Fibrations and lax limits of (∞,2) -categories
aa r X i v : . [ m a t h . A T ] D ec FIBRATIONS AND LAX LIMITS OF (∞ , ) -CATEGORIES ANDREA GAGNA, YONATAN HARPAZ, AND EDOARDO LANARI
Abstract.
We study four types of (co)cartesian fibrations of ∞ -bicategoriesover a given base B , and prove that they encode the four variance flavors of B -indexed diagrams of ∞ -categories. We then use this machinery to set up ageneral theory of 2-(co)limits for diagrams valued in an ∞ -bicategory, capableof expressing lax, weighted and pseudo limits. When the ∞ -bicategory athand arises from a model category tensored over marked simplicial sets, weshow that this notion of 2-(co)limit can be calculated as a suitable form ofa weighted homotopy limit on the model categorical level, thus showing inparticular the existence of these 2-(co)limits in a wide range of examples. Contents
Introduction Acknowledgements ∞ -bicategories 61.3 Gray products of scaled simplicial sets 101.4 Scaled straightening and unstraightening 111.5 Marked-scaled simplicial sets 13 / op-symmetry and cartesian fibrations 353.3 Straightening and unstraightening 36 ∞ -bicategories References Mathematics Subject Classification.
Introduction
A fundamental idea in category theory is that diagrams of ( ∞ -)categories, in-dexed by an ( ∞ -)category B , can be encoded via a suitable form of fibration E → B .This idea, going back to Grothendieck for ordinary categories, has become indis-pensable in the ∞ -categorical realm, where it was developed notably in the extensiveworks of Lurie. In effect, a fibration E → B is often the most efficient option, andsometimes the only practical one, for writing down a diagram of ∞ -categories withall coherence data involved.In the ∞ -categorical context, such fibrations come in two flavors, called cartesian and cocartesian fibrations. The latter encodes the data of a C at ∞ -valued functor on B , while the former the data of a C at ∞ -valued presheaf on B , that is, a functor B op → C at ∞ . The existence of two dual flavors of this type is prevalent in categorytheory, and reflects the Z / C at ∞ given by the involution C ↦ C op . Inparticular, this symmetry sends cartesian fibrations to cocartesian ones, and viceversa.As in ordinary category theory, the study of ∞ -categories often leads to con-sider (∞ , ) -categories , as, for example, the collection of ∞ -categories is itself bestunderstood when organized into one. There are currently many models for (∞ , ) -categories, developed and compared to one another in the works of Lurie [13],Verity [19], Rezk–Bergner [3, 4], Ara [1], Barwick–Schommer-Pries [2], and morerecently in the authors previous work [6], where Lurie’s bicategorical model struc-ture on scaled simplicial sets was compared with the 2-trivial complicial modelstructure on stratified sets developed in [16], yielding the last remaining equivalencebetween the various models. In addition, in [6] we reinterpreted the bicategoricalmodel structure in terms of Cisinski–Olschok’s theory of localizers, a consequenceof which is an identification of the notion of ∞ -bicategories - the fibrant objectin this model structure - with scaled simplicial sets satisfying a suitable extensionproperty.It is natural to ponder the counterpart of the theory of (co)cartesian fibrationsin the case where the base B is now an ∞ -bicategory. On the diagram side, onemay then consider B -indexed diagrams in C at ∞ , where C at ∞ is also considered asan ∞ -bicategory. As an immediate difference from the ∞ -categorical case, oneobserves that the theory of ∞ -bicategories admits not just a Z / ( Z / ) -symmetry: we have the involution C ↦ C op which reverses the directionof all 1-morphisms (without affecting the direction of 2-morphisms), but we alsohave the involution C ↦ C co , which reverses the direction of 2-morphisms, withoutaffecting the direction of 1-morphisms. As a result, one has four variance flavors for B -indexed diagrams in C at ∞ , corresponding respectively to C at ∞ -valued functorsfrom B , B op , B co and B coop = ( B co ) op . As a result, we expect to have four differenttypes of fibrations E → B this time, encoding B -indexed C at ∞ -valued diagramscorresponding to the above four variance flavors.In the first part of this paper we identify these four types of fibrations as innercocartesian , outer cartesian , outer cocartesian and inner cartesian fibrations, re-spectively. The first of these four notions is based on that studied in [13], for whicha straightening-unstraightening Quillen equivalence is constructed. The last onecan be obtained from the first by applying the functor (−) op on the level of scaledsimplicial sets (where we note that this acts on the fibers by (−) op as well). Bothof these are in particular inner fibrations on the level of simplicial sets, hence theirname. On the other hand, since the scaled simplicial set model is not equipped witha convenient point-set model for (−) co , the two flavors designated by the term outer require a more substantial modification of the definition on the simplicial level. Aworking definition was introduced in [6], and in the present paper we prove that IBRATIONS AND LAX LIMITS OF (∞ , ) -CATEGORIES 3 these indeed fulfill the purpose of encoding functors of the form B co → C at ∞ and B op → C at ∞ , respectively. In particular, the main result of the first part affirms thatif we organize the collection of inner/outer (co)cartesian fibrations over a fixed base B , with functors over B which preserves (co)cartesian edges as morphisms, thenthe result is equivalent to the ∞ -bicategory of B -indexed C at ∞ -valued diagrams,with the appropriate variance flavor: Theorem 1 (See Corollary 3.3.2 and Corollary 3.3.3) . For an ∞ -bicategory B ∈ BiCat ∞ there are natural equivalences of ∞ -bicategories coCar inn ( B ) ≃ Fun ( B , C at ∞ ) , Car inn ( B ) ≃ Fun ( B coop , C at ∞ ) , coCar out ( B ) ≃ Fun ( B co , C at ∞ ) and Car out ( B ) ≃ Fun ( B op , C at ∞ ) . Here, coCar inn ( B ) denotes the ∞ -bicategory of inner cocartesian fibrations over B and cocartesian edges preserving functors over B between them, and similarly for Car inn ( B ) , coCar out ( B ) and Car out ( B ) . In the second part of this paper we use the theory of inner/outer (co)cartesianfibrations in order to setup a well-behaved notion of 2-(co)limits for diagrams takingvalues in an ∞ -bicategory. Our notion is sufficiently flexible to accommodate both(op)lax and pseudo-(co)limits, as well as a variety of intermediate variants, and canalso be used to give a notion of weighted (co)limits. As with ordinary 2-categories,this notion comes in principle in four flavors, spanning lax and oplax, limits andcolimits. To enable a systematic treatment we exploit in a crucial manner the iden-tification of the four types of fibrations from the first part of the paper. To keep thenotation tractable, and since the notation of lax and oplax is not completely consis-tent in the literature, we have opted here to call these inner and outer (co)limits ,making each type of (co)limit directly related to the type of fibration that governsit. After giving the definitions and extracting their main properties we proceedto show that our proposed notion of 2-(co)limit is sufficiently flexible to expressa suitable notion of a weighted 2-(co)limit, and that furthermore, every type of2-(co)limit can eventually be viewed as a weighted one for a suitable weight. Wethen exploit this point of view in order to compare our notion of 2-(co)limits withthat of a weighted homotopy (co)limits in the case where the ambient ∞ -bicategorycomes from a model category tensored over marked simplicial sets. This allows us toexhibit a wide range of examples for ∞ -bicategories in which all small 2-(co)limitsexist: Theorem 2 (See Corollary 5.4.11) . Let M be an S et + ∆ -tensored model categorysuch that the projective (resp. injective) model structure exists on M J for any small S et + ∆ -enriched category J ( e . g ., M is a combinatorial model category). Then the ∞ -bicategory M ∞ admits inner and outer limits (resp. colimits) indexed by arbitrarysmall ∞ -bicategories, and these are computed by taking weighted homotopy limits(resp. colimits) in M with respect to a suitable weight. Finally, we also obtain an explicit description of inner and outer limits for di-agrams valued in C at ∞ . In particular, if I is an ∞ -bicategory and χ ∶ I → C at ∞ isa diagram, then by Theorem 1 we may encode χ by an inner cocartesian fibra-tion E inn → I . At the same time, post-composing χ co ∶ I co → C at co ∞ with the functor (−) op ∶ C at co ∞ → C at ∞ we obtain a diagram I co → C at ∞ , which can then be encodedby an outer cocartesian fibration E out → I . We then have the following explicitdescription of the inner and outer (or lax and oplax) limits of χ : Theorem 3 (See Example 5.4.12) . There are natural equivalences lim inn I χ ≃ Fun I ( I , E inn ) and lim out I χ ≃ Fun I ( I , E out ) op ANDREA GAGNA, YONATAN HARPAZ, AND EDOARDO LANARI identifying the inner limit of χ with the ∞ -category of sections of E inn → I , and theouter limit of χ with the opposite of the ∞ -category of sections of E out → I . This paper is organized as follows. In § § § ∞ -categories,and in § § P -fibered model structure, which is the one featuring in thestraightening and unstaightening equivalence, and develop a similar recognitionmechanism for outer fibrations in terms of a suitable extension property againstanodyne maps. Finally, in § § ( Z / ) -symmetry of the theoryof (∞ , ) -categories switches between all four types of fibrations. Unfortunately,the model of scaled simplicial sets does admit a convenient model for (−) co . Tocircumvent this problem we first show that the notions of inner/outer (co)cartesianfibrations can be defined also in the setting of marked simplicial categories , that is,categories enriched in marked simplicial sets. Establishing the equivalence betweenthe scaled and enriched definitions is the main goal of § (−) op and (−) co to reduce the main theorem to the inner cocartesian case.We dedicate § thickened slice construction, which enablesone to construct slice fibrations of all four variance flavors. These play a key role inthe definition of 2-(co)limits, and so we take the time to establish all the propertieswe will need later on. The construction makes use of a variant of the Gray tensorproduct in the setting of simplicial sets with both marking and scaling, which wedefine in § x in an ∞ -bicategory B . In § § f ∶ I → C , togetherwith the auxiliary data consisting of a collection of edges in I . This auxiliary dataroughly encodes along which edges in I the limit is to be “strong”, as apposed tolax. We then prove a characterization of 2-(co)limits in terms of the functors they(co)represent. In § weighted (co)limits. We then show that this particular case is in some sensegeneric: every 2-(co)limit can equivalently be expressed as a weighted (co)limitwith respect to a suitable weight. Finally, in § C arises froma model category M tensored over marked simplicial sets, weighted 2-(co)limits,and consequently all 2-(co)limits, can be computed in terms of weighted homotopy(co)limits in M . We then deduce the main result of the second part, showingthat small 2-(co)limits exist for a wide range of ambient ∞ -bicategories, includingfor example all those associated to combinatorial model categories tensored overmarked simplicial sets. IBRATIONS AND LAX LIMITS OF (∞ , ) -CATEGORIES 5 Acknowledgements
The first and third authors gratefully acknowledge the support of PraemiumAcademiae of M. Markl and RVO:67985840.1.
Preliminaries
In this section we establish notation and recall some preliminary definitionsand results concerning marked and scaled simplicial sets, and the straightening-unstraightening Quillen equivalence of [13].
Notation 1.0.1.
We will denote by ∆ the category of simplices, that is, the cat-egory whose objects are the finite non-empty ordinals [ n ] = { , , , . . . , n } andmorphisms are the non-decreasing maps. We will denote by S et ∆ the category ofsimplicial sets, that is the category of presheaves on sets of ∆, and will employthe standard notation ∆ n for the n -simplex, i . e ., the simplicial set representing theobject [ n ] of ∆. For any subset ∅ ≠ S ⊆ [ n ] we will write ∆ S ⊆ ∆ n to denote the (∣ S ∣ − ) -dimensional face of ∆ n whose set of vertices is S . For 0 ≤ i ≤ n we will de-note by Λ ni ⊆ ∆ n the i -th horn in ∆ n , that is, the subsimplicial set of ∆ n spanned byall the ( n − ) -dimensional faces containing the i -th vertex. For any simplicial set X and any integer p ≥
0, we will denote by deg p ( X ) the set of degenerate p -simplicesof X .By an ∞ -category we will always mean a quasi-category , i . e ., a simplicial set C which admits extensions for all inclusions Λ ni → ∆ n with 0 < i < n (also knownas inner horn inclusions ). Given an ∞ -category C , we will denote its homotopycategory by ho ( C ) . This is the ordinary category having as objects the 0-simplicesof C , and as morphisms x → y the set of equivalence classes of 1-simplices f ∶ x → y of C under the equivalence relation generated by identifying f and f ′ if there is a2-simplex H of C with H ∣ ∆ { , } = f, H ∣ ∆ { , } = f ′ and H ∣ ∆ { , } degenerate on x .1.1. Marked simplicial sets.Definition 1.1.1. A marked simplicial set is a pair ( X, E X ) where X is simplicialset and E X is a subset of the set of 1-simplices of X , called marked -simplices or marked edges , containing the degenerate ones. A map of marked simplicial sets f ∶ ( X, E X ) → ( Y, E Y ) is a map of simplicial sets f ∶ X → Y satisfying f ( E X ) ⊆ E Y .The category of marked simplicial sets will be denoted by S et + ∆ . It is locallypresentable and cartesian closed. Notation 1.1.2.
Let X be a simplicial set. We will denote by X ♭ = ( X, deg ( X )) the marked simplicial set whose marked edges are the degenerate 1-simplices andby X ♯ = ( X, X ) the marked simplicial set where all the edges of X are marked.The assignments X ↦ X ♭ and X ↦ X ♯ are left and right adjoint, respectively, to the forgetful functor S et + ∆ → S et ∆ .Marked simplicial sets can be used as a model for the theory of ( ∞ , ) -categories: Theorem 1.1.3 ([14]) . There exists a model category structure on the category S et + ∆ of marked simplicial sets in which the cofibrations are the monomorphismsand the fibrant objects are the marked simplicial sets ( X, E ) in which X is an ∞ -category and E is the set of equivalences of X , i . e ., -simplices f ∶ ∆ → X whichare invertible in ho ( X ) . ANDREA GAGNA, YONATAN HARPAZ, AND EDOARDO LANARI
The theorem above is a special case of Proposition 3.1.3.7 in [14], when S = ∆ .By [14, Proposition 3.1.5.3] the forgetful functor S et + ∆ → S et ∆ is a right Quillenequivalence, where S et ∆ is endowed with the categorical model structure of Joyal-Lurie. We will refer to the model structure of Theorem 1.1.3 as the marked categor-ical model structure , and its weak equivalences as marked categorical equivalences .We will denote by S et + ∆ - C at the category of categories in enriched in S et + ∆ withrespect to the cartesian product on S et + ∆ . For a S et + ∆ -enriched category C and twoobjects x, y ∈ C we will denote by C ( x, y ) ∈ S et + ∆ to associated mapping markedsimplicial set. By an arrow e ∶ x → y in an S et + ∆ -enriched category C we will simplymean a vertex e ∈ C ( x, y ) .We will generally consider S et + ∆ - C at together with its associated Dwyer-Kanmodel structure (see [14, § A.3.2]). In this model structure the weak equivalencesare the Dwyer-Kan equivalences, that is, the maps which are essentially surjec-tive on homotopy categories and induce marked categorical equivalences on map-ping objects. The fibrant objects are the enriched categories C whose mappingobjects C ( x, y ) are all fibrant, that is, are all ∞ -categories marked by their equiva-lences. The model category S et + ∆ - C at is then a presentation of the theory of ( ∞ , ) -categories, and is Quillen equivalent to other known models, see § Scaled simplicial sets and ∞ -bicategories.Definition 1.2.1 ([13]) . A scaled simplicial set is a pair ( X, T X ) where X issimplicial set and T X is a subset of the set of 2-simplices of X , called thin -simplices or thin triangles , containing the degenerate ones. A map of scaled simplicial sets f ∶ ( X, T X ) → ( Y, T Y ) is a map of simplicial sets f ∶ X → Y satisfying f ( T X ) ⊆ T Y .We will denote by S et sc∆ the category of scaled simplicial sets. It is locally pre-sentable and cartesian closed. Notation 1.2.2.
Let X be a simplicial set. We will denote by X ♭ = ( X, deg ( X )) the scaled simplicial set where the thin triangles of X are the degenerate 2-simplicesand by X ♯ = ( X, X ) the scaled simplicial set where all the triangles of X are thin.The assignments X ↦ X ♭ and X ↦ X ♯ are left and right adjoint, respectively, to the forgetful functor S et sc∆ → S et ∆ . Definition 1.2.3.
Given a scaled simplicial set X , we define its core to be thesimplicial set X th spanned by those n -simplices of X whose 2-dimensional facesare thin triangles. The assignment X ↦ X th is then right adjoint to the functor ( − ) ♯ ∶ S et ∆ → S et sc∆ . Warning . In [15, Tag 01XA], Lurie uses the term pith in place of core, anddenotes it by Pith ( C ) . Notation 1.2.5.
We will often speak only of the non-degenerate thin 2-simpliceswhen considering a scaled simplicial set. For example, if X is a simplicial set and T is any set of triangles in X then we will denote by ( X, T ) the scaled simplicial setwhose underlying simplicial set is X and whose thin triangles are T together withthe degenerate triangles. If L ⊆ K is a subsimplicial set then we use T ∣ L ∶ = T ∩ L to denote the set of triangles in L whose image in K is contained in T . Definition 1.2.6.
The set of generating scaled anodyne maps S is the set of mapsof scaled simplicial sets consisting of:(i) the inner horns inclusions ( Λ ni , { ∆ { i − ,i,i + } }) → ( ∆ n , { ∆ { i − ,i,i + } }) , n ≥ , < i < n ; IBRATIONS AND LAX LIMITS OF (∞ , ) -CATEGORIES 7 (ii) the map ( ∆ , T ) → ( ∆ , T ∪ { ∆ { , , } , ∆ { , , } }) , where we define T def = { ∆ { , , } , ∆ { , , } , ∆ { , , } , ∆ { , , } , ∆ { , , } } ;(iii) the set of maps ( Λ n ∐ ∆ { , } ∆ , { ∆ { , ,n } }) → ( ∆ n ∐ ∆ { , } ∆ , { ∆ { , ,n } }) , n ≥ . A general map of scaled simplicial set is said to be scaled anodyne if it belongs tothe weakly saturated closure of S . Definition 1.2.7. An ∞ -bicategory is a scaled simplicial set C which admits ex-tensions along all maps in S . Warning . The notion of ∞ -bicategory we have just introduced can be provento be equivalent to that of ( ∞ , ) -category given in [15, Tag 01W9]. Remark . If C is an ∞ -bicategory then its core C th is an ∞ -category.To avoid confusion we point out that simplicial sets as in Definition 1.2.7 arereferred to in [13] as weak ∞ -bicategories , while the term ∞ -bicatgory was reservedfor an a priori stronger notion. However, as we have shown in [6], these two notionsin fact coincide. In particular: Theorem 1.2.10 ([13],[6]) . There exists a model structure on S et sc∆ whose cofibra-tions are the monomorphisms and whose fibrant objects are the ∞ -bicategories (inthe sense of Definition 1.2.7). We will refer to the model structure of Theorem 1.2.10 as the bicategorical modelstructure . In [13] Lurie constructs a Quillen equivalence S et sc∆ C sc ' ' N sc g g ⊥ S et + ∆ - C at , in which the right functor N sc is also known as the scaled coherent nerve . A Quillenequivalence(1) S et sc∆ ι ' ' U g g ⊥ Strat , to Verity’s model structure on stratified sets for saturated 2-trivial complicial sets(see [19], [16]) was also established in [6]. We consider the bicategorical model struc-ture as a presentation of the theory of ( ∞ , ) -categories. In addition to the abovetwo comparisons, the bicategorical model structures has been compared in [13] toseveral other models for ( ∞ , ) -categories, which, to our knowledge, have been com-pared to all other known models (see, e.g., [3], [4], [1], [2]; we refer the reader to [6,Figure 1] of a diagrammatic depiction of all equivalences known to us). Definition 1.2.11.
We will denote by C at ∞ the scaled coherent nerve of the (fi-brant) S et + ∆ -enriched subcategory ( S et + ∆ ) ○ ⊆ S et + ∆ spanned by the fibrant markedsimplicial sets. We will refer to C at ∞ as the ∞ -bicategory of ∞ -categories. Definition 1.2.12.
Let C be an ∞ -bicategory. We will say that an edge in C is invertible if it is invertible when considered in the ∞ -category C th , that is, ifits image in the homotopy category of C th is an isomorphism. We will sometimesrefer to invertible edges in C as equivalences . We will denote by C ≃ ⊆ C th the ANDREA GAGNA, YONATAN HARPAZ, AND EDOARDO LANARI subsimplicial set spanned by the invertible edges. Then C ≃ is an ∞ -groupoid (thatis, a Kan complex), which we call the core groupoid of C . It can be consideredas the ∞ -groupoid obtained from C by discarding all non-invertible 1-cells and 2-cells. If X is an arbitrary scaled simplicial set then we will say that an edge in X is invertible if its image in C is invertible for any bicategorical equivalence X → C suchthat C is an ∞ -bicategory. This does not depend on the choice of the ∞ -bicategoryreplacement C . Notation 1.2.13.
Let C be an ∞ -bicategory and let x, y ∈ C be two vertices.In [13, § ∞ -category from x to y in C that we now recall. Let Hom C ( x, y ) be the marked simplicial set whose n -simplices are given by maps f ∶ ∆ n × ∆ → C such that f ∣ ∆ n ×{ } is constant on x , f ∣ ∆ n ×{ } is constant on y , and the triangle f ∣ ∆ {( i, ) , ( i, ) , ( j, )} is thin for every0 ≤ i ≤ j ≤ n . An edge f ∶ ∆ × ∆ → C of Hom C ( x, y ) is marked exactly when thetriangle f ∣ ∆ {( , ) , ( , ) , ( , )} is thin. The assumption that C is an ∞ -bicategory impliesthat the marked simplicial set Hom C ( x, y ) is fibrant in the marked categoricalmodel structure, that is, it is an ∞ -category whose marked edges are exactly theequivalences. Remark . By [13, Remark 4.2.1 and Theorem 4.2.2], if D is a fibrant S et + ∆ -en-riched category and C is an ∞ -bicategory equipped with a bicategorical equivalence ϕ ∶ C ≃ N sc ( D ) , then the mapsHom C ( x, y ) Ð→ Hom N sc ( D ) ( ϕ ( x ) , ϕ ( y )) ←Ð D ( ϕ ( x ) , ϕ ( y )) are marked categorical equivalences for every pair of vertices x, y of C . It thenfollows that a map ϕ ∶ C → C ′ of ∞ -bicategories is a bicategorical equivalence if andonly if it is essentially surjective (that is, every object in C ′ is equivalent to anobject in the image, see Definition 1.2.12) and the induced map Hom C ( x, y ) → Hom C ′ ( ϕ ( x ) , ϕ ( y )) is a marked categorical equivalence of (fibrant) marked simpli-cial sets for every x, y ∈ C . Remark . It follows from Remark 1.2.14 that if ϕ ∶ C → C ′ is a bicategoricalequivalence of ∞ -bicategories then the induced map ϕ th ∶ C th → ( C ′ ) th is an equiva-lence of ∞ -categories.It is shown in [13, Proposition 3.1.8 and Lemma 4.2.6] that the cartesian product × ∶ S et sc∆ × S et sc∆ Ð→ S et sc∆ is a left Quillen bifunctor with respect to the bicategorical model structure, i . e ., S et sc∆ is a cartesian closed model category. In particular, for every two scaled sim-plicial sets X, Y we have a mapping object Fun ( X, Y ) which satisfies (and is deter-mined by) the exponential formulaHom S et sc∆ ( Z, Fun ( X, Y )) ≅ Hom S et sc∆ ( Z × X, Y ) . In addition, when the codomain is an ∞ -bicategory C the mapping object Fun ( X, C ) is an ∞ -bicategory as well, which we can consider as the ∞ -bicategory of functorsfrom X to C . In this case we will denote by Fun th ( X, C ) ⊆ Fun ( X, C ) the associatedcore ∞ -category, which we consider as the ∞ -category of functors from X to C . Definition 1.2.16.
We define BiCat ∞ to be the scaled coherent nerve of the (large) S et + ∆ -enriched category BiCat ∆ whose objects are the ∞ -bicategories and whosemapping marked simplicial set, for C , D ∈ BiCat ∆ , is given by BiCat ∆ ( C , D ) ∶ = Fun th ( C , D ) ♮ . Here by ( − ) ♮ we mean that the associated marked simplicial set inwhich the marked arrows are the equivalences. We will refer to BiCat ∞ as the ∞ -bicategory of ∞ -bicategories. IBRATIONS AND LAX LIMITS OF (∞ , ) -CATEGORIES 9 Since the scaled coherent nerve functor N sc is a right Quillen equivalence itdetermines an equivalence(2) ( S et + ∆ - C at ) ∞ ≃ Ð→ BiCat th ∞ between the ∞ -category associated to the model category S et + ∆ - C at and the core ∞ -category of BiCat ∞ . One of the technical advantages of using S et + ∆ - C at as a modelis that the ( Z / ) -action on the theory of ( ∞ , ) -categories can be realized by anaction of ( Z / ) on S et + ∆ - C at via model category isomorphisms. More precisely,the operation C ↦ C op which inverts only the direction of 1-morphisms is realizedby setting C op ( x, y ) = C ( y, x ) , while the operation C ↦ C co of inverting only thedirection of 2-morphisms is realized by setting C co ( x, y ) = C ( x, y ) op , where the righthand side denotes the operation of taking opposites in marked simplicial sets. Wewill also denote by C coop ∶ = ( C co ) op = ( C op ) co the composition of these operations. Construction . The two commuting involutions ( − ) op and ( − ) co act on S et + ∆ - C at via equivalences of categories which preserve the Dwyer-Kan model structure. Throughthe equivalence (2) these two involutions induce a ( Z / ) -action on the core ∞ -category BiCat th ∞ , which we then denote by the same notation. In particular, wehave involutions ( − ) op ∶ BiCat th ∞ → BiCat th ∞ and ( − ) co ∶ BiCat th ∞ → BiCat th ∞ , the first inverting the direction of 1-morphisms and the second the direction of2-morphisms. Example . The op-action on the core ∞ -category C at th ∞ admits a point-setmodel via the functor ( − ) op ∶ S et + ∆ → S et + ∆ , which is an equivalence of categorieswhich preserves the marked categorical model structure. The functor ( − ) op ishowever not an enriched functor. Instead, it refines to an enriched functor ( − ) op ∶ S et + ∆ → ( S et + ∆ ) co , and hence induces an equivalence ( − ) op ∶ C at ∞ ≃ Ð→ C at co ∞ upon restricting the fibrant objects and taking scaled coherent nerves. This canalso be phrased by saying that the functor ( − ) op endows the ∞ -bicategory C at ∞ ∈ BiCat th ∞ with a fixed point structure under the action of ( − ) co ∶ BiCat th ∞ → BiCat th ∞ ,which is also sometimes called a twisted action . Example . Every Kan complex X admits a canonical zig-zag of equivalences X ≃ ←Ð Tw ( X ) ≃ Ð → X op , where Tw ( X ) denotes the twisted arrow category of X . Inparticular, the restriction of ( − ) op ∶ C at th ∞ → C at th ∞ to the full subcategory spannedby Kan complexes is homotopic to the identity. It then follows that the restrictionof the equivalence ( − ) co ∶ BiCat th ∞ → BiCat th ∞ to the full subcategory C at th ∞ ⊆ BiCat th ∞ (which can be modeled by the full subcategory of [ S et + ∆ - C at ] ○ spanned by the Kan-complex-enriched categories) is homotopic to the identity as well. Remark . The ( Z / ) -action on BiCat th ∞ does not extend to an action of Z / ∞ -bicategory BiCat ∞ . Instead, as in Example 1.2.18, it extends to a twisted action. More precisely, note that by construction, the enrichment of BiCat ∞ in C at th ∞ is the one induced by the closed action of C at th ∞ on BiCat th ∞ via the inclusion C at th ∞ ⊆ BiCat th ∞ and the cartesian product in BiCat ∞ (see [8, §
7] for the relationbetween C at ∞ -enrichment and closed actions of C at ∞ ). In particular, since the co-action on BiCat th ∞ fixes C at th ∞ ⊆ BiCat th ∞ object-wise (see Example 1.2.19) it followsthat it extends to an equivalence of ∞ -bicategories ( − ) co ∶ BiCat ∞ ≃ Ð→ BiCat ∞ . However, since ( − ) op restricts to the usual opposite operation on C at ∞ , its actionon BiCat ∞ is contravariant in 2-morphisms, and it hence extends to an equivalence ( − ) op ∶ BiCat ∞ ≃ Ð→ BiCat co ∞ , similarly to Example 1.2.18.1.3. Gray products of scaled simplicial sets.
In this section we recall from [5]the definition of the
Gray product of two scaled simplicial sets. In what follows,when we say that a 2-simplex σ ∶ ∆ → X degenerates along ∆ { i,i + } ⊆ ∆ (for i = ,
1) we mean that σ is degenerate and σ ∣ ∆ { i,i + } is degenerate. This includesthe possibility that σ factors through the surjective map ∆ → ∆ which collapses∆ { i,i + } as well as the possibility that σ factors through ∆ → ∆ . Definition 1.3.1.
