Enhanced twisted arrow categories
aa r X i v : . [ m a t h . C T ] S e p Enhanced twisted arrow categories
Fernando Abellán García and Walker H. Stern
Abstract
Given an ∞ -bicategory D with underlying ∞ -category D , we construct a Cartesian fibra-tion Tw( D ) → D × D op , which we call the enhanced twisted arrow ∞ -category, classifying therestricted mapping category functor Map D : D op × D → D op × D → C at ∞ . With the aid ofthis new construction, we provide a description of the ∞ -category of natural transformationsNat( F, G ) as an end for any functors F and G from an ∞ -category to an ∞ -bicategory. As anapplication of our results, we demonstrate that the definition of weighted colimits presented inarXiv:1501.02161 satisfies the expected 2-dimensional universal property. Contents
Introduction 2
The twisted arrow category . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2Towards an enhanced twisted arrow category . . . . . . . . . . . . . . . . . . . . . . . . . 3Applications: The category of natural transformations as an end . . . . . . . . . . . . . . 4Structure of the paper . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
Tw( C ) ∞ -categories . . . . . . . . . . . . . . . . . . . . . 321 ntroduction Of the many tools belonging to the study of categories, perhaps the most key is the Yoneda lemma.The fully faithfulness of the functor C Set C x h x : = Hom C ( − , x )means, in particularly, that we can view functors f : C op Set as universal properties , andthereby uniquely specify an object x by requiring h x ∼ = f .In the higher-categorical realm, the good news is that this result still holds. The ( ∞ , Y : C S C is fully faithful (c.f. e.g. [Lur09, 5.1.3.1]). While this is auspicious for the study of universal propertiesas described above, it comes with a significant complication. The standard presentation of the targetcategory S C (which is also written variously as P ( C ) or Fun( C op , S )) is in terms of a model structureon the category Fun( C [ C ] , Set ∆ ) of simplicially enriched functors.The model Fun( C [ C ] , Set ∆ ) is extremely useful in relating the underlying ∞ -category to other ∞ -categories — for example in the proof of the ∞ -categorical Yoneda lemma. The problem arises inthat it is often extremely difficult to write down explicit simplicially-enriched functors, and explicitsimplicially-enriched natural transformations between them. When the initial definition of C is asa quasi-category, it can even be difficult to write down C [ C ] explicitly.As in so many parts of higher category theory, the way out of this dilemma is the Grothendieckconstruction. We can proceed according to the Slogan:
Cartesian fibrations and maps between them are easier to work with thanenriched functors and natural transformations between them.
From this perspective, if we want to study representable functors and universal properties, we needfirst to classify the Yoneda embedding by a fibration.
The twisted arrow category
The canonical solution to this problem is the twisted arrow ( ∞ -)category . In e.g. [Lur11] and [Cis19],it is shown that for each ∞ -category C , there is a right fibration Tw( C ) C × C op which classifies the functor Hom C : C op × C Cat ∞ .The uses of the twisted arrow category are manifold. It appears, as suggested above, in theanalysis of questions of representability throughout the higher categorical literature — e.g. in [Lur11,Lur17]. In addition, it is used to explore E k -monoidal ∞ -categories in [Lur17]. In a completelydifferent direction, there is a fundamental connection between twisted arrow categories and ∞ -categories of spans/correspondences as described in, e.g. [DK19, Ch. 10],[Bar17], and [BGN18].Moreover, this approach has been used to tackle questions related to K -theory in [Bar17]. It is worth commenting that Cisinski and several other authors tend to work with the left fibration associated tothe same functor. The difference between the two definitions amounts to an “op”, in the definition of the simplicesof Tw( C ). Throughout the paper, we will only use the Cartesian/right fibration convention, and will omit anyfurther mention of coCartesian/left fibrations. C ) over a pair ( α, β ) comprising a 1-simplex in C × C op take the form ofcoherent diagrams a ba ′ b ′ fα g β in C . In practice, this means that that the fibers have 1-simplices consisting of diagrams a b fg that commute up to a chosen 2-cell, i.e. the morphisms in the fiber can be easily interpreted as two-cells f ∼ g in C . More generally, the n -simplices of Tw( C ) are given by maps ∆ n ⋆ (∆ n ) op C ,and the projections to C and C op are induced by the inclusions ∆ n ∆ n ⋆ (∆ n ) op (∆ n ) op . Towards an enhanced twisted arrow category
Given an ∞ -bicategory C , presented as a fibrant scaled simplicial set, our aim will be to constructan ∞ -category Tw( C ) together with a Cartesian fibration Tw( C ) → C × C op which classifies thecomposite functor C op × C C op × C C at ∞ , where C at ∞ is the ( ∞ , ∞ -categories. The first step towards this construction is todecide what the 1-simplices of Tw( C ) should be. We would still like these to be something likediagrams a ba ′ b ′ fα g β in C , e.g. 3-simplices.When α and β are identities, we would like these 3-simplices to encode precisely the choiceof a 2-morphism f g . However, heuristically such a 3-simplex should, in fact, encode twofactorizations: a ba ′ b ′ f id g id a ba ′ b ′ f id g id together with the 3-simplex itself, which indicates that the composites — 2-morphisms f g — of both factorizations are equivalent. Fortunately, in the realm of scaled simplicial sets, wecan declare certain 2-simplices to be ‘thin’ — i.e., declare the corresponding 2-morphisms to beinvertible. With this in mind, we can force half of each factorization to be invertible a ba ′ b ′ f id g id (cid:9) a ba ′ b ′ f id g id (cid:9)
3n this case, we obtain two 2-morphisms f g and a 3-simplex showing that they are equivalent— precisely the data that we would like. This suggests a trial definition for the twisted arrow category of an ∞ -bicategory.The twisted arrow ∞ -bicategory T w( C ) should have n -simplices T w( C ) n := Hom Set sc∆ ((∆ n ⋆ (∆ n ) op , T ) , C )where T is the scaling given by requiring that, under the identification ∆ n ⋆ (∆ n ) op ∼ =∆ n +1 , the simplices { i, j, n + 1 − j } and { j, n + 1 − j, n + 1 − i } are thin for i < j .However, we would expect such a construction to yield a fibration over the ( ∞ , C × C op . There are technical difficulties to such a definition, not least the fact that the correspondingGrothendieck construction has not yet appeared in the literature. While we expect this definitionto yield a genuine ( ∞ ,
2) twisted arrow category, we will restrict ourselves to the examination ofthe induced functor C op × C Cat ∞ To restrict to the fibration classifying this functor, we use the expected base-change propertiesof a hypothetical Cartesian Grothendieck construction over an ∞ -bicategorical base. To wit, wedefine Tw( C ) to be the pullback Tw( C ) T w( C ) C × C op C × C op y In terms of the scaling on ∆ n ⋆ (∆ n ) op , This pullback simply amounts to requiring that every 2-simplex contained within ∆ n and every 2-simplex contained within (∆ n ) op is thin. Using pushoutsby scaled anodyne morphisms of the kind described in [GHL20, Rmk. 1.17], we can extend thisscaling to consider a cosimplicial object Q ( n ) := (∆ n ⋆ (∆ n ) op , T ) in scaled simplicial sets, wherethe non-degenerate thin simplices of T are:• 2-simplices which factor through ∆ n or (∆ n ) op .• 2-simplices ∆ { i,j, n +1 − k } and ∆ { k, n +1 − j, n +1 − i } for 0 i j k n .This is the definition of Tw( C ) we adopt throughout the present paper, which is justified by thefollowing result. Theorem 0.1.
Let C be an ∞ -bicategory. Then Tw( C ) → C × C op is a Cartesian fibration classifyingthe restricted mapping category functor Map C : C op × C C op × C C at ∞ This is an amalgam of Theorem 2.5 and Theorem 3.3 from the text.
Applications: The category of natural transformations as an end
Once verified that our definition enjoys the desired properties we turn into our main motivation forthis paper: understanding the category of natural transformations Nat(
F, G ) between functors from On thing we are glossing over is why we choose the “lower” 2-simplices as thin, rather than the “upper” ones. Ina nutshell, the reason is that the lower 2-simplices will encode composites, and thus be unique up to contractiblechoice. A likely candidate for the kind of fibration such a construction would involve is the outer Cartesian fibration of[GHL20]. ∞ -category to an ∞ -bicategory. To do so, we obtain that expected description of the categoryof natural transformations as an end. Theorem 0.2.
Let C be a ∞ -category and D an ∞ -bicategory. Then for every pair of functors F, G : C → D there exists a equivalence of ∞ -categories Nat C ( F, G ) lim
Tw( C ) op Map D ( F ( − ) , G ( − )) which is natural in each variable. This result allows us to analyze in greater detail the theory of weighted colimits of C at ∞ -valuedfunctors exposed in [GHN15], showing that this definition coincides with the definition provided bythe first author in [AG20]. The proof of this fact together with the results of [AG20] constitute apartial answer to a series of conjectures involving ∞ -bicategorical colimits and a categorified theoryof cofinality introduced by the authors in [AGS20]. Structure of the paper
The paper will be laid out as follows. We begin with a preliminary section, which lays out thenotational conventions we follow, and explains several technical constructions and lemmata whichwe use throughout the paper. In particular, we give basic definitions for cosimplicial objects, stateand prove a general lemma on subsets K ⊂ ∆ n † of a scaled n -simplex such that K ∆ n † isscaled anodyne, and define a structure on a poset sufficient for us to give a clean description of thesimplicial mapping spaces in a quotient of its nerve.From there, the work starts in earnest. In section 2, we give the formal definition of Tw( C ), andprove that Tw( C ) → C × C op is a Cartesian fibration, making use of the aforementioned lemmaon simplicial subsets of scaled n -simplices. We then turn to section 3, in which we prove that thisCartesian fibration classifies precisely the enhanced mapping functor C op × C C op × C C at ∞ . This proof is highly technical, and freely uses results from [Lur09a] and [GHL20].In section 4, our attention then turns to the true aim of the paper, a proof of the propositionthat, given two functors
F, G : C → D from an ∞ -category to an ( ∞ , ∞ -categoryof natural transformations between them can be expressed as a limitNat( F, G ) ≃ lim Tw( C ) op Map D ( F ( − ) , G ( − )) , i.e., an end. Once again the proof is highly technical, making use of a wide variety of techniquesnative to the contexts of scaled simplicial sets and marked simplicial sets. In particular, the proofrelies heavily on a sort of dévissage — one in which we reduce from the case of a general ∞ -category(indeed, simplicial set) C to the cases C = ∆ and C = ∆ .We conclude with applications of this theorem, where we upgrade several results appearing in[GHN15]. Acknowledgments
F.A.G. would like to acknowledge the support of the VolkswagenStiftung through the Lichtenberg-Professorship Programme while he conducted this research. W.H.S was supported by Universität5amburg during the early stages of this work, and by the NSF Research Training Group at theUniversity of Virginia (grant number DMS-1839968) during the later stages.
