cell 120595 with codomain 119864 factors through a119901-cartesian lift of 119901120595 along a 2-cell 120574 so that 119901120574 = id The discrete objects of 119974119861 are exactly thoseisofibrations with the property that any 2-cell with codomain 119864 that whiskers with 119901 to an identity isinvertible In particular120574 is an isomorphism and now120595 is isomorphic to a 119901-cartesian transformationand hence by Lemma 514 itself 119901-cartesian
By (i) and Lemma 513 itrsquos now clear that if 119901∶ 119864 ↠ 119861 is a discrete cartesian fibration thenany 2-cell 120573∶ 119887 rArr 119901119890 has an essentially unique lift For the converse note first that any 119901∶ 119864 ↠ 119861satisfying this hypothesis is a discrete object if 120595∶ 119890prime rArr 119890 is so that 119901120595 = id then id ∶ 119890 rArr 119890 isanother lift of 119901120595 and essential uniqueness provides an inverse isomorphism 120595minus1 ∶ 119890 rArr 119890prime
To complete the proof we now show that any 2-cell 120594∶ 119890prime rArr 119890 is cartesian for 119901 and to that endconsider a pair 120591∶ 119890Prime rArr 119890 and 120574∶ 119901119890Prime rArr 119901119890prime so that 119901120591 = 119901120594 sdot 120574 By the hypothesis that every 2-celladmits an essentially unique lift we can construct a lift 120583∶ rArr 119890prime so that 119901120583 = 120574 Now 120591 and 120594 sdot 120583are two lifts of 119901120591 with the same codomain so there exists an isomorphism 120579∶ 119890Prime rArr with 119901120579 = idThe composite 120583 sdot 120579 then defines the desired lift of 120574 to a cell so that 120591 = 120594 sdot 120583 sdot 120579
544 Example (domain projection from an element) For an element 119887 ∶ 1 rarr 119861 the domain-projectionfunctor 1199010 ∶ Hom119861(119861 119887) ↠ 119861 is a discrete cartesian fibration Cartesianness was established in Propo-
sition 5123 and discreteness follows immediately from 2-cell conservativity If 119883 Hom119861(119861 119887)119886
119887
dArr120574
is a natural transformation for which 1199010120574 is an identity then since 1199011120574 is also an identity 120574 must beinvertible
Dually the codomain-projection functor 1199011 ∶ Hom119861(119887 119861) ↠ 119861 is a discrete cocartesian fibration
130
545 Lemma (pullback stability) If
119865 119864
119860 119861
119902
119892
119901
119891
is a pullback square and 119901 is a discrete cartesian fibration then 119902 is a discrete cartesian fibration
Proof In light of Proposition 5120 it remains only to verify that 119902 is discrete Consider a 2-cell
119883 119865119886
119887
dArr120574 so that 119902120574 is invertible Then 119891119902120574 = 119901119892120574 is invertible and conservativity of 119901 implies
that 119892120574 is invertibleBy Lemma 315 the pullback square of functor spaces
Fun(119883 119865) Fun(119883 119864)
Fun(119883119860) Fun(119883 119861)
119902lowast
119892lowast
119901lowast
119891lowast
induces a smothering functor
hFun(119883 119865) rarr hFun(119883 119864) timeshFun(119883119861)
hFun(119883119860)
Wersquove just verified that the image of 120574 is an isomorphism so conservativity implies that 120574 is alsoinvertible
In analogy with Theorem 5111 there is an internal characterization of discrete cartesian fibra-tions which in the discrete case takes a much simpler form Recall any functor 119901∶ 119864 rarr 119861 inducesfunctors 119896 ∶ 119864120794 rarr Hom119861(119861 119901) and 119896 ∶ 119864120794 rarr Hom119861(119901 119861) as in (5110) by applying 119901 to the genericarrow for 119864
546 Proposition (internal characterization of discrete fibrations) An isofibration 119901∶ 119864 ↠ 119861 is adiscrete cartesian fibration if and only if the functor 119896 ∶ 119864120794 rarr Hom119861(119861 119901) is an equivalence and a discretecocartesian fibration if and only if the functor 119896 ∶ 119864120794 rarr Hom119861(119901 119861) is an equivalence
Recall from Theorem 5111(iii) that 119901∶ 119864 ↠ 119861 defines a cartesian fibration if and only if 119896 ∶ 119864120794 rarrHom119861(119901 119861) admits a right adjoint with invertible counit Proposition 546 asserts that 119901 defines a dis-crete cartesian fibration if and only if the unit of that adjunction a natural transformation that definesthe factorization of any natural transformation with codomain119864 through the canonical 119901-cartesian liftof its image under 119901 is an isomorphism in which case that adjunction defines an adjoint equivalenceand all natural transformations with codomain 119864 are 119901-cartesian
Proof Assume first that 119901∶ 119864 ↠ 119861 is a discrete cartesian fibration By Theorem 5111(i)rArr(iii)119896 ∶ 119864120794 rarr Hom119861(119861 119901) then admits a right adjoint with invertible counit ∶ 119896 cong id We will showthat in this case the unit ∶ id rArr 119896 is also invertible proving that 119896 ⊣ defines an adjoint equiva-lence
Since the counit of 119896 ⊣ is invertible 119896 is an isomorphism Thus 1199011119896 = 1199011 and 1199010119896 = 1199011199010are both isomorphisms By conservativity of the discrete fibration 119901∶ 119864 ↠ 119861 proven in Lemma 541this implies that 1199010 is invertible and now 2-cell conservativity for119864120794 reveals that is an isomorphism
131
Conversely if 119896 ∶ 119864120794 ⥲ Hom119861(119861 119901) is an equivalence by Proposition 2111 we may choose a rightadjoint equivalence inverse The counit of this adjoint equivalence is necessarily an isomorphismso by Theorem 5111(iii)rArr(i) we know that 119901∶ 119864 ↠ 119861 is a cartesian fibration Since the unit of119896 ⊣ is also an isomorphism Theorem 5119(vi)rArr(iv) tells us that every natural transformationwith codomain 119864 is 119901-cartesian and now the conservativity property for cartesian transformations ofLemma 515 tells us that 119901∶ 119864 ↠ 119861 defines a conservative fun ctor and in particular is discrete
Since equivalences and simplicial limits in aninfin-cosmos are representably-defined notions it fol-lows immediately from Proposition 546 that
547 Proposition An isofibration 119901∶ 119864 ↠ 119861 in aninfin-cosmos 119974 defines a discrete cartesian fibration ifand only if for all 119883 isin 119974 the functor 119901lowast ∶ Fun(119883 119864) ↠ Fun(119883 119861) defines a discrete cartesian fibration ofquasi-categories
Using the internal characterization it is straightforward to verify that discrete cartesian fibrationscompose and cancel on the left
548 Lemma(i) If 119901∶ 119864 ↠ 119861 and 119902 ∶ 119861 ↠ 119860 are discrete cartesian fibrations so is 119902119901 ∶ 119864 ↠ 119860(ii) If 119901∶ 119864 ↠ 119861 is an isofibration and 119902 ∶ 119861 ↠ 119860 and 119902119901 ∶ 119864 ↠ 119860 are discrete cartesian fibrations
then so is 119901∶ 119864 ↠ 119861
Proof By considering the defining pullback diagrams the map 119864120794 ↠ Hom119860(119860 119902119901) that testswhether 119902119901 ∶ 119864 ↠ 119860 is a discrete cartesian fibration factors as the map 119864120794 ↠ Hom119861(119861 119901) thattests whether 119901∶ 119864 ↠ 119861 is a discrete cartesian fibration followed by a pullback of the map 119861120794 ↠Hom119860(119860 119902) that tests whether 119902 ∶ 119861 ↠ 119860 is a discrete cartesian fibration
119864120794
Hom119861(119861 119901) 119861120794 119860120794
Hom119860(119860 119902119901) Hom119860(119860 119902)
119864 119861 119860
11990111199011
11990111199011
119901 119902
Both parts now follow from the 2-of-3 property
The internal characterization of discrete cartesian fibrations is useful for establishing further ex-amples
549 Lemma A trivial fibration 119901∶ 119864 ⥲rarr 119861 is a discrete bifibration
Proof Recall fromRemark 5113 that the canonical functors 119896 ∶ 119864120794 ↠ Hom119861(119861 119901) and 119896 ∶ 119864120794 ↠Hom119861(119901 119861) can be constructed as the Leibniz cotensor of the monomorphism 1∶ 120793 120794 in the firstcase and 0∶ 120793 120794 in the second with the trivial fibration 119901∶ 119864 ⥲rarr 119861 By Lemma 1210 both mapsare trivial fibrations and in particular equivalences Now Proposition 546 proves that 119901 is a discretecartesian fibration and also a discrete cocartesian fibration
132
A final important family of examples of discrete cartesian fibrations areworth establishing Propo-sition 5123 proves that for any infin-category 119860 the domain-projection functor 1199010 ∶ 119860120794 ↠ 119860 definesa cartesian fibration Thus functor does not define a discrete cartesian fibration in theinfin-cosmos119974but recall that 1199010-cartesian lifts can be constructed to project to identity arrows along 1199011 ∶ 119860120794 ↠ 119860This suggests that we might productively consider the domain-projection functor as a map over119860 inwhich case we have the following result
5410 Proposition The functor
119860120794 119860 times 119860
119860
(11990111199010)
1199011 120587(5411)
defines a discrete cartesian fibration in the sliceinfin-cosmos119974119860
Proof Note that 2-cell conservativity implies that (5411) is a discrete object in (119974119860)120587 ∶ 119860times119860↠119860 cong119974119860times119860 so it remains only to prove that this functor defines a cartesian fibration We prove this usingTheorem 5111(i)hArr(ii) The first step is to compute the right representable comma object for thefunctor (5411) by interpreting the formula (332) in the sliceinfin-cosmos119974119860 using Proposition 1217The 120794-cotensor of the object 120587∶ 119860 times 119860 ↠ 119860 is 120587∶ 119860 times 119860120794 ↠ 119860 so this right representable commais computed by the left-hand pullback in119974119860 below
Hom119860(119860 1199010) 119860 times 119860120794 119860120794
119860120794 119860 times 119860 119860
119860
1199012 id119860 times1199011
120587
1199011
1199011
(11990111199010)
120587
120587
Pasting with the right-hand pullback in119974 we recognize that theinfin-category so-constructed coincideswith the right representable comma object for the functor 1199010 ∶ 119860120794 ↠ 119860 considered as a map in119974 Under the equivalence Hom119860(119860 1199010) ≃ 119860120795 established in the proof of Proposition 5123 theisofibration 1199012 ∶ Hom119860(119860 1199010) ↠ 119860 is evaluation at the final element 2 isin 120795 in the composable pair ofarrows Similarly the canonical functor 119894 ∶ 119860120794 rarr Hom119860(119860 1199010) induced by id1199010 in119974 coincides withthe canonical functor 119894 ∶ 119860120794 rarr Hom119860(119860 1199010) over 119860 induced by id(11990111199010) in119974119860
Now applying Proposition 5123 and Theorem 5111(i)rArr(ii) in 119974 this functor 119894 admits a rightadjoint 119903 over the domain-projection functor
119860120794 perp Hom119860(119860 1199010)
119860
119894
1199010 1199010119903
133
By the proofs of Theorem 5111(iii)rArr(ii) and Proposition 5123 this adjunction can be constructed bycotensoring 119860(minus) the composite adjunction of categories
120794 120794 times 120794 120795 119860120794 perp 119860120795
120793 + 120793 119860 times 119860
120794times0⊤
1205751
120794times⊤ℓ
119896
1205900119894
(11990111199010) (11990121199010)119903
(20)(10)
where ℓ ⊣ 119896 is described in the proof of Proposition 5123 The composite right adjoint is the functor1205900 ∶ 120795 ↠ 120794 that sends 0 and 1 to 0 and 2 to 1 while the composite left adjoint is the functor 1205751 ∶ 120794 ↣120795 that sends 0 to 0 and 1 to 2 In particular this adjunction lies in the strict slice 2-category underthe inclusion of the ldquoendpointsrdquo of 120794 and 120795
It follows that upon cotensoring into 119860 we obtain a fibered adjunction over 119860 times 119860 which byTheorem 5111(ii)rArr(i) implies that (5411) is a cartesian fibration in119974119860 completing the proof
Combining Propositions 5123 and Proposition 5410 we can now generalize both results to ar-bitrary commainfin-categories
5412 Corollary For any functors 119862 119860 119861119892 119891
betweeninfin-categories in aninfin-cosmos119974(i) The domain-projection functor 1199010 ∶ Hom119860(119891 119892) ↠ 119861 is a cartesian fibration Moreover a natural
transformation 120594 with codomain Hom119860(119891 119892) is 1199010-cartesian if and only if 1199011120594 is invertible(ii) The codomain-projection functor 1199011 ∶ Hom119860(119891 119892) ↠ 119862 is a cocartesian fibration Moreover a natu-
ral transformation 120594 with codomain Hom119860(119891 119892) is 1199011-cartesian if and only if 1199010120594 is invertible(iii) The functor
Hom119860(119891 119892) 119862 times 119861
119862
(11990111199010)
1199011 120587
defines a discrete cartesian fibration in119974119862(iv) The functor
Hom119860(119891 119892) 119862 times 119861
119861
(11990111199010)
1199010 120587
defines a discrete cocartesian fibration in119974119861
Proof We prove (i) and (iii) and leave the dualizations to the reader For (iii) we first use thecosmological functor 119892lowast ∶ 119974119860 rarr119974119862 which preserves discrete cartesian fibrations to establish that
Hom119860(119860 119892) 119862 times 119860
119862
(11990111199010)
1199011 120587
134
defines a discrete cartesian fibration in 119974119862 this argument works because 1199011 ∶ Hom119860(119860 119892) ↠ 119862 isthe pullback of 1199011 ∶ 119860120794 ↠ 119860 along 119892 Now
Hom119860(119891 119892) Hom119860(119860 119892)
119862 times 119861 119862 times 119860
119862
(11990111199010)1199011 1199011 (11990111199010)
120587
119862times119891
120587
is a pullback square in119974119862 so Lemma 545 now implies that the pullback is also a discrete cartesianfibration
Using (iii) we can now prove (i) This follows directly from a general claim that if
119864 119862 times 119861
119862119902
(119902119901)
120587
defines a cartesian fibration in 119974119862 then 119901∶ 119864 ↠ 119861 defines a cartesian fibration in 119974 By Theorem5111(i)hArr(ii) this functor defines a cartesian fibration in119974119862 if and only if the functor 119894
119864 perp Hom119861(119861 119901)
119862 times 119861
119894
(119902119901) (11990211990111199010)119903
admits a right adjoint 119903 over 119862 times 119861 Composing with 120587∶ 119862 times 119861 ↠ 119861 this fibered adjunction definesan adjunction over 119861 and Theorem 5111(i)hArr(ii) applied this time in 119974 allows us to conclude that119901∶ 119864 ↠ 119861 is a cartesian fibration
Note that the domain projection 1199010 ∶ Hom119860(119891 119860) ↠ 119861 is the pullback of 1199010 ∶ 119860120794 ↠ 119860 along119891∶ 119861 rarr 119860 so Proposition 5120 proves directly from Proposition 5123 that this functor is a cartesianfibration but 1199010 ∶ Hom119860(119860 119892) ↠ 119860 is not similarly a pullback of 1199010 ∶ 119860120794 ↠ 119860 This is why a morecircuitous argument to the general result is needed
Exercises
54i Exercise Prove Lemma 541
55 The external Yoneda lemma
Let 119887 ∶ 1 rarr 119861 be an element of aninfin-category 119861 and consider its right representation Hom119861(119861 119887)as a comma infin-category In this case there is no additional data given by the codomain-projectionfunctor but Example 544 observes that the domain-projection functor 1199010 ∶ Hom119861(119861 119887) ↠ 119861 hasa special property it defines a discrete cartesian fibration The fibers of this map over an element119886 ∶ 1 rarr 119861 are the internal mapping spaces Hom119861(119886 119887) of Definition 339 In this way the rightrepresentation of the element 119887 encodes the contravariant functor represented by 119887 which is why allalong wersquove been referring to the commainfin-categories Hom119861(119861 119887) as ldquorepresentablerdquo
135
Our aim in this section is to state and prove the Yoneda lemma in this setting where contravari-ant representable functors are encoded as discrete cartesian fibrations A dual statement applies tocovariant representable functors encoded as discrete cocartesian fibration 1199011 ∶ Hom119861(119887 119861) ↠ 119861 butfor ease of exposition we leave the dualization to the reader Informally the Yoneda lemma asserts thatldquoevaluation at the identity defines an equivalencerdquo so the first step towards the statement of the Yonedalemma is to introduce this identity element which in fact is something wersquove already encountered
The identity arrow id119887 induces an element id119887 ∶ 1 rarr Hom119861(119861 119887) which Corollary 349 provesis terminal in the infin-category Hom119861(119861 119887) of arrows in 119861 with codomain 119887 The identity elementinclusion defines a functor over 119861
1 Hom119861(119861 119887)
119861119887
id119887
1199010(551)
Technically this functor does not live in the slicedinfin-cosmos119974119861 because the domain object 119887 ∶ 1 rarr 119861is not an isofibration but nevertheless for any isofibration 119901∶ 119864 ↠ 119861 restriction along id119887 inducesa functor between sliced quasi-categorical functor spaces
Fun119861(Hom119861(119861 119887)1199010minusminusrarrrarr 119861119864
119901minusrarrrarr 119861)
evid119887minusminusminusminusminusrarr Fun119861(1119887minusrarr 119861 119864
119901minusrarrrarr 119861)
Here the codomain is the quasi-category defined by the pullback
Fun119861(119887 119901) Fun(1 119864)
120793 Fun(1 119861)
119901lowast
119887
which is isomorphic to Fun(1 119864119887) the underlying quasi-category of the fiber 119864119887 of 119901∶ 119864 ↠ 119861 over 119887If a discrete cartesian fibration over 119861 is thought of as a 119861-indexed discrete infin-category valued
contravariant functor then maps of discrete cartesian fibrations over 119861 are ldquonatural transformationsrdquothe ldquonaturality in 119861rdquo arises because we only allow functors over 119861 This leads to our first statement ofthe fibrational Yoneda lemma
552 Theorem (external Yoneda lemma discrete case) If 119901∶ 119864 ↠ 119861 is a discrete cartesian fibrationthen
Fun119861(Hom119861(119861 119887)1199010minusminusrarrrarr 119861119864
119901minusrarrrarr 119861)
evid119887minusminusminusminusminusrarr Fun119861(1119887minusrarr 119861 119864
119901minusrarrrarr 119861) cong Fun(1 119864119887)
is an equivalence of Kan complexes
Theorem 552 is subsumed by a generalization that allows 119901∶ 119864 ↠ 119861 to be any cartesian fibra-tion not necessarily discrete In this case 119901 encodes an ldquoinfin-category-valued contravariant 119861-indexedfunctorrdquo as does 1199010 ∶ Hom119861(119861 119887) ↠ 119861 The correct notion of ldquonatural transformationrdquo between twosuch functors is now given by a cartesian functor over 119861 see Exercise 51iii To that end for a pair ofcartesian fibration 119902 ∶ 119865 ↠ 119861 and 119901∶ 119864 ↠ 119861 we write
Funcart119861 (119865119902minusrarrrarr 119861119864
119901minusrarrrarr 119861) sub Fun119861(119865
119902minusrarrrarr 119861119864
119901minusrarrrarr 119861)
136
for the sub quasi-category containing all those simplices whose vertices define cartesian functors from119902 to 119901⁶
553 Theorem (external Yoneda lemma) If 119901∶ 119864 ↠ 119861 is a cartesian fibration then
Funcart119861 (Hom119861(119861 119887)1199010minusminusrarrrarr 119861119864
119901minusrarrrarr 119861)
evid119887minusminusminusminusminusrarr Fun119861(1119887minusrarr 119861 119864
119901minusrarrrarr 119861) cong Fun(1 119864119887)
is an equivalence of quasi-categories
The proofs of these theorems overlap significantly and we develop them in parallel The basicidea is to use the universal property of id119887 as a terminal element of Hom119861(119861 119887) to define a rightadjoint to evid119887 and prove that when 119901∶ 119864 ↠ 119861 is discrete or when the domain is restricted to thesub-quasi-category of cartesian functors this adjunction defines an adjoint equivalence Note that thefunctor evid119887 is the image of the functor id119887 under the 2-functor Fun119861(minus 119901) ∶ 120101(119974119861)op rarr 119980119966119886119905 Ifthe adjunction ⊣ id119887 lived in the sliceinfin-cosmos119974119861 this would directly construct a right adjointto evid119887 The main technical difficulty in following the outline just given is that the adjunction thatwitnesses the terminality of id119887 does not live in the slice of the homotopy 2-category 120101119974119861 but ratherin a lax slice of the homotopy 2-category that we now introduce
554 Definition Consider a 2-category 120101119974 and an object 119861 isin 120101119974 The lax slice 2-category 120101119974⫽119861is the strict 2-category whosebull objects are maps 119891∶ 119883 rarr 119861 in 120101119974 with codomain 119861bull 1-cells are diagrams
119883 119884
119861119891
119896
120572rArr 119892
(555)
in 120101119974 andbull 2-cells from the 1-cell displayed above to the 1-cell below-right are 2-cells 120579∶ 119896 rArr 119896prime so that
119883 119884 119883 119884
119861 119861119891
119896uArr120579119896prime
120572rArr 119892 = 119891
119896prime
120572primerArr 119892
556 Lemma The identity functor (551) is right adjoint to the right comma cone
Hom119861(119861 119887) 1
1198611199010
120601rArr 119887
in 120101119974⫽119861
⁶For any quasi-category 119876 and any subset 119878 of its vertices there is a ldquofullrdquo sub-quasi-category 119876119878 sub 119876 containingexactly those vertices and all the simplices of 119876 that they span
137
Proof Since 1 is the terminalinfin-category we take the counit of the postulated adjunction to bethe identity By Definition 554 to define the unit we must provide a 2-cell
Hom119861(119861 119887) Hom119861(119861 119887) Hom119861(119861 119887) 1 Hom119861(119861 119887)
119861 1198611199010
id119887uArr120578
1199010 = 1199010
id119887
119887120601rArr
1199010
so that 1199010120578 = 120601 This is the defining property of the unit in Lemma 348 The forgetful 2-functor120101119974⫽119861 rarr 120101119974 is faithful on 1- and 2-cells so the verification of the triangle equalities in Lemma 348proves that they also hold in 120101119974⫽119861
Using somewhat non-standard 2-categorical techniques we will transfer the adjunction of Lemma556 to an adjunction between the quasi-categories Fun119861(119887 119901) and Fun119861(1199010 119901) see Proposition 5513Because our initial adjunction lives in the lax rather than the strict slice the construction will besomewhat delicate passing through a pair of auxiliary 2-categories that we now introduce
557 Definition Let 120101119974 be the homotopy 2-category of aninfin-cosmos and write 120101119974⟓ for the strict2-category whosebull objects are cospans
119860 119861 119864119891 119901
in which 119901 is a cartesian fibrationbull 1-cells are diagrams of the form
119860prime 119861prime 119864prime
119860 119861 119864
119886
119891prime
uArr120601 119887
119901prime
119890
119891 119901
(558)
bull and whose 2-cells consist of triples 120572∶ 119886 rArr 120573∶ 119887 rArr and 120598 ∶ 119890 rArr between the verticals ofparallel 1-cell diagrams so that 119901120598 = 120573119901prime and sdot 119891120572 = 120573119891prime sdot 120601
559 Definition Let 120101119974 be the homotopy 2-category of aninfin-cosmos and write 120101119974⊡ for the strict2-category whosebull objects are pullback squares
119865 119864
119860 119861
119902
119892
119901
119891
whose verticals are cartesian fibrations
138
bull 1-cells are cubes
119865prime 119864prime
119865 119864
119860prime 119861prime
119860 119861
119892prime
119902primeℓ
uArr120594 119890
119901prime
119892
119901
119886
119891prime
uArr120601 119887
119891
119902(5510)
whose vertical faces commute and in which 120594∶ 119892ℓ rArr 119890119892prime is a 119901-cartesian lift of 120601119902prime andbull whose 2-cells are given by quadruples 120572∶ 119886 rArr 120573∶ 119887 rArr 120598 ∶ 119890 rArr and 120582∶ ℓ rArr ℓ in which 120598
and 120582 are respectively lifts of 120573119901prime and 120572119902prime and so that 120601 sdot 119891120572 = 120573119891prime sdot 120601 and sdot 119892120582 = 120598119892prime sdot 120594
These definitions are arranged so that there is an evident forgetful 2-functor 120101119974⊡ rarr 120101119974⟓
5511 Lemma The forgetful 2-functor 120101119974⊡ rarr 120101119974⟓ is a smothering 2-functor
Proof Proposition 5120 tells us that 120101119974⊡ rarr 120101119974⟓ is surjective on objects To see that it is fullon 1-cells first form the pullbacks of the cospans in (558) then define 120594 to be any 119901-cartesian lift of120601119902prime with codomain 119890119892prime By construction the domain of 120594 lies strictly over 119891119886119902prime and so this functorfactors uniquely through the pullback leg 119892 defining the map ℓ of (5510)
To prove that 120101119974⊡ rarr 120101119974⟓ is full on 2-cells consider a parallel pair of 1-cells in 120101119974⊡ For oneof these we use the notation of (5510) and for the other we denote the diagonal functors by and ℓ and denote the 2-cells by and the requirement that these 1-cells be parallel implies that thepullback faces are necessarily the same Now consider a triple 120572∶ 119886 rArr 120573∶ 119887 rArr and 120598 ∶ 119890 rArr satisfying the conditions of Definition 557 Our task is to define a fourth 2-cell 120582∶ ℓ rArr ℓ so that119902120582 = 120572119902prime and sdot 119892120582 = 120598119892prime sdot 120594
To achieve this we first define a 2-cell 120574∶ 119892ℓ rArr 119892ℓ using the induction property of the 119901-cartesiancell ∶ 119892ℓ rArr 119892prime applied to the composite 2-cell 120598119892prime sdot 120594 ∶ 119892ℓ rArr 119892prime and the factorization 119901120598119892prime sdot 119901120594 =119902prime sdot 119891120572119902prime By construction 119901120574 = 119891120572119902prime so the pair 120572119902prime and 120574 induces a 2-cell 120582∶ ℓ rArr ℓ so that119902120582 = 120572119902prime and 119892120582 = 120574 The quadruple (120572 120573 120598 120582) now defines the required 2-cell in 120101119974⊡
Finally for 2-cell conservativity suppose 120572 120573 and 120598 as above are isomorphisms By the conser-vativity property for pullbacks to show that 120582 is an isomorphism it suffices to prove that 119902120582 = 120572119902primeis which we know already and that 119892120582 = 120574 is invertible But 120574 was constructed as a factorization120598119892prime sdot 120594 = sdot 120574 with 119901120574 = 119891120572119902prime Since 120598 is an isomorphism 120598119892prime sdot 120594 is 119901-cartesian so Lemma 515proves that 120574 is an isomorphism
5512 Remark While we cannot directly define a pullback 2-functor 120101119974⟓ rarr 120101119974 in the homotopy2-category because the 2-categorical universal property of pullbacks in 120101119974 is weak and not strict thezig zag of 2-functors 120101119974⟓ larr 120101119974⊡ rarr 120101119974 in which the backwards map is a smothering 2-functor andthe forwards map evaluates at the pullback vertex defines a reasonable replacement
Using Lemma 5511 we can now construct the desired adjunction
139
5513 Proposition For any element 119887 ∶ 1 rarr 119861 and any cartesian fibration 119901∶ 119864 ↠ 119861 the evaluation atthe identity functor admits a right adjoint
Fun119861(1199010 119901) Fun119861(119887 119901)
evid119887
perp
119877
defined by the domain-component of the 119901lowast-cartesian lift of the right comma cone over 119887
Fun119861(119887 119901) Fun(1 119864)
Fun119861(1199010 119901) Fun(Hom119861(119861 119887) 119864)
120793 Fun(1 119861)
120793 Fun(Hom119861(119861 119887) 119861)
uArr120594119877
119901lowast
lowast
119901lowastuArr120601
119887lowast
1199010
(5514)
The idea will be to transfer the adjunction of Lemma 556 through a sequence of 2-functors
120101119980119966119886119905⊡ 120101119980119966119886119905
120101119974op⫽119861 120101119980119966119886119905⟓
ev⊤
using Lemma 358 to lift along the middle smothering 2-functor
Proof Fixing a cartesian fibration 119901∶ 119864 ↠ 119861 in119974 we define a 2-functor⁷119974op⫽119861 rarr 120101119980119966119886119905⟓ that
carries a 1-cell (555) to120793 Fun(119884 119861) Fun(119884 119864)
120793 Fun(119883 119861) Fun(119883 119864)
119892
uArr120572 119896lowast
119901lowast
119890
119891 119901lowast
and a 2-cell 120579∶ 119896 rArr 119896prime to the 2-cell that acts via pre-whiskering with 120579 in its two non-identitycomponents By Corollary 5116 the functors 119901lowast are cartesian fibration of quasi-categories
We now apply the 2-functor 119974op⫽119861 rarr 120101119980119966119886119905⟓ to the adjunction of Lemma 556 to obtain an
adjunction in 120101119980119966119886119905⟓ and then use the smothering 2-functor of Lemma 5511 and Lemma 358 to liftthis to an adjunction in 120101119980119966119886119905⊡ As elaborated on in Exercise 35ii the lifted adjunction in 120101119980119966119886119905⊡can be constructed using any lifts of the objects 1-cells and either the unit or counit of the adjunctionin 120101119980119966119886119905⟓
⁷To explain the variance recall that the 2-functor Fun(minus 119861) ∶ 120101119974op rarr 120101119980119966119886119905 is contravariant on 1-cells but covarianton 2-cells Such 2-functors like all 2-functors preserve adjunctions though in this case the left and right adjoints areinterchanged while the units and counits retain the same roles
140
In particular we may take the left adjoint of the lifted adjunction in 120101119980119966119886119905⊡ to be any lift of theimage in 120101119980119966119886119905⟓ of the right adjoint of the adjunction ⊣ id119887 in 120101119974⫽119861 and so our left adjoint is
Fun119861(1199010 119901) Fun(Hom119861(119861 119887) 119864)
Fun119861(119887 119901) Fun(1 119864)
120793 Fun(Hom119861(119861 119887) 119861)
120793 Fun(1 119861)
=evid119887
119901lowast
id119887lowast
119901lowast=
1199010id119887
lowast
119887
We may also take the right adjoint to be any lift of the image of the right adjoint of the adjunctionThis proves that the right adjoint is defined by (5514) Since the counit of ⊣ id119887 is an identitythe counit of the lifted adjunction may also be taken to be an identity
Finally we compose with the forgetful 2-functor 120101119980119966119886119905⊡ rarr 120101119980119966119886119905 that evaluates at the pullbackvertex to project our adjunction in 120101119980119966119886119905⊡ to the desired adjunction in 120101119980119966119886119905
The proof of the discrete case of the Yoneda lemma is now one line
Proof of Theorem 552 If 119901∶ 119864 ↠ 119861 is discrete then Fun119861(1199010 119901) and Fun119861(119887 119901) are Kan com-plexes so the adjunction defined in Proposition 5513 is an adjoint equivalence
Specializing to the case of two right representable discrete cartesian fibrations we conclude thatthe Kan complex of natural transformations is equivalent to the underlying quasi-category of theirinternal mapping space
5515 Corollary (external Yoneda embedding) For any elements 119909 119910 ∶ 1 119860 in aninfin-category 119860evaluation at the identity of 119909 induces an equivalence of Kan complexes
Fun119860(Hom119860(119860 119909)Hom119860(119860 119910)) Fun(1Hom119860(119909 119910))sim
evid119909
It remains to prove the general case of Theorem 553 The next step is to observe that the rightadjoint in the adjunction of Proposition 5513 lands in the sub quasi-category of cartesian functorsfrom 1199010 to 119901 Lemma 5516 proves this after which it is short work to complete the proof of Theorem553 by arguing that this restricted adjunction defines an adjoint equivalence
5516 Lemma For each vertex in Fun119861(119887 119901) below-left
Fun119861(119887 119901) Fun119861(1199010 119901)119877
1 119864 Hom119861(119861 119887) 119864
119861 119861119887
119890
119901 ↦ 1199010
119877119890
119901
the map 119877119890 in Fun119861(1199010 119901) above-right defines a cartesian functor in119974119861
141
Proof From the definition of the right adjoint in (5514) and Lemma 535 we see that 119877119890 is thedomain component of a 119901-cartesian lift 120594 of the composite natural transformation below-left
Hom119861(119861 119887) 1 119864 Hom119861(119861 119887) 119864
119861 119861
uArr1206011199010
119887
119890
119901 =
119890
119877119890
uArr120594
1199010 119901
Since 1199010 ∶ Hom119861(119861 119887) ↠ 119861 is discrete every natural transformation 120595 with codomain Hom119861(119861 119887) is1199010-cartesian so to prove that 119877119890 defines a cartesian functor we must show that 119877119890120595 is 119901-cartesianTo that end consider the horizontal composite
119883 Hom119861(119861 119887) 119864
119910
119909
uArr120595
119890
119877119890
uArr120594
By naturality of whiskering we have 120594119910 sdot 119877119890120595 = 119890120595 sdot 120594119909 = 120594119909 since 1 is the terminal infin-categoryand hence 119890120595 is an identity Now Lemma 5124(ii) implies that 119877119890120595 is 119901-cartesian
Proof of Theorem 553 By Lemma 5516 the adjunction of Proposition 5513 restricts to de-fine an adjunction
Funcart119861 (1199010 119901) Fun119861(119887 119901)
evid119887
perp
119877Since the counit of the original adjunction ⊣ id119887 is an isomorphism and smothering 2-functorsare conservative on 2-cells the counit of the adjunction of Proposition 5513 and hence also of therestricted adjunction is an isomorphism As in the proof of Theorem 552 we will prove that evid119887is an equivalence by demonstrating that the unit of the restricted adjunction is also invertible ByLemma it suffices to verify this elementwise proving that the component of the unit indexed by acartesian functor
Hom119861(119861 119887) 119864
1198611199010
119891
119901
is an isomorphismUnpacking the proof of Proposition 5513 the unit of evid119887 ⊣ 119877 is defined to be a factorization
Fun119861(1199010 119901) Fun(Hom119861(119861 119887) 119864) Fun(1 119864)
Fun119861(1199010 119901) Fun119861(119887 119901) Fun(1 119864) Fun(Hom119861(119861 119887) 119864)
Fun119861(1199010 119901) Fun(Hom119861(119861 119887) 119864)
id119887lowast
lowastevid119887
119877 uArr120594
=
lowast
uArrFun(120578119864)
uArr
142
of the pre-whiskering 2-cell Fun(120578 119864) through the 119901lowast-cartesian lift 120594 The component of the pre-whiskering 2-cell Fun(120578 119864) at the cartesian functor 119891 is 119891120578 Since 1199010 ∶ Hom119861(119861 119887) ↠ 119861 is a discretecartesian fibration any 2-cell such as 120578 which has codomain Hom119861(119861 119887) is 1199010-cartesian and since 119891is a cartesian functor we then see that 119891120578 is 119901-cartesian
By Lemma 538 the components of the 119901lowast-cartesian cell 120594 define 119901-cartesian natural transforma-tions in119974 As is a natural transformation with codomain Fun119861(1199010 119901) its components project along119901 to the identity In this way we see that 119891 is a factorization of the 119901-cartesian transformation 119891120578through a 119901-cartesian lift of 120601 over an identity and Lemma 515 proves that 119891 is an isomorphism asdesired
Exercises
55i Exercise Given an element 119891∶ 1 rarr Hom119860(119909 119910) in the internal mapping space between a pairof elements in aninfin-category 119860 use the explicit description of the inverse equivalence to the map ofCorollary 5515 to construct a map
Hom119860(119860 119909) Hom119860(119860 119910)
1198601199010
119891lowast
1199010
which represents the ldquonatural transformationrdquo defined by post-composing with 119891⁸
⁸Hint this construction is a special case of the construction given in the first half of the proof of Lemma 5516
143
Part II
Homotopy coherent category theory
CHAPTER 6
Simplicial computads and homotopy coherence
Consider a diagram 119889∶ 1 rarr 119860119869 in aninfin-category119860 indexed by a 1-category 119869 Via the isomorphismFun(1 119860119869) cong Fun(1 119860)119869 an element in the infin-category of diagrams 119860119869 equally defines a functor119889∶ 119869 rarr Fun(1 119860) valued in the underlying quasi-category of 119860 Applying h ∶ 119980119966119886119905 rarr 119966119886119905 thisdescends to a diagram h119889∶ 119869 rarr h119860 of shape 119869 in the homotopy category of 119860 such diagrams arecalled homotopy commutative But the original diagram 119889 has a much richer property defining what iscalled a homotopy coherent diagram of shape 119869 in the quasi-category Fun(1 119860)
To make the data involved in defining a homotopy coherent diagram most explicit we first in-troduce a general notion of homotopy coherent diagram as a simplicial functor valued in a simplicialcategory whose hom-spaces are Kan complexes What makes such diagrams ldquohomotopy coherentrdquo andnot just ldquosimplicially enrichedrdquo is that their domains are required to be ldquofreerdquo simplicial categories ofa particular form that we refer to by the name simplicial computads Because every quasi-category canbe presented up to equivalence by a Kan-complex enriched category it will follow that ldquoall diagramsvalued in quasi-categories are homotopy coherentrdquo
To build intuition for the general notion of a homotopy coherent diagram it is helpful to considera special case of diagrams indexed by the category
120654 ≔ 0 1 2 3 ⋯
whose objects are finite ordinals and with a morphism 119895 rarr 119896 if and only if 119895 le 119896 and valued in theKan-complex enriched category of spaces 119982119901119886119888119890 A 120654-shaped graph in 119982119901119886119888119890 is comprised of spaces119883119896 for each 119896 isin 120654 together with continuous maps 119891119895119896 ∶ 119883119895 rarr 119883119896 whenever 119895 lt 119896sup1 This data definesa homotopy commutative diagram just when 119891119894119896 ≃ 119891119895119896 ∘ 119891119894119895 whenever 119894 lt 119895 lt 119896sup2
To extend this data to a homotopy coherent diagram120654 rarr 119982119901119886119888119890 requiresbull Chosen homotopies ℎ119894119895119896 ∶ 119891119894119896 ≃ 119891119895119896 ∘ 119891119894119895 whenever 119894 lt 119895 lt 119896 This amounts to specifying a path
in Map(119883119894 119883119896) from the vertex 119891119894119896 to the vertex 119891119895119896 ∘ 119891119894119895 which is obtained as the composite ofthe two vertices 119891119894119895 isin Map(119883119894 119883119895) and 119891119895119896 isin Map(119883119895 119883119896)
sup1To simplify somewhat we adopt the convention that 119891119895119895 is the identity making this data into a reflexive directedgraph with implicitly designated identities
sup2This data defines a strictly commutative diagram (aka a functor 120654 rarr 119982119901119886119888119890) just when 119891119894119896 = 119891119895119896 ∘ 119891119894119895 whenever119894 lt 119895 lt 119896 Strictly commutative diagrams are certainly homotopy commutative Homotopy coherent category theoryarose from the search for conditions under which something like the converse implication held a homotopy commutativediagram is realized by (ie naturally isomorphic to up to homotopy) a strictly commutative diagram if and only if itextends to a homotopy coherent diagram [18 25]
147
bull For 119894 lt 119895 lt 119896 lt ℓ the chosen homotopies provide four paths in Map(119883119894 119883ℓ)
119891119894ℓ 119891119896ℓ ∘ 119891119894119896
119891119895ℓ ∘ 119891119894119895 119891119896ℓ ∘ 119891119895119896 ∘ 119891119894119895
ℎ119894119896ℓ
ℎ119894119895ℓ 119891119896ℓ∘ℎ119894119895119896
ℎ119895119896ℓ∘119891119894119895
We then specify a higher homotopy mdash a 2-homotopy mdash filling in this squarebull For 119894 lt 119895 lt 119896 lt ℓ lt 119898 the previous choices provide 12 paths and six 2-homotopies in
Map(119883119894 119883119898) that assemble into the boundary of a cube We then specify a 3-homotopy a ho-motopy between homotopies between homotopies filling in this cube
bull EtcEven in this simple case of the category 120654 this data is a bit unwieldy Our task is to define a
category to index this homotopy coherent data arising from120654 the objects119883119894 the functions119883119894 rarr 119883119895the 1-homotopies ℎ119894119895119896 the 2-homotopies and so on This data will assemble into a simplicial categorywhose objects are the same as the objects of 120654 but which will have 119899-morphisms in each dimension119899 ge 0 to index the 119899-homotopies Importantly this simplicial category will be ldquofreely generatedrdquo froma much smaller collection of data We begin by studying such ldquofreely generatedrdquo simplicial categoriesunder the name simplicial computads
61 Simplicial computads
611 Definition (free categories and atomic arrows) An arrow 119891 in a 1-category is atomic if it is notan identity and if it admits no non-trivial factorizations ie if whenever 119891 = 119892 ∘ ℎ either 119892 or ℎ is anidentity
A 1-category is free if every arrow may be expressed uniquely as a composite of atomic arrowswith the convention that empty composites correspond to identity arrowssup3
612 Digression (on free categories and reflexive directed graphs) The category of presheaves onthe truncation
120491le1 ≔ bull bull119904
119905119894 119894119904 = 119894119905 = id
