Gray tensor products and lax functors of (∞,2) -categories
aa r X i v : . [ m a t h . A T ] J un GRAY TENSOR PRODUCTS AND LAX FUNCTORS OF ( ∞ , -CATEGORIES ANDREA GAGNA, YONATAN HARPAZ, AND EDOARDO LANARI
Abstract.
We give a definition of the Gray tensor product in the setting ofscaled simplicial sets which is associative and forms a left Quillen bifunctorwith respect to the bicategorical model category of Lurie. We then introducea notion of oplax functor in this setting, and use it in order to characterize theGray tensor product by means of a universal property. A similar characteri-zation was used by Gaitsgory and Rozenblyum in their definition of the Grayproduct, thus giving a promising lead for comparing the two settings.
Contents
Acknowledgements Introduction ∞ -categories 41.2 Scaled simplicial sets and ∞ -bicategories 4 ∞ -bicategories 193.3 The universal property of the Gray product 21 References Acknowledgements
The first author is supported by GA ˇCR EXPRO 19-28628X. The third authorgratefully acknowledges the support of Praemium Academiae of M. Markl andRVO:67985840, as well as fruitful conversations with Nick Rozenblyum during hisstay at MSRI.
Introduction
The theory of 2-categories provides a powerful framework in which one can de-velop formal category theory, allowing for notions such as adjunctions, Kan ex-tensions, correspondences and lax (co)limits to be studied in an abstract setting.The category of 2-categories carries a monoidal structure, given by the Gray tensorproduct, which makes use of the 2-dimensional structure available. This monoidalstructure comes in a few flavors, as is often the case in the 2-categorical setting:there is a pseudo version, which was the first one to be defined by Gray in [6] (hence
Mathematics Subject Classification. the name), as well as a lax and an oplax version. The Gray tensor product servesthe purpose of representing pseudo/lax natural transformations, and it is thus a re-placement for the cartesian product of 2-categories when one considers these weakernotion of morphisms. Moreover, while the cartesian product is badly-behaved withrespect to the folk model category structure on 2-categories, Lack [7] showed thatthe pseudo Gray tensor is compatible with the folk model category, and recentlyAra and Lucas [1] proved that the same holds for the lax and oplax version of theGray tensor product.In the last decade, the study of ∞ -categories as a building block for homotopy-coherent mathematics has contributed to the development of spectral and derivedalgebraic geometry, culminating in impressive applications, such as the proof byGaitsgory and Lurie in [4] of Weil’s conjecture on Tamagawa numbers for func-tion fields. In the same spirit of organizing ordinary category theory from a 2-dimensional perspective, the framework of ( ∞ , ∞ , ∞ , ∞ -category C at ( ∞ , of ( ∞ , ∞ -category with respect to the Gray product.Versions of the Gray tensor product in different contexts already appear in theliterature. Verity [15] defines a Gray tensor product in stratified simplicial setscompatible with the complicial model category structure for ( ∞ , ∞ )-categories.This can be truncated to be compatible with the model category structure for 2-truncated saturated complicial sets established in [13], giving a Gray tensor productfor this model of ( ∞ , -sets.The main motivation behind our interest in the Gray tensor product comesfrom the recent book [5], where Gaitsgory and Rozenblyum develop a formalism ofcategories of correspondences which makes use of the Gray tensor product. How-ever, there are some unproven claims in the technical section dealing with ( ∞ , ∞ -bicategories.This mirrors the approach of Gaitsgory–Rozenblyum for the definition of the Grayproduct, and reduces the comparison of the respective Gray products that that ofthe two notions of oplax functors. The latter comparison is the subject of currentwork in progress and we hope to settle this question in a future paper. RAY TENSOR PRODUCTS AND LAX FUNCTORS OF ( ∞ , It is worth pointing out that, despite the existence of various models of Graytensor product, none of them have been proven to satisfy the characterizing uni-versal property formulated in [5, Chapter 10, Point 3.2.3], and in particular nonwere ever shown to be equivalent to that of loc. cit. This has been one obstacle inproviding proofs for some of the unproven claims in [5, Chapter 10], and we hopeand expect that the present paper will open a path leading to such proofs.The present paper is organized as follows. In § § § § § ∞ -bicategories. This universalcharacterization allows one in principle to recognize the Gray tensor product in anygiven model for ( ∞ , Preliminaries
In this section we recall the necessary definitions and theorems that will be usedthroughout this paper.
Notation 1.1.
We will denote by ∆ the category of simplices, that is the categorywhose objects are the finite non-empty ordinals [ n ] = { , , , . . . , n } and morphismsare the non-decreasing maps. We will denote by S et ∆ the category of simplicialsets, that is the category of presheaves on sets of ∆, and will employ the standardnotation ∆ n for the n -simplex, i . e ., the simplicial set representing the object [ n ]of ∆. For any subset ∅ 6 = S ⊆ [ n ] we will write ∆ S ⊆ ∆ n to denote the ( | S | − n whose vertices belong to S . For 0 ≤ i ≤ n we will denoteby Λ ni ⊆ ∆ n the i -th horn in ∆ n , that is, the subsimplicial set of ∆ n spanned by allthe ( n − 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 ∞ -category we will always mean a quasi-category , i . e ., a simplicial set X which admits extensions for all inclusions Λ ni → ∆ n , for all n > < i < n (known as inner horns ). Given an ∞ -category X , we will denote its homotopycategory by ho( X ). This is the ordinary category having as objects the 0-simplicesof X , and as morphisms x → y the set of equivalence classes of 1-simplices f : x → y of X under the equivalence relation generated by identifying f and f ′ if there is a2-simplex H of X with H | ∆ { , } = f, H | ∆ { , } = f ′ and H | ∆ { , } degenerate on x .We recall that the functor ho : ∞ - C at → C at is left adjoint to the ordinary nervefunctor N : 1- C at → ∞ - C at . ANDREA GAGNA, YONATAN HARPAZ, AND EDOARDO LANARI
Marked simplicial sets and ∞ -categories.Definition 1.2. A marked simplicial set is a pair ( X, E ) where X is simplicialset and E 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 +∆ . Remark . The category S et +∆ of marked simplicial sets admits an alternativedescription, as the category of models of a limit sketch. In particular, it is areflective localization of a presheaf category and it is a cartesian closed category.In fact, it is a locally cartesian closed category. Theorem 1.4 ([10]) . There exists a model category structure on the category S et +∆ of marked simplicial sets in which cofibrations are exactly the monomorphisms andthe fibrant objects are marked simplicial sets ( X, E ) in which X is an ∞ -categoryand E is the set of equivalences of X , i . e ., -simplices f : ∆ → X which areinvertible in ho( X ) . This is a special case of Proposition 3.1.3.7 in [10], when S = ∆ . We will referto the model structure of Theorem 1.4 as the marked categorical model structure ,and its weak equivalences as marked categorical equivalences . Remark . Marked simplicial sets endowed with the marked categorical modelstructure are a model for ( ∞ , Scaled simplicial sets and ∞ -bicategories.Definition 1.6 ([9]) . A scaled simplicial set is a pair ( X, T ) where X is simplicialset and T 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 often refer to a scaled simplicial set just without explicitly mentioning itsthin 2-simplices, when this causes no ambiguity.We will denote by S et sc∆ the category of scaled simplicial sets. Notation 1.7.
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 the left and right adjoint of the obvious forgetful functor S et sc∆ → S et ∆ . Remark . The category S et sc∆ admits an alternative description, as the categoryof models of a limit sketch. In particular, it is a reflective localization of a presheafcategory and so it is cartesian closed. In fact, we can consider the category ∆ sc having { [ k ] } k ≥ ∪ { [2] t } as set of objects, obtained from ∆ by adding an extra ob-ject and maps [2] → [2] t , σ it : [2] t → [1] for i = 0 , S et sc∆ is then the reflective localization of the category of presheavesPSh(∆ sc ) (of sets) at the arrow [2] t ∐ [2] [2] t → [2] t , where we have identified anobject of ∆ sc with its corresponding representable presheaf. Equivalently, the lo-cal objects are those presheaves X : ∆ opsc → Set for which X ([2] t ) → X ([2]) is amonomorphism. Notation 1.9.