We begin by presenting some background information necessary for the paper, and proving somegeneral lemmata which will help simplify the technical arguments in later sections. We will not,in general, recapitulate material from [Lur09] and [Lur09a], as doing so would greatly extend thelength of the present document for dubious benefit. In particular, we will assume that the reader isfamiliar with the theories of quasi-categories, Cartesian fibrations, and scaled simplicial sets, as wellas the attendant model structures. We will, however, briefly collect the notations and conventionswe will use for these before embarking on the preliminaries proper.
Notation (Model categories).
We denote by Set ∆ the category of simplicial sets, Set +∆ thecategory of marked simplicial sets, and Set sc∆ the category of scaled simplicial sets. We considerthese to be equipped with the Joyal, Cartesian, and bicategorical model structures, respectively.Where context clarifies the meaning, an unadorned Latin capital — e.g. X — may be used todenote an object of any of these categories. When it is necessary to specify a marking or a scalingon X ∈ Set ∆ , we do so by writing a superscript — e.g. X † — for a marking, and a subscript —e.g. X † — for a scaling. In particular, the subscripts ♯ and ♭ will denote the maximal and minimalscalings, respectively. Notation (Rigidification).
We denote by Cat ∆ the category of simplicial set enriched cate-gories, and by Cat Set +∆ the category of marked simplicial set enriched categories. We denote by C : Set ∆ Cat ∆ the rigidification functor, and by C sc : Set sc∆ Cat
Set +∆ its scaled variant. Inthe presence of the sub- and superscript convention above, we will conventionally denote C [ X ]( x, y ) † := C sc [ X † ]( x, y )for any x, y ∈ X . Convention (Fibrant objects).
By an ∞ -category, we will mean an ( ∞ , C — for ∞ -categories.By an ∞ -bicategory, we will mean an ( ∞ , Where possible, we will denote ∞ -bicategories by blackboard-bold capitals — e.g. C . Definition 1.1.
Let C be an ordinary 1-category. A functor F : ∆ C will be called a cosim-plicial object in C . Notation.
Given [ n ] ∈ ∆ we will denote its image under F by F ( n ).In the following sections, we will make extensive use of cosimplicial objects with target a cocom-plete category C . Namely, those that can be “freely extendend” by colimits. Indeed by taking theleft Kan extension along the Yoneda embedding Y : ∆ Set ∆ we can produce a pair of adjointfunctors Y ! F : Set ∆ C : F ∗ The potential for confusion between ∞ -bicategories and weak ∞ -categories created by the terminology of [Lur09a]is obviated by [GHL20, Thm. 5.1]. c ∈ C the n -simplices of F ∗ ( c ) are given by maps F ( n ) → c . Example 1.2.
Let C = Set ∆ and let X ∈ Set ∆ . We define a cosimplicial object( − ) × X : ∆ Set ∆ , [ n ] ∆ n × X. The right adjoint to this cosimplicial object sends each ∞ -category Y to the functor ∞ -categoryFun( X, Y ). Notation.
Let C be a cocomplete category and F a cosimplicial object on C . We set the followingnotation ∂F n = colim ∆ I → ∂ ∆ n F ( I ) . Definition 1.3.
Let P ( n ) denote the power set of [ n ] with n >
0. We say that A ( P ( n ) is dull ifthe following conditions are satisfied:1. It does not contain the empty set, ∅ / ∈ A
2. There exists 0 < i < n such that i / ∈ S for every S ∈ A .3. It contains a pair of singletons { u } , { v } ∈ A such that u < i < v .4. For every S, T ∈ A it follows that S ∩ T = ∅ .We will call the element i in condition (2), the pivot point . Definition 1.4.
Let A ( P ( n ) be a dull subset. Given an scaled n -simplex ∆ n † , we define S A = [ S ∈A ∆ [ n ] \ S ( ∆ n and denote S equipped with the induced scaling by S A† . When the choice of dull subset is clear, wewill use the abusive notation S † . Definition 1.5.
Let A ( P ( n ) be a dull subset. We call X ∈ P ( n ) an A -basal set if it containsprecisely one element from each S ∈ A . We denote the set of all A -basal sets by Bas( A ). Remark 1.6.
Note that our definitions guarantee both that Bas( A ) = ∅ , and that all A -basal setshave the same cardinality. Definition 1.7.
Given a dull subset A , we define M A to be the set of subsets X ∈ P ( n ) satisfyingthe following conditions: A X contains the pivot point, i ∈ X . A
2) The simplex σ X : ∆ X → ∆ n does not factor through S .We set κ A := min {| X | | X ∈ M A } and define, for every κ A j n , the subset M j A ⊂ M A consisting of those sets of cardinality at most j . Remark 1.8.
To ease the notation, when the choice of dull subset it is clear we will drop thesubscript A in M A and κ A . More generally, the right adjoint gives the internal hom of Set ∆ . emma 1.9. Let A be a dull subset of P ( n ) with pivot point i . Then it follows that M κ = { X ∪ { i } | X ∈ Bas( A ) } . Proof.
Left as an exercise.
Notation.
Let A ( P ( n ) be a dull subset with pivot point i . Given an A -basal set X , we willdenote by ℓ Xi − ℓ Xi the pair of consecutive elements in X such that ℓ Xi − < i < ℓ Xi . Lemma 1.10 (The pivot trick).
Let A ( P ( n ) be a dull subset with pivot point i , and let ∆ n † bean scaled simplex. For Z ∈ Bas( A ) suppose that the following condition holds.• For every r, s ∈ [ n ] such that ℓ Zi − r < i < s ℓ Zi the simplex { r, i, s } is scaled in ∆ n † .Then S † → ∆ n † is scaled anodyne.Proof. For ease of notation we will drop the subscript denoting the scaling in this proof, assumingthat all simplicial subsets are equipped with the scaling inherited from ∆ n † . We define for κ j nY j = Y j − ∪ [ X ∈M j σ X , where we set Y κ − = S . This yields a filtration S Y κ · · · Y n − Λ ni ∆ n . where Λ ni = Y n . We will show that each step of this factorization is scaled anodyne.Let X ∈ M j with κ j n −
1. Let us note that as a consequence Lemma 1.9 we obtain apullback diagram Λ Xi ∆ X Y j − Y j y σ X Additionally, the condition of the lemma guarantees that i together with its neighboring elements in∆ X form a scaled 2-simplex. Thus, the map Λ Xi ∆ X is scaled anodyne, allowing us to add ∆ X .It also follows from our definitions that given X, Y ∈ M j such that X = Y then σ X ∩ σ Y ∈ Y j − , sothat we can add the j -simplices ∆ X to Y j − irrespective of their order. This shows that Y j − → Y j is scaled anodyne. Remark 1.11.
It is worth noting that the procedure outlined in Lemma 1.10 only makes use of aspecial subset of the scaled anodyne maps: that generated by the inner horn inclusionsΛ ni ∆ n where ∆ { i − ,i,i +1 } is scaled. Significantly, while the class of scaled anodyne maps is not, in general,self-dual (i.e. f op : X op Y op need not be scaled anodyne when f : X Y is), the classgenerated by these scaled inner horn inclusions is. We will make use of this property to furthersimplify applications of Lemma 1.10. 8 .3 Poset partitions
In section 3 it will be necessary for us to consider mapping spaces in quotients of nerves of posets,as well as their scaled analogues. While these mapping spaces are quite straightforward to describe,we here collect a number of descriptions and notations so as to better facilitate the flow of the latersections of the paper.
Definition 1.12.
Let J be a finite poset, and denote by J its nerve. We call a pair of subsets J , J an ordered partition of J if the following three conditions are satisfied.• J ∪ J = J .• J ∩ J = ∅ .• For every x ∈ J and every y ∈ J , we have either x < y or x and y are incomparable.For such an ordered partition, we denote by J R the quotient J R := J a N( J ) ∆ , and by e J the quotient e J := ∆ a N( J ) J a N( J ) ∆ . We denote the two objects of e J by ∗ and ∗ , and denote the ‘collapse point’ of J R by ∗ . Remark 1.13.
Note that the definition of an ordered partition is symmetric — the opposite of anordered partition is still an ordered partition. It is for this reason that we only consider the quotient J R and not some analogous J L as well. Example 1.14.
In the sequel we will make extensive use of a cosimplicial object Q ( n ) := ∆ n ⋆ (∆ n ) op . Each level of this cosimplicial object admits a canonical ordered partition. Under theidentification Q ( n ) ∼ = ∆ n +1 = N([2 n + 1]), this ordered partition is given by J = [ n ] and J = { n + 1 , . . . , n + 1 } . We will abusively denote each of these ordered partitions by ( J Q , J Q ). Construction 1.15.
