Enhanced bivariant homology theory attached to six functor formalism
aa r X i v : . [ m a t h . C T ] A ug Enhanced bivariant homology theory attached to six functorformalism
Tomoyuki Abe
Abstract
Bivariant theory is a unified framework for cohomology and Borel-Moore homology the-ories. In this paper, we extract an ∞ -enhanced bivariant homology theory from Gaitsgory-Rozenblyum’s six functor formalism. Introduction
Grothendieck’s 6-functor formalism is very powerful in cohomology theory. At the same time,if we want to axiomatize the formalism, it requires a long list of relations between these func-tors, and when we wish to establish a 6-functor formalism for some cohomology theory, we needtremendous amount of work to verify these axioms. After ideas of Lurie, Gaitsgory and Rozen-blyum constructed a very general machinery to construct an ∞ -enhanced 6-functor formalismsfrom minimal amount of data. To proceed, let us recall the category of correspondences usedby Gaitsgory and Rozenblyum. We fix a base scheme S . The ( ∞ , Corr propsep;all ( Corr for short) has the objects the same as the category of S -schemes Sch( S ) (or its subcategory). Amorphism F : X → Y is a diagram of the form Z Fg (cid:15) (cid:15) f / / XY, where g is separated. Given 1-morphisms F, G : X → Y , a 2-morphism F ⇒ G is a diagram Z F α ! ! ❈❈❈❈❈❈❈❈ ! ! Z G / / (cid:15) (cid:15) XY, where α is proper. Let Pres be the ( ∞ , ∞ -categories withcolimit commuting maps as morphisms and natural transforms as 2-morphisms. Gaitsgory andRozenblyum interpret a 6-functor formalism as a 2-functor D : Corr → Pres . In fact, fora correspondence F as above, we have D ( F ) : D ( X ) → D ( Y ). This encodes the data of thefunctor g ! f ∗ . The functoriality of g ! with respect to proper morphism is encoded in 2-morphisms.Since D ( F ) is a map in Pres , it admits a right adjoint, which encodes the data of g ! , f ∗ . Ifwe need ⊗ , H om , we need to consider the ∞ -category of commutative algebra objects in the ∞ -category of presentable ∞ -categories, but we do not go into that far in this introduction.1ven though the data of 6-functors is encoded very beautifully, it is not straightforward toextract concrete data. For example, if we wish to extract an ∞ -functor of cohomology theory H ∗ : Sch( S ) op → Sp , where Sp is the ∞ -category of spectra, this already does not seem to be straightforward from thedefinition. A goal of this paper is to “decode” the data from Gaitsgory-Rozenblyum’s 6-functorformalism so that we can handle more easily.Let us go into more precise statement. In general, when we are given a 6-functor formalism,we can attach 4 kinds of (co)homology theories: cohomology, Borel-Moore homology, homology,compact support cohomology. First two and the last two theories possess essentially the sameinformation via duality theory. Thus, we may focus on the first two theories. Given f : X → S in Sch( S ), cohomology and Borel-Moore homology of X can be defined byH ∗ ( X ) := Map D ( S ) ( S , f ∗ f ∗ S ) , H BM ∗ ( X ) := Map D ( S ) ( S , f ∗ f ! S ) . The cohomology theory is contravariant with respect to any morphism, and Borel-Moore ho-mology is covariant with respect to proper morphism. Seemingly completely different theories,Fulton and MacPherson [FM] unified these two theories into so called the bivariant homologytheory . Let g : X → Y be a morphism in Sch( S ). Then we defineH( g ) := Map D ( Y ) ( Y , g ∗ g ! Y ) ≃ Map D ( Y ) ( g ! X , Y ) . By definition, we have H ∗ ( X ) ≃ H(id X ), H BM ∗ ( X ) ≃ H( X → S ). The main result of this papergives an ∞ -enhancement of the bivariant homology. In order to make this precise, we considerthe category of arrows f Ar. Namely, the objects consist of S -morphisms X → Y . For morphisms,we do not use the evident one: a morphism from f ′ : X ′ → Y ′ to f : X → Y consists of a diagramof the following form: X ′ (cid:15) (cid:15) X × Y Y ′ (cid:15) (cid:15) (cid:3) / / α o o X (cid:15) (cid:15) Y ′ Y ′ / / Y where α is proper. We may check by hand that we have a morphism H( f ) → H( f ′ ). It is evennot too hard to check that bivariant homology theory is a (ordinary) functor H : f Ar op → hSp.Our main result gives an ∞ -enhancement of this functor. A simplified version can be writtenas follows (cf. Theorem 5.3 for the detail): Theorem. —
Given a 6-functor formalism
Corr → Pres , there exists an ∞ -functor H : f Ar op → Sp so that h H ≃ H as a functor f Ar → hSp . One of the obstacles of constructing such a functor is that the functoriality of H ∗ ( X ) comesfrom 1-morphism of Corr , whereas that of H BM ∗ ( X ) comes from 2-morphism of Corr . In orderto combine these two morphisms into one functor as in the theorem, we need to “integrate”these two types morphisms.Our main motivation of the theorem is to construct such a functor for theory of motives.We plan to use the functor above to construct certain elements in Chow groups which appearin ramification theory of ℓ -adic sheaves. Since we need “gluing” of elements in Chow groups, ∞ -enhancement is crucial.Before concluding the introduction, let us see the organization of this paper. Throughoutthis paper, we use the language of ∞ -categories freely. In §
1, we collect some preliminaries on2 -categories. Most of the material in this section should be more or less well-known to experts,but we write here since we could not find references. In §
2, we establish some duality typetheorem. Via straightening/unstraightening construction of Lurie, Cartesian and coCartesianfibrations correspond to each other, and contain essentially the same information, as long as weare considering morphisms which preserve (co)Cartesian edges. However, it is fairly inexplicit innature if we pass through straightening/unstraightening construction. We construct an explicitmodel for such correspondence. This construction naturally appears in §
4. In §
3, we define( ∞ , R -linear ∞ -categories. We heavily use the language of (generalized) ∞ -operads. The construction has already been appeared in [GR]. The main construction iscarried out in § §
5. In § Corr B Sp ⊛ , where B Sp ⊛ isthe ( ∞ , f : X → Y to H( f ), and encodes the complete data of bivariant homology theory. However, togo from this ( ∞ , §
5. Finally, in §
6, we collect some examples of 6-functor formalisms in the senseof Gaitsgory-Rozenblyum. Most of the part of this section has already been appeared elsewhere,but some of the sources are not published and not even available in arXiv, we included this forthe sake of completeness.