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 with RAY TENSOR PRODUCTS AND LAX FUNCTORS OF ( ∞ , the 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 .The following set of inclusions will characterize the model structure on scaledsimplicial sets for ∞ -bicategories. We are implicitly making use of Corollary 3.0.6of [3] in that we assume weak ∞ -bicategories coincide with ∞ -bicategories, asdefined in [9]. Definition 1.10.
Let S be the set of maps of scaled simplicial sets consisting of:(i) the inner horns inclusions (cid:0) Λ ni , ∆ { i − ,i,i +1 } (cid:1) → (cid:0) ∆ n , ∆ { i − ,i,i +1 } (cid:1) , n ≥ , < i < n ;(ii) the map (∆ , T ) → (∆ , T ∪ { ∆ { , , } , ∆ { , , } } ) , where we define T def = { ∆ { , , } , ∆ { , , } , ∆ { , , } , ∆ { , , } , ∆ { , , } } ;(iii) the set of maps (cid:16) Λ n ∐ ∆ { , } ∆ , ∆ { , ,n } (cid:17) → (cid:16) ∆ n ∐ ∆ { , } ∆ , ∆ { , ,n } (cid:17) , n ≥ . We call S the set of generating anodyne morphisms . We make use of it in thefollowing definition. Definition 1.11. An ∞ -bicategory is a scaled simplicial set ( X, T ) which admitsextensions along all maps in S . Diagrammatically, if i : K → L is a map in S , thenfor every map f : K → ( X, T ) there exists an extension as displayed below by thedashed arrow in the following diagram: K ( X, T ) L fi . Putting together the results in [9] and [3], we get the following theorem.
Theorem 1.12.
There exists a model structure on the category of scaled simplicialsets whose cofibrations are the monomorphisms and whose fibrant objects are the ∞ -bicategories. This model structure is proved in [3] to be Quillen equivalent to Verity’s one onstratified sets for saturated 2-trivial complicial sets, as defined in [15] and [13]. Wedepict the Quillen equivalence as follows:(1) S et sc∆ ι ' ' U g g ⊥ Strat , where U denotes the forgetful functor and ι is an inclusion. Furthermore, Lurieshows in [9] that there is a scaled homotopy coherente nerve N sc : S et +∆ - C at −→ S et sc∆ which is a Quillen equivalence, where the category S et +∆ - C at of category enriched inmarked simplicial sets is endowed with the S et +∆ -enriched model category structure(see [10, § A.3.2]).
Definition 1.13.
Let (
X, T ) be a scaled simplicial set. We will say that thecollection of triangles T is saturated if both ( X, T ) and ( X op , T ) satisfy the extensionproperty with respect to the (ii) of Definition 1.10. ANDREA GAGNA, YONATAN HARPAZ, AND EDOARDO LANARI
Example . Any ∞ -bicategory is saturated. Remark . It follows from [9, Remark 3.1.4] that if a scaled simplicial set X is saturated then X satisfies the extension property with respect to the maps(∆ , T i ) → ∆ ♯ for i = 1 , T i is the collection of all triangles in ∆ ex-cept ∆ { ,i, } . Definition 1.16.
Let (
X, T ) be a scaled simplicial set. We define the saturatedclosure of T to be the smallest saturated set of triangles T ′ which contains T (notethat such a set exists because the collection of saturated sets of triangles is closedunder intersection). Lemma 1.17.
Let ( X, T ) be a scaled simplicial set and T the saturated closure of T . Then the map ( X, T ) → ( X, T ) is a trivial cofibration in S et sc∆ .Proof. Applying the small object argument with respect to the set S consisting ofthe map (ii) of Definition 1.10 and its opposite we may find a map ( X, T ) → ( X ′ , T ′ )which is a retract of transfinite compositions of pushouts of maps in S and suchthat T ′ is saturated in X ′ . Since the maps in S are isomorphisms on the underlyingsimplicial set the same holds for X → X ′ , and so we may identify T ′ with a setof triangles in X . Since every map in S is a trivial cofibration the same holds forthe resulting map ( X, T ) → ( X, T ′ ). Now since T ′ is saturated and contains T italso contains its saturated closure T . On the other hand, since ( X, T ) satisfies theextension property with respect to S it also satisfies the extension property withrespect to ( X, T ) → ( X, T ′ ), and so T ′ ⊆ T . We may then conclude that T ′ = T isthe saturated closure of T and hence the desired result follows. (cid:3) Definition 1.18.
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 faces arethin triangles. Remark . Notice that X and X th agree on the 1-skeleton. Moreover, whenever X is an ∞ -bicategory, its core X th (formally, its underlying simplicial set) is an ∞ -category. Definition 1.20.
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, if itsimage 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 thesubsimplicial 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 considered asthe ∞ -groupoid obtained from C by discarding all non-invertible 1-cells and 2-cells.If ( X, T ) is an arbitrary scaled simplicial set then we will say that an edge in X isinvertible if its image in C is invertible for any bicategorical equivalence ( X, T ) → C .This does not depend on the choice of C . Notation 1.21.
Let C be an ∞ -bicategory and let x, y ∈ C be two vertices. Insection 4.2 of [9], Lurie gives an explicit model for the mapping ∞ -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 constanton 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 whenthe triangle f | ∆ { (0 , , (1 , , (1 , } is thin. The assumption that C is an ∞ -bicategoryimplies that the marked simplicial set Hom C ( x, y ) is fibrant in the marked categor-ical model structure, that is, it is an ∞ -category whose marked edges are exactlythe equivalences. RAY TENSOR PRODUCTS AND LAX FUNCTORS OF ( ∞ , Remark . By Remark 4.2.1 and Theorem 4.2.2 of [9], if D is a fibrant Set +∆ -en-riched category and C is an ∞ -bicategory equipped with a bicategorical equivalence ϕ : C ≃ N sc ( D ) then the mapsMap C ( x, y ) −→ Map N sc ( D ) ( ϕ ( x ) , ϕ ( y )) ←− Map D ( ϕ ( x ) , ϕ ( y ))are categorical equivalences of marked simplicial sets for every pair of vertices x, y of C . It then follows that a map ϕ : C → C ′ of ∞ -bicategories is a bicategoricalequivalence if and only if it is essentially surjective (that is, every object in C ′ isequivalent to an object in the image, see Definition 1.20) and the induced mapMap C ( x, y ) → Map C ′ ( ϕ ( x ) , ϕ ( y )) is a categorical equivalence of (fibrant) markedsimplicial sets for every x, y ∈ C . Remark . It follows from Remark 1.22 that if ϕ : C → C ′ is a bicategorical equiv-alence of ∞ -bicategories then the induced map ϕ th : C th → ( C ′ ) th is an equivalenceof ∞ -categories.It is shown in [9, Proposition 3.1.8] that the cartesian product × : S et sc∆ × S et sc∆ −→ S et sc∆ of a scaled anodyne map and a monomorphism is contained in the saturation ofthe class of scaled anodyne maps. Therefore, thanks to Theorem 5.1 of [3], thecartesian product is a left Quillen bifunctor with respect to the bicategorical modelstructure, i.e. S et sc∆ is a cartesian closed model category. In particular, for a everytwo scaled simplicial sets X, Y we have a mapping object Fun(
X, Y ) which satisfies(and is determined by) the exponential formulaHom S et sc∆ ( Z, Fun(
X, Y )) ∼ = Hom S et sc∆ ( Z × X, Y ) . In addition, when Y is an ∞ -bicategory the mapping object Fun( X, Y ) is an ∞ -bi-category as well, which we can consider as the ∞ -bicategory of functors from X to Y . In this case we will denote by Fun th ( X, Y ) ⊆ Fun(
X, Y ) the core ∞ -category andby Fun ≃ ( X, Y ) ⊆ Fun th ( X, Y ) the core ∞ -groupoid of Fun( X, Y ). In particular,Fun th ( X, Y ) is an ∞ -category and Fun ≃ ( X, Y ) is a Kan complex, which we consideras the space of functors from X to Y . We note the following: Lemma 1.24.
