ON THE CONSTRUCTION OF LIMITS AND COLIMITS IN -CATEGORIES

Transcription

1 ON THE CONSTRUCTION OF LIMITS ND COLIMITS IN -CTEGORIES EMILY RIEHL ND DOMINIC VERITY bstract. In previous work, we introduce an axiomatic ramework within which to prove theorems about many varieties o ininite-dimensional categories simultaneously. In this paper, we establish criteria implying that an -category or instance, a quasicategory, a complete Segal space, or a Segal category is complete and cocomplete, admitting limits and colimits indexed by any small simplicial set. Our strategy is to build (co)limits o diagrams indexed by a simplicial set inductively rom (co)limits o restricted diagrams indexed by the pieces o its skeletal iltration. We show directly that the modules that express the universal properties o (co)limits o diagrams o these shapes are reconstructable as limits o the modules that express the universal properties o (co)limits o the restricted diagrams. We also prove that the Yoneda embedding preserves and relects limits in a suitable sense, and deduce our main theorems as a consequence. Contents 1. Introduction Size conventions cknowledgements cosmoi and lexible weighted limits cosmoi and the comma construction Flexible weighted limits in an -cosmos Pseudo limits o homotopy coherent diagrams The ormal theory o -categories bsolute liting diagrams and (relative) adjunctions Fibred equivalences between comma -categories Limits and colimits in an -category Some ormal -category theory Cartesian ibrations and modules Cartesian ibrations and modules cosmoi o (co)cartesian ibrations Specialising to quasi-categories Comprehension and the Yoneda embedding The comprehension unctor 43 Date: September 2, Mathematics Subject Classiication. Primary 1830, 18G55, 55U35, 55U40; Secondary 1805, 18G30, 55U10. 1

2 2 RIEHL ND VERITY 5.2. The Yoneda embedding Internalising the Yoneda embedding Construction o limits and colimits Complete and cocomplete quasi-categories Preservation o limits by generalised Yoneda Colimits o diagrams 59 Reerences Introduction This paper is a continuation o previous work [RV-I, RV-II, RV-III, RV-IV, RV-V, RV-VI, RV-VII] to lay the oundations or the ormal theory o -categories, which model weak higher categories. In contrast with the pioneering work o Joyal [Joy08] and Lurie [Lur09, Lur17], our approach is synthetic in the sense that our proos do not depend on what precisely these -categories are, but rather rely upon an axiomatisation o the universe in which they live. To describe an appropriate universe, we introduce the notion o an - cosmos, a (large) simplicially enriched category K satisying certain axioms. The objects o an -cosmos are called -categories. theorem, e.g. [RV-IV, ] reproduced as Deinition 4.1.2, that characterises a cartesian ibration o -categories in terms o the presence o an adjunction between comma -categories, is a result about the objects o any -cosmos, and thus applies o course to every -cosmos. The prototypical example is the -cosmos whose objects are quasi-categories, a model o (, 1)-categories as simplicial sets satisying the weak Kan condition, and whose unction complexes are the quasi-categories o unctors between them. ut there are other -cosmoi whose objects are complete Segal spaces or Segal categories, each o these being models o (, 1)-categories; and o θ n -spaces, or iterated complete Segal spaces, or n-trivial saturated complicial sets, each modelling (, n)-categories. For any -cosmos K containing an - category, the slice category K / is again an -cosmos. Thus each o these objects are -categories in our sense and our theorems apply to all o them. 1 s a byproduct o our work here, we construct other exotic models o -cosmoi (see Propositions and ) in which our results also apply. For instance, [RV-I] develops the basic theory o limits or colimits o diagrams indexed by a simplicial set X and valued in an -category. In the case where X is the nerve o an ordinary 1-category, this data is traditionally thought o as deining a homotopy coherent diagram o that shape in. In this paper, we shall explain how to construct such limits inductively using the canonical skeletal decomposition o the simplicial set X, in which the cells attached at stage n are indexed by the set L n X X n o non-degenerate 1 This may seem like sorcery but in some sense it is really just the Yoneda lemma. To a close approximation, an -cosmos is a category o ibrant objects enriched over quasi-categories. When the Joyal Lurie theory o quasi-categories is expressed in a suiciently categorical way, it extends to encompass analogous results or the corresponding representably deined notions in a general -cosmos.

3 ON THE CONSTRUCTION OF LIMITS ND COLIMITS IN -CTEGORIES 3 n-simplices: L nx n L nx n (1.0.1) sk 0 X sk 1 X sk 2 X sk n 1 X sk n X colim n sk n X = X The skeletal description gives rise to a presentation o the diagram -category X as the limit in the -cosmos o a countable tower o restriction unctors, each o which is a pullback o a product o maps o the orm n n. We will argue that limits o X-indexed diagrams in can be deined inductively provided that admits products, pullbacks, and sequential inverse limits though since sequential inverse limits may be built rom countable products and pullbacks we are only required to postulate the existence o the irst two o these. 2 This allows us to provide criteria or ascertaining that an -category is complete (or, dually, cocomplete): Theorem. Suppose that κ is a regular cardinal and that is an -category that admits products o cardinality < κ and pullbacks. I X is a κ-presentable simplicial set then admits all limits o diagrams o shape X. To explain the proo strategy, consider a pushout diagram o simplicial sets X Y Z and suppose that an -category admits limits o shape X, Y, and Z and also pullbacks, which are limits o shape := Λ 2,2. diagram d P restricts to sub-diagrams d X X, d Y Y, and d Z X. y hypothesis, these each have limits l X, l Y, and l Z which can be seen to assemble into an internal diagram d := l Y l X l Z in, which by hypothesis also has a limit l. In Proposition 6.3.8, we argue that l deines a limit or the original P -shaped diagram d. To explain why this is the case, we appeal to one o many equivalent deinitions o a limit o a diagram valued in an -category. In general, l deines a limit or a diagram d P i l represents the -category o cones d over d; see 3.3 or precise deinitions. This representability is encoded by an equivalence l d o modules rom 1 to, these modules being the -categories deined by pullbacks in the -cosmos: l 2 (p 1,p 0 ) P d ( P ) 2 (p 1,p 0 ) 1 l id 1 d P P So our hypothesised limits or the sub-diagrams o d provides equivalences d X d X, d Y d Y, and d Z d Z, and similarly, the universal property o l as 2 We thank Tim Campion or pointing out this retrospectively obvious act to us.