Conventions and notations
When we say ∞ -categories, it always mean quasi-categories, in particular, ( ∞ , not abbreviate ∞ -category as category. In principle, we follow the conventions of Luriein [HTT], [HA]. Exceptions are that we call ∞ -operad what Lurie calls planar ∞ -operads, andthat we denote by S pc the ∞ -category of spaces.We denote by ∆ the simplex category, whose objects will be denoted by [ n ] for n ∈ N asusual. A morphism [ n ] → [ m ] corresponds to a function. We denote by σ i : [0] → [ n ] the mapsending 0 to i ∈ [ n ]. We denote by ρ i : [1] → [ n ] for 0 < i ≤ n the map sending 0 to i − i . Both of these are inert maps. We also denote by d i : [ n − → [ n ] increasing map whichavoids i ∈ [ n ].In principle, we use calligraphic fonts ( e.g. C ) for ∞ -categories, and bold fonts ( e.g. C ) for( ∞ , X → S and a vertex s ∈ S , we denote by X s thefiber product X × S,s ∆ . An equivalence of ( ∞ -)categories is denoted by ≃ , and an isomorphismof simplicial sets is denoted by ∼ =. For an ∞ -category C , the space of morphisms is denoted byMap C ( − , − ).We denote by ( − ) × cat( − ) ( − ) for a product in C at ∞ in order to clarify the difference betweenthe fiber product as simplicial sets. If f : D → C be a categorical fibration of ∞ -categories and g : E → C be a functor of ∞ -categories, then the functor D × C E → D × cat C E is a categoricalequivalence. Acknowledgment
The author would like to thank Deepam Patel for continuous discussions and encouragements.Without him, this work would not have appeared. He also thanks Andrew MacPherson, RuneHaugseng, Adeel Khan for some discussions. This work is supported by JSPS KAKENHI GrantNumbers 16H05993, 18H03667, 20H01790.
1. Some preliminaries on ∞ -categories We will fix some notations, and recall some constructions in ∞ -category theory. The expositionsare informal when there are references. 3 .1. Let f : S → T be a map of simplicial sets. Then we have the base change functor f ∗ : ( S et ∆ ) /T → ( S et ∆ ) /S . As in [HTT, 4.1.2.7], f ∗ admits a right adjoint f ∗ . More explic-itly, for X → S , f ∗ X → T is the simplicial set having the following universality: for anysimplicial set K over T , we have the following isomorphism as simplicial sets:Fun T ( K, f ∗ X ) ∼ = Fun S ( K × T S, X ) . Assume we are given a functor of ∞ -categories F : C → C at ∞ . Since C at ∞ ≃ N( C at ∆ ∞ )(where C at ∆ ∞ is the simplicial category of ∞ -categories in [HTT, 3.0.0.1]), we have the simplicialfunctor C [ F ] : C [ C ] → C at ∆ ∞ . Now, we have the simplicial functor MAP : ( C at ∆ ∞ ) op × C at ∆ ∞ → C at ∆ ∞ sending ( E , E ′ ) to Fun( E , E ′ ). Fix D ∈ C at ∆ ∞ . Then we have the functorFun( F, D ) ∆ : C [ C ] op C [ F ] op ×{ D } −−−−−−−→ ( C at ∆ ∞ ) op × C at ∆ ∞ MAP −−−→ C at ∆ ∞ . Taking the adjoint, we get the functor Fun( F, D ) : C op → C at ∞ .The unstraightening of this functor has an alternative description. Let f : X → C op bea Cartesian fibration. By [HTT, 3.2.2.12], f ∗ ( D × X ) → C op is a coCartesian fibration. ThiscoCartesian fibration is denoted by Φ co ( f, D ). By (dual version of) [GHN, 7.3], Φ co (Un C op ( F ) , D )is equivalent to the unstraightening of Fun( F, D ). Dually, given a coCartesian fibration g : Y → C ,we define Φ Cart ( g, D ) := g ∗ ( D × Y ), which is a Cartesian fibration over C . Let Γ be the category whose objects are the pairs ([ n ] , i ) where i ∈ [ n ]. A morphism([ n ] , i ) → ([ n ′ ] , i ′ ) consists of a map α : [ n ′ ] → [ n ] such that i ≤ α ( i ′ ). We have the evidentfunctor γ : Γ → ∆ op sending ([ n ] , i ) to [ n ]. This is a Cartesian fibration. The fiber over [ n ] ∈ ∆ op is ∆ n . We can check easily that this Cartesian fibration is equivalent to the unstraightening ofthe evident functor ∆ • : ∆ → C at ∞ sending [ n ] ∈ ∆ to ∆ n .The coCartesian fibration γ ∨ : Γ ∨ → ∆ with the same straightening can also be definedeasily. It is the category of objects ([ n ] , i ) where i ∈ [ n ] and a map ([ n ] , i ) → ([ n ′ ] , i ′ ) is a map α : [ n ] → [ n ′ ] in ∆ such that α ( i ) ≤ i ′ . We have the evident functor γ ∨ : Γ ∨ → ∆ , which is acoCartesian fibration. We