Let f : X → Y be a map of scaled simplicial sets. Then f is abicategorical equivalence if and only if for every ∞ -bicategory C the induced map (2) f ∗ : Fun ≃ ( Y, C ) −→ Fun ≃ ( X, C ) is an equivalence of Kan complexes.Proof. If f : X → Y is a bicategorical equivalence then Fun( Y, C ) → Fun( X, C ) isa bicategorical equivalence for every ∞ -bicategory C since S et sc∆ is cartesian closedand every object is cofibrant. It then follows from Remark 1.23 that (2) is anequivalence of Kan complexes.Now suppose that (2) is an equivalence of Kan complexes. By the argumentabove the property that (2) is an equivalence of Kan complexes will hold for anyarrow which is levelwise bicategorically equivalent to f ′ in S et sc∆ . We may henceassume without loss of generality that X and Y are fibrant, that is, they are ∞ -bi-categories. Taking C = X in (2) we may conclude there exists a map g : Y → X suchthat gf : X → X is in the same component as Id : X → X in Fun ≃ ( X, X ). There ishence an arrow e : ∆ → Fun ≃ ( X, X ) such that e (0) = Id and e (1) = gf . Let K bea contractible Kan complex which contains a non-degenerate arrow ∆ ⊆ K , whichwe can write as v → u . Then we can extend the map e : ∆ → Fun ≃ ( X, X ) to amap e e : K → Fun ≃ ( X, X ). By adjunction we thus get a map H : X × K → X suchthat H | X ×{ v } = Id and H | X ×{ u } = gf . Since the map X × K → X induced by the ANDREA GAGNA, YONATAN HARPAZ, AND EDOARDO LANARI canonical morphism K → ∆ is a bicategorical equivalence and the compositions X × { v } → X × K → X and X × { u } → X × K → X are both the identity on X ,we may consider X × K as a cylinder object for X in S et sc∆ , so that gf is homotopicto the identity with respect to the bicategorical model structure. We now claimthat f g : Y → Y is homotopic to the identity on Y . To see this, we note that bythe above we have that f gf : X → Y is in the same component as f : X → Y inFun ≃ ( X, Y ). Using that (2) is an equivalence of Kan complexes for C = Y we mayconclude that f g : Y → Y is in the same component as Id : Y → Y in Fun ≃ ( Y, Y ).Arguing as above we get that f g is homotopic to the identity with respect to thebicategorical model structure. We may hence conclude that f is an isomorphism inHo( S et sc∆ ) and hence a bicategorical equivalence, as desired. (cid:3) Gray products and lax natural transformations
The Gray product.
In this section we define the
Gray product of two scaledsimplicial sets. In what follows, when we say that a 2-simplex σ : ∆ → X degener-ates along ∆ { i,i +1 } ⊆ ∆ (for i = 0 ,
1) we mean that σ is degenerate and σ | ∆ { i,i +1 } is degenerate. This includes the possibility that σ factors through the surjectivemap ∆ → ∆ which collapses ∆ { i,i +1 } as well as the possibility that σ factorsthrough ∆ → ∆ . Definition 2.1.
Let (
X, T X ) , ( Y, T Y ) be two scaled simplicial sets. The Gray prod-uct ( X, T X ) ⊗ ( Y, T Y ) is the scaled simplicial set whose underlying simplicial set isthe cartesian product of X × Y and such that a 2-simplex σ : ∆ → X × Y is thinif 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 ∆ { , } . Proposition 2.2.
The natural associativity isomorphisms of the cartesian productof marked simplicial set are also isomorphisms of scaled simplicial sets for the Grayproduct, and hence the Gray product gives a monoidal structure on S et sc∆ .Proof. Let (
X, T X ) , ( Y, T Y ) , ( Z, T Z ) be three scaled simplicial sets. We have tocheck that the thin 2-simplices of (( X, T X ) ⊗ ( Y, T Y )) ⊗ ( Z, T Z ) are the same asthose of ( X, T X ) ⊗ (( Y, T Y ) ⊗ ( Z, T Z )). Indeed, direct inspection shows that bothsets of thin 2-simplices coincide with the set of those ( α, β, γ ) ∈ T X × T Y × T Z suchthat at least one of the following three conditions hold:(1) both α and β degenerate along ∆ { , } ;(2) both β and γ degenerate along ∆ { , } ; or(3) α degenerates along ∆ { , } and γ degenerate along ∆ { , } . (cid:3) Remark . The 0-simplex ∆ can be considered as a scaled simplicial set in aunique way, and serves as the unit of the Gray product. Consequently, if X is any discrete scaled simplicial set then X ⊗ Y ∼ = X × Y and Y ⊗ X ∼ = Y × X for anyscaled simplicial set Y . Remark . The Gray product is not symmetric in general. 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 ∆ { } × ∆ . RAY TENSOR PRODUCTS AND LAX FUNCTORS OF ( ∞ , By definition we see that σ is thin in X but σ is not. If C is an ∞ -bicategorythen a map 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 }× ∆ ). Wethus have an invertible 2-cell h ≃ = ⇒ 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.It is straightforward to verify that the Gray product preserves cofibrations andcolimits separately in each variable. Consequently, one my associate with ⊗ aright and a left mapping objects, which we shall denote by Fun opgr ( X, Y ) andFun gr ( X, Y ) respectively. More explicitly, an n -simplex of Fun opgr ( X, Y ) is givenby a map of scaled simplicial sets∆ n ⊗ X −→ Y. A 2-simplex ∆ ⊗ X → Y of Fun opgr ( X, Y ) is thin if it factors through (∆ ) ♯ ⊗ X .Similarly, an n -simplex of Fun gr ( X, Y ) is given by a map of scaled simplicial sets X ⊗ ∆ n −→ Y and the scaling is determined as above.Let C be an ∞ -bicategory and K a scaled simplicial set. We will see below thatthe scaled simplicial sets Fun opgr ( K, C ) and Fun gr ( K, C ) are in fact ∞ -bicategories.The objects of Fun opgr ( K, C ) correspond to functors K → C and by Example 2.5we may consider morphisms in Fun opgr ( K, C ) as oplax natural transformations . Ifwe take Fun gr ( K, C ) instead then the objects are again functors K → C , but nowthe edges will correspond to lax natural transformations . For example, if K = ∆ and f , f are two morphisms in C then an arrow in Fun opgr (∆ , C ) from f to f is a square in C of the form(3) x yz w f g h g f ≃ with the upper right triangle thin. This can be considered as a 2-cell g ◦ f ⇒ f ◦ g .On the other hand, an arrow in Fun gr (∆ , C ) from f to f is a square of the form 3whose lower left triangle is thin, i.e., a 2-cell in the other direction f ◦ g ⇒ g ◦ f . Remark . Let X , Y and Z be scaled simplicial sets. The associativity isomor-phism of the Gray product yields a natural isomorphismFun opgr ( X, Fun gr ( Y, Z )) ∼ = Fun gr ( Y, Fun opgr ( X, Z )) . Comparison with the Gray product of stratified sets.