4 4 RIEHL ND VERITY the limit o the -shaped diagram d is encoded by an equivalence l d o modules. We must show that l has the stronger universal property o representing cones over the diagram d, i.e., that l is equivalent to d. Since the diagram P is a pushout, it ollows easily that the -category o P -shaped cones d is isomorphic to the pullback: d d Y d Z d X This result appears as Lemma So we may demonstrate the desired equivalence by arguing that l is the pullback o the equivalent cospan l Y l X l Z d Y d X d Z in the large quasi-category o modules rom 1 to. To demonstrate this, and similar results or products and inverse limits o sequences, we prove that Proposition. For any -category, the covariant and contravariant Yoneda embeddings Fun K (1, ) 1 Mod and Fun K (1, ) op Mod 1 co preserves any amily o limits which is stable under precomposition in K. The clause stable under precomposition in K has to do with a subtlety in the statement: the Yoneda embedding appearing there is external, deined as a unctor o quasi-categories, rather than internal to the -cosmos K. The limits it preserves are those arising rom the -cosmos in which is deined, which are detectable as those limits in the underlying quasi-category Fun K (1, ) o the -category that are stable under precomposition. To prove Proposition 6.2.9, in turn, we must irst analyse limits in (large) quasi-categories such as 1 Mod that are deined as homotopy coherent nerves o Kan complex enriched categories. companion paper [RV-VII] does exactly this. There, we consider a general notion o pseudo homotopy limit o a homotopy coherent diagram, deined to be a particular lexible weighted limit whose universal property is satisied up to equivalence o quasicategories. Here we prove a converse to the main result proven there, demonstrating that: 6.1.1, Theorem. Consider homotopy coherent diagram D : C[X] C valued in a Kan complex enriched category C and the transposed diagram d: X C in the homotopy coherent nerve o C. I there exists a pseudo homotopy limit cone Λ: C[ 0 X] C over D in C, then this transposes to deine a limit cone in C, and conversely, i the diagram d: X C admits a limit cone in the quasi-category C, this transposes to deine a pseudo homotopy limit cone in C. Consequently C admits pseudo homotopy limits or all small simplicial sets X i and only i the quasi-category C admits all small limits.

5 ON THE CONSTRUCTION OF LIMITS ND COLIMITS IN -CTEGORIES 5 While the steps in the proo o Theorem certainly contain more subtleties than in the classical case, the construction given here o a general limit out o iterations o simpler limits, is entirely analogous to the proo o the classical 1-category theoretic result presented, or instance, in [Rie16, ]. This paper contains all o the background needed to ill in the details o this outline, with the proos o these results appearing in 6. To concisely cite previous work in this program, we reer to the results o [RV-I, RV-II, RV-III, RV-IV, RV-V, RV-VI, RV-VII] as I.x.x.x., II.x.x.x, III.x.x.x, IV.x.x.x, V.x.x.x, VI.x.x.x, or VII.x.x.x respectively, though the statements o the most important results are reproduced here or ease o reerence. When an external reerence accompanies a restated result, this generally indicates that more expository details can be ound there. In 2, we introduce -cosmoi, lexible weighted limits, and the pseudo homotopy limits appearing in the statement o the previous theorem, and compute some explicit examples o pseudo homotopy limits. In 3, we provide an brie introduction to synthetic -category theory developed in an -cosmos, ocusing on the theory o limits and colimits and the unctors that preserve them. In 4, we deine cartesian ibrations, cocartesian ibrations, and the accompanying notion o modules between -categories. We also prove a new general result in -cosmology, demonstrating that the subcategories cocart(k) / K / and Cart(K) / K / (1.0.2) o (co)cartesian ibrations and cartesian unctors between them deine -cosmoi, as subcategories o the sliced -cosmos. Perhaps the main technical challenge in extending these classical categorical results to the -categorical context is in merely deining the Yoneda embedding that appears in Proposition In [RV-VI], the Yoneda embedding is constructed as an instance o the versatile comprehension construction. This material is reviewed in 5. For any ixed cocartesian ibration p: E in an -cosmos K and -category, the comprehension construction produces a simplicial unctor CFun K (, ) c p, cocart(k) / K / deined on a vertex a: by the pullback: E a l a E p a p a whose codomain is the Kan complex enriched core o the subcategory o K / spanned by the cocartesian ibrations and cartesian unctors. The unctor c p, transposes to deine a unctor rom the quasi-category Fun K (, ) o unctors rom to to the large quasi-category o cocartesian ibrations over. The Yoneda embedding is deined as the restriction o a particular instance o this, obtained by applying this result to the arrow -category, which deines a cocartesian ibration (p 1, p 0 ): 2 in the sliced -cosmos K /.