defines the category 119970119901ℎ of (reflexive directed) graphs for us a graph consists of a set of vertices aset of edges each with a specified source and target vertex and a distinguished ldquoidentityrdquo endo-edgefor each vertex Any category has an underlying reflexive directed graph and this forgetful functoradmits a left adjoint defining the free category whose atomic and identity arrows are the arrows inthe given graph
119966119886119905 119970119901ℎ119880
perp119865
Recall fromDigression 122 that a simplicial category119964may be presented by a family of 1-categories119964119899 of119899-arrows for119899 ge 0 eachwith a common set of objects that assemble into a diagram119964bull ∶ 120491op rarr
sup3Alternatively a 1-category is free if every non-identity arrow may be expressed uniquely as a non-empty compositeof atomic arrows and if identity arrows admit no non-trivial factorizations
148
119966119886119905 comprised of identity-on-objects functors The notion of ldquofreerdquo simplicial category was first in-troduced by Dwyer and Kan [19 45]
613 Definition (simplicial computad) A simplicial category119964 is a simplicial computad if and onlyifbull Each category119964119899 of 119899-arrows is freely generated by the graph of atomic 119899-arrowsbull If 119891 is an atomic 119899-arrow in119964119899 and 120590∶ [119898] ↠ [119899] is an epimorphism in 120491 then the degenerated119898-arrow 119891 sdot 120590 is atomic in119964119898
By the Eilenberg-Zilber lemma⁴ a simplicial category119964 is a simplicial computad if and only if allof its non-identity arrows can be expressed uniquely as a composite
119891 = (1198911 sdot 1205721) ∘ (1198912 sdot 1205722) ∘ ⋯ ∘ (119891ℓ sdot 120572ℓ)in which each 119891119894 is non-degenerate and atomic and each 120572119894 is a degeneracy operator in 120491
614 Example A 1-category 119860 may be regarded as a simplicial category 119860bull in which 119860119899 ≔ 119860 forall 119899 In the construction the hom-spaces of 119860 coincide with the hom-sets of 119860 Such ldquoconstantrdquosimplicial categories define simplicial computads if and only if the 1-category 119860 is free
615 Example For any simplicial set 119880 let 120794[119880] denote the simplicial category with two objectsldquominusrdquo and ldquo+rdquo and with hom-spaces defined by
120794[119880](+ minus) ≔ empty 120794[119880](+ +) ≔ 120793 ≕ 120794[119880](minus minus) 120794[119880](minus +) ≔ 119880This simplicial category is a simplicial computad because there are no composable sequences of arrowsin 120794[119880] containing more than one non-identity arrow Every arrow from minus to + is atomic
616 Definition A simplicial functor 119866∶ 119964 rarr ℬ between simplicial computads defines a sim-plicial computad morphism if it maps every atomic arrow 119891 in 119964 to an arrow 119866119891 which is eitheratomic or an identity in ℬ Write 119982119982119890119905-119966119901119905119889 sub 119982119982119890119905-119966119886119905 for the non-full subcategory of simplicialcomputads and their morphisms
The axioms of Definition 613 assert that the atomic and identity 119899-arrows of a simplicial com-putad assemble into a diagram in 119970119901ℎ120491epi
opand a simplicial computad morphism restricts to define
a natural transformation between the underlying 120491epiop-indexed graphs of atomic and identity ar-
rows in this way restricting to the atomic or identity arrows defines a functor atom ∶ 119982119982119890119905-119966119901119905119889 rarr119970119901ℎ120491epi
op The next lemma tells us that the category 119982119982119890119905-119966119901119905119889 is canonically isomorphic to the
intersection of 119970119901ℎ120491epiop
and 119982119982119890119905-119966119886119905 in 119966119886119905120491epiop
617 Lemma The functor that carries a simplicial computad to its underlying diagram of atomic and identityarrows and the inclusion of simplicial computads into simplicial categories define the legs of a pullback cone
119982119982119890119905-119966119901119905119889 119982119982119890119905-119966119886119905
119970119901ℎ120491epiop
119966119886119905120491epiop
atom
119865120491epiop
⁴The Eilbenberg-Zilber lemma asserts that any degenerate simplex in a simplicial set may be uniquely expressed as adegenerated image of a non-degenerate simplex see [21 II31 pp 26-27]
149
Moreover since 119982119982119890119905-119966119886119905 and 119970119901ℎ120491epiop
have colimits the functors to 119966119886119905120491epiop
preserve them and 119865120491epiop
is an isofibration it follows that 119982119982119890119905-119966119901119905119889 has colimits created by either of the functors to 119982119982119890119905-119966119886119905 or to119970119901ℎ120491epi
op
Proof If119964 is a simplicial category presented as a simplicial object119964bull ∶ 120491op rarr 119966119886119905 then119964 isa simplicial computad if and only if there exists a dotted lift as below-left
120491epiop 119970119901ℎ 120491epi
op 119970119901ℎ
120491op 119966119886119905 120491op 119966119886119905
exist
119865
dArrexist
119865119964bull
119964bull
ℬbull
dArr119866
in which case this lift is necessarily unique Correspondingly as simplicial functor 119866∶ 119964 rarr ℬ de-fines a computad morphism if and only if the restricted natural transformation above-right also liftsthrough the free category functor again necessarily uniquely These facts verify that the category119982119982119890119905-119966119901119905119889 is captured as the stated pullback
Now consider a diagram 119863∶ 119869 rarr 119982119982119890119905-119966119901119905119889 and form its colimit cone in 119982119982119890119905-119966119886119905 and in119970119901ℎ120491epi
op The functors to 119966119886119905120491epi
opcarry these to a pair of isomorphic colimit cones under the same
diagram and since 119865120491epiop
is an isofibration (any category isomorphic to a free category is itself a freecategory and this isomorphism necessarily restricts to underlying graphs of atomic arrows) there existsa colimit cone under 119863 in 119970119901ℎ120491epi
opwhose image under 119865120491epi
opis equal to the image of the colimit
cone under 119863 in 119982119982119890119905-119966119886119905 Now the universal property of the pullback allows us to lift this cone to119982119982119890119905-119966119901119905119889 and a similar argument using the 2-categorical universal property of the pullback diagramof categories demonstrates that the lifted cone is a colimit cone
618 Definition (simplicial subcomputads) A simplicial computad morphism 119964 ℬ that is in-jective on objects and faithful mdash ie carries every atomic arrow in 119964 to an atomic arrow in ℬ mdashdisplays119964 as a simplicial subcomputad of ℬ
The simplicial subcomputad 119878 generated by a set of arrows 119878 in a simplicial computad 119964 is thesmallest simplicial subcomputad of 119964 containing those arrows The objects of 119878 are those objectsthat appear as domains or codomains of arrows in 119878 and its set of morphisms is the smallest subset ofmorphisms containing 119878 that have the following closure properties
bull if 119891 isin 119878 and 120572∶ [119899] rarr [119898] isin 120491 then 119891 sdot 120572 isin 119878bull If 119891 119892 isin 119878 are composable then 119891 ∘ 119892 isin 119878
Lemma 617 proves that simplicial computads and computad morphisms are closed under colimitsformed in the category of simplicial categories For certain special colimit shapes the property of beinga simplicial subcomputad is also preserved
619 Lemma A simplicial subcategory 119964 ℬ of a simplicial computad ℬ displays 119964 as a simplicialsubcomputad of ℬ just when 119964 is closed under factorizations if 119891 and 119892 are composable arrows of ℬ and119891 ∘ 119892 isin 119964 then 119891 and 119892 are in119964
Proof Exercise 61i
150
6110 Lemma Simplicial subcomputads are closed under coproduct pushout and colimit of countable se-quences
Proof Simplicial subcomputad inclusions are precisely those morphisms in 119982119982119890119905-119966119901119905119889 whoseimages in 119970119901ℎ120491epi
opare pointwise monomorphisms Lemma 617 proves that colimits in 119982119982119890119905-119966119901119905119889
are crated in 119970119901ℎ120491epiop
As colimits in this presheaf category are formed pointwise and as monomor-phisms are stable under coproduct pushout and colimit of countable sequences the result follows
6111 Definition (relative simplicial computad) The class of all relative simplicial computads is theclass of all simplicial functors that can be expressed as a countable composite of pushouts of coproductsofbull the unique simplicial functor empty 120793 andbull the simplicial subcomputad inclusion 120794[120597Δ[119899]] 120794[Δ[119899]] for 119899 ge 0
The next lemma reveals that relative simplicial computads differ from simplicial subcomputadinclusions only in the fact that the domains of relative simplicial computads need not be simplicialcomputadsmdashbutwhen they are the codomain is also a simplicial computad and themap is a simplicialsubcomputad inclusion
6112 Lemma If119964 is a simplicial computad then an inclusion of simplicial categories119964 ℬ is a simplicialcomputad inclusion if and only if it is a relative simplicial computad In particular a simplicial category ℬ isa simplicial computad if and only if empty ℬ is a relative simplicial computad
Proof First note that empty 120793 and 120794[120597Δ[119899]] 120794[Δ[119899]] are simplicial subcomputad inclusionsthe first case is trivial and for the latter all the non-identity arrows in 120794[119880] atomic By Lemma 6110to prove that any relative simplicial computad 119964 ℬ whose domain is a simplicial computad is asubcomputad inclusion it suffices to prove that for any simplicial computad 119964 and any simplicialfunctor 119891∶ 120794[120597Δ[119899]] rarr 119964 not necessarily a computad moprhism the pushout of 120794[120597Δ[119899]] 120794[Δ[119899]] along 119891 is a simplicial computad containing119964 as a simplicial subcomputad
The subcomputad inclusion 120794[120597Δ[119899]] 120794[Δ[119899]] is full on 119903-arrows for 119903 lt 119899 for 119903 gt 119899 thecodomain is constructed by adjoining one atomic arrow from minus to+ for each epimorphism [119903] ↠ [119899]It follows that the pushout119964119964prime is similarly full on 119903-arrows for 119903 lt 119899 and for 119903 ge 119899 the categoryof 119903-arrows of 119964prime is obtained from that of 119964 by adjoining one atomic 119903-arrow for each degeneracyoperator [119903] ↠ [119899] with boundary specified by the attaching map 119891 If we adjoin an arrow to a freecategory along its boundary we get a free category with that arrow as an extra generator so each ofthe categories of 119903-arrows of 119964prime are freely generated and it is clear from this description that ondegenerating one of these extra adjoined arrows we just map it to one of the generating arrows wersquoveadjoined at a higher dimension This proves that 119964 119964prime is a simplicial subcomputad inclusionas claimed It follows inductively that the codomain of a relative simplicial computad is a simplicialcomputad whenever its domain is in which case the inclusion is a subcomputad inclusion
For the converse we inductively present any simplicial subcomputad inclusion 119964 ℬ as asequential composite of pushouts of coproducts of the mapsempty 120793 and 120794[120597Δ[119899]] 120794[Δ[119899]] Notethat 120794[Δ[119899]] contains a single non-degenerate atomic arrow not present in 120794[120597Δ[119899]] namely theunique non-degenerate 119899-arrow representing the 119899-simplex
At stage ldquominus1rdquo use the inclusion empty 120793 to attach any object in ℬ119964 to 119964 At stage ldquo0rdquo usethe inclusion 120794[empty] 120794[Δ[0]] to attach each atomic 0-arrows of ℬ that is not in 119964 Iteratively at
151
stage ldquo119903rdquo use the inclusion 120794[120597Δ[119903]] 120794[Δ[119903]] to attach each atomic 119903-arrows of ℬ that is not in119964 There is a canonical map from the codomain of the relative simplicial computad defined in thisway to ℬ which by construction is bijective on objects and sends the unique atomic arrow attachedby each cell to an atomic arrow in ℬ in particular this comparison functor which by constructionlies in 119982119982119890119905-119966119901119905119889 is in fact a simplicial subcomputad inclusion The comparison is surjective onatomic 119899-arrows by construction and hence surjective on all arrows because the unique factorizationany arrow into non-degenerate atomics is present in the colimit at the stage corresponding to thedimension of that arrow Thus we have presented119964 ℬ as a relative simplicial computad inclusion
The morphisms listed in Definition 6111 are the generating cofibrations in the Bergner modelstructure on simplicial categories [4] Hence the relative simplicial computads are precisely the cellu-lar cofibrations those that are built as sequential composites of pushouts of coproducts of generatingcofibrations (without closing under retracts) For non-cofibrant domains the notion of Bergner cofi-bration is more general than the notion of relative simplicial computad However
6113 Lemma Every retract of a simplicial computad is a simplicial computad
Proof A simplicial category 119964 is a retract of a simplicial computad ℬ if there exist simplicialfunctors
119964 ℬ 119964119878 119877
so that 119877119878 = id The category 119964 is then recovered as the coequalizer (or the equalizer) of 119878119877 andthe identity so if we knew that this idempotent defined a computad morphism we could appeal toLemma 617 and be done but we do not know this so we must argue further
First we demonstrate that every retract of a free category is free To see this wersquoll first show thatthe inclusion 119964119899 ℬ119899 satisfies the ldquo2-of-3rdquo property of 119891 and 119892 are composable morphisms of ℬ119899so that two of three of 119891 119892 and 119891 ∘ 119892 lie in119964119899 then so does the third This is clear in the case where119891 119892 isin 119964119899 so suppose that 119891 119891 ∘119892 isin 119964119899 Then 119891 and 119891 ∘119892 are fixed points for the idempotent 119878119877 andso we have 119891 ∘ 119892 = 119878119877(119891 ∘ 119892) = 119878119877(119891) ∘ 119878119877(119892) = 119891 ∘ 119878119877(119892) In the free category ℬ119899 all morphismsare both monic and epic so 119892 = 119878119877(119892) lies in 119964119899 as well Now by induction it is easy to verify thatevery arrow in 119964119899 factors uniquely as a product of composites of atomic arrows in ℬ119899 with each ofthese composites defining an atomic arrow in119964119899
This verifies the first condition of Definition 613 It remains only to verify that degenerate imagesof atomic arrows in 119964119899 are atomic To that end consider an epimorphism 120572∶ [119898] ↠ [119899] and anatomic 119899-arrow 119891 in 119964119899 If 119891 sdot 120572 = 119892 ∘ ℎ is a non-trivial factorization in 119964119898 ℬ119898 then since ℬis a simplicial computad we must have 119878(119891) = 119892prime ∘ ℎprime with 119892prime sdot 120572 = 119878(119892) and ℎprime sdot 120572 = 119878(ℎ) Since119891 = 119877119878(119891) = 119877119892prime ∘119877ℎprime is atomic we must have 119877119892prime or 119877ℎprime an identity but then 119877119892prime sdot 120572 = 119877119878(119892) = 119892or 119877ℎprime sdot 120572 = 119877119878(ℎ) = ℎ is an identity and so the factorization 119891 sdot 120572 = 119892 ∘ ℎ is trivial after all
Consequently
6114 Corollary The simplicial computads are precisely the cofibrant simplicial categories in the Bergnermodel structure
Exercises
61i Exercise Prove Lemma 619
152
61ii Exercise Prove that the 119903-skeleton of a simplicial computad defined by discarding all arrows ofdimension⁵ greater than 119903 is the simplicial subcomputad generated by the atomic arrows of dimension119903
62 Free resolutions and homotopy coherent simplices
The original example of a simplicial computad also due to Dwyer and Kan [19] is given by thefree resolution of a 1-category 119862
621 Definition (free resolutions) Write 119865 ⊣ 119880 for the free category and underlying graph functorsof Digression 612 Note that the components of the counit and comultiplication of the comonad
(119865119880∶ 119966119886119905 rarr 119966119886119905 120598 ∶ 119865119880 rArr id 119865120578119880∶ 119865119880 rArr 119865119880119865119880)define identity-on-objects functors
For any 1-category 119862 we will define a simplicial category 119865119880bull119862 with the same objects and withthe category of 119899-arrows defined to be (119865119880)119899+1119862 A 0-arrow is a sequence of composable arrows in119862 A non-identity 119899-arrow is a sequence of composable arrows in 119862 with each arrow in the sequenceenclosed in exactly 119899 pairs of well-formed parentheses
The simplicial object 120491op rarr 119966119886119905 is formed by evaluating the comonad resolution at 119862 isin 119966119886119905
119865119880119862 (119865119880)2119862 (119865119880)3119862 (119865119880)4119862 ⋯ (622)
For 119895 ge 1 the face maps(119865119880)119896120598(119865119880)119895 ∶ (119865119880)119896+119895+1119862 rarr (119865119880)119896+119895119862
remove the parentheses that are contained in exactly 119896 others while 119865119880⋯119865119880120598 composes the mor-phisms inside the innermost parentheses For 119895 ge 1 the degeneracy maps
119865(119880119865)119896120578(119880119865)119895119880∶ (119865119880)119896+119895minus1119862 rarr (119865119880)119896+119895119862double up the parentheses that are contained in exactly 119896 others while 119865⋯119880119865120578119880 inserts parenthesesaround each individual morphism
623 Example (free resolution of a group) A classically important special case is given by the freeresolution of a 1-object groupoid whose automorphisms are the elements of a discrete group 119866 Inthis case each category of 119899-arrows is again a 1-object groupoid The category of 0-arrows is the groupof words in the non-identity elements of 119866 The category of 1-arrows is the group of words of wordsand so on
We now explain the sense in which free resolutions are ldquoresolutionsrdquo of the original 1-category Asdiscussed in Example 614 a 1-category119862 can be regarded as a constant simplicial category119862bull whosehom-spaces coincide with the hom-sets of119862 There is a canonical ldquoaugmentationrdquo map 120598 ∶ 119865119880bull119862 rarr 119862in 119982119982119890119905-119966119886119905 that is determined by its degree zero component 120598 ∶ 119865119880119862 rarr 119862 which is just given bycomposition in 119862
⁵An 119899-arrow 119891 in a simplicial category119964 has dimension 119903 if there exists a non-degenerate 119903-arrow 119892 and an epimor-phism 120572∶ [119899] ↠ [119903] so that 119891 = 119892 sdot 120572
153
624 Proposition The functor 120598 ∶ 119865119880bull119862 rarr 119862 is a local homotopy equivalence of simplicial sets That isfor any pair of objects 119909 119910 isin 119862 the map 120598 ∶ 119865119880bull119862(119909 119910) rarr 119862(119909 119910) is a simplicial homotopy equivalence119865119880bull119862(119909 119910) is homotopy equivalent to the discrete set 119862(119909 119910) of arrows in 119862 from 119909 to 119910
Proof The augmented simplicial object
119862 (119865119880)119862 (119865119880)2119862 (119865119880)3119862 (119865119880)4119862 ⋯
is split at the level of reflexive directed graphs (ie after applying119880) These splittings are not functorsbut that wonrsquot matter These directed graph morphisms displayed here are all identity on objectswhich means that for any 119909 119910 isin 119862 there is a split augmented simplicial set
119862(119909 119910) (119865119880)119862(119909 119910) (119865119880)2119862(119909 119910) (119865119880)3119862(119909 119910) (119865119880)4119862(119909 119910)⋯
and now some classical simplicial homotopy theory of Meyer [42] reviewed in Appendix C proves that120598 ∶ 119865119880bull119862(119909 119910) rarr 119862(119909 119910) is a simplicial homotopy equivalence
As the name suggests free resolutions are simplicial computads
625 Proposition (free resolutions are simplicial computads) The free resolution of any 1-categorydefines a simplicial computad in which the atomic 119899-arrows are those enclosed in precisely one pair of outermostparentheses
Proof In the free resolution of a 1-category 119862 the category of 119899-arrows is (119865119880)119899+1119862 The cate-gory 119865119880119862 is the free category on the underlying graph of 119862 Its arrows are sequences of composablenon-identity arrows of 119862 the atomic 0-arrows are the non-identity arrows of 119862 An 119899-arrow is asequence of composable arrows in 119862 with each arrow in the sequence enclosed in exactly 119899 pairs ofparentheses The atomic 119899-arrows are those enclosed in precisely one pair of parentheses on the out-side Since composition in a free category is by concatenation the unique factorization property isclear Since degeneracy arrows ldquodouble uprdquo on parentheses these preserve atomics as required
The atomic 119899-arrows in the free resolution of a 1-category index the generating 119899-homotopies ina homotopy coherent diagram such as enumerated for the homotopy coherent 120596-simplex at the startof this chapter
626 Definition (the homotopy coherent 120596-simplex) The homotopy coherent 120596-simplex ℭΔ[120596]is a simplicial category defined to be the free resolution of the category
120654 ≔ 0 1 2 3 ⋯
The objects of ℭΔ[120596] are natural numbers 119896 ge 0 Unpacking Definition 621 we can completelydescribe its arrows
154
bull A non-identity 0-arrow from 119895 to 119896 is a sequence of non-identity composable morphisms from 119895to 119896 the data of which is uniquely determined by the objects being passed through In particularthere are no 0-arrows from 119895 to 119896 if 119895 gt 119896 and the only 0-arrow from 119896 to 119896 is the identity If 119895 lt 119896the non-identity 0-arrows from 119895 to 119896 correspond to subsets
119895 119896 sub 1198790 sub [119895 119896]of the closed internal [119895 119896] = 119905 isin 120654 ∣ 119895 le 119905 le 119896 containing both endpoints
bull A 1-arrow from 119895 to 119896 is a once bracketed sequence of non-identity composable morphisms from119895 to 119896 This data is specified by two nested subsets
119895 119896 sub 1198790 sub 1198791 sub [119895 119896]the larger one 1198791 specifying the underlying unbracketed sequence and the smaller one 1198790 speci-fying the placement of the brackets
bull A 119903-arrow from 119895 to 119896 is an 119903 times bracketed sequence of non-identity composable morphismsfrom 119895 to 119896 the data of which is specified by nested subsets
119895 119896 sub 1198790 sub ⋯ sub 119879 119903 sub [119895 119896] (627)
indicating the locations of all of the parentheses⁶The face and degeneracy maps of (622) are the obvious ones either duplicating or omitting one ofthe sets 119879 119894 In particular the 119903-arrows just enumerated are non-degenerate if and only if each of theinclusions 1198790 sub ⋯ sub 119879 119903 is proper
We now describe the geometry of the mapping spaces ℭΔ[120596](119895 119896)
628 Lemma (homs in the homotopy coherent 120596-simplex are cubes) The mapping spaces of the homo-topy coherent 120596-simplex are defined for 119895 119896 isin 120654 by
ℭΔ[120596](119895 119896) cong
⎧⎪⎪⎪⎨⎪⎪⎪⎩
empty 119895 gt 119896Δ[0] 119896 = 119895 or 119896 = 119895 + 1Δ[1]119896minus119895minus1 119895 lt 119896
Proof Because120654 has no arrows from 119895 to 119896 when 119895 gt 119896 these hom-spaces of ℭΔ[120596] are similarlyempty When 119896 = 119895 or 119896 = 119895 + 1 we have 119895 119896 = [119895 119896] using the notation of Definition 626 soℭΔ[120596](119895 119896) cong Δ[0] is comprised of a single point
For 119896 gt 119895 there are 119896minus119895minus1 elements of [119895 119896] excluding the endpoints and so we see thatℭΔ[120596](119895 119896)has 2119896minus119895minus1 vertices The 119903-simplices ofℭΔ[120596](119895 119896) are given by specifying 119903+1 vertices mdash each a subset119895 119896 sub 119879 119894 sub [119895 119896]mdash that respect the ordering of subsets relation From this we see that
ℭΔ[120596](119895 119896) cong Δ[1]119896minus119895minus1
⁶The nesting is because parenthezations should be ldquowell formedrdquo with open brackets closed in the reverse order to thatin which they were opened
155
is the nerve of the poset of subsets 119895 119896 sub 119879 sub [119895 119896] ordered by inclusion as displayed for instancein the case 119895 = 0 and 119896 = 4
ℭΔ[120596](0 4) ∶=
⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝
0 4 0 1 4
0 3 4 0 1 3 4
0 2 4 0 1 2 4
0 2 3 4 0 1 2 3 4
⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠
Proposition 625 proves that the homotopy coherent 120596-simplex is a simplicial computad and itsproof identifies its atomic arrows
629 Lemma The simplicial category ℭΔ[120596] is a simplicial computad whose atomic 119903-arrows are those witha single outermost parenthesis ie those sequences of subsets
119895 119896 = 1198790 sub ⋯ sub 119879 119903 sub [119895 119896]for which 1198790 = 119895 119896 Geometrically the atomic arrows from 119895 to 119896 are precisely the simplices in the hom-cubeℭΔ[120596](119895 119896) cong Δ[1]119896minus119895minus1 that contain the initial vertex 119895 119896
The finite ordinals define full subcategories of120654 In this way the homotopy coherent 120596-simplexrestricts to define homotopy coherent simplices in each finite dimension
6210 Definition (the homotopy coherent 119899-simplex) The homotopy coherent 119899-simplex ℭΔ[119899]is the full subcategory of the homotopy coherent 120596-simplex ℭΔ[120596] spanned by the objects 0hellip 119899Equivalently it is the free resolution of the ordinal category with 119899 + 1 objects
Explicitly the homotopy coherent 119899-simplex has mapping spaces given for 119895 119896 isin [119899] by cubes
ℭΔ[119899](119895 119896) cong
⎧⎪⎪⎪⎨⎪⎪⎪⎩
empty 119895 gt 119896Δ[0] 119896 = 119895 or 119896 = 119895 + 1Δ[1]119896minus119895minus1 119895 lt 119896
each of which may be understood as the nerve of the poset of subsets 119895 119896 sub 119879 sub [119895 119896] ordered byinclusion An 119903-arrow may be represented as a nested sequence of subsets
119895 119896 sub 1198790 sub ⋯ sub 119879 119903 sub [119895 119896]The homotopy coherent 119899-simplex is a simplicial computad whose atomic 119903-arrows are those se-
quences for which 1198790 = 119895 119896 or those simplices that contain the initial vertex in the hom-cube
Exercises
62i Exercise Compute the free resolution of the commutative square category 120794times120794 and compareit with the product ℭΔ[1] times ℭΔ[1] of two copies of the free resolution of 120794 This computationimplies that the functorℭ∶ 119982119982119890119905 rarr 119982119982119890119905-119966119901119905119889 to be introduced inDefinition 632 does not preserveproducts
156
63 Homotopy coherent realization and the homotopy coherent nerve
Our aim now is to introduce the homotopy coherent realization of any simplicial set 119883 which willdefine a simplicial computad ℭ119883 whose objects are the vertices of 119883 The homotopy coherent real-ization of Δ[119899] will be the homotopy coherent 119899-simplex ℭΔ[119899] of Definition 6210 The homotopycoherent realization of 119883 will be defined by ldquogluing togetherrdquo homotopy coherent 119899-simplices in acanonical way succinctly ℭ119883 is defined as a colimit in 119982119982119890119905-119966119901119905119889 of a diagram of homotopy coher-ent simplices indexed by the category of simplices of 119883 We will then leverage Lemma 629 into anexplicit presentation of ℭ119883 as a simplicial computad stated as Theorem 6310 recovering a result ofDugger and Spivak
The homotopy coherent realization and homotopy coherent nerve functors are determined by thecosimplicial object
120491 119982119982119890119905-119966119886119905
[119899] ℭΔ[119899]
ℭΔ[bull]
where a simplicial operator 120572∶ [119899] rarr [119898] acts on an 119903-arrow from 119895 to 119896 as described in Definition6210 by taking the direct image of the sequence of subsets (627)
We introduce these functors in turn
631 Definition (homotopy coherent nerve) The homotopy coherent nerve of a simplicial category119964 is the simplicial set 120081119964 whose 119899-simplices
120081119964119899 ≔ 119982119982119890119905-119966119886119905(ℭΔ[119899]119964)are defined to be diagrams ℭΔ[119899] rarr 119964 the simplicial operators act contravariantly on 120081119964 by pre-composition
Explicitly a homotopy coherent 119899-simplex in119964 is given bybull a sequence of objects 1198860 hellip 119886119899 isin 119964 andbull a sequence of simplicial maps
119886119894119895 ∶ Δ[1]119896minus119895minus1 rarr119964(119886119895 119886119896)for each 0 le 119895 lt 119896 le 119899
bull satisfying the simplicial functoriality condition
Δ[1]119896minus119895minus1 times Δ[1]119895minus119894minus1 Δ[1]119896minus119894minus1
119964(119886119895 119886119896) times 119964(119886119894 119886119895) 119964(119886119894 119886119896)
119886119895119896times119886119894119895
or119895
119886119894119896∘
where
Δ[1]119896minus119895minus1 times Δ[1]119895minus119894minus1 cong Δ119896minus119894minus2 Δ[1]119896minus119894minus1
ℭΔ[119899](119895 119896) times ℭΔ[119899](119894 119895) ℭΔ[119899](119894 119896)
cong
or119895
cong
∘
is the map that sends a pair of 119903-simplices
119894 119895 sub 1198780 sub ⋯ sub 119878119903 sub [119894 119895] and 119895 119896 sub 1198790 sub ⋯ sub 119879 119903 sub [119895 119896]
157
to their union119894 119896 sub 1198780 cup 1198790 sub ⋯ sub 119878119903 cup 119879 119903 sub [119894 119896]
If the 0 1-valued coordinates of the cubeΔ[1]119896minus119894minus1 are indexed by integers 119894 lt 119905 lt 119896 then the imageof or119895 is the 119895 = 1 face of the cube
632 Definition (homotopy coherent realization) The homotopy coherent realization functor ℭ isthe pointwise left Kan extension of the cosimplicial object ℭΔ[bull] along the Yoneda embedding
119982119982119890119905
120491 119982119982119890119905-119966119886119905
ℭよ
ℭΔ[bull]
uArrcong
The value of a pointwise left Kan extension at an object 119883 isin 119982119982119890119905 can be computed as a colimitindexed by the comma category Hom119982119982119890119905(よ 119883) [48 621] This comma category is better know asthe category of simplices of 119883 whose objects are simplices of 119883 and in which a morphism from an119899-simplex 119909 to an 119898-simplex 119910 is a simplicial operator 120572∶ [119899] rarr [119898] so that 119910 sdot 120572 = 119909 In this casethe colimit formula gives
ℭ119883 ≔ colim[119899]isin120491119909isin119883119899
ℭΔ[119899]
By general abstract nonsense
633 Proposition The homotopy coherent realization functor is left adjoint to the homotopy coherent nerve
119982119982119890119905 119982119982119890119905-119966119886119905
ℭ
perp120081
Proof The homotopy coherent nerve was defined so that this adjoint correspondencewould holdfor the standard simplices and the general result follows since every simplicial set is a colimit indexedby its category of simplices of standard simplices See [48 659] for more details
634 Lemma The homotopy coherent realization functor takes its values in the subcategory of simplicialcomputads and computad morphisms
119982119982119890119905 119982119982119890119905-119966119886119905
119982119982119890119905-119966119901119905119889
ℭ
ℭ
Proof By Lemma 617 which proves that the category of simplicial computads is closed undercolimits in the category of simplicial categories it suffices to demonstrate that the cosimplicial objectℭΔ[bull] is valued in the subcategory of simplicial computads and simplicial computad morphisms Weknow already that the homotopy coherent simplices are simplicial computads so we need only verifythat the simplicial operators act by computad morphisms
A simplicial operator 120572∶ [119899] rarr [119898] acts on the 119903-arrow from 119895 to 119896 described in Definition 6210by taking the direct image of the sequence of subsets (627) The condition that characterizes theatomic arrows 119895 119896 = 1198790 is preserved by direct images so we see that 120572 defines a computad mor-phism ℭ120572∶ ℭΔ[119899] rarr ℭΔ[119898] As the subcategory 119982119982119890119905-119966119901119905119889 119982119982119890119905-119966119886119905 is closed under colimits
158
it follows that every homotopy coherent realization is a simplicial computad and any simplicial map119883 rarr 119884 induces a morphism of simplicial computads ℭ119883 rarr ℭ119884
635 Lemma For any inclusion119883 119884 of simplicial sets the morphism ℭ119883 ℭ119884 is a simplicial subcom-putad inclusion Moreover in the case of 119883 Δ[119899] an atomic 119903-arrow
119895 119896 = 1198790 sub ⋯ sub 119879 119903 sub [119895 119896]from 119895 to 119896 of ℭΔ[119899] lies in the subcomputad ℭ119883 if and only if the simplex spanned by the vertices of 119879 119903 liesin 119883
Proof Recall that everymonomorphism of simplicial sets119883 119884 admits a canonical decomposi-tion as a sequential composite of pushouts of coproducts of the simplex boundary inclusions 120597Δ[119899] Δ[119899] Since all colimits are preserved by the left adjoint ℭ and Lemma 6110 proves that simplicialsubcomputads are closed under colimits of this form it suffices to prove that ℭ120597Δ[119899] ℭΔ[119899] isa simplicial subcomputad inclusion Wersquoll argue more generally that for 119883 sub Δ[119899] ℭ119883 ℭΔ[119899]is a simplicial subcomputad inclusion where the atomic arrows of ℭ119883 are as described in the secondclause of the statement
We argue using ideas from Reedy category theory reviewed in Appendix C Our task is to showthat the image of 119883 Δ[119899] under the functor
119982119982119890119905 119982119982119890119905-119966119901119905119889 119970119901ℎ120491epiopℭ atom
that takes a simplicial set to its 120491epiop-indexed graph of atomic and identity arrows is a pointwise
monomorphismThe simplicial subset 119883 sub Δ[119899] can be described as a colimit of certain faces of Δ[119899] glued along
their common faces the functor just described preserves these colimits Our claim asserts that thecomposite cosimplicial object ℭΔ[bull] ∶ 120491 rarr 119982119982119890119905 rarr 119982119982119890119905-119966119901119905119889 rarr 119970119901ℎ120491epi
opis Reedy monomorphic
meaning that every atomic 119903-arrow 119879bull in the homotopy coherent 119899-simplex is uniquely expressible inthe form 120572 sdot 119878bull where 120572∶ [119898] ↣ [119899] is a monomorphism in 120491 and 119878bull is ldquonon-co-degeneraterdquo ie notin the image of any monomorphism This is clear take 119898 = |119879 119903| minus 1 and define 120572∶ [119898] ↣ [119899] tobe the inclusion with image 119879 119903 sub [119899] Then take 119878bull to be the atomic 119903-arrow from 0 to 119898 in ℭΔ[119898]whose direct image under 120572 is 119879bull It is clear that 119878bull is not in the image of any smaller face map Thisargument also reveals that the atomic 119903-arrows 119879bull of ℭΔ[119899] that are present in the subcomputad ℭ119883are exactly those for which the vertices of 119879bull are contained in one of its faces
Lemma 635 allows us to compute the following subcomputads of the homotopy coherent sim-plex Before stating the results of these computations we introduce notation that suggests the correctgeometric intuition
636 Notation (cubes boundaries and cubical horns) We introduce special notation for the fol-lowing simplicial setsbull Write 119896 for the simplicial cube Δ[1]119896bull Write 120597119896 for the boundary of the 119896-cube Formally 120597119896 is the domain of the iterated Leibniz
product (120597Δ[1] Δ[1])times119896 If an 119903-simplex in 119896 is represented by a 119896-tuple of maps 120588119894 ∶ [119903] rarr[1] then that 119903-simplex lies in 120597119896 if and only if there is some 119894 for which 120588119894 is constant at eithervertex of [1]
159
bull Write ⊓119896119895119890 sub 120597119896 for the cubical horn containing only the face 119890 isin [1] in direction 1 le 119895 le 119896Formally ⊓119896119895119890 is the domain of the iterated Leibniz product
(120597Δ[1] Δ[1])times119895minus1times(Δ[0] 119890minusrarr Δ[1])times(120597Δ[1] Δ[1])times119896minus119895
An 119903-simplex in 119896 represented by a 119896-tuple of map 120588119894 ∶ [119903] rarr [1] lies in ⊓119896119895119890 if and only if forsome 119894 ne 119895 the map 119901119894 is constant or 119901119895 is the constant operator at 119890 isin [1]
637 Lemma (coherent subsimplices) The homotopy coherent realizations of the simplicial sphere and innerand outer horns define subcomputads of the homotopy coherent 119899-simplex containing all of the objects anddefined on homs by
(i) spheres ℭ120597Δ[119899] ℭΔ[119899] is full on all arrows except those from 0 to 119899 and has
ℭ120597Δ[119899](0 119899) ℭΔ[119899](0 119899)
120597119899minus1 119899minus1cong cong
(ii) inner horns For 0 lt 119896 lt 119899 ℭΛ119896[119899] ℭΔ[119899] is full on all arrows except those from 0 to 119899 andhas
ℭΛ119896[119899](0 119899) ℭΔ[119899](0 119899)
⊓119899minus11198961 119899minus1
cong cong(iii) outer horns ℭΛ119899[119899] ℭΔ[119899] is full on all arrows except those from 0 to 119899 minus 1 or 119899 and has
ℭΛ119899[119899](0 119899 minus 1) ℭΔ[119899](0 119899 minus 1) ℭΛ119899[119899](0 119899) ℭΔ[119899](0 119899)
120597119899minus2 119899minus2 ⊓119899minus1119899minus10 119899minus1
cong cong cong cong
Similarly ℭΛ0[119899] ℭΔ[119899] is full on all arrows except those from 0 or 1 to 119899 and has
ℭΛ0[119899](1 119899) ℭΔ[119899](1 119899) ℭΛ0[119899](0 119899) ℭΔ[119899](0 119899)
120597119899minus2 119899minus2 ⊓119899minus110 119899minus1
cong cong cong cong
Proof For (i) the only non-degenerate simplex of Δ[119899] that is not present in 120597Δ[119899] is the topdimensional 119899-simplex Consequently the only atomic 119903-arrows119879bull that are not present inℭ120597Δ[119899] arethose with 119879 119903 = [0 119899] The atomic 119903-arrows must also have 1198790 = 0 119899 and consequently correspondprecisely to those 119903-simplices of the cube 119899minus1 that contain both the first and last vertex Thus wesee that ℭ120597Δ[119899](0 119899) is isomorphic to the cubical boundary 120597119899minus1
For (ii) the only non-degenerate simplex of 120597Δ[119899] that is not present in an inner horn Λ119896[119899] isthe 119896-th face of the top dimensional simplex Consequently the only atomic 119903-arrows 119879bull that are notpresent in ℭΛ119896[119899] but are present in ℭ120597Δ[119899] are those with 119879 119903 = [0 119899]119896 The atomic 119903-arrowsmust also have 1198790 = 0 119899 and consequently correspond precisely to those 119903-simplices of the cube119899minus1 that contain both the first vertex and last vertex of the 119896 = 0 face of the cube 119899minus1 Thus wesee that ℭΛ119896[119899](0 119899) is isomorphic to the cubical horn ⊓119899minus11198961
For (iii) the only non-degenerate simplex of 120597Δ[119899] that is not present in the outer horn Λ119899[119899] isthe 119899-th face of the top dimensional simplex Consequently the only atomic 119903-arrows 119879bull that are not
160
present in ℭΛ119899[119899] but are present in ℭ120597Δ[119899] are those with 119879 119903 = [0 119899 minus 1] note that the source ofsuch 119903-arrows is 0 and the target is 119899minus1 The atomic 119903-arrows must also have1198790 = 0 119899minus1 and so asin the proof of (i) this identifies the inclusionℭΛ119899[119899](0 119899minus1) ℭΔ[119899](0 119899minus1) as 120597119899minus2 119899minus2The subcomputad ℭΛ119899[119899] is also missing non-atomic 119903-arrows that are present in ℭ120597Δ[119899] namelythose composites of the unique arrow from 119899 minus 1 to 119899 with the atomic 119903-arrows from 0 to 119899 minus 1 justdescribed Such non-atomic 119903-arrows are represented by sets of inclusions with 1198790 = 0 119899minus1 119899 and119879 119903 = [0 119899] and thus are found in the 119899minus1 = 1 face of the cube 119899minus1 Thus we see that ℭΛ119899[119899](0 119899)is isomorphic to the cubical horn ⊓119899minus1119899minus10
638 Definition (bead shapes) We call the atomic 119903-arrows of ℭΔ[119899](0 119899) that are not present inℭ120597Δ[119899](0 119899) bead shapes By Lemma 637 an 119903-dimensional bead shape corresponds to a sequenceof subsets
0 119899 = 1198790 sub 1198791 sub ⋯ sub 119879 119903minus1 sub 119879 119903 = [0 119899] (639)and is non-degenerate if and only if each of the inclusions is proper
Putting this together we can now explicitly present the simplicial computad structure on the ho-motopy coherent realization of any simplicial set