In his extensivework [15], Verity constructs a Gray tensor product in the setting of stratified sets.In particular, for two stratified sets (
X, t X ) , ( Y, t Y ), the stratified set ( X, t X ) ⊗ ( Y, t Y ) has as an underlying marked simplicial set the product of the underlyingmarked simplicial sets, while a 2-simplex ( σ X , σ Y ) : ∆ → X × Y is marked in( X, t X ) ⊗ ( Y, t Y ) if and only if (1) σ X and σ Y are marked in X and Y respectively;(2) either ( σ X ) | ∆ { , } is marked in X or ( σ Y ) | ∆ { , } is marked in Y .The marking on higher simplices are defined in a similar manner, though they donot play a significant role if one is only considering stratified sets up to 2-complicialweak equivalence. Our goal in the present subsection is to compare the Grayproduct defined in § S et sc∆ → Strat established in [3]. Remark . The Gray product of stratified sets recalled above is associative, butdoes not preserve colimits in each variable, and in particular cannot be a left Quillenbifunctor. In [15], Verity also considers a variant of the above definition whichpreserve colimits in each variable but is not associative. By contrast, the Graytensor product of scaled simplicial sets is simultaneously associative and a leftQuillen bifunctor, as we will establish in Theorem 2.14 below.In what follows, it will be useful to consider several equivalent variants of theGray tensor product on scaled simplicial sets. Let (
X, T X ) , ( Y, T Y ) be scaled sim-plicial sets and let T gr ⊆ T X × T Y denote the collection of triangles which are thin inthe Gray product ( X, T X ) ⊗ ( Y, T Y ), see Definition 2.1. Let T − ⊆ T X × T Y denotethe subset consisting of those pairs of thin triangles ( σ X , σ Y ) for which either both σ X and σ Y are degenerate or at least one of σ X , σ Y degenerates to a point . On theother hand, let T + ⊆ T X × T Y be the set of those pairs of thin triangles ( σ X , σ Y ) forwhich either ( σ X ) | ∆ { , } is degenerate or ( σ Y ) | ∆ { , } is degenerate. Then we havea sequence of inclusions T − ⊆ T gr ⊆ T + . We claim that these three choices for the collection of thin triangles in X × Y yield equivalent models for the Gray tensor product. More precisely, we have thefollowing: Proposition 2.8.
The maps ( X × Y, T − ) ֒ → ( X × Y, T gr ) ֒ → ( X × Y, T + ) are bothscaled anodyne. The proof of Proposition 2.8 will require a couple of lemmas.
Lemma 2.9.
In the situation of Proposition 2.8, if ( X, T X ) = ∆ ♭ and ( Y, T Y ) = ∆ ♭ then the map ( X × Y, T − ) → ( X × Y, T gr ) is scaled anodyne.Proof. We note that in this case T gr contains exactly one triangle that is not in T − , namely, the triangle σ : ∆ → ∆ × ∆ whose projection to ∆ is the identityand whose projection to ∆ is surjective and degenerates along ∆ { , } . Let ∆ → ∆ × ∆ be the 3-simplex spanned by the vertices (0 , , (1 , , (2 , , (2 , σ | ∆ { , , } , σ | ∆ { , , } and σ | ∆ { , , } are in T − , while σ | ∆ { , , } is exactly the 2-simplex which is in T gr but not in T − . We then get that the map( X × Y, T − ) → ( X × Y, T gr ) is a pushout along the inclusion(∆ , { ∆ { , , } , ∆ { , , } , ∆ { , , } } ) ֒ → ∆ ♯ which is scaled anodyne by [9, Remark 3.1.4]. (cid:3) Lemma 2.10.
Let ( Z , T ) = (cid:18) ∆ a ∆ { , } ∆ (cid:19) ⊗ ∆ and ( Z , T ) = ∆ ⊗ (cid:18) ∆ a ∆ { , } ∆ (cid:19) . RAY TENSOR PRODUCTS AND LAX FUNCTORS OF ( ∞ , For i = 0 , , let S i be the set of -simplices consisting of T i together with the imageof the diagonal -simplex diag : ∆ → ∆ × ∆ . Then the map of scaled simplicialsets ( Z i , T i ) −→ ( Z i , S i ) is scaled anodyne for i = 0 , .Proof. We prove the claim for i = 0. The proof for i = 2 proceeds in a sim-ilar manner. Let σ : ∆ → ∆ × ∆ be the 3-simplex spanned by the vertices(0 , , (1 , , (1 , , (2 , σ ′ : ∆ → Z be its image in Z . By definitionwe have that σ ′| ∆ { , , } , σ ′| ∆ { , , } and σ ′| ∆ { , , } are thin in Z , and σ ′| ∆ { , , } is theimage of the diagonal 2-simplex. We then get that the map ( Z , T ) → ( Z , S ) isa pushout along the inclusion(∆ , { ∆ { , , } , ∆ { , , } , ∆ { , , } } ) ֒ → ∆ ♯ which is scaled anodyne by [9, Remark 3.1.4]. (cid:3) Proof of Proposition 2.8.
The inclusion ( X × Y, T − ) ֒ → ( X × Y, T ) can be obtainedas a sequence of pushouts along the scaled anodyne maps described in Lemma 2.9,while the inclusion ( X × Y, T ) ֒ → ( X × Y, T + ) can be obtained as a sequence ofpushouts along the scaled anodyne maps described in Lemma 2.10. (cid:3) Corollary 2.11 (Comparison of scaled and 2-complicial Gray products) . For scaledsimplicial sets ( X, T X ) , ( Y, T Y ) , the natural inclusion i X,Y : ι (( X, T X ) ⊗ ( Y, T Y )) → ι ( X, T X ) ⊗ ι ( Y, T Y ) is a weak equivalence in Strat .Proof. The map i X,Y coincides with ι (( X × Y, T gr ) ֒ → ( X × Y, T + )). Since ι is a leftQuillen functor and ( X × Y, T gr ) ֒ → ( X × Y, T + ) is scaled anodyne by Proposition2.8 this map is a weak equivalence in the 2-complicial model structure. (cid:3) We finish this section with some additional results concerning the relation be-tween the Gray product of scaled simplicial sets and invertible arrows in ∞ -bicategories(see Definition 1.20). Proposition 2.12.
Let C = ( C , T C ) , D = ( D , T D ) be two ∞ -bicategories. Let T ≃ ⊆ T C × T D denote the subset containing those triangles ( α, β ) such that either α | ∆ { , } is invertible in C or β | ∆ { , } is invertible in D . Then the map C ⊗ D → ( C × D , T ≃ ) is bicategorical equivalence.Proof. Let T gr ⊆ T C × T D be the collection of triangles which are thin in C ⊗ D . ByProposition 2.8 it will suffice to show that T ≃ is contained in the saturated closureof T + . For this, let ( α, β ) ∈ T ≃ be a triangle, so that either α | ∆ { , } is invertible in C or β | ∆ { , } is invertible in D . Assume first that β | ∆ { , } is invertible. Since D isan ∞ -bicategory we have that D th is an ∞ -category (which contains the triangle β ) and hence we may find a map ρ : ∆ → D th such that ρ | ∆ { , , } = β , ρ | ∆ { , } is degenerate on β (0) and ρ | ∆ { , } is degenerate on β (1). Let η : ∆ → C be thecomposed map η : ∆ π → ∆ α → C where π : ∆ → ∆ is the map which is given onvertices by π (0) = 0, π (1) = π (2) = π (3) = 1 and π (4) = 2. Let(4) (∆ , T ) −→ (∆ , T ∪ { ∆ { , , } , ∆ { , , } ) , be the scaled anodyne map of Definition 1.10 (ii). We now claim that the map( η, ρ ) : ∆ → C × D sends T to T + . Indeed, if ∆ { i,j,k } ∈ T then either ( i, j ) ∈{ (0 , , (1 , } , in which case ρ (∆ { i,j } ) is degenerate, or { j, k } ⊆ { , , } , in whichcase η (∆ { j,k } ) is degenerate. It then follows in particular ( α, β ) = ( η, ρ )(∆ { , , } )is contained in the saturated closure of T + . Finally, if we assume that it is α | ∆ { , } that is invertible instead of β | ∆ { , } , then the argument can be carried out in asymmetric manner except that we need to use the opposite of the map (4). (cid:3) Corollary 2.13.
Let C be an ∞ -bicategory and K a Kan complex. Then the maps C ⊗ K ♯ −→ C × K ♯ and K ♯ ⊗ C −→ K ♯ × C are scaled anodyne. The Gray product as a left Quillen bifunctor.
In this section we willprove the main result of the present paper:
Theorem 2.14.
The Gray tensor product is a left Quillen bifunctor − ⊗ − : S et sc∆ × S et sc∆ −→ S et sc∆ with respect to the bicategorical model structure. Combined with Proposition 2.2 this implies that ( S et sc∆ , ⊗ ) is a monoidal modelcategory. Passing to underlying ∞ -categories we conclude: Corollary 2.15.
The Gray product endows the ∞ -category C at ( ∞ , with a monoidalstructure which is compatible with colimits in each variable. We will give the proof of Theorem 2.14 below. The following pushout-productproperty constitutes the principal component of the proof:
Proposition 2.16.