6 6 RIEHL ND VERITY While many o the results herein will be amiliar to the -categorically well inormed reader, in one orm or another, the context in which they are applied and the approach we take to their proos is likely to be more novel. proportion o our arguments are undamentally matter o elementary quasi-category theory, and will thus be amiliar to the quasi-categorical cognoscenti, but they are ultimately pressed into service to explicate certain aspects o the meta-theory o other species o -category. Our guiding light in developing these works has been the pro-arrow equipment [Woo82, Woo85] and Yoneda structure [SW78] based accounts o classical 1-category theory. For example, the preservation result developed in Proposition is a direct analogue o an important component o the pro-arrow axiomatics. In a similar light, the discussion o internal Yoneda embeddings developed in 5.3 will reappear in later work generalising Yoneda structures to the -cosmological realm. The arguments given here also lead, in subsequent work, to independent proos o the exponentiability o (co)cartesian ibrations o quasi-categories (and generalisations to certain higher contexts) and o the density o the point in spaces (in certain -cosmoi o ibred -categories). Furthermore, our oundations will (eventually) make substantial use o the act that the lexible weighted limit-creating inclusions (4.2.17) o Proposition deine monadic unctors between the corresponding large quasi-categories. We might also note, in passing, that the proos leading to Theorem may be generalised to deliver other important results o that kind. For example, when our ambient -cosmos K is cartesian closed then it is natural to study limits o diagrams indexed by -categories in K, rather than by simplicial sets external to it. In that situation, our approach to these results leads to an analogue o Proposition which applies to pseudo homotopy colimits o diagram shapes in K. Indeed, similar comments apply to an endeavour close to our hearts, that o generalising results o this kind to the (, )-categorical theory o complicial sets [Ver07]. The extension o many o the methods we present here to that, much more general, context is largely a matter o taking a little more care to push markings (or stratiications) around our homotopy coherent structures; this, however, is a topic or another work Size conventions. The quasi-categories deined as homotopy coherent nerves are typically large. ll other quasi-categories or simplicial sets, particularly those used to index homotopy coherent diagrams, are assumed to be small. In particular, when discussing the existence o limits and colimits we shall implicitly assume that these are indexed by small categories, and correspondingly, completeness and cocompleteness properties will implicitly reerence the existence o small limits and small colimits. Here, as is typical, small sets will usually reer to those members o a Grothendieck universe deined relative to a ixed inaccessible cardinal. Our intent is to provide a size classiication which allows us state and prove results that require such a distinction or non-triviality, principally those o the orm such and such a large category admits all small limits. Our arguments mostly comprise elementary constructions, so in applications this size distinction need not invoke the ull orce o a Grothendieck universe, indeed it might be as simple as that between the inite and the

7 ON THE CONSTRUCTION OF LIMITS ND COLIMITS IN -CTEGORIES 7 ininite. t the other extreme it might involve the choice o two Grothendieck universes to prove results about large categories. One such result is Theorem which, on interpreting its size distinction relative to a second larger Grothendieck universe, provides a result which applies to large simplicial categories. Indeed, with a little more care, the proo o that result may be adapted to apply to large and locally small simplicial categories without the introduction o a second universe (by restricting arguments only to accessible simplicial presheaves) although we choose not to worry the reader with such arcana here. We use a common typeace e.g., K, 1 Mod(K) to dierentiate small and large quasi-categories rom generic -categories ; see cknowledgements. The authors are grateul or support rom the National Science Foundation (DMS and DMS ) and rom the ustralian Research Council (DP ). This work was commenced when the second-named author was visiting the irst at Harvard and then at Johns Hopkins, continued while the irst-named author was visiting the second at Macquarie, and completed ater everyone inally made their way home. We thank all three institutions or their assistance in procuring the necessary visas as well as or their hospitality. 2. -cosmoi and lexible weighted limits In this section we review the axiomatic ramework or the ormal theory o -categories, introducing the notion o an -cosmos in an abbreviated 2.1. In 2.2, we review some o the more exotic lexible weighted limits that exist in any -cosmos, and in 2.3 we ocus our attention on one important special case: the pseudo homotopy limits which deine limits in the large quasi-categories o -categories in an -cosmos that will make an appearance in cosmoi and the comma construction. n -cosmos is a category K whose objects, we call -categories and whose unction complexes Fun K (, ) are quasicategories o unctors between them. The handul o axioms imposed on the ambient quasi-categorically enriched category K permit the development o a general theory o - categories synthetically, i.e., only in reerence to this axiomatic ramework, as we shall discover in Deinition ( -cosmos). n -cosmos is a simplicially enriched category K whose objects we reer to as the -categories in the -cosmos, whose hom simplicial sets Fun K (, ) are quasi-categories, and that is equipped with a speciied subcategory o isoibrations, denoted by, satisying the ollowing axioms: (a) (completeness) s a simplicially enriched category, K possesses a terminal object 1, small products, cotensors U o objects by all small simplicial sets U, inverse limits o countable sequences o isoibrations, and pullbacks o isoibrations along any unctor.

8 8 RIEHL ND VERITY (b) (isoibrations) The class o isoibrations contains the isomorphisms and all o the unctors!: 1 with codomain 1; is stable under pullback along all unctors; is closed under inverse limit o countable sequences; and i p: E is an isoibration in K and i: U V is an inclusion o simplicial sets then the Leibniz cotensor i p: E V E U U V is an isoibration. Moreover, or any object X and isoibration p: E, Fun K (X, p): Fun K (X, E) Fun K (X, ) is an isoibration o quasi-categories. For ease o reerence, we reer to the limit types listed in axiom (a) as the cosmological limit types, these reerring to diagrams o a particular shape with certain maps given by isoibrations. The underlying category o an -cosmos K has a canonical subcategory o representablydeined equivalences, denoted by, satisying the 2-o-6 property: a unctor : is an equivalence just when the induced unctor Fun K (X, ): Fun K (X, ) Fun K (X, ) is an equivalence o quasi-categories or all objects X K. The trivial ibrations, denoted by, are those unctors that are both equivalences and isoibrations. These axioms imply that the underlying 1-category o an -cosmos is a category o ibrant objects in the sense o rown. Consequently, many amiliar homotopical properties ollow rom Deinition For instance, the axioms o an -cosmos permit us to construct arrow and comma -categories as particular simplicially enriched limits Deinition (arrow -categories). For any -category, the simplicial cotensor 2 := 1 (p 1,p 0 ) 1 = deines the arrow -category 2, equipped with an isoibration (p 1, p 0 ): 2, where p 1 : 2 denotes the codomain projection and p 0 : 2 denotes the domain projection. rrow -categories can be used to deine a general comma -category associated to a cospan o unctors Deinition (comma -categories). ny pair o unctors : and g : C in an -cosmos K has an associated comma -category, constructed by the ollowing pullback in K: g 2 (p 1,p 0 ) C g (p 1,p 0 ) Proposition (maps between commas, VII.3.1.4). natural transormation o cospans on the let o the ollowing display gives rise to the diagram o pullbacks on the