denote by S pc the ∞ -category of spaces. We have the functor S pc → C at ∞ by viewinga spaces as an ∞ -category. Let us see that this inclusion functor admits both left and rightadjoints. Let S be a simplicial set. We put the contravariant model structure on ( S et ∆ ) /S andCartesian model structure on ( S et +∆ ) /S . Consider pairs of adjoint functors:( S et ∆ ) /S ι / / ( S et +∆ ) /S , θ o o ( S et +∆ ) /S µ / / ( S et ∆ ) /S . ι o o Here ι ( X ) := X ♯ , namely all the edges are marked, µ ( X, E ) := X , and θ ( X, E ) be the simplicialsubset of X consisting of all the simplices σ such that every edge of σ belongs to E . We claimthat the above pairs are Quillen adjunctions. The second one is a Quillen adjunction by [HTT,3.1.5.1]. For the first one, since the adjointness is easy to check, it suffices to show that ι preservescofibrations and weak equivalences. Preservation of cofibrations is obvious. The preservationof weak equivalences is shown in the proof of [HTT, 3.1.5.6]: for a morphism of simplicial sets f : X → Y over S , the induced map f ♯ : X ♯ → Y ♯ is a Cartesian equivalence if and only if f is acontravariant equivalence. We also have θ ◦ ι ≃ id, µ ◦ ι ≃ id. Since ι preserves fibrant objects,these imply that R θ ◦ L ι ≃ id, and L µ ◦ R ι ≃ id. In the special case where S = ∆ , we have thefollowing result by [HTT, 5.2.4.6], which is originally due to Joyal [J, 6.15, 6.27]: Lemma. —
The functor ι : S pc → C at ∞ admits a right adjoint θ and a left adjoint µ suchthat θ ◦ ι ≃ id , µ ◦ ι ≃ id . In particular, ι is fully faithful and commutes with small limits andcolimits.
4n the sequel, for a Cartesian or coCartesian fibration X → S we often denote θX by X ≃ /S .When S = ∆ , we omit / ∆ and simply write X ≃ . Consider the following diagrams X h / / f ❅❅❅❅❅❅❅❅ Y g (cid:127) (cid:127) ⑦⑦⑦⑦⑦⑦⑦⑦ S, X × S,f ( e ) ∆ h ′ / / f ′ % % ❑❑❑❑❑❑❑❑❑❑ Y × S,f ( e ) ∆ g ′ y y ssssssssss ∆ , such that f , g are coCartesian fibrations, h is an inner fibration which preserves coCartesianedges, and e is an edge in X . Then e is h -Cartesian if and only if e is h ′ -Cartesian.Proof. The proof is almost a copy of [HTT, 5.2.2.3]. Since we need to check a certain rightlifting property with respect to Λ nn → ∆ n , we may assume that S = ∆ n , in particular, S is an ∞ -category. Let e : x → y . By [HTT, 2.4.4.3], it suffices to show that the following diagram ishomotopy pullback diagram for any z ∈ X :Map( z, x ) / / (cid:15) (cid:15) Map( z, y ) (cid:15) (cid:15) Map( h ( z ) , h ( x )) / / Map( h ( z ) , h ( y )) . If there is no map from f ( z ) to f ( x ), we have nothing to prove, so we may assume that there is a map, in fact a unique map, ǫ : f ( z ) → f ( x ). Let ǫ ′ : z → z be a f -coCartesian edge lifting ǫ .For w ∈ X , consider the following diagrams:Map( z , w ) / / (cid:15) (cid:15) Map( z, w ) (cid:15) (cid:15) Map( f ( z ) , f ( w )) / / Map( f ( z ) , f ( w )) , Map( h ( z ) , h ( w )) / / (cid:15) (cid:15) Map( h ( z ) , h ( w )) (cid:15) (cid:15) Map( f ( z ) , f ( w )) / / Map( f ( z ) , f ( w )) . Both diagrams are homotopy pullback diagram. Indeed, the left one is a homotopy pullbacksince ǫ ′ is f -coCartesian, and the right one is since h ( ǫ ′ ) is g -coCartesian by the assumption.Thus, if w is either x or y , the top horizontal maps are equivalences. Thus, we are reduced toshowing that Map( z , x ) / / (cid:15) (cid:15) Map( z , y ) (cid:15) (cid:15) Map( h ( z ) , h ( x )) / / Map( h ( z ) , h ( y )) . is a homotopy pullback diagram, which follows since e is h ′ -Cartesian. (cid:4) Let f : C → E , g : D → E be functors of ∞ -categories, and assume that g is a categorical fibration. Let ( C , D ) , ( C , D ) be vertices of C × E D . For i = 0 , , put E i := f ( C i ) = g ( D i ) . Then Map C × E D (cid:0) ( C , D ) , ( C , D ) (cid:1) / / (cid:15) (cid:15) Map C ( C , C ) (cid:15) (cid:15) Map D ( D , D ) / / Map E ( E , E ) is a homotopy Cartesian diagram. roof. Since g is a categorical fibration, Fun(∆ , D ) → Fun(∆ , E ) is a categorical fibration aswell by [HTT, 2.2.5.4]. Thus,Fun(∆ , P ) ∼ = Fum(∆ , C ) × Fun(∆ , E ) Fun(∆ , D ) ≃ Fum(∆ , C ) × catFun(∆ , E ) Fun(∆ , D )in S pc. Now, we have Map C ( C , C ) ≃ Hom C ( C , C ) := Fun(∆ , C ) × Fun( ∂ ∆ , C ) { ( C , C ) } by[HTT, 2.2.4.1, 4.2.1.8]. The product is in fact a product in C at ∞ since Fun(∆ , C ) → Fun( ∂ ∆ , C )is a categorical fibration by [HTT, 3.1.4.3]. Thus the square in question is a Cartesian squarein C at ∞ . By Lemma 1.4, the claim follows. (cid:4) Let C i ( i = 0 , , ), D be ∞ -categories and C ′ i , D ′ be its subcategories. Assumewe are given a homotopy commutative diagram (∆ ) → C at ∞ D ′ (cid:15) (cid:15) f ′ { { ①①①①①①① g ′ / / C ′ (cid:15) (cid:15) { { ✇✇✇✇✇✇✇ C ′ / / (cid:15) (cid:15) C ′ (cid:15) (cid:15) D g / / f { { ✇✇✇✇✇✇✇ C z z ✈✈✈✈✈✈ C / / C , where the vertical arrows are inclusions. This induces the diagram D ′ F ′ := f ′ × g ′ / / G (cid:15) (cid:15) C ′ × cat C ′ C ′ =: C ′ H (cid:15) (cid:15) D F := f × g / / C × cat C C =: C where vertical arrows are the canonical functors. Assume the following:1. The functor F is a categorical equivalence;2. The canonical functors C ′ i → C i , D ′ → D are categorical fibrations;3. An object d ∈ D is an object of D ′ precisely when f ( d ) is in C ′ and g ( d ) is in C ′ ;4. A map a : d → d ′ in D such that d, d ′ ∈ D ′ is a map in D ′ precisely when f ( a ) is in C ′ and g ( a ) is in C ′ .Then F ′ is a categorical equivalence as well.Proof. Let A be a (ordinary) category, and B be its subcategory. The functor N B → N A is amonomorphism in C at ∞ (in the sense of [HTT, 5.5.6.13] which coincides with the one in [AFR,A.1]) if B → A is an isofibration ( i.e. if b ∈ B and f : a ∼ −→ b in A , then a and f belongs to B )by [AFR, A.6]. For any ∞ -category E , the functor E → Nh E is a categorical fibration, so by[HTT, 5.5.6.12], the functors D ′ → D , C ′ i → C i are monomorphisms. By [AFR, A.5], the functor C ′ → C is a monomorphism as well. By [AFR, A.4], F ′ is a monomorphism.Let us show that F ′ is essentially surjective. Let p i : C ′ i → C ′ be the given map. Consider atriple ( C , C , α ) where C i ∈ C ′ i and α : p ( C ) ∼ −→ p ( C ). This induces a functor ∆ → C ′ × cat C ′ C ′ (up to contractible choices), and defines an object of the fiber product. Denote the associatedobject by C ′ ( C , C , α ). Any object of the fiber product is equivalent to an object associated6o a triple of the form above, because C ′ → C ′ can be factored into categorically equivalencefollowed by categorical fibration. Since F is a categorical equivalence, there exists an object D such that F ( D ) ≃ H ( C ′ ( C , C , α )). By 3, D belongs to D ′ if f ( D ) and g ( D ) belongs to C ′ and C ′ respectively. Since f ( D ) ≃ C and g ( D ) ≃ C , combining with 2, D ∈ D ′ . Since C ′ → C is amonomorphism, the functor C ′≃ → C ≃ is a monomorphism by [AFR, A.6] and thus fully faithful(cf. [AFR, A.1]). This implies that F ′ ( D ) and C ′ ( C , C , α ) are equivalent, and F ′ is essentiallysurjective.It remains to show the full faithfulness. For a simplicial set, recall Map C at ∞ ( K, − ) ≃ Fun( K, − ) ≃ . If A → B is a categorical fibration, then Fun( K, A ) → Fun( K, B ) is a categoricalfibration for any simplicial set K by [HTT, 2.2.5.4], and so is the map Map( K, A ) → Map( K, B )by Lemma 1.4. Note that Map C at ∞ is a model for the mapping space of C at ∞ by the definition ofBoardman-Vogt weak equivalence (cf. [HTT, A.3.2.1]). Thus, by dual of [HTT, at the beginningof § C at ∞ (∆ , − ) to thediagram in the statement of the lemmaMap(∆ , D ′ ) / / e G (cid:15) (cid:15) Map(∆ , C ′ ) × catMap(∆ , C ′ ) Map(∆ , C ′ ) (cid:15) (cid:15) Map(∆ , D ) / / Map(∆ , C ) × catMap(∆ , C ) Map(∆ , C ) . The map e G is a monomorphism by [AFR, A.6], thus fully faithful (cf. [AFR, A.1]). This is thesame if we replace D , D ′ by C i , C ′ i respectively. The lower horizontal functor is a categoricalequivalence, and the upper horizontal functor is fully faithful. Repeating the argument ofthe essential surjectivity of F ′ , the upper horizontal functor is essentially surjective, and thuscategorical equivalence as required. (cid:4) The following lemma should be well-known to experts, but since we could not find areference, we write here for record.