Given a finite poset J and an ordered partition ( J , J ), we construct a poset P J as follows. The objects of P J are totally ordered subsets S ⊂ J such that min( S ) ∈ J andmax( S ) ∈ J , ordered by inclusion. We will denote the nerve by P J := N( P J ).Let S := ( S ⊂ · · · S k ) be a k -simplex of P J . Set s R := min( S ∩ J ). We define the right truncation of S to be the simplex S R := ( S R ⊂ · · · ⊂ S Rk )where S Rℓ := { s ∈ S ℓ | s s R } . We similarly define s L := max( S ∩ J ) and its correspondig lefttruncation S L where S Lℓ := { s ∈ S ℓ | s > s L } . The ambidextrous truncation S A is obtained by takingboth the left and right truncation of S . We can then define two equivalence relations on P J .1. We say that k -simplices S and T are right equivalent , and we write S ∼ R T , when S R = T R . 9. We say that k -simplices S and T are ambi-equivalent , and we write S ∼ A T , when S A = T A .Note that both of these equivalence relations respect the face and degeneracy maps, so that thequotients of P J by ∼ R and ∼ A are simplicial sets.Finally, for any j ∈ J , we define P j J ⊂ P J to be the full subposet on those sets S with min( S ) = j .Note that ∼ R descends to an equivalence relation on P j J .We can then characterize the desired mapping spaces of C [ J ] in terms of the above posets. Lemma 1.16.
Let J be a finite poset, and ( J , J ) and ordered partition of J . Then1. for every j ∈ J there is an isomorphism C [ J R ]( j, ∗ ) ∼ = ( P j J ) / ∼ R .
2. There is an isomorphism C [ e J ]( ∗ , ∗ ) ∼ = ( P J ) / ∼ A . Proof.
Follows from, e.g., the necklace characterization of [DS11] once the definitions have beenunwound.
Our construction of the enhanced twisted arrow category will depend on an upgrade of the cosim-plicial object ∆ Set ∆ , [ n ] ∆ n ⋆ (∆ n ) op to a cosimplicial object in scaled simplicial sets. For a discussion of the intuition behind this choiceof scaling, see the introduction. To simplify some of the discussion to come, we introduce somenotational conventions surrounding ∆ n ⋆ (∆ n ) op . Note, before we begin, that there is a canonicalidentification ∆ n ⋆ (∆ n ) op ∼ = ∆ n +1 , which we will often use without comment. Notation.
In general, we will denote elements of ∆ n ⋆ (∆ n ) op by i ∈ ∆ n or i ∈ (∆ n ) op . Note thatunder the identification ∆ n ⋆ (∆ n ) op ∼ = ∆ n +1 , i is identified with 2 n + 1 − i . We denote the uniqueduality on ∆ n ⋆ (∆ n ) op by τ n : ∆ n ⋆ (∆ n ) op (∆ n ) op ⋆ ∆ n , i i. When n is clear from context, we will simply denote τ n by τ . Definition 2.1.
We define a cosimplicial object Q : ∆ op Set sc∆ , [ n ] ∆ n ⋆ (∆ n ) op by declaring a non-degenerate 2-simplex σ : ∆ ∆ n ⋆ (∆ n ) op to be thin if:• σ factors through ∆ n ⊂ Q ( n );• σ factors through (∆ n ) op ⊂ Q ( n ); 10 σ = ∆ { i,j,k } , where i < j k ; or• σ = ∆ { k,j,i } , where i < j k .Note that the scaling is symmetric under τ n by definition; i.e. the maps τ n define dualities on thescaled simplicial sets Q ([ n ]).The ‘nerve’ operation associated to Q is a functor Z ∗ : Set sc∆ Set ∆ defined by setting ( Z ∗ X ) n := Hom Set sc∆ ( Z ([ n ]) , X ). Remark 2.2.
We will often abuse notation and denote Q ([ n ]) by Q ( n ). We will adopt a similarconvention for other cosimplicial objects without comment. Definition 2.3.
Let C be an ∞ -bicategory with underlying ∞ -category C . The enhanced twistedarrow category of C is the marked simplicial setTw( C ) := ( Q ∗ C , E )where the edges of E are precisely those corresponding to maps ∆ ♯ C . Note that the inclusions∆ n♯ ⊂ Q ( n ) and (∆ n ) op ♯ ⊂ Q ( n ) induce a canonical mapTw( C ) → C × C op of simplicial sets. Remark 2.4.
It is immediate from the definitions that Tw( C ) is the ∞ -categorical twisted arrowcategory of [Lur11]. With some work it can be shown that this is precisely the simplicial subset ofTw( C ) spanned by the marked morphisms.The immediate aim of this section is to prove the following theorem, which can be seen as an( ∞ , Theorem 2.5.
For any ∞ -bicategory C with underlying ∞ -category C , the canonical map Tw( C ) C × C op is a Cartesian fibration and the marked edges are Cartesian. The proof of Theorem 2.5, while it involves some combinatorial yoga, begins with the usual,straightforward approach: for each 0 < i n , we consider the lifting problems(Λ ni ) ♭ Tw( C )(∆ n ) ♭ C × C op (1)and pass to adjoint lifting problems. It is worth noting that, in the case i = n , we will in factconsider the edge ∆ { n − ,n } ⊂ Λ nn to be marked. 11 onstruction 2.6. The adjoint lifting problem to (1) will be the extension problem( K ni ) † C Q ( n ) (2)where ( K ni ) † ⊂ Q ( n ) is the scaled simplicial subset consisting of those simplices σ : ∆ m → Q ( n )which fulfill one of the following three conditions.• σ factors through ∆ n ⊂ Q ( n ).• σ factors through (∆ n ) op ⊂ Q ( n ).• There exists an integer j = i such that neither j nor j is a vertex of σ . Construction 2.7.
We denote by Q ( n ) ⋄ the scaled simplicial set defined by adding to the scalingof Q ( n ) the triangles of the form { n − , n, j } , { n − , n, j } as well as their duals induced by τ . Itis immediate to observe that solutions to the lifting problem( K nn ) ⋄ C Q ( n ) ⋄ correspond to solutions to (1) with i = n , mapping the last edge in Λ nn to a marked edge in Tw( C ). Construction 2.8.
Let 0 < i n and define K ni to be the simplicial set obtained by adding to K ni the faces d and d n +1 . Denote by ( K ni ) † , ( K nn ) ⋄ the resulting simplicial sets obtained via theinduced scaling.Our proof will proceed by showing that both morphisms in each factorization( K ni ) † ( K ni ) † Q ( n )and ( K nn ) ⋄ ( K nn ) ⋄ Q ( n ) ⋄ are scaled anodyne. Lemma 2.9.
1. For < i < n the morphism ( K ni ) † Q ( n ) is scaled anodyne.2. The morphism ( K nn ) ⋄ Q ( n ) ⋄ is scaled anodyne.Proof. For 0 < i n , we note that unwinding the definition shows that K ni = S A i , where A i ⊂ P (2 n + 1) is the dull subset containing { } , { n + 1 } , and { j, j } for 0 < j n such that j = i . Thelemma follows immediately from Lemma 1.10. Lemma 2.10.
1. For < i < n the morphism ( K ni ) † ( K ni ) † is scaled anodyne.2. For i = n the morphism ( K nn ) ⋄ ( K ni ) ⋄ is scaled anodyne. roof. Let 0 < i n and note that since d ∩ d n +1 ∈ K ni it will suffice to show that the tophorizontal morphism, Y εi ∆ n ( K ni ) ∗ (∆ n +1 ) ∗ y d ε where ε ∈ { , n + 1 } and ∗ ∈ {† , ⋄} , is scaled anodyne.We will first deal with the case ε = 0. Let 1 r n and define σ r : ∆ [ r, n +1] ∆ n +1 to be the obvious inclusion. Let us remark that σ = d and that σ r factors through d for everypossible r . We produce a filtration Y i = X n +1 X n · · · X ∆ [1 , n +1] = X where X r is obtained by adding the simplex σ r to X r .It will thus suffice for us to check that the upper horizontal morphism in the pullback diagram(i.e. the restriction of σ r to X r +1 ) Z r ∆ [ r, n +1] X r +1 ∆ [1 , n +1] y is scaled anodyne. However, we can observe that Z r consists of a union in ∆ [ r, n +1] • The (2 n − r )-dimensional face d r .• The (2 n − d n +1 − j where 0 j < r and j = i .• The (2 n − r − { j, n + 1 − j } with r j n and j = i .That is, Z r = S A r , where A r ( P (2 n + 1 − r ) is the dull subset containing• { } .• The singletons { j } for 2 n + 1 − r < j n + 1 − r with j = 2 n − r − i + 1.• The sets { k, n + 1 − r − k } for 0 k n − r with r + k = i .One can easily verify that the scaling satisfies the conditions of Lemma 1.10, and thus Z r ∆ [ r, n +1] is scaled anodyne. Consequently, each step of the filtration Y i = X n +1 X n · · · X ∆ [1 , n +1] = X is scaled anodyne, completing the proof that Y i ∆ [1 , n +1] is scaled anodyne.We conclude the proof by noting that the case Y n +1 i ∆ [0 , n ] is formally dual, so byRemark 1.11, the proof is complete. Proof of Theorem 2.5.
Combining Lemma 2.9 and Lemma 2.10, it is immediate that (1) Tw( C ) → C × C op is an inner fibration, and (2), the marked edges are Cartesian. It remains only for us to13how that there is a sufficient supply of marked edges. Unwinding the definitions, we find that thiswill be true so long as the lifting problems Sp C ∆ ♯ admit solutions, where Sp := ∆ { , } ` ∆ { } ∆ { , } ` ∆ { } ∆ { , } is the spine of the 3-simplex. How-ever, the left-most map is clearly scaled anodyne so a solution to this problem is guaranteed byfibrancy. The result now follows. Tw( C ) Having now established the Cartesian fibrancy of T w( C ) C , we aim to determine the functorwhich it classifies. It will come as no surprise to those familiar with other twisted-arrow categoryconstructions that the functor in question will be the enhanced mapping functor of [GHN15], i.e., themapping category functor of C restricted to C op × C . The solution to this classification problem willbe quite involved and technical, involving a number of intermediate ∞ -categories. Where possible,we will attempt to elucidate the meaning and function of these constructions in the text. We now turn our attention to the first step in our proof: constructing the comparison map. This partof the proof will be quite straightforward and in total analogy with its ∞ -categorical counterpartin [Lur11]. To construct the desired map, we fix, once an for all, the following data:• An ∞ -bicategory C together with its underlying ∞ -category C .• A fibrant Set +∆ enriched category D , and its maximally marked subcategory (Kan-complexenriched) D , with a commutative diagram C sc [ C ] D C sc [ C ] D ≃≃ such that the horizontal arrows are weak equivalences of Set +∆ -enriched categories.With this data fixed, the enhanced mapping functor is the composite F : D op × D D op × D Map C at ∞ To retain concision, we use the pedestrian notation F for the enhanced mapping functor, ratherthan the more suggestive Map D . Proposition 3.1.