Let ( X, T X ) , ( Y, T Y ) be two scaled simplicial sets. The Grayproduct ( X, T X ) ⊗ ( Y, T Y ) is the scaled simplicial set whose underlying simplicialset is the cartesian product of X × Y and such that a 2-simplex σ ∶ ∆ → X × Y isthin if and only if the following conditions hold:(1) σ belongs to T X × T Y ;(2) either the image of σ in X degenerates along ∆ { , } or the image of σ in Y degenerates along ∆ { , } . Remark . The Gray product of scaled simplicial sets is associative [5, Propo-sition 2.2], and in particular can be iterated in a unique manner. Specifically, forscaled simplicial sets X , ..., X n , the iterated Gray product X ⊗ ⋯ ⊗ X n is given bythe scaled simplicial set whose underlying simplicial set is the cartesian product of X , ..., X n and such that a triangle σ = ( σ , ..., σ n ) ∶ ∆ ♭ → X ⊗ ⋯ ⊗ X n is thin if andonly if the following conditions hold:(1) Each σ i is thin in X i .(2) There exists an j ∈ { , ..., n } such that σ i degenerates along ∆ { , } for i < j and σ i degenerates along ∆ { , } for i > j .The 0-simplex ∆ can be considered as a scaled simplicial set in a unique way, andserves as the unit of the Gray product. In particular ⊗ is a monoidal structure on S et sc∆ . This monoidal structure is however not symmetric. Instead, there is anatural isomorphism X ⊗ Y ≅ ( Y op ⊗ X op ) op . Example . Consider the Gray product X = ∆ ⊗ ∆ . Then X has exactly twonon-degenerate triangles σ , σ ∶ ∆ → X , where σ sends ∆ { , } to ∆ { } × ∆ and∆ { , } to ∆ × ∆ { } , and σ sends ∆ { , } to ∆ × ∆ { } and ∆ { , } to ∆ { } × ∆ . Bydefinition we see that σ is thin in X but σ is not. If C is an ∞ -bicategory then amap p ∶ X → C can be described as a diagram in C of the form x yz w f g h g f ≃ whose upper right triangle is thin (here f i = p ∣ ∆ ×{ i } and g i = p ∣{ i }× ∆ ). We thushave an invertible 2-cell h ≃ Ô⇒ g ○ f and a non-invertible 2-cell h ⇒ f ○ g .Such data is essentially equivalent to just specifying a single non-invertible 2-cell g ○ f ⇒ f ○ g . We may hence consider such a square as a oplax-commutative square, or a square which commutes up to a prescribed 2-cell.One of the main results of [5] is the following: IBRATIONS AND LAX LIMITS OF (∞ , ) -CATEGORIES 11 Proposition 1.3.4 ([5, Theorem 2.16]) . The Gray product ⊗∶ S et sc∆ × S et sc∆ → S et sc∆ is a left Quillen bifunctor with respect to the bicategorical model structure. Proposition 1.3.4 together with Remark 1.3.2 imply that S et sc∆ is a monoidalmodel category with respect to the Gray product. In particular, one may associatewith ⊗ a right and a left mapping objects, which we shall denote by Fun gr ( X, Y ) and Fun opgr ( X, Y ) respectively. More explicitly, an n -simplex of Fun gr ( X, Y ) isgiven by a map of scaled simplicial sets∆ n ♭ ⊗ X Ð→ Y. A 2-simplex ∆ ♭ ⊗ X → Y of Fun gr ( X, Y ) is thin if it factors through ∆ ♯ ⊗ X .Similarly, an n -simplex of Fun opgr ( X, Y ) is given by a map of scaled simplicial sets X ⊗ ∆ n ♭ Ð→ Y and the scaling is determined as above. The compatibility of the Gray product ofthe bicategorical model structure then implies that for a fixed X the functors Y ↦ Fun gr ( X, Y ) and Y ↦ Fun opgr ( X, Y ) are right Quillen functors. In particular, if C is an ∞ -bicategory then Fun gr ( X, C ) and Fun opgr ( X, C ) are ∞ -bicategories as well.The objects of Fun gr ( X, C ) correspond to functors X → C and by Example 1.3.3we may consider morphisms in Fun gr ( X, C ) as lax natural transformations . If wetake Fun opgr ( X, C ) instead then the objects are again functors X → C , but now theedges will correspond to oplax natural transformations .1.4. Scaled straightening and unstraightening.
In [13, §
3] Lurie established astraightening-unstraightening equivalence in the setting of ∞ -bicategories. In thissubsection we recall the setup of [13, §
3] and explain how to obtain from it anequivalence on the level of ∞ -bicategories. Definition 1.4.1.
Let ( S, T S ) be a scaled simplicial set. A marked simplicial set ( X, E X ) equipped with a map of simplicial sets f ∶ X → S is said to be P S -fibered if the following conditions hold:(i) The map f is an inner fibration.(ii) For every edge e ∶ ∆ → S the map e ∗ f ∶ X × S ∆ → ∆ is a cocartesian fibrationand the marked edges of X lying over e are exactly the e ∗ f -cocartesian edges.(iii) For every commutative diagram∆ { , } X ∆ S eσ with e ∈ E X and σ ∈ T S , the edge of X × S ∆ determined by e is σ ∗ f -cocartesian.Let ( S et + ∆ ) / S denote the category of marked simplicial sets ( X, E X ) equippedwith a map of simplicial sets f ∶ X → S . In [13, § ( S et + ∆ ) / S whose cofibrations are the monomorphisms and whose fibrantobjects are exactly the P S -fibered objects. Let us refer to this model structure asthe P S -fibered model structure . Given a weak equivalence of S et + ∆ -enriched cate-gories φ ∶ C ( S, T S ) → C he then proceeds to construct a straightening-unstraightening Quillen equivalence ( S et + ∆ ) / S St sc φ ( ( Un sc φ h h ⊥ ( S et + ∆ ) C where the right hand side denotes the category of S et + ∆ -enriched functors C → S et + ∆ equipped with the projective model structure, and the left hand side is equippedwith the P S -fibered model structure. The straightening functor St sc φ is given bythe explicit formula [ St sc φ ( X, E X )]( v ) = Cone φ ( X, E X )( ∗ , v ) , that is, by the restriction to C of the functor Cone φ ( X, E X ) → S et + ∆ represented bythe cone point ∗ in the scaled cone of X over C , which by is defined byCone φ ( X, E X ) ∶ = C sc ( ∆ ∐ ∆ { } × X ( ∆ × X, T )) ∐ C sc ( ∆ { } × X ♭ ) C . Here, the second pushout is along the composed map C sc ( X ♭ ) → C sc ( S, T S ) → C ,and T denotes the set of all those triangles ( τ, σ X ) ∶ ∆ → ∆ × X such that σ X isdegenerate and either τ ∣ ∆ { , } is degenerate in ∆ or σ X ∣ ∆ { , } belongs to E X . Inthe case where φ ∶ C sc ( S, T S ) → C sc ( S, T S ) is the identity we will generally replacethe subscript φ in St sc φ and Un sc φ and Cone φ by the subscript ( S, T S ) . Remark . When X has no non-degenerate marked edges the cone Cone φ ( X ♭ ) appearing in the straightening construction above can be described in terms of theGray product of § φ ( X ♭ ) = C sc ( ∆ ∐ ∆ { } ⊗ X ♭ [ ♭ ∆ ⊗ X ♭ ]) ∐ C sc ( ∆ { } ⊗ X ♭ ) C . This also holds more generally if one uses a Gray product which takes into accountmarked edges, see Remark 4.2.7.The straightening-unstraightening Quillen equivalence induces an equivalencebetween the ∞ -categories underlying the two sides of the adjunction St sc φ ⊣ Un sc φ .These two sides are both model categories which are tensored over S et + ∆ , that is,they admit a closed action of S et + ∆ in the form of a left Quillen bifunctor, andin particular both acquire en enrichment in S et + ∆ . In addition, the unstraighteningfunctor is lax-compatible with the action of S et + ∆ , in the sense that one has a naturalmap(3) Un sc φ ( F ) × K → Un sc φ ( F ) × Un sc ∗ ( K ) ≅ Un sc φ ( F × K ) , where Un sc ∗ denotes the scaled unstraightening functor with respect to the isomor-phism C ( ∆ ) ≅ ∗ . This structure promotes Un sc φ to a S et + ∆ -enriched functor from ( S et + ∆ ) C to ( S et + ∆ ) / S . Passing to the full subcategories of fibrant-cofibrant objects(and using the simplifying fact that all objects in ( S et + ∆ ) / S are cofibrant) we thenobtain an enriched functor of fibrant S et + ∆ -enriched categories [ Un sc φ ] ○ ∶ [( S et + ∆ ) C ] ○ → [( S et + ∆ ) / S ] ○ . Lemma 1.4.3.
The functor [ Un sc φ ] ○ is a Dwyer-Kan equivalence. Taking scaled nerves we now obtain a form of the unstraightening constructionas an equivalence of ∞ -bicategoriesN sc [( S et + ∆ ) C ] ○ ≃ Ð→ N sc [( S et + ∆ ) / S ] ○ . Proof of Lemma 1.4.3.
To begin, note that this functor is essentially surjectivesince Un sc φ is a right Quillen equivalence. To see that it is also homotopicallyfully-faithful we use the fact that the map (3) is a weak equivalence whenever F and K are fibrant by [13, Proposition 3.6.1] and [12, Corollary 1.4.4(b)]. Then, IBRATIONS AND LAX LIMITS OF (∞ , ) -CATEGORIES 13 for every fibrant F , G ∈ ( S et + ∆ ) C and a fibrant K ∈ S et + ∆ the functor F × K is againfibrant and the induced map of sets [ K, ( S et + ∆ ) C ( F , G )] S et + ∆ ≅ [ F × K, G ] ( S et + ∆ ) C ≅ Ð→ [ Un sc φ ( F × K ) , Un sc φ ( G )] ( S et + ∆ )/ S ≅ Ð→ [ Un sc φ ( F ) × K, Un sc φ ( G )] ( S et + ∆ )/ S [ K, ( S et + ∆ ) / S ( F , G )] S et + ∆ is a bijection, where [ − , − ] denotes sets of homotopy classes of maps with respect tothe relevant model structure. It then follows that Un sc φ induces a weak equivalenceof marked simplicial sets, that is, a marked categorical equivalence ( S et + ∆ ) C ( F , G ) ≃ Ð→ ( S et + ∆ ) / S ( F , G ) , for every F , G ∈ [( S et + ∆ ) C ] ○ . (cid:3) Marked-scaled simplicial sets.
In our treatment of fibrations of ∞ -bicategories,it will be useful to work in a setting where we have both a scaling and a marking. Definition 1.5.1. A marked-scaled simplicial set is a triple ( X, E X , T X ) where X is a simplicial set, E X is a collection of edges containing all the degenerateedges and T X is collection of 2-simplices containing all the degenerate 2-simplices.In particular, if ( X, E X , T X ) is a marked-scaled simplicial set then ( X, E X ) is amarked simplicial set and ( X, T X ) is a scaled simplicial set. A map of marked-scaled simplicial sets ( X, E X , T X ) → ( T, E Y , T Y ) is a map of simplcial sets X → Y such that f ( E X ) ⊆ E Y and f ( T X ) ⊆ T Y .We will denote by S et + , sc∆ the category of marked-scaled simplicial sets. It islocally presentable and cartesian closed. Definition 1.5.2.
For X ∈ S et + ∆ a marked simplicial set we will denote by X ♭ = ( X, E X , deg ( X )) the marked-scaled simplicial set which has the same markingas X and only the degenerate 2-simplices are thin, and by X ♯ = ( X, E X , X ) the marked-scaled simplicial set which has the same marking as X and all 2-simplices are thin. For Y is a scaled simplicial set then we will denote by Y ♭ = ( Y, deg ( Y ) , T Y ) the marked-scaled simplicial set which has the same scaling as Y and only the degenerate edges marked and by Y ♯ = ( Y, Y , T Y ) the marked-scaledsimplicial which has the same scaling as Y and all edges are marked. Finally, for Z a simplicial set we will denote by ♭ Z = ( Z, deg ( Z ) , deg ( Z )) and ♯ Z = ( Z, Z , Z ) the corresponding minimal and maximal marked-scaled simplicial sets as indicated.By abuse of notation we will denote ♯ ∆ ≅ ♭ ∆ simply by ∆ . Definition 1.5.3.
For a marked-scaled simplicial set X we will denote by X theunderlying scaled simplicial set. Definition 1.5.4.
By a marked ∞ -bicategory we will simply mean a marked-scaledsimplicial set whose underlying scaled simplicial set is a weak ∞ -bicategory.2. Inner and outer cartesian fibrations
In this section we will study four types of fibrations between ∞ -categories, whichwe call inner cocartesian, inner cartesian, outer cocartesian and outer cartesian fi-brations. The notion of an inner cocartesian fibration E → B is essentially equivalentto that of a P B -fibered object, as described in § § E → B can be obtainedas the unstraightening of an objectwise fibrant functor C sc ( B ) → S et + ∆ , which we can also encode as a map χ ∶ B → C at ∞ . Using the compatibility of straightening-unstraightening with base change we may informally describe χ by the formula b ↦ E b ∶ = E × B { b } .Dually, a map f ∶ E → B is an inner cartesian fibration if f op ∶ E op → B op is aninner cocartesian fibration. Then f op encodes the data of a diagram B op → C at ∞ ,given informally by the formula b ↦ E op b . Noting (see Example 1.2.18) that thefunctor ( − ) op yields an equivalence of ( ∞ , ) -categories ( − ) op ∶ C at ∞ → C at co ∞ , wemay consider the association b ↦ E b as a functor B coop → C at ∞ .In the paper [6] we have introduced two more notions of fibrations, which wecall outer cartesian and cocartesian fibrations, respectively. Our primary goal, towhich we will arrive in §
3, is to show that the data of an outer cartesian fibration f ∶ E → B encodes a functor B op ↦ C at , while that of an outer cocartesian fibrationencodes a functor B co ↦ C at . In particular, the four types of fibrations mentionedabove correspond exactly to the four variance types a C at -valued diagram can have.For this, we will dedicate the present section to studying the properties of thesefour types of fibrations, establishing the key results about them we will need lateron. In particular, in § ∞ -categories, and show that these are always left and rightfibrations. In § § P B -fibered objects. Finally, in § Recollections.
In this section we recall from [6] the main definitions we willneed, and recall some of their properties which were already established in loc . cit . Definition 2.1.1.
We will say that a map of scaled simplicial sets X → Y is a weakfibration if it has the right lifting property with respect to the following types ofmaps:(1) All scaled inner horn inclusions of the form ( Λ ni , { ∆ { i,i − ,i } } ∣ Λ ni ) ⊆ ( ∆ n , { ∆ { i,i − ,i } }) for n ≥ < i < n .(2) The scaled horn inclusions of the form: ( Λ n ∐ ∆ { , } ∆ , { ∆ { , ,n } } ∣ Λ n ) ⊆ ( ∆ n ∐ ∆ { , } ∆ , { ∆ { , ,n } }) for n ≥ ( Λ nn ∐ ∆ { n − ,n } ∆ , { ∆ { ,n − ,n } } ∣ Λ nn ) ⊆ ( ∆ n ∐ ∆ { n − ,n } ∆ , { ∆ { ,n − ,n } }) for n ≥ Remark . The maps of type (1)-(3) in Definition 2.1.1 are trivial cofibrationswith respect to the bicategorical model structure: indeed, the first two are scaledanodyne and the third is the opposite of a scaled anodyne map. It follows thatevery bicategorical fibration is a weak fibration.
Remark . Let f ∶ X → Y be a weak fibration and suppose in addition that f detects thin triangles, that is, a triangle in X is thin if and only if its image in Y is thin. Then f has the right lifting property with respect to the generating scaledanodyne maps of Definition 1.2.6. In particular, if Y is an ∞ -bicategory then X IBRATIONS AND LAX LIMITS OF (∞ , ) -CATEGORIES 15 is an ∞ -bicategory. In addition, in this case for every y ∈ Y the fiber X y is an ∞ -bicategory in which every triangle is thin, and can hence be considered as an ∞ -category. Definition 2.1.4.
Let f ∶ X → Y be a weak fibration. We will say that an edge e ∶ ∆ → X is f -cartesian if the dotted lift exists in any diagram of the form ( Λ nn , { ∆ { ,n − ,n } } ∣ Λ nn ) σ / / (cid:15) (cid:15) ( X, T X ) f (cid:15) (cid:15) ( ∆ n , { ∆ { ,n − ,n } }) ♠♠♠♠♠♠ / / ( Y, T Y ) with n ≥ σ ∣ ∆ n − ,n = e . We will say that e is f -cocartesian if e op ∶ ∆ → X op is f op -cartesian. Definition 2.1.5.
Let f ∶ X → Y be a weak fibration. We will say that f is(1) an inner fibration if it detects thin triangles and the underlying map of sim-plicial sets is in inner fibration, that is, satisfies the right lifting property withrespect to inner horn inclusions;(2) an outer fibration if it detects thin triangles and the underlying map of simplicialsets satisfies the right lifting property with respect to the inclusionsΛ n ∐ ∆ { , } ∆ ⊆ ∆ n ∐ ∆ { , } ∆ and Λ nn ∐ ∆ { n − ,n } ∆ ⊆ ∆ n ∐ ∆ { n − ,n } ∆ for n ≥ Warning . In [15, Tag 01WF], Lurie uses the term interior fibration to encodewhat we just defined as outer fibrations. Our choice already appeared in Definition2.4 of [6], and it is motivated by the intent of highlighting that special outer horns admit fillers against such maps.
Definition 2.1.7.
Let f ∶ X → Y be a map of scaled simplicial sets. We will saythat f is an outer (resp. inner ) cartesian fibration if the following conditions hold:(1) The map f is an outer (resp. inner) fibration.(2) For every x ∈ X and an edge e ∶ y → f ( x ) in Y there exists a f -cartesian edge ̃ e ∶ ̃ y → x such that f (̃ e ) = e .Dually, we will say that f ∶ X → Y is an outer cocartesian fibration if f op ∶ X op → Y op is an outer cartesian fibration. Remark . The classes of weak fibrations, inner/outer fibrations and inner/outer(co)cartesian fibrations are all closed under base change.It follows from Remark 2.1.3 that if f ∶ X → Y is an inner/outer (co)cartesianfibration and X is an ∞ -bicategory then Y is an ∞ -bicategory as well. In this casewe will say that f is an inner/outer (co)cartesian fibration of ∞ -bicategories. Remark . Let f ∶ E → B be an inner/outer (co)cartesian fibration of ∞ -bicat-egories. Then the base change f ∣ B th ∶ E × B B th → B th (see Definition 1.2.3) is a(co)cartesian fibration of ∞ -categories. In particular, f ∣ B th is a categorical fibration(see [14, Proposition 3.3.1.7]) and so an isofibration. We may hence conclude that f is an isofibration of ∞ -bicategories. Remark . In the setting of Remark 2.1.9, if e ∶ x → y is a f -(co)cartesian edgeof E , then it also (co)cartesian with respect to the (co)cartesian fibration of ∞ -categories E th → B th . This implies, in particular, that any f -(co)cartesian edgewhich lies over an equivalence in B is necessarily an equivalence in E . Remark . An inner/outer (co)cartesian fibration f ∶ E → B of ∞ -bicategoriesis a fibration of scaled simplicial sets. This follows the characterization of thebicategorical model structure established in [6] since f lifts against scaled anodynemaps by virtue of being a weak fibration and is an isofibration by Remark 2.1.9. Remark . It follows from Remarks 2.1.11 and 2.1.8 that if X → Y is aninner/outer fibration then for every y ∈ Y the fiber X y is an ∞ -bicategory in whichevery triangle is thin. Forgetting the scaling, we may simply consider these fibersas ∞ -categories.An important source of examples of outer (co)cartesian fibrations comes from slice fibrations . Let us recall from [6] the relevant definitions. Definition 2.1.13.
Let ( X, E X , T X ) and ( Y, E Y , T Y ) be two marked-scaled sim-plicial sets. We define their join ( X, E X , T X ) ∗ ( Y, E Y , T Y ) = ( X ∗ Y, T X ∗ Y ) tobe the scaled simplicial set whose underlying simplicial set is the ordinary join ofsimplicial sets X ∗ Y , and whose thin triangles are given by T X ∗ Y = T X ∐ ( E X × Y ) ∐ ( X × E Y ) ∐ T Y seen as a subset of ( X ∗ Y ) = X ∐ ( X × Y ) ∐ ( X × Y ) ∐ Y . To avoid confusion, we point out that while the input to the join bifunctor aboveare two marked-scaled simplicial sets, its output is only considered as a scaledsimplicial set.
Construction . Recall that for a marked-scaled simplicial set K we denote by K the underlying scaled simplicial set (see Definition 1.5.3). For a fixed marked-scaled simplicial set K we may regard the association X ↦ X ∗ K as a functor S et + , sc∆ → ( S et sc∆ ) K / . As such, it becomes a colimit preserving functor with a rightadjoint ( S et sc∆ ) K / → S et + , sc∆ by the adjoint functor theorem. Given a scaled simplicial set S and a map f ∶ K → S ,considered as an object of ( S et sc∆ ) K / , we will denote by S / f the marked-scaled sim-plicial set obtained by applying the above mentioned right adjoint, and by S / f theunderlying scaled simplicial set of S / f . In particular, the marked-scaled simplicialset S / f is characterized by the following mapping propertyHom S et + , sc∆ ( X, S / f ) = Hom ( S et sc∆ ) K / ( X ∗ K, S ) , while S / f is characterized by the propertyHom S et sc∆ ( X, S / f ) = Hom ( S et sc∆ ) K / ( X ♭ ∗ K, S ) . Example . If f ∶ ∅ → S is the unique map from the empty scaled simplicialset then S / f = S ♯ = ( S, S , T S ) is the marked-scaled simplicial set having the sameunderlying scaled simplicial set as S and with all edges marked. In this case wewill also use the notation S /∅ . In particular, S /∅ ≅ S canonically. Example . If K = ∆ and f ∶ ∆ → S corresponds to a vertex x ∈ S then we willdenote S / f also by S / x . This can be considered as a version of the slice constructionin the setting of scaled simplicial set. For example, if C is an ∞ -bicategory and x, y ∈ C are two objects, then the fiber of C / y over x ∈ C , which is a marked-scaledsimplicial set in which all triangles are thin, is categorically equivalent as a markedsimplicial set to the mapping ∞ -category Hom C ( x, y ) of Notation 1.2.13, see [6,Proposition 2.24]. We may consider this as a “thinner” model for the mapping ∞ -category, and will make use of it in § IBRATIONS AND LAX LIMITS OF (∞ , ) -CATEGORIES 17 Warning . The notation S / f is somewhat abusive - the marked-scaled sim-plicial set S / f depends not only on the scaled map f ∶ K → S , but also on the givenmarking on K . For example, suppose that K is ∆ with some marking E ⊆ ( ∆ ) and f corresponds to a given edge e ∶ x → y in S . If E consists only of degenerateedges then the vertices of S / f are given by arbitrary triangles of the form z (cid:127) (cid:127) ⑧⑧⑧⑧⑧⑧⑧⑧ (cid:31) (cid:31) ❄❄❄❄❄❄❄❄ x e / / y in S , while if E contains the non-degenerate edge of ∆ then the vertices of S / f correspond only to those triangles as above which are thin .We now recall from [6] the following results concerning the above slice construc-tions: Proposition 2.1.18 ([6, Corollary 2.18]) . Let K be a marked scaled simplicial sets, C an ∞ -bicategory and f ∶ K → C a scaled map. Then the map of scaled simplicialsets p ∶ C / f → C /∅ = C is an outer cartesian fibration such that every marked edge in C / f is p -cartesianand every edge in C admits a marked p -cartesian lift. More generally:
Proposition 2.1.19 ([6, Corollary 2.20]) . Let ι ∶ L ⊆ K be an inclusion of marked-scaled simplicial sets, q ∶ X → S is a weak fibration and f ∶ K → X is a scaled map.Then the map of scaled simplicial sets p ∶ X / f → X / fι × S / qfι × S / qf is an outer fibration such that every marked edge in its domain is p -cartesian andevery marked edge in its codomain admits a marked p -cartesian lift.Remark . In contrast to the case of Proposition 2.1.18, in the situation ofProposition 2.1.19 the map p is an outer fibration which is generally not cartesian:only some of the maps in its codomain admit cartesian lifts.The following result is a reformulation in the present language of [7, Lemma1.2.8], which is stated in loc . cit . without proof. Proposition 2.1.21.
Let C be an ∞ -category and f ∶ X → C ♯ be a weak fibration ofscaled simplicial sets. Then the following are equivalent:(1) f is an inner cartesian fibration.(2) f is an outer cartesian fibration.(3) All triangles in X are thin and the map of simplicial sets underlying f is acartesian fibration.Proof. We first note that both (1) and (2) imply by definition that f detects thintriangles, and since all triangles in C ♯ is thin this is the equivalent to saying thatevery triangle in X is thin. Since this is also stated explicitly in (3), we may simplyassume that all triangles in X are thin. In this case, f is both an inner and an outerfibration as soon as it is a weak fibration, and so (1) and (2) are both equivalent tosaying f is a weak fibration and edges in C admits a sufficient supply of f -cartesianlifts. Using again that all triangles in X and C ♯ are thin we see that this is the same as saying that the map of simplicial sets underlying f is a cartesian fibrationwhich satisfies the right lifting property with respect to the inclusionsΛ n ∐ ∆ { , } ∆ ⊆ ∆ n ∐ ∆ { , } ∆ and Λ nn ∐ ∆ { n − ,n } ∆ ⊆ ∆ n ∐ ∆ { n − ,n } ∆ for n ≥
2. But this holds for any cartesian fibration of ∞ -categories: indeed, anycartesian fibration is also a categorical fibration and the above inclusions are trivialcofibrations in the categorical model structure. (cid:3) Local properties of inner and outer fibrations.
Let C be an ∞ -bicategoryand let x, y be two vertices of C . Recall the explicit model for the mapping ∞ -category Hom C ( x, y ) from x to y discussed in Notation 1.2.13. In [6, § ▷ C ( x, y ) , defined as the underlying marked simplicialset of ( C / y ) x . We have shown the existence of a canonical map (see [6, Construc-tion 2.22]) i ∶ Hom ▷ C ( x, y ) → Hom C ( x, y ) , which is an equivalence of fibrant marked simplicial sets (see [6, Proposition 2.24]).In what follows we will use the term marked left (resp. right ) fibration to denotea map of marked simplicial sets f ∶ X → Y which detects marked edges and whichis a left (resp. right) fibration on the level of underlying simplicial sets. Proposition 2.2.1.
Let f ∶ E → B be an outer (resp. inner) fibration of ∞ -bicategoriesand let x, y be two vertices of E . Then the map of marked simplicial sets f ∗ ∶ Hom ▷ E ( x, y ) → Hom ▷ B ( f ( x ) , f ( y )) is a marked left (resp. right) fibration. Furthermore, if e ∶ x ′ → y is a f -cartesianedge with f ( x ′ ) = f ( x ) then the post-composition with e induces an equivalencebetween the the mapping space Hom E f ( x ) ( x, x ′ ) and the fiber of f ∗ over f ( e ) , where E f ( x ) denotes the fiber of f over f ( x ) .Proof. Assume first that f is an outer cartesian fibration. We need to show thatthe map(4) ( E / y ) x → ( B / f ( y ) ) f ( x ) is a marked left fibration, where abusing of notation we consider both ( E / y ) x and ( B / f ( y ) ) f ( x ) as marked simplicial sets, thereby ignoring their scaling. We firstobserve that since the map E → B has the right lifting property with respect to∆ ⊆ ∆ ♯ , it follows that E / y → B / f ( y ) has the right lifting property with respect to∆ ⊆ ( ∆ ) ♯ . In particular, an arrow in ( E / y ) x is marked if and only if its image in ( B / f ( y ) ) f ( x ) is marked. It will hence suffice to show that the map of simplicial setsunderlying (4) is a left fibration. Unwinding the definitions, we see that in order toshow that (4) has the right lifting property with respect to Λ ni ↪ ∆ n for 0 ≤ i < n we need to prove the existence of a dotted lift in diagrams of the form(5) ∆ n ∗ ∅ ∐ Λ ni ∗∅ Λ ni ∗ ∆ ∆ ∗ ∅ ∐ Λ ni ∗∅ Λ ni ∗ ∆ E ∆ n ∗ ∆ ∆ ∗ ∅ ∐ ∆ n ∗∅ ∆ n ∗ ∆ B f fg where f maps ∆ ∗ ∅ to x , g maps ∆ ∗ ∅ to f ( x ) . Now since the left squareis a pushout square it will suffice to find a lift in the external rectangle of (5).When 0 < i < n the left vertical map is isomorphic to the inner horn inclusionΛ [ n ]∗[ ] i ⊆ ∆ [ n ]∗[ ] and the triangle ∆ { i − ,i,i + } is mapped to a degenerate (andhence thin) triangle of B . On the other hand, when i = IBRATIONS AND LAX LIMITS OF (∞ , ) -CATEGORIES 19 is isomorphic to the 0-horn inclusion Λ [ n ]∗[ ] ⊆ ∆ [ n ]∗[ ] and the edge ∆ { , } ismapped to a degenerate edge of E . In both cases the desired lift exists by virtue ofthe assumption that f ∶ E → B is an outer fibration.For the second part of the claim, we need to show that if e ∶ x ′ → y is f -cartesianedge with f ( x ′ ) = f ( x ) then post-composition with e induces an equivalence be-tween the the mapping space Hom E f ( x ) ( x, x ′ ) and the fiber of f ∗ over f . We notethat this statement is local, i . e ., proving the claim for a given e ∶ x ′ → y only requiresus to consider maps in E lying over either f ( e ) or id f ( x ) . Thus, it is enough to provethe claim for the outer cartesian fibration E × B ∆ → ∆ obtained by pulling back f ∶ E → B along the map ∆ → B corresponding to f ( e ) . Since every triangle in∆ is degenerate this pullback is a cartesian fibration of ∞ -categories. By possiblyreplacing E × B ∆ with an equivalent ∞ -category, we may assume that E × B ∆ is given by the nerve of a map of fibrant simplicial categories C → [ ] , and anapplication of [14, Proposition 2.4.1.10 (2)] then finishes the proof.Finally, the proof in the case where f is an inner cartesian fibration proceedsverbatim, except that in the first part we do not need to consider the case i = i = n . In the latter case the left vertical map in (5)is isomorphic to the inner horn inclusion Λ [ n ]∗[ ] n ↪ ∆ [ n ]∗[ ] , and so the lift existsby the assumption that f is an inner fibration. (cid:3) Corollary 2.2.2.
Let
E E ′ B B ′ rp qf be a diagram of ∞ -bicategories such that p and q are both outer (or both inner)cartesian fibrations and r maps p -cartesian edges to q -cartesian edges. Assumethat f is a bicategorical equivalence. Then the following statements are equivalent:(1) For every x ∈ B the map r x ∶ E x → E ′ p ( x ) is an equivalence of ∞ -categories.(2) The map r ∶ E → E ′ is an equivalence of ∞ -bicategories.Proof. Assume (1) holds. It is clear that r ∶ E → E ′ is essentially surjective. Let usprove that the map r is also fully-faithful. For any pair of vertices x and y of E , wewish to show that the map r ∗ ∶ Hom E ( x, y ) → Hom E ′ ( rx, ry ) of marked ∞ -categoriesis an equivalence. Proposition 2.2.1 tells us that we have a commutative triangleHom ▷ E ( x, y ) Hom ▷ E ′ ( rx, ry ) Hom ▷ B ′ ( px, py ) Hom ▷ B ′ ( f px, f py ) r ∗ p ∗ q ∗ f ∗ of (marked) left fibrations (or right fibrations in the inner case), in which the bottomhorizontal map is an equivalence of ∞ -categories by Remark 1.2.14. It hence sufficesto prove that r ∗ is an equivalence fiber-wise, that is, for any e ∈ Hom ▷ B ( px, py ) , theinduced map on the fibers ( p ∗ ) − ( e ) → ( q ∗ ) − ( f ( e )) is an equivalence. Choose a p -cartesian lift e ′ ∶ x ′ → y of e . Notice that by assumption the edge r ( e ′ ) ∶ r ( x ′ ) → r ( y ) of E ′ is q -cartesian. Using again Proposition 2.2.1, we get a commutative square ( p ∗ ) − ( e ) ( q ∗ ) − ( f ( e )) Hom E p ( x ) ( x, x ′ ) Hom E q ( rx ) ( rx, rx ′ ) ≃ ≃ where the two vertical arrows are equivalences. Statement (1) now tells us thatthe bottom horizontal map is also an equivalence, and so the desired result followsfrom 2-out-of-3.Let us now assume Statement (2). We then obtain a commutative diagram of ∞ -categories(6) E th ( E ′ ) th B th ( B ′ ) th r th ≃ p qf ≃ in which the horizontal maps are categorical equivalences (see Remark 1.2.15) andthe vertical maps are cartesian fibrations (see Remark 2.1.9). Now consider theextended diagram(7) E th B × ( B ′ ) th ( E ′ ) th ( E ′ ) th B th B th ( B ′ ) th f q ≃ f in which the right square is a pullback square and the composition of the twotop horizontal maps is the equivalence r th . By [14, Corollary 3.3.1.4] the rightsquare in (7) is a homotopy pullback square. But the external rectangle is alsohomotopy cartesian (since it contains a parallel pair of equivalences) and so themap E th → B × ( B ′ ) th ( E ′ ) th is a categorical equivalence. By [14, Proposition 3.3.1.5]we now get that the induced map E x → E ′ f ( x ) is a categorical equivalence for everyvertex x of B . (cid:3) Cartesian edges.