6310 Theorem (homotopy coherent realizations explicitly) For any simplicial set 119883 the homotopycoherent realization ℭ119883 is a simplicial computad in whichbull The objects of ℭ119883 are the vertices of the simplicial set 119883bull The atomic 0-arrows are non-degenerate 1-simplices of 119883 with the initial vertex of the simplex defining
the source of the 0-arrow and the final vertex of the simplex defining the target of the 0-arrowbull The atomic 1-arrows are non-degenerate 119899-simplices of 119883 with the initial vertex of the simplex defining
the source of the 0-arrow and the final vertex of the simplex defining the target of the 0-arrowbull The atomic 119896-arrows are pairs comprised of a non-degenerate 119899-simplex in119883 together with a 119896-dimensional
bead shape in ℭΔ[119899] This source of this 119896-arrow is the initial vertex of the 119899-simplex while the targetis the final vertex of the 119899-simplex and the arrow is non-degenerate if and only if the bead shape is non-degenerate
Note that the description of atomic 119896-arrows subsumes those of the atomic 0-arrows and atomic1-arrows as there is a unique 1-dimensional bead shape inℭΔ[119899] and a 0-dimensional bead shape existsonly in the case 119899 = 1 The data of a non-degenerate atomic 119896-arrow from 119909 to 119910 in ℭ119883 is given bya ldquobeadrdquo that is a non-degenerate 119899-simplex in 119883 from 119909 to 119910 together with the additional data of aldquobead shaperdquo sequence of proper subset inclusions (639) which Dugger and Spivak refer to as a ldquoflagof vertex datardquo [16] Non-atomic 119896-arrows are then ldquonecklacesrdquo that is strings of beads in 119883 joinedhead to tail together with accompanying ldquovertex datardquo for each simplex
Proof As observed in the proof of Lemma 635 using the canonical skeletal decomposition ofthe simplicial set 119883
∐119883119899119871119899119883
120597Δ[119899] ∐119883119899119871119899119883
Δ[119899]
empty sk0119883 ⋯ sk119899minus1119883 sk119899119883 ⋯ colim = 119883
161
the homotopy coherent realization ℭ119883 is constructed iteratively by a process that adjoints one copyof ℭΔ[119899] along a map of its boundary ℭ120597Δ[119899] for each non-degenerate 119899-simplex of 119883 Thus eachatomic 119896-arrow arises from a unique pushout of this form as the image of an atomic 119896-arrow in ℭΔ[119899]that is not present in the subcomputad ℭ120597Δ[119899]
6311 Remark From the description of Theorem 6310 the subcomputad inclusions ℭ119883 ℭ119884induced by monomorphisms of simplicial sets 119883 119884 are easily understood an 119903-arrow in ℭ119884 lies inℭ119883 if and only if each bead in its representing necklace in119884 lies in119883 This generalizes the descriptiongiven to subcomputads of homotopy coherent simplices in Lemma 635
Our convention is to identify 1-categories with their nerves The homotopy coherent realizationof these simplicial sets then produces a simplicial computad But we have already encountered a wayto produce a simplicial computad from a 1-category namely via the free resolution of Definition 621This would lead to a potential source of ambiguity were it not for the happy coincidence that thesetwo constructions are isomorphic⁷
6312 Proposition (free resolutions are homotopy coherent realizations) For any 1-category the freeresolution is naturally isomorphic to the homotopy coherent realization of its nerve
Proof Proposition 625 and Theorem 6310 present both simplicial categories as simplicial com-putads We will argue that they have the same objects and non-degenerate atomic 119896-arrows
Both have the same set of objects the objects of the 1-category coinciding with the vertices in itsnerve Atomic 0-arrows of the free resolution are morphisms in the category while atomic 0-arrowsin the coherent realization are non-degenerate 1-simplices of the nerve mdash these are the same thingAtomic non-degenerate 1-arrows of the free resolution are sequences of at least two morphisms (en-closed in a single set of outer parentheses) while atomic 1-arrows of the coherent realization are non-degenerate simplices of dimension at least two mdash again these are the same Finally a non-degenerateatomic 119896-arrow is a sequence of 119899 composable morphisms with (119896minus1) non-repeating bracketings thisnon-degeneracy necessitates 119899 gt 119896 This data defines a 119899-simplex in the nerve together with a non-degenerate atomic 119896-arrow in ℭΔ[119899](0 119899) ie an atomic 119896-arrow in the coherent realization
We now exploit the description of the subcomputads of the homotopy coherent 119899-simplex inLemma 637 to prove a key source of examples of quasi-categories a result first stated in this form in[13]
6313 Theorem The homotopy coherent nerve 120081119982 of a Kan-complex enriched category 119982 is a quasi-category
Proof By adjunction to extend along an inner horn inclusion Λ119896[119899] Δ[119899] mapping into thehomotopy coherent nerve120081119982 is to extend along simplicial subcomputad inclusionsℭΛ119896[119899] ℭΔ[119899]mapping into the Kan complex enriched category 119982 By Lemma 637(ii) the only missing 119903-arrows
⁷Note the isomorphism between the homotopy coherent realization of the 119899-simplex and the free resolution of theordinal category [119899] is tautologous The left Kan extension along the Yoneda embedding is defined so as to agree withℭΔ[bull] ∶ 120491 rarr 119982119982119890119905-119966119901119905119889 on the subcategory of representables Many arguments involving simplicial sets can be reducedto a check on representables with the extension to the general case following formally by ldquotaking colimitsrdquo This resulthowever is not one of them since we are trying to prove something for all categories and the embedding 119966119886119905 119982119982119890119905 doesnot preserve colimits
162
are in the mapping space from 0 to 119899 so we are asked to solve a single lifting problem
ℭΛ119896[119899](0 119899) cong ⊓119899minus11198961 Map(1198830 119883119899)
ℭΔ[119899](0 119899) cong 119899minus1
Cubical horn inclusions can be filled in the Kan complex Map(1198830 119883119899) completing the proof
We conclude by giving a precise meaning to the notion that motivated this chapter
6314 Definition Let 119883 be a simplicial set A homotopy coherent diagram of shape 119883 in a Kancomplex enriched category119982 is a simplicial functorℭ119883 rarr 119982 from the homotopy coherent realizationof 119883 to 119982
Exercise 62i reveals that there are two possible notions of natural transformation between homo-topy coherent diagram We opt for the more structured of the two
6315 Definition Let 119865119866∶ ℭ119883 119982 be homotopy coherent diagrams of shape 119883 in a Kan com-plex enriched category 119982 A homotopy coherent natural transformation 120572∶ 119865 rArr 119866 is a homotopycoherent diagram of shape 119883 times 120794
ℭ119883
ℭ[119883 times 120794] 119982
ℭ119883
1198650120572
1198661
that restricts to 119865 and to 119866 along the edges of the cylinder
These notions are originally due to Boardman and Vogt in [8] who observed that homotopy coher-ent natural transformations do not define a the morphisms of a 1-category Instead as they observedthey define the 1-arrows of a quasi-category
6316 Corollary Homotopy coherent diagrams of shape 119883 valued in a Kan complex enriched category119982 and homotopy coherent natural transformations between them define the objects and 1-arrows of a quasi-category Coh(119883 119982) whose 119899-simplices are homotopy coherent diagram ℭ[119883 times Δ[119899]] rarr 119982
Proof By adjunctionℭ ⊣ 120081 the simplicial set Coh(119883 119982) is isomorphic to (120081119982)119883 As the quasi-categories form an exponential ideal in the category of simplicial sets this follows immediately fromTheorem 6313
6317 Remark (all diagrams in homotopy coherent nerves are homotopy coherent) Corollary 6316explains that any homotopy coherent diagram ℭ119883 rarr 119982 of shape 119883 in a Kan complex enriched cat-egory 119982 transposes to define a map of simplicial sets 119883 rarr 120081119982 valued in the quasi-category definedas the homotopy coherent nerve of 119982 While not every quasi-category is isomorphic to a homotopycoherent nerve of a Kan complex enriched category once consequence of the internal Yoneda lemmato be proven much later is that every quasi-category is equivalent to a homotopy coherent nerve Thisexplains the slogan introduced at the beginning of this chapter and the title for this part of the bookall diagrams in quasi-categories are homotopy coherent thus quasi-category theory can be understoodas ldquohomotopy coherent category theoryrdquo
163
Exercises
63i Exercise Compare the simplicial computad structure of the homotopy coherent 120596-simplex asgiven by Theorem 6310 with the simplicial computad structure of Lemma 629
164
CHAPTER 7
Weighted limits ininfin-cosmoi
71 Weighted limits and colimits
Let (119985times 120793) denote a complete and cocomplete cartesian closed monoidal category The ex-amples we have in mind are (119982119982119890119905 times 120793) its cartesian closed subcategory (119966119886119905 times 120793) or its furthercartesian closed subcategory (119982119890119905 times 120793)
Ordinary limits and colimits are objects representing the functor of cones with a given apex over orunder a fixed diagram Weighted limits and colimits are defined analogously except that the cones overor under a diagram might have exotic ldquoshapes rdquo which are allowed to vary with the objects indexingthe diagram More formally in the119985-enriched context the weight defining the ldquoshaperdquo of a cone overa diagram indexed by119964 or under a diagram indexed by119964op takes the form of a functor in119985119964
Before introducing the general notion of weighted limit and colimit we first reacquaint ourselvesan example that we have seen already in Digression 124
711 Example (tensors and cotensors) A diagram indexed by the category 120793 valued in a119985-enrichedcategoryℳ is just an object119860 inℳ In this case the weight is just an object119880 of119985 The119880-weightedlimit of the diagram119860 is an object ofℳ denoted119860119880 mdash or denoted 119880119860 whenever superscripts areinconvenient mdash called the the cotensor of 119860 isin ℳ with 119880 isin 119985 defined by the universal property
ℳ(119883119860119880) cong 119985(119880ℳ(119883119860))this isomorphism between mapping spaces in 119985 Dually the 119880-weighted colimit of 119860 is an object119880 otimes 119860 isin ℳ called the tensor of 119860 isin ℳ with 119880 isin 119985 defined by the universal property
ℳ(119880 otimes 119860119883) cong 119985(119880ℳ(119860119883))this isomorphism again between mapping spaces in119985 Assuming such objects exist the cotensor andtensor define119985-enriched bifunctors
119985op timesℳ ℳ 119985timesℳ ℳminusminus minusotimesminus
in a unique way making the defining isomorphisms natural in 119880 and 119860 as wellSince 119985 is cartesian closed it is tensored and cotensored over itself with 119880 otimes 119881 ≔ 119880 times 119881
and 119881119880 ≔ 119985(119880119881) In particular the defining natural isomorphisms characterizing tensors andcotensors inℳ can be rewritten as
ℳ(119883119860119880) cong ℳ(119883119860)119880 and ℳ(119880 otimes 119860119883) cong ℳ(119860119883)119880
The fact that the natural isomorphisms defining tensors and cotensors are required to exist in119985(and not merely in 119982119890119905) has the following consequence
712 Lemma (associativity of tensors and cotensors) If ℳ is a 119985-category with tensors and cotensorsthen for any 119880119881 isin 119985 and 119860 isin ℳ there exist natural isomorphisms
119880 otimes (119881 otimes 119860) cong (119880 times 119881) otimes 119860 and (119860119881)119880 cong 119860119880times119881165
Proof By the defining universal property
ℳ(119883 (119860119881)119880) cong 119985(119880ℳ(119883119860119881)) cong 119985(119880119985(119881ℳ(119883119860)))cong 119985(119880 times 119881ℳ(119883119860)) cong ℳ(119883119860119880times119881)
for all 119883 isin ℳ By the Yoneda lemma (119860119881)119880 cong 119860119880times119881 The case for tensors is similar
We now introduce the general notions of weighted limit and weighted colimit from three differentviewpoints We introduce these perspectives in the reverse of the logical order because we find thisroute to be the most intuitive We first describe the axioms that characterize the weighted limit andcolimit bifunctors whenever they exist We then explain how weighted limits and colimits can beconstructed again assuming these exist We then finally introduce the general universal property thatdefines a particular weighted limit or colimit which tells us when the notions just introduced do infact exist
713 Definition (weighted limits and colimits axiomatically) For a small 119985-enriched category 119964and a large119985-enriched categoryℳ the weighted limit and weighted colimit bifunctors
lim119964minus minus∶ (119985119964)op timesℳ119964 rarrℳ and colim119964
minus minus∶ 119985119964 timesℳ119964op rarrℳare characterized by the following pair of axioms whenever they exist
(i) Weighted (co)limits with representable weights evaluate at the representing object
lim119964119964(119886minus) 119865 cong 119865(119886) and colim119964
119964(minus119886)119866 cong 119866(119886)
(ii) The weighted (co)limit bifunctors are cocontinuous in the weight for any diagrams 119865 isin ℳ119964
and 119866 isin ℳ119964op the functor colim119964
minus 119866 preserves colimits while the functor lim119964minus 119865 carries
colimits to limitssup1We interpret axiom (ii) to mean that weights can be ldquomade-to-orderrdquo a weight constructed as a colimitof representables mdash as all119985-valued functors are mdash will stipulate the expected universal property
714 Definition (weighted limits and colimits constructively) The limit of 119865 isin ℳ119964 weighted by119882 isin 119985119964 is computed by the functor cotensor product
lim119964119882 119865 ≔ 1114009
119886isin119964119865(119886)119882(119886) ≔ eq
⎛⎜⎜⎜⎜⎝ prod119886isin119964
119865(119886)119882(119886) prod119886119887isin119964
119865(119887)119964(119886119887)times119882(119886)⎞⎟⎟⎟⎟⎠ (715)
where the product and equalizer should be interpreted as conical limits see Digression 124 or Defini-tion 7114 below The maps in the equalizer diagram are induced by the actions 119964(119886 119887) times 119882(119886) rarr119882(119887) and 119865(119886) rarr 119865(119887)119964(119886119887) of the hom-object 119964(119886 119887) on the 119985-functors 119882 and 119865 the latter case
makes use of the natural isomorphism 119865(119887)119964(119886119887)times119882(119886) cong (119865(119887)119964(119886119887))119882(119886)
of Lemma 712
sup1More precisely as will be proven in Proposition 7112 the weighted colimit functor colim119964minus 119866 preserves weighted
colimits while the weighted limit functor lim119964minus 119865 carries weighted colimits to weighted limits
166
Dually the colimit of 119866 isin ℳ119964opweighted by 119882 isin 119985119964 is computed by the functor tensor
product
colim119964119882119866 ≔ 1114009
119886isin119964119882(119886) otimes 119866(119886) ≔ coeq
⎛⎜⎜⎜⎜⎝ ∐119886119887isin119964
(119882(119886) times 119964(119886 119887)) otimes 119866(119887) ∐119886isin119964
119882(119886) otimes 119866(119886)⎞⎟⎟⎟⎟⎠
(716)where the coproduct and coequalizer should be interpreted as conical colimits One of the maps in thecoequalizer diagram is induced by the action119964(119886 119887) otimes 119866(119887) rarr 119866(119886) of119964(119886 119887) on the contravariant119985-functor 119866 and the natural isomorphism (119882(119886) times 119964(119886 119887)) otimes 119866(119887) cong 119882(119886) otimes (119964(119886 119887) otimes 119866(119887)) ofLemma 712 the other uses the covariant action of119964(119886 119887) on119882 as before
717 Definition (weighed limits and colimits the universal property) The limit lim119964119882 119865 of the dia-
gram 119865 isin ℳ119964 weighted by119882 isin 119985119964 and the colimit colim119964119882119866 of119866 isin ℳ119964op
weighted by119882 isin 119985119964
are characterized by the universal properties
ℳ(119883 lim119964119882 119865) cong 119985119964(119882ℳ(119883 119865)) and ℳ(colim119964
119882119866119883) cong 119985119964(119882ℳ(119866119883)) (718)
each of these defining an isomorphism between objects of119985
When the indexing category 119964 is clear from context as is typically the case we frequently dropit from the notation for the weighted limit and weighted colimit We now argue that these threedefinitions characterize the same objects Along the way we obtain results of interest in their ownright that we record separately
719 Lemma The category119985 admits all weighted limits as defined by the formula of (715) satisfying thenatural isomorphism of (718) Explicitly for a weight 119882∶ 119964 rarr 119985 and a diagram 119865∶ 119964 rarr 119985 theweighted limit
lim119964119882 119865 ≔ 119985119964(119882 119865)
is the119985-object of119985-natural transformations from119882 to 119881
Proof The119985-functor119985(120793 minus) ∶ 119985 rarr 119985 represented by the monoidal unit is naturally isomor-phic to the identity functor So taking119883 = 120793 in the universal property of (718) in the case where thediagram 119865 isin 119985119964 is valued in the119985-category119985 we have
lim119964119882 119865 cong 119985119964(119882 119865)
Simultaneously the formula (715) computes the119985-object119985119964(119882 119865) of119985-natural transformationsfrom119882 to 119865 defined in Definition A38
The119985-object of119985-natural transformations satisfies the natural isomorphism119985(119881119985119964(119882 119865)) cong119985119964(119882119985(119881 119865)) for any 119881 isin 119985 Applying the observation that 119882-weighted limits of 119985-valuedfunctors are 119985-objects of natural transformations to the functor ℳ(119883 119865minus) and ℳ(119866minus119883) in thecase of 119865 isin ℳ119964 and 119866 isin ℳ119964op
we may re-express the natural isomorphism (718) as
7110 Corollary The weighted limits and weighted colimits of (718) are representably defined as weightedlimits in119985 for119882 isin 119985119964 and 119865 isin ℳ119964 and 119866 isin ℳ119964op
the weighted limit and colimit are characterizedby isomorphisms
ℳ(119883 lim119882 119865) cong lim119882ℳ(119883 119865) and ℳ(colim119882119866119883) cong lim119882ℳ(119866119883) (7111)
natural in 119883 in119985
167
We now unify the Definitions 713 714 and 717
7112 Proposition When the limits and colimits of (715) and (716) exist they define objects satisfyingthe universal properties (718) or equivalently (7111) and bifunctors satisfying the axioms of Definition 713
Proof The proofs are dual so we confine our attention to the limit case The general case of theimplication Definition 714 rArr 717 mdash for either weighted limits or weighted colimits mdash is a directconsequence of the special case of this implication for weighted limits valued in ℳ = 119985 proven asLemma 719 and Corollary 7110 The limits of (715) inℳ are also defined representably in terms ofthe analogous limits in119985 So the objected defined by (715) represents the119985-functor lim119882ℳ(minus 119865)that defines the weighted limit lim119882 119865
It remains to prove that the weighted limits of Definitions 714 and 717 satisfy the axioms ofDefinition 713 In the case of a119985-valued diagram 119865 isin 119985119964 axiom (i) is the119985-Yoneda lemma
119985119964(119964(119886 minus) 119865) cong 119865(119886)proven in Theorem A311 Once again the general case for 119865 isin ℳ119964 follows from the special casefor 119985-valued diagrams for to demonstrate an isomorphism lim119964(119886minus) 119865 cong 119865(119886) in ℳ it suffices todemonstrate an isomorphismℳ(119883 lim119964(119886minus) 119865) cong ℳ(119883 119865(119886)) in119985 for all 119883 isin ℳ and have such anatural isomorphism by applying (7111) and the observation just made to the functor ℳ(119883 119865minus) isin119985119964
For the axiom (ii) consider a diagram 119870∶ 119973op rarr 119985119964 of weights and a weight 119881 isin 119985119973 so thatcolim119973
119881 119870 cong 119882 For any 119865 isin ℳ119964 we will show that the 119985-functor lim119964minus 119865∶ (119985119964)op rarr ℳ carries
the 119881-weighted colimit of 119870 to the 119881-weighted limit of the composite diagram lim119964119870 119865∶ 119973 rarrℳ
The universal property (718) applied first to the colim119973119881 119870-weighted limit of the diagram 119865 and
the object 119883 and then to the 119881-weighted colimit of the diagram 119870 and the objectℳ(119883 119865) suppliesisomorphisms
ℳ(119883 lim119964colim119973119881 119870
119865) cong 119985119964(colim119973119881 119870ℳ(119883 119865)) cong 119985119973(119881119985119964(119870ℳ(119883 119865)))
Applying (718) twice more first for the weights 119870119895 for each 119895 isin 119973 and then for the weight119881 and thediagram lim119964
119870 119865∶ 119973 rarrℳ we have
cong 119985119973(119881ℳ(119883 lim119964119870 119865)) cong ℳ(119883 lim119973
119881 lim119964119870 119865)
By the Yoneda lemma this proves that
lim119964colim119973119881 119870
119865 cong lim119973119881 lim119964
119867 119865
ie that the weighted limit functor lim119964minus 119865 is carries a weighted colimit of weights to the analogous
weighted limit of weights
7113 Remark (for unenriched indexing categories) When the indexing category is unenriched thelimit and colimit formulas from Definition 714 simplify
lim119964119882 119865 cong eq
⎛⎜⎜⎜⎜⎝ prod119886isin119964
119865(119886)119882(119886) prod119964(119886119887)
119865(119887)119882(119886)⎞⎟⎟⎟⎟⎠
colim119964119882119866 cong coeq
⎛⎜⎜⎜⎜⎝ ∐119964(119886119887)
(119882(119886) otimes 119866(119887) ∐119886isin119964
119882(119886) otimes 119866(119886)⎞⎟⎟⎟⎟⎠
168
and in fact it suffices to consider only non-identity arrows or even just atomic arrows
7114 Definition (conical limits and colimits) The unit for the cartesian product defines a terminalobject 120793 isin 119985 The constant diagram at the terminal object then defines a terminal object 120793 isin 119985119964 Alimit weighted by the terminal weight is called a conical limit and a colimit weighted by the terminalweight is called a conical colimit It is common to use the simplified notation lim 119865 ≔ lim119964
120793 119865 andcolim119866 ≔ colim119964
120793 119866Conical limits and colimits satisfy the defining universal properties
ℳ(119883 lim 119865) cong 119985119964(120793ℳ(119883 119865)) and ℳ(colim119866119883) cong 119985119964(120793ℳ(119866119883))which say that lim 119865 and colim119866 represent the functors of119985-enriched conical cones over 119865 or under119866 respectively
We can now properly understand the formulae for weighted limits and colimits given in Definition714 In particular these formulae give criteria under which weighted limits or colimits are guaranteedto exist
7115 Corollary Ifℳ is a119985-enriched category that admits cotensors and conical limits of all unenricheddiagram shapes then ℳ admits all weighted limits Dually if ℳ admits tensors and conical colimits of allunenriched diagram shapes thenℳ admits all weighted colimits
7116 Remark Ifℳ is a119985-category whose underlying unenriched category admits all small limitsthen if ℳ admits cotensors and tensors over 119985 then ℳ admits all weighted limits Via the Yonedalemma the presence of tensors suffices to internalize the isomorphism of sets expressing the unen-riched universal property of limits to an isomorphism in 119985 that expresses the universal property ofconical limits See Exercise 71i
7117 Example (commas) The comma infin-category is the limit in the infin-cosmos 119974 of the diagram⟓rarr 119974 given by the cospan
119862 119860 119861119892 119891
weighted by the diagram ⟓rarr 119982119982119890119905 given by the cospan
120793 120794 1207931 0
Under the simplification of Remark 7113 the formula for the weighted limit reduces to the equalizer
eq
⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝
119860120794
119862 times 119860120794 times 119861 119860 times 119860119862 times 119861
(11990111199010)120587
120587 119892times119891
⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠
which computes the pullback of (332) The universal property (718) says that functors119883 rarr Hom119860(119891 119892)in119974 correspond to simplicial natural transformations the data of which is given by the three dashedvertical maps that fit into two commutative squares
120793 120794 120793
Fun(119883 119862) Fun(119883119860) Fun(119883 119861)
1
119888 120572
0
119887
119892lowast 119891lowast
169
7118 Example (Bousfield-Kan homotopy limits) In their classic book on homotopy limits and col-imits [11] Bousfield and Kan define the homotopy limit of a diagram indexed by a 1-category 119860 andvalued in a Kan-complex enriched categoryℳ to be the limit weighted by the functor
119860 119982119982119890119905
119886 119860119886which carries each object 119886 isin 119860 the the nerve of the slice category over 119886
7119 Example (Kan extensions as weighted colimits) The usual colimit or limit formula that com-putes the value of a pointwise left or right Kan extension of an unenriched functor 119865∶ 119862 rarr 119864 along119870∶ 119862 rarr 119863 at an object 119889 isin 119863 can be succinctly expressed by the weighted colimit or weighted limit
lan119870 119865(119889) ∶= colim119863(119870minus119889) 119865 and ran119870 119865(119889) ∶= lim119863(119889119870minus) 119865
We conclude with a few results from the general theory of weighted limits and colimits Immedi-ately from their defining universal properties it can be verified that
7120 Lemma (weighted limits of restricted diagrams) Suppose given a119985-functor119870∶ 119964 rarr ℬ a weight119882∶ 119964 rarr 119985 and diagrams 119865∶ ℬ rarr ℳ and 119866∶ ℬop rarrℳ Then the119882-weighted limit or colimit of therestricted diagram is isomorphic to the lan119870119882-weighted limit or colimit of the original diagram
lim119964119882 (119865 ∘ 119870) cong limℬ
lan119870119882 119865 and colim119964119882 (119866 ∘ 119870) cong colimℬ
lan119870119882119866
Proof Exercise 71iii
An enriched adjunction is comprised of a pair of 119985-functors 119865∶ ℬ rarr 119964 and 119880∶ 119964 rarr ℬtogether with a family of isomoprphisms119964(119865119887 119886) cong ℬ(119887119880119886) that are119985-natural in both variablessee Definition A315 The usual Yoneda-style argument enriches to show
7121 Proposition (weighted RAPLLAPC) A119985-enriched right adjoint functor119880∶ 119964 rarr ℬ preservesall weighted limits that exist in119964 while itrsquos119985-enriched left adjoint 119865∶ ℬ rarr 119964 preserves all weighted colimitsthat exist in ℬ
Proof Exercise 71iv
By the axioms ofDefinition 121infin-cosmoiwill admit a large class of simplicially-enrichedweight-ed limits built from the simplicial cotensors and conical simplicial limits enumerated in 121(i) Inpractice infin-cosmoi often arise as subcategories (of ldquofibrant objectsrdquo) in a larger category that is alsoadmits simplicially-enriched weighted colimits which can then be reflected back into theinfin-cosmosto defined weighted bicolimits This is a story for much later so we will confine our attention to thecase of weighted limits for the rest of this chapter
Exercises
71i Exercise Suppose ℳ is a tensored and 119985-enriched category whose underlying unenrichedcategory admits limits of all unenriched diagram shapes Show that ℳ admits conical limits of allunenriched diagram shapes proving the extension of Corollary 7115 described in Remark 7116
71ii Exercise Taking the base for enrichment119985 to be 119982119890119905 compute the following weighted limitsof a simplicial set 119883 regarded as a diagram in 119982119890119905120491
op weighted by
170
(i) the standard 119899-simplex Δ[119899] isin 119982119890119905120491op
(ii) the spine of the 119899-simplex the simplicial subset Γ[119899] Δ[119899] obtained by gluing together the
119899 edges from 119894 to 119894 + 1 into a composable path(iii) the 119899-simplex boundary 120597Δ[119899] isin 119982119890119905120491
opsup2
71iii Exercise Prove Lemma 7120
71iv Exercise Prove Proposition 7121
72 Flexible weighted limits and the collage construction
Our aim in this section is to introduce a special class of 119982119982119890119905-valued weights whose associatedweighted limit notions are homotopically well-behaved Borrowing a term from 2-category theorywe refer to these weights as flexible All of the limits enumerated in 121(i) are flexible limits In factwe will prove that infin-cosmoi admit all flexible weighted limits because these can be built out of thelimits enumerated in 121(i) In sect74 we will use this observation to help us verify the limit axiom fornewly constructedinfin-cosmoi in a more systematic way
In this section we will characterize the class of flexible weights as precisely those whose associatedcollages define relative simplicial computads which will allow us to readily produce examples Insect73 we will establish the homotopical properties of flexible weighted limits and make precise therelationships between this class of the limits and limits assumed present in any infin-cosmos by axiom121(i)
721 Definition (flexible weights and projective cell complexes) Let119964 be a simplicial categorybull A simplicial natural transformation of the form
120597Δ[119899] times 119964(119886 minus) Δ[119899] times 119964(119886 minus)is called a projective 119899-cell at 119886 isin 119964
bull A natural transformation 120572∶ 119881 119882 in 119982119982119890119905119964 that can be expressed as a countable compositeof pushouts of coproducts of projective cells is called a projective cell complex
bull A weight119882 isin 119982119982119890119905119964 is flexible just when empty 119882 is a projective cell complex
722 Remark (on generalized projective cells) Since any monomorphism of simplicial sets 119880 119881 can be decomposed as a sequential composite of pushouts of coproducts of boundary inclusions120597Δ[119899] Δ[119899] the class of projective cell complexes may be also be described as the class of maps in119982119982119890119905119964 that can be expressed as a countable composite of pushouts of coproducts of monomorphisms119880 times119964(119886 minus) 119881 times119964(119886 minus) for some 119886 isin 119964
723 Example Since any simplicial set can be decomposed as a sequential composite of pushouts ofcoproducts of boundary inclusions 120597Δ[119899] Δ[119899] simplicial cotensors are flexible weights
724 Example Conical products also define flexible weighted limits built by attaching one projec-tive 0-cell for each object in the indexing set
725 Non-Example Conical limits indexed by any 1-category that contains non-identity arrows arenot flexible because the legs of a conical cone over the domain and codomain of each arrow in thediagram are required to define a strictly commutative triangle of 0-arrows The specifications for a
sup2The limit of a simplicial object weighted by 120597Δ[119899] is called the 119899th-matching object see Appendix C
171
flexible weight allow us to free attach 119899-arrows of any dimension but do not provide a mechanism fordemanding strict commutativity of any diagram of 119899-arrows mdash only commutativity up to the presenceof a higher cell
726 Digression (on flexible limits in 2-category theory) Simplicial limits weighted by flexibleweights should be thought of as analogous to flexible 2-limits ie 2-limits built out of products insert-ers equifiers and retracts (splittings of idempotents) [6] More precisely simplicial limits weightedby flexible weights are analogous to the PIE limits those built just from products inserters and equi-fiers but we choose to adopt the moniker from the slight larger class of weights because we find itto be more evocative The PIE limits also include iso-inserters descent objects comma objects andEilenberg-Moore objects as well as all pseudo lax and oplax limits Many important 2-categoriessuch as the 2-category of accessible categories and accessible functors fail to admit all 2-categoricallimits but do admit all PIE-limits [40]
The weights for flexible limits are the cofibrant objects in a model structure on the diagram2-category 119966119886119905119964 that is enriched over the folk model structure on 119966119886119905 the PIE weights are exactlythe cellular cofibrant objects Correspondingly the projective cell complexes of Definition 721 areexactly the cellular cofibrations in the projective model structure on 119982119982119890119905119964
Recall that a weight is intended to describe the ldquoshaperdquo of cones over diagrams indexed by a par-ticular category In the case a weight 119882∶ 119964 rarr 119982119982119890119905 valued in simplicial sets we can describe theshape of119882-cones directly as a simplicial category called the collage of the weight
727 Definition (collage construction) The collage of a weight119882∶ 119964 rarr 119982119982119890119905 is a simplicial cat-egory coll119882 which contains 119964 as a full subcategory together with one additional object ⊤ whoseendomorphism space is the point The mapping spaces from any object 119886 isin 119964 to ⊤ are empty whilethe mapping spaces from ⊤ to the image of119964 coll119882 are defined by
coll119882(⊤ 119886) ≔ 119882(119886)with the action maps119964(119886 119887) times119882(119886) rarr 119882(119887) from the simplicial functor119882 used to define compo-sition This defines a simplicial category together with a canonical inclusion 120793 +119964 coll119882 that isbijective on objects and fully faithful on 120793 and119964 separately
728 Proposition (collage adjunction)(i) The collage construction defines a fully faithful functor
119982119982119890119905119964 120793+119964119982119982119890119905-119966119886119905coll
from the category of 119964-indexed weights to the category of simplicial categories under 120793 + 119964 whoseessential image is comprised of those ⟨119890 119865⟩ ∶ 120793 + 119964 rarr ℰ that are bijective on objects fully faithfulwhen restricted to 120793 and119964 and have the property that there are no arrows in ℰ from the image of 119865to 119890
(ii) The collage functor admits a right adjoint
119982119982119890119905119964 120793+119964119982119982119890119905-119966119886119905
coll
perpwgt
which carries a pair ⟨119890 119865⟩ ∶ 120793 +119964 rarr ℰ to the weight ℰ(119890 119865minus) ∶ 119964 rarr 119982119982119890119905172
Proof The construction of the collage functor is straightforward and left to the reader Thecharacterization of its essential image follows from the observation that to define a simplicial functor119882∶ 119964 rarr 119982119982119890119905 requires no more and no less thanbull the specification of simplicial sets119882(119886) for each 119886 isin 119964 andbull and the specification of simplicial maps119964(119886 119887) times 119882(119886) rarr 119882(119887) for each 119886 119887 isin 119964
so that this action is associative in sense that the diagram
119964(119887 119888) times 119964(119886 119887) times 119882(119886) 119964(119886 119888) times 119882(119886)
119964(119887 119888) times 119882(119887) 119882(119888)
∘
commutes This is the same as what is required to extend 120793 +119964 to a simplicial category in which allof the additional maps start at ⊤ and end at an object in119964
The adjunction asserts that simplicial functors
120793 +119964
coll119882 ℰ⟨119890119865⟩
119866
from coll119882 to ℰ under 120793 + 119964 stand in natural bijective correspondence with simplicial naturaltransformations 120574∶ 119882 rArr ℰ(119890 119865minus) Since the inclusion 1 + 119964 coll119882 is bijective on objectsand full on most homs the data of the simplicial functor requires only the specification of the mapscoll119882(⊤ 119886) = 119882(119886) rarr ℰ(119890 119865119886) These define the components of the simplicial natural transforma-tion 120574 and functoriality of 119866 corresponds to naturality of 120574
The collage adjunction has a useful and important interpretation
729 Corollary The collage of a weight119882∶ 119964 rarr 119982119982119890119905 realizes the shape of119882-weighted cones in thesense that simplicial functors 119866∶ coll119882 rarr ℰ with domain coll119882 stand in bijection to 119882-cones over thediagram 119866|119964 with summit 119866(⊤)
On account of Lemma 7120 wersquoll be interested in computing left Kan extensions of weightsencoded as collages
7210 Lemma For anyweight119882∶ 119964 rarr 119982119982119890119905 and simplicial functor119870∶ 119964 rarr ℬ the pushout of simplicialcategories
120793 +119964 120793 + ℬ
coll119882 coll(lan119870119882)
120793+119870
computes the collage of the weight lan119870119882∶ ℬ rarr 119982119982119890119905
Proof By the defining universal property a simplicial functor out of the pushout is given by apair of functors ⟨119890 119865⟩ ∶ 120793 + ℬ rarr ℰ and 119866∶ coll119882 rarr ℰ so that
120793 +119964
coll119882 ℰ⟨119890119865119870⟩
119866
173
By Corollary 729 this data defines a 119882-cone with summit 119890 over the diagram 119865119870∶ 119964 rarr ℰ ByLemma 7120 such data equivalently describes a lan119870119882-cone with summit 119890 over the diagram 119865∶ ℬ rarrℰ Applying Corollary 729 again we conclude that the pushout is given by the simplicial categorycoll(lan119870119882) as claimed
In analogy with Corollary 435 we can encode simplicial 119882-weighted limits as a right Kan ex-tension from the indexing simplicial category to the simplicial category that describes the shape of119882-cones
7211 Lemma For any simplicial functor 119865∶ 119964 rarr ℰ and any weight119882∶ 119964 rarr 119982119982119890119905 the weighted limitlim119882 119865 exists if and only if the pointwise right Kan extension of 119865 along 119964 coll119882 exists in which caselan 119865(⊤) cong lim119882 119865
Proof Since119964 coll119882 is fully faithful pointwise right Kan extensionsmay be chosen to definegenuine extensions
119964
coll119882 ℰ119865
ran 119865ByCorollary 729 this data defines a119882-cone over119865with summit ran 119865(⊤) thatwe denote by120582∶ 119882 rArrℰ(ran 119865(⊤) 119865(minus))
By Corollary 729 again the data of a cone over the right Kan extension diagram displayed below-left
119964 119964
coll119882 ℰ coll119882 ℰ119865 = 119865
119866
uArr120572
119866existuArr120578
ran 119865
defines a119882-cone119882 ℰ(119866(⊤) 119866|119964(minus)) ℰ(119866(⊤) 119865(minus))
120574 120572lowast
over 119865 The universal property of the right Kan extension depicted above-right says this cone factorsuniquely through the 119882-cone 120582 along a map 120578⊤ ∶ 119866(⊤) rarr ran 119865(⊤) Thus the right Kan extensionof 119865 along 119964 coll119882 equips the resulting 119882-cone with the universal property of the 119882-weightedlimit
A particularly convenient aspect of the collage construction is that it allows us to detect the classof flexible weights
7212 Theorem (flexible weights and collages) A natural transformation 120572∶ 119881 119882 between weightsin119982119982119890119905119964 is a projective cell complex if and only if coll(120572) ∶ coll119881 coll119882 is a relative simplicial computadIn particular119882 is a flexible weight if and only if 120793 +119964 coll119882 is a relative simplicial computad
Proof If 120572∶ 119881 119882 is a projective cell complex then it can be presented as a countable com-posite of pushouts of coproducts of projective cells of varying dimensions indexed by the objects119886 isin 119964 Since the collage construction is a left adjoint it preserves these colimits and hence themap coll(120572) ∶ coll119881 coll119882 as a transfinite composite of pushouts of coproducts of simplicial func-tors coll(120597Δ[119899] times 119964(119886 minus)) coll(Δ[119899] times 119964(119886 minus)) in 120793+119964119982119982119890119905-119966119886119905 This composite colimit dia-gram is connected mdash note collempty = 120793 + 119964 so this cell complex presentation is also preserved by the
174
forgetful functor 120793+119964119982119982119890119905-119966119886119905 rarr 119982119982119890119905-119966119886119905 and the simplicial functor coll(120572) ∶ coll119881 coll119882can be understood as a transfinite composite of pushouts of coproducts of coll(120597Δ[119899] times 119964(119886 minus)) coll(Δ[119899] times 119964(119886 minus)) in 119982119982119890119905-119966119886119905
This is advantageous because there is a pushout square in 119982119982119890119905-119966119886119905
120794[120597Δ[119899]] coll(120597Δ[119899] times 119964(119886 minus)) 120597Δ[119899] 120597Δ[119899] times 119964(119886 119886)
120794[Δ[119899]] coll(Δ[119899] times 119964(119886 minus)) Δ[119899]] Δ[119899] times 119964(119886 119886)
id119886
id119886
(7213)