Let f : X → Y be a monomorphism of scaled simplicial setsand g : Z → W be a scaled anodyne map. Then the pushout-products f ˆ ⊗ g : (cid:0) X ⊗ W (cid:1) a X ⊗ Z (cid:0) Y ⊗ Z (cid:1) −→ Y ⊗ W and g ˆ ⊗ f : (cid:0) W ⊗ X (cid:1) a Z ⊗ X (cid:0) Z ⊗ Y (cid:1) −→ W ⊗ Y are scaled anodyne maps of scaled simplicial sets.Proof. We adapt the argument of [9, Proposition 3.1.8] to the context of Grayproducts. We can assume that f is either the inclusion ∂ ∆ n♭ ֒ → ∆ n♭ for n ≥ ♭ ⊆ ∆ ♯ , and that g is one of the generating scaledanodyne maps appearing in Definition 1.10. The case where f is the inclusion ∂ ∆ ֒ → ∆ is trivial since in this case both f ˆ ⊗ g and g ˆ ⊗ f are isomorphic to g , seeRemark 2.3. If f is the inclusion ∆ ♭ ⊆ ∆ ♯ then, since all generating anodyne mapsin Definition 1.10 are bijective on vertices, the map f ˆ ⊗ g becomes an isomorphismif we replace the collection of thin triangles in the Gray product by its minimalistvariant T − as in § f ˆ ⊗ g is scaled anodyne can then bededuced from Proposition 2.8 (and the same argument works for g ˆ ⊗ f ). We mayhence assume that f is the map ∂ ∆ n♭ ֒ → ∆ n♭ for n ≥
1. We now need to addressthe three different possibilities for g appearing in Definition 1.10.(A) Suppose that g is the inclusion (Λ mi , T ′ ) ֒ → (∆ m , T ) for 0 < i < m , where T denotes the union of all degenerate edges and { ∆ { i − ,i,i +1 } } and T ′ = T | (Λ mi ) .We argue the case of g ˆ ⊗ f . The proof for f ˆ ⊗ g proceeds in a similar manner.Let ( Z , M ) = [(∆ m , T ) ⊗ ∂ ∆ n♭ ] a (Λ mi ,T ′ ) ⊗ ∂ ∆ n♭ [(Λ mi , T ′ ) ⊗ ∆ n♭ ]We will extend ( Z , M ) to a filtration of (∆ m , T ) ⊗ ∆ n♭ as follows. Let S denotethe collection of all simplices σ : ∆ k σ → ∆ m × ∆ n with the following properties: RAY TENSOR PRODUCTS AND LAX FUNCTORS OF ( ∞ , (i) the simplex σ is non-degenerate and induces surjections ∆ k σ → ∆ n and∆ k σ → ∆ m along the projections;(ii) there exist integers 0 < p σ < k σ and 0 < j σ ≤ m (necessarily unique) suchthat σ ( p σ −
1) = ( i − , j σ ) and σ ( p σ ) = ( i, j σ ).We make the following observation, which is useful to keep in mind during thearguments below: if τ : ∆ k → ∆ m × ∆ n is an arbitrary simplex, then τ belongsto Z unless its projection to ∆ n is surjective and its projection to ∆ m containsthe face opposite i . In the latter case, if the image of τ in ∆ m is exactly theface opposite i then there is a unique ( k + 1)-simplex σ ∈ S of which τ is a face:indeed, if 0 ≤ p ≤ k + 1 is the maximal number such that τ ( p −
1) = ( i − , j )for some j ∈ [ m ], then the only ( k + 1)-simplex in S which has τ as a face mustsend p to ( i, j ) and have σ as its face opposite p . Otherwise, if the projectionof τ to ∆ m is surjective and p is defined in the same manner then either τ itselfbelongs to S or τ is the face of exactly two simplices σ, σ ′ which belong to S , onewith σ ( p ) = ( i, j ) so that p σ = p and j σ = j and one with σ ′ ( p ) = ( i − , j + 1)so that p σ ′ = p + 1 and j σ = j + 1. In particular, ∆ m × ∆ n is obtained from Z by adding all the simplices in S . To proceed, we will need to identify theright order in which to add them.Choose an ordering σ < ... < σ ℓ on S such that a < b whenever dim( σ a ) < dim( σ b ) or dim( σ a ) = dim( σ b ) and j σ a < j σ b . We then abbreviate k a := k σ a , p a = p σ a and j a := j σ a . We now observe that for a = 1 , ..., ℓ we have p a < k a , otherwise the projection of σ a to the ∆ m coordinate will not besurjective (since it sends p a to i < n ). We then denote by T a ⊆ ∆ k a theunion of all degenerate 2-simplices and the 2-simplex ∆ { p a − ,p a ,p a +1 } , and set T ′ a = T a ∩ (cid:0) Λ k a p a (cid:1) . We now claim that each σ a ∈ S maps T a into the set of thintriangles of (∆ m , T ) ⊗ ∆ n♭ . Indeed, it suffice to observe that σ a sends the 2-simplex ∆ { p a − ,p a ,p a +1 } to either a degenerate simplex in ∆ m or to ∆ { i − ,i,i +1 } ,and on the other hand sends the same triangle to a 2-simplex of ∆ n♭ whichdegenerates along ∆ { p a ,p a +1 } . In particular, we may view each σ a as a map ofscaled simplicial sets σ a : (∆ k a , T a ) → (∆ m , T ) ⊗ ∆ n♭ . Now for a = 1 , ..., ℓ let Z a ⊆ ∆ m × ∆ n to be the union of Z and the images ofthe simplices σ a ′ for a ′ ≤ a , and let M a be the union of the images σ a ′ ( T a ′ ) for a ′ ≤ a . We claim that for a = 1 , ..., ℓ we have a pushout square of the form (cid:0) Λ k a p a , T ′ a (cid:1) / / (cid:15) (cid:15) ( Z a − , M a − ) (cid:15) (cid:15) (cid:0) ∆ k a , T a (cid:1) σ a / / ( Z a , M a )To prove this, it will suffice to show that all the faces of σ a are contained in Z a − . Indeed, for k ′ = p a − , p a the restriction of σ a to the face opposite k ′ is either contained in Z or is a ( k a − S . In the latter case itwill correspond to an index a ′ < a and will be contained in Z a ′ ⊆ Z a − . Nowsuppose that k ′ = p a −
1. In this case either σ a sends the face opposite k ′ to Z or p a ≥ j ′ < j a such that σ a ( p a −