The notion of a (co)cartesian edge admits equivalent de-scriptions in various contexts. To fully exploit this it will be convenient to introducethe following two variants:
Definition 2.3.1.
Let f ∶ X → S be a weak fibration of scaled simplicial sets. Wewill say that an edge e ∶ x → y in X is weakly f -cartesian if the dotted lift exists inany square of the form ( Λ nn , T ∣ Λ nn ) X ( ∆ n , T ) S σ f with n ≥ σ ∣ ∆ n − ,n = e , where T is the collection of all triangles in ∆ n whichare either degenerate or contain the edge ∆ { n − ,n } . We will say that e ∶ x → y is strongly f -cartesian if the dotted lift exists in any square of the form ( Λ n ) ♭ X ∆ n ♭ S σ f with n ≥ σ ∣ ∆ n − ,n = e . Similarly, we define the notions of weak and strongcocartesian edges by using the opposite scaled simplicial sets. Example . Let B be an ∞ -bicategory and f ∶ E → B be a weak fibration. Thenevery equivalence in E is both weakly f -cartesian and weakly f -cocartesian. Thisfollows by applying [6, Lemma 5.2] to f and f op . IBRATIONS AND LAX LIMITS OF (∞ , ) -CATEGORIES 21 Proposition 2.3.3.
Given a weak fibration of ∞ -bicategories f ∶ E → B and an edge e ∶ y → z in E , the following statements are equivalent: ● e is weakly f -cartesian. ● The following square is a homotopy pullback in the marked categorical modelstructure: (8) Hom ▷ E ( x, y ) Hom ▷ E ( x, z ) Hom ▷ B ( f x, f y ) Hom ▷ B ( f x, f z ) f x,y e ○− f x,z f ( e )○− For the proof of Proposition 2.3.3 we introduce the following piece of notation:
Notation 2.3.4.
Given a scaled simplicial set X and an edge e ∶ x → y in X , wewill denote by X / e ♯ ∈ S et + , sc∆ the result of the slice Construction 2.1.14 applied tothe marked-scaled simplicial set ( ∆ ) ♯ and the map ∆ → X determined by e .Explicitly, the set of n -simplices in X / e ♯ is given by: ( X / e ♯ ) n def = { α ∶ ♭ ∆ n ∗ ♯ ∆ → X ∣ α ∣ ∆ { n + ,n + } = e } , the marked edges are those which factor through ♯ ∆ ∗ ♯ ∆ and the thin trianglesare those which factor through ( ∆ ) ♭♯ ∗ ♯ ∆ . We will write X / e ♯ for the underlyingscaled simplicial set of X / e ♯ Remark . We note that for an ∞ -bicategory E , it follows from Corollary 2.1.18that the projection E / e ♯ → E is an outer cartesian fibration such that every markededge in E / e ♯ is cartesian. On the other hand, from the dual version of [6, Lemma2.17] (see also Lemma 2.4.6 below) E / e ♯ → E / x is a trivial fibration. Lemma 2.3.6.
Let f ∶ X → S be a weak fibration of scaled simplicial sets. An edge e ∶ x → y in X is weakly f -cartesian if and only if the map (9) X / e ♯ → X / y × S / fy S / fe ♯ is a trivial fibration of scaled simplicial sets.Proof. For any n ≥
0, we have the following correspondence of lifting problems: ♭ ∂ ∆ n ∗ ♯ ∆ ∪ ♭ ∆ n ∗ { } X ♭ ∆ n ∗ ♯ ∆ S ↭ ( ∂ ∆ n ) ♭ X / e ♯ ∆ n ♭ X / y × S / py S / pe ♯ . It is clear from the definition that the scaled simplicial set ♭ ∆ n ∗ ♯ ∆ is isomorphicto ( ∆ n + , T ) , where T is the scaling of Definition 2.1.1, for any n ≥
0. Identifying∆ n ∗ ∆ with ∆ n + , one immediately checks that ∂ ∆ n ∗ ∆ is the subsimplicial setgiven by the union of Λ n + n + and Λ n + n + and that ∆ n ∗ { } is the face ∆ { , ,...,n,n + } .Hence, the left vertical map in the left square of the previous correspondence isisomorphic to the map ( Λ n + n + , T ∣ Λ n + n + ) ↪ ( ∆ n + , T ) appearing in Definition 2.1.1,for any n ≥
0. Finally, the lifting against the map ∆ ♭ ↪ ∆ ♯ holds without anyassumption on e since thin triangles on both sides of (9) are detected by f . (cid:3) Proof of Proposition 2.3.3.
By Lemma 2.3.6 we have that an edge e ∶ y → z is weakly f -cartesian if and only if the map E / e ♯ → E / z × B / fz B / fe ♯ is a trivial fibration. This can be viewed as a morphism between outer cartesianfibrations over E , thanks to Proposition 2.1.18. Therefore, according to Corol-lary 2.2.2 it is an equivalence if and only if it induces an equivalence between therespective fibers E / e ♯ × E { x } → ( E / z × B / fz B / fe ♯ ) × E { x } , for any vertex x of E . At the same time, the square in (8) can be modeled by thefollowing commutative square of marked-scaled simplicial sets, after ignoring the(maximal) scaling everywhere: E / e ♯ × E { x } E / z × E { x } B / fe ♯ × B { f x } B / fz × B { f x } . We note that the underlying marked simplicial sets in the above square are allfibrant (see [6, Remark 2.23]). To finish the proof it will hence suffice to showthat the map of marked simplicial sets underlying the bottom horizontal map isa fibration in the marked categorical model structure. Indeed, it follows fromProposition 2.1.19 that the map in question satisfies the right lifting property withrespect to all cartesian anodyne maps (in the sense of [14, Definition 3.1.1.1]).Since its domain and target are fibrant this map is a marked categorical fibrationby [10, Lemma 4.40]. (cid:3)
Clearly any cartesian edge is weakly cartesian, and any strongly cartesian edgeis cartesian. The following proposition offers a partial inverse to this statement:
Proposition 2.3.7.
Let f ∶ E → B be a weak fibration of ∞ -bicategories. Then anedge of E is f -cartesian if and only if it is weakly f -cartesian. If f is an outerfibration then an edge in E is f -cartesian if and only if it is strongly f -cartesian.In particular, in the latter case all three classes coincide. The proof of Proposition 2.3.7 will rely on the following lemmas, which will alsogive us useful 2-out-of-3 type properties for cartesian edges. To formulate it, let B be an ∞ -bicategory and let f ∶ E → B be a weak fibration, that we will assume tobe an outer fibration when considering statements about strong f -cartesian edges.Let σ ∶ ∆ → E be a thin triangle, depicted as a commutative diagram(10) y x y te e . Lemma 2.3.8. If e is f -cartesian (resp. strongly f -cartesian) and t is weakly f -cartesian then e is f -cartesian (resp. strongly f -cartesian). Lemma 2.3.9. If e is weakly f -cartesian and t is f -cartesian (resp. strongly f -cartesian) then e is f -cartesian (resp. strongly f -cartesian).Proof of Lemma 2.3.8. We will prove the f -cartesian case; the proof for strongly f -cartesian edges is similar (and easier). We need to show that the dotted lift existin any square of the form(11) ( Λ nn , { ∆ { ,n − ,n } } ∣ Λ nn ) E ( ∆ n , { ∆ { ,n − ,n } }) B σ fτ IBRATIONS AND LAX LIMITS OF (∞ , ) -CATEGORIES 23 with n ≥ σ ∣ ∆ { n − ,n } = e . Let us view Λ nn and ∆ n as the subsimplicial sets ∂ ∆ { ,...,n − } ∗ ∆ { n } and ∆ { ,...,n − } ∗ ∆ { n } of ∆ n + = ∆ { ,...,n − } ∗ ∆ { n,n + } , andconsider in addition the subsimplicial set Z ∶ = ∂ ∆ { ,...,n − } ∗ ∆ { n,n + } ⊆ ∆ n + . Let T ⊆ T ′ ⊆ ∆ n + be the following two sets of triangles: T consists of the triangle∆ { ,n − ,n } as well as all the triangles which contain the edge ∆ { n,n + } , while T ′ = T ∪ { ∆ { ,n − ,n + } } . The maps σ and the triangle (10) now fit to form a map ofscaled simplicial sets(12) ( Λ nn , T ∣ Λ nn ) ∐ ∆ { n − ,n }♭ ∆ { n − ,n,n + }♯ → E . Applying (the dual version of) [6, Lemma 2.17] to the inclusions ∆ { n − } ⊆ ∂ ∆ { ,...,n − }♭ and ∆ { n } ⊆ ♯ ∆ { n,n + } we deduce that the map ( Λ nn , { ∆ { ,n − ,n } } ∣ Λ nn ) ∐ ∆ { n − ,n }♭ ∆ { n − ,n,n + }♯ ↪ ( Z, T ∣ Z ) is scaled anodyne. We may hence extend the map (12) to a map ρ ∶ ( Z, T ∣ Z ) → E .The maps τ and f ρ now combine to form a map η ∶ ( ∆ n , { ∆ { ,n − ,n } }) ∐ ( Λ nn , { ∆ { ,n − ,n } } ∣ Λ nn ) ( Z, T ∣ Z ) → B . Applying again the dual of [6, Lemma 2.17], this time to the inclusions ∂ ∆ { ,...,n − }♭ ⊆ ∆ { ,...,n − }♭ and ∆ { n } ⊆ ♯ ∆ { n,n + } we deduce that the map ( ∆ n , { ∆ { ,n − ,n } }) ∐ ( Λ nn , { ∆ { ,n − ,n } } ∣ Λ nn ) ( Z, T ∣ Z ) → ( ∆ n + , T ) is scaled anodyne. We may hence extend the square (11) to a square(13) ( Z, T ∣ Z ) E ( ∆ n + , T ) B . ρ fτ We now observe that the 3-simplex ∆ { ,n − ,n,n + } ⊆ ∆ n + has the property thatall its faces except ∆ { ,n − ,n + } are contained in T , while the face ∆ { ,n − ,n + } isexactly the one triangle that is in T ′ but not in T . We also note that Z contains∆ { ,n − ,n,n + } unless n =
2, in which case Z does not contain ∆ { ,n − ,n + } either.Since E and B are ∞ -bicategories we may now conclude from [13, Remark 3.1.4]that the square (13) extends to a square(14) ( Z, T ′∣ Z ) E ( ∆ n + , T ′ ) B . ρ ′ fτ ′ To finish the proof we now produce a lift in (14). For this, notice thatΛ n + n + = Z ∐ Λ { ,...,n − ,n + } n + ∆ { ,...,n − ,n + } so that we can write ∆ n + as∆ n + = Z ∐ Λ { ,...,n − ,n + } n + ∆ { ,...,n − ,n + } ∐ Λ n + n + ∆ n + . We can then construct a lift in two steps, the first by using the assumption that ρ ′∣ ∆ { n − ,n + } = e is f -cartesian (and that T ′ contains ∆ { ,n − ,n + } ), and the second byusing the assumption that ρ ′∣ ∆ { n,n + } = t is weakly f -cartesian (and that T ′ containsall the triangles with the edge ∆ { n,n + } ). (cid:3) Proof of Lemma 2.3.9.
The proof is very similar to that of Lemma 2.3.8, but wespell out the differences for the convenience of the reader. As in the case ofLemma 2.3.8, we will prove only the f -cartesian case, as the proof for strongly f -cartesian edges proceeds verbatim, with just less details to verify.Let I = { , ..., n − , n + } ⊆ [ n + ] , and let us identify Λ nn and ∆ n with thesubsimplicial setsΛ In + = ∂ ∆ { ,...,n − } ∗ ∆ { n − ,n + } ∐ ∂ ∆ { ,...,n − } ∗ ∆ { n + } ∆ { ,...,n − ,n + } and ∆ I = ∆ { ,...,n − } ∗ ∆ { n − ,n + } of ∆ n + , respectively. In addition, we considerthe subsimplicial sets Z ∶ = ∂ ∆ { ,...,n − } ∗ ∆ { n − ,n,n + } and W ∶ = ∂ ∆ { ,...,n − } ∗ ∆ { n − ,n + } ⊆ Z. Let T ⊆ T ′ ⊆ ∆ n + be the following two sets of triangles: T consists of the triangle∆ { ,n − ,n + } , as well as all the triangles which contain the edge ∆ { n − ,n } , while T ′ = T ∪ { ∆ { ,n,n + } } . We need to show that the dotted lift exist in any square ofthe form(15) ( Λ In + , { ∆ { ,n − ,n + } } ∣ Λ In + ) E ( ∆ I , { ∆ { ,n − ,n + } }) B σ fτ with n ≥ σ ∣ ∆ { n − ,n + } = e . Similarly to the proof of Lemma 2.3.8, the maps σ and the triangle (10) fit to form a map of scaled simplicial sets(16) ( Λ In + , T ∣ Λ In + ) ∐ ∆ { n − ,n + }♭ ∆ { n − ,n,n + }♯ → E . We now apply [6, Lemma 2.17] to the map ∅ ⊆ ∂ ∆ { ,...,n − }♭ and the composedinclusion∆ { n − ,n + }♭ ⊆ ( Λ { n − ,n,n + } n − , { ∆ { n − ,n } } , ∅ ) ⊆ ( ∆ { n − ,n,n + } , { ∆ { n − ,n } } , { ∆ { n − ,n,n + } }) to deduce that the map ( W, T ∣ W ) ∐ ∆ { n − ,n + }♭ ∆ { n − ,n,n + }♯ ↪ ( Z, T ∣ Z ) is scaled anodyne. We may hence extend the map (16) to a map ρ ∶ ( Λ In + , T ∣ Λ In + ) ∐ ( W,T ∣ W ) ( Z, T ∣ Z ) → E . The maps τ and f ρ now combine to form a map η ∶ ( ∆ I , { ∆ { ,n − ,n + } }) ∐ ( W,T ∣ W ) ( Z, T ∣ Z ) → B . Applying again [6, Lemma 2.17], now to the inclusions ∂ ∆ { ,...,n − }♭ ⊆ ∆ { ,...,n − }♭ and∆ { n − ,n + }♭ ⊆ ( ∆ { n − ,n,n + } , { ∆ { n − ,n } } , { ∆ { n − ,n,n + } }) , we deduce that the map ( ∆ I , { ∆ { ,n − ,n + } }) ∐ ( W,T ∣ W ) ( Z, T ∣ Z ) → ( ∆ n + , T ) IBRATIONS AND LAX LIMITS OF (∞ , ) -CATEGORIES 25 is scaled anodyne. Setting ( Y, T ∣ Y ) = ( ∆ I , { ∆ { ,n − ,n + } }) ∐ ( W,T ∣ W ) ( Z, T ∣ Z ) , we may hence extend the square (15) to a square(17) ( Y, T ∣ Y ) E ( ∆ n + , T ) B . ρ fτ We now observe that the 3-simplex ∆ { ,n − ,n,n + } ⊆ ∆ n + has the property that allits faces except ∆ { ,n,n + } are contained in T , while the face ∆ { ,n,n + } is exactlythe one triangle that is in T ′ but not in T . As in the proof of Lemma 2.3.8, wededuce that the square (17) extends to a square(18) ( Y, T ′∣ Y ) E ( ∆ n + , T ′ ) B . ρ ′ fτ ′ To finish the proof we now produce a lift in (18). For this, we note thatΛ n + n = Y ∐ Λ { ,...,n − ,n,n + } n + ∆ { ,...,n − ,n,n + } ∐ Λ { ,...,n } n ∆ { ,...,n } so that we can write ∆ n + as∆ n + = Z ∐ Λ { ,...,n − ,n,n + } n + ∆ { ,...,n − ,n,n + } ∐ Λ { ,...,n } n ∆ { ,...,n } ∐ Λ n + n ∆ n + . We can then construct a lift in three steps, the first by using the assumption that ρ ′∣ ∆ { n,n + } = t is f -cartesian (and that T ′ contains ∆ { ,n,n + } ), the second by usingthe assumption that ρ ′∣ ∆ { n − ,n } = e is weakly f -cartesian (and that T ′ contains allthe triangles with the edge ∆ { n − ,n } ), and the third by using the fact that f is aweak fibration and T ′ contains the triangle ∆ { n − ,n,n + } . (cid:3) Proof of Proposition 2.3.7.
Apply Lemma 2.3.9 to the degenerate triangle y x y y e e . (cid:3) Combining Proposition 2.3.7 with Example 2.3.2 we now conclude:
Corollary 2.3.10.
Let f ∶ E → B be a weak fibration of ∞ -bicategories. Then everyequivalence in E is both f -cartesian and f -cocartesian. Corollary 2.3.11.
Let p ∶ E → B be a weak fibration of ∞ -bicategories and let yx z te e be a thin triangle in E such that t is p -cartesian (e.g., p is an equivalence, seeCorollary 2.3.10). Then e is p -cartesian if and only if e is p -cartesian. Remark . It follows from Corollary 2.3.11 that the property of being a carte-sian edge is closed under equivalence. More precisely, suppose that p ∶ E → B is aweak fibration and e ∶ x → y, e ′ ∶ x ′ → y ′ two edges which are equivalent in the ∞ -bicategory Fun ( ∆ , E ) . We claim that e is p -cartesian if and only if e ′ is p -cartesian.Indeed, suppose that e ′ is p -cartesian and let η ∶ ∆ × ∆ → E encode x x ′ y y ′≃ e e ′ ≃ an equivalence from e = η ∣ ∆ { } × ∆ to e ′ = η ∣ ∆ { } × ∆ . Applying Corollary 2.3.11 oncewe get that the diagonal arrow x → y ′ is p -cartesian, and applying it a second timegives that e ∶ x → y is p -cartesian. Taking the equivalence in the other direction wecan similarly deduce that if e is p -cartesian then e ′ is p -cartesian.2.4. Marked fibrations and anodyne maps.
As we have alluded to in the begin-ning of §
2, the notion of inner cocartesian fibration over a base ( S, T S ) is essentiallya reformulation of the notion of P S -fibered objects studied in [13, § § Proposition 2.4.1.
Let ( S, T S ) be a scaled simplicial set and let f ∶ X → S aninner fibration of simplicial sets. Let T X = f − ( T S ) to be the set of all triangles in X whose image in S belongs to T S and let E X be the set of edges in X which arelocally f -cocartesian. Then the following are equivalence:(1) f ∶ ( X, T X ) → ( S, T S ) is an inner cocartesian fibration.(2) f exhibits ( X, E X ) as P S -fibered.In addition, when these two equivalent conditions hold the set E X identifies withthe set of f -cocartesian arrows in ( X, T X ) (in the sense of Definition 2.1.4).Remark . In the situation of Proposition 2.4.1, if ( X, E ) is P S -fibered for someset of marked edges E , then E is necessarily the set of locally f -cocartesian edgesby Condition (ii) above. In particular, the condition that ( X, E X ) is P S -fiberedcould be replaced by the a-priori weaker condition that ( X, E ) is P S -fibered for some set of marked edges E . Proof of Proposition 2.4.1.
Suppose (1) holds. We need to verify conditions (i)-(iii) of Definition 1.4.1. Condition (i) holds since f is an inner fibration of scaledsimplicial sets. To see (ii), note that for every edge e ∶ ∆ → S the base change e ∗ f ∶ ( X, T X ) × ( S,T S ) ∆ ♭ → ∆ ♭ is an inner cocartesian fibration. Since all the trianglesin ∆ ♭ are thin we get that the same holds for the domain of e ∗ f and so e ∗ f can beviewed as a cocartesian fibration of ∞ -categories. The e ∗ f -cocartesian edges arethen by definition the locally cocartesian edges lying over e , and they are also bydefinition the marked edges lying over e . This shows (ii).To prove (iii), let now σ ∶ ∆ → S be a thin triangle and let e ∶ ∆ → X be a markededge lying over e ∶ = σ ∣ ∆ { , } , with domain x ∶ = e ∣ ∆ { } and codomain y ∶ = e ∆ { } . Wewish to show that e is σ ∗ f -cocartesian. For this, we note that σ ∗ f ∶ ( X, T X ) × ( S,T S ) ∆ ♯ → ∆ ♯ is an inner cocartesian fibration, and hence a cocartesian fibration on thelevel of the underlying simplicial sets, since all the triangles in ∆ ♯ are thin. We maythen conclude that there exists a σ ∗ f -cocartesian edge e ′ ∶ ∆ → X lying over e suchthat e ′∣ ∆ { } = x . Then e and e ′ both determine cocartesian edges of X × S ∆ → ∆ with the same domain x , and hence there exists a commutative diagram in X × S ∆ (∞ , ) -CATEGORIES 27 of the form yx y ue e ′ where u is an equivalence which covers the identity id f ( y ) . Since e ′ and u are σ ∗ f -cocartesian it follows that e is σ ∗ f -cocartesian, as desired.Let us now assume that (2) holds. We first show that f ∶ ( X, T X ) → ( S, T S ) isa weak fibration. Since f is already assumed to be an inner fibration and we alsoassume that T X = f − ( T S ) , it will suffice to show that f ∶ ( X, T X ) → ( S, T S ) has theright lifting property with respect to f ∶ ( A, T A ) → ( B, T B ) , where f is one of themaps appearing in (2) and (3) of Definition 2.1.1. Since T X = f − ( T S ) it will sufficeto check the lifting problem of X → S against A → B on the level of underlyingsimplicial sets. Now by assumption the map f ∶ X → S exhibits ( X, E X ) as P S -fibered. In particular, the object ( X, E X ) of ( S et + ∆ ) / S is fibrant with respect to the P S -fibered model structure. In order to prove the lifting property against A → B itwill consequently suffice to prove that the induced map A ♭ → B ♭ , when consideredas a map in ( S et + ∆ ) / S , is a P S -fibered weak equivalence. Consider the straighteningfunctor St sc ( S,T S ) ∶ ( S et + ∆ ) / S → ( S et + ∆ ) C ( S,T S ) associated to the identity id ∶ C sc ( S, T S ) → C sc ( S, T S ) . Since St scid is a left Quillenequivalence and all the objects of ( S et + ∆ ) / S are cofibrant it is enough to show thatSt sc ( S,T S ) ( A ♭ ) → St sc ( S,T S ) ( B ♭ ) is a weak equivalence in the projective model structure on ( S et + ∆ ) C ( S,T S ) . Unwind-ing the definitions, let Z = ∆ × B and let T denote the set of those triangles ( τ, σ B ) ∶ ∆ → ∆ × B such that σ B is degenerate and either τ ∣ ∆ { , } is degeneratein ∆ or σ B ∣ ∆ { , } is degenerate in B , together with the triangles in T B × ∆ { } and T B × ∆ { } . Let Z = ( ∆ × A ) ∐ ∂ ∆ × A ( ∂ ∆ × B ) ⊆ Z and let T be the collection of 2-simplices of Z whose image in Z belongs to T .Consider the commutative rectangle ∂ ∆ × ( B, T B ) ( Z , T ) ( Z, T ) ∆ { } ∐ [ ∆ { } × ( S, T S )] C A C B in which C A and C B are defined by the condition that the left square and theexternal rectangle are pushout squares. By the definition of S t scid (recalled in § C A → C B is a bicategorical weak equivalence. By thepasting lemma the right square is a pushout square as well, and so it suffices toshow that the top horizontal map in the right square is scaled anodyne. Inspectingthe set of thin triangles T we observe that we have a commutative diagram ( Z , T ) ( Z, T )( ∆ ♭ ⊗ ( A, T A )) ∐ ∂ ∆ ♭ ⊗( A,T A ) ( ∂ ∆ ♭ ⊗ ( B, T B )) ∆ ♭ ⊗ ( B, T B ) ≃ ≃ where ⊗ denotes the Gray product of scaled simplicial sets, see § we now just need to show that the lower horizontal map in the last square is abicategorical equivalence. Now in the case where f ∶ ( A, T A ) → ( B, T B ) is as in (2)of Definition 2.1.1 this follows from [5, Proposition 2.16] since f is scaled anodyne.In the case where f ∶ ( A, T A ) → ( B, T B ) is as in (3) of Definition 2.1.1 we havethat f op ∶ ( A op , T A ) → ( B op , T B ) is scaled anodyne and hence the lower horizontalmap is the opposite of a scaled anodyne map, then in particular a bicategoricalequivalence, by [5, Remark 2.4 and Proposition 2.16].We thus proved that f ∶ ( X, T X ) → ( S, T S ) is a weak fibration. To finish theproof we need to show that every arrow f ∶ x → y in S admits f -cocartesian liftsstarting from any object x ′ ∈ X lying over x . For this we invoke [13, Proposition3.2.16] which implies that the object ( X, E X ) of ( S et + ∆ ) / S satisfies the right liftingproperty with respect to the P S -anodyne maps listed in [13, Definition 3.2.10]. Inparticular, the right lifting property with respect to the maps of type ( C ) of thislist implies that every edge in E X is f -cocartesian, and the right lifting propertywith respect to the maps of type ( B ) implies that every arrow f ∶ x → y in S admits a marked lift starting from any vertex x ′ ∈ X lying over x . We may henceconclude that ( X, T X ) → ( S, T S ) is an inner cocartesian fibration, as desired. Toobtain the last statement, note that the last argument shows that when (2) holdsevery marked edge is f -cocartesian, while every f -cocartesian edge is in particularlocally cocartesian, and hence marked by the definition of E X . (cid:3) One of the advantages of describing inner cocartesian fibrations in terms of P S -fibered object is that the latter can be characterized using a right lifting prop-erty with respect to a suitable collection of anodyne maps, see [13, Proposition3.2.16]. Our next goal is to show that a similar statement holds in the case of outer(co)cartesian fibrations. A step in that direction was already taken in [6] using thecollection of maps given in Definition 2.13 of loc . cit ., but that collection was onlypartial. In what follows we identify the complete list of outer cartesian anodynemaps and show that the lifting property against them characterizes outer cartesianfibrations. Notation 2.4.3.
Given a scaled simplicial set ( S, T S ) , we write ( S et + , sc∆ ) /( S,T S ) todenote the category of marked scaled simplicial sets ( X, E X , T X ) equipped with amap of scaled simplicial sets ( X, T X ) → ( S, T S ) . Definition 2.4.4.
Let ( S, T S ) be a scaled simplicial set. We call outer cartesiananodyne maps the smallest weakly saturated class of maps in ( S et + , sc∆ ) /( S,T S ) con-taining the following maps:(1) The inclusion ( Λ ni , ∅ , { ∆ { i − ,i,i + } } ∣ Λ ni ) ↪ ( ∆ n , ∅ , { ∆ { i − ,i,i + } }) for every 0 < i < n and every map ( ∆ n , { ∆ { i − ,i,i + } }) → ( S, T S ) .(2) The inclusion ( Λ nn , { ∆ { n − ,n } } , ∅ ) ⊆ ( ∆ n , { ∆ { n − ,n } } , ∅ ) for every n ≥ n → S (when n = { } ⊆ ( ∆ ) ♯ ).(3) The inclusion ( Λ n ∐ ∆ { , } ∆ , ∅ , ∅ ) ⊆ ( ∆ n ∐ ∆ { , } ∆ , ∅ , ∅ ) for every n ≥ ( ∆ n ∐ ∆ { , } ∆ , ∅ ) → ( S, T S ) .(4) The inclusion ♭ ∆ ⊆ ( ∆ , ∅ , { ∆ }) for every map ∆ ♯ → ( S, T S ) . IBRATIONS AND LAX LIMITS OF (∞ , ) -CATEGORIES 29 (5) The inclusion ( Q, ∅ , Q ) ⊆ ( Q, E, Q ) for every map Q ♯ → ( S, T S ) , where Q = ∆ ∐ ∆ { , } ∆ ∐ ∆ { , } ∆ and E contains all the degenerate edges and in addition the edges ∆ { , } and∆ { , } .(6) The inclusion ( ∆ , { ∆ { , } , ∆ { , } } , { ∆ }) ⊆ ♯ ∆ for every map ∆ ♯ → ( S, T S ) .Dually, we let the collection of outer cocartesian anodyne maps to be the weaklysaturated class generated by the opposites of the above maps.The following proposition extends [6, Proposition 2.14]: Proposition 2.4.5.
Let B be an ∞ -bicategory, ( X, E X ) a marked simplicial setand f ∶ ( X, T X ) → B a map which detects thin triangles. The object of ( S et + , sc∆ ) / B determined by ( X, E X , T X ) and f has the right lifting property with respect to outercartesian anodyne maps if and only if f ∶ ( X, T X ) → B is an outer cartesian fibrationand E X is the collection of f -cartesian edges of X .Proof. We first prove the “only if” direction. Since every degenerate edge in X belongs to E X the right lifting property with respect to outer cartesian anodynemaps of type (1),(2), (3) and (4) implies that f is an outer fibration and that everyedge in E X is f -cartesian. In addition, the case n = e ∶ x → y in B and for every ̃ y ∈ X such that f (̃ y ) = y thereexists a marked (and hence f -cartesian) edge ̃ e ∶ ̃ x → ̃ y in X such that f (̃ e ) = e . Wemay hence conclude that f ∶ ( X, T X ) → B is an outer cartesian fibration with allmarked edges being f -cartesian. Let us now show that every f -cartesian edge ismarked. Let e ∶ x → y be a f -cartesian edge lying over an edge e ∶ x → y of B . Thenthere exists a marked edge e ′ ∶ x ′ → y in X such that f ( e ′ ) = e . By the above e ′ is f -cartesian, and so we may factor e through e ′ , in the sense that we may find athin triangle in X of the form x ′ e ′ (cid:31) (cid:31) ❄❄❄❄❄❄❄❄ x u ? ? ⑧⑧⑧⑧⑧⑧⑧⑧ e / / y which lies over the degenerate triangle x e (cid:30) (cid:30) ❂❂❂❂❂❂❂❂ x id @ @ ✁✁✁✁✁✁✁✁ e / / y By Corollary 2.3.11 we may conclude that u is f -cartesian, hence an equivalence byRemark 2.1.10. We now observe that the right lifting property with respect to outercartesian anodyne maps of type (5) implies in particular that every equivalence in X is marked, and so in particular u is marked. Finally, since u and e ′ are markedan application of the right lifting property against maps of type (6) implies that e is marked, as desired.We now prove the “if” direction, and so we assume that f is an outer cartesianfibration and E X consists of the f -cartesian edges. Proposition 2.3.7 implies thatevery f -cartesian edge in X is strongly f -cartesian. This together with the factthat f is an outer fibration implies that ( X, E X , T X ) has the right lifting propertywith respect to outer cartesian anodyne maps of type (1), (2), (3) and (4). The right lifting property with respect to maps of type (6) follows directly from Corol-lary 2.3.11. In order to conclude the proof we wish to show that the map f hasthe right lifting property with respect to maps of type (5). For this we note that ( X, T X ) is an ∞ -bicategory in this case by Remark 2.1.3, and that any edge in Q is necessarily sent to an equivalence in ( X, T X ) , which is therefore f -cartesian byvirtue of Corollary 2.3.10. (cid:3) We finish this subsection by discussing the compatibility of outer (co)cartesiananodyne maps with ∗ -pushout-products. In particular, the following lemma ex-tends [6, Lemma 2.17]: Lemma 2.4.6.