whose horizontals sends the two objects minus and+ of the simplicial computads defined in Example 615to ⊤ and 119886 and act on the non-trivial hom-spaces via the inclusions whose component in 119964(119886 119886) isconstant at the identity element at 119886 The fact that coll(120597Δ[119899] times 119964(119886 minus)) coll(Δ[119899] times 119964(119886 minus)) isa pushout of 120794[120597Δ[119899]] 120794[Δ[119899]] can be verified by transposing across the adjunction of Proposition728 and applying the Yoneda lemma From this we see that coll(120572) ∶ coll119881 coll119882 is a transfinitecomposite of pushouts of coproducts of simplicial functors 120794[120597Δ[119899]] 120794[Δ[119899]] which proves thatthis map is a relative simplicial computad
Conversely if coll(120572) ∶ coll119881 coll119882 is a relative simplicial computad then it can be presentedas a countable composite of pushouts of coproducts of simplicial functors 120794[120597Δ[119899]] 120794[Δ[119899]]since this inclusion is bijective on objects the inclusion empty 120793 is not needed Since the only arrowsof coll119882 that are not present in coll119881 have domain ⊤ and codomain 119886 isin 119964 the characterization ofthe essential image of the collage functor of Proposition 728(i) allows us to identify each stage of thecountable composite
coll119881 coll(1198821) ⋯ coll(119882 119894) coll(119882 119894+1) ⋯ coll119882
as the collage of some weight 119882 119894 ∶ 119964 rarr 119982119982119890119905 Each attaching map 120794[120597Δ[119899]] rarr coll119882 119894 in the cellcomplex presentation acts on objects by mapping minus and + to ⊤ and 119886 for some 119886 isin 119964 and hencefactors through the top horizontal of the pushout square (7213) Hence the inclusion coll(119882 119894) coll(119882 119894+1) is a pushout of a coproduct of the maps coll(120597Δ[119899] times 119964(119886 minus)) coll(Δ[119899] times 119964(119886 minus))one for each cell 120794[120597Δ[119899]] 120794[Δ[119899]] whose attaching map sends + to 119886 isin coll(119882 119894) As the collagefunctor is fully faithful we have now expressed coll(120572) ∶ coll119881 coll119882 as a countable compositeof pushouts of coproducts of simplicial functors coll(120597Δ[119899] times 119964(119886 minus)) coll(Δ[119899] times 119964(119886 minus)) Afully faithful functor that preserves colimits also reflects them so in this way we see that 120572∶ 119881 119882is a countable composite of pushouts of coproducts of projective cells proving that it is a projectivecell complex as claimed
As the first of many applications we introduce the weights for pseudo limits by constructing theircollages and observe immediately that this class of weights is flexible
7214 Definition (weights for pseudo limits) For any simplicial set 119883 the coherent realization ofthe canonical inclusion 120793 + 119883 119883◁ defines a collage 120793 + ℭ119883 ℭ(119883◁) By Lemma 635 120793 +ℭ119883 ℭ(119883◁) is a simplicial subcomputad inclusion and hence by Lemma 6112 a relative simplicialcomputad Thus Theorem 7212 tells us that the collage 120793+ℭ119883 ℭ(119883◁) encodes a flexible weight119882119883 ∶ ℭ119883 rarr 119982119982119890119905 which we call the weight for the pseudo limit of a homotopy coherent diagramof shape 119883 The 119882119883-weighted limit of a homotopy coherent diagram of shape 119883 is then referred toas the pseudo weighted limit of that diagram
175
Since the left adjoint of the collage adjunction is fully faithful its unit is an isomorphism and thispermits us to define the weight119882119883 explicitly for a vertex 119909 isin 119883
119882119883(119909) ≔ ℭ(119883◁)(⊤ 119909)
We call119882119883 the weight for a ldquopseudordquo limit because we anticipate considering homtopy cohererentdiagrams valued in Kan-complex enriched categories in which all arrows in positive dimension areautomatically invertible By Corollary 733 119882119883-weighted limits also exist for diagrams valued ininfin-cosmoi which are only quasi-categorically enriched In such contexts it would bemore appropriateto refer to119882119883 as the weight for lax limits since in that context the 1-arrows of ℭ(119883◁) will likely mapto non-invertible morphisms
Exercises
72i Exercise Compute the collage of the weight 119880∶ 120793 rarr 119982119982119890119905 and use Theorem 7212 to give asecond proof that simplicial cotensors are flexible weighted limits Compare this argument with thatgiven in Example 723
72ii Exercise The inclusion into the join 120793+119883 119883◁ cong 120793⋆119883 is bijective on vertices The com-plement of the image contains a non-degenerate (119899 + 1)-simplex for each non-degenerate 119899-simplexof 119883 whose initial vertex is ⊤ and whose 0th face is isomorphic to the image of that simplex UseTheorem 6310 to describe the atomic arrows of the simplicial computad ℭ(119883◁) that are not in theimage of 120793 + ℭ119883
73 Homotopical properties of flexible weighed limits
In a119985-model categoryℳ the fibrant objects are closed under weighted limits whose weights areprojective cofibrant For instance the fibrant objects in a 119966119886119905-enriched model structure are closedunder flexible weighted limits [33 54] in the sense of [6] Specializing this argument to the case ofinfin-cosmoi we obtain the following result
731 Proposition (flexible weights are homotopical) Let119882∶ 119964 rarr 119982119982119890119905 be a flexible weight and let119974 be aninfin-cosmos
(i) The weighted limit lim119964119882 119865 of any diagram 119865∶ 119964 rarr 119974 may be expressed as a countable inverse limit
of pullbacks of products of isofibrations
119865119886Δ[119899] 119865119886120597Δ[119899] (732)
one for each projective 119899-cell at 119886 in the given projective cell complex presentation of119882(ii) If 119881 119882 isin 119982119982119890119905119964 is a projective cell complex between flexible weights then for any diagram
119865∶ 119964 rarr 119974 the induced map between weighted limits
lim119882 119865 ↠ lim119881 119865is an isofibration
(iii) If 120572∶ 119865 rArr 119866 is a simplicial natural transformation between a pair of diagrams 119865119866∶ 119964 119974 whosecomponents 120572119886 ∶ 119865119886 ⥲ 119866119886 are equivalences then the induced map
lim119882 119865 lim119882119866sim120572
is an equivalence
176
Proof To begin observe that the axioms of Definition 713 imply that the limit of 119865 weightedby the weight 119880 times 119964(119886 minus) for 119880 isin 119982119982119890119905 and 119886 isin 119964 is the cotensor 119865119886119880 Consequently the mapof weighted limits induced by the projective 119899-cell at 119886 is the isofibration (732) By definition anyflexible weight is built as a countable composite of pushouts of coproducts of these projective cellsand the weighted limit functor lim119964
minus 119865 carries each of these conical colimits to the correspondinglimit notion So it follows that lim119964
119882 119865 may be expressed as a countable inverse limit of pullbacks ofproducts of the maps (732) This proves 92
The same argument proves (ii) By definition a relative cell complex119881 119882 is built as a countablecomposite of pushouts of coproducts of these projective cells and the weighted limit functor lim119964
minus 119865carries each of these conical colimits to the corresponding limit notion So it follows that lim119964
119882 119865 isthe limit of a countable tower of isofibrations whose base in lim119964
119881 119865 where each of these isofibrationsis the pullback of products of the maps (732) appearing in the projective cell complex decompositionof119881 119882 As products pullbacks and limits of towers of isofibrations are isofibrations (ii) follows
In 92 we have decomposed each weighted limit lim119964119882 119865 as the limit of a tower of isofibrations
in which each of these isofibrations is the pullback of a product of the isofibrations (732) We argueinductively that if 120572∶ 119865 rArr 119866 is a componentwise equivalence then the map induced between thetowers of isofibrations for 119865 and for 119866 by the projective cell complex presentation of 119882 is a level-wise equivalence It follows from the standard argument in abstract homotopy theory reviewed inAppendix C that the inverse limit is then an equivalence proving (iii)
The bottom of the tower of isofibrations is limempty 119865 cong 1 cong limempty119866 which is certainly an equiva-lence For the inductive step observe that upon taking the map of weighted limits induced by eachprojective 119899-cell at 119886 in119882 we obtain a commutative square
119865119886Δ[119899] 119866119886Δ[119899]
119865119886120597Δ[119899] 119866119886120597Δ[119899]
sim120572Δ[119899]119886
sim120572120597Δ[119899]119886
defining a pointwise equivalence between the isofibrations the simplicial cotensor as cosmologi-cal functor preserves equivalences Now the product of squares of this form gives a commutativesquare whose horizontals are isofibrations and whose verticals are equivalences A pullback of thissquare forms the next layer in the tower of isofibrations by the inductive hypothesis the map be-tween the codomains of the pulled back isofibrations is already known to be an equivalence Nowthe equivalence-invariance of pullbacks of isofibrations established in Appendix C completes theproof
Immediately from the construction of Proposition 73192
733 Corollary (infin-cosmoi admit all flexible weighted limits) infin-cosmoi admit all flexible weightedlimits and cosmological functors preserve them
Our aim is now to describe a converse of sorts to Proposition 73192 which proves that the flexibleweighted limit of any diagram in an infin-cosmos can be constructed out of the limits of diagrams ofisofibrations axiomatized in 121(i) Over a series of lemmas we will construct each of the limitslisted there as instances of flexible weighted limits It will follow that any quasi-categorically enrichedcategory equipped with a class of representably-defined isofibrations that possesses flexible weighted
177
limits will admit all of the simplicial limits of 121(i) This will help us identify new examples ofinfin-cosmoi
To start simplicial cotensors are flexible weighted limits For any simplicial set 119880 the collage of119880∶ 120793 rarr 119982119982119890119905 is the simplicial computad 120794[119880] As 120793 + 120793 120794[119880] is a simplicial subcomputadinclusion Theorem 7212 tells us that 119880∶ 120793 rarr 119982119982119890119905 is a flexible weight this solves Exercise 72i
This leaves only the conical limits The weights for products are easily seen to be flexible directlyfrom Definition 721 However the weights for conical pullbacks or limits of towers of isofibrationsare not flexible because the definition of a cone over either diagram shape imposes composition rela-tions on 0-arrows
734 Example (the collage of the conical pullback) Let ⟓ denote the 1-category 119888 rarr 119886 larr 119887 Itscollage is the 1-category with four objects and five non-identity 0-arrows as displayed
⊤ 119887
119888 119886regarded as a constant simplicial category as in Example 614 Because the square commutes thiscategory is not free and hence does not define a simplicial computad though the subcategory 120793+ ⟓is free and hence is a simplicial computad Lemma 6111 tells us that the inclusion is not a relativesimplicial computad and so by Theorem 7212 the weight for the conical pullback is not flexible
Our strategy is to modify the weights for pullbacks and for limits of countable towers so that eachcomposition equation involved in defining cones over such diagrams is replaced by the insertion ofan ldquoinvertiblerdquo arrow of one dimension up where we must also take care to define this ldquoinvertibilityrdquowithout specifying any equations between arrows in the next dimension We have a device for speci-fying just this sort of isomorphism recall from Exercise 11iv(i) a diagram 120128 rarr Fun(119860 119861) specifies aldquohomotopy coherent isomorphismrdquo between a pair of 0-arrows 119891 and 119892 from 119860 to 119861 given bybull a pair of 1-arrows 120572∶ 119891 rarr 119892 and 120573∶ 119892 rarr 119891bull a pair of 2-arrows
119892 119891
119891 119891 119892 119892
120573Φ
120572Ψ
120572 120573
bull a pair of 3-arrows whose outer faces are Φ andΨ and whose inner faces are degeneratebull etc
We now introduce the weight for pullback diagrams whose cone shapes are given by squares in-habited by a homotopy coherent isomorphism
735 Definition (iso-commas) The iso-comma object 119862 ⨰119860119861 of a cospan
119862 119860 119861119892 119891
in a simplicially-enriched and cotensored category ℳ is the limit weighted by a weight 119882⨰ ∶ ⟓rarr119982119982119890119905 defined by the cospan
120793 120128 1207931 0
178
Under the simplification of Remark 7113 the formula for the weighted limit reduces to the equalizer
eq
⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝
119860120128
119862 times 119860120128 times 119861 119860 times 119860119862 times 119861
(11990211199020)120587
120587 119892times119891
⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠
where the maps (1199021 1199020) ∶ 119860120128 rarr 119860times119860 are defined by restricting along the endpoint inclusion 120793+120793 =120597120128 120128 In aninfin-cosmos this map is an isofibration and the equalizer defining the iso-comma objectis computed by the pullback
119862 ⨰119860119861 119860120128
119862 times 119861 119860 times 119860
(11990211199020)
(11990211199020)
119892times119891
(736)
737 Lemma Iso-comma objects are flexible weighted limits and in particular exist in anyinfin-cosmos
Proof Reprising the notation for the category ⟓ used in Example 734 the weight 119882⨰ is con-structed by the pushout
120597120128 times ⟓(119886 minus) ⟓ (119887 minus)⊔ ⟓(119888 minus)
120128 times ⟓(119886 minus) 119882⨰
where the attaching map picks out the two arrows in the cospan ⟓ As a projective cell complex119882⨰ is built from a project 0-cell at 119887 a projective 0-cell at 119888 and two projective 119896-cells at 119886 for each119896 gt 0 correspnoding to the non-degenerate simplices of 120128 As described by Remark 722 these maybe attached all at once In this way we see that119882⨰ is a flexible weight so Corollary 733 tells us thatiso-comma objects exist in anyinfin-cosmos a fact that is also evident from the pullback (736)
738 Remark In the homotopy 2-category of an infin-cosmos there is a canonical invertible 2-celldefining the iso-comma cone
119862 ⨰119860119861
119862 119861
119860
1199021 1199020120601cong
119892 119891
that has a weak universal property analogous to that of the comma cone presented in Proposition 336The proof like the proof of that result makes use of the fact that 119860120128 is the weak 120128-cotensor in thehomotopy 2-category The proof of this fact is somewhat delicate making use of marked simplicialsets as appeared already in the proof of Corollary 1116 which gives 1-cell induction
Our notation for iso-commas is deliberately similar to the usual notation for pullbacks In aninfin-cosmos iso-commas can be used to compute ldquohomotopy pullbacksrdquo of diagrams in which neithermap is an isofibration When at least one map of the cospan is an isofibration these constructions areequivalent
179
739 Lemma (iso-commas and pullbacks) In an infin-cosmos 119974 pullbacks and iso-commas of cospans inwhich at least one map is an isofibration are equivalent More precisely given a pullback square as below-leftand an iso-comma square as below-right
119875 119861 119862 ⨰119860119861 119861
119862 119860 119862 119860
119888
119887
119891 1199021 120601cong
1199020
119891
119892 119892
119875 ≃ 119862 ⨰119860119861 over 119862 and up to isomorphism over 119861
Proof Applying Lemma 1212 to the functor 119887 ∶ 119875 rarr 119861 we can replace the span (119888 119887) ∶ 119875 rarr119862times119861 by a span (119888119902 119901) ∶ 119875119887 ↠ 119862times119861whose legs are both isofibrations that is related via an equivalence119904 ∶ 119875 ⥲ 119875119887 that lies over 119862 on the nose and over 119861 up to isomorphism We will show that under thehypothesis that 119891 is an isofibration this new span is equivalent to the iso-comma span
To see this note that the factorization constructed in (1213) is in fact defined using an iso-commaconstructed via the pullback in the top square of the diagram below-left Since the map 119887 is itselfdefined by a pullback the bottom square of the diagram below-left is also a pullback defining theleft-hand pullback rectangle
119875119887 119861120128
119875 times 119861 119861 times 119861
119862 times 119861 119860 times 119861
(119902119901) (11990211199020)
119887times119861
119888times119861 119891times119861
119892times119861
119862 ⨰119860119861 119860 ⨰
119860119861 119860120128
119862 times 119861 119860 times 119861 119860 times 119860
(11990211199020)
(11990211199020)
(11990211199020)
119892times119861 119860times119891
Now the iso-comma is constructed by a similar pullback rectangle displayed above-right And because119891 is an isofibration Lemma 1210 tells us that the Leibniz tensor 1198940 1114023⋔ 119891∶ 119861120128 ⥲rarr 119860⨰
119860119861 of 1198940 ∶ 120793 120128
with 119891∶ 119861 ↠ 119860 is a trivial fibration This equivalence commutes with the projections to 119860 times 119861 andhence the maps (119888119902 119901) ∶ 119875119887 ↠ 119862 times 119861 and (1199021 1199020) ∶ 119862 ⨰119860 119861 ↠ 119862 times 119861 defined as pullbacks of anequivalence pair of isofibrations along 119892 times 119861 are equivalent as claimed
We now introduce a flexible weight diagrams given by a countable tower of 0-arrows whose coneshapes will have a homotopy coherent isomorphism in the triangle over each generating arrow in thediagram
7310 Definition (iso-towers) Recall the category120654 whose objects are natural numbers and whosemorphisms are freely generated by maps 120580119899119899+1 ∶ 119899 rarr 119899 + 1 for each 119899
180
The iso-tower of a diagram 119865∶ 120654op rarrℳ in a simplicially enriched and cotensored categoryℳis the limit weighted by the diagram119882larr ∶ 120654op rarr 119982119982119890119905 defined by the pushout
∐119899isin120654
120597120128 times 120654(minus 119899) ∐119899isin120654
120654(minus 119899)
∐119899isin120654
120128 times 120654(minus 119899) 119882larr
(id119899120580119899119899+1)
(7311)
in 119982119982119890119905120654op
By Definition 713(ii) in aninfin-cosmos the iso-tower of a diagram
119865 ≔ ⋯ 119865119899+1 119865119899 ⋯ 1198651 1198650119891119899+2119899+1 119891119899+1119899 119891119899119899minus1 11989121 11989110
is constructed by the pullback
lim119882larr 119865 prod119899isin120654
119865120128119899
prod119899isin120654
119865119899 prod119899isin120654
119865119899 times 119865119899
120588
120601
prod(11990211199020)
(119891119899+1119899id119865119899)
(7312)
The limit cone is generated by a 0-arrow120588119899 ∶ lim119882larr 119865 rarr 119865119899 for each119899 isin 120654 together with a homotopycoherent isomorphism 120601119899 in each triangle over a generating arrow 119865119899+1 rarr 119865119899 in the 120654op-indexeddiagram
7313 Lemma Iso-towers are flexible weighted limits and in particular exist in anyinfin-cosmos
Proof The weight 119882larr is a projective cell complex built by attaching one projective 0-cell ateach 119899 isin 120654 mdash forming the coproduct appearing in the upper right-hand corner of (7311) mdash andthen by attaching a projective 119896-cell at each 119899 isin 120654 for each non-degenerate 119896-simplex of 120128 Ratherthan attach each projective 119896-cell for fixed 119899 isin 120654 in sequence by Remark 722 these can all beattached at once by taking a single pushout of the ldquogeneralized projective cell at 119899rdquo defined by themap 120597120128 times 120654(minus 119899) 120128 times 120654(minus 119899) These are the maps appearing as the left-hand vertical of (7311)Now Corollary 733 or the formula (7312) make it clear that such objects exist in anyinfin-cosmos
7314 Lemma (iso-towers and inverse limits) In aninfin-cosmos 119974 the inverse limit of a countable towerof isofibrations is equivalent to the iso-pullback of that tower
Proof We will rearrange the formula (7312) to construct the iso-tower lim119882larr 119865 as an inverselimit of a countable tower of isofibrations 119875∶ 120654op rarr 119974 that is pointwise equivalent to the diagram119865∶ 120654op rarr119974 In the case where the diagram 119865 is also given by a tower of isofibrations
lim119875 cong ⋯ 119875119899+1 119875119899 ⋯ 1198751 1198750
lim 119865 cong ⋯ 119865119899+1 119865119899 ⋯ 1198651 1198650
sim
119901119899+2119899+1 119901119899+1119899
sim 119890119899
119901119899119899minus1
sim 119890119899
11990121 11990110
sim 1198901 sim 1198900
119891119899+2119899+1 119891119899+1119899 119891119899119899minus1 11989121 11989110
(7315)
181
the equivalence invariance of the inverse limit of a diagram of isofibrations will imply that the limitslim119882larr 119865 cong lim119875 and lim 119865 are equivalent as claimed
Theinfin-categories 119875119899 will be defined as conical limits of truncated versions of the diagram (7312)To start define 1198750 ≔ 1198650 and 1198900 to be the identity then define 1198751 11990110 and 1198901 via the pullback
1198751 1198651201280 1198650
1198651 1198650
11990110
sim1198901
sim 1199021
sim1199020
11989110
Note that 1198751 cong 1198651 ⨰11986501198650 computes the iso-comma objects of the cospan given by id1198650 and 11989110
Now define 1198752 11990121 and 1198902 using the composite pullback
1198752 bull 1198751 1198651201280 1198650
bull 1198651201281 1198651 1198650
1198652 1198651
11990121
sim1198902
sim
sim
sim
sim1198901
sim 1199021
sim1199020
sim
sim 1199021
sim1199020 11989110
11989121
Continuing inductively 119875119899 119901119899119899minus1 and 119890119899 are defined by appending the diagram
119865120128119899minus1 119865119899minus1
119865119899 119865119899minus1
sim1199020
sim 1199021
119891119899119899minus1
to the limit cone defining 119875119899minus1 and taking the limit of this composite diagramThere is one small problem with the construction just given it defines a diagram (7315) in which
each square commutes up to isomorphism mdash the isomorphism encoded by the map 119875119899 rarr 119865120128119899minus1 mdashnot on the nose But because the maps 119891119899+1119899 are isofibrations this is no problem The isomorphisminhabiting the square 119890011990110 cong 119891101198901 can be lifted along 11989110 to define a new map 119890prime1 ∶ 1198751 ⥲ 1198651isomorphic to 1198901 as observed in the proof of Theorem 147 this 119890prime1 is then also an equivalence sowe replace 1198901 with 119890prime1 and then continue inductively to lift away the isomorphisms in the square119890prime111990121 cong 119891211198902
Since inverse limits of towers of isofibrations are equivalence-invariant it follows that lim119875 ≃lim 119865 By construction lim119875 cong lim119882larr 119865 so it follows that lim119882larr 119865 ≃ lim 119865 which is what wewanted to show
Exercises
73i Exercise Verify mdash either directly from Definition 721 or by applying Theorem 7212 mdash thatconical products are flexible weighted limits
182
74 Moreinfin-cosmoi
Our aim in this section is to introduce further examples ofinfin-cosmoiOur first example is a special case of a more general result that will appear in Appendix E that we
nonetheless spell out in detail to illustrate the ideas involved in this sort of argument The walkingarrow category 120794 is an inverse Reedy category where the domain of the non-identity arrow is assignedldquodegree 1rdquo and the codomain is assigned ldquodegree zerordquo This Reedy structure motivates the definitionsin theinfin-cosmos of isofibrations that we now introduce
741 Proposition (infin-cosmoi of isofibrations) For anyinfin-cosmos119974 there is aninfin-cosmos119974120794 whose(i) objects are isofibrations 119901∶ 119864 ↠ 119861 in119974(ii) functor-spaces say from 119902 ∶ 119865 ↠ 119860 to 119901∶ 119864 ↠ 119861 are defined by pullback
Fun(119865119902minusrarrrarr 119860119864
119901minusrarrrarr 119861) Fun(119865 119864)
Fun(119860 119861) Fun(119865 119861)
119901lowast
119902lowast
(iii) isofibrations from 119902 to 119901 are commutative squares
119865 119864bull
119860 119861
119902
119892
119901
119891
in which the horizontals and the induced map from the initial vertex to the pullback of the cospan areisofibrations in119974
(iv) limits are defined pointwise in119974(v) and in which a map
119865 119864
119860 119861
119902
sim
119892
119901
sim119891
is an equivalence in theinfin-cosmos119974120794 if and only if 119892 and 119891 are equivalences in119974Relative to these definitions the domain codomain and identity functors
119974120794 119974dom
cod
id
are all cosmological
Proof The diagram category119974120794 inherits its simplicially enriched limits defined pointwise from119974 The functor-spaces described in (ii) are the usual ones for an enriched category of diagrams Thisverifies 121(i)
For axiom 121(ii) note that the product and simplicial cotensor functors carry pointwise isofibra-tions to isofibrations The pullback of an isofibration as in (iii) along a commutatative square from an
183
isofibration 119903 to 119901 may be formed in119974 Our task is to show that the induced map 119905 is an isofibrationand also that the square from 119905 to 119903 is an isofibration in the sense of (iii)
119866 times119864119865 119865
119866 119864bull bull
119862 times119861119860 119860
119862 119861
119905 119902
119911
119901 119903
(742)
The map 119905 factors as a pullback of 119911 followed by a pullback of 119903 as displayed above and is thus an isofi-bration as claimed This observation also verifies that the square from 119905 to 119903 defines an isofibrationA similar argument verifies the Leibniz stability of the isofibrations and that the limit of a tower ofisofibration is an isofibartion This proves that 119974120794 defines an infin-cosmos in such a way so that thedomain codomain and identity functors are cosmological
Finally since pullbacks of isofibrations in 119980119966119886119905 are invariant under equivalences a pair of equiv-alences (119892 119891) induces an equivalence between the functor-spaces defined in (ii) The converse that anequivalence in119974120794 defines a pair of equivalences in119974 follows from the fact that domain and codomain-projection functors are cosmological and Lemma 132
In close analogy with Proposition 353 we have a smothering 2-functor that relates the homo-topy 2-category of 119974120794 to the 2-category of isofibrations commutative squares and parallel naturaltransformations in the homotopy 2-category of119974
743 Lemma There is an identity on objects and 1-cells smothering 2-functor 120101(119974120794) rarr (120101119974)120794 whosecodomain is the 2-category whosebull objects are isofibrations in119974bull 1-cells are commutative squares between suchbull 2-cells are pairs of 2-cells in 120101119974
119865 119864
119860 119861
119892
119892prime119902
dArr120572
119901119891
119891prime
dArr120573
Proof Exercise 74i
For anyinfin-cosmos119974 and any subcategory of its underlying 1-category mdash that is for any subset ofits objects and subcategory of its 0-arrows mdash one can form a quasi-categorically enriched subcategoryℒ sub 119974 that contains exactly those objects and 0-arrows and all higher dimensional arrows that theyspan We call such subcategoriesℒ full on positive-dimensional arrows note the functor spaces ofℒare quasi-categories because all inner horn inclusions are bijective on vertices We will take particularinterest in subcategories that satisfy a further ldquorepletenessrdquo condition
184
744 Definition Let 119974 be an infin-cosmos A subcategory ℒ sub 119974 is replete in 119974 if it is full onpositive-dimensional arrows and moreover
(i) Everyinfin-category in119974 that is equivalent to an object inℒ lies inℒ(ii) Any equivalence in119974 between objects inℒ lies inℒ(iii) Any arrow in119974 that is isomorphic in119974 to an arrow inℒ lies inℒ
745 Lemma Suppose ℒ sub 119974 is a replete subcategory of an infin-cosmos Then any map 119901∶ 119864 rarr 119861 inℒ that defines an isofibration in 119974 is a representably-defined isofibration in ℒ that is for all 119883 isin ℒ119901lowast ∶ Funℒ(119883 119864) ↠ Funℒ(119883 119861) is an isofibration of quasi-categories
Proof Since 119974 is an infin-cosmos axiom 121(ii) requires that 119901lowast ∶ Fun119974(119883 119864) ↠ Fun119974(119883 119861)is an isofibration of quasi-categories Because the inner horn inclusions are bijective on vertices andFunℒ(119883 119864) Fun119974(119883 119864) is full on positive-dimensional arrows it follows immediately that therestricted map 119901lowast ∶ Funℒ(119883 119864) ↠ Funℒ(119883 119861) lifts against the inner horn inclusions Thus it remainsonly to solve lifting problems of the form displayed below-left
120793 Funℒ(119883 119864) Fun119974(119883 119864)
120128 Funℒ(119883 119861) Fun119974(119883 119861)
119890
119901lowast 119901lowast
120573
The lifting problem defines a 0-arrow 119890 ∶ 119883 rarr 119864 in ℒ and an isomorphism 120573∶ 119887 cong 119901119890 in ℒ Itssolution in119974 defines a 0-arrow 119890prime ∶ 119883 rarr 119864 in119974 so that 119901119890prime = 119887 together with an isomorphism 119890 cong 119890primein119974 By fullness on positive-dimensional arrows to show that this lift factors through the inclusionFunℒ(119883 119864) Fun119974(119883 119864) we need only argue that the map 119890prime lies in ℒ but this is the case bycondition (iii) of Definition 744
The following result describes a condition under which a replete subcategory ℒ sub 119974 inherits aninfin-cosmos structure created from119974
746 Proposition Suppose ℒ sub 119974 is a replete subcategory of aninfin-cosmos If ℒ is closed under flexibleweighted limits in 119974 then ℒ defines an infin-cosmos with isofibrations equivalences trivial fibrations andsimplicial limits created by the inclusionℒ 119974 which then defines a cosmological functor
When these conditions hold we refer to ℒ as a replete sub infin-cosmos of 119974 and ℒ 119974 as acosmological embedding
Proof To say that a replete subcategory ℒ 119974 is closed under flexible weighted limits meansthat for any diagram in ℒ and any limit cone in119974 that limit cone lies in ℒ and satisfies appropriatesimplicially-enriched universal property of Definition 717 in there We must verify that each of thelimits of axiom 121(i) exist in ℒ Immediately ℒ has a terminal object products and simplicialcotensors since all of these are flexible weighted limits By Lemmas 737 and 7313ℒ also admits theconstruction of iso-comma objects and of iso-towers
Define the class of isofibrations in ℒ to be those maps in ℒ that define isofibrations in 119974 ByLemmas 739 and 7314 pullbacks and limits of towers of isofibrations are equivalent in 119974 to theiso-commas and iso-towers formed over the same diagrams Since these latter limit cones lie in ℒ byhypothesis so do the equivalence former cones by repleteness ofℒ in119974
185
There is a little more still to verify namely that pullbacks and limits of towers of isofibrationssatisfy the simplicially-enriched universal property as conical limits in ℒ In the case of a pullbackdiagram
119875 119861
119862 119860
119888
119887
119891
119892
inℒ we must show that for each 119883 isin ℒ the functor-space Funℒ(119883 119875) is isomorphic to the pullbackFunℒ(119883 119862)timesFunℒ(119883119860)Funℒ(119883 119861) of functor spaces We have such an isomorphism for functor spacesin119974 and on account of the commutative diagram
Funℒ(119883 119875) Funℒ(119883 119862) timesFunℒ(119883119860)
Funℒ(119883 119861)
Fun119974(119883 119875) Fun119974(119883 119862) timesFun119974(119883119860)
Fun119974(119883 119861)cong
and fullness on positive-dimensional arrows we need only verify surjectivity of the dotted map on0-arrows So consider a cone (ℎ ∶ 119883 rarr 119861 119896 ∶ 119883 rarr 119862) over the pullback diagram inℒ By the universalproperty of the isocomma119860⨰
119861119864 there exists a factorization 119910∶ 119883 rarr 119862⨰
119860119861 inℒ Composing with the
equivalence119862⨰119860119861 ≃ 119875 this map is equivalent to the factorization 119911 ∶ 119883 rarr 119875 of the cone (ℎ 119896) through
the limit cone (119887 119888) in119974 that exists on account of the strict universal property of the pullback in thereBy repleteness the isomorphism between 119911 and the composite of 119910 with the equivalence suffices toshow that 119911 lies in ℒ Hence the functor spaces in ℒ are isomorphic A similar argument invokingLemma 7314 proves that inverse limits of towers of isofibrations define conical limits in ℒ Thiscompletes the proof of the limit axiom 121(i)
Since the isofibrations in ℒ are a subset of the isofibrations in 119974 and the limit constructionsin both contexts coincide most of the closure properties of 121(ii) are inherited from the closureproperties in119974 The one exception is the requirement that the isofibrations inℒ define isofibrationsof quasi-categories representably which was proven for any replete subcategory in Lemma 745 Thisproves thatℒ defines aninfin-cosmos
Finally we argue that the equivalences in ℒ coincide with those of 119974 which will imply that thetrivial fibrations inℒ coincide with those of119974 as well Condition (ii) of Definition 744 implies thatfor any arrow inℒ that defines an equivalence in119974 its equivalence inverse and witnessing homotopiesof Lemma 1215 lie inℒ Because we have already shown thatℒ admits cotensors with 120128 preserved bythe inclusion ℒ 119974 Lemma 1215 implies that this data defines an equivalence in ℒ Converselyany equivalence inℒ extends to the data of (1216) and sinceℒ 119974 preserves 120128-cotensors this datadefines an equivalence in 119974 Thus by construction the infin-cosmos structure of ℒ is preserved andreflected by the inclusionℒ 119974 as claimed
In practice the repleteness condition of Definition 744 is satisfied by any subcategory of objectsand 0-arrows that is determined by someinfin-categorical property so the main task in verifying that asubcategory defines aninfin-cosmos is verifying the closure under flexible weighted limits
747 Proposition For anyinfin-cosmos119974 let119974⊤ denote the quasi-categorically enriched category whose(i) objects areinfin-categories in119974 that possess a terminal object
186
(ii) functor spaces Fun⊤(119860 119861) sub Fun(119860 119861) are the sub-quasi-categories whose 0-arrows preserve termi-nal objects and containing all 119899-arrows they span
Then the inclusion119974⊤ 119974 creates aninfin-cosmos structure on119974⊤ from119974 and moreover for each object of119974⊤ defined as a flexible weighted limit of some diagram in119974⊤ its terminal element is created by the 0-arrowlegs of the limit cone
Proof We apply Proposition 746 Lemma 226 and Proposition 2110 verify the repletenesscondition so it remains only to prove closure under flexible weighted limits which we do by inductionover the tower of isofibrations constructed in Proposition 73192 which expresses a flexible weightedlimit lim119882 119865 as the inverse limit of a tower of isofibrations
lim119882 119865 ⋯ lim119882119896+1 119865 lim119882119896 119865 ⋯ lim1198820 119865 1
each of which is a pullback of products of maps of the form (732) indexed by the projective cells ofthe flexible weight119882 Wersquoll argue inductively that eachinfin-category in this tower possesses a terminalelement thatrsquos created by the legs of the tower of isofibrations
For the base case note that if (119860119894)119894isin119868 is a family of infin-categories possessing terminal elements119905119894 ∶ 1 rarr 119860119894 then the product of the adjunctions ⊣ 119905119894 defines an adjunction
1 cong prod119894isin119868 1 prod119894isin119868119860119894perp
(119905119894)119894isin119868congprod119894 119905119894
exhibiting (119905119894)119894isin119868 as a terminal element of prod119894isin119868119860119894 By construction this terminal element is jointlycreated by the legs of the limit cone Note that by construction the product-projection functors pre-serve this terminal element and the map into the productinfin-categoryprod119894isin119868119860119894 induced by any familyof terminal element preserving functors (119891 ∶ 119883 rarr 119860119894)119894isin119868 will preserve terminal elements This verifiesthat the subcategory119974⊤ is closed under products
For the inductive step consider a pullback diagram
lim119882119896+1 119865 119860Δ[119899]
lim119882119896 119865 119860120597Δ[119899]
ℓ
that arises from the attaching map for a projective 119899-cell The inductive hypothesis tells us thatlim119882119896 119865 admits a terminal element 119905119896 and for each vertex of 119894 isin 120597Δ[119899] the corresponding com-ponent ℓ119894 ∶ lim119882119896 119865 rarr 119860 of the limit cone preserves it Since 119865 is a diagram valued in119974⊤ and119860 is aninfin-category in its image we know that 119860 must possess a terminal element 119905 ∶ 1 rarr 119860 By Proposition217(iii) the constant diagram at 119905 then defines a terminal element in 119860120597Δ[119899] and 119860Δ[119899] which wealso denote by 119905 By terminality there is a 1-arrow 120572∶ ℓ(119905119896) rarr 119905 isin 119860120597Δ[119899] whose components at each119894 isin 120597Δ[119899] are isomorphisms in 119860 By Lemma it follows that 120572 is also an isomorphism which tellsus that ℓ(119905119896) is also a terminal element of 119860120597Δ[119899] The same argument demonstrates that terminalelements in simplicial cotensors in this case by 120597Δ[119899] are jointly created by the 0-arrow componentsof the limit cone namely by evaluation on each of the vertices of the cotensoring simplicial set Theproof is completed now by the following lemma
187
748 Lemma Consider a pullback diagram
119865 119864
119860 119861
119892
119902
119901
119891
in which the infin-categories 119860 119861 and 119864 possess a terminal element and the functors 119891 and 119901 preserve themThen 119865 possesses a terminal element that is created by the legs of the pullback cone 119902 and 119892
Proof If 119890 ∶ 1 rarr 119864 and 119886 ∶ 1 rarr 119860 are terminal then this implies that 119891(119886) cong 119901(119890) isin 119861 Using thefact that 119901 is an isofibration there is a lift 119890prime cong 119890 of this isomorphism along 119901 that then defines anotherterminal element of 119864 The pair (119886 119890prime) now induces an element 119905 of 119865 that we claim is terminal
To see this wersquoll apply Proposition 4310 which proves that 119905 is a terminal element of 119865 if andonly if the domain-projection functor 1199010 ∶ Hom119865(119865 119905) ↠ 119865 is a trivial fibration By construction of119905 we know that the domain-projection functors for the elements 119892119905 119902119905 and 119901119892119905 = 119891119902119905 are all trivialfibrations and moreover the top and bottom faces of the cube
Hom119865(119865 119905) Hom119864(119864 119892119905)
Hom119860(119860 119902119905) Hom119861(119861 119891119902119905)
119865 119864
119860 119861
1199010
sim 1199010
sim1199010119902
119892
119901
119891
1199010≀
are pullbacks Since homotopy pullbacks are homotopical the fact that the three maps between thecospans are equivalences implies that the map between their pullbacks is also an equivalence as re-quired
Applying the result of Proposition 747 to 119974co constructs an infin-cosmos 119974perp whose objects areinfin-categories in119974 that possess an initial object and 0-arrows are initial-element-preserving functorsCombining these we get an infin-cosmos for the pointed infin-categories of Definition 441 those thatpossess a zero element
749 Proposition For anyinfin-cosmos119974 let119974lowast denote the quasi-categorically enriched category of pointedinfin-categories ieinfin-categories that possess a zero element and functors that preserve them Then the inclusion119974lowast 119974 creates aninfin-cosmos structure on119974⊤ from119974
Proof The natural inclusions define a pullback diagram of quasi-categorically enriched cate-gories
119974perp⊤ 119974⊤
119974perp 119974
where the objects of 119974perp⊤ are infin-categories that possess both an initial and terminal element andfunctors that preserve them separately
188
Now repleteness of 119974perp⊤ follows from repleteness of 119974⊤ and 119974perp as does closure under flexibleweighted limits Given a diagram in 119974perp⊤ admitting a flexible weighted limit in 119974 that limit conelies in both 119974⊤ and 119974perp and hence in their intersection Since the functor spaces of 119974perp⊤ are theintersections of the functor spaces of 119974⊤ and 119974perp in 119974 the simplicial universal property in thoseinfin-cosmoi restricts to the required simplicially-enriched universal property in119974perp⊤