9 ON THE CONSTRUCTION OF LIMITS ND COLIMITS IN -CTEGORIES 9 right g 2 C c g a C g b C c b g g a 2 a a ( ) 2 C g in which the uniquely induced dashed map completing the commutative cube is denoted (b, a, c): g g. Moreover, (b, a, c) is an isoibration (resp. trivial ibration, equivalence) whenever the components a, b and c are all maps o that kind Proposition ( -cosmoi o isoibrations, VII.2.1.5). For any -cosmos K, there is an -cosmos K which has: objects all isoibrations p: E in K; unctor space rom p: E to q : F deined by taking the pullback Fun K 2(p, q) Fun K (E, F ) Fun K (E,q) Fun K (, ) Fun K (E, ) FunK (p,) in simplicial sets so, in particular, the 0-arrows rom p to q are commutative squares E g F q p (2.1.6) in K; equivalences those squares (2.1.6) whose components and g are equivalences in K and isoibrations (resp. trivial ibrations) those squares or which the map and the induced map E F (and thus also g) are isoibrations (resp. trivial ibrations) in K. The cosmological limits are deined object-wise in K, or in other words are jointly created by the domain and codomain projections dom, cod: K 2 K. cosmological unctor is a simplicial unctor K L between -cosmoi that preserves the classes o isoibrations and all the cosmological limits. For instance, the domain and codomain projections dom, cod: K 2 K are both cosmological. The cosmological unctor cod: K 2 K has a special property: namely its ibres deine -cosmoi: the sliced - cosmoi K / o Example IV

10 10 RIEHL ND VERITY 2.2. Flexible weighted limits in an -cosmos. The basic simplicially-enriched limit notions enumerated in axiom 2.1.1(a) imply that an -cosmos K possesses a much larger class o simplicially enriched limits. eore we introduce them, we warm up with the ollowing elementary observation: Deinition. wide pullback diagram in an -cosmos K is a countable diagram o the ollowing orm n n 1 p n 1 n 1 p n 2 n 2 p 1 1 p 0 n 1 n in which right acing arrows are isoibrations, which may be truncated to the let o some n or it may extend ininitely. We now observe that -cosmoi admit wide pullbacks: limits o the diagrams just described Lemma. -cosmoi admit wide pullbacks and their construction is invariant under pointwise equivalence between diagrams. Proo. We may orm limits o such diagrams in any -cosmos by working rom the right to pullback successive isoibrations as ar as possible and then, in the ininite case, taking the limit o the countable sequence o isoibrations proceeding down the right-hand diagonal o the resulting diagram. To see that wide pullbacks are homotopical, in the sense that any natural transormation o wide pullback diagrams (o the same cardinality) whose components are all equivalences induces a map between the wide pullbacks o those diagrams which is also an equivalence, we appeal to the corresponding acts or the pullbacks and limits o countable sequences o isoibrations which were used to construct them. Now we introduce the general notion o a weighted limit o a simplicially enriched unctor valued in a simplicial category C with hom-spaces dented by Map C (X, Y ) Deinition. weight or a diagram indexed by a small simplicial category is a simplicial unctor W : SSet. W -cone over a diagram F : C in a simplicial category C is comprised o an object L C together with a simplicial natural transormation λ: W Map C (L, F ). Such a cone displays L as a W -weighted limit o F i and only i or all X C the simplicial map Map C (X, L) = Map SSet (W, Map C (X, F )) (2.2.4) given by post-composition with λ is an isomorphism, in which case the limit object L is typically denoted by {W, F } or simply {W, F }. In this notation, the universal property (2.2.4) o the weighted limit asserts an isomorphism Map C (X, {W, F } ) = {W, Map C (X, F )}.

11 ON THE CONSTRUCTION OF LIMITS ND COLIMITS IN -CTEGORIES Deinition (lexible weights). For a small simplicial category and pair o objects [n] and, the projective n-cell associated with is the simplicial natural transormation: n Map (, ) n Map (, ). weight W : SSet is a lexible i the inclusion W may be expressed as a countable composite o pushouts o coproducts o projective cells Proposition (VII.4.1.5). Let K be an -cosmos and let be a small simplicial category. (i) For any diagram F : K and lexible weight W : SSet, the weighted limit {W, F } exists in K. (ii) I κ: F G is a simplicial natural transormation between two such diagrams whose components are equivalences, isoibrations, or trivial ibrations in K and W is a lexible weight, then the induced map {W, F } {W,κ} {W, G} is an equivalence, isoibration, or trivial ibration (respectively) in K. When working in a quasi-categorically enriched category K it is oten the case that we are only interested in weighted (co)limits that are deined up to equivalence rather than isomorphism. To that end we have the ollowing deinition: Deinition (lexible weighted homotopy limits). Suppose that W : SSet is a lexible weight and that F : K is a diagram in a quasi-categorically enriched category K. We say that a W -cone λ: W Fun K (L, ) displays an object L K as a lexible weighted homotopy limit o F weighted by W i or all objects X K the map Fun K (X, L) {W, Fun K (X, F )}. (2.2.8) induced by post-composition with λ is an equivalence o quasi-categories, 3 in which case we denote the limit object by {W, F }. In particular, lexible weighted homotopy limits exist in the (, 1)-categorical core g K o an -cosmos K, the subcategory with the same objects and with unctor spaces Fun g K(, ) := gfun(, ) deined to be the groupoid cores o the unctor quasi-categories; see Deinition VII Proposition (VII.4.2.7). The (, 1)-core o an -cosmos admits lexible weighted homotopy limits. See VII.4.2 or more details about their construction. 3 Here we show that the codomain o the comparison map in (2.2.8) is a quasi-category by applying Proposition in the -cosmos o quasi-categories.