There is an map β : Tw( C ) → Un + C × C op ( F ) of Cartesian fibrations over C × C op . roof. The proof proceeds along the same lines as the analogous argument in [Lur11]. We definean ancillary simplicial category E with objects either the objects of D op × D , or a "cone point" v .The mapping spaces will be those of D op × D if they don’t involve v , and will be defined byMap E ( v, ( D, D ′ )) := ∅ Map E (( D, D ′ ) , v ) := Map D ( D, D ′ )otherwise.As in [Lur11], a map over C × C op preserving markings — β : Tw( C ) Un C × C op ( F ) — will beequivalent to giving a map γ : Tw( C ) ⊲ N( E )such that the diagramTw( C ) ⊲ Tw( C ) C × C op N( D ) × N( D ) op N( E )commutes, and such that, for every f : ∆ → Tw( C ) which is marked, the two-simplex f ∗ id ∆ :∆ ⋆ ∆ → Tw( C ) ⊲ → N( E ) is sent to a scaled 2-simplex in N sc ( E ). (Equivalently, if the adjointmap C [∆ ] → E determines a marked edge in the mapping space from 0 to 2.)We now define the map in question: Given an n -simplex σ : ∆ n → Tw( C ), we obtain by definitionand adjunction a map ν σ : C [∆ n +1 ] C [ C ] D . We now define a map γ σ : C [∆ n +1 ] E On C [∆ n ] ⊂ C [∆ n +1 ], this is completely determined by the commutativity condition above. Formapping spaces involving the ( n + 1) st -vertex, we define the maps ζ i : O n +1 ( i, n + 1) O n +1 ( i, n + 1 − i ) S ∪ { n + 1 } S ∪ τ ( S )where S is considered as a subset of [ n ], and τ is the involution on vertices of ∆ n +1 . We thendefine γ σ : O n +1 ( i, n + 1) ζ O n +1 ( i, n + 1 − i ) ν σ Map D ( ν σ ( i ) , ν σ (2 n + 1 − i ))Completing our definition of γ σ , and thus of γ . It is obvious from our definitions that γ respectsthe marking/scalings. Remark 3.2.
The definition of the maps ζ i which allow us to define the map above are quitead-hoc in appearance, as indeed are their analogues in [Lur11]. Once we pass to fibers, the map canbe much more elegantly defined: in terms of a composite with a map of posets (see Remark 3.19).The goal of the remainder of this section will be the proof of the following. Note that (1) this follows directly from unwinding the characterization of the marked 1-simplices of the straighteningin [Lur09, 3.2.1.2], and (2) this is in nearly precise analogy with the definition of the scaling on the scaled cone in[Lur09a, 3.5.1.] heorem 3.3. The map β is an equivalence of Cartesian fibrations over C × C op . In the sections which follow, there will be a variety of cosimplicial objects in play, each relating to aspecific construction necessary for the proof. For ease of reference, we list these here, and describeadditional structures (in particular ordered partitions) which will come into play in their study.
Definition 3.4 (The compendium).
We fix, for the rest of the section, the following cosimplicialobjects, along with ordered partitions of their n th levels.1. A cosimplicial object ⋆ : ∆ Set sc∆ [ n ] ∆ n ⋆ ∆ where the scaling on ⋆ ( n ) = ∆ n ⋆ ∆ is given by declaring every 2-simplex in ∆ n ⊂ ⋆ ( n ) tobe thin.• We define an ordered partition ( J ⋆ , J ⋆ ) of ⋆ ( n ) for each n by setting J ⋆ = [ n ] and J ⋆ = { n + 1 } , where we have identified ⋆ ( n ) with N([ n + 1]).2. A cosimplicial object ⊠ : ∆ Set sc∆ [ n ] ∆ n ⋆ (∆ n ) op ⋆ ∆ We use our existing notational conventions for objects of Q ( n ) ⊂ ⊠ ( n ), and denote the finalvertex by v . We equip ⊠ ( n ) with a scaling by (1) requiring the inclusions Q ( n ) ⊂ ⊠ ( n ) and((∆ n ) op ⋆ ∆ ) ♯ ⊂ ⊠ ( n ) to be maps of scaled simplicial sets; and (2) declaring any simplex ofthe form ∆ { j,i,v } , where 0 i j n , to be thin.• We define an ordered partition ( J ⊠ , J ⊠ ) of ⊠ ( n ) for each n by setting J ⊠ = [ n ] and J ⊠ = { n + 1 , n + 2 , . . . n + 2 } under the identification of ⊠ ( n ) with N([2 n + 2]).3. A cosimplicial object (cid:3) : ∆ Set sc∆ [ n ] ∆ n × ∆ where the scaling consists of those triangles factoring through ∆ n × ∆ { } and those specifiedin [Lur09a, 4.1.5].• We define an ordered partition ( J (cid:3) , J (cid:3) ) of (cid:3) ( n ) by setting J (cid:3) := [ n ] × { } and J =[ n ] × { } under the identification of (cid:3) with N([ n ] × [1]). Having established the existence of a comparison map β : Tw( C ) F of Cartesian fibrations,we now must pause and circumnavigate our way to a proof that it is an equivalence. The windingroute we take will make use of a Cartesian fibration C /y defined in [GHL20]. The utility of C /y for us lies in the fact that, as established in [GHL20, §2.3], C /y classifies the contravariant Yonedaembedding Y y on D . In spite of the fact that C /y C is a Cartesian fibration, we will refer to C /y as the outer Cartesian slice category , in recognition of the fact that our C /y C is a pullbackalong the inclusion C C of an outer Cartesian fibration as defined in [GHL20]. We begin byrecalling the definition of C /y . 16 efinition 3.5. For an object y ∈ C , we define the outer Cartesian slice category C /y whose n -simplices are given by maps σ : ⋆ ( n ) C , such that σ | n +1 = y. We equip C /y with a marking by declaring an edge to be marked precisely when it can be representedby a map ∆ ♯ C . Note that the canonical inclusion ∆ n♯ ⊂ ⋆ ( n ) induces a map C /y C . Proposition 3.6 ([GHL20, Cor. 2.18]).
The functor C /y C is a Cartesian fibration, andan edge of C /y is Cartesian if and only if it is marked. Since our mode of proof is so circuitous, let us take a moment to sketch the path we will take.We begin by showing that there is a span( C /y ) x M x,y Tw( C ) ( x,y ) ∼∼ displaying a weak equivalence of the fibers of C /y and Tw( C ).We then show that there is a weak equivalenceUn + ∗ ( C sc [ D ]( x, y )) Un sc ∗ ( C sc [ D ]( x, y )) . From [GHL20], there is an equivalence f : ( C /y ) x Un sc ∗ ( C sc [ D ]( x, y )) . The final step to showing that β is an equivalence is therefore establishing that the diagram( C /y ) x M x,y Tw( C ) ( x,y ) Un sc ∗ ( C sc [ D ]( x, y )) Un + ∗ ( C sc [ D ]( x, y )) f ≃ ∼∼ β ≃ commutes up to equivalence.We begin this journey in the present section by defining the span( C /y ) x M x,y Tw( C ) ( x,y ) ∼∼ and showing that its legs are weak equivalences. Notation.
Let x, y ∈ C . We denote by Tw( C ) y the pullbackTw( C ) y Tw( C ) C C × C opid ×{ y } and by Tw( C ) ( x,y ) the fiber over ( x, y ) ∈ C × C op . Definition 3.7.
For y ∈ C , we define a simplicial set M y whose n -simplices are given by maps σ : ⊠ ( n ) C such that σ | N ( J ⊠ ) = y. ⊠ ( n ) R → C . The inclusion ∆ n♯ = N (cid:0) J ⊠ (cid:1) ♯ ⊂ ⊠ ( n ) inducesa map M y C . Remark 3.8.
We will view ⋆ ( n ) , ⊠ ( n ), and Q ( n ) as equipped with their ordered partitions fromDefinition 3.4 and Example 1.14, and consider their right quotients ⋆ ( n ) R , ⊠ ( n ) R , and Q ( n ) R ,each of which piece together to form a cosimplicial object in Set sc∆ . The obvious natural inclusions ⋆ R ⊠ R Q R then induce maps C /y M y Tw( C ) yπρ over C . Proposition 3.9.