Now theinfin-cosmos119974lowast of interest is the full subcategory of119974perp⊤ spanned by thoseinfin-categoriesin which the initial and terminal element coincide To prove that 119974lowast is an infin-cosmos created bycosmological embedding 119974lowast 119974perp⊤ it remains only to show that this subcategory is closed un-der flexible weighted limits To that end consider a diagram 119865∶ 119964 rarr 119974perp⊤ and a flexible weight119882∶ 119964 rarr 119982119982119890119905 We have shown that the limit lim119882 119865 admits an initial element 119894 ∶ 1 rarr lim119882 119865 andalso a terminal element 119905 ∶ 1 rarr lim119882 119865 each of which is preserved the the 0-arrow legs of the limitcone The elements 119894 and 119905 are vertices in the underlying quasi-category Fun(1 lim119882 119865) and remainrespectively initial and terminal in there Appealing to the universal property of either there is a1-simplex 120572∶ 119894 rarr 119905 isin Fun(1 lim119882 119879) representing a natural transformation 120572∶ 119894 rArr 119905 in 120101119974 andlim119882 119865 is a pointedinfin-category if and only if 120572 is invertible
Invertibility of 1-simplices in quasi-categories can be detected by applying the cosmological func-tor (minus) ∶ 119980119966119886119905 rarr 1-119966119900119898119901 of Example 137 a 1-simplex in Fun(1 lim119882 119865) is an isomorphism if andonly if its image in Fun(1 lim119882 119865) is marked Corollary 733 proves that the cosmological functorFun(1 minus) ∶ 119974 rarr 1-119966119900119898119901 presrves flexible weighted limits so
Fun(1 lim119882 119865) cong lim119882 Fun(1 119865minus)and thus it remains only to show that a 1-simplex in a flexibleweighted limit of a diagramof 1-complicialsets whose image under each of the legs of the limit cone is marked is marked in the weighted limitIndeed the markings for all simplicially enriched limits in 1-119966119900119898119901 are defined in this manner a1-simplex is marked if and only if each of its components is marked So this proves that119974lowast 119974perp⊤is closed under flexible weighted limits completing the proof
Applying the result of Proposition 747 or its dual to the infin-cosmos 119974119861 of isofibrations over119861 isin 119974 we obtain two newinfin-cosmoi of interest
7410 Corollary For anyinfin-category119861 in aninfin-cosmos119974 the slicedinfin-cosmos119974119861 admits subinfin-cosmoi
ℛ119886119903119894(119974)119861 119974119861 ℒ119886119903119894(119974)119861whosebull objects are isofibrations over 119861 admitting a right adjoint right inverse or left adjoint right inverse respec-
tively andbull 0-arrows are functors over 119861 that commute with the respective right or left adjoints up to fibered isomor-
phismwith theinfin-cosmos structures created by the inclusions
Proof Theseinfin-cosmoi are defined byℛ119886119903119894(119974)119861 ≔ (119974119861)⊤ andℒ119886119903119894(119974)119861 ≔ (119974119861)perp
Leveraging Corollary 7410 we can establish similar cosmological embeddings
ℛ119886119903119894(119974) 119974120794 ℒ119886119903119894(119974)
189
The quasi-categorically enriched subcategoriesℛ119886119903119894(119974) andℒ119886119903119894(119974) are replete in119974120794 so by Propo-sition 746 we need only check closure under flexible weighted limits We argue separately for coten-sors which are easy and for the conical limits which are harder For this we make use of a general1-categorical result making use of the fact that the codomain-projection functor cod ∶ ℛ 119886119903119894(119974) rarr 119974is a Grothendieck fibration of underlying 1-categories defined by restricting the Grothendieck fibra-tion cod ∶ 119974120794 rarr119974
7411 Lemma Let 119875∶ ℰ rarr ℬ be a Grothendieck fibration between 1-categories Suppose that 119973 is a smallcategory that 119863∶ 119973 rarr ℰ is a diagram and that
(i) the diagram 119875119863∶ 119973 rarr ℬ has a limit 119871 in ℬ with limiting cone 120582∶ Δ119871 rArr 119863(ii) the diagram 120582lowast119863∶ 119973 rarr ℰ119871
119973 ℰ 119973 ℰ
ℬ ℬ
119863
Δ119871uArr120582 119875 =
119863
120582lowast119863
uArr120594
119875
constructed by lifting the cone 120582 to a cartesian natural transformation 120594∶ 120582lowast119863 rArr 119863 has a limit119872in the fibre ℰ119871 with limiting cone 120583∶ Δ119872 rArr 120582lowast119863 and
(iii) the limit 120583∶ Δ119872 rArr 120582lowast119863 is preserved by the re-indexing functor 119906lowast ∶ ℰ119871 rarr ℰ119861 associated with anyarrow 119906∶ 119861 rarr 119871 in ℬ
Then the composite cone
Δ119872 120582lowast119863 119863120583 120594
displays119872 as a limit of the diagram 119863 in ℰ
Proof Any arrow 119891∶ 119864 rarr 119864prime in the domain of a Grothendieck fibration 119875∶ ℰ rarr ℬ factorsuniquely up to isomorphism through a ldquoverticalrdquo arrow in the fiber ℰ119875119864 followed by a ldquohorizontalrdquocartesian lift of 119875119891 with codomain 119864prime
Given a cone 120572∶ Δ119864 rArr 119863 with summit 119864 isin ℰ over 119863 by (i) its image 119875120572∶ Δ119875119864 rArr 119875119863 factorsuniquely through the limit cone 120582∶ Δ119871 rArr 119863 via a map 119887 ∶ 119875119864 rarr 119871 isin ℬ By the universal property ofthe cartesian lift 120594 of 120582 constructed in (ii) it follows that 120572 factors uniquely through 120594 via a naturaltransformation 120573∶ Δ119864 rArr 120582lowast119863 so that 119875120573 = Δ119887 This arrow factors uniquely up to isomorphism vialdquoverticalrdquo natural transformation 120574∶ Δ119864 rarr 120572lowast119863 cong 119887lowast120574lowast119863 followed by a ldquohorizontalrdquo cartesian lift of119887 By (iii) the limit cone 120583∶ Δ119872 rArr 120582lowast119863 in ℰ119871 pulls back along 119887 to a limit cone in ℰ119875119864 throughwhich the pullback of 120573 factors via a map 119896 ∶ 119864 rarr 119887lowast119872 Thus 120573 itself factors uniquely through 120583 viathe composite of this map 119896 ∶ 119864 rarr 119887lowast119872 with the cartesian arrow 119887lowast119872rarr119872 lifting 119887 ∶ 119875119864 rarr 119871
7412 Proposition For anyinfin-cosmos119974 theinfin-cosmos of isofibrations admits subinfin-cosmoi
ℛ119886119903119894(119974) 119974120794 ℒ119886119903119894(119974)
whosebull objects are isofibrations admitting a right adjoint right inverse or left adjoint right inverse respectively
andbull 0-arrows are commutative squares between the right or left adjoints respectively whose mates are isomor-
phisms
190
with theinfin-cosmos structures created by the inclusions
We refer to a commutative square between right adjoints whose mate is an isomorphism as anexact square
Proof The quasi-categorically enriched subcategoriesℛ119886119903119894(119974) andℒ119886119903119894(119974) are replete in119974120794so by Proposition 746 we need only check that ℛ119886119903119894(119974) 119974120794 is closed under flexible weightedlimits the argument forℒ119886119903119894(119974) 119974120794 is dual We argue separately for cotensors and for the conicallimits
If 119901∶ 119864 ↠ 119861 is an isofibration admitting a right adjoint right inverse in 119974 and 119880 is a simplicialset then the cosmological functor (minus)119880 ∶ 119974 rarr 119974 carries this data to a right adjoint right inverseto 119901119880 ∶ 119864119880 ↠ 119861119880 which proves that the simplicial cotensor in 119974120794 of an object in ℛ119886119903119894(119974) liesin ℛ119886119903119894(119974) The limit cone for the cotensor is given by the canonical map of simplicial sets 119880 rarr
Fun(119864119880119901119880minusminusrarrrarr 119861119880 119864
119901minusrarrrarr 119861) defined on each vertex 119906∶ 120793 rarr 119880 by the commutative square
119864119880 119864
119861119880 119861
119901119880
119906lowast
119901
119906lowast
(7413)
The maps 119906lowast define the components of a simplicial natural transformation from (minus)119880 to the iden-tity functor and thus the mate of this commutative square is an identity so the limit cone for the119880-cotensor lies inℛ119886119903119894(119974) Finally to verify the universal property of the cotensor inℛ119886119903119894(119974) wemust show that for any commutative square whose domain is an isofibration admitting a right adjointright inverse
119865 119864119880
119860 119861119880119902 119901119880
that composes with each of the squares (7413) to an exact square is itself exact To see this takethe mate to define a 1-arrow in Fun(119860 119864119880) cong Fun(119860 119864)119880 and note that the hypothesis says thatthe components of this 1-arrow are invertible for each vertex of 119880 Lemma then tells us that this1-arrow is invertible as required
Taking 119880 to be a set the argument just given proves also that ℛ119886119903119894(119974) is closed in 119974120794 underproducts It remains only to show that it is closed under the remaining conical limits By pullbackstability of fibered adjunctions the Grothendieck fibration of 1-categories cod ∶ 119974120794 rarr 119974 restrictsto cod ∶ ℛ 119886119903119894(119974) rarr 119974 so we may appeal to Lemma 7411 to calculate 1-categorical limit cones inℛ119886119903119894(119974) sub 119974120794 as composites of cartesian cells with limit cones of fiberwise diagrams By Corol-lary 7410 these fiberwise limits in 119974119861 of diagrams in ℛ119886119903119894(119974)119861 lie in ℛ119886119903119894(119974)119861 ℛ119886119903119894(119974)Moreover these 1-categorical limits are preserved by the simplicial cotensor which by PropositionA56 implies that their universal property enriches to define conical limits In this way we see thatℛ119886119903119894(119974) 119974120794 is closed under flexible weighted limits and thus defines a cosmological embeddingas claimed
Proposition 7412 allows us to construct furtherinfin-cosmoi of interest
191
7414 Proposition For anyinfin-cosmos119974 and simplicial set 119869 there exist subinfin-cosmoi
119974⊤119869 119974 119974perp119869
whosebull objects areinfin-categories in119974 that admit all limits of shape 119869 or all colimits of shape 119869 respectivelybull 0-arrows are the functors that preserve them
with theinfin-cosmos structures created by the inclusions Moreover for each object of 119974⊤119869 or 119974perp119869 defined asa flexible weighted limit of some diagram in that infin-cosmos its 119869-shaped limits or colimits arecreated by the0-arrow legs of the limit or colimits cones respectively
Proof First note that the quasi-categorically enriched subcategories119974⊤119869 and119974perp119869 are replete in119974 so by Proposition 746 we need only confirm that the inclusions are closed under flexible weightedlimits We prove this in the case of colimits the other case being dual
For any fixed simplicial set 119869 there is a cosmological functor 119865119869 ∶ 119974 rarr 119974120794 defined on objectsby mapping an infin-category 119860 to the isofibration 119860119869▷ ↠ 119860119869 in the notation of 426 and a functor119891∶ 119860 rarr 119861 to the commutative square
119860119869▷ 119861119869▷
119860119869 119861119869
119891119869▷
119891119869
By Corollary 435 119860 admits colimits of shape 119869 if and only if this isofibration admits a left adjointright inverse and now it is clear that 119891∶ 119860 rarr 119861 preserves these colimits if and only if the squaredisplayed above is exact In summary the quasi-categorically enriched subcategory119974perp119869 is defined bythe pullback
119974perp119869 ℒ119886119903119894(119974)
119974 119974120794
119865119869
Proposition 7412 proves that ℒ119886119903119894(119974) 119974120794 is closed under flexible weighted limits and 119865119869 ∶ 119974 rarr119974120794 preserves them so it follows as in the proof of Lemma 617 that119974perp119869 is closed in119974 under flexibleweighted limits Now Proposition 746 proves that the inclusion 119974perp119869 119974 creates an infin-cosmosstructure
7415 Proposition Theinfin-cosmos of isofibrations admits subinfin-cosmoi
119966119886119903119905(119974) 119974120794 119888119900119966119886119903119905(119974)
whose objects are cartesian or cocartesian fibrations respectively and whose 0-arrows are cartesian functorswith theinfin-cosmos structures created by the inclusions Similarly for anyinfin-category 119861 in aninfin-cosmos119974the slicedinfin-cosmos119974119861 admits subinfin-cosmoi
119966119886119903119905(119974)119861 119974119861 119888119900119966119886119903119905(119974)119861
192
whose objects are cartesian or cocartesian fibrations over 119861 respectively and whose 0-arrows are cartesianfunctors with theinfin-cosmos structures created by the inclusions
Proof The quasi-categorically enriched subcategories119966119886119903119905(119974) and 119888119900119966119886119903119905(119974) are replete in119974120794
by Corollary 5117 By Theorems 5111 and 5119 the quasi-categorically enriched category 119966119886119903119905(119974)is defined by the pullback
119966119886119903119905(119974) ℛ 119886119903119894(119974)
119974120794 119974120794
119870
along the simplicial functor that sends an isofibration 119901∶ 119864 ↠ 119861 to the isofibration 119896 ∶ 119864120794 ↠Hom119861(119861 119901) defined by Remark 5113 The simplicial functor 119870 is constructed out of weighted limitsand thus preserves all weighted limits and the replete subcategory inclusion ℛ119886119903119894(119974) 119974120794 cre-ates flexible weighted limits by Proposition 7412 Hence as in the proof of Lemma 617 119966119886119903119905(119974)is closed in 119974120794 under flexible weighted limits and now Proposition 746 proves that the inclusion119966119886119903119905(119974) 119974120794 creates aninfin-cosmos structure
The result for cartesian fibrations with a fixed base can be proven directly by a similar argumentor deduced by considering 119974119861 sub 119974120794 as the (non-replete) subcategory whose 119899-arrows have id119861 astheir codomain components
Exercises
74i Exercise Prove Lemma 743
74ii Exercise Use an argument similar to that given in the proof of Proposition 749 to prove thatanyinfin-cosmos119974 admits a subinfin-cosmos119982119905119886119887(119974) 119974 whose objects are the stableinfin-categories ofDefinition 445 and whose morphisms are the exact functors which preserve the zero elements andthe exact triangles
74iii Exercise Consider a functor between isofibrations
119864 119865
119861
119901
119892
119902
119903⊢
119904⊣
in which 119901 admits a right adjoint right inverse 119903 and 119902 admits a right adjoint right inverse 119904 Prove thatif 119892119903 cong 119904 over119861 then the mate of the identity 119902119892 = 119901 is an isomorphism This proves that the 0-arrowsin theinfin-cosmosℛ119886119903119894(119974)119861 are exact transformations between right adjoint right inverse adjunctions
74iv Exercise Prove thatℒ119886119903119894(119974) cong ℛ 119886119903119894(119974co)co
74v Exercise Use Exercise 12iv to show that ifℒ 119974 is a replete subinfin-cosmos then an object119860 isin ℒ is discrete if and only if 119860 is discrete as an object of119974
75 Weak 2-limits revisited
To wrap up this chapter on weighted limits we briefly switch the base of enrichment from sim-plicial sets to categories and reconsider the weak 2-limits introduced in Chapter 3 We give a general
193
definition that unifies the weak 2-limits introduced in special cases there and prove their essentialuniqueness in a uniform manner
Before turning our attention to weak 2-limits we describe the explicit construction of the weightedlimit of any 119966119886119905-valued diagram Let 119964 be a small 2-category and consider any pair of 2-functors119865119882∶ 119964 119966119886119905 the first regarded as the diagram and the second as the weight
751 Lemma (on the construction of weighted limits in119966119886119905) For any diagram 119865∶ 119964 rarr 119966119886119905 and weight119882∶ 119964 rarr 119966119886119905 the weighted limit lim119882 119865 isin 119966119886119905 exists defined to be the category whosebull objects are 2-natural transformations 120572∶ 119882 rArr 119865 andbull morphisms are modifications
Proof By Definition 717 the 119882-weighted limit of 119865 is a category lim119882 119865 isin 119966119886119905 characterizedby a natural isomorphism of categories
119966119886119905(119883 lim119882 119865) cong 119966119886119905119964(119882119966119886119905(119883 119865minus))
for any category119883 The 2-functor 119966119886119905(120793 minus) ∶ 119966119886119905 rarr 119966119886119905 is the identity so taking119883 = 120793 this tells usthat
lim119882 119865 cong 119966119886119905119964(119882 119865)
the category of 2-natural transformations and modifications from119882 to 119865
752 Definition (weak 2-limits in a 2-category) Consider a 2-functor 119865∶ 119964 rarr 119966 indexed by a small2-category119964 and a weight119882∶ 119964 rarr 119966119886119905 A119882-cone with summit 119875 isin 119966
119882 119966(119875 119865minus)120582
displays 119875 as a weak 2-limit of 119865 if and only if for all 119883 isin 119966 the functor induced by composition with120582
119966(119883 119875) lim119882 119966(119883 119865minus)120582lowast
to the119882-weighted limit of the diagram119966(119883 119865minus) ∶ 119964 rarr 119966119886119905 is smothering in the sense of Definition312
The weak universal property encoded by a smothering functor is sufficiently strong to characterizethe limit objects up to equivalence in the ambient 2-category
753 Proposition (uniqueness of weak 2-limits) For any fixed diagram and fixed weight any pair ofweak 2-limits are equivalent via an equivalence that commutes with the legs of the limit cones
Proof If119882 119966(119875 119865minus) and 119882 119966(119875prime 119865minus)120582 120582prime
both define weak 2-limit 119882-weighted cones over a 2-functor 119865∶ 119964 rarr 119966 then for any 119883 isin 119966 wehave a pair of smothering functors
119966(119883 119875) lim119882 119966(119883 119865minus) 119966(119883 119875prime)120582lowast 120582primelowast
Taking 119883 = 119875 the identity id119875 isin 119966(119875 119875) maps to the cone 120582 isin lim119882 119966(119875 119865minus) which then liftsalong the right-hand smothering functor to define a 1-cell 119906∶ 119875 rarr 119875prime a 1-cell 119907∶ 119875prime rarr 119875 is definedsimilarly as the lift of 120582prime isin lim119882 119966(119875prime 119865minus) along 120582lowast By construction both 119906 and 119907 commute withthe legs of the limit cones 120582 and 120582prime
194
Now 120582lowast carries the composite 119907119906 isin 119966(119875 119875) to the cone 119882 rArr 119966(119875 119865minus) whose component at119886 isin 119964 is the composite
119875 119875prime 119875 119865119886119906
120582119886
119907
120582prime119886
120582119886
which equals simply the cone leg 120582119886 Thus id119875 and 119907119906 lie in the same fiber of the smothering functor120582lowast and so by Lemma 313 must be isomorphic via an isomorphism that whiskers to identities alongthe legs of the limit cone Similarly 119906119907 cong id119875prime proving that 119875 ≃ 119875prime as claimed
Exercises
75i Exercise An inserter is a limit of a diagram indexed by the parallel pair category bull bullweighted by the weight
120793 120794 isin 1199661198861199050
1Prove that the homotopy 2-category of aninfin-cosmos has weak inserters of any parallel pair of functors119891 119892 ∶ 119860 119861 constructed by the pullback
Ins(119891 119892) 119861120794
119860 119861 times 119861
(11990111199010)
(119892119891)
195
CHAPTER 8
Homotopy coherent adjunctions and monads
Bar and cobar resolutions are ubiquitous in modern homotopy theory defining for instances var-ious completions of spaces and spectra [11] and free resolutions such as given in Definition 621 For-mally these bar or cobar constructions are associated to the monad or comonad of an adjunction
119860 119861119906perp119891
120578∶ id119861 rArr 119906119891 120598 ∶ 119891119906 rArr id119860
between infin-categories The monad and comonad resolutions associated to an adjunction are dualBy the triangle equalities the unit and counit maps give rise to a coaugmented cosimplicial object inhFun(119861 119861) the ldquomonad resolutionrdquo
id119861 119906119891 119906119891119906119891 119906119891119906119891119906119891 ⋯120578120578119906119891
119906119891120578
119906120598119891 (801)
an augmented simplicial object in hFun(119860119860) the ldquocomonad resolutionrdquo
id119860 119891119906 119891119906119891119906 119891119906119891119906119891119906 ⋯120598 119891120578119906
119891119906120598
120598119891119906
(802)
and augmented simplicial objects in hFun(119860 119861) and hFun(119861119860) admitting forwards and backwardscontracting homotopies
119906 119906119891119906 119906119891119906119891119906 119906119891119906119891119906119891119906 ⋯
119891 119891119906119891 119891119906119891119906119891 119891119906119891119906119891119906119891 ⋯
120578119906
119906120598 119906119891120578119906
120578119906119891119906
119906120598119891119906
119906119891119906120598
120598119891
119891120578 119891119906120598119891
120598119891119906119891
119891120578119906119891
119891119906119891120578
(803)
A classical categorical observation tells a richer story relating the four resolutions displayed aboveThere is a strict 2-category 119964119889119895 containing two objects and an adjunction between them mdash the free2-category containing an adjunction mdash and collectively these four diagrams display the image of a2-functor whose domain is119964119889119895 [53] More precisely each diagram is the image of one of the four hom-categories of this two object 2-category a 2-functor 119964119889119895 rarr 120101119974 extending the adjunction 119891 ⊣ 119906 isdefined by a pair of objects119860119861 isin 120101119974 the monad and comonad resolutions in the functor categorieshFun(119861 119861) and hFun(119860119860) and the dual pair of split augmented simplicial objects in hFun(119860 119861) andhFun(119861119860) The fact that these resolutions assemble into a 2-functor says that eg that the image of
197
the comonad resolution under 119906 is an augmented simplicial object in hFun(119860 119861) that admits ldquoextradegeneraciesrdquo
In this chapter we will prove that any adjunction in the homotopy 2-category of an infin-cosmosmdash that is any adjunction between infin-categories mdash can be lifted to a homotopy coherent adjunction intheinfin-cosmos The data of a homotopy coherent adjunction is indexed by a simplicial computad thatis uncannily closely related to the free adjunction 119964119889119895 In fact we define the free homotopy coherentadjunction to be the 2-category119964119889119895 regarded as a simplicial category by identifying its hom-categorieswith their nerves Section 81 is spent justifying this definition by introducing a graphical calculusthat allows us to precisely understand homotopy coherent adjunction data and prove that 119964119889119895 is asimplicial computad
In a homotopy coherent adjunction the resolutions (801) (802) and (803) lift to homotopycoherent diagrams
120491+ rarr Fun(119861 119861) 120491op+ rarr Fun(119860119860) 120491⊤ rarr Fun(119860 119861) 120491perp rarr Fun(119861119860)
indexed by the 1-categories introduced in Definition 239 and valued in the functor quasi-categories oftheinfin-cosmos In the case of the split augmented simplicial objects the contracting homotopies alsocalled ldquosplittingsrdquo or ldquoextra degeneraciesrdquo are given by the bottom and top 120578rsquos respectively ApplyingProposition 2311 it follows that the geometric realization or homotopy invariant realization of thesimplicial objects spanned by maps in the image of 119891119906119891 and 119906119891119906 are simplicial homotopy equivalentto 119891 and 119906 Dual results apply to the (homotopy invariant) totalization of the cosimplicial objectspanned by these same objects in this case the ldquoextra codegeneraciesrdquo are given by the top and bottom120598rsquos
The main theorem of this chapter proves that homotopy coherent adjunctions are abundant in-deed any adjunction ofinfin-categories extends to a homotopy coherent adjunction Homotopy coherentadjunctions extending a subcomputad of generating adjunction data are not unique on the nose How-ever whenever the subcomputad of generating adjunction is ldquoparentalrdquo mdash loosely generating from theuniversal property of either the right adjoint or the left adjoint exclusively mdash then extensions to a fullhomotopy coherent adjunction define the vertices of a contractible Kan complex proving appropri-ately generated extensions are ldquohomotopically uniquerdquo
All of the results in this chapter apply to any adjunction defined in (the homotopy 2-category of) aquasi-categorically enriched category119974 Yet since wersquoll typically apply these results toinfin-cosmoi weretain the usual notation Fun(119860 119861) for the hom quasi-categories of119974 to trigger the correct intuitionin these contexts
81 The free homotopy coherent adjunction
In this section we present a strict 2-category 119964119889119895 introduced by Schanuel and Street under thename ldquothe free adjunctionrdquo [53] which has the universal property that it is the free 2-category contain-ing an adjunction Immediately after introducing this classical object we take the unorthodox stepof reconsidering it as a simplicial category via a mechanism that we shall describe We develop a newpresentation of 119964119889119895 by introducing a graphical calculus that allows us to prove the surprising factthat this simplicial category is a simplicial computad This justifies referring to it as the free homotopycoherent adjunction The remainder of this chapter will explore the consequences of this definition
198
811 Definition (the free adjunction) Let119964119889119895 denote the 2-category with two objects + and minus andthe four hom-categories
119964119889119895(+ +) ≔ 120491+ 119964119889119895(minus minus) ≔ 120491op+ 119964119889119895(minus +) ≔ 120491⊤ 119964119889119895(+ minus) ≔ 120491perp
displayed in the following cartoon
+ minus120491perpcong120491⊤op
120491+ ⟂ 120491op+
120491⊤cong120491perpop
Here 120491⊤ 120491perp sub 120491 sub 120491+ are the subcategories of order-preserving maps that preserve the top orbottom elements respectively in each ordinal as described in Definition 239 Their intersection
120491perp⊤ ≔ 120491perp cap 120491⊤ cong 120491op+
is the subcategory of order-preserving maps that preserve both the top and bottom elements in eachordinal This identifies 120491op
+ with the subcategory 120491perp⊤ sub 120491+ of ldquointervalsrdquo as is elaborated upon inDigression 818
The horizontal composition maps in 119964119889119895 are defined in 119964119889119895(minus minus)op cong 119964119889119895(+ +) cong 120491+ by theordinal sum operation
120491+ times 120491+ 120491+
[119899] [119898] [119899 + 119898 + 1]
[119899prime] [119898prime] [119899prime + 119898prime + 1]
oplus
120572120573 ↦ 120572oplus120573 120572 oplus 120573(119894) ≔ 1114108120572(119894) 119894 le 119899120573(119894 minus 119899 minus 1) + 119899prime + 1 119894 gt 119899
The object [minus1] isin 120491+ serves as the identity for ordinal sum and thus represents the identity 1-cellson minus and + in119964119889119895 Ordinal sum restricts to the subcategories 120491perp 120491⊤ sub 120491+ to give bifunctors
120491+ times 120491⊤ 120491⊤ 120491perp times 120491+ 120491perp
119964119889119895(+ +) times 119964119889119895(minus +) 119964119889119895(minus +) 119964119889119895(+ minus) times 119964119889119895(+ +) 119964119889119895(+ minus)
oplus
cong cong
oplus
cong cong∘ ∘
defining these horizontal composition operations that wersquoll later come to think of as ldquoactionsrdquo of119964119889119895(+ +) on the left and right of119964119889119895(minus +) and119964119889119895(+ minus) The opposites of these functors definesthe action of119964119889119895(minus minus) on the right and left of119964119889119895(minus +) and119964119889119895(+ minus)
812 Lemma The 2-category119964119889119895 contains a distinguished adjunction
+ minus[0]
⟂[0]
with unit ∶ [minus1] rarr [0] isin 120491+ cong 119964119889119895(+ +) and counit given by the same map in 120491op+ cong 119964119889119895(minus minus)
Proof We must verify the triangle equalities in 119964119889119895(minus +) and 119964119889119895(+ minus) these categories areopposites and the calculation in each case is dual so we focus on the case of119964119889119895(minus +) The whiskeredcomposite of the unit ∶ [minus1] rarr [0] isin 119964119889119895(+ +) with the right adjoint [0] isin 119964119889119895(minus +) is the map1205750 ∶ [0] rarr [1] isin 119964119889119895(minus +) which is indeed top-element preserving The whiskered composite of thecounit in 119964119889119895(minus minus) with the right adjoint in 119964119889119895(minus +) is defined by whiskering the opposite map
199
∶ [minus1] rarr [0] isin 119964119889119895(+ +) cong 119964119889119895(minus minus)op with [0] isin 119964119889119895(+ minus) cong 119964119889119895(minus +)op This composite isthe map 1205751 ∶ [0] rarr [1] isin 119964119889119895(+ minus) cong 119964119889119895(minus +)op which is indeed top-element preserving Underthe isomorphism119964119889119895(minus +) cong 119964119889119895(+ minus)op this corresponds to the map 1205900 ∶ [1] rarr [0] isin 119964119889119895(minus +)Now the composite 1205900 sdot 1205750 ∶ [0] rarr [0] isin 119964119889119895(minus +) is the identity as required
The 2-categorical universal property of 119964119889119895 mdash that 2-functors 119964119889119895 rarr 119966 correspond to adjunc-tions in the 2-category119966mdashis statedwithout proof in [53] Wewill take a roundabout route to verifyingit in Proposition 8113 that uses the simplicial computads of Chapter 6
Throughout this text we have found it convenient to identify 1-categories with their nerves whichdefine simplicial sets The 1-categories can be characterized as those simplicial sets that admit uniqueextensions along any inner horn inclusion or spine inclusion or as those 2-coskeletal simplicial setsthat admit unique extensions along inner horn or spine inclusions in dimensions 2 and 3 see Remark115 Similarly in developing homotopy coherent category theory it will be convenient to identity2-categories with the simplicial categories obtained by identifying each of the hom-categories withits nerve mdash a categorification of the previous construction As a corollary of the characterization ofnerves of 1-categories we obtain a characterization of the simplicially enriched categories that arisein this way
813 Lemma (2-categories as simplicial categories) A 2-category119964may be regarded as a quasi-categoricallyenriched category whosebull objects are the objects of119964bull 0-arrows in119964(119909 119910) are the 1-cells of119964 from 119909 to 119910
bull 1-arrows in119964(119909 119910) from 119891 to 119892 are the 2-cells of119964 of the form 119909 119910119891
119892dArr
bull and in which there exists a 2-arrow 120590 in119964(119909 119910) whose faces 120590119894 ≔ 120590 sdot 120575119894 are 1-arrows in119964(119909 119910)
119892
119891 ℎ
12059001205902
1205901
120590 119909 119910 = 119909 119910119892
119891
dArr1205902
ℎdArr1205900
119891
ℎdArr1205901
if and only if 1205901 is the vertical composite 1205900 sdot 1205902
with the higher-dimensional arrows determined by the property that each of the hom-spaces is 2-coskeletalConversely a simplicially-enriched category 119964 is isomorphic to a 2-category if and only if each of its hom-spaces are 2-coskeletal simplicial sets that admit unique extensions along the spine inclusions in dimensions 2and 3
We now give the first of two presentations of the free homotopy coherent adjunction Since we usethe same notation for 1-categories and their nerves we also adopt the same notation for a 2-categoryand its corresponding simplicial category under the embedding of Lemma 813
814 Definition (the free homotopy coherent adjunction as a 2-category) The free homotopy co-herent adjunction to be the free adjunction119964119889119895 regarded as a simplicial category Explicitly119964119889119895 hastwo objects + and minus and the four hom quasi-categories defined by
119964119889119895(+ +) ≔ 120491+ 119964119889119895(minus minus) ≔ 120491op+ 119964119889119895(minus +) ≔ 120491⊤ 119964119889119895(+ minus) ≔ 120491perp
with the composition maps defined in 811200
This presentation of the free homotopy coherent adjunction is not particularly enlightening ViaLemma 813 and Definition 811 we could in principle describe the 119899-arrows in 119964119889119895 but itrsquos trickyto get a real feel for them We will now reintroduce this simplicial category in a different guise thatachieves just this Before doing so note
815 Observation To specify a simplicial category thought of as an identity-on-objects simplicialobject in 119966119886119905 it suffices to specifybull a set of objectsbull for each 119899 ge 0 a set of 119899-arrows whose domains and codomains are among the specified object
setbull a right action of the morphisms in 120491 on this graded set of arrows that preserves domains and
codomainsbull a ldquohorizontalrdquo composition operation for the 119899-arrows with compatible (co)domains that pre-
serves the simplicial action
We will now reintroduce119964119889119895 following this outline by exhibiting its graded set of 119899-arrows be-tween the objects + and minus
816 Definition (strictly undulating squiggles) Define a graded set of arrows between objects minus and+ whose 119899-arrows are strictly undulating squiggles on 119899 + 1 lines such as displayed below in the case119899 = 5
12345
minus
+
012345
+ 2 3 1 4 2 + minus 4 minus 3 2 + minus
The lines are labeled 0 1 hellip 119899 and the gaps between them are labeled minus 1 hellip 119899 + A squiggle muststart on the right-hand side and end on the left-hand side in either the gap minus or + The right-handstarting gap becomes the domain of the squiggle and the left-hand ending gap becomes its codomainthese conventions chosen to follow the usual composition order Each turning point of the squigglemust lie entirely within a single gap The qualifier ldquostrict undulationrdquo refers to the requirement thatadjacent turning points should be distinct and that they should oscillate up and down as we proceedfrom right to left
Formally the data of a strictly undulating squiggle on 119899 + 1 lines can be encoded by a string119886 = (1198860 1198861 hellip 119886119903minus1 119886119903) of letters in the set minus 1 2 hellip 119899 + corresponding to the gaps in which eachsuccessive turning point occurs whose width is the integer 119908(119886) ≔ 119903 subject to the following condi-tions
(i) The domain 119886119908(119886) and codomain 1198860 of 119886 are both in minus +(ii) If 1198860 = minus then for all 0 le 119894 lt 119908(119886) we have 119886119894 lt 119886119894+1 whenever 119894 is even and 119886119894 gt 119886119894+1
whenever 119894 is odd and if 1198860 = + then for all 0 le 119894 lt 119908(119886) we have 119886119894 gt 119886119894+1 whenever 119894 iseven and 119886119894 lt 119886119894+1 whenever 119894 is odd
817 Lemma The graded set of strictly undulating squiggles admits a right action of the morphisms in 120491 thatpreserves domains and codomains and a ldquohorizontalrdquo composition operation for arrows in the same degree with
201
compatible (co)domains that preserves the simplicial action Relative to the horizontal composition an 119899-arrowis atomic if and only if there are no instances of + or minus occurring in its interior While the faces of an atomicarrow need not be atomic the degeneracies of an atomic arrow always are
Proof The horizontal composition of 119899-arrows is given by horizontal juxtaposition of strictlyundulating squiggles on 119899 + 1 lines which produces a well-formed squiggle just when the codomainof the right-hand squiggle matches the domain of the left-hand squiggle
1234
minus
+
01234
+ 3 4 1 3 2 +
∘
1234
minus
+
01234
+ 2 3 1 3 minus
=
1234
minus
+
01234
+ 3 4 1 3 2 + 2 3 1 3 minus
This operation is clearly associative Moreover any strictly undulating squiggle admits a unique de-composition into squiggles that do not contain+ orminus in their interior sequences of gaps which provesthat any squiggle 119886 = (1198860 1198861 hellip 119886119903minus1 119886119903) with the property that 1198861 hellip 119886119903minus1 isin 1 hellip 119899 is atomic
The right action of the simplicial operators on strictly undulating squiggles is best described intwo cases If 120572∶ [119898] ↠ [119899] is an epimorphism then a strictly undulating squiggle on lines 0hellip 119899becomes a strictly undulating squiggle on lines 0hellip 119898 by replacing each line labeled 119894 isin [119899]with lineslabeled by each element of the fiber 120572minus1(119894) and then ldquopulling these lines apartrdquo to create new gaps
[5] [2]
0 1 0
2 3 4 1
5 2
120572
119886 ≔12
minus
+
012
+ 1 + 2 + minus 1 minus
119886 sdot 120572 ≔
12345
minus
+
012345
+ 2 + 5 + minus 2 minus
Note that ldquopulling apart linesrdquo does not create instances of + or minus in the interior sequence of gaps sothe action by degeneracy operators preserves atomic arrows
If 120572∶ [119898] ↣ [119899] is a monomorphism then a strictly undulating squiggle on lines 0hellip 119899 becomesa strictly undulating squiggle on lines 0hellip 119898 by removing the lines labelled by elements 119894 isin [119899] notin the image of 120572 and renumbering the lines that are in the image in sequence The original squigglewill still undulate between the new lines but may not do so ldquostrictlyrdquo mdash it is possible for the squiggleto turn around mutliple times in the same gap mdash but this is easily corrected by ldquopulling the string
202
tautrdquo
[2] [5]
0 11 42 5
120572
119886 ≔
12345
minus
+
012345
+ 2 3 1 4 2 + minus 5 2 4 3 + minus
119886 sdot 120572 ≔12
minus
+
012
+ 1 1 minus 1 1 + minus 2 1 1 1 + minus
≔12
minus
+
012
+ minus + minus 2 1 + minus
Note that in both of these cases the actions by epimorphisms and monomorphisms preserve domainsand codomains of squiggles and respect horizontal concatenations
Now in general a simplicial operator 120572∶ [119898] rarr [119899] can be factored uniquely in 120491 as an epimor-phism followed by a monomorphism so it acts on a strictly undulating squiggle on 119899+1 lines by firstldquoremoving lines and pulling tautrdquo and then by ldquoduplicating lines and pulling apartrdquo
We leave the formalization of these geometric descriptions of the actions of the simplicial opera-tors on strictly undulating squiggles presented as sequences satisfying the axioms of Definition 816(i)and (ii) to Exercise 81iii with the hint that a combinatorial description of this action can easily bedefined using the ldquointerval representationrdquo of 120491op
818 Digression (the interval representation) There is a faithful interval representation of the cat-egory 120491op
+ as a subcategory of 120491 in the form of a functor
120491op+ 120491
[119899 minus 1] [119899]
[119899] [119899 + 1]
[119899 + 1] [119899 + 2]
ir
120575119894 ↦
↦
120590119894
120575119894+1120590119894
whose image is the subcategory 120491perp⊤ ≔ 120491perp cap 120491⊤ sub 120491 of simplicial operators that preserve both thetop and bottom elements in each ordinal
If we think of the elements of [119899] as labelling the lines in the graphical representation of an119899-arrow then the elements of ir[119899] ≔ [119899 + 1] label the gaps with the bottom element 0 relabeled asminus and the top element 119899+1 relabeled as + If the elementary face operators 120575119894 and elementary degen-eracy operators 120590119894 act by removing and duplicating the lines in a squiggle diagram then the functorsir 120575119894 and ir 120590119894 describe the corresponding actions on the gaps
The graphical description of its 119899-arrows of119964119889119895 clearly exhibits a simplicial computad structureon a simplicial category we now more formally introduce following the outline of Observation 815
819 Definition (the free homotopy coherent adjunction as a simplicial computad) The free ho-motopy coherent adjunction119964119889119895 is the simplicial computad with
203
bull two objects + and minusbull whose 119899-arrows are strictly undulating squiggles on 119899 + 1 lines of Definition 816 oriented from
right to left with the right-hand starting position defining the domain object and the left-handending position defining the codomain object
bull with the simplicial operators acting as described in Lemma 817 andbull with horizontal composition defined by horizontal juxtaposition of squiggle diagrams
which means that the atomic 119899-arrows are those squiggles each of whose interior undulations occursbetween the lines 0 and 119899
We now reconcile our two descriptions of the free homotopy coherent adjunction comparingDefinition 819 with Definition 814 Zaganidis discovered a similar proof in his PhD thesis [65sect233] that improves upon some aspects of the authorsrsquo original argument
8110 Proposition The simplicial computad119964119889119895 is isomorphic to the 2-category119964119889119895