Let f ∶ X → Y and g ∶ A → B be injective maps of marked-scaledsimplicial sets. If either f is outer cartesian anodyne or g is outer cocartesiananodyne then the map of scaled simplicial sets (19) X ∗ B ∐ X ∗ A Y ∗ A → Y ∗ B is a bicategorical trivial cofibration.Proof. Since the collection of trivial cofibrations is closed under taking opposites, itwill suffice to verify the case where f is outer cartesian anodyne. For this, one maycheck the claim on generators, and so we may assume that g is either the inclusion ♭ ( ∂ ∆ n ) ↪ ♭ ∆ n , the inclusion ♭ ∆ ↪ ( ∆ , ∅ , { ∆ }) , or the inclusion ♭ ∆ ↪ ♯ ∆ , and f is one of the generating anodyne maps appearing in Definition 2.4.4. We first notethat when g is the map ♭ ∆ ↪ ( ∆ , ∅ , { ∆ }) then (19) is an isomorphism. When g is the map ♭ ∆ ↪ ♯ ∆ the map (19) is an isomorphism except if f is ∆ { } ↪ ( ∆ ) ♯ ,in which case the map (19) takes the form ( ∆ , { ∆ { , , } , ∆ { , , } , ∆ { , , } }) → ∆ ♯ , which is scaled anodyne by [13, Remark 3.1.4].We may hence assume that g is the inclusion ♭ ( ∂ ∆ n ) ↪ ♭ ∆ n . For the first fourtypes of generating anodyne maps this was proven in [6, Lemma 2.17]. We nowverify the remaining two cases: ● When f is the inclusion ( ∆ , { ∆ { , } , ∆ { , } } , { ∆ }) ⊆ ♯ ∆ the map (19) is iso-morphic to the map ( ∆ [ ]∗[ n ] , T ) → ( ∆ [ ]∗[ n ] , T ′ ) , where T contains all the triangles of the form ∆ { , ,i } and ∆ { , ,i } while T ′ contains T plus all the triangles of the form ∆ { , ,i } . In this case we see that ( ∆ [ ]∗[ n ] , T ′ ) can be obtained from ( ∆ [ ]∗[ n ] , T ) by performing a sequence ofpushouts along the maps ( ∆ , { ∆ { , , } , ∆ { , , } , ∆ { , , } } , ∆ ) → ♯ ∆ , which are scaled anodyne (see [13, Remark 3.1.4]). ● When f is the inclusion ( Q, ∅ , Q ) ⊆ ( Q, { ∆ { , } , ∆ { , } } , Q ) , we set W = ∆ [ ]∗[ n ] ∐ ∆ { , }∗[ n ] ∆ [ ]∗[ n ] ∐ ∆ { , }∗[ n ] ∆ [ ]∗[ n ] . The map (19) is then isomorphic to the map ( W, T ) → ( W, T ′ ) where T containsall the triangles which are contained in ∆ { , , , } and T ′ contains T and moreoverall the triangles of the form ∆ { , ,i } and ∆ { , ,i } . In this case the map (19) canbe realized as a sequence of pushouts along the scaled anodyne maps ( ∆ , ∆ , T ) → ( ∆ , ∆ , T ∪ { ∆ { , , } , ∆ { , , } }) as in Definition 1.2.6(ii). (cid:3) IBRATIONS AND LAX LIMITS OF (∞ , ) -CATEGORIES 31 Corollary 2.4.7.
Let C be an ∞ -bicategory and let f ∶ K → C be a map of scaledsimplicial sets. Then the map of scaled simplicial sets f ∶ C / f → C is an outer cartesian fibration and the marked edges of C / f are exactly the f -cartesian edges. In particular, an edge e ∶ ∆ ∗ K → C in C / f is f -cartesian if andonly if for every vertex x of K the triangle e ∣ ∆ ∗{ x } is thin. Dually, the map f ∶ C f / → C is an outer cocartesian fibration and the marked edges of C f / are precisely the f -cocartesian edges.Remark . The previous result also appears in [15, Tag 01WT], in a weakerform that only deals with the “outer fibration” part.2.5.
Cartesian lifts of lax transformations.
In this section we relate the theoryof outer cartesian fibrations as developed so far in this work with the notion of laxtransformations defined via the Gray product, see § Proposition 2.5.1 (Lifting lax transformations) . Let f ∶ E → B be an outer fibrationof ∞ -bicategories and K ⊆ L an inclusion of scaled simplicial sets. Consider a liftingproblem of the form (20) ∆ { } ⊗ L ∐ ∆ { } ⊗ K ∆ ♭ ⊗ K f / / (cid:15) (cid:15) E f (cid:15) (cid:15) ∆ ♭ ⊗ L H / / ̃ H ♣♣♣♣♣♣♣♣ B such that f sends every edge of the form ∆ × { v } (for v ∈ K ) to a f -cartesianedge. Suppose that for every u ∈ L ∖ K there exists a f -cartesian edge with target f ( ∆ { } × { u }) which lifts H ( ∆ × { u }) . Then the dotted lift ̃ H ∶ ∆ ♭ ⊗ B → E exists.Furthermore, ̃ H can be chosen so that the edges ̃ H ( ∆ × { u }) for u ∈ L ∖ K areany prescribed collection of f -cartesian lifts.Proof. Since f ∶ E → B is an outer fibration it detects thin triangles, and so a dottedlift in (20) with the desired properties exists if and only if it exists on the level ofunderlying simplicial sets. We may hence assume without loss of generality thatthe L and K have only degenerate triangles thin. Arguing simplex by simplex itwill suffice to prove the claim for L ⊆ K being the inclusion ∂ ∆ n ♭ ⊆ ∆ n ♭ . In thecase n = n ≥ ∂ ∆ n ♭ ⊆ ∆ n ♭ is bijective on vertices and so we just needto construct a lift without the additional constraints on the edges. Consider thefiltration ∆ ♭ ⊗ ∂ ∆ n ♭ ∐ ∆ { } ⊗ ∂ ∆ n ♭ ∆ { } ⊗ ∆ n ♭ = X ⊆ X ⊆ ⋅ ⋅ ⋅ ⊆ X n + = ∆ ♭ ⊗ ∆ n ♭ , where X i + is the union of X i and the image of the map τ i ∶ ( ∆ n + , T + i ) → ∆ ♭ ⊗ ∆ n ♭ given on vertices by the formula τ i ( m ) = { ( , m ) m ≤ i ( , m − ) m > i , and T + i is the collection of all triangles in ∆ n + which are either degenerate orof the form ∆ { i,i + ,k } for k > i +
1. We then observe that for i = , ..., n − X i ⊆ X i + is a pushout along τ i of the scaled inner horn ( Λ n + i + , T + i ∣ Λ n + i + ) ↪ ( ∆ n + , T + i ) , while at the last step of the filtration the inclusion X n ⊆ X n + is apushout along τ n of the outer horn ( Λ n + n + ) ♭ ⊆ ∆ n + ♭ , but since τ n sends ∆ { n,n − } to ∆ × { n } its image in E is f -cartesian by assumption, and hence strongly f -cartesian by Proposition 2.3.7. Since f is an outer fibration it then follows that thelift ̃ H ∶ ∆ ♭ ⊗ ∆ n ♭ → E exists, as desired. (cid:3) Remark . Passing to opposites, Proposition 2.5.1 yields a dual statement forthe case where the lift is taken against the map L ⊗ ∆ { } ∐ K ⊗ ∆ { } K ⊗ ∆ ♭ → L ⊗ ∆ ♭ , assuming as above that edges of the form { v } × ∆ are sent f -cocartesian edges. Inother words, we need to change ∆ { } to ∆ { } but also switch the order of the Grayproduct. In particular, we obtain cocartesian lifts for oplax natural transformationsgiven a lift of their domains. On the other hand, the analogue for inner cocartesian fibrations, which is proven in [13, Lemma 4.1.7], states that such fibrations admitcocartesian lifts against ∆ { } ⊗ L ∐ ∆ { } ⊗ K ∆ ♭ ⊗ K → ∆ ♭ ⊗ L, assuming again that edges of the form ∆ × { v } are sent f -cocartesian edges. Inparticular, they admit cocartesian lifts for lax transformations given a lift of theirdomains. Finally, passing to opposites one obtains that inner cartesian fibrationsadmit cartesian lifts against L ⊗ ∆ { } ∐ K ⊗ ∆ { } K ⊗ ∆ ♭ → L ⊗ ∆ ♭ assuming that edges of the form { v } × ∆ are sent to f -cartesian edges. Thelast claim can also be proven using exactly the same filtration as in the proofof Proposition 2.5.1, which this time will involve a slightly different scaling (cf. theproof of [13, Lemma 4.1.7]).3. The bicategorical Grothendieck–Lurie correspondence
In this section we will prove one of the principal results of the present paper byshowing that the four types of fibrations studied in §
2, over a fixed base B , encodethe four variance flavors of B -indexed C at ∞ -valued diagrams, a phenomenon we call the bicategorical Grothendieck–Lurie correspondence . Our approach is as follows.First, in § S et + ∆ -enriched categories, and show that these are equivalent to the ones defined inthe setting of scaled simplicial sets via the Quillen equivalence S et sc∆ C sc ' ' N sc g g ⊥ S et + ∆ - C at . The advantage of S et + ∆ - C at as a model for ( ∞ , ) -categories is that it admits point-set models for the ( Z / ) -symmetry of the theory of ( ∞ , ) -categories, a fact we willexploit in § § IBRATIONS AND LAX LIMITS OF (∞ , ) -CATEGORIES 33 Fibrations of enriched categories.
In this section we will study analo-gous of the notions of inner/outer (co)cartesian fibrations in the setting of C at ∞ -categories, by which we mean fibrant objects in S et + ∆ - C at with respect to theDwyer-Kan model structure. Explicitly, this means that their mapping objectsare ∞ -categories marked by their equivalences, which is at the origin of the aboveterm. Definition 3.1.1.
Let f ∶ C → D be a map of C at ∞ -categories. An arrow e ∶ x → y in C is said to be f -cartesian if for every z ∈ C the induced square C ( z, x ) C ( z, y ) D ( f z, f x ) D ( f z, f x ) e ○− f z,x f z,y f ( e )○− is a homotopy pullback square in Set + .Recall that we use the term marked left (resp. right ) fibration to indicate a mapof marked simplicial sets f ∶ X → Y which detects marked edges and which is a left(resp. right) fibration on the level of underlying simplicial sets. We say that a mapof C at ∞ -categories f ∶ C → D is locally a marked left (resp. right ) fibration if for anypair of objects x, y of C , the induced map C ( x, y ) → D ( f x, f y ) is a marked left(resp. right) fibration. Definition 3.1.2.
Let f ∶ C → D be a map of C at ∞ -categories. We say that f is an inner cartesian fibration (resp. outer cartesian fibration ) if it satisfies the followingproperties:(1) Given y ∈ C and an arrow e ∶ x → f ( y ) in D , there exists a f -cartesian arrow ̃ e ∶ x ′ → y such that f (̃ e ) = e .(2) The map f ∶ C → D is locally a marked right (resp. left) fibration.We say f ∶ C → D is an inner (resp. outer) cocartesian fibration if f op ∶ C op → D op is an inner (resp. outer) cartesian fibration, where the operation ( − ) op is definedby E op ( x, y ) = E ( y, x ) (see Construction 1.2.17 and the discussion preceding it in § fibration of C at ∞ -categories we will simply mean a fibration between fibrantobjects with respect to the Dwyer-Kan model structure. Proposition 3.1.3.
Let f ∶ C → D be a fibration of C at ∞ -categories. Then f is aninner (resp. outer) cartesian fibration in the above sense if and only if N sc f ∶ N sc C → N sc D is an inner (resp. outer) cartesian fibration in the sense of Definition 2.1.7. Inaddition, an arrow in C is f -cartesian if and only if the corresponding edge in N sc ( C ) is f -cartesian.Remark . Since N sc ( C op ) ≅ N sc ( C ) op the statement of Proposition 3.1.3 impliesthe same statement for inner/outer cocartesian fibrations.In the proof that follows we will use the following notation. We will denote by ✷ n = ( ∆ ) n the n -cube and by ∂ ✷ n its boundary, so that the inclusion ∂ ✷ n ⊆ ✷ n can be identified with the pushout-product of ∂ ∆ ↪ ∆ with itself n times. Wealso denote by ⊓ n − ,i ↪ ✷ n − the iterated pushout-product [ ∂ ∆ ↪ ∆ ] ✷ ⋯ ✷ [ ∆ { ε } ↪ ∆ ] ✷ ⋯ ✷ [ ∂ ∆ ↪ ∆ ] , where ε ∈ { , } and [ ∆ { ε } ↪ ∆ ] appears in the i ’th factor. Proof of Proposition 3.1.3.
To begin, we recall from Remark 1.2.14 that we havecanonical marked categorical equivalencesHom N sc C ( x, y ) ≃ C ( x, y ) and Hom N sc D ( z, w ) ≃ D ( z, w ) . By Proposition 2.2.1 we then see that if N sc ( f ) is an inner (resp. outer) fibrationthen f x,y ∶ C ( x, y ) → D ( f x, f y ) is weakly equivalent as an arrow to a marked right(resp. left) fibration. Since f is a fibration in the Dwyer-Kan model structure wehave that each f x,y is a fibration between fibrant objects. Since the condition ofbeing a marked right (resp. left) fibration is given in terms of a suitable right liftingproperty this is equivalent to f x,y itself being a marked right (resp. left) fibration.Finally, Proposition 2.3.3 implies that every N sc ( f ) -cartesian edge of N sc C is also f -cartesian as an edge of C . Since the objects and arrows of N sc C are in bijectionwith the objects and arrows of C , and the same goes for D , we now conclude that ifN sc ( f ) is an inner (resp. outer) cartesian fibration then f is an inner (resp. outer)cartesian fibration of C at ∞ -categories.Now assume that f is an inner (resp. outer) cartesian fibration of C at ∞ -categories.Since f was assumed to be a fibration between fibrant objects it follows thatN sc ( f ) ∶ N sc C → N sc D is a bicategorical fibration of ∞ -bicategories, and in particulara weak fibration (Remark 2.1.2). As above, Proposition 2.3.3 implies that every f -cartesian edge of C is at least weakly f -cartesian as an edge of N sc C , and hence f -cartesian by Proposition 2.3.7. Using again the bijection between objects andarrows of C at ∞ -categories and their scaled nerves we conclude that N sc ( f ) satisfiesCondition (2) of Definition 2.1.7, that is, arrows in N sc D admit N sc ( f ) -cartesianlifts.We now show that f is an inner fibration (resp. outer) fibration. In the innercase, consider a lifting problem of the form ( Λ ni ) ♭ N sc C ( ∆ n ) ♭ N sc D . N sc ( f ) with 0 < i < n . By adjunction, this corresponds to a lifting problem of the form(21) C sc Λ ni C C sc ∆ n D h fg As a straightforward calculation shows, the lifting problem in (21) corresponds, atthe level of simplicial sets, to the following lifting problem: ( ⊓ n − ,i ) ♭ C ( h ( ) , h ( n ))( ✷ n − ) ♭ D ( g ( ) , g ( n )) . f h ( ) ,h ( n ) The last square then admits a lift since f is assumed to be locally a marked rightfibration and ⊓ n − ,i ↪ ✷ n − is right anodyne.In the outer case, we have to solve any lifting problem of the form: ( Λ n ∐ ∆ { , } ∆ ) ♭ N sc C ( ∆ n ∐ ∆ { , } ∆ ) ♭ N sc D N sc ( f ) IBRATIONS AND LAX LIMITS OF (∞ , ) -CATEGORIES 35 Arguing as above we see that this amounts to solving two lifting problems in thecategory of marked simplicial sets, namely:(22) ( ∂ ✷ n − ) ♭ C ( h ( ) , h ( n ))( ✷ n − ) ♭ D ( g ( ) , g ( n )) f h ( ) ,h ( n ) and(23) ( ⊓ n − , ) ♭ C ( h ( ) , h ( n ))( ✷ n − ) ♭ D ( g ( ) , g ( n )) f h ( ) ,h ( n ) Since we collapsed the edge ∆ { , } , we get that pre-composition with the image ofthe map 0 = → C ( f ( ) , f ( n )) C ( f ( ) , f ( n )) D ( g ( ) , g ( n )) D ( g ( ) , g ( n )) = f f ( ) ,f ( n ) f f ( ) ,f ( n ) = Under this identification, the lifting problem (22) corresponds to filling the missing ( i, ) -face of ✷ n − in (23). Therefore, solving (23) also produces a solution for (22).A solution to (23) then exists since ⊓ n − , ⊆ ✷ n − is left anodyne and f is nowassumed to be locally a marked left fibration. (cid:3) Corollary 3.1.5.
For a map f ∶ E → B in BiCat ∞ the following are equivalent:(1) f can be represented by an inner/outer (co)cartesian fibration of ∞ -bicategories.(2) Under the equivalence ( S et + ∆ - C at ) ∞ ≃ Ð→ BiCat th ∞ the arrow f can be representedby an inner/outer (co)cartesian fibration of C at ∞ -enriched categories. The co / op -symmetry and cartesian fibrations. In the previous section wedefined inner/outer (co)cartesian fibrations for C at ∞ -categories, and showed thatthese coincide with the corresponding notion of inner/outer (co)cartesian fibrationsunder the scaled nerve functor N sc ∶ S et + ∆ - C at → S et sc∆ , which is a right Quillen equiv-alence. Both S et + ∆ - C at and S et sc∆ are models for the theory of ( ∞ , ) -categories, andCorollary 3.1.5 suggests to consider the notions of inner/outer (co)cartesian fibra-tions model independently: Definition 3.2.1.
We will refer to arrows in BiCat ∞ which satisfy the equivalentconditions of Corollary 3.1.5 as inner/outer (co)cartesian maps . In addition, givena map f ∶ E → B in BiCat ∞ and a 1-morphism e ∶ ∆ → E in E , we will say that e is f -(co)cartesian if we can represent f by a map of ∞ -bicategories such that e isrepresented by a f -(co)cartesian edge. Equivalently, by Proposition 3.1.3 this is thesame as saying that we can represent f by a map of C at ∞ -enriched categories suchthat e is represented by a f -(co)cartesian arrow. Lemma 3.2.2.
Let f ∶ C → D be a map of C at ∞ -categories and e ∶ [ ] → C an arrowin C . Then the following are equivalent:(1) e ∶ [ ] → C is f -cartesian.(2) e op ∶ [ ] op ≅ [ ] → C op is f op -cocartesian.(3) e co ∶ [ ] co ≅ [ ] → C co is f co -cartesian.(4) e coop ∶ [ ] coop → C coop is f coop -cocartesian. In addition, f is locally a marked right fibration if and only f co is locally a markedleft fibration, while the operation ( − ) op preserves locally marked left/right fibrations.Proof. All the claims follow directly from the definitions and the fact that the op-eration ( − ) op on the level of marked simplicial sets preserves and detects homotopycartesian squares and switches between marked left fibrations and marked rightfibrations. (cid:3) The following two corollaries directly follow:
Corollary 3.2.3.
Let f ∶ C → D be a map of C at ∞ -categories. Then the followingare equivalent:(1) f ∶ C → D is an inner cartesian fibration.(2) f op ∶ C op → D op is an inner cocartesian fibration.(3) f co ∶ C co → D co is an outer cartesian fibration.(4) f coop ∶ C coop → D coop is an outer cocartesian fibration. Corollary 3.2.4.
Let f ∶ B → E be a map in BiCat ∞ . Then the following areequivalent:(1) f ∶ E → B is an inner cartesian map.(2) f op ∶ E op → B op is an inner cocartesian map.(3) f co ∶ E co → B co is an outer cartesian map.(4) f coop ∶ E coop → B coop is an outer cocartesian map.In addition, an edge e ∶ ∆ → E is f -cartesian if and only if e op is f op -cocartesian,if and only if e co is f co -cartesian, and if and only if e coop is f coop -cocartesian. Definition 3.2.5.
For an ∞ -bicategory B , let us denote by Car inn ( B ) (resp. Car out ( B ) ,coCar inn ( B ) , coCar out ( B )) the sub-bicategories of ( BiCat ∞ ) / B spanned by the in-ner cartesian fibrations (resp. outer cartesian fibrations, inner cocartesian fibra-tions, outer cocartesian fibrations) over B and the 1-morphisms which preserve(co)cartesian edges.The following corollary is the main conclusion of the present section. To formu-late it, we recall from Remark 1.2.20 that the equivalence ( − ) co ∶ BiCat th ∞ ≃ Ð→ BiCat th ∞ extends to a bicategorical equivalence ( − ) co ∶ BiCat ∞ → BiCat ∞ , while the equiv-alence ( − ) op ∶ BiCat th ∞ ≃ Ð→ BiCat th ∞ becomes a bicategorical equivalence of the form ( − ) op ∶ BiCat ∞ → BiCat co ∞ . Corollary 3.2.6.
For a fixed B ∈ BiCat ∞ , the induced bicategorical equivalence ( − ) co ∶ ( BiCat ∞ ) / B ≃ Ð→ ( BiCat ∞ ) / B co restricts to give bicategorical equivalences Car inn ( B ) ≃ Ð→ Car out ( B co ) and coCar inn ( B ) ≃ Ð→ coCar out ( B co ) . Similarly, the induced bicategorical equivalence ( − ) op ∶ ( BiCat ∞ ) / B ≃ Ð→ ( BiCat ∞ ) co / B op restricts to give bicategorical equivalences Car inn ( B ) ≃ Ð→ coCar inn ( B op ) co and Car out ( B ) ≃ Ð→ coCar out ( B op ) co . Straightening and unstraightening.
In light of Proposition 2.4.1, the ∞ -bicategory coCar inn ( B ) of Definition 3.2.5 can be identified with the scaled coherentnerve of the fibrant S et + ∆ -enriched category [( S et + ∆ ) / B ] ○ ⊆ ( S et + ∆ ) / B spanned by thefibrant(-cofibrant) objects with respect to the P B -fibered model structure. In thissubsection we will use the connection, together with the results of the previoussubsection, in order to extract from Lurie’s straightening-unstraightening theoremthe bicategorical Grothendieck–Lurie correspondence for all variance flavors. IBRATIONS AND LAX LIMITS OF (∞ , ) -CATEGORIES 37 Let C be a S et + ∆ -enriched category and φ ∶ C sc ( B ) → C a Dwyer-Kan equivalence.By Lemma 1.4.3, Lurie’s scaled unstraightening functor induces a Dwyer-Kan equiv-alence of C at ∞ -categories [( S et + ∆ ) C ] ○ ≃ Ð→ [( S et + ∆ ) / B ] ○ , and hence an equivalence of ∞ -bicategories(24) N sc [( S et + ∆ ) C ] ○ ≃ Ð→ N sc [( S et + ∆ ) / B ] ○ ≃ coCar inn ( B ) We now claim that the ∞ -bicategory N sc [( S et + ∆ ) C ] ○ is naturally equivalent to the ∞ -bicategory of functors N sc ( C ) → N sc ([ S et + ∆ ] ○ ) ≃ C at ∞ . To see this let us firstconstruct a map(25) N sc ([( S et + ∆ ) C ] ○ ) → Fun ( B , N sc [ S et + ∆ ] ○ ) . Given a scaled simplicial set K , maps from K to the left hand side in (25) correspondby adjunction to enriched functors C sc ( K ) → [( S et + ∆ ) C )] ○ , which in turn correspondto enriched functors C sc ( K ) × C → [ S et + ∆ ] ○ satisfying a certain condition. On theother hand, maps from K to the right hand side in (25) correspond to maps K × B → N sc [ S et + ∆ ] ○ , and hence to enriched functors C sc ( K × B ) → [ S et + ∆ ] ○ . The map (25) isthen obtained by restriction along C sc ( K × B ) → C sc ( K ) × C sc ( B ) → C sc ( K ) × C . Proposition 3.3.1.
The map (25) is an equivalence of ∞ -bicategories.Proof. By the (enriched) Quillen equivalences of [14, Proposition A.3.3.8(1)] we mayas well assume that C is fibrant. We now argue as in the proof of [14, Proposition4.2.4.4]. In particular, writing S et + ∆ as a sufficiently filtered colimit of small C -chunks U in the sense of [14, Definition A.3.4.9] (see also [14, Definition A.3.4.1] forthe notion of a chunk of an enriched model category), we may reduce to showingthat for every small C -chunk U the mapN sc [ U C ] ○ ≃ Ð→ Fun ( B , N sc ( U ○ )) is an equivalence of ∞ -bicategories. Consider the composed map(26) B × N sc [ U C ] ○ → B × Fun ( B , N sc ( U ○ )) → N sc ( U ○ ) , the second one being the evaluation map. Since N sc ( U ○ ) is an ∞ -bicategory and thebicategorical model structure is cartesian closed the second map exhibits Fun ( B , N sc ( U ○ )) as an internal mapping object in the homotopy category of S et sc∆ (with respect tothe bicategorical model structure). It will hence suffice to show that the composedmap (26) exhibits N sc [ U C ] ○ as the same internal mapping object. Unwinding thedefinitions, this composed map identifies with the composed map B × N sc [ U C ] ○ ≃ Ð→ N sc ( C ) × N sc [ U C ] ○ ≅ N sc ( C × [ U C ] ○ ) → N sc ( U ○ ) , where the first map is a bicategorical equivalence since C is now assumed fibrantand the second map is the image under N sc of the evaluation map C × [ U C ] ○ → U ○ . Since C sc ⊣ N sc is a Quillen equivalence and this last evaluation map is betweenfibrant objects, it will now suffice to verify that it exhibits [ U C ] ○ as an internal map-ping object in S et + ∆ - C at . Indeed, this last statement is established in [14, CorollaryA.3.4.14]. (cid:3) Combining the map (25) with the inverse of the equivalence (24) we now obtainthe following conclusion:
Corollary 3.3.2 (Lurie) . For an ∞ -bicategory B ∈ BiCat ∞ there is a naturalequivalence of ∞ -bicategories coCar inn ( B ) ≃ Fun ( B , C at ∞ ) . Corollary 3.3.3.
For an ∞ -bicategory B ∈ BiCat ∞ there are natural equivalencesof ∞ -bicategories coCar out ( B ) ≃ Fun ( B co , C at ∞ ) , Car inn ( B ) ≃ Fun ( B coop , C at ∞ ) , and Car out ( B ) ≃ Fun ( B op , C at ∞ ) . Proof.
Combining Corollary 3.2.6 and Corollary 3.3.2 we obtain the first equivalenceabove as a composite of equivalencescoCar out ( B ) (−) co Ð→ ≃ coCar inn ( B co ) ≃ Fun ( B co , C at ∞ ) , The same argument deduces the third desired equivalence from the second one. Toobtain the second equivalence we again invoke Corollary 3.2.6 and Corollary 3.3.2to obtain equivalences of ∞ -bicategoriesCar inn ( B ) ≃ coCar inn ( B op ) co ≃ Fun ( B op , C at ∞ ) co ≃ Fun ( B coop , C at co ∞ ) and finish by identifying C at co ∞ ≃ C at ∞ via the functor ( − ) op , see Example 1.2.18. (cid:3) Remark . By [13, Remark 3.5.16 and Remark 3.5.17] the scaled unstraighteningfunctor intertwines base change with restriction. In particular, given a map f ∶ B → B ′ of ∞ -categories, the base change and restriction functors fit in a commutativesquare of right quillen functors ( S et + ∆ ) C sc ( B ′ ) B × B ′ (−) / / (cid:15) (cid:15) ( S et + ∆ ) C sc ( B ) (cid:15) (cid:15) ( S et + ∆ ) / B ′ C sc ( f ) ∗ / / ( S et + ∆ ) / B , whose vertical arrows are the respective unstraightening functors. Applying theoperation N sc ([ − ] ○ ) and taking into account the identification of Proposition 3.3.1we obtain a commutative square of ∞ -bicategoriesFun ( B ′ , C at ∞ ) / / ≃ (cid:15) (cid:15) Fun ( B , C at ∞ ) ≃ (cid:15) (cid:15) coCar inn ( B ′ ) / / coCar inn ( B ) , expressing the fact that the Grothendieck–Lurie correspondence intertwines be-tween base change on the level of fibrations and restriction on the level of di-agrams. Since all the four flavors of the Grothendieck–Lurie correspondence inCorollary 3.3.3 are deduced from the inner cocartesian one by acting with the ( Z / ) -symmetry spanned by ( − ) op and ( − ) co (which certainly preserves the notionof base change) it follows that all four flavors enjoy the exact same base-change–restriction compatibility. IBRATIONS AND LAX LIMITS OF (∞ , ) -CATEGORIES 39 Lax transformations and thick slice fibrations
In the setting of ∞ -categories, the usual slice construction, featuring prominentlyin the theory of ∞ -categories, is given a “thickened” counterpart in [14, § ∞ -bicategories. An important difference is however present: while in [14] the thickenedjoint and slice constructions constitute a completely equivalent alternative, whoserole is mostly technical, here they offer an important conceptual advantage. Moreprecisely, while there is only one type of join X ∗ Y of two marked-scaled simplicialsets given the order of factors, the thick join comes in two flavors, which we call theinner and outer thick join. This means that, given a marked-scaled simplicial set K and diagram f ∶ K → C in an ∞ -bicategory C , there are now not only two differentslice constructions C / f and C f / , but four different (thickened) slice constructions,which we will denote by C / f inn , C / f out , C f / inn and C f / out . This allows to incorporate into theslice construction the ( Z / ) -symmetry of the theory of ( ∞ , ) -categories, which weheavily relied on in § C constituteexamples of the four types of fibrations we studied above, that is, inner/outercartesian fibrations and inner/outer cocartesian fibrations, respectively.In addition to the theoretical advantage of avoiding a break of symmetry, theframework of four slice constructions allows one to define and study the correspond-ing four types of ( ∞ , ) -cateogrical limits, namely, the lax limit, the oplax limit, thelax colimit and the oplax colimit, as well as their marked (or partially lax) versions,see § § § § §
3. Finally,in § ∞ -categories of lax transformations.4.1. Gray products of marked-scaled simplicial sets.
In this section we con-sider a version of the Gray product in the setting of marked-scaled simplicial sets,which we will use in § § X , ..., X n of marked-scaled simplicial sets, so as to avoidthe need to iterate binary Gray products. Definition 4.1.1.