The map π : M y Tw( C ) y is a trivial Kan fibration.Proof. We first aim to show that the inclusions i n : Q ( n ) R ⊠ ( n ) R are scaled trivial cofibrations.To this end, we define a map r n : ⊠ ( n ) Q ( n ) i i i < n + 2 i − i = 2 n + 2We see immediately that r n descends to a map r n : ⊠ ( n ) R Q ( n ) R , and that r n ◦ i n = id.Moreover, one can check that the natural transformation i n ◦ r n ⇒ id descends to a transformation∆ ♭ × ⊠ ( n ) R ⊠ ( n ) R whose components are degenerate. Consequently, we see that i n is an equivalence of scaled simplicialsets. We then consider the boundary lifting problem and its associated adjoint problem ∂ ∆ n M y ∆ n Tw( C ) yπ K n C ⊠ ( n ) R K n = ∂ ( ⊠ R ) n a ∂ ( Q R ) n Q ( n ) R Examining Definition 1.12, we note that we can extend our conventions to Q ( ∅ ) R = ∆ and ⊠ ( ∅ ) R = ∆ . Consequently, we can write ∂ ( ⊠ R ) n = hocolim I ( [ n ] ⊠ ( I ) R and ∂ ( Q R ) n = hocolim I ( [ n ] Q ( I ) R . Since the two diagrams are naturally equivalent, this yields an equivalence ∂ ( Q R ) n ≃ ∂ ( ⊠ R ) n .Since this map is a trivial cofibration it follows that Q ( n ) R → K n is an equivalence. Finally weconsider the factorization Q ( n ) R K n ⊠ ( n ) R and we conclude by 2-out-of-3 that the map K n → ⊠ ( n ) R is a trivial cofibration. This finishes theproof. Corollary 3.10.
The map M y → C is a Cartesian fibration, and π : M y Tw( C ) y is an equiv- lence of Cartesian fibrations over C . Lemma 3.11.
The Cartesian edges of M y over C are precisely those which can be represented byscaled maps ⊠ † → C , where † is the extension of the scaling on ⊠ to include1. all 2-simplices in ∆ ⋆ (∆ ) op , and2. the 2-simplex ∆ { ,v } .Proof. Left as an exercise to the reader.
Corollary 3.12.
The map ρ : M y C /y is a map of naturally-marked Cartesian fibrations over C . Proposition 3.13.
For any x ∈ C , denote the fiber of M y over x by M x,y . Then the induced map ρ : M x,y ( C /y ) x is a trivial Kan fibration.Proof. We follow effectively the same method as in the proof of Proposition 3.9, now using thetwo-sided quotients f ⋆ ( n ) and e ⊠ ( n ) of the defining cosimplicial objects. By the same homotopycolimit argument, it will suffice for us to show that i n : f ⋆ ( n ) e ⊠ ( n )is an equivalence. However, i n is already a bijection on objects, so it will suffice for us to show that i n induces an equivalence on the single non-trivial mapping space. To this end, we make use of thecharacterization of Lemma 1.16. It will thus suffice to to show that the maps of marked simplicialsets e s : (cid:16) P ⋆ ( n ) (cid:17) / ∼ A (cid:16) P ⊠ ( n ) (cid:17) / ∼ A are equivalences for any n . For the rest of the proof we will abuse notation and denote the nervesof these posets by P ⋆ and P ⊠ .We will work with the unquotiented simplicial sets, and define maps which descend to quotients.Before we can do this, however, we must fix some notation. We denote object S ∈ P ⊠ by triples( S , S , S ) of subsets of each of the three joined components in ⊠ ( n ) = ∆ n ⋆ (∆ n ) op ⋆ ∆ . We willsimilarly denote objects of P ⋆ by pairs ( S , v ) of sets. Note that with this new coordinates theunquotient version of e s can be described as s : P ⋆ P ⊠ , ( S , v ) ( S , ∅ , v )We define P G ⊂ P ⊠ as the nerve of the full subposet on those objects of the form• ( S , ∅ , v ).• ( S , S , v ) such that – S = ∅ – S ∪ { v } contains all elements of ⊠ ( n ) greater than min( S ), and – τ ( S ) ⊂ S . 19e equip P G with the induced marking producing a factorization P ⋆ s α P G s β P ⊠ . In light ofthis fact, we will turn our efforts into showing that s α , s β descend to equivalences e s α and e s β . Wedefine a marking-preserving map of posets r α : P G P ⋆ , ( S , S , v ) ( S , v )such that r α ◦ s α = id. We, moreover, observe that there is a natural transformation ε α : s α ◦ r α ⇒ idwhose components are marked in P G . To check that ε α (and consequently r α ) factors through thequotient it is enough to note that given k -simplices S ∼ A T in P G then it follows that S ∼ L T . Wecan now conclude that e s α is an equivalence.We define a map of posets r β : P ⊠ P G ( S , S , S ) ( S , ∅ , v ) if S = ∅ ( S ∪ τ n ([min( S ) , v )) , [min( S ) , v ) , v ) otherwise.such that r β ◦ s β = id and note that there is a map S → r β ( S ) inducing a natural transformation ε β : id ⇒ s β ◦ r β . To show that P G P ⊠ is an equivalence, it is sufficient to check that r β preserves markings, that ε β descends to quotients, and that the components of the naturaltransformation become equivalences in the fibrant replacement of the localizations. We will provehere that ε β descends to the quotient leaving the rest of the checks as exercises for the interestedreader. Let S ∼ A T be k -simplices and denote by β S and β T their images under r β . Let s R , s L bethe truncation points for S and denote by β s R , β s L the truncation points for β S . It is immediate tosee that β s R = s R , β s L = max (cid:8) s L ,τ ( β s R ) (cid:9) if ( S ) = ∅ s L otherwise.This implies that, in order to show our claim, it suffices to check that for every ℓ ∈ [ k ] theambidextrous truncations of β S ℓ , β T ℓ with respect to s L , s R coincide. If ( S ℓ ) = ∅ the conclusionfollows immediately. We will also assume that κ = τ (min(( S l ) )) < max(( S l ) ), since otherwise wewould have β S ℓ = β T ℓ . Denote by β b S Aℓ the truncation with respect to our chosen points. Then weobserve that β b S Aℓ = [ s L , κ ] ∪ ( S ℓ ) > κ ∪ [ τ ( κ ) , v ] if s L κ ( S ℓ ) ∪ [ τ ( κ ) , v ] otherwisewhere ( S ℓ ) > κ stands for the obvious notation. Since this only depends on S Aℓ it follows that β b S Aℓ = β b T Aℓ .We thus have completed the first step of the proof: Corollary 3.14.
The maps C /y M y Tw( C ) yπρ are equivalences of naturally marked Cartesian fibrations over C . Remark 3.15.
This would already be sufficient, in light of [GHL20, §2.3], for us to conclude thatTw( C ) y classifies the restriction to C of the representable functor defined by y . It is not, however,sufficient to show that Tw( C ) classifies the enhanced mapping functor. We still have work to do.20 .4 Comparing the comparisons By [GHL20], there are equivalences of marked simplicial sets( C /y ) x ∼ ( C x/ ) y ∼ Un sc ∗ ( C [ C ]( x, y ))where Un sc ∗ is the scaled coCartesian unstraightening, and C x/ denotes the scaled slices definedusing the fat join in [Lur09a, 4.1.5]. We also have, by Proposition 3.1, a comparison map β : Tw( C ) x,y → Un + ∗ ( C [ C ]( x, y )) , where Un + ∗ is the marked Cartesian unstraightening.We now aim to compare these two comparison maps, using the equivalence between Tw( C ) y and C /y of Corollary 3.14. The first step is to relate the scaled coCartesian and marked Cartesianstraightenings over the point. Construction 3.16.
We denote the former by St sc and the latter by St + , leaving the point implicit.These give us two functors St sc , St + : Set ∆ + Set +∆ By [ADS20, Lem. 4.3.3], to display a natural equivalence between them it will suffice to display it onsimplices. By definition, we have that St sc (∆ n ) = C sc [ e (cid:3) ( n )]( x, y ) and St + (∆ n ) = C sc [ f ⋆ ( n )]( x, y ) . Since the collapse map ∆ n × ∆ ∆ n ⋆ ∆ ( i, k ) i k = 0 n + 1 k = 1preserves the scaling and ordered partitions, we thus obtain compatible maps θ n : St sc ((∆ n ) ♭ )St + ((∆ n ) ♭ ) and St sc ((∆ ) ♯ ) St + ((∆ n ) ♯ ) Moreover, the trianglesSt sc ((∆ n ) ♭ ) St + ((∆ n ) ♭ )(∆ n ) ♭θ n p q commute, where q is the map π of [Lur09, Prop 3.2.1.14], and p is the map α of [Lur09a, Prop.3.6.1]. Since both of these are marked equivalences, we have that θ n is as well. Thus, θ extends toa natural equivalence θ : St sc St + .It immediately follows that Lemma 3.17.
The natural transformation µ : Un sc → Un + adjoint to θ is an equivalence. It now remains only for us to show Technically speaking, in [Lur09a] Lurie defines a scaled coCartesian fibration C x/ C . We will make use of thepullback along C C , and denote it by C x/ . roposition 3.18. The diagram ( C /y ) x M x,y Tw( C ) ( x,y ) Un sc ∗ ( C sc [ D ]( x, y )) Un + ∗ ( C sc [ D ]( x, y )) f ≃ ∼∼ β ≃ (3) commutes up to natural equivalence. To effect a proof, we first note that the maps f and β are both induced by maps of posets. Forthe reader’s convenience, we briefly unwind how in the case of β . For f , we merely state the posetmap in question, and leave it to the interested reader to unwind the definitions. Remark 3.19.
Given an n -simplex σ of Tw( C ) ( x,y ) , the simplex β ( σ ) of Un + ( C sc [ D ]( x, y )) is givenby pulling back the rigidification C [˜ σ ] of the adjoint map˜ σ : e Q ( n ) C along the maps ζ i constructed in Proposition 3.1. Using the poset-quotient description of the map-ping spaces, however, one can easily check that the ζ i ’s combine to define a map B : P ⋆ ( n ) P Q ( n ) , ( S , v ) ( S , τ ( S ))with associated map on the quotient e B : C sc [ f ⋆ ( n )]( ∗ , ∗ ) C sc [ e Q ( n )]( ∗ , ∗ ). That is, β ( σ ) isdefined by pulling back C [˜ σ ] along e B .More generally, let σ be a simplex of M x,y and ˜ σ : e ⊠ ( n ) C its adjoint. The right-handcomposite γ : M x,y Un sc ∗ ( C sc [ D ]( x, y )) in (3) is given by pulling C [˜ σ ] back along a map C sc [( (cid:3) ( n )) R ]( ∗ , ∗ ) → C sc [ e ⊠ ( n )]( ∗ , ∗ ) induced by G : P (cid:3) ( n ) P ⊠ ( n ) , ( S , S ) ( S , τ ( S ) , v ) . The left hand composite η : M x,y Un sc ∗ ( C sc [ D ]( x, y )) in (3) is given by pulling C [˜ σ ] back alonga map C sc [( (cid:3) ( n )) R ]( ∗ , ∗ ) → C sc [ e ⊠ ( n )]( ∗ , ∗ ) induced by H : P (cid:3) ( n ) P ⊠ ( n ) , ( S , S ) ( S , ∅ , { v } ) . With these definitions in place, we can proceed to the final step of our proof.