Lemma. —
Let C be an ∞ -category. Consider a diagram F : K := K × K → C where K , K are simplicial sets. For any simplicial subsets K ′ ⊂ K , assume that the functor F | K ′ admitsa limit. Let F : K → Fun( K , C ) and F : K → Fun( K , C ) be functors induced by F . Thenwe have a canonical equivalence lim ←− K (lim ←− K F ) ≃ lim ←− K F ≃ lim ←− K (lim ←− K F ) . Proof.
Let us show the first equivalence. By taking the opposite category, we show the equiv-alence for colimits instead of limits. By [HTT, 4.2.3.15], there exists a (left) cofinal mapN( I ) → K from an partially ordered set I . In view of [HTT, 4.1.1.13], we may replace K by N( I ). For each I ∈ I , let K I := { I } × K and we have the functor G : N( I ) → ( S et ∆ ) /K sending I to K I . In [HTT, 4.2.3.1], the simplicial set K G is defined. In view of [HTT, 4.2.3.9],the hypotheses of [HTT, 4.2.3.8] is satisfied. Now, by construction, we have the evident inclusion K G → K ⋄ N( I ) N( I ). By using [HTT, 4.2.2.7], F admits an extension e F : K ⋄ N( I ) N( I ) → C . Since e F | K G satisfies the hypotheses of [HTT, 4.2.3.4], and we invoke [HTT, 4.2.3.10] to conclude. (cid:4) Let C be an ∞ -category, and consider a diagram F : (Λ ) ⊲ → C and a map t → F ( ∞ ) , where ∞ is the cone point. Then we have the canonical equivalence ( F (0) × F (2) F (1)) × F ( ∞ ) t ≃ ( F (0) × F ( ∞ ) t ) × ( F (2) × F ( ∞ ) t ) ( F (1) × F ( ∞ ) t ) .Proof. Let us construct a functor e F : Λ × Λ → C as follows. Let D := (Λ ) ⊲ ` ∞ , {∗} , [1] ∆ . Wehave the functor F ′ : D → C sending ∆ to t → F ( ∞ ). Let i : D → Λ × Λ be the inclusion. Let e F to be a right Kan extension of F along i . Now, the claim follows by applying the lemma. (cid:4) .10. The map Fun(∆ , C at ∞ ) → C at ∞ induced by ∆ { } ֒ → ∆ is a Cartesian fibration since C at ∞ admits limits. Let C art ∞ be the full subcategory of Fun(∆ , C at ∞ ) spanned by Cartesianfibrations. Then the induced map θ : C art ∞ → C at ∞ is a Cartesian fibration as well. Indeed,since C art ∞ ֒ → Fun(∆ , C at ∞ ) is a full subcategory, it is an inner fibration. Thus, θ is aninner fibration. Since Cartesian fibration is stable under base change, and Cartesian fibrationis a categorical fibration by [HTT, 3.3.1.7], θ is a Cartesian fibration. For an ∞ -category C ,we denote C art ∞ × C at ∞ { C } by C art( C ). Note that since θ is a categorical fibration, given acategorical equivalence C ∼ −→ C ′ , the base change functor C art( C ′ ) → C art( C ) is a categoricalequivalence as well by [HTT, 3.3.1.3] applied to the case where S is the category with twoobjects and one isomorphism and T is an inclusion from ∆ to an object of S . Dually, we putco C art ∞ to be the full subcategory of Fun(∆ , C at ∞ ) spanned by coCartesian fibrations, anddefine co C art( C ) to be the fiber.Let co C art str ∞ be the subcategory of co C art ∞ consisting of simplices ∆ n → co C art ∞ such thatall the edges are of the form D r / / q (cid:15) (cid:15) D ′ p (cid:15) (cid:15) C / / C ′ such that r sends q -coCartesian edges to p -coCartesian edges. The functor co C art str ∞ → C at ∞ isCartesian as well. The fiber over C ∈ C at ∞ is denoted by co C art str ( C ). We have the equivalencesFun( C , C at ∞ ) ∼ −−→ Un C N(( S et +∆ ) ◦ / C ) ∼ −→ co C art str ( C ) , where the first one is the unstraightening functor. The category ( S et +∆ ) ◦ / C is endowed with co-Cartesian model structure, and let us construct the second equivalence. The functor of simplicialcategories ( S et +∆ ) ◦ / C → (( S et +∆ ) / ∗ ) ◦ / C sending X to X × C ♯ C ♮ → C ♮ , where C is considered to bea coCartesian fibered over ∆ , induces the functor N(( S et +∆ ) ◦ / C ) → ( C at ∞ ) / C ≃ ( C at ∞ ) / C , using[HTT, 6.1.3.13, 4.2.1.5]. By definition, C art str ( C ) is a subcategory of ( C at ∞ ) / C , and the abovefunctor factors through C art str ( C ), which is the desired functor. This functor is essentially sur-jective by definition. It remains to show that it is fully faithful. Let D , D ′ be coCartesianfibrations over C . Let Fun C ( D , D ′ ) coCart be the full subcategory of Fun C ( D , D ′ ) spanned byfunctors preserving coCartesian edges. Then we haveMap ♯ C ( D ♮ , D ′ ♮ ) ∼ = Map ♭ C ( D ♮ , D ′ ♮ ) ≃ ≃ (cid:0) Fun C ( D , D ′ ) coCart (cid:1) ≃ , where the first isomorphism follows by [HTT, 3.1.3.1]. In view of [HTT, 3.1.4.4], we get theclaim. Let D ≃ C ` C C be a pushout in C at ∞ . Then we have an equivalence α : C art( D ) ≃ C art( C ) × C art( C ) C art( C ) in C at ∞ .Proof. The author learned the proof from Lysenko’s notes (1) . First, for any ∞ -category C , wehave Map C at ∞ (∆ , C art( C )) ≃ C art str ( C ) ≃ ≃ Map C at ∞ ( C , C at ∞ ) , where the second equivalence is the straightening/unstraightening equivalence. Since the pushoutis in C at ∞ , we haveMap C at ∞ ( D , C at ∞ ) ≃ Map C at ∞ ( C , C at ∞ ) × catMap C at ∞ ( C , C at ∞ ) Map C at ∞ ( C , C at ∞ ) (1)