Proof We explain how the 119899-arrows in each of the four homs of the simplicial category119964119889119895 cor-respond to sequences of 119899 composable arrows in the corresponding hom-categories 120491+ 120491perp 120491⊤ and120491perp⊤ of the 2-category119964119889119895 Here it is most convenient to think of120491perp120491⊤ and120491perp⊤ as subcategoriesof 120491 the latter via the interval representation of Digression 818
To easily visualize the isomorphism shade the region under a strictly undulating squiggle on 119899+1lines to define a topological space 119878 embedded in the plane In the case of a squiggle from minus to minusthis shaded region includes both the left and right boundary regions of the squiggle diagram Foreach 119894 isin [119899] the connected components of the intersection 119878119894 of the shaded region with the regionabove the labelled 119894 define a finite linearly ordered set ordered from left to right by the order in whichthe intervals defined as the intersection of each component with the line 119894 appear on that line Theseordinals define the objects in the composable sequence of arrows in120491+ or one of its subcategory120491perp⊤120491perp or 120491⊤ corresponding to the squiggle diagramsup1 Formally the ordinal representing the 119894th objectin the sequence of 119899 arrows counts the number of ldquomaximal convex subsequencesrdquo of the sequence119886 = (1198860 hellip 119886119908(119886)) that encodes the squiggle a maximal convex subsequence is a maximal sequence ofconsecutive entries 119886119895 satisfying the condition 119886119895 le 119894
The intersections of the shaded region 119878 with the regions above the lines 119894 = 0hellip 119899 define anested sequence of subspaces
1198780 1198781 ⋯ 119878119899minus1 119878119899Taking path components yields a composable sequence
12058701198780 rarr 12058701198781 rarr⋯rarr 1205870119878119899minus1 rarr 1205870119878119899As explained above each of the sets 1205870119878119894 is linearly ordered from left to right by the positions inwhich each component of 119878119894 intersects the line labeled 119894 and the functions induced by the inclusionsare order-preserving This defines the composable sequence of arrows in 120491+ 120491perp 120491⊤ or 120491perp⊤ corre-sponding to a strictly undulating squiggle on 119899 + 1-lines
For instance suppose the shaded region under the squiggle diagram intersects the line labelled119894 in 119898 + 1 components corresponding to the ordinal [119898] and similarly suppose the shaded region
sup1In the case of a squiggle from + to + it is possible that the squiggle diagram does not intersect the line labeled 0 inwhich case the subspace 1198780 is empty if this is the case the squiggle may also fail to cross the next several lines In each ofthe other three hom-categories the ordinals defined by taking the connected components of the intersection of each linewith the shaded region are non-empty because either the left or right boundary of the squiggle diagram is also shaded
204
under the squiggle diagram intersects the line labelled 119894+1 in 119896+1 components corresponding to theordinal [119896] We identify the corresponding simplicial operator 120572∶ [119898] rarr [119896] by taking the fiber over119905 isin [119896] to be the subset of shaded regions 119904 isin [119898] above the 119894th line that belong to the same connectedcomponent as 119905 in the shaded region above the line 119894 + 1 Formally the element 119905 isin [119896] representsa maximal convex subsequence of 119886 = (1198860 hellip 119886119908(119886)) comprised of those 119886119895 le 119894 + 1 This convexsubsequence is partitioned into possibly smaller maximally convex subsequences satisfying the morerestrictive condition 119886119895 le 119894 and the elements 119904 isin [119898] indexing these subsequences form the fiber of 120572over 119905 Note that if the domain of the squiggle is minus then the top element of each ordinal is necessarilypreserved because the right-hand boundary of the squiggle diagram defines a connected shaded regionsimilarly if the codomain of the squiggle is minus then the bottom element of each ordinal is preservedThe constructs a map from the simplicial computad119964119889119895 of Definition 819 to the 2-category119964119889119895 ofDefinition 814
The converse map can be constructed by iterating a splicing operation that we now introduceThis splicing operation proves that each of the hom-spaces of the simplicial computad 119964119889119895 satisfiesa ldquostrict Segal conditionrdquo that says that the set of 119899 + 119898-arrows is isomorphic to the set of pairs of119899-arrows and 119898-arrows whose last and first vertices coincide In more detail if 119886 and 119887 are strictlyundulating squiggles on 119899 + 1 and 119898 + 1 lines that lie in the same hom-space with the property thatthe 119899th vertex of 119886 coincides with the 0th vertex of 119887 then these squiggle diagrams can be uniquelyspliced to form a strictly undulating squiggle 119888 on 119899 +119898+ 1 lines whose face spanned by the vertices0hellip 119899 is 119886 and whose face spanned by the vertices 119899hellip 119898 + 119899 is 119887
119886 ≔123
minus
+
0123
minus + 1 3 minus 2 1 + 2 +
119887 ≔
1234
minus
+
01234
minus 2 1 3 minus 4 1 + 2 3 minus +
119888 ≔
1234567
minus
+
01234567
minus 5 4 6 1 3 minus 2 1 7 4 + 5 6 2 +
This splicing operation is defined graphically by separating the squiggle diagrams for 119886 and for 119887 intoan ordered sequence of components by cutting the squiggle 119886 each time it enters or leaves the gapmarked ldquo+rdquo removing the dotted arcs at the bottom of the squiggle diagram and cutting the squiggle119887 each time it enters or leaves the gap marked ldquominusrdquo removing the dotted arcs at the top of the depictedsquiggle diagram This requires two cuts for each occurrence of ldquo+rdquo or ldquominusrdquo in the interior of 119886 or 119887respectively and one cut for each occurrence of ldquo+rdquo or ldquominusrdquo as the source or target of 119886 or 119887 respectivelyThe condition that the 119899th vertex of 119886 coincides with the 0th vertex of 119887 ensures that the numbers ofcuts made to 119886 (the number of times 119886 intersects the 119899th line) and 119887 (the number of times 119887 intersectsthe 0th line) are the same Now form 119888 by sewing together these squiggle components in order to formthe crossings of the line labeled ldquo119899rdquo
By iterating the splicing operation we can construct a strictly undulating squiggle on 119899 + 1 linesfrom a sequence of 119899 composable arrows once we know how to encode a single 1-arrow of 119964119889119895 as astrictly undulating squiggle on 2 lines We give full details in the case of a 1-arrow from + to + A
205
simplicial operator 120572∶ [119898] rarr [119896] defines a strictly undulating squiggle on 2 lines from + to + by asequence 119886 with ldquominusrdquo occurring 119898 + 1 times (each occurrence of which corresponds to an intersectionof the shaded region and the 0th line) and ldquo+rdquo occurring 119896 + 2 times (each consecutive pair boundingan intersection of the shaded region and the 1st line) The strings between each consecutive pair of+s correspond to the elements 119894 isin [119896] If the fiber over 119894 is empty the sequence is ldquo+1+rdquo If the fibercontains a single element the sequence is ldquo+ minus 1 minus +rdquo and if the fiber contains 119904 gt 1 elements thesequence is ldquo+ minus (1minus)119904minus1+rdquo
Note that the planar orientation of a strictly undulating squiggle diagram makes the numerical la-bels for the lines gaps and the sequence of undulation points redundant Going forward we typicallyomit them
8111 Example (adjunction data in 119964119889119895) For later use we name some of the low-dimensional non-degenerate atomic arrows in119964119889119895 There exist just two atomic 0-arrows which we call
119891 ≔ and 119906 ≔
Since119964119889119895 is a simplicial computad all of its other 0-arrows may be obtained as a unique alternatingcomposite of these two for example
119891119906119891119906119891 ≔
There are exactly two non-degenerate atomic 1-arrows in119964119889119895 these being
120578 ≔ and 120598 ≔
Again since 119964119889119895 is a simplicial computad all of its other 1-arrows are uniquely expressible as hori-zontal composites of the 1-arrows 120578 and 120598 with degenerated 1-arrows obtained from 119891 and 119906 such asfor example
119906119891120578 ≔ and 120598120598 ≔
There exist two non-degenerate atomic 2-arrows with minimal width whose faces are easily com-puted
120572 ≔ 120572 sdot 1205752 ≔ 119891120578 ≔ 120572 sdot 1205751 ≔ 119891 ≔ 120572 sdot 1205750 ≔ 120598119891 ≔
120573 ≔ 120573 sdot 1205752 ≔ 120578119906 ≔ 120573 sdot 1205751 ≔ 119906 ≔ 120573 sdot 1205750 ≔ 119906120598 ≔
For each width119908 ge 3 there exist exactly two non-degenerate atomic 2-arrows the description ofwhich we leave to Exercise 81v There are countably many atomic non-degenerate 119899-arrows in eachdimension 119899 ge 2 which we shall partially enumerate in the proof of Proposition 8211
In the ldquohomotopy 2-categoryrdquo of the simplicial category119964119889119895 which is isomorphic to the 2-category119964119889119895 the atomic 2-arrows 120572 and 120573witness the triangle equalities proving that this 2-category containsan adjunction (119891 119906 120578 ∶ id+ rArr 119906119891 120598 ∶ 119891119906 rArr idminus)
206
8112 Lemma The simplicial computad119964119889119895 contains a distinguished adjunction
+ minus
119891
⟂119906
with unit 120578∶ id+ rarr 119906119891 isin 119964119889119895(+ +) and counit 120598 ∶ 119891119906 rarr idminus isin 119964119889119895(minus minus)
Proof By Proposition 8110 this follows from Lemma 812 though the reader may prefer to provethis result directly
We can now prove the universal property of the free adjunction 119964119889119895 claimed by Schanuel andStreet
8113 Proposition For any 2-category119966 2-functors119964119889119895 rarr 119966 correspond to adjunctions in the 2-category119966
Proof Lemma 812 identifies an adjunction in 119964119889119895 that we denote in the notation of Lemma8112 by (119891 119906 120578 ∶ id+ rArr 119906119891 120598 ∶ 119891119906 rArr idminus) A 2-functor119964119889119895 rarr 119966 carries this data to an adjunctionin the 2-category 119966
Conversely we suppose that119966 is equipped with an adjunction (119891 119906 120578 ∶ id119861 rArr 119906119891 120598 ∶ 119891119906 rArr id119860)To construct a 2-functor 119964119889119895 rarr 119966 it suffices because the embedding 2-119966119886119905 119982119982119890119905-119966119886119905 is fullyfaithful to consider both 2-categories as simplicial categories via Lemma 813 and instead define asimplicial functor119964119889119895 rarr 119966 see Exercise 81i Now since119964119889119895 is a simplicial computad and the homsof 119966 are 2-coskeletal it suffices to define a simplicial functor sk2119964119889119895 rarr 119966 from the subcomputad of119964119889119895 generated by its atomic 0- 1- and 2-arrows This functor is given by the mappingbull + ↦ 119861 and minus ↦ 119860 on objectsbull 119891 ↦ 119891 and 119906 ↦ 119906 on atomic 0-arrowsbull 120578 ↦ 120578 and 120598 ↦ 120598 on atomic non-degenerate 1-arrows
and for each of the atomic non-degenerate 2-arrows of Exercise 81v we must by Lemma 813 verifythat the 2-cells in 119966 defined by their boundaries compose vertically as indicated
For 120572 and 120573 the unique non-degenerate atomic 2-arrows of minimal width 3 the required com-position relations are
120598119891 sdot 119891120578 = id119891 and 119906120598 sdot 120578119906 = id119906which hold by the triangle equalities for the adjunction in 119966 For the atomic 2-arrows of odd width2119903 + 1 the required composition relations are the ldquohigher order triangle equalitiesrdquo
120598119903119891 sdot 119891120578119903 = id119891 and 119906120598119903 sdot 120578119903119906 = id119906which can easily be seen to hold by depicting the left-hand expressions as pasting diagramssup2 For theatomic 2-arrows of even with 2119903 the required composites are closely related ldquohigher order triangleequaltiesrdquo
120598119903+1 sdot 119891120578119903119906 = 120598 and 119906120598119903 sdot 120578119903+1 = 120578which again can easily be seen to hold by depicting the left-hand expressions as pasting diagrams
sup2Here 120578119903 refers to the horizontal composite of 119903 copies of the unit and the ldquosdotrdquo expresses a vertical composite of awhiskered copy of this with a whiskered copy of 120598119903 the horizontal composite of 119903-copies of the counit The 1-cell codomainof the former and 1-cell domain of the latter fit together like a cartoon depiction of a closed mouth of pointy teeth
207
Motivated by this result and the fact that the simplicial category119964119889119895 is a simplicial computad weuse it to define a notion of homotopy coherent adjunction in any quasi-categorically enriched categoryand in particular in anyinfin-cosmos
8114 Definition A homotopy coherent adjunction in a quasi-categorically enriched category119974 isa simplicial functor119964119889119895 rarr 119974 Explicitly it picks outbull a pair of objects 119860119861 isin 119974bull together with four homotopy coherent diagrams
120491+ rarr Fun(119861 119861) 120491op+ rarr Fun(119860119860) 120491⊤ rarr Fun(119860 119861) 120491perp rarr Fun(119861119860) (8115)
that are functorial with respect to the composition action of119964119889119895The 0- and 1-dimensional data of the these diagrams has the form displayed in (801) (802) and(803) We interpret the homotopy coherent diagrams (8115) as defining homotopy coherent versionsof the bar and cobar resolutions of the adjunction (119891 ⊣ 119906 120578 120598)
Exercises
81i Exercise Prove that the embedding 2-119966119886119905 119982119982119890119905-119966119886119905 defined by Lemma 813 is fully faith-ful prove that simplicial functors119964rarr ℬ between 2-categories define 2-functors
81ii Exercise Use Lemma 813 to prove that the homotopy coherent 120596-simplex ℭΔ[120596] is a 2-cat-egorysup3
81iii Exercise Describe the action of a general simplicial operator 120572∶ [119898] rarr [119899] on a strictlyundulating squiggle on 119899 + 1 lines represented by a sequence 119886 = (1198860 1198861 hellip 119886119903minus1 119886119903) of ldquogapsrdquo 119886119894 isinminus 1 hellip 119899 +
81iv Exercise Give a graphical interpretation of the dualities
119964119889119895(minus minus) cong 119964119889119895(+ +)op and 119964119889119895(minus +) cong 119964119889119895(+ minus)op
of Definition 814
81v Exercise(i) Describe the non-degenerate atomic 2-arrows of 119964119889119895 and compute their faces and compare
your results with the composition relations appearing in the last paragraph of the proof ofProposition 8113
(ii) Describe the degenerate atomic 2-arrows of119964119889119895 and ldquocomputerdquo their faces
81vi Exercise Use the graphical calculus presented in Definition 819 to verify the following ob-servation of Karol Szumiło the simplicial category 119964119889119895 is isomorphic to the Dwyer-Kan hammocklocalization [17] of the category consisting of two objects + and minus and a single non-identity arrow+ ⥲ minus that is a weak equivalence
81vii Exercise For any homotopy coherent adjunction as in Definition 8114 define internal ver-sions of monad resolution the comonad resolution the bar resolution and the comonad resolution
119861(119906119891)bullminusminusminusminusrarr 119861120491+ 119860
(119891119906)bullminusminusminusminusrarr 119860120491op+ 119860 barminusminusrarr 119861120491⊤ 119861 cobarminusminusminusminusrarr 119860120491perp
sup3In light of Lemma 813 Theorem 64 of [46] proves more generally that the simplicial computads defined as freeresolutions of strict 1-categories are always 2-categories
208
and explore the relationships between these functors⁴
82 Homotopy coherent adjunction data
Any homotopy coherent adjunction in an infin-cosmos or general quasi-categorically enriched cat-egory has an underlying adjunction in its homotopy 2-category Remarkably this low-dimensionaladjunction data may always be extended to give a full homotopy coherent adjunction by repeated in-voking the universal property of the unit as expressed in Proposition 832 In this section we filterthe free homotopy coherent adjunction 119964119889119895 by a sequence of ldquoparentalrdquo subcomputads which mustcontain for each atomic 119899-arrow with codomain ldquominusrdquo its ldquofillable parentrdquo the atomic 119899+1-arrow withcodomain ldquo+rdquo obtained by whiskering with 119906 and the ldquoprecomposingrdquo with 120578
In sect83 we then use this filtration to prove that any adjunction in aninfin-cosmos mdash or more preciselyany diagram indexed by a parental subcomputad mdash extends to a homotopy coherent adjunction Ourproof is essentially constructive enumerating the choices necessary tomake each stage of the extensionIn sect84 we give precise characterizations of the homotopical uniqueness of such extensions provingthat the appropriate spaces of extensions are contractible Kan complexes Via the 2-categorical self-duality of 119964119889119895 described in Remark 8212 there is a dual proof that instead exploits the universalproperty of the counit the main steps of which are alluded to in the exercises
Our proof that any adjunction extends to a homotopy coherent adjunction inductively specifiesthe data in the image of a homotopy coherent adjunction by choosing fillers for horns correspondingto ldquofillablerdquo arrows
821 Definition An arrow 119887 of119964119889119895 is fillable ifbull it is non-degenerate and atomicbull its codomain 1198870 = + andbull 119887119894 ne 1198871 for 119894 gt 1
001
012
0123
0123
0123
01234
Write Fill119899 sub Atom119899 for the subset of fillable 119899-arrows of the subset of atomic and non-degenerate119899-arrows
822 Definition (distinguished faces of fillable arrows) On account of the graphical calculus werefer to ℎ(119887) ≔ 1198871 minus 1 an integer in 0 hellip 119899 minus 1 that labels the line immediately above the positionof the left-most turn-around as the height of the fillable arrow 119887⁵ The fillability of 119887 implies that noldquotauteningrdquo is required in computing the distinguished codimension-one face 119887 sdot 120575ℎ(119887) which then isnon-degenerate and has the same width as 119887 Further analysis of this face differs by case
⁴For instance there is a commutative diagram119860 119861120491⊤
119861 119861120491+
bar
119906 res
(119906119891)bull
⁵The unique fillable 0-arrow 119906 behaves somewhat differently but nonetheless it is linguistically convenient to includeit among the fillable arrows
209
bull case ℎ(119887) gt 0 The face 119887 sdot 120575ℎ(119887) is non-degenerate and atomic but it is not fillable since non-degeneracy implies that there is some 119894 gt 1 with 119887119894 = ℎ(119887) whence the entries of 119887 sdot 120575ℎ(119887) at 1 and119894 both equal ℎ(119887)
bull case ℎ(119887) = 0 The face 119887 sdot 120575ℎ(119887) decomposes as 119906 sdot 119886 where 119886 is non-degenerate atomic has widthone less than the width of 119887 and has codomain minus
We write
119887⋄ ≔ 1114108119887 sdot 120575ℎ(119887) ℎ(119887) gt 0119886 ℎ(119887) = 0
for the non-degenerate atomic ℎ(119887)th face in the positive height case and for the non-degenerateatomic factor of the 0th face 119906119886 in the height 0 case and refer to the atomic 119899 minus 1-arrow definedin either case as the distinguished face of the fillable arrow
We now argue that any non-degenerate and atomic 119899-arrow of119964119889119895 that is not fillable arises as thedistinguished face of a unique fillable 119899+ 1-arrow in the form of the first case just described when itscodomain is + and in the form of the second case just described when its codomain is minus
823 Lemma (identifying fillable parents)(i) If 119887 is a non-degenerate and atomic 119899-arrow of119964119889119895 with codomain + that is not fillable then it is the
codimension-one face of exactly two fillable (119899 + 1)-arrows with the same width both of which has 119887as its 1198871th face one which has height 1198871 that we refer to as its fillable parent and denote by 119887
dagger and theother which has height 1198871 minus 1
(ii) If 119886 is a non-degenerate and atomic 119899-arrow of119964119889119895 with codomain minus then the composite arrow 119906119886 isa codimension-one face of exactly one fillable (119899 + 1)-arrow which we call the fillable parent of 119886 anddenote by 119886dagger The fillable parent 119886dagger has width one greater than the width of 119886 has height 0 and has119906119886 as its 0th face
Together these cases define a ldquofillable parentdistinguished facerdquo bijection
Atom119899Fill119899 Atom119899+1
(minus)dagger
cong(minus)⋄
between fillable 119899 + 1-arrows and non-degenerate atomic 119899-arrows which are not fillable
Proof For 82 if 119887 is a non-degenerate atomic 119899-arrow with domain + that is not fillable thenit must be the case that 119887119894 = 1198871 for some 119894 gt 1 This arrow is then a codimension-one face of thetwo atomic 119899 + 1-arrows that are formed by inserting an extra line into the gap labeled 1198871 separatethe entry 1198871 from the other turn-arounds that occur in the same gap the 119899 + 1-arrows obtained inthis way will then clearly have 119887 as their 1198871th face There are exactly two ways to do this as illustratedbelow
119887 ≔012
119887dagger ≔0123
or
0123
For (ii) given a non-degenerate atomic 119899-arrow 119886 with codomain minus there is a unique way to make119906119886 into the face of an atomic 119899 + 1-arrow with domain + whose width is only one greater by adding
210
an extra line in the upper-most space above the squiggle diagram
119886 ≔012
119906119886 ≔012
119886dagger ≔0123
824 Definition A subcomputad119964 of119964119889119895 is parental if it contains at least one non-identity arrowand satisfies the conditionbull if 119888 is a non-degenerate atomic arrow in119964 then either it is fillable or its fillable parent 119888dagger is also
in119964
The condition implies that any parental subcomputad 119964 sub 119964119889119895 contains at least one fillablearrow The last vertex of any fillable arrow has the form 119906119886 for some 0-arrow 119886 so it follows that the0-arrow 119906 is contained in any parental subcomputad
Recall from Definition 618 that any collection of arrows 119878 in a computad generates a minimalsubcomputad 119878
825 Example (notable parental subcomputads)
bull The unique fillable 0-arrow119906 generates theminimal parental subcomputad 119906 sub 119964119889119895 containingonly 119906 and its degenerate copies
bull The unit 1-arrow 120578 is fillable and the subcomputad 120578 sub 119964119889119895 it generates has 119906 119891 and 120578 as itsonly non-degenerate atomic arrows Of these 119906 and 120578 are parental and 120578 is the fillable parent of119891 so this subcomputad is parental
bull The triangle identity witness 120573 of Example 8111 is fillable and the subcomputad 120573 sub 119964119889119895 itgenerates has 119906 119891 120578 120598 and 120573 as its only non-degenerate atomic arrows Since 120573 is the fillableparent of 120598 this subcomputad is parental
bull The pair of fillable 3-arrows
120596 ≔ and 120591 ≔
generate a subcomputad 120596 120591 sub 119964119889119895 that has 119906 119891 120578 120598 both triangle identity witnesses 120573 and
120572 of Example 8111 and 120584 ≔ 120596 sdot 1205751 = 120591 sdot 1205751 as its only non-degenerate atomic arrows Since 120596is the fillable parent of 120572 and 120591 is the fillable parent of
120584 ≔
this subcomputad is parentalbull 119964119889119895 is trivially a parental subcomputad of itself
826 Non-Examplebull The subcomputad 120598 has 119906 119891 and 120598 as its only non-degenerate atomic arrows Since both 119891 and120598 are missing their fillable parents this subcomputad is not parental
211
bull The subcomputad 120578 120598 that has 119906 119891 120578 and 120598 as its only non-degenerate atomic arrows still failsto be parental since the fillable parent of 120598 is missing
bull The subcomputad 120573 120572 that has 119906 119891 120578 120598 and both triangle identity witnesses 120573 and 120572 as itsatomic non-degenerate arrows is not parental since the fillable parent 120596 of 120572 is missing
These examples establish a chain of parental subcomputad inclusions
119906 sub 120578 sub 120573 sub 120596 120591 sub 119964119889119895
Our aim in the remainder of this section is to filter a general parental subcomputad inclusion 119964 sub119964prime as a countable tower of parental subcomputad inclusions with each sequential inclusion pre-sented as the pushout of an explicit simplicial subcomputad inclusion The subcomputad inclusions120794[Λ119896[119899]] 120794[Δ[119899]] from Example 615 will feature where Λ119896[119899] is an inner horn A second familyof inclusions will also be needed which we now introduce
827 Definition For any simplicial set119880 let 120795[119880] denote the simplicial category whose three non-trivial homs are displayed in the following cartoon
minus
⊤ +
120793119880
120793⋆119880
That is 120795[119880] has objects ldquo⊤rdquo ldquominusrdquo and ldquo+rdquo and non-trivial hom-sets
120795[119880](⊤ minus) ≔ 119880 120795[119880](minus +) ≔ 120793 120795[119880](⊤ +) ≔ 120793 ⋆ 119880and whose only non-trivial composition operation is defined by the canonical inclusion
120795[119880](minus +) times 120795[119880](⊤ minus) 120795[119880](⊤ +)
119880 120793 ⋆ 119880
cong cong
Here we define the endo hom-spaces to contain only the respective identities and the remaining hom-spaces to be empty
828 Lemma A simplicial functor 120795[119880] rarr 119974 is uniquely determined by the databull a pair of 0-arrows 119906∶ 119860 rarr 119861 and 119887 ∶ 119883 rarr 119861bull a cone with summit 119880 over the cospan
119880 119887Fun(119883 119861)
Fun(119883119860) Fun(119883 119861)
120587
119906lowast
or equivalently a map 119880 rarr 119887119906lowast whose codomain is its pullback
Proof The objects119883119860 and 119861 are the images of⊤ minus and+ respectively The map 119906∶ 119860 rarr 119861 isthe image of the unique 0-arrow from minus to + Simplicial functoriality then demands the specification
212
of the vertical maps below making the square commute
119880 120793 ⋆ 119880
Fun(119883119860) Fun(119883 119861)119906lowast
By the join ⊣ slice adjunction of Proposition 425 the simplicial map 120793 ⋆ 119880 rarr Fun(119883 119861) may bedefined by specifying a 0-arrow 119887 ∶ 119883 rarr 119861 the image of the cone point of 120793⋆119880 together with a map119880 rarr 119887Fun(119883 119861) Now the above commutative square transposes to the one of the statement
Similarly a simplicial functor120794[119880] rarr 119974 is uniquely determined by the data119880 rarr Fun(119883119860) of asimplicial map valued in one of the functor spaces of119974 In particular simplicial functors 120794[Δ[119899]] rarr119974 correspond to 119899-arrows in119974
829 Notationbull For each fillable 119899-arrow 119887 of positive height let 119865119887 ∶ 120794[Δ[119899]] rarr 119964119889119895 be the simplicial functor
classified by the 119899-arrow 119887 of119964119889119895bull For each fillable 119899-arrow 119887 of height 0 let 119865119887 ∶ 120795[Δ[119899 minus 1]] rarr 119964119889119895 be the simplicial functor
defined on objects by minus ↦ minus + ↦ + and ⊤ ↦ 119887119908(119887) the domain of 119887 and on the threenon-trivial homs by
120795[Δ[119899 minus 1]](⊤ minus) cong Δ[119899 minus 1] 119964119889119895(119887119908(119887) minus) 120795[Δ[119899 minus 1]](minus +) cong Δ[0] 119964119889119895(minus +)119887⋄ 119906
120795[Δ[119899 minus 1]](⊤ +) cong Δ[119899] 119964119889119895(119887119908(119887) +)119887
8210 Lemma (extending parental subcomputads) Suppose119964 sub 119964119889119895 is a parental subcomputad and 119887is a fillable 119899-arrow of height 119896 that is not a member of119964 but whose faces 119887 sdot 120575119894 are in119964 for all 119894 ne 119896 Thenthe subcomputad119964prime ≔ 119964 119887 generated by119964 and 119887 is defined by the pushout on the left below in the case119896 gt 0 and on the right below in the case 119896 = 0
120794[Λ119896[119899]] 119964 120795[120597Δ[119899 minus 1]] 119964
120794[Δ[119899]] 119964prime 120795[Δ[119899 minus 1]] 119964prime
119865119887
119865119887
119865119887 119865119887
and in both cases119964prime is again a parental subcomputad
Proof Because 119887 is the fillable parent of its distinguished face 119887⋄ this non-degenerate atomicnon-fillable (119899 minus 1)-arrow cannot be a member of the parental subcomputad119964 Since the other facesof 119887 are assumed to belong to 119964 119887 and 119887⋄ are the only two atomic arrows that are in 119964prime but notin 119964 Since the first is fillable and the second has the first as its fillable parent it is clear that thesubcomputad119964prime is again parental
To verify the claimed pushouts we considerwhat is required to extend a simplicial functor119865∶ 119964 rarr119974 to a simplicial functor 119865∶ 119964prime rarr119974 In the positive height case all that is needed for such an exten-sion is an 119899-arrow 119891 in119974 with the property that 119891 sdot 120575119894 = 119865(119887 sdot 120575119894) for all 119894 ne 119896 which may be specified
213
by a simplicial functor 119891∶ 120794[Δ[119899]] rarr 119974 that makes the following square commute
120794[Λ119896[119899]] 119964
120794[Δ[119899]] 119974
119865119887
119865
119891
If the height of 119887 is zero then both its 0th face 119887 sdot 1205750 = 119906 sdot119886 and its atomic non-degenerate factor 119886are missing from119964 So to extend a simplicial functor 119865∶ 119964 rarr 119974 to a simplicial functor 119865∶ 119964prime rarr119974requires an (119899 minus 1) arrow 119892 and an 119899-arrow 119891 in 119974 so that 119892 sdot 120575119894 = 119865(119886 sdot 120575119894) for all 119894 isin [119899 minus 1] and119891 sdot 120575119894 = 119865(119887 sdot 120575119894) for all 119894 ne 0 isin [119899] so that 119891 sdot 1205750 = 119865119906 sdot 119892 By Lemma 828 this data may be specifiedby a simplicial functor 119891∶ 120795[Δ[119899 minus 1]] rarr 119974 that makes the following square commute
120795[120597Δ[119899 minus 1]] 119964
120795[Δ[119899 minus 1]] 119974
119865119887
119865
119891
8211 Proposition Any inclusion119964119964prime of parental subcomputads of119964119889119895 may be filtered as a count-able tower of parental subcomputad inclusions
119964 = 1199640 1199641 ⋯1113996119894ge0
119964119894 = 119964prime
in such a way that for each 119894 ge 1 there is a finite non-empty set 119878119894 of fillable arrows that are not themselvescontained in but which have all faces except the distinguished one contained in 119964119894minus1 so that the parentalsubcomputad119964119894 is generated by119964119894minus1 cup 119878119894 Hence the inclusion119964 119964prime may be expressed as a countablecomposite of inclusions which are constructed by pushouts of the form
⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝∐119887isin119878119894ℎ(119887)gt0
120794[Λℎ(119887)[dim(119887)]]
⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠⊔
⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝∐119887isin119878119894ℎ(119887)=0
120795[120597Δ[dim(119887)]]
⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠
119964119894minus1
⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝∐119887isin119878119894ℎ(119887)gt0
120794[Δ[dim(119887)]]
⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠⊔
⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝∐119887isin119878119894ℎ(119887)=0
120795[Δ[dim(119887)]]
⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠
119964119894
⟨119865119887⟩119887isin119878119894
⟨119865119887⟩119887isin119878119894
Proof Let 119878 denote the set of fillable arrows in119964prime which are not in119964 and let 119878119908119899119896 denote thesubset of arrows with width 119908 dimension 119899 and height 119896 Note that any non-degenerate arrow of119964119889119895must have dimension strictly less than its width and there are only finitely many non-degeneratearrows of any given width The height is strictly less than the dimension so each set 119878119908119899119896 is finiteand if non-empty we must have 119896 lt 119899 lt 119908 Now we order the triples (119908 119899 119896) that index non-emptysubsets of fillable arrows lexicographically by increasing width increasing dimension and decreasingheight and let 119878119894 ≔ 119878119908119894119899119894119896119894 denote subset of fillable arrows in the 119894th triple in this ordering for 119894 ge 1
214
Let 119964119894 be the subcomputad of 119964119889119895 generated by 119964 cup (⋃119894119895=1 119878119894) By construction this family filters
the inclusion of119964 into119964prime = 119964 cup (⋃119894 119878119894)We complete the proof by induction on the index starting from the parental subcomputad1199640 =
119964 For the inductive step we must verify that all but the distinguished face of each 119887 isin 119878119894 ≔ 119878119908119894119899119894119896119894lie in119964119894minus1 By iterating Lemma 8210 it will follow that119964119894 is again parental
By construction119964119894minus1 is the smallest subcomputad of119964119889119895 which contains119964 and all of the fillablearrows of119964prime which havebull width less that 119908119894 orbull width 119908119894 and dimension less than 119899119894 orbull width 119908119894 dimension 119899119894 and height greater than 119896119894
Each fillable arrow 119887 isin 119878119894 has width119908119894 dimension 119899119894 and height 119896119894 We must show that its faces 119887 sdot 120575119895lie in119964119894minus1 for each 119895 ne 119896119894 which we verify by a tedious but straightforward case analysis
case 119895 ne 119896119894 + 1 If 119895 ne 119896119894 + 1 then since 119895 ne 119896119894 the line numbered 119895 is not one of the onesseparating the gap 1198871 from the other entries of 119887 which means that this entry will not be eliminatedwhen computing the 119895th face Consider the unique factorization
119887 sdot 120575119895 = (1198871 sdot 1205721) ∘ ⋯ ∘ (119887119903 sdot 120572119903)of the face 119887 sdot120575119895 into non-degenerate and atomic arrows of119964prime acted upon by degeneracy operators Bythe analysis just given 1198871 is fillable with width at most 119908119894 height 119896119894 or 119896119894 minus 1 depending on whether119895 gt 119896119894 +1 or 119895 lt 119896119894 and dimension less than 119899119894 Thus 1198871 it is contained in119964119894minus1 by the hypothesis thatthis subcomputad contains all fillable arrows of width at most 119908119894 and dimension less than 119899119894 Since1198871 has width at least 2 the other atomic factors have width less than or equal to 119908119894 minus 2 Thus each ofthese is either a fillable arrow with width at most 119908119894 minus 2 which means that it is in119964119894minus1 or its fillableparent in119964prime has width at most 119908119894 minus 1 which means that this fillable parent is also in119964119894minus1 As 119887 sdot 120575119895is a composite of degenerate images of arrows in119964119894minus1 it too lies in119964119894minus1
case 119895 = 119896119894+1 Since 119887 notin 119964119894minus1 and all parental subcomputads contain the unique fillable 0-arrow119906 we know that 119887 ne 119906 and so has width at least 2 and dimension at least 1 The only fillable arrowof width 2 is 120578 which has depth 0 and face 120578 sdot 1205751 = id+ which certainly lies in119964119894minus1 so we can safelyassume that the width of 119887 is at least 3 which forces the height 119896119894 to be at most 119899119894 minus 2 otherwise 119887would not be atomic Thus 119895 is at most 119899119894 minus 1 which tells us that 119887 sdot 120575119895 is again atomic Now we havetwo subcasesbull case 1198872 = 1198871 + 1 In this case the line 119895 = 1198961 + 1 = 1198871 separates the gaps 1198871 and 1198872 so the face119887 sdot 120575119895 will have width at least two less than the width of 119887 This face may also be degenerate but inany case the fillable parent of the non-degenerate arrow that it represents has width less than 119908119894and so this face is present in119964119894minus1
bull case 1198872 gt 1198871 + 1 In this case the line 119895 = 1198961 + 1 = 1198871 separates the gap 1198871 from the gapimmediately below it which contains neither 1198870 nor 1198872 So the face 119887 sdot 120575119895 has width 119908119894 and isnon-degenerate Because 119887 is non-degenerate there is some 119904 gt 2 so that 119887119904 = 1198871 + 1 so 119887 sdot 120575119895is not fillable Now Lemma 823 tells us that its fillable parent has width 119908119894 dimension 119899119894 andheight 119896119894 + 1 Thus this fillable parent lies in119964119894minus1 by our hypothesis that that119964119894minus1 contains allfillable arrows of119964prime of width 119908119894 dimension 119899119894 and height greater than 119896119894 so 119887 sdot 120575119895 must also bein there as desired
215
8212 Remark (a dual form) In the 2-category 119964119889119895co the 0-arrow 119906 is left adjoint to the 0-arrow119891 with unit 120598 and counit 120578 The 2-functor 119964119889119895 rarr 119964119889119895co that classifies this adjunction is an iso-morphism Via this duality we could have introduced a variant notion of ldquofillable 119899-arrowrdquo (a subsetof those 119899-arrows with codomain minus) and ldquoparental subcomputadrdquo of 119964119889119895 so that every parental sub-computad contained the 0-arrow 119891 Dualizing the argument of Example 825 the subcomputads
119891 sub 120598 sub 120572 would all be parental but the subcomputads of Example 825 would no longer be so
Exercises
82i Exercise Give a graphical description of the dual fillable 119899-arrows discussed in Remark 8212
83 Building homotopy coherent adjunctions
We now employ Proposition 8211 to prove that every adjunction in aninfin-cosmos119974 extends to ahomotopy coherent adjunction119964119889119895 rarr 119974which carries the canonical adjunction in119964119889119895 to the chosenadjunction in119974 The data used to present an adjunction in a quasi-categorically enriched category119974in the sense of Definition 211 mdash a pair of infin-categories 119860 and 119861 a pair of 0-arrows 119906 isin Fun(119860 119861)and 119891 isin Fun(119861119860) a pair of 1-arrows 120578∶ id119861 rarr 119906119891 isin Fun(119861 119861) and 120598 ∶ 119891119906 rarr id119860 isin Fun(119860119860)and witnesses to the triangle equalities given by a pair of 2-arrows
119906119891119906 119891119906119891
119906 119906 119891 119891119906120598 isin Fun(119860 119861)
120598119891isin Fun(119861119860)
120578119906120573
119891120578120572
mdash determines a simplicial functor119879∶ 120573 120572 rarr 119974whose domain is the subcomputad of119964119889119895 generatedby the triangle identity 2-arrows This functor then is defined by mapping minus to 119860 and + to 119861 andacting on the non-degenerate and atomic arrows in the way suggested by their syntax 119879(119906) ≔ 119906119879(119891) ≔ 119891 119879(120578) ≔ 120578 119879(120598) ≔ 120598 119879(120573) ≔ 120573 and 119879(120572) ≔ 120572 In Theorem 834 we prove that it isalways possible to extend the data (119891 ⊣ 119906 120578 120598) to a homotopy coherent adjunction 119964119889119895 rarr 119974 butunless 120573 and 120572 satisfy a coherence condition that will be described doing so might require making adifferent choice for one of these triangle equality witnesses
By Proposition 8211 extensions along parental subcomputad inclusions may be built inductivelyfrom two types of extension problem one of which involves attaching fillable arrows with positiveheight and the other of which involves attaching fillable arrows of height zero We prove that bothsimplicial subcomputad extension problems can be solved relatively against any simplicial functor119875∶ 119974 rarr ℒ between quasi-categorically enriched categories that defines a ldquolocal isofibrationrdquo mean-ing that the action of 119875 on functor spaces is by isofibrations This relative lifting result will be usedin the proof of the homotopical uniqueness of such extensions in sect84
831 Lemma Let 119875∶ 119974 rarr ℒ be a simplicial functor that defines a local isofibration of quasi-categoriesThen any lifting problem against the directed suspension of an inner horn inclusion
120794[Λ119896[119899]] 119974
120794[Δ[119899]] ℒ
119875
has a solution
216
Proof The lifting problem of the statement is equivalent to asking for a lift of the inner horninclusion
Λ119896[119899] Fun(119860 119861)