Proof of Proposition 3.18.
We will define an explicit homotopy (∆ ) ♯ × M x,y Un sc ∗ ( C sc [ D ]( x, y ))between γ and η .Given an n -simplex ( ρ, σ ) : ∆ n (∆ ) ♯ × M x,y , we note that ρ is uniquely specified by 0 i n + 1: (0 , , . . . , , i z}|{ , , . . . , . We define, for S ⊂ [ n ], the subset S > i := { s ∈ S | s > i } and then define a map h ρ : P (cid:3) ( n ) P ⊠ ( n ) , ( S , S ) ( S , τ ( S > i ) , v ) In point of fact, unraveling the definitions would lead one to believe that map is induced by ( S , S )( S , τ ( S ) , ∅ ), however, both this map and G lead to the same map on quotients, so the distinction is irrele-vant. i = n + 1 (i.e. ρ is constant on 0) we have that τ ( S > i ) = ∅ , so that the mapspecializes to H . Similarly, when i = 0 (i.e. ρ is constant on 1) S > i = S , so that the map specializesto G .Let us check that h ρ descend to quotients. Note that the case where ρ is constant on 0. In orderto do so, given a k -simplex S we compute its ambidextrous truncation in P ⊠ ( n ) . Let l ∈ [ k ] anddenote ( S l ) ∩ [ s L , n ] = b S l . Then we obtain h ρ ( S ) Al = ( b S l , τ ( b S l )) if i s L (cid:16) b S l , τ ( b S l ) ∩ [ n + 1 , τ ( i )] (cid:17) if s L < i < n + 1( b S l , ∅ , v ) if i = n + 1since this only depends on the truncation of S the claim follows. It is immediate that the mapsrespect the simplicial identities, so sending a simplex ( ρ, σ ) ∈ ∆ × M x,y to the simplex h ∗ ρ ( C [˜ σ ]) ∈ Un sc ( C [ C ]( x, y )) defines a homotopy η γ .To see that it is a marked homotopy, consider the 1-simplex (0 ,
1) in ∆ , and a degenerate 1-simplex σ : ⊠ (1) ♯ C in M x,y . This corresponds to a map ( P ⊠ (1) ) ♯ C [ C ]( x, y ) , and sopulling back along h { , } : P (cid:3) (1) → P ⊠ (1) yields a map h ∗{ , } ( γ ) : ( P (cid:3) (1) ) ♯ C [ C ]( x, y ) , i.e., a marked morphism in Un sc ( C [ C ]( x, y )). We have thus defined a marked homotopy as desired,and the proof is complete.For completeness, we can now give Proof of Theorem 3.3.
Using Proposition 3.18, the theorem follows immediately by 2-out-of-3 fromCorollary 3.14, the equivalence of [GHL20, Prop. 2.24], and Lemma 3.17.
In this section we will denote by X be a maximally scaled simplicial set and by D an ∞ -bicategorythat will remain fixed throughout. Given a pair of functors F, G : X D we will denote the asso-ciated mapping category in D X by Nat X ( F, G ). The aim of this section is to show that Nat X ( F, G )can be expressed as the limit of the functor N ( F,G ) : Tw( X ) op X op × X D op × D Cat ∞ . F op × G Map D ( − , − ) That is to say, the ∞ -category of natural transformations can be obtained as an end, in the termi-nology of [GHN15]. Definition 4.1.
Let ℓ : D X × (cid:16) D X (cid:17) op Fun(Tw( X ) , D × D op ) be the functor that maps asimplex of the product σ : X × ∆ n D , σ : X × (∆ n ) op D to the compositeTw( X ) × ∆ n X × X op × ∆ n D × D op . ( σ ,σ op2 ) We define a marked simplicial set L X equipped with Cartesian fibration to D X × (cid:16) D X (cid:17) op via the23ullback square L X Fun(Tw( X ) ♯ , Tw( D ) † ) D X × (cid:16) D X (cid:17) op Fun(Tw( X ) , D × D op ) . y ℓ Remark 4.2.
Given a pair of functors
F, G we see that the fiber ( L X ) ( F,G ) is given by the categoryof “ways of completing the commutative diagram”Tw( X ) Tw( D ) X × X op D × D op F × G op In other words, it is precisely the category of Cartesian sections associated to N ( F,G ) . In particularit follows from [Lur09, Cor. 3.3.3.2] that ( L X ) ( F,G ) is a model for the limit lim Tw(X) op N ( F,G ) . We willabuse notation and denote the fiber by L ( F,G ) when the diagram category is clear from the context.Recall the canonical map D X × X D and note that since Tw( − ) preserves limits we can usethe tensor-hom adjunction to produce u : Tw (cid:16) D X (cid:17) Fun(Tw( X ) ♯ , Tw( D ) † )fitting into the commutative diagramTw (cid:16) D X (cid:17) Fun(Tw( X ) ♯ , Tw( D ) † ) D X × (cid:16) D X (cid:17) op Fun(Tw( X ) , D × D op ) . θ ℓ This in turn yields a map of Cartesian fibrations Θ X : Tw( D X ) L X which we call the canonicalcomparison map . The rest of this section is devoted to showing that Θ X is a fiberwise equivalencefor every simplicial set X . Our first observation is that both constructions behave contravariantlyin the simplicial set X thus producing functorsTw (cid:16) D ( − ) (cid:17) , L ( − ) : Set op∆ Set +∆ equipped with a natural transformation Θ : Tw (cid:16) D ( − ) (cid:17) L ( − ) . We can now state the mainresult of the section. Theorem 4.3.
For every simplicial set X ∈ Set ∆ , the map Θ X : Tw (cid:16) D X (cid:17) L X is an equiva-lence of Cartesian fibrations over D X × (cid:16) D X (cid:17) op . Our proof strategy will consist in reducing the problem to the case X = ∆ n with n = 0 ,
1. Inorder to achieve this we will show that the both functors are homotopically well-behaved.
Proposition 4.4.
Let α : X Y be a cofibration of simplicial sets. Then for every pair of unctors F, G ∈ D Y the induced maps Tw (cid:16) D Y (cid:17) ( F,G ) Tw (cid:16) D X (cid:17) ( α ∗ F,α ∗ G ) , ( L Y ) ( F,G ) ( L X ) ( α ∗ F,α ∗ G ) are fibrations in the Joyal model structure.Proof. Let us observe that due to Theorem 2.5 and [Lur09, Cor. 2.4.6.5], to check that the firstmap is a Joyal fibration it will suffice to solve the lifting problems(Λ ni ) ♭ Tw( D Y ) ( F,G ) (∆ n ) ♭ Tw( D X ) ( α ∗ F,α ∗ G ) (∆ ) ♯ Tw( D Y ) ( F,G ) (∆ ) ♯ Tw( D X ) ( α ∗ F,α ∗ G ) with n > < i < n . These lifting problems can be easily seen to be equivalent to their adjointproblems (where we are using the notation of the proof of Theorem 2.5)( K ni ) † × Y ` ( K ni ) † × X Q ( n ) × X D Q ( n ) × Y Sp × Y ` Sp × X ∆ ♯ × X D ∆ ♯ × Y which admit solution in virtue of [Lur09a, Proposition 3.1.8]. The proof for the other functor isalmost analogous. First we note that the induced map Tw( X ) → Tw( Y ) is a cofibration of markedsimplicial sets. Let A ⋄ → B ⋄ be a marked anodyne morphism, then using [Lur09, Prop. 3.1.2.3] wesee that lifting problems of the formTw( X ) ♯ × B ⋄ ` Tw( X ) ♯ × A ⋄ Tw( Y ) ♯ × A ⋄ Tw( D ) † Tw( Y ) ♯ × B ⋄ D × D op admit a solution. The claim follows immediately from this fact coupled with [Lur09, Cor. 2.4.6.5]. Proposition 4.5.
Let P i : O i Set ∆ with i = 1 , , be two diagrams of simplicial sets such that1) P is a cotower diagram such that for every ℓ → k in D the induced morphism P ( ℓ ) → P ( k ) is a cofibration.2) P is a pushout diagram such there exists a morphism a → b such that P ( a ) → P ( b ) is acofibration.Denote by X i the colimit of P i and by { β j } j ∈O i the canonical cone of X i . Given F, G ∈ D X i then itfollows that we have equivalences of ∞ -categories Tw( D X ) ( F,G ) ≃ holim j ∈O op i Tw (cid:16) D P i ( j ) (cid:17) ( β ∗ j F,β ∗ j G ) , L ( F,G ) ≃ holim j ∈O op i L ( β ∗ j F,β ∗ j G ) Proof.
Observe that the functors Tw( D − ) and L ( − ) preserve the ordinary limits of shape O i . Since25aking fibers commutes with limits we observe that it is enough to show that both diagrams areinjectively fibrant. This follows immediately from our hypothesis and Proposition 4.4. Lemma 4.6.
Let ι : Λ ni → ∆ n be an inner horn inclusion. Then for every F, G ∈ D ∆ n we haveequivalences of ∞ -categories Tw( D ∆ n ) ( F,G ) ≃ Tw( D Λ ni ) ( ι ∗ F,ι ∗ G ) , L ( F,G ) ≃ L ( ι ∗ F,ι ∗ G ) . Proof.