12 12 RIEHL ND VERITY 2.3. Pseudo limits o homotopy coherent diagrams. In classical homotopy theory, a homotopy coherent diagram is a simplicial unctor CX C whose domain is the homotopy coherent realisation o the simplicial set X, a simplicial category ormed as the let adjoint to the homotopy coherent nerve: sset-cat For diagrams o this shape, there is a lexible weight W X o particular interest, with the property that a homotopy coherent diagram o shape X and a W X -shaped cone over that diagram together assemble into a simplicial unctor C[ 0 X] C (see VII.5.2) Deinition (VII.5.2.8, VII.5.2.9). For any simplicial set X, the weight or the pseudo limit o a homotopy coherent diagram o shape X is the unctor CX C N sset W X SSet given by W X (x) := Fun C[ 0 X](, x). Lemma VII veriies that this is a lexible weight. The W X -weighted limit o a homotopy coherent diagram o shape X is then reerred to as the pseudo limit o that diagram. y Propositions and 2.2.9, -cosmoi and their groupoidal cores admit pseudo (homotopy) limits o homotopy coherent diagrams. In this section, we calculate pseudo homotopy limits applying Deinition to the weight o Deinition or simple but important diagram shapes. Consider as the indexing 1-category J either: a discrete category, the pullback shape, or the category ω op indexing inverse sequences. In each case, J is a ree category on an underlying graph G J o atomic arrows, which we regard as a 1-skeletal simplicial set. s the ollowing lemma explains, in such contexts, strictly diagrams J C are automatically homotopy coherent Lemma. Let J be a 1-categorically reely generated by the graph G J. (i) The homotopy coherent realisation C[G] is isomorphic to J, regarded as a simplicial category with discrete hom sets. Hence diagrams J C in a Kan complex enriched category, correspond bijectively to diagrams G N C in the homotopy coherent nerve. (ii) For any Kan complex enriched category C, the quasi-categories NC J and NC G o diagrams are equivalent. Hence up to equivalence, we can represent a quasicategorical diagram J NC by a point-set diagram J C. Proo. The isomorphism C[G] = J o (i) is easily recognised rom the explicit description o the homotopy coherent realisation unctor given in Proposition VII : the homotopy coherent realisation o any 1-skeletal simplicial set is the ree discrete category generated by this graph. Hence, diagrams J = C[G] C in a Kan complex enriched category, correspond bijectively to diagrams G NC in the homotopy coherent nerve.

13 ON THE CONSTRUCTION OF LIMITS ND COLIMITS IN -CTEGORIES 13 For (ii), since NC is a quasi-category and G J is inner anodyne when considered as a monomorphism o simplicial sets, the quasi-categories NC J and NC G o diagrams are equivalent. So, up to equivalence, we can represent any diagram J NC by its restriction G J NC, which transposes to a strictly commuting diagram in C by (i) Deinition. When J is the ree category generated by a graph G, a strictly commuting pseudo cone over a diagram F : J = C[G] C is ormed by restricting a strict cone α: L F ( ), presenting as a simplicial natural transormation α: 1 Map C (L, F ( )), along the unique map W G 1 o weights: We have the ollowing result: W G! 1 α Map C (L, F ) Proposition. In the case where, J is a discrete category, the pullback shape, or ω op with generating subgroup G J, the strictly commuting pseudo cone ormed rom the limit cone π : lim(f ) F ( ) over a diagram F : J K valued in an -cosmos K presents presents lim(f ) as a pseudo homotopy limit o the diagram F in the (, 1)-categorical core g K. Proo. Given X K, post-composition by the weighted cone π!: W G Fun gk (lim(f ), F ) determines a map Fun g K(X, lim(f )) {W G, Fun g K(X, F )} (2.3.5) o Kan complexes, and our task is to show that this is an equivalence. Notice, however, that we have Fun g K(X, lim(f )) = g(fun K (X, lim(f ))) = g(lim(fun K (X, F ))) = lim(g(funk (X, F ))) = lim(fun g K(X, F )) = {1, Fun g K(X, F )} in which the irst isomorphism ollows because lim(f ) is a simplicially enriched limit in K and the second because the groupoid core unctor is a right adjoint on underlying categories. Under this isomorphism it is easily checked that the map in (2.3.5) is isomorphic to the map {1, Fun g K(X, F )} {W G, Fun g K(X, F )} induced by the unique map o weights!: W G 1. It ollows that it is enough to show that or any diagram F : J Kan o the appropriate kind in Kan complexes the induced map {!, F }: {1, F } {W G, F } rom the strict limit o F to the pseudo limit o F is an equivalence, which is achieved by the next three lemmas Lemma. For any amily o objects { i } in a simplicial category with products, the strict limit cone π : i i i deines a pseudo homotopy limit cone. Proo. In this case the result is trivial because the weight or the pseudo limit o a discrete diagram is isomorphic to the terminal weight.