Δ[119899] Fun(119875119860 119875119861)against the action of 119875 on some functor spaces of119974 andℒ By hypothesis this action is an isofibrationof quasi-categories and so the claimed lift exists
832 Proposition (the relative universal property of the unit) If 119974 has an adjunction (119891 ⊣ 119906 120578)119875∶ 119974 rarr ℒ is a local isofibration of quasi-categorically enriched categories and if 119879∶ 120795[120597Δ[119899]] rarr 119974 isdefined to carry the unique 0-arrow minus rarr + to 119906∶ 119860 rarr 119861 the cone point of 120793 ⋆ 120597Δ[119899] cong Λ0[119899] to119887 ∶ 119883 rarr 119861 and the 1-arrow from 0 to 1 in Λ0[119899] to 120578119887∶ 119887 rarr 119906119891119887 for some 119899 ge 1 then any lifting problem
120795[120597Δ[119899]] 119974
120795[Δ[119899]] ℒ
119879
119875
has a solution
Proof By Lemma 828 the lifting problem of the statement translates to a lifting problem ofthe following form
120597Δ[119899] 119887119906lowast
Δ[119899] 119875119887119875119906lowast
119905
where the upper horizontal map carries the initial vertex 0 isin 120597Δ[119899] to the object 120578119887 isin 119887119906lowast which isinitial in there by Proposition 414 applied to the adjunction between functor spaces
Fun(119883 119861) Fun(119883119860)
119891lowast
perp
119906lowast
and Appendix C which proves that Joyalrsquos slices are fibered equivalent to comma quasi-categoriesThe right-hand vertical is an isofibration that preserves this initial element since it carries it to acomponent of the unit for the adjunction (119875119891 ⊣ 119875119906 119875120578) inℒ Lemma proven in Appendix F nowproves that the desired lift exists
Our theorems about extensions to homotopy coherent adjunctions will follow as special cases ofthe following relative extension and lifting result for parental subcomputads containing the unit ofan adjunction
833 Theorem Suppose 120578 sub 119964 sub 119964prime sub 119964119889119895 are parental subcomputads Then if 119875∶ 119974 rarr ℒ is alocal isofibration between quasi-categorically enriched categories and if119974 has an adjunction (119891 ⊣ 119906 120578) then
217
we may solve any lifting problem
119964 119974
119964prime ℒ
119879
119875
so long as 119879(119891) = 119891 119879(119906) = 119906 and 119879(120578) = 120578
Proof By Proposition 8211 the inclusion119964 119964prime can be fillered as a sequential composite ofpushouts of coproducts of maps of two basic forms so it suffices to demonstrate that we may solvelifting problems of the following two forms
120794[Λ119896[119899]] 119964 119974 120795[120597Δ[119899 minus 1]] 119964 119974
120794[Δ[119899]] 119964prime ℒ 120795[Δ[119899 minus 1]] 119964prime ℒ
119865119887
119879
119875
119865119887
119879
119875
119865119887 119865119887
for some fillable 119899-arrow 119887 isin 119964119889119895 of positive height in the left-hand case and of height zero in theright-hand case In the case of the left-hand lifting problem such lifts exist immediately from Lemma831
In the case of the right-hand lifting problem we must verify that the hypotheses of Proposition832 are satisfied which amounts to verifying that for any fillable 119899-arrow 119887 of height zero the functor119865119887 ∶ 120795[Δ[119899 minus 1]] rarr 119964119889119895 defined in Notation 829 in relavant part by the map
120795[Δ[119899 minus 1]](⊤ +) cong Δ[119899] 119964119889119895(119887119908(119887) +)119887
carries the 1-arrow from 0 to 1 in Δ[119899] to a component of the unit in 119964 sub 119964119889119895 This is easilyverified using the graphical calculus The image of the 1-arrow from 0 to 1 under the map 119887 ∶ Δ[119899] rarr119964119889119895(119887119908(119887) +) is simply the initial edge of 119887 which may be computed by removing all of the other linesfrom the strictly undulating squiggle diagram By the specifications of a fillable arrow with height 0there are only two possibilities depending on whether the domain of 119887 is + or minus as illustrated by thefollowing diagrams
119887 ≔
01234
11988712057501 ≔ 01 = 0
1
119887 ≔
01234
11988712057501 ≔ 01 = 0
1
Thus we see that 119887 sdot 12057501 = 120578119906 or 119887 sdot 12057501 = 120578 both of which satisfy the conditions required forProposition 832 to provide the desired lift
218
834 Theorem (homotopy coherent adjunctions) If 119974 is a quasi-categorically enriched category con-taining an adjunction (119891 ⊣ 119906 120578 120598 120573 120572)
(i) There exists a simplicial functor120120∶ 119964119889119895 rarr 119974 for which120120(119906) = 119906120120(119891) = 119891 and120120(120578) = 120578(ii) There exists a simplicial functor 120120∶ 119964119889119895 rarr 119974 for which 120120(119906) = 119906 120120(119891) = 119891 120120(120578) = 120578
120120(120598) = 120598 and120120(120573) = 120573(iii) If there exist a pair of 3-arrows 120596 and 120591 in119974
120596 ≔
119906119891
id119861 119906119891
119906119891119906119891
120578
120578
120578sdot120578 119906120598119891119906119891120578
120591 ≔
119906119891
id119861 119906119891
119906119891119906119891
120578
120578
120578sdot120578 119906120598119891120578119906119891
witnessing the coherence relations 1205961205750 = 119906120572 1205911205750 = 120573119891 1205961205751 = 1205911205751 1205961205752 = 1205911205752 = 1205781205901 with the3rd faces of these simplices defined by the pair of 2-arrows given by the horizontal composite
Δ[1] times Δ[1] Fun(119861 119861) times Fun(119861 119861) Fun(119861 119861)120578times120578 ∘
then there exists a simplicial functor120120∶ 119964119889119895 rarr 119974 for which120120(119906) = 119906 120120(119891) = 119891 120120(120578) = 120578120120(120598) = 120598120120(120573) = 120573120120(120572) = 120572120120(120596) = 120596 and120120(120591) = 120591
Coherence conditions of the form stated in ((iii)) appear in the definition of a biadjoint pair in astrongly bicategorically enriched category in [62 138]
Exercises
83i Exercise Unpack Proposition 832 in the case 119899 = 1 and 119899 = 2
83ii Exercise Use Remark 8212 to formulate the dual of Theorem 834 describing homotopycoherent adjunctions generated by a given counit
84 Homotopical uniqueness of homotopy coherent adjunctions
The homotopy 2-category of an infin-cosmos can be regarded as a simplicial category of the formdescribed in Lemma 813 in which context it is equipped with a canonical quotient functor119876∶ 119974 rarr120101119974 of simplicial categories Formally 119876 is a component of the counit of the adjunction described inDigression 142 It follows easily from the characterization of 2-categories in Lemma 813 that119876 is alocal isofibration
By Proposition 8113 any adjunction in the homotopy 2-category of aninfin-cosmos is representedby a unique simplicial functor119879∶ 119964119889119895 rarr 120101119974 To say that a homotopy coherent adjunction120120∶ 119964119889119895 rarr119974 lifts the adjunction in the homotopy 2-category means that120120 is a lift of 119879 along119876 Theorem 834proves that a lift of any adjunction in the homotopy 2-category 120101119974 can be constructed by specifyinga lift 119879∶ 120573 rarr 119974 of 119879∶ 119964119889119895 rarr 120101119974mdash this amounting to a choice of 1-arrows representing the unit
219
and counit and a 2-arrow witnessing one of the triangle equalities mdash and then extending along theparental subcomputad inclusion to define the homotopy coherent adjunction120120
120573 119974
119964119889119895 120101119974
119879
119876120120
119879
Note that the commutativity of the top left triangle implies the commutativity of the bottom rightone so instead of thinking of 120120 as a lift of 119879∶ 119964119889119895 rarr 120101119974 to a homotopy coherent adjunction wecan equally think of120120 as an extension of 119879∶ 120573 rarr 119974 to a homotopy coherent adjunction
In this section we will define the space of extensions of given adjunction data to a homotopycoherent adjunction Our main theorem in this section is that if the base diagram to be extended isindexed by a parental subcomputad then the space of lifts is a contractible Kan complex
The first step is to construct a (possibly large) simplicial hom-space between two simplicial cate-gories
841 Definition Define the cotensor ℒ119880 of a simplicial category ℒ with a simplicial set 119880 to bethe simplicial category with objℒ119880 ≔ objℒ and hom-spaces
ℒ119880(119883 119884) ≔ ℒ(119883 119884)119880Any simplicial set119880 defines a comonoid (119880 ∶ 119880 rarr 120793Δ∶ 119880 rarr 119880times119880)with respect to the cartesianproduct and diagonal maps so the endofunctor (minus)119880 ∶ 119982119982119890119905-119966119886119905 rarr 119982119982119890119905-119966119886119905 is a monad whose unitand multiplication are defined by restricting along these maps
For simplicial categories 119974 and ℒ let Icon(119974ℒ) denote the large simplicial sets defined by thenatural isomorphism
Icon(119974ℒ)119880 cong Icon(119974ℒ119880) ie Icon(119974ℒ)119899 ≔ Icon(119974ℒΔ[119899])The composition map
Icon(ℒℳ) times Icon(119974ℒ) Icon(119974ℳ)∘
is given by defining the composite of a pair of 119899-arrows 119865∶ 119974 rarr ℒΔ[119899] and 119866∶ ℒ rarr ℳΔ[119899] to bethe Kleisli composite 119899-arrow
119974 ℒΔ[119899] (ℳΔ[119899])Δ[119899] cong ℳΔ[119899]timesΔ[119899] ℳΔ[119899]119865 119866Δ[119899] Δlowast
842 Remark The vertices of Icon(119974ℒ) are simplicial functors119974rarr ℒ The name ldquoiconrdquo is chosenbecause the 1-simplices are analogous to the ldquoidentity component oplax natural transformationsrdquo in2-category theory as defined by Lack [34] In particular each 1-simplex 119974 rarr ℒΔ[1] or 119899-simplex119974rarr ℒΔ[119899] spans simplicial functors119974rarr ℒ that agree on objects
843 Lemma If 119875∶ 119974 ↠ ℒ is a local isofibration between quasi-categorically enriched categories and119868 ∶ 119964 ℬ is a simplicial subcomputad inclusion then if either 119875 is bijective on objects or 119868 is injective onobjects then the Leibniz map
1114024Icon(119868 119875) ∶ Icon(ℬ119974) ↠ Icon(119964119974) timesIcon(119964ℒ)
Icon(ℬℒ)
220
is an isofibration between quasi-categories
Proof A lifting problem of simplicial sets as below-left transposes into a lifting problem of sim-plicial categories below-right
119880 Icon(ℬ119974) 119964 119974119881
119881 Icon(119964119974) timesIcon(119964ℒ)
Icon(ℬℒ) ℬ 119974119880 timesℒ119880ℒ119881
119894 1114024Icon(119868119875) 119868 1114024119894119875
When 119875 is surjective on objects so is the simplicial functor 1114024119894 119875 and in general the action on homs isgiven by the Leibniz map
1114024119894 119875 ∶ Fun(119883 119884)119881 ↠ Fun(119883 119884)119880 timesFun(119875119883119875119884)119880
Fun(119875119883 119875119884)119881
which is a trivial fibration whenever 119880 119881 is an inner horn inclusion or is the inclusion 120793 120128By Definition 6111 to solve the right-hand lifting problem it suffices to solve the two lifting
problems
empty 119974119881 120794[120597Δ[119899]] 119974119881
120793 119974119880 timesℒ119880ℒ119881 120794[Δ[119899]] 119974119880 times
ℒ119880ℒ119881
1114024119894119875 1114024119894119875
where the left-hand lifting problem is not needed if 119868 is bijective on objects The right-hand liftingproblem can be solved because 1114024119894 119875 is a local trivial fibration and the left-hand lifting problem canbe solved whenever 119875 is surjective on objects
844 Corollary If 119974 is a quasi-categorically enriched category if 119964 is a simplicial computad and if119868 ∶ 119964 ℬ is a simplicial subcomputad inclusion then
Icon(119868119974) ∶ Icon(ℬ119974) ↠ Icon(119964119974)is an isofibration between quasi-categories
845 Lemma Suppose119875∶ 119974 rarr ℒ is a simplicial functor between quasi-categories that is locally conservativein the sense that each Fun(119860 119861) rarr Fun(119875119860 119875119861) reflects isomorphisms Then if119964 is a simplicial computadand119964 ℬ is a bijective-on-objects simplicial subcomputad inclusion then the Leibniz map
1114024Icon(119868 119875) ∶ Icon(ℬ119974) ↠ Icon(119964119974) timesIcon(119964ℒ)
Icon(ℬℒ)
is a conservative functor of quasi-categories
Proof In the language of marked simplicial sets we must show that any lifting problem below-left has a solution
120794 Icon(ℬ119974) 119964 119974120794♯
120794♯ Icon(119964119974) timesIcon(119964ℒ)
Icon(ℬℒ) ℬ 119974120794 timesℒ120794ℒ120794♯
1114024Icon(119868119875)
221
where 120794♯ represents invertible 1-arrows By adjunction this transposes to the lifting problem above-right As in the proof of Lemma 843 it suffices to consider the case where119964 ℬ is 120794[120597[Δ[119899]]] 120794[Δ[119899]] for some 119899 ge 0 in which case the lifting problem of simplicial categories reduces to one ofsimplicial sets
120597Δ[119899] Fun(119860 119861)120794♯ (120597Δ[119899] times 120794♯) cup (Δ[119899] times 120794) Fun(119860 119861)
Δ[119899] Fun(119860 119861)120794 timesFun(119875119860119875119861)120794
Fun(119875119860 119875119861)120794♯ Δ[119899] times 120794♯ Fun(119875119860 119875119861)
The marked simplicial sets on the left have the same underlying sets differing only in their markingsso by the hypothesis that Fun(119860 119861) rarr Fun(119875119860 119875119861) is conservative the result follows
846 Definition The space of homotopy coherent adjunctions in a quasi-categorically enrichedcategory119974 is
cohadj(119974) ≔ Icon(119964119889119895119974)
847 Proposition The space of homotopy coherent adjunctions in119974 is a Kan complex possibly a large one
Proof Since 119964119889119895 is a simplicial computad Corollary 844 implies that cohadj(119974) is a quasi-category By Corollary 1115 we need only show that all of its arrows are isomorphism Since a 1-arrowin a functor space of 119974 is an isomorphism if and only if it represents an invertible 2-cell in 120101119974 thequotient simplicial functor 119876∶ 119974 rarr 120101119974 is locally conservative and by Lemma 845 it suffices toshow that cohadj(120101119974) is a Kan complex
From Proposition 8113 we can extract an explicit description of cohadj(120101119974) that reveals that itis actually isomorphic the nerve of a 1-category itsbull objects are adjunctions (119891 ⊣ 119906 120578 120598) in 120101119974bull arrows (120601 120595) ∶ (119891 ⊣ 119906 120578 120598) rarr (119891prime ⊣ 119906prime 120578prime 120598prime) consist of a pair of 2-cells 120601∶ 119891 rArr 119891prime and120595∶ 119906 rArr 119906prime so that 120598prime sdot (120601120595) = 120598 and 120578prime = (120595120601) sdot 120578 and
bull identities and composition are given componentwiseThe isomorphisms in cohadj(120101119974) are those pairs (120601 120595) whose components 120601 and 120595 are both
invertible From the defining equations 120598prime sdot (120601120595) = 120598 and 120578prime = (120595120601) sdot 120578 it follows that the mate of 120595is an inverse to 120601 and the mate of 120601 is an inverse to 120595 so every arrow is in fact an isomorphism
848 Proposition (homotopical uniqueness of parental subcomputad extensions) Suppose 120578 sub119964 sub 119964prime sub 119964119889119895 are parental subcomputads Suppose 119879∶ 119964 rarr 119974 is a simplicial functor so that 119879(119891) = 119891119879(119906) = 119906 and 119879(120578) = 120578 define an adjunction in119974 Then the fiber of the isofibration
Icon(119964prime 119974) ↠ Icon(119964119974)over 119879 is a contractible Kan complex
Proof The fibers over 119879 are those simplicial functors 119879∶ 119964prime rarr 119974 extending 119879∶ 119964 rarr 119974 ByTheorem 833 this fiber is non-empty We must show for any inclusion 119880 119881 of simplicial sets
222
the lifting problem119880 119864119879 Icon(119964prime 119974)
119881 120793 Icon(119964119974)
119894
119879
has a solution Transposing we obtain a lifting problem of simplicial categories
119964 119974 119974119881
119964prime 119974119880
119879 119974
against the local isofibration 119974119881 ↠ 119974119880 The simplicial functor 119974 ∶ 119974 rarr 119974119881 like all simplicialfunctors preserves the adjunction (119891 ⊣ 119906 120578) in119974 so Theorem 833 applies to provide a solution
Taking 119964prime = 119964119889119895 Proposition 848 tells us that the space of homotopy coherent adjunctionsextending a simplicial functor119964rarr 119974 indexed by a parental subcomputad whose image specifies theunit of an adjunction is a contractible Kan complex This proves that such extensions are ldquohomotopi-cally uniquerdquo We conclude this section with more refined presentations of this kind of result for twoinstances of basic adjunction data of interest
849 Definition (the space of units) Simplicial functors 120578 rarr 119974 correspond bijectively to thechoice of a pair of 0-arrows 119891 isin Fun(119861119860) 119906 isin Fun(119860 119861) together with a 1-arrow 120578∶ id119861 rarr 119906119891 isinFun(119861 119861) We refer to the simplicial subset unit(119974) sub Icon(120578119974) of those triples (119891 119906 120578) thatspecify the unit of an adjunction and those 1-simplices that define isomorphisms between them as thespace of units and denote its objects by (119891 ⊣ 119906 120578) Since Icon(120578119974) is a quasi-category unit(119974) isa Kan complex
8410 Lemma There is an isomorphism of quasi-categories
Icon(120578119974) cong 1114018119860119861isin119974
HomFun(119861119861)(id119861 ∘)
HomFun(119861119861)(id119861 ∘)
Fun(119860 119861) times Fun(119861119860) 120793
Fun(119861 119861)
1199011 1199010
120601lArr
∘ ∘
Proof Exercise 84ii
8411 Theorem (uniqueness of homotopy coherent extensions of a unit)(i) The space 119864120578 of homotopy coherent adjunctions extending the counit 120578 is a contractible Kan complex(ii) The forgetful functor 119901119880 ∶ cohadj(119974) ⥲rarr unit(119974) is a trivial fibration of Kan complexes
Proof Both statements follow from specializing Proposition 848 to the parental subcomputad120578 sub 119964119889119895 If 120120∶ 119964119889119895 rarr 119974 is a homotopy coherent adjunction its restriction to 120578 rarr 119974 defines
223
an object of unit(119974) sub Icon(120578119974) The fiber of the isofibration 119901119880 ∶ cohadj(119974) ⥲rarr unit(119974) over(119891 ⊣ 119906 120578) coincides with the fiber considered in Proposition 848 and is thus a contractible Kancomplex
By Corollay 844 the map 119901119880 ∶ cohadj(119974) ⥲rarr unit(119974) is an isofibration between Kan complexesand is thus a Kan fibration By Proposition any Kan fibration between Kan complexes with con-tractible fibers is a trivial fibration proving (ii)
8412 Definition (the space of right adjoints) Simplicial functors 119906 rarr 119974 correspond bijectivelyto the choice of a 0-arrow 119906 in119974 Indeed
Icon(119906119974) cong 1114018119860119861isin119974
Fun(119860 119861)
We refer to the simplicial subset rightadj(119974) sub Icon(119906119974) of those 0-arrows that possess a leftadjoint and the isomorphisms between them as the space of right adjoints As a quasi-category whose1-arrows are all isomorphisms rightadj(119974) is a Kan complex
The isofibration Icon(120578119974) ↠ Icon(119906119974) restricts to define an isofibration 119902119877 ∶ unit(119974) ↠rightadj(119974) of Kan complexes
8413 Proposition The isofibration 119902119877 ∶ unit(119974) ↠ rightadj(119974) is a trivial fibration of Kan complexes
Proof Since both unit(119974) and rightadj(119974) are Kan complexes automatically 119902119877 is a Kan fibra-tion By Proposition we need only show that its fibers are contractible The fiber of 119902119877 over119906∶ 119860 rarr 119861 is isomorphic to the sub-quasi-category of the fiber of the isofibration Icon(120578119974) ↠
Icon(119906119974) over 119906 whose objects are pairs (119891 120578) which have the property that 119891 is a left adjoint tothe fixed 0-arrow 119906 with unit represented by 120578 and whose 1-simplices are all invertible By Lemma8410 this isofibration is isomorphic to the coproduct of the family of projections
1114018119860119861isin119974
HomFun(119861119861)(id119861 ∘)1199011minusminusrarrrarr Fun(119860 119861) times Fun(119861119860) 120587minusrarrrarr Fun(119860 119861)
whose fiber over 119906 is isomorphic to HomFun(119861119861)(id119861 119906lowast)Applying Proposition 414 to the adjunction between functor spaces
Fun(119861 119861) Fun(119861119860)
119891lowast
perp
119906lowast
and the element id119861 isin Fun(119861 119861) reveals that (119891 120578) is a initial in HomFun(119861119861)(id119861 119906lowast) So the fiberof 119902119877 over 119906 is isomorphic to the sub-quasi-category of HomFun(119861119861)(id119861 119906lowast) spanned by its initialelements and as such is a contractible Kan complex⁶
8414 Theorem (uniqueness of homotopy coherent extensions of a right adjoint)
⁶In any quasi-category in which every element is initial all 1-arrows are isomorphisms since the initial elements arealso initial in its homotopy category which must then be a groupoid This proves that the quasi-category spanned by initialelements is in fact a Kan complex Any adjunction between Kan complexes is an adjoint equivalence so in particular theright adjoint ∶ 119860 ⥲rarr 120793 defines an equivalence and hence a trivial fibration
224
(i) The space 119864119906 of homotopy coherent adjunctions with right adjoint 119906 is a contractible Kan complex(ii) The forgetful functor 119901119877 ∶ cohadj(119974) ⥲rarr rightadj(119974) is a trivial fibration of Kan complexes
Proof The space 119864119906 is defined as the fiber of the composite fibration
119901119877 ∶ cohadj(119974) unit(119974) rightadj(119974)sim
119901119880 sim
119902119877
By Theorem 8411 and Proposition 8413 both maps are trivial fibrations of Kan complexes hence sois the composite and thus its fiber is a contractible Kan complex
Exercises
84i Exercise State and prove a relative version of Proposition 848 establishing the homotopicaluniqueness of solutions to lifting problems of parental subcomputad inclusions against local isofibra-tions of quasi-categorically enriched categories
84ii Exercise Prove Lemma 8410
225
CHAPTER 9
The formal theory of homotopy coherent monads
91 Homotopy coherent monads
The free homotopy coherent monad is defined as a full subcategory ℳ119899119889 119964119889119895 on the object+ Via this definition it inherits a graphical calculus from the graphical calculus established for119964119889119895in sect81
911 Definition (the free homotopy coherent monad) The free homotopy coherent monad ℳ119899119889is the full subcategory of the free homotopy coherent adjunction 119964119889119895 on the object + Proposition8110 gives us two definitions ofℳ119899119889bull It is the 2-category regarded as a simplicial category with one object with the hom-category 120491+
and with horizontal composition given by ordinal sum oplus∶ 120491+ times 120491+ rarr 120491+bull It is the simplicial category with one object whose 119899-arrows are strictly undulating squiggles on(119899 + 1)-lines that start and end in the gap labeled +
912 Lemma The simplicial categoryℳ119899119889 is a simplicial computad thoughℳ119899119889 119964119889119895 is not a simpli-cial subcomputad inclusion
Proof Horizontal composition inℳ119899119889 is given by horizontal juxtaposition of squiggle diagramsthat start and end at+ Thus an 119899-arrow is atomic if and only if it has no instances of+ in its interiorThis proves thatℳ119899119889 is a simplicial computad but note thatℳ119899119889 includes atomic arrows such as
119905 ≔ 119906119891 ≔minus+ and 120583 ≔ 119906120598119891 ≔ 1
minus
+(913)
that do have minus in their interiors and thus fail to be atomic in119964119889119895
Employing the graphical calculus we discover another characterization of the atomic 119899-arrows ofℳ119899119889 in reference to the atomic 0-arrow 119905 defined in (913)
914 Lemma An 119899-arrow inℳ119899119889 is atomic if and only if its final vertex is 119905
In particular 119905 is the unique atomic 0-arrow
Proof The 119899-arrows are strictly undulating squiggles on (119899 + 1)-lines that start and end at thespace labeled + these are atomic if and only if there are no instances of + in their interiors
227
This condition implies that if all the lines are removed except the bottom one a process that computesthe final vertex of the 119899-simplex the resulting squiggle looks like a single hump over one line whichis the graphical representation of the 0-arrow 119905
915 Definition A monad in a 2-category is given bybull an object 119861bull an endofunctor 119905 ∶ 119861 rarr 119861bull and a pair of 2-cells 120578∶ id119861 rArr 119905 and 120583∶ 1199052 rArr 119905
so that the ldquounitrdquo and ldquoassociativityrdquo pasting equalities hold
119861 119861 119861 119861 119861 119861 119861
119861 119861 119861 119861 119861 119861 119861 119861 119861
119905dArr120583
= 119905119905 = =119905dArr120578
dArr120583
119905
119905dArr120583
119905dArr120583 =
119905
dArr120583 119905dArr120583119905
dArr120578
119905 119905
119905 119905
119905
119905
119905119905
When these conditions are satisfied we say that (119905 120578 120583) defines a monad on 119861
916 Lemma The atomic arrows
119905 ≔ 120578 ≔ and 120583 ≔ 119906120598119891 ≔
define a monad (119905 120578 120583) on + in the 2-categoryℳ119899119889
Proof The unit pasting identities of Definition 915 are witnessed by the atomic 2-arrows
119906120572 ≔ and 120573119891 ≔
The pair of atomic 2-arrows
and
demonstrate that the left hand side and right-hand side of the associativity pasting equality have acommon composite namely the common first face
As a 2-categoryℳ119899119889 has a familiar universal property Lawverersquos characterization of (120491+ oplus [minus1])as the free strict monoidal category containing a monoid ([0] ∶ [minus1] rarr [0] 1205900 ∶ [1] rarr [0]) tells usthatℳ119899119889 is the free 2-category containing a monad [35]
917 Proposition For any 2-category 119966 2-functorsℳ119899119889 rarr 119966 correspond to monads in 119966
These considerations motivate the following definition
918 Definition (homotopy coherent monad) A homotopy coherentmonad in a quasi-categoricallyenriched category 119974 is a simplicial functor 120139∶ ℳ119899119889 rarr 119974 Explicitly a homotopy coherent monadconsists of
228
bull an object 119861 isin 119974 andbull a homotopy coherent diagram 120139∶ 120491+ rarr Fun(119861 119861) that we refer to as the monad resolution of120139
so that the diagram
120491+ times 120491+ Fun(119861 119861) times Fun(119861 119861)
120491+ Fun(119861 119861)
oplus
120139times120139
∘
120139commutes This simplicial functoriality condition implies that the generating 0- and 1-arrows of themonad resolution have the following form
id119861 119905 119905119905 119905119905119905 ⋯ isin Fun(119861 119861)120578120578119905
119905120578
120583 119905120578119905
120578119905119905
119905119905120578
119905120583
120583119905(919)
where (119905 120578 120583) denotes the image of the monad (119905 120578 120583) of Lemma 916 under 120139∶ ℳ119899119889 rarr 119974 Werefer to the 0-arrow 119905 ∶ 119861 rarr 119861 as the functor part of the homotopy coherent monad 120139 and refer tothe 1-arrows 120578 and 120583 as the unit and associativity maps
Note that for any generalized element 119887 ∶ 119883 rarr 119861 the monad resolution (919) restricts to definea monad resolution
119887 119905119887 119905119905119887 119905119905119905119887 ⋯ isin Fun(119883 119861)120578119887
120578119905119887
119905120578119887
120583119887 119905120578119905119887
120578119905119905119887
119905119905120578119887
119905120583119887
120583119905119887(9110)
9111 Example (free monoid monad) Letℳ be a Kan-complex (or topologically) enriched categoryequipped with an enriched monoidal structure minus otimes minus∶ ℳtimesℳrarrℳ that admits countable conicalcoproducts that are preserved by the monoidal product separately in each variable Then there existsa simplicially enriched endofunctor 119879∶ ℳ rarrℳ defined on objects by
119879(119883) ≔1114018119899ge0
119883otimes119899
equipped with simplicial natural transformations 120578∶ idℳ rArr 119879 and 120583∶ 1198792 rArr 119879 defined by includingat the degree-one component and ldquodistributingrdquo the coproduct
1198792(119883) cong1114018119899ge0
11141021114018119898ge0
119883otimes1198981114105otimes119899
The monad resolution of this simplicially enriched monad defines a simplicial functor120491+timesℳrarrℳ regarding120491+ as a simplicial category whose hom-spaces are sets Applying the homotopy coherentnerve120081∶ 119974119886119899-119982119982119890119905 rarr 119980119966119886119905 this simplicially enriched monad defines the left action120491+times120081ℳrarr120081ℳ of a homotopy coherent monad in 119980119966119886119905 on 120081ℳ
229
More generally any topologically enriched monad on a topologically enriched category defines ahomotopy coherent monad on its homotopy coherent nerve
Any homotopy coherentmonad120139∶ ℳ119899119889 rarr 119974 defines amonad in the homotopy 2-category sim-ply by composing with the canonical quotient functor discussed at the beginning of sect84 and applyingProposition 917
ℳ119899119889 119974 120101119974120139 119876
However ldquomonads up to homotopyrdquo mdash that is monads in the homotopy 2-category of an infin-cosmosmdash cannot necessarily be made homotopy coherent
9112 Non-Example (a monad in the homotopy 2-category that is not homotopy coherent) Stasheffidentifies homotopy associative 119867-spaces that do not extend to 119860infin-spaces that is monoids up tohomotopy cannot necessarily be rectified into homotopy coherent monoids Let
(119872 120578 ∶ lowast rarr 119872 120578∶ 119872 times119872 rarr119872)denote such an up-to-homotopy monoid This structure defines a monad up to homotopy on the(large) quasi-category of spaces by applying the homotopy coherent nerve to the endofunctor 119872 timesminus∶ 119974119886119899 rarr 119974119886119899 and natural transformations induced by 120578 and 120598 This monad in 120101119980119966119886119905 cannot bemade homotopy coherent
Exercises
91i Exercise(i) Show thatℳ119899119889 contains a unique atomic 1-arrow 120583
119899∶ 119905119899 rarr 119905 for each 119899 ge 0 120583
1= id119905 being
degenerate but each of the other 120583119899
being non-degenerate
(ii) Identify the images of these atomic 1-arrows in the monad resolution (919)(iii) Given an interpretation for the 1-arrow 120583
119899that acknowledges the role played by 120583
3in the
proof of Lemma 916
92 Homotopy coherent algebras and the monadic adjunction
Homotopy coherent monads can be defined in any quasi-categorically (or merely simplically) en-riched category but we are particularly interested in homotopy coherent monads valued ininfin-cosmoibecause the flexible weighted limits guaranteed by Corollary 733 permit us to construct the monadicadjunction which relates theinfin-category on which the monad acts to theinfin-category of algebras
The universal property of the monadic adjunction associated to a homotopy coherent monad120139∶ ℳ119899119889 rarr 119974 is very easy to describe though some more work will be required to demonstratethat any infin-cosmos 119974 admits such a construction The monadic adjunction is the terminal adjunc-tion extending the homotopy coherent monad which means that it is given by the right Kan extensionalong the inclusion
119964119889119895
ℳ119899119889 119974
120120120139dArrcong
120139Sinceℳ119899119889 119964119889119895 is fully faithful the value of the right Kan extension at + isin ℳ119899119889 is isomorphicto 119861 ≔ 120139(+) By Example 7119 the value of the right Kan extension at minus isin ℳ119899119889 is computed as the
230
limit of 120139∶ ℳ119899119889 rarr 119974 weighted by the restriction of the covariant representable of119964119889119895 at minus alongℳ119899119889 119964119889119895 which is how we will now define the weight119882minus for theinfin-category of algebras
921 Observation (weights onℳ119899119889) A weight onℳ119899119889 is a simplicial functor119882∶ ℳ119899119889 rarr 119982119982119890119905Explicitly to specify a weight onℳ119899119889 is equivalent to specifyingbull a simplicial set119882 ≔119882(+)bull equipped with a left action of the simplicial monoidsup1 (120491+ oplus [minus1])
120491+ times119882 119882sdot so that120491+ times 120491+ times119882 120491+ times119882 119882 120491+ times119882
120491+ times119882 119882 119882
idtimessdot
oplustimesid sdot
[minus1]timesid
sdot
sdot
Frequently the simplicial set119882 happens to be a quasi-category in which case the weight119882 onℳ119899119889is precisely a homotopy coherent monad on the quasi-category119882
Relative to the encoding of weights onℳ119899119889 as simplicial sets with a left120491+-action amap119891∶ 119881 rarr119882 isin 119982119982119890119905ℳ119899119889 is given by a map 119891∶ 119881 rarr 119882 of simplicial sets that is 120491+-equivariant in the sensethat the diagram
119881 119882
120491+ times 119881 120491+ times119882
119881 119882
119891
[minus1]timesid [minus1]timesid
sdot
idtimes119891
sdot
119891commutes
922 Lemma Let 119882∶ ℳ119899119889 rarr 119982119982119890119905 be a weight on ℳ119899119889 and let 119879∶ ℳ119899119889 rarr 119974 be a homotopycoherent monad on 119861 isin 119974 Then a 119882-shaped cone over 120139 with summit 119883 is specified by a simplicial map120574∶ 119882 rarr Fun(119883 119861) which makes the square
120491+ times119882 Fun(119861 119861) times Fun(119883 119861)
119882 Fun(119883 119861)
sdot
120139times120574
∘
120574
Proof By Observation 921 a simplicial natural transformation 120574∶ 119882 rarr Fun(119883120139minus) is givenby its unique component 120574∶ 119882 rarr Fun(119883 119861) subject to the equivariance condition
120491+ times119882 120491+ times Fun(119883 119861)
119882 Fun(119883 119861)
sdot
idtimes120574
sdot
120574
sup1The strict monoidal category (120491+ oplus [minus1]) is a monoid in (119982119982119890119905 times 120793) Applying the nerve functor (120491+ oplus [minus1])also defines a monoid in (119982119982119890119905 times 120793)
231
The right-hand action map is the transpose of the action map of the composite simplicial functorFun(119883120139minus) ∶ ℳ119899119889 rarr 119980119966119886119905 this being
120491+ Fun(119861 119861) Fun(Fun(119883 119861)Fun(119883 119861))120139 Fun(119883minus)
which transposes to
120491+ times Fun(119883 119861) Fun(119861 119861) times Fun(119883 119861) Fun(119883 119861)120139timesid ∘
Now the equivariance square coincides with the commutative square of the statement
923 Example (notable weights onℳ119899119889) We fix notation for a few notable weights onℳ119899119889(i) Write119882+ ∶ ℳ119899119889 rarr 119982119982119890119905 for the unique represented functor onℳ119899119889 which is given by the
quasi-category 120491+ = ℳ119899119889(+ +) = 119964119889119895(+ +) acted upon the left by itself via the ordinalsum map
120491+ times 120491+ 120491+oplus
(ii) Write119882minus ∶ ℳ119899119889 rarr 119982119982119890119905 for the restriction of the covariant representable functor119964119889119895 rarr119982119982119890119905 on minus along ℳ119899119889 119964119889119895 This weight is presented by the quasi-category 120491⊤ =119964119889119895(minus +) acted upon the left by 120491+ by the ordinal sum map
120491+ times 120491⊤ 120491⊤oplus
which defines the horizontal composition in119964119889119895 in Definition 811There is a natural inclusion
120491+ 120491⊤
119964119889119895(+ +) 119964119889119895(minus +)
minusoplus[0]
cong cong
minus∘119906
that ldquofreely adjoins a top element in each ordinalrdquo or in the graphical calculus of Definition 819ldquoprecomposes a strictly undulating squiggle with 119906 ≔ (+ minus)rdquo This commutes with the left120491+-actionsand so defines an inclusion of weights119882+ 119882minus by Observation 921sup2
Since119882+ is the representable weight it is automatically flexible By the first axiom of Definition713 the119882+-weighted limit of a homotopy coherent monad120139 recovers theinfin-category lim119882+ 120139 cong 119861on which 120139 acts
924 Lemma The inclusion119882+ 119882minus is a projective cell complex in119982119982119890119905ℳ119899119889 built by attaching projective119899-cells 120597Δ[119899] times119882+ Δ[119899] times119882+ in dimensions 119899 gt 0 In particular119882minus is a flexible weight
Proof We apply Theorem 7212 and prove that119882+ 119882minus is a projective cell complex by verify-ing that coll(119882+) coll(119882minus) is a relative simplicial computad Since119882+ is flexible it follows thenthat119882minus is too The collage coll(119882minus) can be identified with the non-full simplicial subcategory of119964119889119895containing two objects minus (aka ⊤) and + and the hom-spaces 119964119889119895(minus +) and 119964119889119895(+ +) but with thehom-space from minus to minus trivial and the hom-space from + to minus empty Via the graphical calculus ofDefinition 819 we see that coll(119882minus) is a simplicial category whose 119899-arrows are strictly undulatingsquiggles from minus to + or from + to + with composition defined by concatenation at + Its atomic119899-arrows are then those that have no instances of + in their interiors
sup2A ldquoright adjointrdquo to the inclusion119882+ 119882minus will be described in the proof of Proposition 9211
232
Similarly coll(119882+) is the simplicial category with two objects ⊤ and + and with the hom-spacesfrom ⊤ to + and from + to + both defined to be119964119889119895(+ +) with the hom space from ⊤ to ⊤ trivialand the hom-space from + to ⊤ empty This is also a simplicial computad in which the only atomicarrow from ⊤ to + is the identity 0-arrow corresponding to [minus1] isin 119964119889119895(+ +) = 120491+ as beforethe atomic arrows from + to + are the strictly undulating squiggles which have no instances of + intheir interiors To provide intuition for this simplicial computad structure on coll(119882+) recall thatsince the representable 119882+ defines a projective cell complex empty 119882+ built by attaching a singleprojective 0-cell at + the proof of Theorem 7212 tells us that the simplicial subcomputad inclusion120793 +ℳ119899119889 coll(119882+) is defined by adjoining a single atomic 0-arrow from ⊤ to + to the simplicialcomputad 120793 +ℳ119899119889
The inclusion coll(119882+) coll(119882minus) is bijective on the common subcategory120793+ℳ119899119889 and definedby sending each119899-arrow from⊤ to+ in coll(119882+) represented as a squiggle from+ to+ to the squigglefrom minus to + defined by precomposing with 119906 This function carries the unique atomic 0-arrow from⊤ to + in coll(119882+) to 119906 which is the unique atomic 0-arrow in coll(119882minus) from⊤ to + Now Theorem7212 proves that119882+ 119882minus is a projective cell complex Furthermore since coll(119882+) coll(119882minus)is surjective on atomic 0-arrows only projective cells of positive dimension are needed to present119882+ 119882minus as a sequential composite of pushouts of projective 119899-cells120597Δ[119899]times119882+ Δ[119899]times119882+
Now let119974 be aninfin-cosmos
925 Definition The infin-category of 120139-algebras for a homotopy coherent monad 120139∶ ℳ119899119889 rarr 119974in aninfin-cosmos119974 is the flexible weighted limit lim119882minus 120139 When 120139 acts on theinfin-category 119861 via themonad resolution (919) with functor part 119905 ∶ 119861 rarr 119861 we write
Alg120139(119861) ≔ lim119882minus 120139for theinfin-category of algebras By Proposition 731(ii) the projective cell complex119882+ 119882minus inducesan isofibration
lim119882minus 120139 ↠ lim119882+ 120139upon taking weighted limits defining a map that we denote by 119906119905 ∶ Alg120139(119861) ↠ 119861 and refer to as themonadic forgetful functor This map is the leg of the 119882minus-shaped limit cone indexed by the uniqueobject + isin ℳ119899119889
By Corollary 733 and Lemma 924
926 Proposition Any homotopy coherent monad in aninfin-cosmos admits aninfin-category of algebras
Wenow introduce the generic bar resolution120491⊤ rarr Fun(Alg120139(119861) 119861) associated to theinfin-categoryof 120139-algebras for a homotopy coherent monad acting on 119861
927 Definition (generic bar resolution) The limit cone 120573∶ 119882minus rArr Fun(Alg120139(119861) 120139(minus)) defines thegeneric bar resolution of a homotopy coherent monad120139 acting on aninfin-category 119861 By Lemma 922and Example 923 a 119882minus-cone with summit Alg120139(119861) ≔ lim119882minus 120139 over a homotopy coherent monadacting on 120139 is given by a simplicial map 120573∶ 120491⊤ rarr Fun(Alg120139(119861) 119861) so that the square
120491+ times 120491⊤ Fun(119861 119861) times Fun(Alg120139(119861) 119861)
120491⊤ Fun(Alg120139(119861) 119861)
∘
120139times120573
∘
120573
233
commutes Under the identification 120491⊤ cong 119964119889119895(minus +) we write 119906119905 and 120573119905 ∶ 119905119906119905 rarr 119906119905 for the 0- and1-arrows of Fun(Alg120139(119861) 119861) defined to be the images of 119906 and 119906120598 under 120573∶ 120491⊤ rarr Fun(Alg120139(119861) 119861)respectively This 0-arrow 119906119905 is the monadic forgetful functor of Definition 925 Then in the notationof (919) the generic bar resolution has the form of a homotopy coherent diagram