First, let us observe that D ∆ n → D Λ ni is a trivial fibration in the scaled model structure.After noticing this, the result follows immediately for Tw. To show the claim for the second functorwe just need to show that the inclusion ι : Tw(Λ ni ) → Tw(∆ n ) is cofinal. Then the result will followfrom the fact that restriction along ι op preserves limits. We left as an exercise to the reader thislast check, that follows easily from Quillen’s Theorem A. Proposition 4.7.
Suppose the map Θ X in Theorem 4.3 is an equivalence of Cartesian fibrationsfor X = ∆ n with n = 0 , . Then for every X ∈ Set ∆ the map Θ X is an equivalence of Cartesianfibrations.Proof. We will say that a simplicial set X satisfies the property ( > ) if Θ X is an equivalence ofCartesian fibrations. First we will assume that the simplicial sets ∆ n with n > > ).As a direct consequence of Proposition 4.5 2), we deduce that boundaries ∂ ∆ n fullfil condition( > ) for n >
0. Let X be an arbitrary simplicial set. We claim that given n > n -skeleton sk n ( X )satisfies ( > ). It is clear that the claim holds for sk ( X ) since it is just a disjoint union of points.Suppose that the claim holds for sk l − ( X ) and let I be the set of non degenerate simplices containedin sk l ( X ) \ sk l − ( X ). Given i ∈ I we can attach that non-degenerate simplex via a pushout square ∂ ∆ l ∆ l sk l − ( X ) P p Proposition 4.5 implies that Θ P is an equivalence. Now let us pick a linear order on I and attachone by one all the simplices in I . We can then produce a functor P : I Set ∆ , such that colim I P ∼ = sk l ( X ) . which is an instance of Proposition 4.5 1) and therefore the inductive step is proved. The sameproposition now applied to X ∼ = colim N sk n ( X ) finally shows that Θ X is an equivalence of Cartesianfibrations provided Θ ∆ n is an equivalence for n > ∆ n is an equivalence for n >
0. Our ground casesare n = 0 ,
1. Now assume the claim holds for ( n − > ι : Tw(Λ ni ) → ∆. Then we have a commutative diagramTw( D ∆ n ) ( F,G ) ( L ∆ n ) ( F,G ) Tw( D Λ ni ) ( ι ∗ F,ι ∗ G ) (cid:16) L Λ ni (cid:17) ( ι ∗ F,ι ∗ G ) ≃ ≃≃ where the vertical morphisms are equivalences due to Lemma 4.6. It is easy to see that the bottom26orizontal morphism is an equivalence due to the induction hypothesis. The result follows from2-out-of-3.At this point we have made a drastic reduction in complexity and we are left to show thatthe object ∆ satisfies ( > ), the case of ∆ being obvious. We will tackle this last case by a directcomputational approach. Before diving into the proof of Theorem 4.3 we will take a small detour toanalyze the relevant combinatorics. Throughout the rest of this section we will use the coordinates a b for ∆ instead of the standard 0 ) by ab → aa , ab → bb . Definition 4.8.
We define a cosimplicial object R : ∆ Set sc∆ , [ n ] ( R ( n ) , T ) , R ( n ) = (∆ n × ∆ ) ⋆ (∆ n × ∆ ) op a ∆ n +1 (∆ n × ∆ ) ⋆ (∆ n × ∆ ) op We describe the scaling using the notation of Remark 4.9. T is the scaling which is (1) identicalon the two summands and (2) such that the non-degenerate thin 2-simplices of the first summand(∆ n × ∆ ) ⋆ (∆ n × ∆ ) op are those σ such that• σ factors through either (∆ n × ∆ ) or (∆ n × ∆ ) op .• i p < j q < k r is a simplex in ∆ n × ∆ , and σ = ( i p < j q < k r ).• k r < j q < i p is a simplex in ∆ n × ∆ and σ = ( i p < j q < k r ).• i j k is a simplex of ∆ n and – σ = i ab < j aa < k a ; – σ = k ab < j aa < i ab ; – σ = i aa < j aa < k ab ; – σ = k ab < j aa < i aa ; – σ = i ab < j ab < k aa ; or – σ = k aa < j ab < i ab . Remark 4.9.
We can describe the underlying simplicial set of R ( n ) as the nerve of a poset R n asfollows• The set of objects is given by symbols ℓ ε where ℓ ∈ [ n ] and ε ∈ { ab, aa, bb } together with theirformal duals ℓ ε .• We declare ℓ ab k ε where ε ∈ { ab, aa, bb } if and only if ℓ k . Dually we declare ℓ ab k ε ifand only if k ℓ . Finally we declare ℓ ε < ℓ ε . The ordering on R n is the minimal one generatedby the inequalities above.We provide graphical representations of the posets for n Remark 4.10.
We observe that the posets above come equipped with an isomorphism ( R n ) op ∼ = R n given by applying the “bar operator” ( − ). It is worth pointing out that our scaling is symmetricwith respect to this duality. 27 aa ab bb aa ab bb R aa ab bb aa ab bb aa ab bb aa ab bb R aa ab bb aa ab bb aa ab bb aa ab bb aa ab bb aa ab bb Figure 1: The posets R n for n ≤ Definition 4.11.
We define a cosimplicial object Q : ∆ Set sc∆ , [ n ] Q (∆ n × Tw(∆ ) ♯ )where Q was already introduced in Definition 2.1 and the functoriality is the obvious one. Since Q preserves colimits we see that Q ( n ) splits into Q (∆ n × Tw(∆ )) ∼ = Q (∆ n × ∆ ) a Q (∆ n ) Q (∆ n × ∆ ) . Remark 4.12.
Recall that our definitions imply that a map Q ( n ) → D corresponds precisely to afunctor ∆ n × Tw(∆ ) Tw( D ) . We see that a simplex in L ( F,G ) is given by a map Q ( n ) → D satisfying the obvious conditions after restriction to ∆ n × Tw(∆ ) , (∆ n × Tw(∆ )) op ⊂ Q ( n ). Definition 4.13.
Let n > R ( n ) fits into a cocone for the colimit defining Q ( n ). Then the induced cofibrations ε n : Q ( n ) R ( n ), assemble into map of cosimplicial objects ξ : Q R . 28 efinition 4.14.
We define a cosimplicial object T : ∆ Set sc∆ , [ n ] Q ( n ) × ∆ . Remark 4.15.
Analogously to Remark 4.12, we can identify a simplex ∆ n → Tw( D ∆ ) ( F,G ) witha map T ( n ) → D such that the restrictions to ∆ n × ∆ and (∆ n ) op × ∆ are constant on F and G op respectively. Definition 4.16.
Define a map of posets µ n : T ( n ) R ( n ) , ( ℓ, a ) ℓ ab , ( ℓ, a ) ℓ aa , ( ℓ, b ) ℓ bb , ( ℓ, b ) ℓ ab Then the maps µ n assemble into a map of cosimplicial objects µ : T R . Remark 4.17.
At this juncture it is worth noting that the scaling on R ( n ) is the minimal scalingsuch that ξ : Q R and µ : T R respect the scaling, and such that the scaling on R ( n ) hasthe two symmetries previously mentioned.Let us take a small break to put the previous definitions into perspective. We have defined threecosimplicial objects R , Q and T , the last two of which define the simplices of the ∞ -categoriesTw( D ∆ ) ( F,G ) and L ( F,G ) respectively. The proof of Theorem 4.3 will rely on identifying R as aninterpolating cosimplicial object between Q and T . In the next proposition, we will show an equiv-alence of cosimplicial objects between Q and R thus providing a key technical ingredient for theproof of the main theorem. Readers unwilling to join us for this combinatorial ride can safely skipthe next proof. Proposition 4.18.
The map of cosimplicial objects ξ : Q R is a levelwise trivial cofibrationin the scaled model structure.Proof.
We will prove something stronger, namely, for every n > ξ n is scaled anodyne.Using the description of both Q ( n ) and R ( n ) as pushouts, we deduce that it will suffice to showthat the map Q (∆ n × (∆ ) ♯ ) (cid:16) (∆ n × ∆ ) ⋆ (∆ n × ∆ ) op (cid:17) ⋄ is scaled anodyne, where the subscript ⋄ indicates the scaling induced by that of R ( n ). Beforeembarking upon the proof of our claim we will set some notation Q (∆ n × (∆ ) ♯ ) = A n ⋄ , (cid:16) (∆ n × ∆ ) ⋆ (∆ n × ∆ ) op (cid:17) ⋄ = B n ⋄ . Let ( r, s ) be a pair of non-negative integers such that r, s n . We define a simplex σ ( r,s ) : ∆ n +3 B n ⋄ ℓ ℓ a if ℓ rℓ b if r + 1 ℓ n + 1 ℓ b if n + 2 ℓ n + 2 − sℓ a if 2 n + 3 − s ℓ n + 3and note that B n ⋄ = S ( r,s ) σ ( r,s ) . We further divide the simplices σ ( r,s ) into three families parametrizedby r − s = α . To illuminate our claims let us include some examples for n = 3.29 > α = 0 α < a b a b a b a b a b a b a b a b a b a b a b a b a b a b a b a b a b a b a b a b a b a b a b a b σ (3 , σ (2 , σ (2 , We define B + ⋄ (resp. B −⋄ , resp. B ⋄ ) as the union of the simplices σ ( r,s ) such that α > α
0, resp. α = 0) with the induced scaling. It follows from unwinding the definitions that A n ⋄ = B ⋄ and that B + ⋄ ∩ B −⋄ = B ⋄ . We have thus produced a pushout square A n ⋄ B + ⋄ B −⋄ B n ⋄ . p We turn now to show that A n ⋄ → B ±⋄ is scaled anodyne. First let us tackle the case α >
0. To thisend we produce a filtration A n ⋄ = X X · · · X n − X n = B + ⋄ where X j is the scaled simplicial subset consisting in those simplices contained in some σ ( r,s ) with α j . We claim that in order to show that X j − → X j is scaled anodyne it suffices to show thattop horizontal morphism f ( r,s ) in the pullback diagram below W ( r,s ) ∆ n +3 X j − X + j . f ( r,s ) y σ ( r,s ) is scaled anodyne with respect to the induced scaling. Indeed, we observe that given ( r, s ) , ( u, v )such that r − s = u − v = j then it follows that σ ( r,s ) ∩ σ ( u,v ) ∈ X j − and the claim follows. Aftersome contemplation we discover that W ( r,s ) = d r ( σ ( r,s ) ) ∪ d n +2 − s ( σ ( r,s ) ) . Consequently we can define a dull subset consisting of the sets { r } , { n + 2 − s } with pivot point2 n + 2 − r . Using Lemma 1.10 we conclude that A n ⋄ → B + ⋄ is scaled anodyne.30he case α < R ( n ) restricts to ( B + ⋄ ) op ∼ = B −⋄ and that our scaling is symmetric. The case α < Corollary 4.19.