8y [HTT, at the beginning of § C at ∞ (∆ , α ) is anequivalence. It remains to show that Map C at ∞ (∆ , α ) is an equivalence. By [GR, Ch.12, 2.1.3],we have an equivalenceMap C at ∞ (∆ , C art( C )) ≃ Map C at ∞ ( C op , co C art(∆ ))for any ∞ -category C . Since ( − ) op is an auto-equivalence of C at ∞ , we have D op ≃ C op1 ` C op0 C op2 ,we haveMap C at ∞ ( D op , co C art(∆ )) ≃ Map C at ∞ ( C op1 , co C art(∆ )) × catMap C at ∞ ( C op0 , co C art(∆ )) Map C at ∞ ( C op1 , co C art(∆ )) . Combining these equivalences, Map(∆ , α ) is an equivalence as required. (cid:4) Let C be an ∞ -category. We define the ∞ -category of arrows to be Ar C := Fun(∆ , C ).On the other hand, the ∞ -category of twisted arrows denoted by Tw C is studied extensively in[HA, § c → c ′ in C , with a map ( c → c ′ ) → ( c → c ′ ) given by a diagram ∆ → C depicted as c (cid:15) (cid:15) / / c (cid:15) (cid:15) c ′ c ′ . o o We put Tw op C := (Tw C ) op . For a functor F : K → C from a simplicial set K , we denote byTw op F C := Tw op C × C K . We use two types of operads in this paper: ∞ -operads and planar ∞ -operads in thesense of [HA]. In this paper, after [GH], we call operad what Lurie calls planar ∞ -operad, and symmetric operad what Lurie calls ∞ -operad. We only recall ∞ -operads very briefly.Recall that a function a : [ n ] → [ m ] is said to be inert if there exists i ∈ [ m ] such that a ( j ) = i + j . An active map is a map a such that a (0) = 0, a ( n ) = m . The correspondingmaps in ∆ op are also called inert and active maps. A generalized ∞ -operad is an inner fibration f : C ⊛ → ∆ op satisfying the following three conditions: 1. for any X ∈ C ⊛ and an inert edge f ( x ) → y in ∆ op , there exists a f -coCartesian edge X → Y lifting the inert edge; 2. the inducedmap C ⊛ [ n ] → C ⊛ { , } × cat C ⊛ { } C ⊛ { , } × cat C ⊛ { } · · · × cat C ⊛ { n − } C ⊛ { n − ,n } is a categorical equivalence (Segal condition); 3. for any C ∈ C ⊛ [ n ] , we have a map from C to thediagram C { , } ( ( PPPPPP C { , } v v ♥♥♥♥♥♥ ( ( PPPPPP . . . C { n − ,n } u u ❦❦❦❦❦❦ C { } C { } . . . C { n − } which exhibits C as a π -limit. Here the functor C ⊛ { i,i +1 } → C ⊛ { j } ( j = i, i + 1) is induced bythe assumption that f is coCartesian over inert edges in ∆ op . An f -coCartesian edges in C ⊛ over an inert map in ∆ op are called inert edges . A generalized ∞ -operad is an ∞ -operad if C ⊛ [0] is contractible. A map of generalized ∞ -operads O ⊛ , C ⊛ is a functor O ⊛ → C ⊛ over ∆ op which preserves inert edges. The ∞ -category of maps between generalized ∞ -operads is denotedby Alg O ( C ). As C at ∞ , ∞ -operads forms an ∞ -category. The ∞ -category of (generalized) ∞ -operads is denoted by Op ns , (gen) ∞ (cf. [GH, § C ⊗ for (generalized) symmetric ∞ -operads and C ⊛ for (generalized) ∞ -operads. 9 efinition. — A map M ⊛ → N ⊛ of generalized ∞ -operads is said to be base preserving if theinduced map M ⊛ [0] → N ⊛ [0] is a categorical equivalence. We have the bifunctor of symmetric ∞ -operads N F in ∗ × N F in ∗ → N F in ∗ . As in theproof of [HA, 3.2.4.3], we have the left Quillen bifunctor ( S et +∆ ) / P × ( S et +∆ ) / P → ( S et +∆ ) / P ,where P is the categorical pattern defining the ∞ -category of symmetric ∞ -operads Op ∞ (cf.[HA, proof of 2.1.4.6]), which is identical to ⊙ in [HA, 2.2.5.5]. Thus, if we fix a fibration ofsymmetric ∞ -operads C ⊗ → N F in ∗ , we have a functor Op op ∞ → Op ∞ sending O ⊗ to Alg O ( C ) ⊗ .In particular, considering the embedding C at ∞ → Op ∞ (cf. [HA, 2.1.4.11]), we have the functorFun( − , C ⊗ ) : C at op ∞ → Op ∞ sending D to Alg D ( C ) ⊗ ≃ Fun( D , C ⊗ ). Let Op co , pres ∞ be the subcate-gory of Op ∞ spanned by coCartesian fibration O ⊗ → N F in ∗ which comes from CAlg( P r L ), andcolimit preserving morphisms which preserve coCartesian edges. If C ⊗ → N F in ∗ is a coCartesianfibration coming from CAlg( P r L ), then [HA, 3.2.4.3] further implies that the functor Fun( − , C ⊗ )factors through C at ∞ → Op co , pres ∞ ≃ CAlg( P r L ). In this paper, we follow [GR] for the terminology for ( ∞ , ∞ -category model for ( ∞ , ∞ , ∞ -categories and complete Segal spaces.First, we have the pair of adjoint functorsFun( ∆ op , S pc) JT / / C at ∞ . Seq • o o Indeed, by [JT, 4.11] taking [Hn, 1.5.1] into account, we have an equivalence C at ∞ ≃ CSS ,where