14 14 RIEHL ND VERITY In particular, Lemma applies to the -cosmos Kan o Kan complexes o Example VII Lemma. The strict pseudo cone ormed rom the pullback cone over a diagram o Kan complexes and Kan ibrations P C deines a pseudo homotopy limit cone in Kan. Proo. Unpacking Deinition 2.3.1, the weight W : C = SSet or pseudo limits over the pullback shape is given by the simplicial unctor which maps the outer objects o to 0 and the middle object to op = Λ 2,0. From the pullback diagram in the statement, we derive the ollowing diagram C C p p op p Here the upper row is a wide pullback diagram whose limit is simply the pullback o the original diagram. The lower row is the wide pullback diagram whose limit is the end that computes the limit weighted by W. The middle component o the transormation rom top to bottom is an equivalence because is contractible in the Kan model structure and is a Kan complex; it ollows by Lemma that the induced map between the wide pullbacks o these diagrams is an equivalence as required Lemma. The strict pseudo cone ormed rom the limit cone over a sequence o Kan ibrations between Kan complexes deines a pseudo homotopy limit cone. p n n p n 1 p 1 1 p 0 0 Proo. The diagram shape in the statement is the ordered set N op with objects n and nonidentity edges n + 1 n and the weight W N op : CN op = ω op SSet maps each object n to the 1-skeletal simplicial set N with connecting map rom one integer to its predecessor given by the successor map s: N N. From the given sequence o Kan ibrations we may derive the ollowing commutative diagram: p n 1 p n p N n N p N 2 N 1 N 1 N 0 N 0 n 1 s n s 2 p N 1 s 1 p N 0 s 0

15 ON THE CONSTRUCTION OF LIMITS ND COLIMITS IN -CTEGORIES 15 Here the upper row is a wide pullback diagram whose limit is simply the limit o the original diagram. The lower row is the wide pullback diagram whose limit is the end that computes the limit weighted by W N op. The component o the transormation rom top to bottom are equivalences because N is contractible in the Kan model structure and each n is a Kan complex. Now it ollows by Lemma that the induced map between the wide pullbacks o these diagrams is an equivalence as required. 3. The ormal theory o -categories The motivation or -cosmology is that it enables us to develop the theory o - categories ormally"; in particular, independently o the semantics o any particular model. In this section, which is part review and part new material, we introduce those aspects o the ormal theory o -categories that we will need later in this paper. We start in 3.1 with an introduction to absolute liting diagrams, which encode universal properties o -categories using the structure o a strict 2-category o -categories, - unctors, and -natural transormations constructed as a quotient o an -cosmos. The universal properties expressed by absolute liting diagrams can also be encoded internally to the -cosmos as a ibred equivalence between comma -categories, which are the subject o 3.2. In 3.3, we deine limits and colimits o diagrams valued in an -category, introducing the main subject o this paper. Finally, in 3.4 we use an elementary lemma about composition and cancellation o absolute liting diagrams to deduce myriad ormal consequences: the act that right adjoints preserve limits and ully aithul unctors relect them among others. We also deine what it means or a unctor o -categories to be ully aithul or strongly generating, two important properties o the Yoneda embeddings o bsolute liting diagrams and (relative) adjunctions. Recall the quotient homotopy 2-category o an -cosmos: Deinition (the homotopy 2-category o an -cosmos). The homotopy 2-category o an -cosmos K is deined by applying the homotopy category unctor h: QCat Cat to the unctor spaces o the -cosmos: The objects o h K are the objects o K, i.e., the -categories. The 1-cells : o h K are the vertices Fun K (, ) in the unctor spaces o K, i.e., the -unctors. 2-cell α g in h K is represented by a 1-arrow α: g Fun K (, ), where a parallel pair o 1-arrows in Fun K (, ) represent the same 2-cell i and only i they bound a 2-arrow whose remaining outer ace is degenerate Deinition (dual -cosmoi). For any -cosmos K, write K co or the -cosmos with the same objects but with the opposite unctor spaces Fun K co(, ) := Fun K (, ) op.

16 16 RIEHL ND VERITY The homotopy 2-category o K co is the co dual o the homotopy 2-category o K, reversing the 2-cells but not the 1-cells Deinition. n adjunction between -categories, K is simply an adjunction in the homotopy 2-category h K: i.e., is comprised o a pair unctors : and u:, together with a pair o 2-cells η : id u and ε: u id satisying the triangle identities. We shall also make use o the ollowing partial adjunction notion: Deinition (absolute right liting). Given a cospan C g, a unctor l: C and a 2-cell (3.1.5) C l g deine an absolute right liting o g through i any 2-cell as displayed below-let actors uniquely through λ as displayed below-right X c b α C g = λ X c C b! ᾱ l g λ (3.1.6) When this property holds, we say that the triangle displayed in (3.1.5) as an absolute right liting diagram Remark. In category theory, the term absolute typically means preserved by all unctors. In that spirit, an absolute right liting diagram is a right liting diagram λ: l g with the property that the restriction o λ along any generalised element c: X C again deines a right liting diagram Example. Importantly, u is an adjunction with counit ε: u id i and only i the triangle is an absolute right liting diagram. u ε

17 ON THE CONSTRUCTION OF LIMITS ND COLIMITS IN -CTEGORIES Deinition. Transormations C g w u v C g between diagrams which admit absolute right litings give rise to the ollowing diagram C l w g λ u v C g = C l w v τ l λ C g (3.1.10) in which the triangles are absolute right litings and the 2-cell τ is induced by the universal property o the triangle on the right. We say that the transormation (3.1.10) is right exact i and only i the induced 2-cell τ is an isomorphism. This right exactness condition holds i and only i, in the diagram on the let, the whiskered 2-cell uλ displays vl as the absolute right liting o g w through Observation. Unpacking the deinitions in any 2-category, λ: l g deines an absolute right liting diagram i and only i the induced unctor hom(x, ) hom(x, l) = hom(x, ) hom(x, g) (3.1.12) deines an isomorphism o comma categories, natural in X. From the 2-categorical universal property o these comma categories, it is now clear that λ: l g deines an absolute right liting diagram i and only i (i) or each object X, the diagram hom(x,l) hom(x,λ) hom(x, ) hom(x,) hom(x, C) hom(x, ) hom(x,g) deines an absolute right liting diagram in Cat, and (3.1.13)