Let D be an ∞ -bicategory and let X be the simplicial set obtained via the cosim-plicial object R . Consider the induced map ξ ∗ : X L ∆ . Then the ξ ∗ is a trivial Kan fibration. In particular, after passing to fibers we obtain an equivalenceof ∞ -categories X ( F,G ) ≃ L ( F,G ) . Proof.
Note that as an immediate consequence of Proposition 4.18 we obtain an scaled anodynemap ∂ Q n → ∂ R n . Consider the morphisms Q ( n ) Q ( n ) a ∂ Q n ∂ R n R ( n ) , and note the last map is a trivial cofibration by 2-out-of-3. The reader will observe that the boundarylifting problems are in bijection with lifting problems of the form Q ( n ) ` ∂ Q n ∂ R n D R ( n )and hence the result. aa ab bb aa ab bb aa ab bb aa ab bb Figure 2: T (1) pictured in blue as a subset of R (1) under the inclusion µ . The map ψ can bealternately characterized as the unique map such that ψ ◦ µ = id and ψ preserves ( − )and its dual. Construction 4.20.
We define a map R ( n ) Q ( n )by requiring i xy i and i xy i . We further define a map R ( n ) ∆ i xy x, i xy y. Note that both of these maps can be easily checked to preserve thescalings. Together, they thus define a map ψ n : R ( n ) T ( n ) such that ψ n ◦ µ n = id (see Figure 2).Moreover, the ψ n yield a natural transformation ψ : R → T . We denote by ψ ∗ : Tw( D ∆ ) X the induced map. Lemma 4.21.
The diagram
Tw( D ∆ ) X L ∆ Θ ∆1 ψ ∗ ξ ∗ commutes.Proof. Left as an exercise to the reader.
Proof of Theorem 4.3.
By virtue of Proposition 4.7, it will suffice to show that Θ X is an equivalenceof Cartesian fibrations for X = ∆ n with n = 0 ,
1. The case n = 0 is obvious. To show the case n = 1we observe that due to Corollary 4.19 and Lemma 4.21 it will suffice to show ψ ∗ is an equivalenceof ∞ -categories upon passage to fibers. We further note that since µ ∗ ◦ ψ ∗ = id it will be enoughto show that ϕ ∗ = ψ ∗ ◦ µ ∗ is a fiberwise equivalence. Let σ : ∆ n → ∆ and let j ∈ [ n ] be the firstobject such that σ ( j ) = 1 if σ is constant on 0 we set the convention j = n + 1. Now we can definea map of scaled simplicial sets ϕ σ : R ( n ) R ( n )which leaves every object invariant except those of the form ℓ aa with ℓ < j which are sent to ℓ ab .Given ρ : ∆ n → X ( F,G ) we define a simplex H ( σ, ρ ) : ∆ n → X ( F,G ) given by the composite R ( n ) ϕ σ R ( n ) ρ D This assignment extends to a homotopy H : ∆ × X ( F,G ) X ( F,G ) which is component-wisean equivalence. This exhibits an equivalence of morphisms id ∼ (cid:0) ϕ (cid:1) ∗ ( F,G ) where ϕ denotes thepreviously defined map with respect with the constant simplex at 0.Let σ : ∆ n → ∆ . Then we define a map of scaled simplicial sets ϕ σ : R ( n ) R ( n )that leaves every object invariant except those of the form ℓ aa which are sent to ℓ ab and those ofthe form ℓ bb with ℓ < j with are sent to ℓ ab . We can now define, in perfect analogy to the situationabove, a natural equivalence H : ∆ × X ( F,G ) X ( F,G ) between ϕ ∗ ( F,G ) and (cid:0) ϕ (cid:1) ∗ ( F,G ) , hence theresult. ∞ -categories We conclude this section (and thereby the paper) with several corollaries of Theorem 4.3, and theirapplication to the 2-dimensional universal property of weighted colimits. Because of the techni-cal complexities shunted into the proofs of the properties of Tw( D ), the proof of this 2-universalproperty is extremely straightforward.Throughout this section we will fix an ∞ -category C and a pair of functors F : C C at ∞ , W : C op C at ∞ that we will refer of as the diagram and the weight functors respectively. Wewill denote by C at ∞ the ∞ -bicategory of ∞ -categories.32 efinition 4.22. Let D be an ∞ -bicategory. We say that the underlying ∞ -category D is tensored over C at ∞ with respect to D if for every d ∈ D the mapping functor Map D ( d, − ) has a left adjoint − ⊗ d : C at ∞ → D ; in this case these adjoints determine an essentially unique functor C at ∞ × D → D . Corollary 4.23.
Let D be an ∞ -bicategory such that the underlying ∞ -category D is tensored over C at ∞ with respect to D . Then for every ∞ -category C the functor category D C is tensored over C at ∞ with respect to D C .Proof. Combine Theorem 4.3 with [GHN15, Lem. 6.7].
Corollary 4.24.
Let C be an ∞ -category and let E C , E ′ C be Cartesian fibrations. Wedenote by Fun cart C ( E , E ′ ) the ∞ -category of maps of Cartesian fibrations. Then there is a naturalequivalence of ∞ -categories Fun cart C ( E , E ′ ) ≃ Nat C (St( E ) , St( E )) where St denotes the straightening functor.Proof. Combine Theorem 4.3 with [GHN15, Prop. 6.9].
Remark 4.25.
It is worth noting that Corollary 4.24 can be interpreted as very compelling evidencesuggesting that an enhanced version of the straightening functor St, will yield an equivalence of ∞ - bicategories between the category of Cartesian fibrations over C and the category of C at ∞ -valuedfunctors on C .Recall that in [GHN15, Def. 2.7], the authors define the weighted colimit of F with weight W asthe coend colim Tw( C ) W × F. According to this definition the universal property of the weighted colimit is purely 1-dimensional.Our aim in this section is to show that the previous definition is just a shadow of a bicategoricaluniversal property and thus find a bridge between the classical theory of weighted colimits in2-categories and the realm of ∞ -bicategories. Definition 4.26.
The weighted colimit of F with weight W is the unique (up to equivalence) ∞ -category representing the functor C at ∞ C at C at op ∞ ∞ C at C op ∞ C at ∞Y F ∗ Nat C op ( W , − ) where Y denotes the bicategorical Yoneda embedding. We will denote weighted colimit by W ⊗ F .More compactly, this definition means that there is an equivalence Nat C op ( W, Fun( F ( − ) , X )) ≃ Fun( W ⊗ F, X ), natural in X . Remark 4.27.
This definition of weighted colimits was already considered in more generality in[AG20]. We are here ignoring some substantial set-theoretic complexities. We should, more properly, fix a nested pair ofGrothendieck universes, and consider variants of C at ∞ based on size. In the interest of concision, we will sweepsuch set-theoretic concerns under the rug, leaving their contemplation to the interested reader. heorem 4.28. Consider a pair of functors F : C C at ∞ , W : C op C at ∞ . Then there isan equivalence of ∞ -categories W ⊗ F ≃ colim Tw( C ) W × F. Proof.
Let X be an ∞ -category. We trace out a chain of equivalences, natural in X . By Theorem 4.3,we have Nat C op ( W, Fun( F ( − ) , X )) ≃ lim Tw( C ) op Fun( W ( − ) , Fun( F ( − ) , X )) . A standard chain of manipulations then yieldslim
Tw( C ) op Fun( W ( − ) , Fun( F ( − ) , X )) ≃ lim Tw( C ) op Fun( W ( − ) × F ( − ) , X )) ≃ Fun colim
Tw( C ) W × F, X ! so that colim Tw( C ) W × F satisfies the universal property defining W ⊗ F , completing the proof.34 eferences [ADS20] Abellán García, F., Dyckerhoff, T., and Stern, W. H. “A relative 2-nerve”. Algebraic &Geometric Topology (To appear)[AGS20] Abellán García, Fernando and Stern, Walker H. “Theorem A for marked 2-categories.”2020 arXiv:2002.12817.[AG20] Abellán García, F. “Marked cofinality for ∞ -functors”. 2020 arXiv: 2006.12416.[BGN18] Barwick, Clark, Glasman, Saul, and Nardin, Denis. “Dualizing cartesian and cocartesianfibrations”. Theory and Applications of Categories . Vol. 33 No. 4 (2018), pp. 67-94.[DK19] Dyckerhoff, Tobias and Kapranov, Mikhail. “Higher Segal Spaces”. Springer Lecture Notesin Mathematics 2244.
Springer , 2019.[Bar17] Barwick, Clark. “Spectral Mackey Functors and equivariant Algebraic K-Theory (I).”
Ad-vances in Mathematics
Vol. 304 (Jan. 2017), pp. 646-727.[Cis19] Cisinski, Dennis-Charles. “Higher Categories and Homotopical Algebra.” Cambridge Uni-versity Press, 2019.[DS11] Dugger, Daniel and Spivak, David I. “Rigidification of quasi-categories”.