CSS denotes the ∞ -category of complete Segal spaces. By definition of the complete Segalspace model structure, CSS is a localization of Fun( ∆ op , S pc), and we get the adjoint functorsabove. The construction shows that, for an ∞ -category C , the adjunction JT(Seq • ( C )) → C isa categorical equivalence. For C ∈ C at ∞ , we can compute Seq • ( C ) as follows. Let ∆ • : ∆ → C at ∞ be the cosimplicial object such that ∆ • ([ n ]) := ∆ n . For an ∞ -category C , we haveSeq • ( C ) := θ ◦ Fun(∆ • , C ), where θ is the functor in Lemma 1.4, by [JT, 4.10]. Explicitly,Seq n ( C ) ≃ Fun(∆ n , C ) ≃ ≃ Map C at ∞ (∆ n , C ).The observation above shows that we may think of ∞ -category as an object of Fun( ∆ op , S pc)which is a complete Segal space. In our treatment, following [GR], we upgrade this picture, anduse complete Segal ∞ -category model for a model of ( ∞ , ∞ , -category C isa functor C • : ∆ op → C at ∞ satisfying the following conditions: • The ∞ -category C is a space; • (Segal condition) The functor C n → C × cat C C × cat C · · · × cat C C induced by inert maps is anequivalence for n ≥ • (completeness) There exists an ∞ -category C such that Seq • ( C ) ≃ θ ◦ C • .By definition the composition ∆ op C • −→ C at ∞ θ −→ S pc is a complete Segal space, and yields an ∞ -category. This ∞ -category is called the underlying ∞ -category of C . Example. —
Let p : A ⊛ → ∆ op be a monoidal ∞ -category. By straightening, this coCartesianfibration corresponds to a functor ∆ op → C at ∞ . Since p is a an ∞ -operad, it satisfies the Segalcondition, and A is contractible Kan complex. Unfortunately, this Segal ∞ -category may notbe complete. By [L1, 1.2.13], we can localize the Segal ∞ -category into a complete Segal ∞ -category. This complete Segal space is called the classifying ( ∞ , A ⊛ denoted by B A ⊛ . 10 .16. We recall the ( ∞ , C be a category (2) .We need 3 classes of morphisms in C denoted by vert , horiz , adm satisfying certain axioms(cf. [GR, Ch.7, 1.1.1]). To define the ( ∞ , Corr admvert ; horiz ( C ), we should define itsassociated Segal space Seq • ( Corr admvert ; horiz ( C )). For n ≥
0, Seq n ( Corr admvert ; horiz ( C )) is the categoryof diagrams of the form X n / / (cid:15) (cid:15) (cid:3) . . . / / (cid:3) X (cid:15) (cid:15) / / (cid:3) X (cid:15) (cid:15) / / X .X n (cid:15) (cid:15) / / . . . / / X / / (cid:15) (cid:15) X X n / / (cid:15) (cid:15) . . . / / X ... (cid:15) (cid:15) . . .X nn , where horizontal arrows are in the class horiz and the vertical arrows are in the class vert .A morphism in Seq n ( Corr admvert ; horiz ( C )) is a morphism of diagrams X •• → Y •• such that eachmorphism X ij → Y ij is in adm and X kk → Y kk is an equivalence. We may check that this is an( ∞ ,
2. Dualizing coCartesian fibrations
Let f : X → S be a Cartesian fibration. Via straightening/unstraightening construction, thereexists a coCartesian fibration f ′ : X ′ → S op with the same straightening as f . The existence ofsuch coCartesian fibration readily follows from straightening/unstraightening theorem, but theconstruction is far from explicit. As far as the author knows, there are two models for f ′ . Oneis in [L2, 14.4.2], and the other is in [BGN]. In this section, we construct yet another model of f ′ at least when S is an ∞ -category. This model naturally appears in a construction in § Let C be an ∞ -category. Using the notation of 1.10, we have the auto-functor D : co C art str ( C op ) ≃ Fun( C op , C at ∞ ) ≃ C art str ( C ) . When C = ∆ , the functor D is equivalent to the identity functor. Let f : D → C op be inco C art str ( C op ). We have a Cartesian fibration D ( f ) → C . Then by the functoriality of straight-ening/unstraightening functor, we have an equivalence D v ∼ = D ( D ) v for each object v ∈ C . Inthe following we sometimes denote D ( f ) by D ( D ) or D C ( D ) if no confusion may arise. Notethat, by construction, D ( f ) op ∼ = D − ( f op ). Our goal of this section is to compare some Cartesian fibration with D ( f ). For a prepa-ration, we give a criterion to detect D ( f ). A diagram of ∞ -categories C ← M → D is said tobe a weak pairing if it is an object of CPair (cf. [HA, 5.2.1.14, 5.2.1.15]). In other words, weakpairing is a diagram which is equivalent to a pairing (cf. [HA, 5.2.1.5]), namely a diagram suchthat the induced map M → C × D is equivalent to a right fibration. We often say M → C × D is a weak pairing without