119906119905 119905119906119905 119905119905119906119905 119905119905119905119906119905 ⋯ isin Fun(Alg120139(119861) 119861)120578119906119905
120573119905 119905120578119906119905
120578119905119906119905
120583119906119905
119905120573119905
(928)that restricts along the embedding 120491+ 120491⊤ that freely adjoins the top element in each ordinal tothe monad resolution (9110) applied to 119906119905
For any generalized element119883 rarr Alg120139(119861) of theinfin-category of120139-algebras associated to a homo-topy coherent monad acting on 119861 an 119883-family of 120139-algebras in 119861 the generic bar resolution (928)restricts to define a bar resolution
119887 119905119887 119905119905119887 119905119905119905119887 ⋯120578119887
120573 119905120578119887
120578119905119887
120583119887
119905120573
(929)
9210 Proposition The monadic forgetful functor 119906119905 ∶ Alg120139(119861) ↠ 119861 is conservative for any 2-cell 120574with codomain Alg120139(119861) if 119906
119905120574 is invertible then so is 120574
Proof Conservativity of the functor 119906119905 asserts that for all 119883 the isofibration of quasi-categories119906119905lowast ∶ Fun(119883Alg120139(119861)) ↠ Fun(119883 119861) reflects invertible 1-cells Working with marked simplicial sets
this is the case just when this map has the right lifting property with respect to the inclusion 120794 120794♯of the walking arrow into the walking marked arrow
By Definition 925 the monadic forgetful functor is defined by applying limminus120139 to the projectivecell complex 119882+ 119882minus of Lemma 924 By Proposition 731 the isofibration 119906119905 ∶ Alg120139(119861) ↠ 119861then factors as the inverse limit of a tower of isofibrations each layer of which is constructed asthe pullback of products of projective cells 119861Δ[119899] ↠ 119861120597Δ[119899] for 119899 ge 1 The cosmological functorFun(119883 minus) ∶ 119974 rarr 119980119966119886119905 preserves this limit so 119906119905lowast ∶ Fun(119883Alg120139(119861)) ↠ Fun(119883 119861) is similarly the
inverse limit of pullbacks of products of maps Fun(119883 119861)Δ[119899] ↠ Fun(119883 119861)120597Δ[119899] for 119899 ge 1 Sinceconservativity of a functor between quasi-categories may be captured by a lifting property it sufficesto show that the maps Fun(119883 119861)Δ[119899] ↠ Fun(119883 119861)120597Δ[119899] reflective invertibility of 1-simplices Sincefor 119899 ge 1 the inclusion 120597Δ[119899] Δ[119899] is bijective on vertices this is immediate from Lemma which says that invertibility in exponentiated quasi-categories is detected pointwise
We now show that any homotopy coherent monad 120139∶ ℳ119899119889 rarr 119974 on an infin-category 119861 in aninfin-cosmos extends to a homotopy coherent adjunction120120120139 ∶ 119964119889119895 rarr 119974 whose right adjoint is 119906119905
9211 Proposition (the monadic adjunction) For any homotopy coherent monad 120139∶ ℳ119899119889 rarr 119974 on119861 the monadic forgetful functor 119906119905 ∶ Alg120139(119861) rarr 119861 is the right adjoint of a homotopy coherent adjunction
234
120120120139 ∶ 119964119889119895 rarr 119974
119861 Alg120139(119861)
119891119905
perp
119906119905
120578119905 ∶ id119861 rArr 119906119905119891119905 120598119905 ∶ 119891119905119906119905 rArr idAlg120139(119861)
whose underlying homotopy coherent monad is 120139 This constructs the monadic adjunction of the homotopycoherent monad
In particular the triple (119906119905119891119905 120578119905 119906119905120598119905119891119905) recovers the monad (119905 120578 120583) on 119861 defined in 918
Proof Recall the weights 119882+ and 119882minus are defined in Example 923 to be restrictions of therepresentable functors on119964119889119895 in the case of119882+ this restriction defines the representable functor forℳ119899119889 since the inclusionℳ119899119889 119964119889119895 is full on + The weight for the monadic homotopy coherentadjunction is defined to be the composite of the Yoneda embedding with the restriction functor
119964119889119895op 119982119982119890119905119964119889119895 119982119982119890119905ℳ119899119889
+ 119882+
minus 119882minus
119988 res
119891 ↦⊣ minus∘119906 minus∘119891 ⊢
which can be interpreted as defining an adjunction of weights whose left and right adjoints in theencoding of Observation 921 are given by the maps
120491+ 120491⊤
minus∘119906
perpminus∘119891
that act on strictly undulating 119899-arrows by precomposing with 119906 = (+ minus) or 119891 = (minus +) as appro-priate these maps commute with the left 120491+-actions by postcomposition with a strictly undulatingsquiggle from + to +
Composing with the weighted limit functor limminus120139 defines a simplicial functor
120120120139 ≔ limres119988(minus)120139∶ 119964119889119895 rarr 119974ie a homotopy coherent adjunction between lim119882+ 120139 cong 119861 and lim119882minus 120139 cong Alg120139(119861) whose rightadjoint is given by the action of the 0-arrow 119906 which is the monadic forgetful functor 119906119905 ∶ Alg120139(119861) rarr119861 is defined in 925
Finally the underlying homotopy coherent monad of the homotopy coherent adjunction just con-structed is defined to be the limit of 120139∶ ℳ119899119889 rarr 119974 weighted by
ℳ119899119889op 119964119889119895op 119982119982119890119905119964119889119895 119982119982119890119905ℳ119899119889119988 res
which is just the Yoneda embedding for ℳ119899119889 By the first axiom of Definition 713 this functor isisomorphic to 120139
In sect94 we give a characterization of the monadic adjunction of a homotopy coherent monad Tobuild towards this result we spend the next section establishing important special properties of the
235
monadic forgetful functor 119906119905 ∶ Alg120139(119861) ↠ 119861 and its left adjoint 119891119905 ∶ 119861 rarr Alg120139(119861) whose essentialimage identifies the free 120139-algebras
Exercises
92i Exercise Prove that the infin-category of algebras associated to the homotopy coherent monad119882+ ∶ ℳ119899119889 rarr 119980119966119886119905 on 120491+ is 120491perp What is the monadic adjunction
93 Limits and colimits in theinfin-category of algebras
The key technical insight enabling Beckrsquos proof of the monadicity theorem [1] is the observationthat any algebra is canonically a colimit of a particular diagram of free algebras In the case of a monad(119905 120578 120583) acting on a 1-category 119861 the data of a 119905-algebra in 119861 is given by a 119906119905-split coequalizer diagram
1199052119887 119905119887 119887120583119887
119905120573
119905120578119887
120578119905119887
120573
120578119887
(931)
Here the solid arrows are maps which respect the 119905-algebra structure where the dotted splittings donot Split coequalizers are examples of absolute colimits which are preserved by any functor and inparticular by 119905 ∶ 119861 rarr 119861 a fact we may exploit to show that the underlying fork of (931) defines areflexive coequalizer diagram in the category of 119905-algebras
In the infin-categorical context we require a higher-dimensional version of the diagram (931)namely the bar resolution constructed in (929) for any generalized element 119883 rarr Alg120139(119861) of theinfin-category of 120139-algebras for a homotopy coherent monad acting on 119861 This replaces the 119906119905-splitcoequalizer by a canonically-defined 119906119905-split augmented simplicial object
Before defining this special class of colimits we establish a more general result
932 Proposition Let 120139∶ ℳ119899119889 rarr 119974 be a homotopy coherent monad on aninfin-category 119861 with functorpart 119905 ∶ 119861 rarr 119861 Then if 119861 admits and 119905 preserves colimits of shape 119869 then the monadic forgetful functor119906119905 ∶ Alg120139(119861) rarr 119861 creates colimits of shape 119869
Proof The 0-arrows in the image of a homotopy coherent monad 120139∶ ℳ119899119889 rarr 119974 are given bythe identity functor at 119861 the ldquofunctor partrdquo 119905 ∶ 119861 rarr 119861 defined as the image of the unique atomic0-arrow ofℳ119899119889 and finite composites 119905119899 ∶ 119861 rarr 119861 for each 119899 ge 1 If 119905 preserves colimits of shape 119869 in119861 then so does 119905119899 Thus in the case where 119861 admits and 119905 preserves 119869-shaped colimits the homotopycoherent monad lifts to homotopy coherent monad120139∶ ℳ119899119889 rarr 119974perp119869 in theinfin-cosmos of Proposition7414 Since the inclusion119974perp119869 119974 creates flexible weighted limits such as those weighted by119882minusit follows that the limit cone 119906119905 ∶ Alg120139(119861) ↠ 119861 lifts to 119974perp119861 This monadic forgetful functor is theunique 0-arrow component of the limit cone so by Proposition 7414 this tells us that Alg120139(119861) admitsand 119906119905 ∶ Alg120139(119861) ↠ 119861 creates 119869-shaped colimits
A dual argument employing the infin-cosmos 119974⊤119869 of infin-categories that admit and functors thatpreserve 119869-indexed limits proves that if 119861 admits and 119905 preserves limits of shape 119869 then the monadicforgetful functor 119906119905 ∶ Alg120139(119861) ↠ 119861 creates limits of shape 119869 We donrsquot explicitly consider this dual ver-sion here however because we will prove a stronger result in Theorem 939 that drops the hypotheseson 119905
236
933 Definition (119906-split simplicial objects) The image of the embedding
120491op+ 120491⊤
119964119889119895(minus minus) 119964119889119895(minus +)
[0]oplusminus
cong cong
119906∘minus
is the subcategory of 120491⊤ generated by all of its elementary operators except for the face operators1205750 ∶ [119899 minus 1] rarr [119899] for each 119899 ge 1 We refer to these extra face maps as splitting operators By Defini-tion 239 a simplicial object 119883 rarr 119861120491op
in 119861 admits an augmentation if it lifts along the restrictionfunctor 119861120491
op+ ↠ 119861120491op
and an augmented simplicial object 119883 rarr 119861120491op+ in 119861 admits a splitting if it
lifts along the restriction functor 119861120491⊤ ↠ 119861120491op+ Thus given any functor 119906∶ 119860 rarr 119861 ofinfin-categories
the infin-categories 119878120491op(119906) and 119878120491op+ (119906) of 119906-split simplicial objects and 119906-split augmented simplicial
objects in 119860 are defined by the pullbacks
119878120491op(119906) 119861120491⊤ 119878120491op+ (119906) 119861120491⊤
119860120491op 119861120491op 119860120491op+ 119861120491
op+
res
res
119906120491op
119906120491op+
and there exists a forgetful functor
119860120491op+ 119878120491
op+ (119906)
119861120491⊤
119860120491op 119878120491op(119906)
res
Our interest in these notions is explained by the following example if 119906∶ 119860 rarr 119861 is a right adjointfunctor between infin-categories any homotopy coherent adjunction extending 119906 defines a canonical119906-split augmented simplicial object
934 Lemma Let120120∶ 119964119889119895 rarr 119974 be a homotopy coherent adjunction with right adjoint 119906∶ 119860 rarr 119861 Thenthe comonad resolution and bar resolution
119860 119861120491⊤
119860120491op+ 119861120491
op+
119896bull
bar
res
119906120491op+
jointly define a 119906-split augmented simplicial object 119860 rarr 119878120491op+ (119906)
Proof Functoriality of120120 supplies a commutative diagram below-left
120491op+ cong 119964119889119895(minus minus) Fun(119860119860)
120491⊤ cong 119964119889119895(minus +) Fun(119860 119861)
120120
119906∘minus 119906∘minus
120120
237
which internalizes to the commutative diagram of the statement By the definition of theinfin-categoryof 119906-split augmented simplicial objects in 933 this induces the claimed functor 119860 rarr 119878120491
op+ (119906) Thus
the comonad resolution 119896bull ∶ 119860 rarr 119860120491op+ defines an augmented simplicial object in119860 that is 119906-split by
the bar resolution for120120
935 Proposition The monadic forgetful functor 119906119905 ∶ Alg120139(119861) rarr 119861 creates colimits of 119906119905-split simplicialobjects Moreover for any 119906119905-split augmented simplicial object the augmentation defines the colimit cone forthe underlying simplicial object in Alg120139(119861)
Proof Theinfin-category of 119906119905-split simplicial objects is defined by the pullback
119878120491op(119906119905) 119861120491⊤
Alg120139(119861)120491op 119861120491op
res
(119906119905)120491op
By Proposition 2311 the canonical 2-cell (2310) defined by the initial object in120491+ defines an absoluteleft lifting diagram
119861
119878120491op(119906119905) 119861120491⊤ 119861120491op+ 119861120491op
uArr119861120584op Δ
res res
evminus1 (936)
that is also an absolute colimit in 119861 preserved by all functors and in particular by 119905 ∶ 119861 rarr 119861Now Proposition 932 tells us that 119906119905 ∶ Alg120139(119861) ↠ 119861 creates this colimit which means that there
exists an absolute left lifting diagram as below-left
Alg120139(119861)
119878120491op(119906119905) Alg120139(119861)120491op
uArr120582Δcolim
Alg120139(119861) 119861 119861
119878120491op(119906119905) Alg120139(119861)120491op 119861120491op 119878120491op(119906119905) 119861120491
op+ 119861120491op
uArr120582Δ
119906119905
Δ =uArr119861120584
opΔcolim
(119906119905)120491op res
evminus1
that when postcomposed with (119906119905)120491op ∶ Alg120139(119861)120491op ↠ 119861120491op
recovers the absolute left lifting diagram(936) in the sense expressed by the pasting equality above-right Thus the monadic forgetful functorcreates colimits of 119906119905-split simplicial objects in Alg120139(119861)
Upon precomposing with the 119878120491op+ (119906119905) rarr 119878120491op(119906119905) the fact that Proposition 932 tells us that
119906119905 ∶ Alg120139(119861) ↠ 119861 creates the colimit (936) also means that whenever there exists a pasting equality
Alg120139(119861) 119861 119861
119878120491op+ (119906119905) Alg120139(119861)
120491op+ Alg120139(119861)
120491op 119861120491op 119878120491op+ (119906119905) 119861120491⊤ 119861120491
op+ 119861120491op
uArrAlg120139(119861)120584op Δ
119906119905
Δ =uArr119861120584
opΔ
res
evminus1
(119906119905)120491op res res
evminus1
238
such as arises here by 2-functoriality of the simplicial cotensor construction the 2-cell
Alg120139(119861)
119878120491op+ (119906119905) Alg120139(119861)
120491op+ Alg120139(119861)
120491opuArrAlg120139(119861)
120584op Δ
res
evminus1
is an absolute left lifting diagram This proves that 119906119905 ∶ Alg120139(119861) ↠ 119861 creates colimits of 119906119905-splitsimplicial objects
Now as displayed by the bar resolution (928) any 120139-algebra in 119861 canonically gives rise to a119906119905-split simplicial object towhich Proposition 935 applies the bar resolution120491⊤ rarr Fun(Alg120139(119861) 119861)internalizes to a diagrambar ∶ Alg120139(119861) rarr 119861120491⊤ The colimit cone inAlg120139(119861) is given by the120491op
+ -shapedsubdiagram of the bar resolution that omits the dashed maps
119906119905 119905119906119905 119905119905119906119905 119905119905119905119906119905 ⋯120578119906119905
120573119905 119905120578119906119905
120578119905119906119905
120583119906119905
119905120573119905
This subdiagram admits a concise description it is the comonad resolution for the comonad inducedby the monadic adjunction 119891119905 ⊣ 119906119905 on Alg120139(119861) this being a functor 120491op
+ rarr Fun(Alg120139(119861)Alg120139(119861))that internalizes to a functor 119896119905bull ∶ Alg120139(119861) rarr Alg120139(119861)
120491op+
937 Theorem (canonical colimit representation of algebras) For any homotopy coherent monad 120139on 119861 the induced comonad resolution 119896119905bull ∶ Alg120139(119861) rarr Alg120139(119861)
120491op+ on the infin-category of 120139-algebras in 119861
encodes an absolute left lifting diagram
Alg120139(119861) Alg120139(119861)
Alg120139(119861) Alg120139(119861)120491op
Alg120139(119861) Alg120139(119861)120491op+ Alg120139(119861)
120491opuArr120573
Δ ≔uArrAlg120139(119861)
120584opΔ
119896119905bull 119896119905bullres
evminus1 (938)
created from the 119906119905-split simplicial object in 119861
Alg120139(119861) 119861 119861
Alg120139(119861) Alg120139(119861)120491op+ Alg120139(119861)
120491op 119861120491opAlg120139(119861) 119861120491⊤ 119861120491
op+ 119861120491op
uArrAlg120139(119861)120584op Δ
119906119905
Δ =uArr119861120584
opΔ
119896119905bullres
evminus1
(119906119905)120491op bar res res
evminus1
Thus (938) exists the algebras for a homotopy coherent monad as colimits of canonical simplicial objects of freealgebras
Proof Applying Lemma 934 to the monadic adjunction of Proposition 9211 we see that thecomonad resolution 119896119905bull ∶ Alg120139(119861) rarr Alg120139(119861)
120491op+ on Alg120139(119861) and the bar resolution bar ∶ Alg120139(119861) rarr
239
119861120491⊤ defined in (928) together define a canonical 119906119905-split augmented cosimplicial object
Alg120139(119861)
119878120491op+ (119906119905) 119861120491⊤
Alg120139(119861)120491op+ 119861120491
op+
bar
119896119905bull
res
(119906119905)120491op+
Now the claimed result follows immediately from Proposition 935
Our final task for this section is to generalize the dual of Proposition 932 proving that themonadic forgetful functor creates all limits that 119861 admits whether or not 119905 preserves them
939 Theorem Let120139∶ ℳ119899119889 rarr 119974 be a homotopy coherent monad on aninfin-category119861 Then the monadicforgetful functor 119906119905 ∶ Alg120139(119861) rarr 119861 creates all limits that 119861 admits
Proof See [50 sect5] for now
Exercises
94 The monadicity theorem
Consider an adjunction
119860 119861119906perp119891
120578∶ id119861 rArr 119906119891 120598 ∶ 119891119906 rArr id119860
between infin-categories that is in the homotopy 2-category of an infin-cosmos Theorem 834 provesthat this data lifts to a homotopy coherent adjunction 120120∶ 119964119889119895 rarr 119974 which then restricts to definea homotopy coherent monad 120139∶ 119964119889119895 rarr 119974 on 119861 Proposition 9211 then constructs a new homo-topy coherent adjunction with 120139 as its underlying homotopy coherent monad namely the monadicadjunction 119891119905 ⊣ 119906119905 between 119861 and the infin-category of 120139-algebras Alg120139(119861) Immediately from itsconstruction as a right Kan extension mdash there is a simplicial natural transformation from the firsthomotopy coherent adjunction to the second whose component at + is the identity and whose com-ponent at minus defines a functor that we call 119903 ∶ 119860 rarr Alg120139(119861) commuting strictly with all of the data ofeach homotopy coherent adjunction
119860 Alg120139(119861)
119861
119903
119906⊤
119906119905
⊤
119891
119891119905
This monadicity theorem originally proven for 1-categories by Beck [1] and first proven for quasi-categories by Lurie [37] supplies conditions under which this comparison functor 119903 is an equivalenceso that theinfin-category 119860 can be identified with theinfin-category of 120139-algebras
240
To construct this simplicial natural transformation we re-express theinfin-category of algebras as aweighted limit of the full homotopy coherent adjunction diagram not merely as a weighted limit ofits underlying homotopy coherent monad
941 Proposition The infin-category of algebras associated to the homotopy coherent monad underlying ahomotopy coherent adjunction120120∶ 119964119889119895 rarr 119974 is the limit weighted by the weight lan119882minus defined by the leftKan extension
119964119889119895
ℳ119899119889 119982119982119890119905
lan119882minusuArrcong
119882minus
Explicitly lan119882minus ∶ 119964119889119895 rarr 119982119982119890119905 is the homotopy coherent adjunction displayed on the top below
120491op 120491⊤ 119964119889119895 119982119982119890119905
119964119889119895(minus minus) 119964119889119895(minus +) 119964119889119895 119982119982119890119905
119906∘minus
perp
119891∘minus
lan119882minus
119906∘minus
perp
119891∘minus
119964119889119895minus
defined by restricting the domain of the right adjoint and codomain of the left adjoint of the representableadjunction119964119889119895minus along the canonical inclusion 120491op 120491op
+
Proof Recall Lemma 7120 which says that the weighted limit of a restricted diagram can becomputed as the limit of the original diagram weighted by the left Kan extension of the weight Thus
lim119964119889119895lan119882minus
120120 cong limℳ119899119889119882minus res120120
recovers theinfin-category of algebras for the homotopy coherent monad underlying120120All that remains is to compute the functor lan119882minus ∶ 119964119889119895 rarr 119982119982119890119905 explicitly Because the inclusion
ℳ119899119889 119964119889119895 is full on + lan119882minus(+) cong 119882minus(+) cong 119964119889119895(minus +) since119882minus was defined as the restrictionof the covariant representable functor 119964119889119895minus ∶ 119964119889119895 rarr 119982119982119890119905 along ℳ119899119889 119964119889119895 By the standardformula for left Kan extensions reviewed in Example 7119 presented in the form of (716) the valueof lan119882minus at the object minus is computed by
lan119882minus(minus) cong 1114009ℳ119899119889
119964119889119895(+ minus) times 119964119889119895(minus +)
cong coeq 1114102 119964119889119895(+ minus) times 119964119889119895(+ +) times 119964119889119895(minus +) 119964119889119895(+ minus) times 119964119889119895(minus +)∘timesid
idtimes∘1114105
By associativity of composition in119964119889119895 the composition map
119964119889119895(+ minus) times 119964119889119895(minus +) 119964119889119895(minus minus)∘
defines a cone under the coequalizer diagram By the graphical calculus and Proposition 8110 theimage of this map in119964119889119895(minus minus) cong 120491op
+ is comprised of those strictly undulating squiggles from minus to minus
241
that pass through + This is the subcategory 120491op 120491op+ In fact we claim that
119964119889119895(+ minus) times 119964119889119895(+ +) times 119964119889119895(minus +) 119964119889119895(+ minus) times 119964119889119895(minus +) 120491op∘timesid
idtimes∘∘ (942)
is a coequalizer diagram The map from the coequalizer to 120491op is surjective for the reason just de-scribed a strictly undulating squiggle from minus to minus that passes through + can be decomposed as ahorizontal composite of a squiggle in119964119889119895(minus +) followed by a squiggle in119964119889119895(+ minus) To see that themap from the coequalizer to 120491op is injective consider two distinct subdivisions of a squiggle from minusto minus into a pair of squiggles from minus to + and from + to minus The subsquiggle between the two chosen+ symbols in this an element of119964119889119895(+ +) and thus this pair of elements of119964119889119895(+ minus) times 119964119889119895(minus +)are identified in the coequalizer diagram
943 Lemma For any homotopy coherent adjunction120120∶ 119964119889119895 rarr 119974 there exists a simplicial natural trans-formation from120120 to the monadic adjunction lim119964119889119895
lan119882minus120120∶ 119964119889119895 rarr 119974 whose components at + and minus defined
on weights by the counit of the adjunction
119982119982119890119905119964119889119895 119982119982119890119905ℳ119899119889
res
perp
lanlan119882minus = lan res119964119889119895minus 119964119889119895minus
119964119889119895+ cong lan res119964119889119895+ 119964119889119895+
120598minus
120598+cong
Proof Consider the diagram of weights in 119982119982119890119905119964119889119895
119964119889119895minus lan res119964119889119895minus
119964119889119895+ cong lan res119964119889119895+
minus∘119891
perp
120598
perplan res(minus∘119891)minus∘119906
lan res(minus∘119906)
Applying theseweights to a homotopy coherent adjunction120120∶ 119964119889119895 rarr 119974with underlying adjunction119891 ⊣ 119906∶ 119860 rarr 119861 yields
119860 Alg120139(119861)
119861
119903
119906⊤
119906119905
⊤
119891
119891119905
with the component 120598minus inducing the non-identity component of the canonical comparison functorwith the monadic adjunction
944 Example Returning to Example 9111 there is a Kan-complex enriched categoryℳ119900119899(ℳ) ofmonoids inℳ equipped with a simplicially enriched adjunction
ℳ119900119899(ℳ) ℳ119880
perp119865
242
Applying the homotopy coherent nerve this defines a homotopy coherent adjunction between thequasi-categories 120081ℳ119900119899(ℳ) and 120081ℳ By Lemma 943 there is a canonical comparison map to themonadic homotopy coherent adjunction
120081ℳ119900119899(ℳ) Alg120139(120081ℳ)
120081ℳ
119903
119906
⊤119906119905
⊤
119891
119891119905
that is not an equivalence Elements of 120081ℳ119900119899(ℳ) are strict monoids in ℳ while elements ofAlg120139(120081ℳ) are homotopy coherent monoids objects 119883 isin ℳ equipped with 119899-ary multiplicationmaps 120583119899 ∶ 119883otimes119899 rarr 119883 for all 119899 that are coherently associative up to higher homotopy
945 Lemma Let 120120∶ 119964119889119895 rarr 119974 be a homotopy coherent adjunction with right adjoint 119906∶ 119860 rarr 119861 andlet120120120139 ∶ 119964119889119895 rarr 119974 be the associated monadic adjunction Then there is a canonical functor 119871∶ Alg120139(119861) rarr119860120491op
that(i) is 119906-split by the bar resolution bar ∶ Alg120139(119861) rarr 119861120491op
(ii) is so that the composite 119871 ∘ 119903 ∶ 119860 rarr 119860120491op
is the simplicial object underlying the comonad resolution119896bull ∶ 119860 rarr 119860120491
op+ and
(iii) is so that the composite 119903120491op ∘ 119871 ∶ Alg120139(119861) rarr Alg120139(119861)120491op
is the simplicial object underlying the
comonad resolution 119896119905bull ∶ Alg120139(119861) rarr Alg120139(119861)120491op+
Proof The first two statements ask for a functor 119871 that fits into a commutative diagram below-left
119860 119860120491op+ 119964119889119895minus 119964119889119895minus times 120491
op+
Alg120139(119861) 119860120491op lan res119964119889119895minus 119964119889119895minus times 120491op
119861120491⊤ 119861120491op 119964119889119895+ times 120491⊤ 119964119889119895+ times 120491op
119896bull
119903 res
∘
119871
bar 119906120491op
120598
∘
res
∘
idtimes(119906∘minus)
(minus∘119906)timesid
in which each of the objects and all but the map 119871 have been described as maps induced by takingweighted limits of the homotopy coherent adjunction diagram with the weights in119982119982119890119905119964119889119895 displayedabove-right By the Yoneda lemma each of the three maps of weights labeled ldquo∘rdquo are defined by asingle map of simplicial sets In the case of ∘ ∶ 119964119889119895minus times120491
op+ rarr119964119889119895minus the Yoneda lemma says it suffices
to define a map 120491op+ rarr 119964119889119895minus(minus) = 120491op
+ we take this map to be the identity which implies that∘ ∶ 119964119889119895minustimes120491
op+ rarr119964119889119895minus acts in both components by composing over minus in119964119889119895 In light of the explicit
description of the adjunction lan res119964119889119895minus given in Proposition 941 the other two maps labelled ldquo∘rdquomay be defined similarly by identity maps Since the dashed map makes the right-hand diagram ofweights commute the induced functor on weighted limits has the desired properties (i) and (ii)
243
The final statements demands commutativity of the diagram below left which again follows fromthe commutativity of the corresponding diagram of weights below-right
Alg120139(119861) lan res119964119889119895minus
119860120491opAlg120139(119861)
120491op 119964119889119895minus times 120491op lan res119964119889119895minus times 120491
op
119896119905bull119871
119903120491op
∘ ∘
120598timesid
this just amounting to the simple observation that the counit component 120598 ∶ lan res119964119889119895minus 119964119889119895minus isjust given by the natural inclusion 120491op 120491op
+
946 Lemma Given any homotopy coherent adjunction with left adjoint 119891∶ 119861 rarr 119860 the diagram definedby restricting the canonical cone (2310) built from the interalized comonad resolution 119896bull ∶ 119860 rarr 119860120491
op+ along
119891∶ 119861 rarr 119860119860 119860
119861 119860120491op 119861 119860 119860120491op+ 119860120491op
uArr120577Δ ≔
uArr119860120584op Δ119891
119896bull119891 119891 119896bull res
evminus1
displays 119891 as an absolute colimit of the family of diagrams 119896bull119891∶ 119861 rarr 119860120491op
Proof The homotopy coherent adjunction provides a commutative diagram below-left
120491op+ Fun(119860119860) 119861 119860
120491perp Fun(119861119860) 119860120491⊤ 119860120491op+
120120
119891∘minus 119891∘minus cobar
119891
119896bull
120120 res
which transposes to the commutative diagram above-right which tells us that when the internalizedcomonad resolution 119896bull ∶ 119860 rarr 119860120491
op+ is restricted along 119891 it extends to a split augmented simplicial
object with the splittings on the opposite side as usual this is no matter since120491op+ considered as a full
sub 2-category of 119982119982119890119905 spanned by finite ordinals is isomorphic to its co-dual via an isomorphismthat commutes with 120491⊤co cong 120491perp This tells us that the colimit cone of the statement is the one ofProposition 2311
There are many versions of the monadicity theorem For expediencyrsquos sake we prove just onefor now We break its statement into two parts first constructing a left adjoint to the canonicalcomparison functor which under additional hypotheses we prove defines an adjoint equivalence
947 Theorem Let120120∶ 119964119889119895 rarr 119974 be a homotopy coherent adjunction with right adjoint 119906∶ 119860 rarr 119861 withunderlying homotopy coherent monad 120139∶ ℳ119899119889 rarr 119974 If 119860 admits colimits of 119906-split simplicial objects thenthe canonical comparison functor admits a left adjoint
119860 Alg120139(119861)119903
perpℓ
244
Proof If 119860 admits colimits of 119906-split simplicial objects then there exists an absolute left liftingof the 119906-split simplicial object 119871∶ Alg120139(119861) rarr 119860120491op
defined in Lemma 945
119860
Alg120139(119861) 119860120491opuArr120582
Δ
119871
ℓ (948)
whose functor part we take to be the definition of the left adjoint ℓ ∶ Alg120139(119861) rarr 119860 By Lemma945(ii) the diagram defined by restricting along 119903 agrees with the cosimplicial object underlying thecomonad resolution which has a canonical cone (2310) as displayed below-left
119860 119860
119860 119860120491op+ 119860120491op 119860 Alg120139(119861) 119860120491op
uArr119860120584op Δ = existuArr120598
uArr120582Δ
119896bull
evminus1
res 119903
ℓ
119871
(949)
By the universal property of the absolute left lifting diagram (ℓ119903 120582119903) this induces a unique 2-cell120598 ∶ ℓ119903 rArr id119860
The unit is induced from the absolute left lifting diagram (938) By Lemma 945(iii) the comonadresolution 119896119905bull ∶ Alg120139(119861) rarr Alg120139(119861)
120491opfactors as 119903120491op sdot 119871 so the pasted composite below left factors
through the absolute left lifting diagram as below right
119860 Alg120139(119861) 119860 Alg120139(119861)
Alg120139(119861) 119860120491opAlg120139(119861)
120491opAlg120139(119861) Alg120139(119861)
120491opuArr120582
Δ
119903
Δ =
119903
existuArr120578
uArr120573Δ
119871
ℓ
119903120491op 119896119905bull
ℓ (9410)
To verify the triangle equalities note that by construction
119860 Alg120139(119861) 119860 Alg120139(119861)
119860 Alg120139(119861) Alg120139(119861)120491op 119860 Alg120139(119861) 119860120491op
Alg120139(119861)120491op
uArr120598
119903
uArr120578
uArr120573Δ = uArr120598
uArr120582Δ
119903
Δ
119903 119896119905bull
ℓ
119903 119871
ℓ
119903120491op
119860 Alg120139(119861) Alg120139(119861)
119860 119860120491opAlg120139(119861)
120491op 119860 Alg120139(119861) Alg120139(119861)120491op
=uArr119860120584
opΔ
119903
Δ = uArr120573Δ
119896bull 119903120491op 119903 119896119905bull
the last equality following from simplicial naturality of 119903 and the definition of 120573 as Alg120139(119861)120584op in
Theorem 937 Thus the triangle equality composite 119903120598 sdot 120578119903 = id119903It follows that the other triangle equality composite 120601 ≔ 120598ℓ sdot ℓ120578 is an idempotent
120601 sdot 120601 ≔ (120598ℓ sdot ℓ120578) sdot (120598ℓ sdot ℓ120578) = 120598ℓ sdot 120598ℓ119903ℓ sdot ℓ119903ℓ120578 sdot ℓ120578 = 120598ℓ sdot ℓ119903120598ℓ sdot ℓ120578119903ℓ sdot ℓ120578 = 120598ℓ sdot ℓ120578 ≕ 120601
245
so to prove that 120598ℓ sdot ℓ120578 = idℓ it suffices to show that 120601 is an isomorphism To demonstrate this wewill show
(i) that 120601119891119905 is invertible ie that 120601 is an isomorphism when restricted to free 120139-algebras(ii) and that the putative left adjoint ℓ preserves the canonical colimit (938) that expresses every
120139-algebra as a colimit of free 120139-algebrasWe then combine (i) and (ii) to argue that 120601 is invertible
To this end we first observe by Lemma 946 and the definition of ℓ above that we have a pair ofabsolute left lifting diagrams
119860 119860
119861 119860120491op 119861 Alg120139(119861) 119860120491opuArr
Δ conguArr120582
Δ119891
119896bull119891 119891119905 119871
ℓ
By simplicial naturality of the canonical comparison map 119903119891 = 119891119905 and by Lemma 945(ii) 119871119891119905 =119871119903119891 = 119896bull119891 Thus the absolute left lifting problems coincide and we obtain a canonical natural iso-morphism 120574∶ ℓ119891119905 cong 119891
Now to prove the claim of (i) that 120601119891119905 is an isomorphism it suffices to prove that 120578119891119905 is an iso-morphism and that 120598ℓ119891119905 is an isomorphism mdash and by naturality of whiskering and the isomorphism120574∶ ℓ119891119905 cong 119891 just constructed 120598ℓ119891119905 is an isomorphism if and only if 120598119891 is an isomorphism
By (9410) the construction of 120574 and simplicial naturality of 119903 which implies that 119903120491op119896bull = 119896119905bull119903
119860 Alg120139(119861) 119860 Alg120139(119861)
119861 Alg120139(119861) Alg120139(119861)120491op 119861 Alg120139(119861) 119860120491op
Alg120139(119861)120491op
conguArr120574
119903
uArr120578
uArr120573Δ = conguArr120574
uArr120582Δ
119903
Δ
119891119905
119891
119896119905bull
ℓ
119891119905
119891
119871
ℓ
119903120491op
=
119860 Alg120139(119861) Alg120139(119861)
119861 119860120491opAlg120139(119861)
120491op 119861 Alg120139(119861) Alg120139(119861)120491op
uArrΔ
119903
Δ = uArr120573 Δ119891
119896bull119891 119903120491op 119891119905 119896119905bull
So 120578119891119905 is the inverse of the isomorphism 119903120574 and is consequently invertibleSimilarly by (949)
119860 119860 119860
119861 119860 Alg120139(119861) 119860120491op 119861 119860120491op 119861 Alg120139(119861) 119860120491op
uArr120598uArr120582
Δ = uArr120577Δ =
uArr120582uArrcong120574 Δ
119891 119903
ℓ
119871 119896bull119891
119891
119891119905
119891
119871
ℓ
so 120598119891 = 120574 is also an isomorphism Thus we conclude that 120601119891119905 is invertible as claimed in (i)
246
To prove (ii) we must show that
Alg120139(119861) 119860
Alg120139(119861) Alg120139(119861)120491op 119860120491op
uArr120573 Δ
ℓ
Δ
119896119905bull ℓ120491op
(9411)
is an absolute left lifting diagram Of course we expect this to be true because left adjoints preservecolimits by Theorem but as we have not yet shown that ℓ is a left adjoint this requires a directargument
By the equational characterization of Theorem the cosmological functor (minus)120491op ∶ 119974 rarr 119974 pre-serves the absolute left lifting diagram (948) thus
119860120491op
Alg120139(119861)120491op (119860120491op)120491op
uArr120582120491op Δ120491
op
119871120491op
ℓ120491op
is absolute left lifting Since 120582120491op sdot Δ = Δ120491op sdot 120582 there are two equivalent ways to compute thehorizontal composite of this 2-cell with 120573 displayed below-left and below-right
119860 119860
Alg120139(119861) 119860120491opAlg120139(119861) 119860120491op
Alg120139(119861) Alg120139(119861)120491op (119860120491op)120491op
Alg120139(119861) Alg120139(119861)120491op (119860120491op)120491op
ΔuArr120582
Δ
uArr120573
ℓ
ΔuArr120582120491
op Δ120491op
=
uArr120573
ℓ
119871
Δ Δ120491op
119896119905bull 119871120491op
ℓ120491op
119896119905bull 119871120491op
By Lemma 241 to show that (9411) is absolute left lifting ie that ℓ preserves the absolute left liftingdiagram 120573 it suffices to prove that 119871 preserves the absolute left lifting diagram 120573 By Proposition4315 to show that this diagram is absolute left lifting it suffices to show that for each [119899] isin 120491 thatthe diagram
Alg120139(119861) 119860120491op 119860
Alg120139(119861) Alg120139(119861)120491op (119860120491op)120491op 119860120491op
uArr120573
119871
Δ Δ120491op
ev119899
Δ
119896119905bull 119871120491op ev119899
is absolute left liftingBy the construction of 119871 in Lemma 945 ev119899 119871∶ Alg120139(119861) rarr 119860 is the map induced by taking
the weighted limits of the homotopy coherent adjunction 120120∶ 119964119889119895 rarr 119974 by the map of weightsminus ∘ (119891119906)119899+1 ∶ 119964119889119895minus rarr lan res119964119889119895minus Thus we see that ev119899 119871 is the map 119891119905119899119906119905 ∶ Alg120139(119861) rarr 119860 and in
particular factors through 119906119905 ∶ Alg120139(119861) rarr 119861 Since the canonical colimit of Theorem 937 is 119906119905-split(119906119905 119906119905120573) is an absolute left lifting diagram preserved under postcomposition by all functors and in
247
particular by 119891(119906119891)119899 Thus the above diagram is absolute left lifting as claimed which tells us that 119871and thus ℓ preserves the colimit (938)
It remains only to combine (i) and (ii) to argue that 120601 is invertible For this we consider thepasting equality
Alg120139(119861) 119860 Alg120139(119861) 119860
Alg120139(119861) Alg120139(119861)120491op 119860120491op
Alg120139(119861) Alg120139(119861)120491op 119860120491op
uArr120573 Δ
ℓ
Δ =uArr120573 Δ
ℓ
ℓ
uArr120601
Δ
119896119905bull
ℓ120491op
uArr120601120491op
ℓ120491op
119896119905bull ℓ120491op
By the definition of 119896119905bull the components of the whiskered natural transformation120601120491op sdot119896119905bull at [119899] isin 120491op
is 120601(119891119905119906119905)119899+1 which is an isomorphism by (i) By Lemma this proves that 120601120491op sdot 119896119905bull is invertibleThus by (ii) the left hand diagram is isomorphic to an absolute left lifting diagram and thus is absoluteleft lifting The pasting equality describes a factorization of the left hand absolute left lifting diagramthrough the absolute left lifting diagram of (ii) via 120601 so by the uniqueness in the universal property ofabsolute left lifting diagrams we conclude that 120601 is invertible as desired This proves that (ℓ ⊣ 119903 120578 120598)defines an adjunction as claimed
We now describe conditions under which the adjunction just constructed defines an adjoint equiv-alence As the proof will reveal condition (ii) implies that the unit is an isomorphism while conditions(ii) and (iii) together imply that the counit is an isomorphism
9412 Theorem (monadicity) Let120120∶ 119964119889119895 rarr 119974 be a homotopy coherent adjunction with right adjoint119906∶ 119860 rarr 119861 with underlying homotopy coherent monad 120139∶ ℳ119899119889 rarr 119974 If
(i) 119860 admits colimits of 119906-split simplicial objects(ii) 119906∶ 119860 rarr 119861 preserves them and(iii) 119906∶ 119860 rarr 119861 is conservative
then the canonical comparison functor 119903 ∶ 119860 rarr Alg120139(119861) admits a left adjoint ℓ ∶ Alg120139(119861) rarr 119860 defining anadjoint equivalence
Note that Theorem 937 and Proposition 9210 establish these properties for the monadic adjunc-tion
Proof Theorem 947 constructs an adjunction (ℓ ⊣ 119903 120578 120598) under the hypothesis (i) with the leftadjoint ℓ ∶ Alg120139(119861) rarr 119860 defined as the colimit of the 119906-split family of diagrams 119871∶ Alg120139(119861) rarr 119860120491op
with colimit cone 120582∶ Δℓ rArr 119871 It remains to show that this defines an adjoint equivalenceBy hypothesis (ii) 119906 preserves the 119906-split colimit diagram that defines ℓ By Lemma 945(i)
119906120491op119871∶ Alg120139(119861) rarr 119861120491opis the monadic bar resolution so the absolute left lifting diagrams below-left
and below-center are isomorphic
119860 119861 119861 Alg120139(119861) 119861
Alg120139(119861) 119860120491op 119861120491opAlg120139(119861) 119861120491⊤ 119861120491op
Alg120139(119861) Alg120139(119861)120491op 119861120491op
uArr120582Δ
119906
Δ conguArr119861120584
opΔ =
uArr120573
119906119905
Δ Δ
119871
ℓ
119906120491op
119906119905
bar res
ev0
119896119905bull (119906119905)120491op
248