For n ≥ X , ..., X n ∈ S et + , sc∆ we define their associated Gray product X ⊗ ⋯ ⊗ X n ∈ S et sc∆ to be the scaled simplicial set whose underlying simplicial set is the cartesian prod-uct of X , ..., X n and such that a triangle σ = ( σ , ..., σ n ) ∶ ∆ ♭ → X ⊗ ⋯ ⊗ X n is thinif and only if the following conditions hold: (1) Each σ i is thin in X i .(2) There exists a j ∈ { , ..., n } such that σ i is degenerate for i ≠ j , σ i ∣ ∆ { , } ismarked for i > j and σ i ∣ ∆ { , } is marked for i < j . Warning . To avoid confusion, we explicitly point out that the above Grayproduct takes as input a sequence of marked-scaled simplicial sets, and outputs justa scaled simplicial set, to which we associate no particular marking.
Remark . If X , .., X n have only their degenerate edges marked then the Grayproduct of Definition 4.1.1 coincides with the iteration of the binary Gray productof scaled simplicial sets defined in [5, § Remark . For marked-scaled simplicial sets X , ..., X n , Y there is a naturalmap ( X ⊗ ⋯ ⊗ X n ) ♭ ⊗ Y → X ⊗ ⋯ ⊗ X n ⊗ Y which is an isomorphism on underlying simplicial sets, but is generally only aninclusion on thin triangles. This map is however an isomorphism in the particularcase where X , ..., X n have no non-degenerate marked edges. Remark . For fixed marked-scaled simplicial sets X , ..., X n the functors Y ↦ X ⊗ ⋯ ⊗ X n ⊗ Y and Y ↦ Y ⊗ X ⊗ ⋯ ⊗ X n are colimit preserving functors from S et + , sc∆ to S et sc∆ . The proof proceeds exactly as the proof of [18, Lemma 142]. Remark . The Gray product is not symmetric in general. For marked-scaledsimplicial sets X , ..., X n we however have the relation X ⊗ ⋯ ⊗ X n = ( X op n ⊗ ⋯ ⊗ X op1 ) op . Example . While the Gray product ♭ ∆ ⊗ ♭ ∆ is a square ( , ) ( , )( , ) ( , ) ≃ in which exactly one of the triangles is thin, the Gray products ♯ ∆ ⊗ ♭ ∆ , ♭ ∆ ⊗ ♯ ∆ and ♯ ∆ ⊗ ♯ ∆ are all squares in which both triangles are thin.As in the unmarked case (see [5, § X, Y be marked-scaled simplicial sets and let T gr ⊆ ( X × Y ) denotethe collection of triangles which are thin in X ⊗ Y , as described in Definition 4.1.1.Let T − ⊆ T gr denote the subset of those triangles σ = ( σ X , σ Y ) ∈ T gr for which eitherboth σ X and σ Y are degenerate or at least one of σ X , σ Y degenerates to a point .On the other hand, let T + be the set of those triangles ( σ X , σ Y ) ∶ ♭ ∆ → X × Y forwhich both σ X and σ Y are thin and such that either σ X ∣ ∆ { , } is marked or σ Y ∣ ∆ { , } is marked. Then we have a sequence of inclusions T − ⊆ T gr ⊆ T + . Proposition 4.1.8.
Let ( X, E X , T X ) and ( Y, E Y , T Y ) be two marked-scaled simpli-cial sets and let T gr be the collection of thin triangles in ( X, E X , T X ) ⊗ ( Y, E Y , T Y ) .Then the maps (27) ( X × Y, T − ) ↪ ( X × Y, T gr ) ↪ ( X × Y, T + ) are bicategorical trivial cofibrations. IBRATIONS AND LAX LIMITS OF (∞ , ) -CATEGORIES 41 Proof.
We will show that for every triangle σ ∈ T + there is a 3-simplex ρ ∶ ∆ → X × Y and an i ∈ { , } such that η ∣ ∆ { ,i, } = σ while the three other faces of ρ lie in T gr .This will imply that the second map in (27) is a sequence of pushouts along mapsof the form ( ∆ , T i ) → ( ∆ ) ♯ , where T i denotes all triangles except ∆ { ,i, } , and ishence scaled anodyne by [13, Remark 3.1.4]. We will then apply the same argumentto show that the first map in (27) is scaled anodyne.Given a triangle σ = ( σ X , σ Y ) ∶ ∆ → X × Y , let us denote by ρ i,jσ = ( s i σ X , s j σ Y ) ∶ ∆ → X × Y the 3-simplex in X × Y whose X component is the degenerate 3-simplex obtained bypre-composing σ X with the surjective map [ ] → [ ] hitting i ∈ [ ] twice, and whose Y component is obtained by pre-composing σ Y with the surjective map [ ] → [ ] hitting j twice. We note in particular that d ρ , σ = d ρ , σ = σ and d ρ , σ = d ρ , σ = σ. Now suppose that σ belongs to T + . Then σ X and σ Y are thin, and either σ X ∣ ∆ { , } is marked in X or σ Y ∣ ∆ { , } is marked in Y . If σ X ∣ ∆ { , } is marked in X then the3-simplex ρ , σ has the property that all its faces are in T gr except possibly its faceopposite the vertex 2, which is σ . Similarly, if σ Y ∣ ∆ { , } is marked in Y then ρ , σ hasthe property that all its faces are in T gr except possibly its face opposite the vertex1, which is again σ . We may hence conclude that the map ( X × Y, T gr ) ↪ ( X × Y, T + ) is scaled anodyne.Now suppose that σ belongs to T gr . Then σ X and σ Y are thin, and either σ X isdegenerate and σ X ∣ ∆ { , } is marked or σ Y is degenerate and σ Y ∣ ∆ { , } is marked in Y . We now separate into four cases:(1) If σ X degenerates along ∆ { , } then the 3-simplex ρ , σ has the property thatall its faces are in T − except possibly its face opposite 1, which is σ .(2) If σ X degenerates along ∆ { , } and σ X ∣ ∆ { , } is marked and then the 3-simplex ρ , σ has the property that all its faces are in T − except possibly its face opposite2, which is σ .(3) If σ Y degenerates along ∆ { , } then the 3-simplex ρ , σ has the property thatall its faces are in T − except possibly its face opposite 2, which is σ .(4) If σ Y degenerates along ∆ { , } and σ Y ∣ ∆ { , } is marked and then the 3-simplex ρ , σ has the property that all its faces are in T − except possibly its face opposite1, which is σ .We may hence conclude that the map ( X × Y, T − ) ↪ ( X × Y, T gr ) is scaled anodyne,and so the proof is complete. (cid:3) The following proposition extends [5, Proposition 2.16]:
Proposition 4.1.9.
Let f ∶ X → Y be a monomorphism of marked-scaled simplicialsets and let g ∶ Z → W be a scaled anodyne map of scaled simplicial sets . Then thepushout-products f ✷ gr g ♭ ∶ [ X ⊗ W ♭ ] ∐ X ⊗ Z ♭ [ Y ⊗ Z ♭ ] → Y ⊗ W ♭ and g ♭ ✷ gr f ∶ [ W ♭ ⊗ X ] ∐ Z ♭ ⊗ X [ Z ♭ ⊗ Y ] → W ♭ ⊗ Y are scaled anodyne maps.Proof. To prove this statement we can assume that that g is one of the generat-ing scaled anodyne maps appearing in Definition 1.2.6 and that f is either theinclusion ♭ ( ∂ ∆ n ) ↪ ♭ ∆ n for n ≥
0, the inclusion ♭ ∆ ↪ ♯ ∆ , or the inclusion ♭ ∆ ⊆ ( ∆ , ∅ , { ∆ }) . Now the first and last cases follow, in light of Remark 4.1.3,from the unmarked analogue of the present proposition, see [5, Proposition 2.16].On the other hand, if f is the inclusion ♭ ∆ ↪ ♯ ∆ , then f ✷ gr g ♭ and g ♭ ✷ gr f areisomorphisms except when g is the inclusion ( Λ ) ♭ ⊆ ∆ ♯ . In this last case, the map g ♭ ✷ gr f identifies with the inclusion ( ∆ × ∆ , T ) ⊆ ( ∆ × ∆ ) ♯ , where T is the collection of all triangles except∆ {( , ) , ( , ) , ( , )} and ∆ {( , ) , ( , ) , ( , )} . To see that this is scaled anodyne, it suffices by [13, Remark 3.1.4] to note that the3-simplex ρ ∶ ∆ → ∆ × ∆ spanned by the vertices ( , ) , ( , ) , ( , ) , ( , ) has theproperty that all its faces except the one opposite ( , ) are in T , while the faceopposite ( , ) is ∆ {( , ) , ( , ) , ( , )} , while the 3-simplex ρ ∶ ∆ → ∆ × ∆ spanned bythe vertices ( , ) , ( , ) , ( , ) , ( , ) has the property that all its faces except theone opposite ( , ) are in T ∪ { ∆ {( , ) , ( , ) , ( , )} } , while the face opposite ( , ) is∆ {( , ) , ( , ) , ( , )} . The case of f ✷ gr g ♭ admits a completely analogous argument. (cid:3) Corollary 4.1.10.
For every marked-scaled simplicial set X the functors X ⊗ ( − ) ♭ ∶ S et sc∆ → S et sc∆ and ( − ) ♭ ⊗ X ∶ S et sc∆ → S et sc∆ are left Quillen functors with respect to the bicategorical model structure.Proof. It is straightforward to verify that the functors in question preserve colimitsand monomorphisms, and so it is left to verify that they preserve trivial cofibrations.We prove this for the first functor, the proof for the second one proceeds in acompletely analogous manner. In light of Proposition 4.1.9 and the collection ofgenerating trivial cofibrations established in [6], it will suffice to check that for everymarked-scaled simplicial set X the map X = X ⊗ { } → X ⊗ J ♭♯ is a bicategorical equivalence, where J = cosk ({ , }) is the nerve of the walkingisomorphism and X is the underlying scaled simplicial set of X . Then X ⊗ J ♭♯ and X ⊗ J ♯ have isomorphic underlying simplicial sets, with the former having potentiallymore thin triangles than the latter. However, since J is a Kan complex [5, Corollary2.17] tells us that the map(28) X ⊗ J ♯ → X × J ♯ from the Gray product to the cartesian product is a trivial cofibration. Since thethin triangles of X ⊗ J ♭♯ are also thin in X × J ♯ it follows that the map X ⊗ J ♭♯ → X × J ♯ is a pushout of (28), and is hence a trivial cofibration as well. It will hence sufficeto check that X × { } → X × J ♯ is a trivial cofibration, which is a consequence ofthe bicategorical model structure being cartesian. (cid:3) Notation 4.1.11.
For a marked-scaled simplicial set X , we denote by Y ↦ Fun gr ( X, Y ) and Y ↦ Fun opgr ( X, Y ) the right adjoints of the left Quillen functors of Corollary 4.1.10. To avoid confusionwe point out that while X is a marked-scaled simplicial set, Y ,Fun gr ( X, Y ) and IBRATIONS AND LAX LIMITS OF (∞ , ) -CATEGORIES 43 Fun opgr ( X, Y ) are scaled simplicial sets. Explicitly, an n -simplex of Fun gr ( X, Y ) isgiven by a map of scaled simplicial sets ♭ ∆ n ⊗ X → Y and a 2-simplex ♭ ∆ ⊗ X → Y is thin if it factors through ♯ ∆ ⊗ X . Similarly, an n -simplex of Fun opgr ( X, Y ) is given by a map of scaled simplicial sets X ⊗ ♭ ∆ n → Y and the scaling is determined as above. Remark . It follows from Remark 4.1.3 that if
X, Y are scaled simplicialsets then Fun gr ( X ♭ , Y ) and Fun opgr ( X ♭ , Y ) coincide with the scaled simplicial setsFun gr ( X, Y ) and Fun opgr ( X, Y ) recalled in § Remark . By Remark 4.1.6 we have natural isomorphismsFun gr ( X op , Y op ) ≅ Fun opgr ( X, Y ) op and Fun opgr ( X op , Y op ) ≅ Fun gr ( X, Y ) op Being right adjoints to left Quillen functors, the functorsFun gr ( K, − ) , Fun opgr ( K, − ) ∶ S et sc∆ → S et sc∆ are right Quillen functors for any scaled simplicial set K with respect to the bicat-egorical model structure. In particular, if C is an ∞ -bicategory then Fun gr ( K, C ) and Fun opgr ( K, C ) are ∞ -bicategories. The objects of the ∞ -bicategory Fun gr ( K, C ) correspond to functors K → C , where we recall from Definition 1.5.3 that K standsfor the underlying scaled simplicial set of K . If all the edges in K are marked thenFun gr ( K, C ) ≃ Fun opgr ( K, C ) and both coincide with the ∞ -bicategory Fun ( K, C ) of functors K → C . On the other hand, if only the degenerate edges in K aremarked then, as in § gr ( K, C ) correspond to lax naturaltransformations . Dually, in the case of Fun opgr ( K, C ) we obtain functors and oplaxnatural transformations between them.4.2. The thick join and slice constructions.Definition 4.2.1.
Let X and Y be two marked-scaled simplicial sets. We definethe inner thick join X ◇ inn Y ∈ S et sc∆ by the formula X ◇ inn Y = X ∐ X ⊗ ∆ { } ⊗ Y ( X ⊗ ♭ ∆ ⊗ Y ) ∐ X ⊗ ∆ { } ⊗ Y Y , and the outer thick join X ◇ out Y ∈ S et sc∆ by the formula X ◇ out Y = X ∐ Y ⊗ ∆ { } ⊗ X ( Y ⊗ ♭ ∆ ⊗ X ) ∐ Y ⊗ ∆ { } ⊗ X Y .
Here, we use triple Gray products as in Definition 4.1.1. In particular, the input ofthe thick join consists of marked-scaled simplicial sets, while its output is a scaledsimplicial set.For a fixed marked-scaled simplicial K with underlying scaled simplicial set K ,we may consider the assignment X ↦ X ◇ inn K ( resp . X ↦ X ◇ out K ) as a functor Set + , sc∆ → Set sc K / . As such, it is a colimit preserving functor which admits a right adjointSet sc K / → Set + , sc∆ by the adjoint functor theorem. Given a map f ∶ K → S of scaled simplicial set,considered as an object of Set sc K / , we will denote by S / f inn (resp. S / f out ) the marked-scaled simplicial set obtained by applying the above right adjoint. In particular, themarked-scaled simplicial sets S / f inn and S / f out are characterized by mapping propertiesof the form Hom S et + , sc∆ ( X, S / f inn ) ≅ Hom ( S et sc∆ ) K / ( X ◇ inn K, S ) and Hom S et + , sc∆ ( X, S / f out ) ≅ Hom ( S et sc∆ ) K / ( X ◇ out K, S ) , respectively. In a similar manner, we may consider the right adjoint to the functorSet + , sc∆ → Set sc K / given by the assignment X ↦ K ◇ inn X ( resp . X ↦ K ◇ out X ) . For any map f ∶ K → S of scaled simplicial set, considered as an object of Set sc K / ,we will denote by S f / inn (resp. S f / out ) the marked-scaled simplicial set obtained byapplying this right adjoint. In particular, the marked-scaled simplicial sets S f / inn and S f / out are characterized by mapping properties of the formHom S et + , sc∆ ( X, S f / inn ) ≅ Hom ( S et sc∆ ) K / ( K ◇ inn X, S ) and Hom S et + , sc∆ ( X, S f / out ) ≅ Hom ( S et sc∆ ) K / ( K ◇ out X, S ) , respectively. We will then denote by S / f inn , S / f out , S f / inn and S f / out the underlying scaledsimplicial sets of S / f inn , S / f out , S f / inn and S f / out , respectively. Remark . By Remark 4.1.6 we have canonical isomorphisms ( X ◇ inn Y ) op ≅ Y op ◇ inn X op and ( X ◇ out Y ) op ≅ Y op ◇ out X op . As a result, if f ∶ K → S is a map of scaled simplicial sets then we have canonicalisomorphisms ( S f / inn ) op ≅ ( S op ) / f op inn and ( S f / out ) op ≅ ( S op ) / f op out . Remark . It follows from Corollary 4.1.10 that for a fixed marked-scaled sim-plicial set K the functors ( − ) ♭ ◇ inn K ∶ S et sc∆ → ( S et sc∆ ) K / ( − ) ♭ ◇ out K ∶ S et sc∆ → ( S et sc∆ ) K / K ◇ inn ( − ) ♭ ∶ S et sc∆ → ( S et sc∆ ) K / and K ◇ out ( − ) ♭ ∶ S et sc∆ → ( S et sc∆ ) K / , are left Quillen functors, where ( S et sc∆ ) K / is considered with the slice model struc-ture associated to the bicategorical model structure. It then follows that their rightadjoints ( S et sc∆ ) K / ∋ [ f ∶ K → S ] ↦ S / f inn ∈ S et sc∆ ( S et sc∆ ) K / ∋ [ f ∶ K → S ] ↦ S / f out ∈ S et sc∆ , ( S et sc∆ ) K / ∋ [ f ∶ K → S ] ↦ S f / inn ∈ S et sc∆ and ( S et sc∆ ) K / ∋ [ f ∶ K → S ] ↦ S f / out ∈ S et sc∆ are right Quillen functors. IBRATIONS AND LAX LIMITS OF (∞ , ) -CATEGORIES 45 It follows from Remark 4.2.3 that if C is an ∞ -bicategory, K is a marked-scaled simplicial set and f ∶ K → C is a diagram, then C / f inn , C / f out , C f / inn and C f / out are ∞ -bicategories. If all the edges in K are marked then the objects of these ∞ -bicategories correspond to pseudo-natural cones on f , while if only the degenerateedges are marked they correspond to lax (or op-lax) cones. In general, we may con-sider them as partially lax cones , with the amount of “pseudo-naturality” encodedby the collection of marked edges. Regardless of the marked edges, the morphismsin these slice ∞ -bicategories always correspond to lax (or op-lax) transformationof cones, with the marked edges in C / f inn , C / f out , C f / inn and C f / out respectively indicatingthose lax transformation which are pseudo-natural. Example . Let C be an ∞ -bicategory. When K = ∆ and f ∶ ∆ → C is theinclusion of the vertex x ∈ C then we will denote C f / inn and C f / out by C x / inn and C x / out respectively, and similarly for C / x inn and C / x out . In this case the vertices of C x / inn and C x / out are just arrows x → y of C with source x . An edge in C x / inn from x → y to x → z is a diagram x yx z ≃ . On the other hand, an edge of C x / out from x → y to x → z is a diagram x yx z ≃ . Example . When K = ♭ ∆ and f ∶ ♭ ∆ → C is given by an edge e ∶ x → y in C then we will denote C f / inn and C f / out by C e / inn and C e / out respectively, and similarly for C / e inn and C / e out . The vertices of C e / inn are then given by diagrams of the form x zy z e ≃ , while the vertices of C e / out are diagrams of the form x zy z e ≃ . Remark . Let C be a ∞ -bicategory. For any vertex x in C the underlyingmarked simplicial set of C x / inn is isomorphic to the marked simplicial set denoted by C x / in [13, Notation 4.1.5]. In particular, the fiber ( C x / inn ) y of the projection C x / inn → C over y ∈ C is the marked simplicial set Hom C ( x, y ) used in [13] as a model for themapping ∞ -category from x to y , see Notation 1.2.13. The same holds for the fiberof C / y out over x (since these two fibers are naturally isomorphic). By Remark 4.2.2these fibers are also isomorphic to (( C op ) / x inn ) op y and (( C op ) y / out ) op x . Similarly, we havenatural isomorphisms ( C x / out ) y ≅ ( C / y inn ) x ≅ (( C op ) y / inn ) op x ≅ (( C op ) / x out ) op y ≅ Hom C op ( y, x ) op and the latter is categorically equivalent (though generally not isomorphic) as amarked simplicial set to Hom C ( x, y ) op by Remark 1.2.14. Remark . The thick join of Definition 4.2.1 can be used to express the coneconstruction appearing in the definition of the scaled straightening functor, see § ( S, T S ) is a scaled simplicial set, φ ∶ C sc ( S, T S ) → C a Dwyer-Kanequivalence, ( X, E X ) is a marked simplicial set and f ∶ X → S is a map, then [ St φ ( X, E X )]( v ) = Cone φ ( X, E X )( ∗ , v ) , where Cone φ ( X, E X ) can be described in terms of the thick join asCone φ ( X, E X ) = C sc ( ∆ ◇ inn ( X, E X , ∅ )) ∐ C sc ( X ♭ ) D . It then follows, for example, that if D is a fibrant S et + ∆ -enriched category and x ∈ D is an object, then the underlying marked simplicial set of N sc ( D ) x / inn is naturallyisomorphic to the unstraightening with respect to the counit map ψ ∶ C sc ( N sc ( D )) → D of the functor D ( x, − ) ∶ D → S et + ∆ corepresented by D . Indeed, for a marked simpli-cial set ( X, E X ) and a map p ∶ X ♭ → N sc ( D ) with adjoint p ad ∶ C sc ( X ♭ ) → D , we getfrom the above that enriched natural transformations S t sc ψ ( X, E X ) → D ( x, − ) arein bijection with enriched functors C sc ( ∆ ◇ inn ( X, E X , ∅ )) → D which restrict to p ad on C sc ( X ♭ ) and send the cone point to x . By adjunction, thesecorrespond to maps ∆ ◇ inn ( X, E X , ∅ ) → N sc ( D ) extending p and sending the conepoint to x , and hence to maps ( X, E X , ∅ ) → N sc ( D ) x / inn over D .Let us now compare the outer thick join construction to the standard join con-struction. Proposition 4.2.8.
For marked-scaled simplicial sets
X, Y let (29) r ∶ X ◇ out Y → X ∗ Y be the unique map which is compatible with the canonical inclusions X ↪ X ◇ out Y ↩ Y and X ↪ X ∗ Y ↩ Y . Then r is a bicategorical equivalence. Before we give the proof of Proposition 4.2.8 let us take a minute to verifythat the map r is indeed well-defined. First, on the level of underlying simplicialsets, note that n -simplices of X ∗ Y corresponds to a triple ( i, σ i − , σ n − i ) where i ∈ { , ..., n + } corresponds to a choice of a partition [ n ] = { , ..., i − } ∗ { i, ..., n } (the twoextreme options corresponding to partitions in which one part is empty), σ i − is an { , ..., i − } -simplex of X , and σ n − i is an { i, ..., n } -simplex of Y . By convention theset of {} -simplices of any simplicial set is a singleton, and these extreme partitionscorrespond to the simplices of X ∗ Y which are in the image of the inclusions X ↪ X ∗ Y ↩ Y . On the other hand, an n -simplex of X ◇ out Y is given by anequivalence class of triples ( ρ Y , τ, ρ X ) , of n -simplices of Y , ∆ , and X , respectively.The n -simplex τ ∶ ∆ n → ∆ then determines a partition [ n ] = τ − ( ) ∗ τ − ( ) , andthe map r sends ( ρ Y , τ, ρ X ) to ( min ( τ − ( )) , ρ X ∣ τ − ( ) , ρ Y ∣ τ − ( ) ) . One may thenverify that this is the only option that is compatible with the simplicial face anddegeneracy maps, and that behaves in the prescribed manner when the partitionis one of the two extreme cases. It is clear from this description that the map r issurjective on n -simplices for every n .Let us now verify that this map sends thin triangles of X ◇ out Y to thin triaglesof X ∗ Y . Unwinding the definitions we see that a triangle given by a class of atriple of triangles ( ρ Y , τ, ρ X ) is thin in X ◇ out Y if and only if one of the followingpossibilities hold: ● τ sends all vertices to 0 and ρ X is thin in X . IBRATIONS AND LAX LIMITS OF (∞ , ) -CATEGORIES 47 ● τ sends all vertices to 1 and ρ Y is thin in Y . ● τ is given on vertices by 0 , , ↦ , , ρ Y is thin, ρ X is degenerate and ρ X ∣ ∆ { , } is marked in X . ● τ is given on vertices by 0 , , ↦ , , ρ X is thin, ρ Y is degenerate and ρ Y ∣ ∆ { , } is marked in Y .Examining Definition 2.1.13 we then see that in all these cases the associated tri-angle ( min ( τ − ( )) , ρ X ∣ τ − ( ) , ρ Y ∣ τ − ( ) ) is thin in X ∗ Y . In addition, one directlyverifies that the map r is surjective on thin triangles.The proof of Proposition 4.2.8 will require the next lemma. In what follows, forintegers p, q ≥ p ◇ out ∆ q ∶ = ∆ p ∐ ∆ q × ∆ { } × ∆ p ( ∆ q × ∆ × ∆ p ) ∐ ∆ q × ∆ { } × ∆ p ∆ q for the underlying simplicial set of ♭ ∆ p ◇ out ♭ ∆ q . Lemma 4.2.9.
Let T denote the collection of triangles in ∆ p ◇ out ∆ q which arethin in ♭ ∆ p ◇ out ♭ ∆ q , and T ′ the collection of all triangles in ∆ p ◇ out ∆ q whose imagein ∆ p ∗ ∆ q is degenerate. Then T ⊆ T ′ and the inclusion (30) ( ∆ p ◇ out ∆ q , T ) ⊆ ( ∆ p ◇ out ∆ q , T ′ ) is scaled anodyne.Proof. We will denote the vertices of ∆ p ◇ out ∆ q by triples [ v, ε, u ] with v ∈ [ q ] , ε ∈ [ ] , u ∈ [ p ] , under the equivalence relation in which [ v, , u ] ∼ [ v ′ , , u ] for every v, v ′ ∈ [ q ] , u ∈ [ p ] and [ v, , u ] ∼ [ v, , u ′ ] for every v ∈ [ q ] , u, u ′ ∈ [ p ] . Using theidentification ∆ p ∗ ∆ q ≅ ∆ p + q + on the level of the underlying simplicial sets, themap ∆ p ◇ out ∆ q → ∆ p ∗ ∆ q can be written on vertices by r ([ v, ε, u ]) = ⎧⎪⎪⎨⎪⎪⎩ u ε = v + p + ε = ♭ ∆ p ∗ ♭ ∆ q are degenerate, while the non-degenerate thin triangles of ♭ ∆ p ◇ out ♭ ∆ q are given by the classes of those triples ( ρ ∆ q , τ, ρ ∆ p ) such that τ is surjective, ρ ∆ q degenerates along ∆ { , } and ρ ∆ p degen-erates along ∆ { , } . In particular, these all map to degenerate triangles in ∆ p ∗ ∆ q ,and so we have T ⊆ T ′ . On the other hand, the non-degenerate triangles in T ′ aregiven by the classes of those triples ( ρ ∆ q , τ, ρ ∆ p ) such that τ is surjective, and suchthat either both τ and ρ ∆ q degenerate along ∆ { , } or both τ and ρ ∆ p degeneratealong ∆ { , } .We will now show that for every non-degenerate triangle σ ∈ T ′ there is a 3-simplex η ∶ ∆ → ∆ p ◇ out ∆ q and an i ∈ { , } such that η ∣ ∆ { ,i, } = σ while the threeother faces of η lie in T . This will imply that (30) is a sequence of pushouts alongmaps of the form ( ∆ , T i ) → ( ∆ ) ♯ , where T i denotes all triangles except ∆ { ,i, } ,and is hence scaled anodyne by [13, Remark 3.1.4]. Now if we take a triangle in T ′ of the form ( ρ ∆ q , τ, ρ ∆ p ) such that both τ and ρ ∆ q degenerate along ∆ { , } then welet η be the 3-simplex ( ρ ∆ q ○ α, τ ○ β, ρ ∆ p ○ γ ) , where α ∶ ∆ → ∆ is given on verticesby 0 , , , ↦ , , , β ∶ ∆ → ∆ is given on vertices by 0 , , , ↦ , , , γ ∶ ∆ → ∆ is given on vertices by 0 , , , ↦ , , ,
2. Notice that the restriction of ρ ∆ p ○ γ to ∆ { , , } is quotiented to the point in ∆ p ◇ out ∆ q . Similarly, if we take atriangle in T ′ of the form ( ρ ∆ q , τ, ρ ∆ p ) such that both τ and ρ ∆ p degenerate along∆ { , } then we let η be the 3-simplex ( ρ ∆ q ○ α, τ ○ β, ρ ∆ p ○ γ ) , where α ∶ ∆ → ∆ is given on vertices by 0 , , , ↦ , , , β ∶ ∆ → ∆ is given on vertices by0 , , , ↦ , , , γ ∶ ∆ → ∆ is given on vertices by 0 , , , ↦ , , ,
2. Noticethat the restriction of ρ ∆ q ○ α to ∆ { , , } is quotiented to the point in ∆ p ◇ out ∆ q . (cid:3) Proof of Proposition 4.2.8.