2) = ( i − , j ′ ). In thelatter case the face opposite k ′ is also the face of a simplex σ a ′ : ∆ k a ′ → ∆ m × ∆ n in S with k a ′ = k a , p a ′ = p a − j a ′ = j ′ < j a . Then a ′ < a and this facewill again belong Z a − . Finally, if k ′ = p a then we claim that the image under σ a of the face opposite l will not belong to Z a − . To see this, we note that σ a ( p a + 1) must be either ( i, j + 1) , ( i + 1 , j ) or ( i + 1 , j + 1). We then seethat in all three cases the the restriction of σ a to the face opposite p a does not belong to Z and does not belong to S . In addition, by the observationfollowing the definition of S above, when σ a ( p a + 1) = ( i + 1 , j ) , ( i + 1 , j + 1) the( k a − S , namely, σ a ,and when σ a ( p a + 1) = ( i, j + 1) it is the face of exactly two simplices in S , oneof which is σ a and the other is σ a ′ for which j a ′ = j + 1 and so a ′ > a . We maythus conclude that the the restriction of σ a to the face opposite p a does notbelong to Z a . We may hence conclude that the inclusion ( Z , M ) ֒ → ( Z ℓ , M ℓ )is scaled anodyne.Now when m ≥ m , T ) is contained in (Λ mi , T ),and when n ≥ n is contained in ∂ ∆ n . In either ofthese cases we have that every thin triangle in (∆ m , T ) ⊗ ∆ n♭ is contained in Z and so ( Z ℓ , M ℓ ) = (∆ m , T ) ⊗ ∆ n♭ as scaled simplicial sets and the proof iscomplete. In the special case n = 1 and m = 2 we have S = { σ < τ < τ < τ } , where σ is the triangle of ∆ × ∆ with vertices (cid:0) (0 , , (1 , , (2 , (cid:1) and τ j , j = 0 , , τ j ( l ) = ( (0 , l ) l ≤ j, (1 , l − l > j. We then see that M ℓ contains all the triangles which are thin in ∆ ♯ × ∆ ♭ exceptthe one with vertices (0 , , (2 , , (2 , , { ∆ { , , } , ∆ { , , } , ∆ { , , } } ) / / (cid:15) (cid:15) ( Z ℓ , M ℓ ) (cid:15) (cid:15) ∆ ♯ / / ∆ ♯ ⊗ ∆ where the map ∆ ♯ → ∆ ♯ ⊗ ∆ is the one determined by the chain of vertices(0 , , (1 , , (2 , , (2 , Z ℓ , M ℓ ) → ∆ ♯ ⊗ ∆ is scaled anodyne.(B) The case where g is the inclusion (cid:0) ∆ , T (cid:1) ֒ → (cid:0) ∆ , T ∪ { ∆ { , , } , ∆ { , , } } (cid:1) where T is the set of triangles specified in Definition 1.10(B). If n ≥ f ˆ ⊗ g and g ˆ ⊗ f are isomorphisms of scaled simplicial sets and so we mayassume that n = 1.Let us denote the domain of f ˆ ⊗ g by (∆ × ∆ , T ′ ) and the domain of g ˆ ⊗ f by (∆ × ∆ , T ′′ ). Let p : ∆ → ∆ be the unique map which sends 0 to 0 and1 , , q : ∆ → ∆ be the unique map which sends 0 , , f ˆ ⊗ g and g ˆ ⊗ f are scaled anodyne we then observe thatthere are pushout diagrams of the form (cid:0) ∆ , T (cid:1) (∆ × ∆ , T ′ ) (cid:0) ∆ , T ∪ { ∆ { , , } , ∆ { , , } } (cid:1) ∆ ⊗ (cid:0) ∆ , T ∪ { ∆ { , , } , ∆ { , , } } (cid:1) p × Id f ˆ ⊗ g RAY TENSOR PRODUCTS AND LAX FUNCTORS OF ( ∞ , and (cid:0) ∆ , T (cid:1) (∆ × ∆ , T ′′ ) (cid:0) ∆ , T ∪ { ∆ { , , } , ∆ { , , } } (cid:1) (cid:0) ∆ , T ∪ { ∆ { , , } , ∆ { , , } } (cid:1) ⊗ ∆ × q g ˆ ⊗ f . (C) The case where g is in the inclusion (cid:0) Λ m ` ∆ { , } ∆ , T (cid:1) ֒ → (cid:0) ∆ m ` ∆ { , } ∆ , T (cid:1) for m ≥
3, where T denotes the union of all degenerate edges together with thetriangle ∆ { , ,n } . We prove the case of g ˆ ⊗ f . The proof for f ˆ ⊗ g proceeds in asimilar manner.We argue as in the proof of case (A). Let S denote the collection of allsimplices σ : ∆ k σ → ∆ m × ∆ n with the following properties:(i) the simplex σ is non-degenerate, and induces surjections ∆ k σ → ∆ m and∆ k σ → ∆ n ;(ii) there exist integers 0 ≤ p σ < k σ and 0 < j σ ≤ m (necessarily unique) suchthat σ ( p σ ) = (0 , j σ ) and σ ( p σ + 1) = (1 , j σ ).We may then choose an ordering σ < ... < σ q on S such that a < b wheneverdim( σ a ) < dim( σ b ) or dim( σ a ) = dim( σ b ) and j σ a > j σ b . We then abbreviate k a := k σ a , p a = p σ a and j a := j σ a . For every index a , let T a ⊆ ∆ k a denotethe collection of all 2-simplices which are either degenerate, have the form∆ { p a − ,p a ,p a +1 } if p a >
0, or have the form ∆ { , ,m } if p a = 0. Let T ′ a ⊆ T a be the subset of those 2-simplices in T a which lie in Λ k a p a . We claim that each σ a ∈ S maps T a into the set of thin triangles in (∆ m ` ∆ { , } ∆ , T ) ⊗ ∆ n♭ . Tosee this it suffice to observe that σ a always sends the 1-simplex ∆ { p a ,p a +1 } toa degenerate 1-simplex in both ∆ m ` ∆ { , } ∆ and ∆ n .Now define a sequence of scaled simplicial sets( Z , M ) ⊆ ( Z , M ) ⊆ · · · ⊆ ( Z ℓ , M ℓ ) ⊆ (cid:18) ∆ m a ∆ { , } ∆ , T (cid:19) ⊗ ∆ n♭ where ( Z , M ℓ ) is the domain of g ˆ ⊗ f and for every a = 1 , ..., ℓ we let Z a be theunion of Z with the images of σ a ′ for a ′ ≤ a and M a the union of M with theimages σ a ′ ( T a ′ ) for all a ′ ≤ a . Arguing as in the case (A) we now observe that( Z ℓ , M ℓ ) = (cid:18) ∆ m ` ∆ { , } ∆ , T (cid:19) ⊗ ∆ n♭ (note that m ≥
3) and for a = 1 , ..., ℓ we have a pushout diagram (cid:0) Λ k a p a , T ′ a (cid:1) / / (cid:15) (cid:15) ( Z a − , M a − ) (cid:15) (cid:15) (cid:0) ∆ k a , T a (cid:1) σ a / / ( Z a , M a )where in the case that p a = 0 the simplex σ a sends ∆ { , } to a degenerate edgeof Z a − . We may then conclude that each ( Z a − , M a − ) → ( Z a , M a ) is scaledanodyne and so the desired result follows. (cid:3) Corollary 2.17.
Let ( X, T X ) be a scaled simplicial set and K a Kan complex.Then the maps (5) ( X, T X ) ⊗ K ♯ −→ ( X, T X ) × K ♯ and (6) K ♯ ⊗ ( X, T X ) −→ K ♯ × ( X, T X ) are trivial cofibrations.Proof. We prove that (5) is a bicategorical equivalence, the proof for (6) proceedsin a similar manner. Let C be an ∞ -bicategory equipped with a scaled anodynemap ( X, T X ) ֒ → C . We then obtain a commutative square( X, T X ) ⊗ K ♯ / / (cid:15) (cid:15) ( X, T X ) × K ♯ (cid:15) (cid:15) C ⊗ K ♯ / / C × K ♯ in which the vertival maps are bicategorical equivalences by Proposition 2.16 and [9,Proposition 3.1.8] and the bottom horizontal map is a bicategorical equivalenceby Corollary 2.17. It then follows that the top horizontal map is a bicategoricalequivalence as well. (cid:3) Proof of Theorem 2.14.
Thanks to Proposition 2.16 and the description of the bi-categorical model structure provided in [3], we are left with proving that the maps( ∂ ∆ n ֒ → ∆ n ) ˆ ⊗ ( { ǫ } → J ♯ ) for n ≥ ǫ = 0 , ֒ → ∆ ♯ ) ˆ ⊗ ( { ǫ } → J ♯ )are weak bicategorical equivalences. But by Corollary 2.17 these maps are equiva-lent to the corresponding ones having ˆ × in place of the Gray tensor product, sinceboth ∆ and J ♯ are maximally marked Kan complexes. Therefore, the result followsfrom the the fact that the bicategorical model structure on scaled simplicial sets iscartesian closed, as previously observed. (cid:3) Corollary 2.18.