18 18 RIEHL ND VERITY (ii) moreover, each morphism e: Y X induces a right exact transormation hom(x, ) hom(x,) hom(x, C) hom(x,g) hom(x, ) hom(e,x) hom(y, ) hom(e,x) hom(e,x) hom(y,) hom(y, C) hom(y,g) hom(x, ) In act, by the Yoneda lemma, it suices to assume in (i) that hom(x, g) admits any absolute right liting along hom(x, ) or which the transormation induced rom any e: Y X is right exact. Specialising to the case X = C reveals that this absolute let litings are represented by a morphism l: C and 2-cell λ: l g Fibred equivalences between comma -categories. In general the homotopy 2-category o an -cosmos will admit ew 2-dimensional limit notions. Nonetheless, the 2-cell representing the horizontal unctor in the deining pullback o the comma -category o a cospan: g 2 (p 1,p 0 ) C g enjoys a weak universal property: (p 1,p 0 ) C p 1 g g p 0 φ,g (3.2.1) Proposition (IV.3.4.6). 2-cells in the homotopy 2-category h K o the orm depicted on the let in the ollowing diagram D b c α C g D (c,b) ᾱ C g (p 1,p 0 ) stand in bijective correspondence to isomorphism classes o 1-cells in the slice 2-category (h K) /C as shown on the right, with the action o this bijection, rom right to let, is given by composition with the comma square depicted in (3.2.1). We reer the curious reader interested in more details to I.3.3 or IV Observation. We may generalise the construction o Proposition to diagrams in h K o the ollowing orm: C c β g C g a α b (3.2.4)

19 ON THE CONSTRUCTION OF LIMITS ND COLIMITS IN -CTEGORIES 19 This may be glued onto the square that displays the comma g (3.2.1) to give the pasted square on the let o the ollowing diagram C c C p 1 g β g g p 0,g φ b α a = C p 1 g g α β g φ,g p 0 (3.2.5) which induces a unctor α β as shown on the right, by the weak 2-universal property o g. Indeed, the Proposition tells us that α β is a representative o a uniquely determined isomorphism class o such unctors in K /C. This construction is unctorial in the ollowing sense, suppose that we are given a second diagram o the orm given in (3.2.4): C g c β C g a α We may juxtapose these two diagrams vertically to give the ollowing diagram C c g p 1 p 0,g φ g b β α a b β α a C g c C g = p 1 g α β g C φ,g g β α c C g a p 0 b = b C p 1 g g α β g α β g φ,g in which we ve applied the induction process depicted in (3.2.5) twice. lternatively, take the same diagram on the let and start by orming the pasting composites o each column p 0

20 20 RIEHL ND VERITY o squares C c g p 1 p 0,g φ g b β α a b β α a C g c C g = C p 1 g g g p 0,g φ c c b b p α α q β r p β C a a = C p 1 g g g φ,g (p α α q) (β r p β) then apply the induction process depicted in (3.2.5) only once. It ollows that the unctors (α β )(α β) and (p α α q) (β r p β) are both induced by the same diagram on the let under the weak 2-universal property o g, consequently there exists an isomorphism between them in K /C. For instance, suppose that we are given triangle as in (3.1.5) o Deinition 3.1.4, then we may apply the construction o Observation to the diagram C l λ C g to give a unctor λ: l g ibred over C, which detects whether the diagram (3.1.5) is absolute right liting: Proposition (I.5.1.3). Given a cospan C g then there exists an equivalence g l over C i and only i there exists an absolute right liting diagram o the ollowing orm C l g Remark. The construction in Observation may clearly be regarded as being a generalisation o that discussed in Proposition There is, however, a good reason or distinguishing them in the way we have. On the one hand, the construction in Proposition relies only upon the strict universal property o the comma construction, and so it delivers a uniquely determined map. On the other hand, the construction in Observation depends upon a choice o unctors representing a 2-cell and so is only deined up to isomorphism. It is, nevertheless, the case that when both constructions apply the irst λ p 0

21 ON THE CONSTRUCTION OF LIMITS ND COLIMITS IN -CTEGORIES 21 provides a speciic choice which is certainly a member o the isomorphism class determined by the second. This distinction becomes particularly important in situations where it is important to iner that the map induced by a transormation o cospans is an isoibration. s we have seen such results hold or the construction o Proposition 2.1.4, but they cannot reasonably be expected to hold or that o Observation 3.2.3, simply because the class o isoibrations is not closed under isomorphisms at the 2-cell level Limits and colimits in an -category. Via the nerve embedding, diagrams indexed by small categories are among the diagrams indexed by small simplicial sets. The simplicial cotensors o axiom 2.1.1(a) are used to deine -categories o diagrams Deinition (diagram -categories). I J is a small simplicial set and is an - category, then the -category J is naturally thought o as being the -category o J-indexed diagrams in Deinition. n -category admits all limits o shape J i the constant diagram unctor : J,constructed by applying the contravariant unctor ( ) to the unique simplicial map J 0, has a right adjoint: J lim For many purposes, however, this deinition is insuiciently general since many - categories will have some, but not all, limits o diagrams o a particular indexing shape. One useul such generalisation, based upon the absolute liting notion introduced in Deinition 3.1.4, ollows: Deinition (limits o amilies). I J is a small simplicial set and and D are -categories then we can regard a unctor d: D J as being an amily o J-indexed diagrams in. We say that the members o such a amily admits a amily o limits l: D i there exists an absolute right liting diagram: l D d λ J (3.3.4) In the case where D is taken to be the terminal -category 1, we think o d as being a single diagram o shape J, l as its limit, and λ as the limiting cone Deinition ( -categories o cones). For any diagram d: 1 J o shape J in an -category, the -category o cones over d is the comma -category p 0 : d