Let us say that a pair ( X, Y ) is good if the map (29) isan equivalence. We then observe that for a fixed X the operations Y ↦ X ◇ out Y and Y ↦ X ∗ Y preserve monomorphisms, pushout squares and filtered colimits. On theother hand, pushout squares with parallel legs cofibrations are always homotopypushout squares (since all objects in S et sc∆ are cofibrant), and filtered colimits arealways homotopy colimits since bicategorical equivalences are closed under filteredcolimits. We then conclude that for a fixed X , the collection of Y for which ( X, Y ) is good is closed under pushouts with parallel legs cofibrations and filtered colimits.It will hence suffice to prove the claim for Y = ♭ ∆ q , Y = ∆ ♯ and Y = ( ∆ ) ♯ . Applyingthis argument for X instead of Y we may equally suppose that X is either ♭ ∆ p , ∆ ♯ or ( ∆ ) ♯ . We now observe that for every marked-scaled simplicial set there arepushout squares of scaled simplicial sets X ◇ out ♭ ∆ (cid:15) (cid:15) / / X ◇ out ( ∆ ♯ ) ♭ (cid:15) (cid:15) X ∗ ♭ ∆ / / X ∗ ( ∆ ♯ ) ♭ X ◇ out ♭ ∆ (cid:15) (cid:15) / / X ◇ out ( ∆ ♭ ) ♯ (cid:15) (cid:15) X ∗ ♭ ∆ / / X ∗ ( ∆ ♭ ) ♯ . Indeed, since the horizontal maps are isomorphisms on the level of underlying sim-plicial sets, this follows from the fact that the vertical maps are surjective on thintriangles. These squares are then also homotopy pushout squares with respect tothe bicategorical model structure since they have parallel legs cofibrations and allobjects cofibrant. We may consequently assume without loss of generality that X = ♭ ∆ p and Y = ♭ ∆ q . In light of Lemma 4.2.9 it will now suffice to show that themap ( ∆ p ◇ out ∆ q , T ′ ) → ( ∆ p ∗ ∆ q ) ♭ is a bicategorical equivalence.As in the proof of Lemma 4.2.9 we will denote the vertices of ∆ p ◇ out ∆ q by triples [ v, ε, u ] with v ∈ [ q ] , ε ∈ [ ] , u ∈ [ p ] , under the equivalence relation [ v, , u ] ∼ [ v ′ , , u ] for every v, v ′ ∈ [ q ] , u ∈ [ p ] and [ v, , u ] ∼ [ v, , u ′ ] for every v ∈ [ q ] , u, u ′ ∈ [ p ] , sothat the map ∆ p ◇ out ∆ q → ∆ p ∗ ∆ q can be written on vertices by r ([ v, ε, u ]) = ⎧⎪⎪⎨⎪⎪⎩ u , ε = ,v + p + , ε = . We now define maps of scaled simplicial sets ( ∆ q × ∆ × ∆ p , T ′′ )( ∆ p ∗ ∆ q ) ♭ ( ∆ p ◇ out ∆ q , T ′ ) ̃ s s where T ′′ is the preimage of T ′ and ̃ s is induced by the order preserving map onvertices ̃ s ( i ) = ⎧⎪⎪⎨⎪⎪⎩[ , , i ] , i ≤ p, [ i − p − , , p ] , i > p. It is immediate to check that rs = id. On the other hand, if we denote by ̃ r ∶ ( ∆ q × ∆ × ∆ p , T ′′ ) → ( ∆ p ◇ out ∆ q , T ′ ) → ( ∆ p ∗ ∆ q ) ♭ so that the composite ̃ s ○ ̃ r is given on vertices by the order preserving map ̃ s ○ ̃ r ([ v, ε, u ]) = ⎧⎪⎪⎨⎪⎪⎩[ , , u ] , ε = , [ v, , p ] , ε = . We will now exhibit a zig-zag of natural transformations from sr to the identity.More precisely, we construct a map of scaled simplicial sets u ∶ ( ∆ p ◇ out ∆ q , T ′ ) → ( ∆ p ◇ out ∆ q , T ′ ) IBRATIONS AND LAX LIMITS OF (∞ , ) -CATEGORIES 49 and a pair of transformations h, k ∶ ( ∆ p ◇ out ∆ q , T ′ ) × ∆ ♭ → ( ∆ p ◇ out ∆ q , T ′ ) such that h ∣{ } = k ∣{ } = u, h ∣{ } = sr and k ∣{ } = id . Furthermore, these transformations will satisfy the additional property that h ∣ x × ∆ and k ∣ x × ∆ are degenerate in ∆ p ◇ out ∆ q for every vertex x of ∆ p ◇ out ∆ q . Themap (29) is then a bicategorical equivalence by [6, Corollary 7.8].We now finish the proof by constructing h, k and u as above. For u , we define itto be the map induced on quotients by the endomorphism ̃ u of ( ∆ q × ∆ × ∆ p , T ′′ ) given on vertices by the order preserving map ̃ u ([ v, ε, u ]) = ⎧⎪⎪⎨⎪⎪⎩[ , , u ] , ε = , [ v, , u ] , ε = . This order preserving map satisfies pointwise the inequalities ̃ u ([ v, ε, u ]) ≤ [ v, ε, u ] and ̃ u ([ v, ε, u ]) ≤ ̃ s ̃ r ([ v, ε, u ]) . Therefore there are natural transformations of sim-plicial sets ̃ h, ̃ k ∶ ( ∆ q × ∆ × ∆ p ) × ∆ → ∆ q × ∆ × ∆ p with ̃ h ∣{ } = ̃ k ∣{ } = u and ̃ h ∣{ } = ̃ s ○ ̃ r , ̃ k ∣{ } = id. Both these homotopies have theproperty that when projected down to ∆ p ∗ ∆ q they yield the identify transformationfrom r to itself. Since T ′ consists by definition of those triangles whose image in∆ p ∗ ∆ q is degenerate we see that ̃ h and ̃ k refine to scaled maps ̃ h, ̃ k ∶ ( ∆ q × ∆ × ∆ p , T ′′ ) × ∆ ♭ → ( ∆ q × ∆ × ∆ p , T ′′ ) . Finally, by direct inspection they also pass to the quotient, so we get the desiredmaps h, k ∶ ( ∆ p ◇ out ∆ q , T ′ ) × ∆ ♭ → ( ∆ p ◇ out ∆ q , T ′ ) . We are left with checking that both h and k are constant along all edges of theform w × ∆ for w ∈ ∆ p ◇ out ∆ q . We have four distinct cases to analyze: ● the edge ̃ h ∣[ v, ,u ]× ∆ is the degenerate edge on [ , , u ] ; ● the edge ̃ h ∣[ v, ,u ]× ∆ is [ v, , u ] → [ v, , p ] , whose image in ∆ p ◇ out ∆ q is degenerate; ● the edge k ∣[ v, ,u ]× ∆ is the edge [ , , u ] → [ v, , u ] , whose image in ∆ p ◇ out ∆ q isdegenerate; ● the edge k ∣[ v, ,u ]× ∆ is the degenerate edge on [ v, , u ] .We may finally conclude that the map r is bicategorical equivalence. (cid:3) As a first corollary of Proposition 4.2.8 we obtain the following analogue ofLemma 4.2.9 for the thick outer join:
Corollary 4.2.10.
Let f ∶ X → Y and g ∶ A → B be injective map of marked-scaledsimplicial sets. If either f is outer cartesian anodyne or g is outer cocartesiananodyne then the map of scaled simplicial sets (31) [ X ◇ out B ] ∐ X ◇ out A [ Y ◇ out A ] → Y ◇ out B is a bicategorical trivial cofibration.Proof. The map (31) is clearly a cofibration and so it will suffice to show that it is abicategorical equivalence. This follows from the analogous claim for the ∗ -pushout-product in Lemma 2.4.6, together with the comparison of Proposition 4.2.8. (cid:3) Corollary 4.2.11.
Let C be an ∞ -bicategory and f ∶ K → C a map of scaled simpli-cial sets. Then we have a bicategorical equivalence C / f C / f out C ≃ p q of ∞ -bicategories over C . In addition, q is an outer cartesian fibration which isfiberwise equivalent to p . Similarly, we have a bicategorical equivalence of the form C f / C f / out C ≃ p q which is also a fiberwise equivalence of outer cocartesian fibrations over C .Proof. By Remark 4.2.3 for every marked-scaled simplicial set K the functors ( − ) ♭ ◇ out K ∶ S et sc∆ → ( S et sc∆ ) K / and K ◇ out ( − ) ♭ ∶ S et sc∆ → ( S et sc∆ ) K / are left Quillen functors. On the other hand, the functors ( − ) ♭ ∗ K ∶ S et sc∆ → S et sc∆ and K ∗ ( − ) ♭ ∶ S et sc∆ → S et sc∆ preserves colimits and cofibrations, and hence also trivial cofibration by the com-parison of ◇ and ∗ of Proposition 4.2.8. We may hence consider the natural transfor-mation appearing in that proposition as a transformation between two left Quillenfunctors, which is then shown to be a levelwise weak equivalence. By [12, Corol-lary 1.4.4(b)] we may conclude that the adjoint transformations C / f → C / f out and C f / → C f / out between the corresponding right adjoints are bicategorical equivalenceswhenever C is an ∞ -bicategory. In light of Corollary 2.2.2 it will now suffice to showthat C / f out → C is an outer cartesian fibration and C f / out → C is an outer cocartesianfibration. Both these claims follow as in Corollary 2.4.7 from Corollary 4.2.10. (cid:3) Proposition 4.2.12.
Let K be a marked-scaled simplicial set, C an ∞ -bicategoryand f ∶ K → C a map of scaled simplicial sets. Then the maps C / f inn → C and C f / inn → C are inner cartesian and cocartesian fibrations respectively.Proof. We prove the claim for C f / inn . The case of C / f inn then follows by taking oppo-sites, see Remark 4.2.2. Let [ f ] ∶ ∆ → Fun opgr ( K, C ) be the map corresponding to f . By Remarks 4.1.3 and 4.1.4 we have for a scaled simplicial set Z an isomorphism K ⊗ ( ∆ ♭ ⊗ Z ) ♭ ≅ K ⊗ ♭ ∆ ⊗ Z ♭ , which induces an isomorphism K ⊗ ( ∆ ◇ inn Z ) ♭ ∐ K ⊗ Z ♭ Z ≅ K ◇ inn Z ♭ of functors S et sc∆ → ( S et sc∆ ) K / natural in Z . Passing to right adjoints, we obtain apullback square C f / inn Fun opgr ( K, C ) [ f ]/ inn C Fun opgr ( K, C ) where the top right corner stands for the underlying scaled simplicial set of themarked-scaled simplicial set Fun opgr ( K, C ) [ f ]/ inn . Replacing C with Fun opgr ( K, C ) we IBRATIONS AND LAX LIMITS OF (∞ , ) -CATEGORIES 51 may hence assume without loss of generality that K = ∆ and f ∶ K → C is given bya vertex x ∈ C .By Remark 4.2.6 and [13, Proposition 4.1.6] the underlying marked simplicial setof C x / inn constitutes a P C -fibrant object of ( S et + ∆ ) / C . Applying Proposition 2.4.1 andRemark 2.4.2 it will hence suffice to prove that the thin triangles in C x / inn are exactlythose whose image in C is thin. This follows from Lemma 4.2.13 just below. (cid:3) Lemma 4.2.13.
Let T denote the collection of those triangles in ∆ × ∆ whichare either thin in ∆ ♭ ⊗ ∆ ♭ or are contained in ∂ ∆ × ∆ . Then the inclusion ( ∆ × ∆ , T ) → ∆ ♭ ⊗ ∆ ♯ is scaled anodyne.Proof. Direct inspection shows that the only thin 2-simplex of ∆ ⊗ ∆ ♯ which isnot in T is the triangle σ = ∆ {( , ) , ( , ) , ( , )} . Let ρ ∶ ∆ → ∆ × ∆ be the 3-simplexspanned by ( , ) , ( , ) , ( , ) , ( , ) . Then ρ sends the triangles ∆ { , , } , ∆ { , , } and ∆ { , , } to triangles in T , while ∆ { , , } maps to σ . The desired map is thenscaled anodyne by [13, Remark 3.1.4]. (cid:3) Representable fibrations.
Given an ∞ -bicategory C , we may then constructa model for the Yoneda embedding of C by picking a fibrant S et + ∆ -category D equipped with a bicategorical equivalence η ∶ C ≃ Ð→ N sc ( D ) , and considering the com-posed functor j C ∶ C op ≃ Ð→ N sc ( D op ) N sc ( j D ) ÐÐÐÐ→ N sc [( S et + ∆ ) D ] ○ ≃ Ð→ Fun ( C , C at ∞ ) , where j D ∶ D op → ( S et + ∆ ) D is the enriched Yoneda embedding of C (which takesvalues in N sc [( S et + ∆ ) D ] ○ when D is fibrant), and the last map is the bicategoricalequivalence of Proposition 3.3.1. The functor j C can then be morally described assending x ∈ C to the functor Hom C ( x, − ) ∶ C op → C at ∞ corepresented by x . Since j D is fully-faithful in the enriched sense we have that j C is fully-faithful in thebicategorical sense.By definition, the Yoneda image j C ( x ) of an object x is the vertex of Fun ( C , C at ∞ ) determined by the enriched functor j C ( η ( x )) = D ( η ( x ) , − ) ∶ D → S et + ∆ . We may alsoencode the latter by unstraightening it to an inner cocartesian fibration over C . Bythe second part of Remark 4.2.7 and the compatibility of unstraightening with basechange we see that the this inner cocartesian fibration is given explicitly by thefibration C × N sc ( D ) N sc ( D ) x / inn → C , where N sc ( D ) x / inn is the underlying scaled simplicial set of N sc ( D ) x / inn . Since η ∶ C → N sc ( D ) is a bicategorical equivalence the induced map C x / inn C × N sc ( D ) N sc ( D ) x / inn C ≃ is an equivalence of inner cocartesian fibrations over C by Corollary 2.2.2 and Re-mark 4.2.6, and hence we conclude that the functor corepresented by x classifiesthe associated inner slice fibration. In this section we will see that the analogous statements hold for all four types of slice fibrations C x / inn C x / out C / x inn C / x out C . We will do this by showing that they all admit the same type of a universal prop-erty, exhibiting them as freely generated by x ∈ C . We will deduce from this thatthe ( Z / ) -symmetry of BiCat th ∞ switches between these four fibrations, and con-sequently that they are all classified by the functors (co)represented by x , withrespect to the appropriate variance flavor.To facilitate the discussion, let us work with a variable var ∈ { out , inn } , whichwe will call the variance parameter . We will then refer to inner/outer (co)cartesianfibrations as var-(co)cartesian fibrations. Notation 4.3.1.
Let p ∶ X → Y be a bicategorical fibration of scaled simplicial setsand K a marked-scaled simplicial set equipped with a map f ∶ K → Y . We willdenote byFun car Y ( K, X ) , Fun coc Y ( K, X ) ⊆ Fun th Y ( K, X ) = Fun th ( K, X ) × Fun th ( K,Y ) { f } the full subcategory spanned by those maps K → X over Y which send the markededges of K to p -cartesian (resp. p -cocartesian) edges of X . Remark . In the situation of Notation 4.3.1, the assumption that p ∶ X → Y is abicategorical fibration implies that Fun ( K, X ) → Fun ( K, Y ) is a bicategorical fibra-tion, and hence that Fun Y ( K, X ) ∶ = Fun ( K, X ) × Fun ( K,Y ) { f } is an ∞ -bicategory.We may then identify Fun th Y ( K, X ) with the core ∞ -category of Fun Y ( K, X ) . Moregenerally, if K ↪ L is an inclusion of marked-scaled simplicial sets thenFun Y ( L, X ) → Fun Y ( K, X ) is a bicategorical fibration of ∞ -bicategories, and henceFun th Y ( L, X ) → Fun th Y ( K, X ) is a categorical fibration of ∞ -categories. Since the condition of being a (co)cartesianedge is closed under equivalences (see Remark 2.3.12) it follows thatFun car Y ( L, X ) → Fun car Y ( K, X ) and Fun coc Y ( L, X ) → Fun coc Y ( K, X ) are categorical fibrations as well. Definition 4.3.3.
Let Y be a scaled simplicial set, h ∶ K → L a map of marked-scaled simplicial sets and f ∶ L → Y a map of scaled simplicial sets. For a varianceparamter var ∈ { inn , out } , we will say that h is a var -cartesian equivalence over Y if for every var-cartesian fibration p ∶ X → Y the restriction map(32) h ∗ ∶ Fun car Y ( L, X ) → Fun car Y ( K, X ) is an equivalence of ∞ -categories. Similarly, we define var-cocartesian equivalencein the same manner using mapping ∞ -categories into var-cocartesian fibrationsover Y . Example . In the situation of Definition 4.3.3, if the induced map h ∶ K → L isa bicategorical equivalence and the marked edges in L are the images of the markededges in K then h is both a var-cartesian and a var-cocartesian equivalence over X . Indeed, in this case the map (32) is a base change of the trivial fibrationFun th Y ( L, X ) → Fun th Y ( K, X ) . IBRATIONS AND LAX LIMITS OF (∞ , ) -CATEGORIES 53 Remark . In the situation of Definition 4.3.3, suppose that Y is an ∞ -bicategoryand that K h / / (cid:31) (cid:31) ❅❅❅❅❅❅❅❅ L (cid:127) (cid:127) ⑧⑧⑧⑧⑧⑧⑧⑧ Y is a map of var-cartesian fibrations over Y such that the marked edges of K and L are exactly the cartesian edges over Y . Then h ∶ K → L is a var-cartesian equiv-alence over Y if and only if h is an equivalence in the sub-bicategory Car var ( Y ) of ( BiCat ∞ ) / Y , which is the same as saying that h is a bicategorical equivalence.We now turn to establishing the universal properties of the slice fibrations overan object. We start with the inner cocartesian case, which follows directly from thework of [13]. Proposition 4.3.6.
Let C be an ∞ -bicategory. Then for every x ∈ C the inclusionof marked-scaled simplicial sets { id x } ⊆ C x / inn is an inner cocartesian equivalenceover C .Proof. This follows directly from [13, Proposition 4.1.8], which says that the mapof marked simplicial sets underlying { id x } ⊆ C x / inn is P C -anodyne, and in particulara weak equivalence in the P -fibered model structure on ( S et + ∆ ) / C . We also notethat this gives the result on the level of mapping ∞ -categories, since the P -fiberedmodel structure is compatible with the action of S et + ∆ on ( S et + ∆ ) / C . (cid:3) Passing the opposites and using Remark 4.2.2 we obtain:
Corollary 4.3.7.
Let C be an ∞ -bicategory. Then for every x ∈ C the inclusion { id x } ⊆ C / x inn is an inner cartesian equivalence over C . We now establish the analogous statements for outer slice fibrations.
Proposition 4.3.8.
Let C be an ∞ -bicategory. Then for every x ∈ C the inclusion { id x } ⊆ C / x out is an outer cartesian equivalence over C . We reproduce the argument of [13, Proposition 4.1.8] in the outer cartesiansetting. For this, we will require the following lemma:
Lemma 4.3.9.
Let C be an ∞ -bicategory and K → L an inclusion of marked scaledsimplicial sets. Let E be the collection of those edges ( e, e ′ ) ∶ ♭ ∆ → ♭ ∆ ⊗ K suchthat e ′ is marked and either e or e ′ are degenerate, and define E ′ in a similarmanner for ♭ ∆ ⊗ L . Then for any map ♭ ∆ ⊗ L → C the inclusion [ ∆ { } ⊗ L ] ∐ ∆ { } ⊗ K ( ♭ ∆ ⊗ K, E ) → ( ♭ ∆ ⊗ L, E ′ ) is an outer cartesian equivalence over C , where ( ♭ ∆ ⊗ K, E ) denotes the marked-scaled simplicial set whose underlying scaled simplicial set is ♭ ∆ ⊗ K and whoseset of marked edges is E , and similarly for ( ∆ ♭ ⊗ L, E ′ ) .Proof. We need to show that for every outer cartesian fibration E → C the restrictionfunctorFun car C (( ♭ ∆ ⊗ L, E ′ ) , E ) → Fun car C ([ ∆ { } ⊗ L ] ∐ ∆ { } ⊗ K ( ♭ ∆ ⊗ K, E ) , E ) is an equivalence of ∞ -categories. For this we first note that by Corollary 2.3.11we have a pullback squareFun car C (( ♭ ∆ ⊗ L, E ′ ) , E ) / / (cid:15) (cid:15) Fun car C ([ ∆ { } ⊗ L ] ∐ ∆ { } ⊗ K ( ♭ ∆ ⊗ K, E ) , E ) (cid:15) (cid:15) Fun car C (( ♭ ∆ ⊗ L, E ′ ) , E ) / / Fun car C ([ ∆ { } ⊗ L ] ∐ ∆ { } ⊗ K ( ♭ ∆ ⊗ K, E ) , E ) where E denotes the set of edges of the form ∆ × { v } for v ∈ K and E ′ the set ofedges of the form ∆ × { u } for u ∈ L . It will hence suffice to prove that the bottomhorizontal map is a trivial fibration. In particular, we may ignore the marking on K and L and work with their underlying scaled simplicial sets K and L . Now, toshow that the map in question has the right lifting property with respect to anyinclusion Z ⊆ W of scaled simplicial sets translates to finding a lift in a square ofthe form(33) [ ∆ ♭ ⊗ L ] × Z ∐ [ ∆ { } ⊗ L ∐ ∆ { }⊗ K ∆ ♭ ⊗ K ]× Z [ ∆ { } ⊗ L ∐ ∆ { } ⊗ K ∆ ♭ ⊗ K ] × W / / (cid:15) (cid:15) E p (cid:15) (cid:15) [ ∆ ♭ ⊗ L ] × W / / B which sends the edges in the set E ′ × W to p -cartesian edges.Let K ′ → L ′ = ( K → L ) ✷ ( Z → W ) be the pushout product of K → L and Z → W with respect to the cartesian product of scaled simplicial set. To carry onthe proof we would like to replace the left vertical map in the above square withthe Gray pushout-product of ∆ { } ↪ ∆ ♭ and K ′ → L ′ . These are identical on thelevel of the underlying simplicial sets (since the pushout-product of simplicial setsis associative), but have different scaling, since for general scaled simplicial sets A, B, C one has A ⊗ ( B × C ) / ≅ ( A ⊗ B ) × C . Explicitly, a triangle ( σ A , σ B , σ C ) in thecartesian product of A, B and C is thin in A ⊗ ( B × C ) if and only if σ A , σ B and σ C are all thin and in addition either σ A degenerates along ∆ { , } or both σ B and σ C degenerates along ∆ { , } . On the other hand, ( σ A , σ B , σ C ) is thin in ( A ⊗ B ) × C if and only if σ A , σ B and σ C are all thin and either σ A degenerates along ∆ { , } or σ B degenerates along ∆ { , } . In particular, we have a canonical inclusion of scaledsimplicial sets A ⊗ ( B × C ) ⊆ ( A ⊗ B ) × C, and so the square (33) restricts to a square(34) ∆ { } ⊗ L ′ ∐ ∆ { } ⊗ K ′ ∆ ♭ ⊗ K ′ E ∆ ♭ ⊗ L ′ B f pH ̃ H Since the map E → B is an outer fibration it detects thin triangles, and so thedotted lift exists in (33) if and only if it exists in (34). The existence of a lift in (34)is then given by Proposition 2.5.1. (cid:3) Proof of Proposition 4.3.8.
Consider the composite e ∶ ♭ ∆ ⊗ ♭ ∆ ⊗ C / x out → ♭ ∆ ⊗ C / x out → C / x out ◇ out ∆ → C , IBRATIONS AND LAX LIMITS OF (∞ , ) -CATEGORIES 55 where the first map is induced by the map ♭ ∆ ⊗ ♭ ∆ → ♭ ∆ given on vertices by i, j ↦ max ( i, j ) , the second is the quotient map, and the third is obtained fromthe counit of the adjunction between outer join and slice. The restriction of e to∆ { } ⊗ ♭ ∆ ⊗ C / x out is then constant with value x ∈ C and so descends to give a map ( ♭ ∆ ⊗ C / x out ) ♭ ◇ out ∆ → C . The last map then transposes to a map r ∶ ♭ ∆ ⊗ C / x out → C / x out , which we read as a lax transformation from the identity on C / x out to the composite C / x out → { id x } → C / x out . Furthermore, this transformation is constant when restrictedto { id x } ⊆ C / x out , and sends every edge of the form ♭ ∆ ⊗ { v } in ♭ ∆ ⊗ C / x out to anedge which is marked in C / x out . The map r then fits into a commutative diagram ofmarked-scaled simplicial sets∆ { } × { id x } / / (cid:15) (cid:15) ∆ { } × C / x out (cid:15) (cid:15) [ ♯ ∆ × { id x }] ∐ ∆ { } ×{ id x } [ ∆ { } × C / x out ] (cid:15) (cid:15) / / ( ♭ ∆ ⊗ C / x out , E ) r (cid:15) (cid:15) { id x } / / C / x out where ( ♭ ∆ ⊗ C / x out , E ) denotes the marked-scaled simplicial set whose underlyingscaled simplicial set is ♭ ∆ ⊗ C / x out and whose set of marked edges is the set E consisting of those edges ( e, e ′ ) ∶ ∆ ♭ → ♭ ∆ ⊗ C / x out such that e ′ is marked and either e or e ′ is degenerate. This shows that the inclusion { id x } ⊆ C / x out is a retract, over C , of the inclusion(35) [ ♯ ∆ × { id x }] ∐ ∆ { } ⊗{ id x } [ ∆ { } × C / x out ] → ( ♭ ∆ ⊗ C / x out , E ) , where to avoid confusion we emphasize that we consider the object on the right asliving over C via the map ♭ ∆ ⊗ C / x out r Ð→ C / x out → C . It will hence suffice to show that (35) is an outer cartesian equivalence over C .Indeed, this is a particular case of Lemma 4.3.9. (cid:3) Passing the opposites and using Remark 4.2.2 we obtain:
Corollary 4.3.10.
Let C be an ∞ -bicategory. Then for every x ∈ C the inclusion { id x } ⊆ C x / out is an outer cocartesian equivalence over C . Corollary 4.3.11.
Under the equivalences of Corollary 3.3.2 and Corollary 3.3.3the fibrations C x / inn C x / out C / x inn C / x out C correspond to the functors (op)(co)represented by x ∈ C , respectively. In addition,the ( Z / ) -action on BiCat ∞ switches between these four fibrations, so that we haveequivalences (36) C x / inn ≃ (( C co ) x / out ) co ≃ (( C op ) / x inn ) op ≃ (( C coop ) / x out ) coop C . Proof.
The equivalences (36), two of which are already visible on the level of the sim-plicial construction as described in Remark 4.2.2, are implied by Propositions 4.3.6and 4.3.8 and Corollaries 4.3.7 and 4.3.10, which characterize each of these fibrationsby the same type of universal mapping property. By these equivalences and theway that the Lurie-Grothendieck correspondence is constructed in Corollary 3.3.3,to prove the first claim it is enough to consider the case of C x / inn . Indeed, as explainedabove, it follows from Remark 4.2.7 that C x / inn is equivalent to the unstraighteningof the functor corepresented by x . (cid:3) We now consider the question of identifying when a given inner/outer (co)cartesianfibration E → C is (co)representable by an object x ∈ C . To fix ideas, let us considerthe cocartesian case, and fix a variance parameter var ∈ { out , inn } as above. Definition 4.3.12.
Let p ∶ E → C be a var-cocartesian fibration of ∞ -bicategories.We will say that an object x ∈ E is p -universal if the inclusion { x } ⊆ E ♮ is a var-cocartesian equivalence over C , where E ♮ denotes the marked-scaled simplicial setwhose underlying scaled simplicial set is E and whose marked edges are the p -cocartesian ones. In this case we will also say that x ∈ E exhibits E as corepresented by p ( x ) . Remark . In the situation of Definition 4.3.12, the inclusion { x } ⊆ E ♮ canalways be extended to a map C p ( x )/ var → E ♮ which sends id p ( x ) to x : indeed, therestriction Fun coc C ( C p ( x )/ inn , E ) → Fun coc C ({ id p ( x ) } , E ) is trivial fibration of ∞ -categories by Proposition 4.3.6 and a Remark 4.3.2. Usingagain Proposition 4.3.6 it now follows that x is p -universal if and only if the re-sulting map C / p ( x ) var → E ♮ is a var-cocartesian equivalence over C , or equivalently, anequivalence of var-cocartesian fibrations over C (see Remark 4.3.5). Remark . In the situation of Definition 4.3.12, the collection of p -universalobjects in E is closed under equivalence. Indeed, if x ≃ y are two equivalent objectsin E then there exists a map η ∶ J ♯ → E such that η ( ) = x and η ( ) = y . Since both { } ⊆ J ♯ and { } ⊆ J ♯ are bicategorical equivalences it follows from Example 4.3.4that { x } ⊆ E ♮ is a var-cocartesian equivalence over C if and only if J ♭♯ → E ♮ is avar-cocartesian equivalence over C , and the same goes for y . It then follows that x is p -universal if and only if y is p -universal. Remark . If f ∶ E → E ′ is an equivalence of var-cocartesian fibrations over C then f preserves and detects universal objects by Example 4.3.4.The fully-faithfulness of the Yoneda embedding suggests that if a var-cocartesianfibration E → C is classified by a corepresentable functor, then the corepresentingobject x is essentially unique. The following proposition makes this statementprecise: IBRATIONS AND LAX LIMITS OF (∞ , ) -CATEGORIES 57 Proposition 4.3.16.
Let p ∶ E → C be a var -cocartesian fibration of ∞ -bicategories.Let X ⊆ E be the sub-bicategory spanned by the p -universal objects and the p -cocartesian morphisms between them. Then X is either empty or a contractibleKan complex.Proof. Suppose that X is non-empty, so that there exists a p -universal object x ∈ E .By Remark 4.3.13 the inclusion { x } ⊆ E extends to an equivalence C p ( x )/ var f ≃ / / q ! ! ❇❇❇❇❇❇❇❇❇ E p (cid:1) (cid:1) ✄✄✄✄✄✄✄✄ C of var-cocartesian fibrations over C . Let Y ⊆ C p ( x )/ var be the full sub-bicategoryspanned by the q -universal objects and the q -cocartesian edges between them.Combining Remark 4.3.14 and Remark 4.3.15 we may deduce that f induces anequivalence Y ≃ Ð→ X . It will hence suffice to prove that Y is a contractible Kancomplex.We now claim that an object [ α ∶ p ( x ) → y ] ∈ C p ( x )/ var is q -universal if and only if α is invertible in C . To see this, extend the inclusion {[ α ∶ p ( x ) → y ]} ⊆ C p ( x ) var to amap of var-cocartesian fibrations C y / var g ≃ / / q ′ (cid:31) (cid:31) ❅❅❅❅❅❅❅❅ C p ( x )/ var q } } ⑤⑤⑤⑤⑤⑤⑤⑤⑤ C so that [ α ] is universal if and only if g is a bicategorical equivalence. By Corol-lary 2.2.2 this is equivalent to the induced map ( C y / var ) z → ( C p ( x )/ var ) z being a categorical equivalence on underlying simplicial sets for every z ∈ C . Nowby Remark 4.2.6 we may identify this map with the induced map ( − ) ○ α ∶ Hom C ( y, z ) → Hom C ( p ( x ) , z ) when var = inn, and with the opposite of this map (up to categorical equivalence)when var = out. These maps are all equivalences precisely when α is invertible.Now if α ∶ p ( x ) → y and β ∶ p ( x ) → z are two q -universal objects then by Corol-lary 2.4.7 a q -cocartesian arrow from α to β corresponds to a commutative square p ( x ) yp ( x ) z , αβ ≃≃ and since α and β are invertible the arrow y → z is invertible as well. In particular,every arrow in Y is invertible. We now claim that every triangle in Y is thin. Tosee this, note that a triangle in Y corresponds to a map ρ ∶ ∆ ♭ ⊗ ∆ ♭ → C such that ρ ∣ ∆ { } × ∆ ♭ is constant with image p ( x ) , and by the above we also have that ρ sendsevery edge of ∆ ♭ ⊗ ∆ ♭ to an equivalence in C and every triangle in ∆ ♭ ⊗ ∆ ♭ whoseprojection to ∆ ♭ is degenerate to a thin triangle in C . By [6, Corollary 3.5] thetriangle ρ ∣ ∆ { }♭ ⊗ ∆ ♭ is also thin in C . In fact, the proof of that corollary actuallyshows that ρ sends every triangle in ∆ ♭ ⊗ ∆ ♭ to a thin triangle in C . We may consequently identify Y with the subgroupoid ( C ≃ ) p ( x )/ ⊆ C p ( x )/ var . We now finish theproof by noting that ( C ≃ ) p ( x )/ is a contractible Kan complex, being an ∞ -groupoidwith an initial object. (cid:3) Categories of lax transformations.