Let C be an ∞ -bicategory and K a scaled simplicial set. Then Fun opgr ( K, C ) and Fun gr ( K, C ) are ∞ -bicategories.Remark . Given a pair of vertices x, y in an ∞ -bicategory C , the fiber of theprojection Fun gr (∆ , C ) → C × C induced by ∂ ∆ → ∆ over ( x, y ) is naturallyisomorphic to the underlying simplicial set of Hom C ( x, y ) (as defined in Notation1.21). Remark . The Gray tensor product and the cartesian product do not “asso-ciate”, that is K × ( X ⊗ Y ) = ( K × X ) ⊗ Y. for general scaled simplicial sets K, X, Y . For this reason we also haveMap( X ⊗ Y, Z ) = Map( X, Fun opgr ( Y, Z ))in general (and similarly for Fun gr ( − , − )). However, by Corollary 2.17 and Propo-sition 2.16 we have that if K is a Kan complex then the maps K ♯ × ( X ⊗ Y ) ←− K ♯ ⊗ X ⊗ Y −→ ( K ♯ × X ) ⊗ Y, are scaled anodyne (and isomorphisms on the level of the underlying simplicialsets). This means that for every ∞ -bicategory C we have an isomorphism of Kancomplexes Fun ≃ ( X ⊗ Y, C ) ∼ = Fun ≃ ( X, Fun opgr ( Y, C )) . RAY TENSOR PRODUCTS AND LAX FUNCTORS OF ( ∞ , Oplax functors and the universal property of Gray products
The goal of this section is to characterize the Gray tensor product defined in § ∞ -category of ( ∞ , ∞ -bicategories. Wewill then show that for two ∞ -bicategories C and D , maps from the Gray product C ⊗ D to any other ∞ -bicategory E can be identified as a suitable subspace of thespace of oplax functor C × D → E . This opens the door to comparing the Grayproduct defined here with that of [5] by comparing the two notions of oplax maps.3.1. Normalised oplax functors of 2-categories.
Before we begin, let us recallthe classical notion of normalised oplax -functor for 2-categories. We shall denotethe horizontal composition of 1-cells and 2-cells by ∗ and the vertical compositionof 2-cells by ∗ .Let A and B be two 2-categories. A normalised oplax -functor F : A → B isgiven by:- a map Ob( A ) → Ob( B ) that to any object x of A associates an object F ( a ) of B ;- a map Cell ( A ) → Cell ( B ) that to any 1-cell f : x → y of A associates a 1-cell F ( f ) : F ( x ) → F ( y ) of B ;- a map Cell ( A ) → Cell ( B ) that to any 2-cell α : f → g of A associates a 2-cell F ( α ) : F ( f ) → F ( g ) of B ;- a map that to any composable 1-cells x f −→ y g −→ z of A associates a 2-cell F ( g, f ) : F ( g ∗ f ) → F ( g ) ∗ F ( f )of B .These data are subject to the following coherences: normalisation: for any object x of A (resp. any 1-cell f of A ) we have F (1 x ) =1 F ( x ) (resp. F (1 f ) = 1 F ( f ) ); moreover for any 1-cell f : x → y of A we have F (1 y , f ) = 1 F ( f ) = F ( f, x ) ; cocycle: for any triple x f −→ y g −→ z h −→ t of composable 1-cells of A we have (cid:0) F ( h ) ∗ F ( g, f ) (cid:1) ∗ F ( h, g ∗ f ) = (cid:0) F ( h, g ) ∗ F ( f ) (cid:1) ∗ F ( h ∗ g, f ) ; vertical compatibility: for any pair a a ′ fghαβ of 1-composable 2-cells α and β of A , we have F ( β ∗ α ) = F ( β ) ∗ F ( α ); horizontal compatibility: for any pair • • • ff ′ α gg ′ β of 0-composable 2-cells α and β of A , we have F ( g ′ , f ′ ) ∗ F ( β ∗ α ) = (cid:0) F ( β ) ∗ F ( α ) (cid:1) ∗ F ( g, f ) . Remark . With the notations of the previous paragraph, consider a diagram x yy g fα ≃ in A where the 2-cell α is invertible, that is there is a 2-cell β : g → f such that β ∗ α = 1 f and α ∗ β = 1 g . The conditions of normalisation and the verticalcompatibility then imply that the normalised oplax 2-functor F maps the abovediagram to a diagram F x F yy
F g F fF α in B , where F α is invertible. Said otherwise, the normalised oplax functor F mapsa 2-simplex of the Duskin nerve of A of the form above, to a 2-simplex of the Duskinnerve of B with the same properties. Remark . Normalised oplax functors of 2-categories are not invariant underbiequivalence of 2-categories. For example, suppose that C is a 2-category whosemapping categories are all singletons (2-categories with this property are called codiscrete ), and D is a 2-category with a single object ∗ whose endomorphismcategory is a monoidal category V := Map D ( ∗ , ∗ ). Then it is a well-known thatnormalised oplax functors C D correspond to V op -enriched categories havingOb( C ) as class of objects, and such that the identity maps I V → Map( x, x ) areall isomorphisms, where I V denotes the monoidal unit of V . Let us refer to suchenriched categories as normalized categories. Since the 2-category C is codiscreteit is biequivalent to a point. Nonetheless, every normalized V op -enriched categorywith one object is necessarily trivial (its unique mapping object is I V ) while thisis not true in general if C has more than one object (take for example C to bethe codiscete 2-category on two objects, V = Set op , and the normalized categorycorresponding to the discrete category on two objects). This phenomenon can beconsidered as a consequence of the fact that, in general, normalised oplax 2-functorsfail to preserve invertible 1-morphisms.Given two normalised oplax 2-functors F : A → B and G : B → C , there is anobvious candidate for the composition GF : A → C and one checks that this is stilla normalised oplax 2-functor; furthermore, the identity functor on a category isclearly an identity element for normalised oplax 2-functor too. Hence, there is acategory ^ C at with small 2-categories as objects and normalised oplax 2-functorsas morphisms.There is also a standard cosimplicial object ∆ → ^ C at inducing a nerve functor f N : ^ C at → S et ∆ . For any n ≥
0, the normalised oplax 2-functors [ n ] → A correspond precisely to 2-functors O n → A (see, for instance, [11, Tag 00BE]),where by O n we denote the 2-truncated Street n -th oriental (see [14]).Hence, we get a triangle diagram of functors S et ∆ C at ^ C at N f N , where N is the Duskin nerve, which is commutative (up to a canonical isomor-phism). RAY TENSOR PRODUCTS AND LAX FUNCTORS OF ( ∞ , Remark . It is a standard fact that the functor f N : ^ C at → S et ∆ is fully faithful(see, for instance [2], [8] or [11, Tag 00AU]). With this at hand, one sees that thepreservation of triangles of the kind described in Remark 3.1 can be realized asan immediate consequence of the simplicial interpretation of normalised oplax 2-functors.3.2. Oplax functors of ∞ -bicategories. The notion of a normalize oplax functorwas generalized to the ( ∞ , ∞ -categoriesto model ( ∞ , ∞ , C is encoded via acocartesian fibration C H → ∆ op whose classifing functor ∆ op → C at ∞ is a completeSegal object in C at ∞ in the sense of Definition 1.2.7 and 1.2.10 of [9]. In this modela functor of ( ∞ , C H D H ∆ op φ over ∆ op which preserves cocartesian edges. They then define the notion of anoplax functor of ( ∞ , ρ : [ m ] → [ n ] of ∆ op idle if the imageof ρ is a segment { i ∈ [ n ] | a ≤ i ≤ b } for some a ≤ b in [ n ]. The authors of [5] thendefine the notion of an oplax functor of ( ∞ , op asin (7) which is only assumed to preserve cocartesian edges lying over idle maps.In the present section we offer a definition of oplax functors in the setting ofscaled simplicial sets. We expect the two definitions to be equivalent, proof ofwhich will be the topic of future work. The definition of oplax functors given belowwill serve us in § Definition 3.4.