22 22 RIEHL ND VERITY ormed by the pullback d J 2 (p 1,p 0 ) 1 J J d (p 1,p 0 ) Dually, the -category o cones under d is the comma -category p 1 : d. The ollowing result recasts Deinition in terms o ibred equivalences o comma -categories: Proposition (I.5.1.8). Given a small simplicial set J and -categories D and, then a amily d: D J o J-indexed diagrams admits a amily o limits l: D i and only i the the -category o cones d is equivalent to l over D. Colimits are characterised dually by absolute let liting diagrams; that is to say a triangle C l g λ (3.3.7) in which the direction o the 2-cell is switched relative to that in (3.1.5) and which enjoys a universal property akin to that in (3.1.6) but with the sense o all 2-cells reversed. Propositions and dualise to characterise absolute let liting diagrams and colimits as ibred equivalences between comma -categories Some ormal -category theory. In this subsection we collect an assortment o -categorical variants o amiliar categorical results or application in later sections. Our arguments here are entirely ormal and we work entirely within a ixed -cosmos K. Indeed, all o these results are amenable to 2-categorical arguments in the homotopy 2- category h K. Our irst two results arise as consequences o the ollowing composition and cancellation result or absolute right liting diagrams in a 2-category: Lemma (composition and cancellation o absolute right liting diagrams). In any 2-category, suppose we are given a diagram D h λ k µ l and assume that the lower triangle is an absolute right liting diagram. Then the upper triangle is an absolute right liting diagram i and only i the composite triangle displays h as an absolute right liting o l along g. C g

23 ON THE CONSTRUCTION OF LIMITS ND COLIMITS IN -CTEGORIES 23 Proo. This is a matter o routine veriication, directly rom the universal property o absolute right liting diagrams, which we leave to the reader. Our irst application o this result proves the ollowing chestnut: Proposition (I ). Right adjoints preserve limits and let adjoints preserve colimits. Proo. We prove the preservation result or limits; the result or colimits ollows dually. Suppose we are given a unctor u: with let adjoint : displayed by a unit η : id u and counit ε: u id. Our aim is to show that i we are given an absolute right liting diagram as depicted on the let o the ollowing diagram l λ λ u D J D J J d d u J l (3.4.3) then the pasted triangle on the right displays ul as an absolute right liting o u J d along. To that end, observe that the operation o cotensoring with J provides us with a 2- unctor ( ) J : h K h K and so it carries the adjunction u to an adjunction J u J with unit η J and counit ε J. It ollows, by Example and the act that absolute liting diagrams are stable under restriction along any unctor, that the whiskered triangle D d J u J id J J ε J J J (3.4.4) is an absolute right liting diagram, which we may paste onto the diagram on the right o (3.4.3) to give a triangle whose 2-cell is given by the vertical composite: This we may reduce to the composite J ul J u J l J u J λ J u J d ε J d d J ul J u J l ε J l l λ d by application o the middle-our interchange rule. On noting that J = and ε J = ε this reduces, in turn, to the 2-cell obtained by pasting the ollowing diagram: D ul εl l d λ J (3.4.5)

24 24 RIEHL ND VERITY Here the lower triangle is an absolute right liting diagram by assumption and the upper triangle is also such by Example and Remark We conclude by applying Lemma twice, irstly to show that the pasted triangle in (3.4.5) is an absolute right liting diagram and then to show that we may cancel the absolute right liting in (3.4.4) to show that the pasted triangle on the right o (3.4.3) is an absolute right liting as required. Inormally we say that a unctor k : I J quasi-categories is initial i it enjoys the property that an object is a limit o a J indexed diagram i and only i it is a limit o the I indexed diagram obtained by restricting along. The next proposition establishes the amiliar result that let adjoint unctors are initial in this sense: Proposition. Suppose k : I J is a unctor between small quasi-categories that admits a right adjoint r : J I, with unit η : id I rk and counit ε: kr id J, and consider a amily o diagrams d: D J valued in any -category. Then a unctor l: D is a limit o d i and only i it also a limit o the restricted amily d I : D d J k I. Proo. Our proo is similar to that o the last proposition. In the ollowing diagram we must show that the let-hand diagram is an absolute right liting diagram l λ D J D J I d d k l λ (3.4.7) i and only i the right-hand pasted diagram is such. Cotensoring into provides a contravariant 2-unctor ( ) : h K op h K, so it carries the adjunction k r to an adjunction r k with unit η : id I k r and counit ε : r k id J. It ollows, by Example and Remark 3.1.7, that the whiskered triangle D d J k id J I ε r J (3.4.8) is an absolute right liting diagram, which we may paste onto the diagram on the right o (3.4.7) to give a triangle whose 2-cell is given by the vertical composite: r l r k l r kλ r k d ε d d This we may reduce to the composite λ ( ε l) by application o the middle-our interchange rule. On noting that r = and ε = id we see that this reduces urther to λ. In other words, we have shown that the triangle on the let o (3.4.7) is equal to the pasting the triangle on the right o (3.4.7) with the absolute right liting diagram in (3.4.8). Consequently Lemma applies to propagate absolute right liting properties rom one side o (3.4.8) to the other, with the passage rom right to let by composition and that rom let to right by cancellation.