In this subsection we study the relationbetween slice fibrations and certain ∞ -categories of lax transformations. By thelatter we mean the following: Definition 4.4.1.
Let C be an ∞ -bicategory and K a marked-scaled simplicial set.For two diagrams f, g ∶ K → C we define Nat gr K ( f, g ) and Nat opgr K ( f, g ) to be the map-ping ∞ -categories from f to g in the ∞ -bicategories Fun gr ( K, C ) and Fun opgr ( K, C ) ,respectively.Our main goal in the present subsection is to identify the fibers of the slicefibrations over a diagram in C in terms of suitable spaces of lax natural transfor-mations. We also deduce in the end a useful invariance property with respect tothe restriction along inner/outer (co)cartesian equivalences K → L . We begin withthe following statement, identifying ∞ -categories of lax transformations to/from adiagram which is constant on an object x ∈ C in terms cartesian lifts to the slicefibration over/under x : Proposition 4.4.2.
Let C be an ∞ -bicategory, K a marked-scaled simplicial setand f ∶ K → C be a functor. Then for x ∈ C there are natural equivalences Nat gr K ( x, f ) ≃ Fun coc C ( K, C x / inn ) , Nat gr K ( f, x ) ≃ Fun car C ( K, C / x out ) , Nat opgr K ( f, x ) ≃ Fun car C ( K, C / x inn ) op and Nat opgr K ( x, f ) ≃ Fun coc C ( K, C x / out ) op , where x denotes the constant map K → C with value x .Proof. Let us prove the first pair of equivalences. The second pair can then be de-duced by replacing f ∶ K → C with f op ∶ K op → C op using Remarks 4.2.6 and 4.1.13.To obtain the first equivalence it will suffice by Remark 4.2.6 to produce an iso-morphism(37) ( Fun gr ( K, C ) / f out ) x ≅ Fun coc C ( K, C x / inn ) . Indeed, using Remark 4.1.4 we see that the scaled simplicial set on the left handside represents the sub-functor of Z ↦ Fun ( ♭ ∆ ⊗ Z ♭ ⊗ K, C ) spanned by those maps ♭ ∆ ⊗ Z ♭ ⊗ K → C whose restriction to ∆ { } ⊗ Z ♭ ⊗ K isgiven by Z ♭ ⊗ K → K f Ð→ C and whose restriction to ∆ { } ⊗ Z ♭ ⊗ K is given by Z ♭ ⊗ K → ∆ x Ð→ C . On the other hand, the scaled simplicial set on the right handside of (37) represents the sub-functor of Z ↦ Fun ( ♭ ∆ ⊗ ( Z ♭ × K ) , C ) defined bythe same conditions on the values at ∆ { } and ∆ { } . We now observe that we havenatural inclusions of scaled simplicial sets (which are isomorphisms on underlyingsimplicial sets)(38) ♭ ∆ ⊗ Z ♭ ⊗ K ← ♭ ∆ ⊗ ( Z ♭ ⊗ K, E ) → ♭ ∆ ⊗ ( Z ♭ × K ) where E is the set of edges which are marked in the cartesian product Z ♭ × K . Wehence obtain a zig-zag of maps relating the two sides of (37). But this zig-zag is infact a zig-zag of isomorphisms since the left map in (38) is a bicategorical trivialcofibrations by Proposition 4.1.8 and the right map becomes scaled anodyne aftercollapsing ∂ ∆ × Z × K to ∆ { } ∐ ∆ { } × K by Lemma 4.2.13.For the second equivalence, we invoke again Remark 4.2.6 and produce insteadan isomorphism ( Fun gr ( K, C ) f / inn ) x ≅ Fun coc C ( K, C / x out ) . IBRATIONS AND LAX LIMITS OF (∞ , ) -CATEGORIES 59 The argument in this case then proceeds exactly as above by noting that bothsides represent again a common sub-functor of Z ↦ Fun ( ♭ ∆ ⊗ Z ♭ ⊗ K, C ) and Z ↦ Fun ( ♭ ∆ ⊗ ( Z ♭ × K ) , C ) defined in a similar manner, where this time theconditions at ∆ { } and ∆ { } are switched. (cid:3) Applying Proposition 4.4.2 in the case of C = C at ∞ and x = ∗ we obtain: Corollary 4.4.3.
Let K be a marked-scaled simplicial set and χ ∶ K → C at ∞ be afunctor. Let E → K be the inner cocartesian fibration classified by χ and E ′ → K the outer cocartesian fibration classified by K co χ co ÐÐ→ C at co ∞ op Ð→ C at ∞ . Then there are natural equivalences
Nat gr K ( ∗ , χ ) ≃ Fun coc C at ∞ ( K, ( C at ∞ ) ∗/ inn ) ≃ Fun coc K ( K, E ) , and Nat opgr K ( ∗ , χ ) ≃ Fun coc C at ∞ ( K, ( C at ∞ ) ∗/ out ) op ≃ Fun coc K ( K, E ′ ) op . Combining Proposition 4.4.2 and Corollary 4.4.3 we then conclude:
Corollary 4.4.4.
Let C be an ∞ -bicategory, K a marked-scaled simplicial set and f ∶ K → C be a functor. Then for x ∈ C there are natural equivalences Nat gr K ( x, f ) ≃ Nat gr K ( ∗ , Hom C ( x, f ( − ))) , Nat opgr K ( x, f ) ≃ Nat opgr K ( ∗ , Hom C ( x, f ( − ))) and Nat opgr K ( f, x ) ≃ Nat gr K op ( ∗ , Hom C ( f ( − ) , x )) , Nat gr K ( f, x ) ≃ Nat opgr K op ( ∗ , Hom C ( f ( − ) , x )) . To avoid confusion, we note that in Corollary 4.4.4 the transformations on theleft side of each equivalence concern C -valued diagrams indexed by K , whereas thetransformations on the right of each equivalence concern C at ∞ -valued diagrams,indexed by either K or K op . The notations Hom C ( x, f ( − ))) and Hom C ( f ( − ) , x )) refer to the post composition of f ∶ K → C with the functors represented and corep-resented by x . Proof of Corollary 4.4.4.
The second pair of equivalences can be deduced from thefirst by replacing f ∶ K → C with f op ∶ K op → C op , using the equivalencesNat opgr K ( f, x ) ≃ Nat gr K op ( f op , x ) and Nat gr K ( f, x ) ≃ Nat opgr K op ( f op , x ) , see Remark 4.1.13. The first pair of equivalences follows by combining Proposi-tion 4.4.2 with Corollary 4.4.3 (applied to χ ∶ = Hom C ( x, f ( − )) ). (cid:3) We now turn to discussing the slice fibrations C f / inn C f / out C / f inn C / f out C and the functors that classify them under the bicategorical Grothendieck–Luriecorrespondence. Proposition 4.4.5.
Let K be a marked-scaled simplicial set and f ∶ K → C a dia-gram. ● The inner cocartesian fibration C f / inn → C is classified by the functor x ↦ Nat opgr K ( f, x ) ≃ Nat gr K op ( ∗ , Hom C ( f ( − ) , x )) . ● The outer cocartesian fibration C f / out → C is classified by the co -functor x ↦ Nat gr K ( f, x ) op ≃ Nat opgr K op ( ∗ , Hom C ( f ( − ) , x )) op . ● The inner cartesian fibration C / f inn → C is classified by the co -presheaf x ↦ Nat gr K ( x, f ) op ≃ Nat gr K ( ∗ , Hom C ( x, f ( − ))) op . ● The outer cartesian fibration C / f out → C is classified by the presheaf x ↦ Nat opgr K ( x, f ) ≃ Nat opgr K ( ∗ , Hom C ( x, f ( − ))) . Here, in all cases x denotes the constant diagram K → C with value x .Proof. As in the proof of Proposition 4.2.12 we have a pullback square C f / inn Fun opgr ( K, C ) [ f ]/ inn C Fun opgr ( K, C ) where Fun opgr ( K, C ) [ f ]/ inn denotes the underlying scaled simplicial set of the marked-scaled simplicial set Fun opgr ( K, C ) [ f ]/ inn . The desired result then follows from Corol-lary 4.3.11 and the compatibility of the straightening-unstraightening equivalencewith base change, see [13]. The equivalence between lax transformations of C -valuedand C at ∞ -valued diagrams is then given by Corollary 4.4.4.The proof of the other three variances is the same: the analogous pullbacksquare exists in all four cases, and the compatibility with base change of the Lurie-Grothendieck correspondence in the inner cocartesian cases implies all other vari-ances, see Remark 3.3.4. (cid:3) Corollary 4.4.6 (Invariance of slice fibrations) . Let h ∶ K → L be a map of marked-scaled simplicial sets. Let C be an ∞ -bicategory and f ∶ L → C be a scaled map. Fora variance parameter var ∈ { inn , out } , if h is a var -cocartesian (resp. var -cartesian)equivalence over C then the projections C / f var → C / fh var ( resp. C f / inn → C fh / inn ) are equivalences of ∞ -bicategories, respectively.Remark . In the situation of Corollary 4.4.6, if the induced map h ∶ K → L is abicategorical equivalence and the marked edges in L are the images of the markededges in K then h is in particular an inner/outer (co)cartesian equivalence over C (see Example 4.3.4). We then deduce that in this case the four restriction maps C / f inn → C / fh inn , C / f out → C / fh out , C f / inn → C fh / inn and C f / out → C fh / out are all equivalence of ∞ -bicategories. Proof of Corollary 4.4.6.
Replacing h ∶ K → L and f ∶ L → C by h op ∶ K op → L op and f op ∶ L op → C op switches between cartesian and cocartesian fibrations, and so it willsuffice to prove the case where h is a var-cocartesian equivalence.Applying Corollary 2.2.2 it will suffice to show that for every x ∈ C the map ( C / f var ) x → ( C / fι var ) x IBRATIONS AND LAX LIMITS OF (∞ , ) -CATEGORIES 61 is an equivalence of ∞ -categories. Using Proposition 4.4.5 and Proposition 4.4.2 wemay identify this map with the mapFun coc C ( L, C x / var ) op → Fun coc C ( K, C x / var ) op . This, in turn, is an equivalence by the assumption that K → L is a var cocartesianequivalence over C . (cid:3) Limits and colimits in ∞ -bicategories In this section we define and study a notion of 2-(co)limit suitable for ∞ -bicategories. These (when exist) are associated to a diagram f ∶ K → C in an ∞ -bicategory C , indexed by the underlying scaled simplicial set of a marked-scaledsimplicial set K . They come in four flavors, depending on a variance parametervar ∈ { inn , out } , and on whether we take limits or colimits. We give the basicdefinitions and extract some of their properties in § § § § § ∞ -bicategories whichcome from model categories M tensored over marked simplicial sets exist and canbe computed in terms of weighted homotopy colimits, under mild assumptions on M .5.1. Universal cones.Notation 5.1.1.
Let K be a marked-scaled simplicial set. We will denote by K ◁ inn the marked-scaled simplicial set whose underlying scaled simplicial set is given by K ◁ inn ∶ = { ∗ } ◇ inn K , and whose marked edges are the union of the marked edgesof K as well as every edge that contains ∗ . Similarly, we will denote by K ◁ out themarked-scaled simplicial set whose underlying scaled simplicial set is { ∗ } ◇ out K and whose marked edges are defined in the analogous way.For K a marked-scaled simplicial set and C and ∞ -bicategory, we shall call adiagram K ◁ inn → C an inner cone diagram and a diagram K ◁ out → C an outer conediagram . Definition 5.1.2.
Let C be an ∞ -bicategory, K a marked-scaled simplicial set and g ∶ K ◁ inn → C be an inner cone diagram. We will say that g is an inner limit cone on f ∶ = g ∣ K if the projection C / g inn → C / f inn is an equivalence of ∞ -bicategories. Similarly, we will say that an outer cone dia-gram g ∶ K ◁ out → C is an outer limit cone on f if the projection C / g out → C / f out is an equivalence of ∞ -bicategories. The definition of inner and outer colimit cones is defined in a similar way using the right cones K ▷ inn and K ▷ out . To facilitate the following discussion, let us fix a variance parameter var ∈ { out , inn } . We will refer to var-(co)limits as in definition 5.1.2 as .We note that while the diagram f ∶ K → C does not take into account the markingon K , the slice ∞ -bicategories appearing in Definition 5.1.2, and hence the associ-ated notion of 2-(co)limit, critically depend on it. In particular, when all edges in K are marked the associated notion of 2-(co)limit should be considered as a formof pseudo-(co)limits , while if only the degenerate edges are marked it should beconsidered rather as a (op)lax (co)limit , see also Remark 5.1.10 below. Example . Suppose that K = ∅ . Then K ◁ var = ∆ and the data of a var-conein C is simply the data of an object x ∈ C . By definition, this objects determinesa var-limit cone over ∅ if and only if the map C / x var → C is an equivalence of ∞ -bicategories. Since this map is a var-cartesian fibration Corollary 2.2.2 tells us thatit is an equivalence if and only if its fibers are categorically equivalent to ∆ . ByRemark 4.2.6 this is the same as saying that the mapping ∞ -categories Hom C ( y, x ) are categorically equivalent to ∆ for every y ∈ x . In this case we say that x isa final object of C . We note that in this case it does not matter if the varianceparameter is inn or out. Dually, inner and outer colimits of the empty diagram aregiven by objects x ∈ C such that Hom C ( x, y ) is categorically equivalent to ∆ forevery y . We will then say that such an x is an initial object of C . Warning . In contrast to the case of Example 5.1.3 above, in general var-limitcones are not final objects in C / f var . They can however be characterized by a suitable cofinality property, see Proposition 5.2.5 below. Remark . Let C be an ∞ -bicategory, K a marked-scaled simplicial set and g ∶ K ◁ var → C a cone diagram for some variance parameter var ∈ { inn , out } . Then g is a var-limit cone if and only if g op ∶ ( K ◁ var ) op ≅ ( K op ) ▷ var → C op is a var-colimit cone. This follows directly from the definition in light of the behaviorof slice fibrations under opposites described in Remark 4.2.2.Let now C be an ∞ -bicategory, K a marked-scaled simplicial set, and g ∶ K ◁ var → C a var-cone in C extending f = g ∣ K . Consider the diagram of ∞ -bicategories(39) C / g var C / g (∗) var C / f var . By definition we have that the map g is a var-limit cone if and only if the rightdiagonal map is an equivalence of ∞ -bicategories. We now claim that the leftdiagonal map is always an equivalence: Lemma 5.1.6.
Let C be an ∞ -bicategory and K a marked-scaled simplicial setequipped with a map g ∶ K → C . Then the inclusion { ∗ } ⊆ K ◁ var is a var -cocartesianequivalence over C .Proof. When var = out this map is a pushout of the map K ⊗ ∆ { } ⊆ K ⊗ ♭ ∆ , whichis an outer cocartesian equivalence by (the dual of) Lemma 4.3.9. When var = innit is a pushout of the map ∆ { } ⊗ K ⊆ ♭ ∆ ⊗ K which is an inner cocartesianequivalence by [13, Lemma 4.1.7]. (cid:3) IBRATIONS AND LAX LIMITS OF (∞ , ) -CATEGORIES 63 Corollary 5.1.7.
Let C be an ∞ -bicategory and K a marked-scaled simplicial set.Let var ∈ { inn , out } be a variance parameter. Then for any diagram g ∶ K ◁ var → C theprojection C / g var → C / g (∗) var is a trivial fibration.Proof. This follows follows from Lemma 5.1.6 and Corollary 4.4.6. (cid:3)
Combining Corollary 5.1.7 with Corollary 4.3.11 we find that C / g var → C is amodel for the var-cartesian fibration represented by g ( ∗ ) . In particular, the cone g determines a map from the var-functor represented by g ( ∗ ) and the var-functorassociated to the var-cocartesian fibration C / f var → C , which we have identified inProposition 4.4.5 in terms of (partially) lax natural transformations. By var-functorhere we mean a functor C ε → C at ∞ , where the variance ε = ∅ , co of C is determinedby the variable var. The condition that g is a var-limit cone is exactly the conditionthat this map is an equivalence. In particular, we may conclude the following: Corollary 5.1.8.
Let K be a marked-scaled simplicial set and C an ∞ -bicategory.(1) An inner cone g ∶ K ◁ inn → C extending f = g ∣ K is an inner limit cone if and onlyif it determines a natural equivalence of ∞ -categories Hom C ( x, g ( ∗ )) ≃ Nat gr K ( ∗ , Hom C ( x, f ( − ))) . (2) An outer cone g ∶ K ◁ out → C extending f = g ∣ K is an outer limit cone if and onlyif it determines a natural equivalence of ∞ -categories Hom C ( x, g ( ∗ )) ≃ Nat opgr K ( ∗ , Hom C ( x, f ( − ))) . Arguing in a dual manner for colimit cones using Remark 5.1.5 we may equallydeduce:
Corollary 5.1.9.
Let K be a marked-scaled simplicial set and C an ∞ -bicategory.(1) An inner cone g ∶ K ▷ inn → C extending f = g ∣ K is an inner colimit cone if andonly if it determines a natural equivalence of ∞ -categories Hom C ( g ( ∗ ) , x ) ≃ Nat gr K op ( ∗ , Hom C ( f ( − ) , x )) . (2) An outer cone g ∶ K ▷ out → C extending f = g ∣ K is an outer colimit cone if andonly if it determines a natural equivalence of ∞ -categories Hom C ( g ( ∗ ) , x ) ≃ Nat opgr K op ( ∗ , Hom C ( f ( − ) , x )) . Remark . To avoid confusion, let us recall that the notions of gr- and opgr-natural transformations appearing in Corollaries 5.1.8 and 5.1.9, strongly dependon the marking on K . In particular, when all edges in K are marked this coincideswith the usual notion of a natural transformation (that is, maps in the functorbicategories Fun ( K, C at ∞ ) and Fun ( K op , C at ∞ ) ). On the other hand, if only thedegenerate edges are marked these correspond instead to lax (or oplax) transfor-mations. In general, we have that 2-(co)limits represent ∞ -categories of “partiallylax” natural transformation, where the precise level of pseudo-naturality dependson the marking on K .The following corollary is essentially an equivalent reformulation of the state-ments of Corollaries 5.1.8 and 5.1.9, which is more convenient for certain applica-tions: Corollary 5.1.11.
Let C be an ∞ -bicategory, K a marked-scaled simplicial set and f ∶ K → C a diagram. Then a var -cone g ∶ K ◁ var is a var -limit cone if and only if forevery x ∈ C the restriction map Fun coc C ( K ◁ var , C x / var ) → Fun coc C ( K, C x / var ) is an equivalence of ∞ -categories. Dually, a var -cone g ∶ K ▷ var is a var -colimit coneif and only if for every x ∈ C the restriction map Fun car C ( K ▷ var , C / x var ) → Fun car C ( K, C / x var ) is an equivalence of ∞ -categories.Proof. We prove the first claim. The dual version is obtained in a similar manner.Applying Corollary 4.4.6 we deduce that g is a var-limit cone if and only if for every x ∈ C the induced map ( C / g var ) x → ( C / f var ) x is an equivalence of ∞ -categories on the underlying simplicial sets. By Proposi-tion 4.4.5 this is equivalent to saying that for every x ∈ C the induced mapNat εK ◁ ( x, g ) → Nat εK ( x, f ) is an equivalence of ∞ -categories, where ε = gr if var = inn and ε = opgr if var = out.This statement then translates to the desired results by using the identification ofProposition 4.4.2. (cid:3) We finish this subsection by discussing an analogue of Remark 5.1.5 where ( − ) op is replaced with ( − ) co . We note that latter does not admit a convenient modelwith scaled simplicial sets, and so we will need to formulate the claim using the ∞ -bicategory BiCat ∞ . For this, we note that by Remark 4.2.3 the functors ( − ) ♭ ◇ inn ∆ , ( − ) ♭ ◇ out ∆ , ∆ ◇ inn ( − ) ♭ and ∆ ◇ out ( − ) ♭ are left Quillen functors and henceinduce functors on the level of ∞ -categoriesBiCat th ∞ → ( BiCat th ∞ ) ∆ / , which we will denote by the same name. Lemma 5.1.12.
For an ∞ -bicategory I there are natural equivalences ( I ◇ inn ∆ ) co ≃ I co ◇ out ∆ and ( ∆ ◇ inn I ) co ≃ ∆ ◇ out I co , where both sides are considered as functors BiCat th ∞ → ( BiCat th ∞ ) ∆ / in the input I .Proof. This follows from the identification of the right adjoints of these functorsgiven in Corollary 4.3.11. (cid:3)
Corollary 5.1.13.
Let C be an ∞ -bicategory, I a marked-scaled simplicial set whoseunderlying scaled simplicial set I is an ∞ -bicategory and g ∶ I ◁ inn → C an inner cone.Then g is an inner limit cone if and only if g co ∶ ( I ◁ inn ) co ≃ ( I co ) ◁ out → C co is an outer limit cone, where the identification is done using Lemma 5.1.12. Asimilar statement holds for colimits.Proof. Taking into account the compatibility of cones under taking ( − ) co describedin Lemma 5.1.12 and the compatibility of slice fibrations with ( − ) co appearing in thelast part of Corollary 4.3.11, the desired claim follows directly from the equivalentcriterion for limit cones of Corollary 5.1.11. (cid:3) IBRATIONS AND LAX LIMITS OF (∞ , ) -CATEGORIES 65 Cofinality.
Throughout this section var ∈ { out , inn } is a fixed variance pa-rameter . Definition 5.2.1.
Let h ∶ K → L be a map of marked-scaled simplicial sets. Wewill say that h is var -cofinal if it is a var-cartesian equivalence over L . We will saythat h is var -coinitial if it is a var-cocartesian equivalence over L . Lemma 5.2.2.
Let h ∶ K → L be a map of marked scaled simplicial sets, Y a scaledsimplicial set and g ∶ L → Y a map. Then the following statements hold:(1) If h is var -cofinal then h is a var -cartesian equivalence over Y .(2) If h is a var -cartesian equivalence over Y and g ∶ L → Y is a var -cartesianfibration such that the marked arrows in L are g -cartesian, then h is var -cofinal.The same holds if we replace var -cofinal by var -coinitial and var -cartesian equiva-lence/fibration by var -cocartesian equivalence/fibration.Proof. The first claim is clear since any var-cartesian fibration X → Y restricts toa var-cartesian fibration X ′ = X × Y L → L , such that functors to X ′ over L are inbijection with functors into X over Y . To see the second claim let X → L be avar-cartesian fibration. Since L → Y is now assumed a var-cartesian fibration thecomposed map X → L → Y is also a var-cartesian fibration. We may then considerthe commutative diagramFun car L ( L, X ) / / (cid:15) (cid:15) Fun car L ( K, X ) (cid:15) (cid:15) Fun car Y ( L, X ) / / (cid:15) (cid:15) Fun car Y ( K, X ) (cid:15) (cid:15) Fun car Y ( L, L ) / / Fun car Y ( K, L ) in which the top row can be identified with the induced map from the fiber of thebottom left vertical map over id ∶ L → L to the fiber of the bottom right vertical mapover h ∶ K → L (where we note that id and h indeed send marked edges to carte-sian edges by assumption). In addition, both these vertical arrows are categoricalfibrations, and so their fibers are also homotopy fibers. But under the assumptionthat h is a var-cartesian equivalence and L → Y is var-cartesian fibration we havethat the bottom and middle horizontal arrows are equivalence of ∞ -categories, andhence the top horizontal arrow is an equivalence as well, so that h is var-cofinal. (cid:3) Proposition 5.2.3.
Let h ∶ K → L be a map of marked-scaled simplicial sets and f ∶ L → C a diagram. If h is a var -cartesian equivalence over C then the map C f / var → C fh / var preserves and detects var -colimit cones. In particular, if h is var -cofinal then thisholds for any f ∶ L → C . Dually, if h is a var -cocartesian equivalence over C (e.g., if h is var -coinitial) then the map C / f var → C / fh var preserves and detects var -limit cones.Proof. We prove the case of coinitiality and limits. The dual case is proven in asimilar manner. Let ̃ h ∶ L ◁ var → K ◁ var be the induced map on var-cones. For g ∶ L ◁ var → C extending f consider the diagram C / g (∗) ≅ (cid:15) (cid:15) C / g ≃ o o / / (cid:15) (cid:15) C / f (cid:15) (cid:15) C / g ̃ h (∗) C / g ̃ h ≃ o o / / C / fh , where the left facing horizontal arrows are trivial fibrations by Corollary 5.1.7 andthe left vertical arrow is an isomorphism since ̃ h ∶ L ◁ var → K ◁ var sends the cone pointto the cone point. It then follows that the middle vertical map is an bicategoricalequivalence. Now by definition g is a var-limit cone if and only if the top righthorizontal map is a bicategorical equivalence and that g ̃ h is a var-limit cone if andonly if the bottom right horizontal map is a bicategorical equivalence. To finish theproof it will hence suffice to verify that the rightmost vertical map is a bicategoricalequivalence. Indeed, this follows from Corollary 4.4.6 since h is a var-cocartesianequivalence over C . (cid:3) Remark . In light of Proposition 5.2.3, there is no real restriction of general-ity in considering limits and colimits just for diagrams indexed by marked-scaledsimplicial sets whose underlying scaled simplicial sets are ∞ -bicategories. Indeed,if K is a marked-scaled simplicial set and f ∶ K → C is a diagram valued in an ∞ -bicategory C , then we can factor f as K → E → C where E is an ∞ -bicategory and themap K → E is bicategorical trivial cofibration. Setting E to be the marked-scaledsimplicial set whose underlying scaled simplicial set is E and whose marked edgesare the images of the marked edges in K , we get that K → E is a var-(co)cartesianequivalence over C (see Remark 4.4.7), and hence by Proposition 5.2.3 one may aswell replace K with E when considering 2-(co)limits of f .The notion of cofinality can also be used to obtain an equivalent characterizationof 2-(co)limit cones: Proposition 5.2.5.
Let C be an ∞ -bicategory, K a marked-scaled simplicial setand f ∶ K → C a diagram. Then for a var -cone g ∶ K ◁ var → C extending f the followingare equivalent:(1) g is a var -limit cone.(2) g is a universal object of C / f var with respect to the projection C / f var → C .(3) The inclusion { g } ⊆ C / f var is var -cofinal.Proof. We first note that ( ) ⇔ ( ) by Lemma 5.2.2. We now show that ( ) ⇔ ( ) .Let id g ∈ C / g var be the vertex determined, in the var = inn case, by the composed map ♭ ∆ ⊗ K ◁ inn → K ◁ inn → C where the first map is induced by the map ♭ ∆ ⊗ ♭ ∆ → ♭ ∆ given on vertices by ( i, j ) ↦ max ( i, j ) and the second determined by g , and in the var = out case by thecomposed map K ◁ out ⊗ ♭ ∆ → K ◁ out → C where the first map is induced by the map ♭ ∆ ⊗ ♭ ∆ → ♭ ∆ given on verticesby ( i, j ) ↦ min ( i, j ) and the second determined by g . Combining Corollary 4.3.7and Proposition 4.3.8 with Corollary 5.1.7 we may conclude that the inclusion { id g } ⊆ C / g var is a var-cartesian equivalence over C . It then follows that g ∈ C / f var isuniversal over C if and only if C / g var → C / f var is a var-cartesian equivalence over C . Butthe latter is a map of var-cartesian fibrations over C (with marked arrows being thecartesian ones), and is hence a var-cartesian equivalence over C if and only if it is a IBRATIONS AND LAX LIMITS OF (∞ , ) -CATEGORIES 67 bicategorical equivalence on underlying scaled simplicial sets. This shows that (1)and (2) are equivalent. (cid:3) Corollary 5.2.6.
Let C be an ∞ -bicategory, K a marked-scaled simplicial set and f ∶ K → C a diagram. Then the sub-bicategory of C / f var spanned by var -limit conesand cartesian arrows between them is a contractible Kan complex.Proof. Combine Proposition 5.2.5 and Proposition 4.3.16. (cid:3)
We consider Proposition 5.2.5 as an analogue for 2-(co)limit of the fact that inthe ∞ -categorical setting, limit cones are final objects in the ∞ -category of cones,and in particular the singleton inclusion they determine is cofinal. We note howevertwo importance differences between that case and the present one:(1) In the ∞ -bicategorical setting, 2-limit cones are not necessarily final objects in C / f var (in particular, being final is not implied by the inclusion { g } ⊆ C / f var beingvar-cofinal).(2) The ∞ -bicategorical notion of cofinality is defined for marked-scaled simplicialsets, and hence depends on a choice of marked edges. In particular, while inthe ∞ -categorical case the collection of limit cones can be recovered just fromthe information of the ∞ -category of cones (as the collection of final objects),here one has to take into account the ∞ -bicategory C / f var of cones together withits collection of marked edges.5.3. Weighted (co)limits.
In this section we will consider a particular case of2-(co)limits which corresponds to the classical notion of weighted (co)limits . Wewill then show that this particular case is in some sense completely general: everytype of 2-colimit can be described as a weighted colimit with respect to a suitableweight.
Definition 5.3.1.
Let C be an ∞ -bicategory C and I be an ∞ -bicategory equippedwith a map f ∶ I → C . Let var ∈ { inn , out } be a variance parameter. Let I var = I ifvar = inn and I var = I co if var = out. We define weighted var-(co)limits as follows:(1) For a weight w ∶ I var → C at ∞ classified by a var-cocartesian fibration p ∶ ˜ I → I wedefine the w -weighted var -limit of f to be the var-limit of f ○ p ∶ ˜ I → C .(2) For a weight w ∶ ( I var ) coop → C at ∞ classified by a var-cartesian fibration p ∶ ˜ I → I we define the w -weighted var -colimit of f to be the var-colimit of f ○ p ∶ ˜ I → C .In all the above cases the 2-(co)limit is taken with respect to the marking ˜ I ♮ con-sisting of the p -(co)-cartesian edges . Remark . It follows from Corollary 5.1.13 that the notion of an I -indexedinner limit in C with respect to a weight w ∶ I → C at ∞ is the same as the notion ofan I co -indexed outer limit in C co with respect to the weight w ∶ I = ( I co ) co → C at ∞ .Similarly, by Remark 5.1.5 these are equivalent to the notions of I op -indexed innercolimits in C op and I coop -indexed outer colimits in C coop with respect to the weight ( − ) op ○ w ∶ I co → C at ∞ . Proposition 5.3.3.