Let C = ( C , T C ) and E = ( E , T E ) be two ∞ -bicategories. We willdenote by L C ⊆ T C the collection of those thin triangles σ ∈ T C such that either σ | ∆ { , } or σ | ∆ { , } is invertible in C . By an oplax functor from C to E we will meana map of scaled simplicial sets ϕ : ( C , L C ) → E . The collection of oplax functorscan be organized into an ∞ -bicategory Fun oplax ( C , E ) := Fun(( C , L C ) , E ) using theinternal mapping objects of S et sc∆ , see § C → D of ∞ -bicategories sends invertible edges to invertibleedges and hence maps L C into L D . We then have the following: Lemma 3.5. If ϕ : C → D is an bicategorical equivalence of ∞ -bicategories thenthe induced map ( C , L C ) → ( D , L D ) is a bicategorical equivalence. In particular, inthis case for every ∞ -bicategory the restriction functor Fun oplax ( D , E ) → Fun oplax ( C , E ) is an equivalence of ∞ -bicategories.Proof. By Lemma 1.24 any bicategorical equivalence of ∞ -bicategories admits a ho-motopy inverse ϕ ′ : D → C such that ϕ ◦ ϕ ′ and ϕ ′ ◦ ϕ are equivalent to the identitiesin the ∞ -groupoids Fun ≃ ( C , C ) and Fun ≃ ( D , D ) respectively. These equivalences areencoded by maps of scaled simplicial sets η : ∆ ♭ × C → C and η ′ : ∆ ♭ × D → D , in which η (∆ ♭ × { c } ) (resp. η ′ (∆ ♭ × { d } )) is invertible in C (resp. D ) for every vertex c in C (resp. every vertex d in D ).As mentioned above, since ϕ and ϕ ′ preserve invertible edges they preserve theoplax scaling L C and L D on both sides. We now claim that the homotopies η, η ′ also preserve the oplax scaling in the sense that they extend to maps of scaledsimplicial sets(8) ∆ ♭ × ( C , L C ) → ( C , L C ) and ∆ ♭ × ( D , L D ) → ( D , L D ) . We prove the claim for the map on the left, the argument for that on the rightproceeds in a similar manner. Observe that by the definition of L C it will sufficeto show that η sends every arrow in ∆ × C whose C -component is invertible to aninvertible arrow in C . Indeed, let f : x → y be an invertible arrow encoded by amap e : ∆ → C . Consider the composite σ = (Id , e ) : ∆ ♭ × ∆ ♭ → ∆ ♭ × C → C . We note that since all triangles in ∆ ♭ are degenerate it follows that all triangles in∆ ♭ × ∆ ♭ are thin. In particular σ determines a commutative square in C th of theform x f / / (cid:15) (cid:15) y (cid:15) (cid:15) x ′ / / y ′ in which the top horizontal arrow and both vertical arrows are invertible. By the2-out-of-3 property for invertible arrows we deduce that all arrows in this square(including the diagonal arrow x → y ′ ) are invertible in C . We may then concludethat η sends every arrow in ∆ ♭ × C whose C -component is invertible to an invertibleedge in C , and in particular restrict to maps of scaled simplicial sets (8).To finish the proof, we now note that every invertible arrow in C is also invertiblewhen considered in the non-fibrant scaled simplicial set ( C , L C ) (in the sense ofDefinition 1.20), since the triangles exhibiting their inverses are included in L C by definition. In particular, these edges are sent to invertible edges by any mapof scaled simplicial sets ( C , L C ) → E . We may thus conclude that for every ∞ -bicategory E the inverse functor ϕ ′ : D → C and the homotopies η and η ′ determinea homotopy inverse for the restriction functorFun(( D , L D ) , E ) → Fun(( C , L C ) , E )which is consequently an equivalence of ∞ -bicategories. (cid:3) Remark . If we restrict to scaled simplicial sets which are the Duskin nervesof 2-categories then we recover those normalised oplax 2-functors that preserveequivalences (rather than just identities), which is what one might expect in lightof the homotopy soundness established in Lemma 3.5. On the other hand, thefully-faithfulness of the Duskin nerve (see Remark 3.3) might suggests that, for ∞ -bicategories C = ( C , T C ) and D = ( D , T D ), the direct analogue of the notion ofa normalised oplax functor of 2-categories should simply be maps C → D betweenthe underlying simplicial sets. In fact, these automatically send thin triangles in C with one external legs degenerate to thin triangles in D (cf. Remark 3.1), but notnecessarily all thin triangles with one external leg invertible. However, this notionof an oplax functor C D is not homotopically sound, since the operation C C does not send equivalences of ∞ -bicategories to bicategorical equivalences of scaledsimplicial sets. For example, if J is the nerve of the walking isomorphism with two RAY TENSOR PRODUCTS AND LAX FUNCTORS OF ( ∞ , objects then J ♯ → ∆ is an equivalence of ∞ -bicategories but J ♭ → ∆ is not abicategorical equivalence.3.3. The universal property of the Gray product.
In this section we willendow the Gray tensor product of § universal mapping property defined interms of the ∞ -bicategory of lax functors above. To formulate it, let C = ( C , T C )and D = ( D , T D ) be two ∞ -bicategories. Let T ≃ ⊆ T C × T D be the subset consistingof those ( α, β ) such that either α | ∆ { , } is invertible in C or β | ∆ { , } is invertible in D . By Corollary 2.12 the inclusion C ⊗ D ֒ → ( C × D , T ≃ )is a bicategorical equivalence. Since this map is also an isomorphism on the level ofthe underlying simplicial sets we get that for every ∞ -bicategory E the associatedrestriction map gives an isomorphism of (fibrant) scaled simplicial setsFun(( C × D , T ≃ ) , E ) ∼ = Fun( C ⊗ D , E ) . On the other hand, the collection of triangles T ≃ also contains the set of triangles L C × D ⊆ T C × T D of Definition 3.4. Given an ∞ -bicategory E we then get arestriction(9) Fun( C ⊗ D , E ) ∼ = Fun(( C × D , T ≃ ) , E ) → Fun oplax ( C × D , E ) . The universal mapping property of the Gray product can now be formulated asfollows:
Theorem 3.7.
For an ∞ -bicategory E the restriction functor (9) is fully-faithfuland its essential image consists of those oplax functors ϕ : C × D E which satisfythe following conditions:(i) for all objects x ∈ C , y ∈ D , the restrictions of ϕ to { x } × D and C × { y } aremaps of scaled simplicial sets.(ii) for all arrows f : x → x ′ in C and g : y → y ′ in D , the 2-simplex in C × D depicted by (cid:0) x, y (cid:1) (cid:0) x ′ , y (cid:1)(cid:0) x ′ , y ′ (cid:1) ( f,y )( f,g ) ( x ′ ,g ) is mapped by f to a thin triangle in E .Proof. Since the map ( C × D , L C × D ) → (( C × D , T ≃ ) is an isomorphism on theunderlying simplicial sets it follows that the restriction functor (9) is an inclusionof simplicial sets whose image is completely determined by the image on the levelof vertices, and so as a functor between ∞ -bicategories it is indeed fully-faithful.Let L C × D ⊆ G C × D ⊆ T C × T D be the intermediate set of triangles consisting of L C × D as well as all those 2-simplices( α, β ) such that either α degenerates to a point, or β degenerates to a point, or α degenerates along ∆ { , } and β degenerates along ∆ { , } . Unwinding the definitionswe see that a map ( C × D , L C × D ) → E satisfies conditions (i) and (ii) above if andonly if it sends all the triangles in G C × D to thin triangles in E . To finish the proofit will hence suffice to show that every thin 2-simplex T ≃ is in the saturated closureof G C × D . ST Figure 1.
The 4-simplex P (magenta) and the diagonal (cyan) Pick a 2-simplex ( α, β ) ∈ T ≃ . Then either α | ∆ { , } is invertible in C or β ∆ { , } isinvertible in D . To fix ideas assume we are in the former case (the argument in thelatter case is entirely similar). Consider the 4-simplex B : ∆ → ∆ × ∆ spanned bythe vertices ((0 , , (1 , , (1 , , (1 , , (2 , P def = ( α × β ) ◦ B : ∆ → C × D as depicted in Figure 1.Here, P | ∆ { , , } , P | ∆ { , , } and P ∆ { , , } belong to G C × D . Using P | ∆ { , , , } wethen get that P | ∆ { , , } is in the saturated closure of G C × D . Considering now theface P | ∆ { , , , } , we see that since P | ∆ { , , } and P | ∆ { , , } belong to L C × D ⊆ G C × D it follows that P | ∆ { , , } is also in the saturated closure of G C × D , as desired. (cid:3) References
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