Let C be an ∞ -bicategory and f ∶ I → C and w ∶ I → C at ∞ twofunctors of ∞ -bicategories. Then the limit ℓ ∈ C of f weighted by w is characterizedby a natural equivalence of ∞ -categories Hom C ( x, ℓ ) ≃ Nat I ( w, Hom C ( x, f ( − ))) for x ∈ C , where the right hand side denotes the mapping category in the ∞ -bicategory Fun ( I , C ) between w and the restriction along f of the functor representedby x . Remark . Taking into account the behavior of weighted (co)limits under changeof variance we may dualize the statement of Proposition 5.3.3 and obtain that theouter limit ℓ of a diagram f ∶ I → C weighted by w ∶ I co → C at is characterized by anatural equivalenceHom C ( x, ℓ ) ≃ Nat I ( w ( − ) op , Hom C ( x, f ( − ))) Similarly, the outer colimit c of f weighted by w ∶ I op → C at and the inner colimit c ′ of f weighted by w ′ ∶ I coop → C at are characterized by equivalencesHom C ( c, x ) ≃ Nat I op ( w ( − ) , Hom C ( f ( − ) , x )) and Hom C ( c, x ) ≃ Nat I op ( w ′ ( − ) op , Hom C ( f ( − ) , x )) , respectively. Remark . When the indexing ∞ -category I is an ∞ -category and the ∞ -bicategory C is obtained from a presentable ∞ -category tensored over C at ∞ , weightedlimits were previously defined and studied by Gepner–Haugseng–Nikolaus [9]. Themapping property of Proposition 5.3.3 can then be used to show that the presentdefinition coincides with the one of [9], whenever the latter is defined, by expressingspaces of natural transformations in terms of limits over the twisted arrow category.We will prove Proposition 5.3.3 below. Before that, let us establish some ter-minology that will be convenient for handling weighted (co)limits. Following Rov-elli [17], we generalize the definition of the thick join to allow for weights to beconsidered. In what follows, we fix a variance parameter var ∈ { inn , out } . Definition 5.3.6.
Given a var-cocartesian fibration p ∶ ˜ I → I we define the p -weighted var -cone of I , to be the object at the bottom right corner in the pushoutdiagrams displayed below: ˜ I ∆ ◇ var ˜ I ♮ I ∆ ◇ p var I . p where ˜ I ♮ is the marked-scaled simplicial set whose underlying simplicial set is I and whose marked edges are the p -cocartesian ones. Similarly, for a var-cartesianfibration p ∶ ˜ I → I we define I ◇ p var ∆ using a similar pushout square.Now suppose that C is an ∞ -bicategory, f ∶ I → C is a diagram and p ∶ ˜ I → I is avar-cocartesian fibration, considered as a weight. By definition, a candidate for thecorresponding weighted var-limit of f is given by a cone of the form g ∶ ∆ ◇ var ˜ I ♮ → C such that g ∣ ˜ I = f p . In particular, such a g always factors through the weighted cone∆ ◇ p var I . Definition 5.3.7.
We will say that a map g ∶ ∆ ◇ p var I → C is a weighted var -limitcone if its restriction to ∆ ◇ var ˜ I ♮ is a var-limit cone with respect to the markingof ˜ I ♮ . Similarly, we say that a map g ∶ I ◇ p var ∆ → C is a weighted var -colimit cone ifits restriction to ˜ I ♮ ◇ var ∆ is a var-colimit cone with respect to the marking of ˜ I ♮ .We now observe that the weighted inner cone ∆ ◇ p inn I also appears in the formulafor the scaled straightening functor recalled in § S t I ( ˜ I ) ∶ C sc ( I ) → S et + ∆ is defined as the restriction to C sc ( I ) of the functor C sc ( ∆ ◇ p inn I ) → S et + ∆ corepre-sented by the cone point ∗ . Let B be an ∞ -bicategory equipped with a bicategorical IBRATIONS AND LAX LIMITS OF (∞ , ) -CATEGORIES 69 trivial cofibration ∆ ◇ p inn I ↪ B . Since representable functors unstraighten to slicefibrations (see Remark 4.2.7) and using the compatibility of unstraightening withbase change we conclude that the derived unstraightening of the straightening of˜ I → I can be identified, up to weak equivalence, with the base change B ∗/ inn × B I → I of the slice fibration B ∗/ inn → B . In particular, there is a natural equivalence˜ I B ∗/ inn × B II ≃ . We may then recover the functor w ∶ I → C at ∞ classifying ˜ I → I as the composite I → B Hom B (∗ , −) ÐÐÐÐÐÐ→ C at ∞ where the second map is the functor corepresented by ∗ . Any weighted inner cone∆ ◇ p inn I → C taking values in an ∞ -bicategory C must extend to a map B → C inan essentially unique fashion. Definition 5.3.8.
We will say that a map g ∶ B → C exhibits g ( ∗ ) as the w -weightedlimit of f if its restriction to ∆ ◇ inn ˜ I ♮ is an inner limit cone. Proposition 5.3.9.
Keeping the above notations, a map g ∶ B → C extending f ∶ I → C exhibits g ( ∗ ) as the w -weighted limit of f if and only if for every x ∈ C therestriction map Fun coc B ( B ∗/ inn , C x / inn × C B ) → Fun coc I ( ˜ I ♮ , C x / inn × C I ) is an equivalence of ∞ -categories.Proof. Consider the composite g ′ ∶ ( ˜ I ♮ ) ◁ inn → B → C . By Corollary 5.1.11 we havethat g ′ is an inner colimit cone if and only if the mapFun coc C (( ˜ I ♮ ) ◁ C x / inn ) → Fun coc C ( ˜ I ♮ , C x / inn ) is an equivalence of ∞ -categories for every x ∈ C . Since g ′ factors by constructionthrough g ∶ B → C we may identify this map with the map(40) Fun coc B (( ˜ I ♮ ) ◁ , C x / inn × C B ) → Fun coc B ( ˜ I ♮ , C x / inn × C B ) . Now as in the proof of [13, Proposition 4.1.8] (whose outer counterpart was spelledout in the proof of Proposition 4.3.6), the ∞ -bicategory B ∗/ inn admits a lax contrac-tion ( B ∗/ inn ) ◁ inn → B ∗/ inn to the the point id ∗ . We may thus extend the map˜ I ♮ ≃ Ð→ B ∗/ inn × B I → B ∗/ inn to a map ( ˜ I ♮ ) ◁ inn → B ∗/ inn sending the cone point to id ∗ . By Lemma 5.1.6, Proposition 4.3.6 and 2-out-of-3property the last map is an inner cocartesian equivalence over B . We may thusidentify the map (40) with the mapFun coc B ( B ∗/ inn , C x / inn × C B ) → Fun coc I ( ˜ I ♮ , C x / inn × C I ) , and so the desired statement follow. (cid:3) Proof of Proposition 5.3.3.
The map in Proposition 5.3.9 fits in a zig-zag diagramFun coc B ( B ∗/ inn , C x / inn × C B ) C x / inn × C { g ( ∗ )} Fun coc I ( ˜ I ♮ , C x / inn × C I ) ≃ where the left diagonal map is an equivalence by Proposition 4.3.6. Applying thestraightening-unstraightening equivalence we obtain a zig-zagNat B ( Hom B ( ∗ , − ) , Hom C ( g ( − ) , x )) Hom C ( x, g ( ∗ )) Nat I ( w, Hom C ( f ( − ) , x )) ≃ , where the right diagonal map is induced by the identification w ≃ Hom B ( ∗ , − ) ∣ I .Since all these maps are natural in x we may now conclude that g exhibits g ( ∗ ) asthe w -weighted limit of f ∶ I → C if and only if it exhibits g ( ∗ ) as representing thefunctor x ↦ Nat I ( w, Hom C ( f ( − ) , x )) , as desired. (cid:3) Our next goal is to show that any type of 2-(co)limit can be replaced with anequivalent weighted (co)limit. We will argue this first for inner 2-limits, but willexplain afterwards how one can obtain the statement for all variances using the ( Z / ) -symmetry on BiCat ∞ .Let I be an ∞ -bicategory and E a collection of edges in I . Denote by I + themarked-scaled simplicial set having I as underlying scaled simplicial set and E asa set of marked edges. Suppose that f ∶ I → C is a diagram in an ∞ -bicategory C , to which we can associate the corresponding inner 2-limit with respect to themarking E . Let us now consider the underlying marked-simplicial set of I + as anobject in the P I -fibered model structure on ( S et + ∆ ) / I . Taking a fibrant replacementwith respect to this model structure, we may find an inner cocartesian fibration p ∶ ˜ I → I and a map of marked-scaled simplicial sets ι ∶ I + → ˜ I ♮ whose underlying mapof marked simplicial sets is a P I -fibered trivial cofibration. Let w ∶ I → C at ∞ be thefunctor classifying p . We then have the following: Proposition 5.3.10.
A map g ∶ ∆ ◇ p inn I → C exhibits g ( ∗ ) as the w -weighted innerlimit of f if and only if its restriction to ∆ ◇ inn I + is an inner limit cone.Proof. This follows directly from Proposition 5.2.3 since the map ι ∶ I + → ˜ I ♮ is aninner cocartesian equivalence over C . (cid:3) We note that while Proposition 5.3.10 is phrased for inner limits, the same ideaapplies to all types of 2-(co)limits. This might not be clear at first sight, since wehave used the P I -fibered model structure to construct the map ι ∶ I + → ̃ I ♮ , and thismodel structure has been elaborated only in the inner cocartesian case. Nonethe-less, the only property of ι that we actually needed is that it is an inner cocartesianequivalence whose target is an inner cocartesian fibration. The P I -fibered modelstructure was used to show the existence of such a map, for any given markingon I . However, given that it exists in the inner cocartesian setting implies thatit exists in all four variances. Indeed, it follows from Corollary 3.2.4 that for avariance parameter var ∈ { inn , out } , the functor ( − ) op sends var-cocartesian equiv-alences/fibrations over I to var-cartesian equivalences/fibrations over I op , and viceversa. Similarly, the functor ( − ) co sends inner (co)cartesian equivalences/fibrationsover I to outer (co)cartesian equivalences/fibrations over I co . The existence of amap of the form ι in the inner cocartesian context implies its existence for the other IBRATIONS AND LAX LIMITS OF (∞ , ) -CATEGORIES 71 variances as well. We may summarize the resulting statement for all four variancesas follows: Corollary 5.3.11.
Let var ∈ { inn , out } be a variance parameter. Let I be an ∞ -bicategory and I + a marked-scaled simplicial set whose underlying scaled sim-plicial set is I . Then there exists a var -(co)cartesian fibration p ∶ ̃ I → I and a var -(co)cartesian equivalence ι ∶ I + → ̃ I ♮ over I . Let w denote the weight associatedto p . Then a map g ∶ ∆ ◇ p var I → C exhibits g ( ∗ ) as the w -weighted var -limit of g ∣ I if and only if its restriction to ∆ ◇ var I + is a var -limit cone. Dually, a map g ∶ I ◇ p var ∆ → C exhibits g ( ∗ ) as the w -weighted var -colimit of g ∣ I if and only if itsrestriction to I + ◇ var ∆ is a var -colimit cone. Corollary 5.3.12. An ∞ -bicategory C admits all small -(co)limits if and only ifit admits all small weighted limits. Comparison with model categorical weighted limits.
In this sectionwe consider the case where the ∞ -bicategory C comes from a model category M tensored over the marked-categorical model structure on S et + ∆ , that is, M admits aclosed action of S et + ∆ via a left Quillen bifunctor S et + ∆ × M → M ( K, X ) ↦ K ⊗ X, whose adjoints in each variable give a cotensor operation ( K, X ) ↦ X K ∈ M and anenrichment ( X, Y ) ↦ M ( X, Y ) ∈ S et + ∆ . In particular, we may consider M as a S et + ∆ -enriched category. The full subcategory M ○ ⊆ M spanned by the fibrant-cofibrantobjects is then fibrant as an S et + ∆ -enriched category, and we may consider its scalednerve M ∞ ∶ = N sc ( M ○ ) , which is an ∞ -bicategory.Our goal in this section is to show that M ∞ admits small inner and outer(co)limits and that, furthermore, these inner and outer (co)limits can be expressedas weighted homotopy (co)limit of a suitable S et + ∆ -enriched diagram in M . Toformulate our statement we will need a few preliminaries. Recall that the cate-gory S et + ∆ is cartesian closed, and so in particular for every two marked simplicialsets X, Y we have an internal mapping object Map ( X, Y ) ∈ S et + ∆ determined by auniversal property of the formHom S et + ∆ ( K, Map ( X, Y )) ≅ Hom S et + ∆ ( K × X, Y ) . If J is a small S et + ∆ -enriched category then the category ( S et + ∆ ) J of S et + ∆ -enrichedfunctors J → S et + ∆ inherits a tensor structure over S et + ∆ given by ( K × F )( i ) = K × F ( i ) , with a compatible enrichment Nat J ( F , G ) ∈ S et + ∆ determined by the universalproperty Hom S et + ∆ ( K, Nat J ( F , G )) ≅ Hom ( S et + ∆ ) J ( K × F , G ) . We may consider Nat J ( F , G ) as the marked simplicial set of natural transformationsfrom F to G . It also admits an explicit description as the equalizer of the followingpair of parallel maps ∏ i ∈ J Map ( F ( i ) , G ( i )) ∏ i,j ∈ J Map ( Hom J ( i, j ) × F ( i ) , G ( j )) . Let us now recall the definition of weighted limits and colimits in the setting of S et + ∆ -enriched categories. Definition 5.4.1.
Let C be a S et + ∆ -enriched category, F ∶ J → C be an enrichedfunctor and W ∶ J → S et + ∆ an enriched functor. For an object X ∈ C let us denoteby F X ∶ J → S et + ∆ the functor given by F X ( i ) = C ( X, F ( i )) . Given an object Z ∈ C we will say that a natural transformation τ ∶ W ⇒ F Z exhibits Z as the W -weightedlimit of F if for every object Y ∈ C the composed map(41) Hom C ( Y, Z ) Nat J ( F Z , F Y ) Nat J ( W, F Y ) τ ∗ is an isomorphism of marked simplicial sets. In this case we will also say that τ exhibits Z as the W -weighted colimit of F op ∶ J op → C op . Remark . In the setting of Definition 5.4.1, let J ◁ W denote the S et + ∆ -enrichedcategory whose objects are { ∗ } ∪ Ob ( J ) , and such that J ◁ W ( i, j ) = J ( i, j ) , J ◁ W ( ∗ , i ) = W ( i ) , J ◁ W ( i, ∗ ) = ∅ and J ◁ W ( ∗ , ∗ ) = ∆ , and where the composition is given by thefunctorial dependence of W ( i ) on i . Then the data of a natural transformation ofthe form τ ∶ W ⇒ F X is equivalent to the data of a functor G ∶ J ◁ W → C which sends ∗ to X . Furthermore, in this case we may identify the map (41) with the map C ( Y, X ) ≅ Nat J ◁ W ( J ◁ W ( ∗ , − ) , G Y ) → Nat J ( W, F Y ) where the isomorphism is given by the Yoneda lemma and the map is obtained byrestriction along J ↪ J ◁ W .Let us now fix a S et + ∆ -enriched category J such that the projective model struc-ture on M J exists. In this case, we have a left Quillen bifunctor ( S et + ∆ ) J × M → M J ( G , X )( i ) ↦ G ( i ) ⊗ X where ( S et + ∆ ) J is endowed as well with the projective model structure (this onealways exists since S et + ∆ is combinatorial). Given enriched functors G ∶ J → S et + ∆ and F ∶ J → M , let us denote by F G ∈ M the image of F under the right adjoint of G ⊗ ( − ) ∶ M → M J . Similarly, given an object X ∈ M and a functor F ∶ J → M let us denote by F X ∈ ( S et + ∆ ) J the image of F under the right adjoint of ( − ) ⊗ X ∶ ( S et + ∆ ) J → M J . We note that F X is given by F X ( i ) = Hom M ( X, F ( i )) and so this notation isconsistent with the notation of Definition 5.4.1.For an object W of ( S et + ∆ ) J and an object F of M J , we have a canonical naturaltransformation W → F F W given by the image of the identity morphism via thenatural isomorphismsHom M ( F W , F W ) ≅ Hom M J ( W ⊗ F W , F ) ≅ Nat J ( W, F F W ) . Replacing the first argument of Hom M ( F W , F W ) with any object Y of M , we getthe isomorphism Hom M ( Y, F W ) ≅ Nat J ( W, F Y ) . Unwinding the definitions, this shows the following standard result:
Lemma 5.4.3.
Let F ∶ J → M and W ∶ J → S et + ∆ be enriched functors. Then thenatural transformation W ⇒ F F W exhibits F W ∈ M as the W -weight limit of F . When W is projectively cofibrant the assignment F ↦ F W is a right Quillenfunctor. In this case we will refer to ( F fib ) W (where ( − ) fib denotes a projectivefibrant replacement) as the W -weighted homotopy limit of F . More generally, itwill be useful to adopt the following more flexible terminology: IBRATIONS AND LAX LIMITS OF (∞ , ) -CATEGORIES 73 Definition 5.4.4.
We will say that a natural transformation τ ∶ W ⇒ F Z exhibits Z as the W -weighted homotopy limit of F if for every object Y ∈ M the composedmap M ( Y, Z ) → Nat J ( W, F Y ) → Nat J ( W, ( F fib ) Y ) is a weak equivalence in S et + ∆ (that is, a marked categorical equivalence). Remark . In the situation of Definition 5.4.4, it does not matter which fibrantreplacement F fib is used. In particular, if F is already projectively fibrant then wecan take F fib = F , in which case a natural transformation τ ∶ W ⇒ F Z exhibits Z asthe W -weighted homotopy limit of F if and only if the map M ( Y, Z ) → Nat J ( W, F Y ) is a marked categorical equivalence.We now consider the following setup. Let J be a fibrant S et + ∆ -enriched categorysuch that the projective model structure on M J exists, and I an ∞ -bicategoryequipped with a bicategorical equivalence φ ∶ C sc ( I ) ≃ Ð→ J . Let W ∶ J → S et + ∆ be a enriched functor which is fibrant and cofibrant with respectto the projective model structure, p ∶ ˜ I → I an inner cocartesian fibration and ψ ∶ S t sc φ ( ˜ I ♮ ) → W a weak equivalence in ( S et + ∆ ) J . Finally, fix a diagram F ∶ J → M ○ , and let f ∶ I → M ∞ be the adjoint of F φ ∶ C sc ( I ) → M ○ . Let w ∶ I → C at ∞ be the functor classifying p ,which in light of the equivalence ψ we can identify with the restriction to I of thefunctor N sc ( W ) ∶ N sc ( J ) → N sc ( M ○ ) = M ∞ induced by W . In this situation we would like to obtain a comparison between the W -weighted limit of F and the w -weighted inner limit of f .Let J ◁ W be as in Remark 5.4.2. Then the weak equivalences φ ∶ C sc ( I ) → J and ψ ∶ S t sc φ ( ˜ I ♮ ) → W determine a commutative square C sc ( I ) φ ≃ / / (cid:15) (cid:15) J (cid:15) (cid:15) C sc ( ∆ ◇ p inn I ) C sc ([ ∆ ◇ inn ˜ I ♮ ] ∐ ˜ I I ) ≃ / / J ◁ W with horizontal legs weak equivalences. Now since W is assumed fibrant I ◁ W isa fibrant S et + ∆ -enriched category. Let B ∶ = N sc ( J ◁ W ) be its scaled nerve, so that B is an ∞ -bicategory and the lower horizontal map in the above square gives abicategorical equivalence ∆ ◇ p inn I = [ ∆ ◇ inn ˜ I ♮ ] ∐ ˜ I I ≃ Ð→ B . Our comparison statement then takes the following form:
Proposition 5.4.6.
Keeping the assumptions and notations above, let F ∶ J → M ○ be a levelwise fibrant functor, Z ∈ M ○ an object and τ ∶ W ⇒ F Z a natural trans-formation, corresponding to an enriched functor G ∶ J ◁ W → M ○ . Then τ exhibits Z as the W -weighted homotopy limit of F if and only if the adjoint map g ∶ B → M ∞ exhibits Z as the w -weighted inner limit (in the sense of Definition 5.3.8) of thecomposite f ∶ I Ð→ N sc ( J ) N sc ( F ) ÐÐÐÐ→ M ∞ . Proof.
Since F is levelwise fibrant Remark 5.4.5 and Remark 5.4.2 tell us that τ exhibits Z as the W -weighted homotopy limit of F if and only if the map(42) M ( Y, Z ) ≅ Nat J ◁ W ( J ◁ W ( ∗ , − ) , M ( Y, G ( − ))) → Nat J ( W, M ( Y, F ( − ))) is a marked categorical equivalence for every Y ∈ M ○ . We note that since Y, Z and W are fibrant-cofibrant the marked simplicial sets appearing in (42) are fibrant.Unstraightening along φ ∶ C sc ( I ) → J and the counit map C sc ( B ) → I ◁ W we mayidentify the map of simplicial sets underlying (42), up to categorical equivalence,with the map Fun coc B ( B ∗/ inn , M Y /∞ × M ∞ B ) → Fun coc I ( ˜ I ♮ , M Y /∞ × M ∞ I ) . This last map is a categorical equivalence if and only if g ∶ B → M ∞ exhibits Z asthe w -weighted inner limit of f by Proposition 5.3.9. (cid:3) Using Proposition 5.3.10 we would now like to deduce a result for general 2-limits. Let I , J and φ ∶ C sc ( I ) ≃ Ð→ J be as above, and suppose we are given a markingon I , that is, a marked-scaled simplicial set I + whose underlying scaled simplicialset is I . As above, we then let W ∶ J → S et + ∆ be a fibrant-cofibrant functor equippedwith a weak equivalence ψ ∶ S t sc φ ( I + ) ≃ Ð→ W. Let F ∶ J → M ○ be an enriched diagram and f ∶ I → N sc ( J ) → M ∞ the resulting ∞ -bicategorical diagram. We would like to describe inner 2-limits of f with respectto the marking I + in terms of the W -weighted homotopy limit of F . In particular,given an object Z ∈ M ○ and a natural transformation τ ∶ W ⇒ F Z , we may considerthe associated functor G ∶ I ◁ W → M ○ . On the other hand, the map ψ above encodesa weak equivalence of S et + ∆ -enriched categories ψ ◁ ∶ C sc ( ∆ ◇ inn I + ) ≃ Ð→ I ◁ W , by which G determines an inner cone∆ ◇ inn I + / / g N sc ( I ◁ W ) N sc ( G ) / / M ∞ Proposition 5.4.7.
Keeping the assumptions and notations above, the naturaltransformation τ exhibits Z as the W -weighted homotopy limit of F if and only if g is an inner limit cone.Proof. Proceeding similarly to § p ∶ ˜ I → I equipped with a map ι ∶ I + → I ♮ whose underlying map of marked simplicial sets isa P I -fibered trivial cofibration. The S t sc φ ( I + ) → S t sc φ ( I ♮ ) is then a trivial cofibrationin ( S et + ∆ ) J , and hence the map ψ can be factored as S t sc φ ( I + ) ≃ Ð→ S t sc φ ( I ♮ ) ψ ′ Ð→ W. This factorization then translates to a factorization of ψ ◁ as C sc ( ∆ ◇ inn I + ) ≃ Ð→ C sc ( ∆ ◇ p inn I ) ≃ Ð→ I ◁ W . Setting B ∶ = N sc ( I ◁ W ) we then obtain a sequence of maps∆ ◇ inn I + ≃ Ð→ ∆ ◇ p inn I ≃ Ð→ B → M ∞ . By Proposition 5.4.6 we then have that τ exhibits Z as the W -weighted limit of F if and only if the map ∆ ◇ p inn I → M ∞ is a weighted inner limit cone. On the otherhand, by Proposition 5.3.10, the latter statement is equivalent to g ∶ ∆ ◇ inn I + → M ∞ being an inner limit cone, and so the proof is complete. (cid:3) IBRATIONS AND LAX LIMITS OF (∞ , ) -CATEGORIES 75 Remark . If M is a model category tensored over S et + ∆ , then we may considerthe S et + ∆ -tensored model category M co , whose underlying model category is thesame as M , and whose tensor structure S et + ∆ × M co → M co is given by ( K, M ) ↦ K op × M . Then the fibrant S et + ∆ -enriched category ( M co ) ○ isobtained from M ○ by applying ( − ) op on all mapping objects, and so ( M co ) ∞ ≃ ( M ∞ ) co . In addition, for every J -enriched category we have a natural isomorphism of S et + ∆ -enriched model categories ( M co ) J co ≃ ( M J ) co and so the projective model structure on ( M co ) J co exists if and only if the projec-tive model structure on M J -exists. When J is fibrant we may in this case applyProposition 5.4.7 to M co and J co (and take I ≃ N sc ( J co ) ). Interpreting the resultingstatement in terms of M ∞ and using Corollary 5.1.13 we get that outer limits in M ∞ can be computed as weighted limits in M co . Remark . If M is a model category tensored over S et + ∆ then the model category M op (equipped with the opposite model structure in the which the fibrations arewhat used to be cofibrations, and vice versa), is also canonically tensored over S et + ∆ ,whose tensor operation S et + ∆ × M op → M op is now given by the cotensor operation ( K, X ) ↦ X K we had before. Similarly,the cotensor operation on M op is given by the tensor operation on M , and theenrichment in the two cases coincides to the extent that M op ( X, Y ) ≅ M ( Y, X ) . Inparticular, M op is also the opposite of M as an S et + ∆ -enriched category.In the situation of Proposition 5.4.7, if the projective model structure on ( M op ) J exists (equivalently, if the injective model structure on M J exists), then we mayapply that proposition to M op . Interpreting the resulting statement in terms of M ∞ and using Remark 5.1.5 we get that inner colimits in M ∞ can be computed as weighted homotopy colimits in M . Remark . Combining Remarks 5.4.9 and 5.4.8 we may similarly identify outercolimits in terms of weighted homotopy colimits in M co , assuming the relevantinjective model structure exists. Corollary 5.4.11.
Let M be an S et + ∆ -tensored model category such that the pro-jective (resp. injective) model structure exists on M J for any small S et + ∆ -enrichedcategory J (e.g, M is a combinatorial model category). Then the ∞ -bicategory M ∞ admits inner and outer limits (resp. colimits) indexed by arbitrary small marked-scaled simplicial sets, and these are computed by taking weighted homotopy limits(resp. colimits) in M .Proof. By Remark 5.4.9, 5.4.8 and 5.4.10 it will suffice to prove the case of innerlimits. Let f ∶ I → M ∞ be a diagram. We may then find a fibrant S et + ∆ -enrichedcategory J and a trivial cofibration φ ∶ C sc ( I ) → J . Since M ○ is fibrant as a S et + ∆ -enriched category the transposed enriched functor C sc ( I ) → M ○ then factors throughan enriched functor F ∶ J → M ○ . We now choose a trivial cofibration S t sc φ ( I + ) ↪ W with W a projectively fibrant (and cofibrant, since S t sc φ ( I + ) is cofibrant) enrichedfunctor J → S et + ∆ . Since F is levelwise fibrant by construction, its strict W -weightedlimit is also a weighted homotopy limit. Any τ ∶ Z → F W exhibiting such a W -weighted homotopy limit gives rise to an inner limit cone on f by Proposition 5.4.7,and so the desired result follows. (cid:3) Example ∞ -categories) . A fundamental example of an S et + ∆ -tensored model category is S et + ∆ itself, which is in particular combinatorial andhence admits all projective and injective model structures. We may then concludefrom Corollary 5.4.11 that C at ∞ ≃ ( S et + ∆ ) ∞ admits all small 2-(co)limits, and thatthose can be computed as weighted homotopy limits and colimits in S et + ∆ . On theother hand, given that we know that 2-limits exists, an explicit description of themcan be deduced from their universal property of Corollary 5.1.8. Indeed, taking x = ∆ in that corollary we deduce that if K is a marked-scaled simplicial set and f ∶ K → C at ∞ a diagram thenlim inn K f ≃ Nat gr K ( ∗ , f ) and lim out K f ≃ Nat opgr K ( ∗ , f ) , where for var ∈ { inn , out } we use the notation lim var K f to indicate the image ofthe cone point under any var-limit cone extending f . Alternatively, using thedescription of Corollary 4.4.3, we may write this identification aslim inn K f ≃ Fun coc K ( K, E inn f ) and lim out K f ≃ Fun coc K ( K, E out f ) op , where E inn f → K is the inner cocartesian fibration classified by f and E out f → K isthe outer cocartesian fibration classified by f ∶ K co → C at co ∞ (−) op ÐÐÐ→ C at ∞ . References
1. Dimitri Ara,
Higher quasi-categories vs higher Rezk spaces , J. K-Theory (2014), no. 3,701–749.2. Clark Barwick and Christopher Schommer-Pries, On the unicity of the homotopy theory ofhigher categories , arXiv:1112.0400, 2011, preprint.3. Julia E. Bergner and Charles Rezk,
Comparison of models for (∞ , n ) -categories, I , Geom.Topol. (2013), no. 4, 2163–2202.4. Julia E. Bergner and Charles Rezk, Comparison of models for (∞ , n ) -categories, ii ,arXiv:1406.4182, 2014.5. Andrea Gagna, Yonatan Harpaz, and Edoardo Lanari, Gray tensor products and lax functorsof (∞ , ) -categories , arXiv:2006.14495, 2020.6. , On the equivalence of all models for (∞ , ) -categories , arXiv:1911.01905, 2020.7. Dennis Gaitsgory and Nick Rozenblyum, A study in derived algebraic geometry I & II , Math-ematical Surveys and Monographs, vol. 221, American Mathematical Soc., 2017.8. David Gepner and Rune Haugseng,
Enriched ∞ -categories via non-symmetric ∞ -operads ,Advances in mathematics (2015), 575–716.9. David Gepner, Rune Haugseng, and Thomas Nikolaus, Lax colimits and free fibrations in ∞ -categories , Doc. Math. (2017), 1225–1266.10. Yonatan Harpaz, Joost Nuiten, and Matan Prasma, Quillen cohomology of ( infty, ) -categories , Higher Structures (2019), no. 1, 17—-66.11. Rune Haugseng, On (co)ends in ∞ -categories , arXiv:2008.03758, 2020.12. Mark Hovey, Model categories , Mathematical Surveys and Monographs, vol. 63, AmericanMathematical Society, 1999.13. Jacob Lurie, (∞ , ) -categories and the Goodwillie Calculus I , Preprint.Available at author’s website.14. , Higher Topos Theory , Annals of Mathematics Studies, vol. 170, Princeton UniversityPress, Princeton, NJ, 2009.15. Jacob Lurie,
Kerodon , https://kerodon.net , 2018.16. Victoriya Ozornova and Martina Rovelli, Model structures for (∞ , n ) -categories on(pre)stratified simplicial sets and prestratified simplicial spaces , 2020, pp. 1543–1600.17. Martina Rovelli, Weighted limits in an (∞ , ) -category , arXiv:1902.00805, 2019, preprint.18. Dominic Verity, Complicial sets characterising the simplicial nerves of strict ω -categories ,Mem. Amer. Math. Soc. (2008), no. 905, xvi+184.19. , Weak complicial sets. I. Basic homotopy theory , Adv. Math. (2008), no. 4,1081–1149.
IBRATIONS AND LAX LIMITS OF (∞ , ) -CATEGORIES 77 Institute of Mathematics, Czech Academy of Sciences, ˇZitn´a 25, 115 67 Praha 1,Czech Republic
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