referring to the diagram. A weak pairing is said to be perfect if it is (2) We may assume C to be an ∞ -category, but for simplicity, we assumed this. For details see [GR]. perf (cf. [HA, 5.2.1.20]), namely a paring which is equivalentto the pairing Tw C → C × C op for some ∞ -category C . Definition [HA, 5.2.1.8] makes sense alsofor weak pairings, so we may talk about left universality etc. Let λ : M → C × D be a weakpairing and take an equivalence M ∼ γ / / λ (cid:15) (cid:15) M ′ λ ′ (cid:15) (cid:15) C × D ∼ α × β / / C ′ × D ′ where λ ′ is a pairing. Then M ∈ M , such that λ ( M ) = ( C, D ), is left universal if and only if γ ( M ) is left universal because M × C { C } ∼ −→ M ′ × C ′ { α ( C ) } and [HTT, 1.2.12.2]. Lemma. —
Let S be a simplicial set, and consider the following diagram M p (cid:31) (cid:31) ❅❅❅❅❅❅❅❅ ν / / C × S D q = f × g { { ✇✇✇✇✇✇✇✇✇ S Assume that p , f , g are Cartesian fibrations, and ν sends p -Cartesian edges to q -Cartesianedges. If the following conditions are satisfied, then D ∼ = D ( f op ) . • For any vertex s ∈ S , ν s := ν × S s is a perfect weak pairing (resp. ν op s is a perfect weakpairing); • For any p -Cartesian edge x → y in M , if y is right universal (resp. right universal withrespect to ν op p ( y ) ), then so is x .Proof. First, let us show the non-resp claim. Recall that the ∞ -category CPair is a full sub-category of Fun(Λ , C at ∞ ). Thus, ν corresponds to a functor S → CPair by straightening.By the second condition, this functor induces a functor M : S → CPair R (cf. [HA, 5.2.1.16]for the notation). By [HA, 5.2.1.19], the functor φ : Pair R → C at ∞ sending ν to C admitsa right adjoint Tw Pair such that Tw
Pair C ≃ (Tw C → C × C op ). Thus, we have the naturaltransform id → Tw Pair ◦ φ . Thus, we have the natural transform M → Tw Pair ◦ φ ◦ M . PutTw( f ) := Un S (Tw Pair ◦ St S ( f )) and recall that D ( f op ) ≃ Un S ( χ ◦ St S ( f )), where χ is the uniquenon-trivial automorphism of C at ∞ , by [L2, 14.4.2.4]. By unstraightening, the natural transforminduces a diagram of Cartesian fibrations over S M / / (cid:15) (cid:15) Tw( f ) (cid:15) (cid:15) C × S D / / C × S D ( f op )where horizontal functors send p -Cartesian edges to Cartesian edges of Tw( f ). Invoking [HTT,3.3.1.5], horizontal functors are equivalences if and only if they are equivalences for each fibersof S . Since the construction is functorial with respect to S , the perfectness of ν s implies that thehorizontal functors are in fact equivalences, which implies that D ≃ D ( f op ) as required. Finally,let us show the resp claim. Consider the following diagram D ( p op ) p ′ ❋❋❋❋❋❋❋❋❋ D ( ν op ) / / D ( q op ) q ′ { { ①①①①①①①①① S. p ′ , q ′ are Cartesian fibrations by construction, D ( ν op ) s ≃ ν op s , and D ( q op ) ≃ D ( f op ) × S D ( g op ). Since a p -Cartesian edge x → y yields a p ′ -Cartesian edge x ′ → y ′ by constructionof D , we may apply the non-resp claim, which implies that D ( g op ) ≃ D ( f ′ op ) ≃ C , where f ′ : D ( f op ) → S is the Cartesian fibration. Thus, taking D − , we get the claim. (cid:4) Let f : A → B be an inner fibration, and let g : Tw op A → Tw op B be theinduced map. Let e : ∆ → Tw op A be an edge, and let e e : ∆ → A be the associated map defining e depicted as follows: e e (1) (cid:15) (cid:15) e e (0) α o o (cid:15) (cid:15) e e (2) β / / e e (3) .
1. The map g is an inner fibration. If, moreover, f is a categorical fibration, so is g .2. Assume that α := e e (∆ { , } ) is an f -coCartesian edge and β := e e (∆ { , } ) is an f -Cartesianedge. Then e is a g -Cartesian edge.3. Assume that α is an f -Cartesian edge and β is an f -coCartesian edge. Then e is a g -coCartesian edge.Proof. Let us show the first claim. It suffices to show that the induced map Tw A → Tw B × ( B × B op ) ( A × A op ) is a right fibration. Indeed, we need to show the right lifting property of the mapwith respect to the inclusion Λ ni → ∆ n for 0 < i ≤ n . Unwinding the definition, it suffices tosolve the right lifting problem of f with respect to K ֒ → ∆ n +1 , where K is the same simplicialsubset of ∆ n +1 appearing in the proof of [HA, 5.2.1.3]. Since K ֒ → ∆ n +1 is shown to be aninner anodyne in ibid. , the claim follows.Let us prove the second claim. It amounts to solving the lifting problem on the left for 2and right for 3:∆ { n − ,n } (cid:31) (cid:127) / / e * * Λ nn (cid:15) (cid:15) / / Tw op A g (cid:15) (cid:15) ∆ n / / ; ; ✇✇✇✇✇ Tw op B , ∆ { , } (cid:31) (cid:127) / / e ) ) Λ n (cid:15) (cid:15) / / Tw op A g (cid:15) (cid:15) ∆ n / / ; ; ✈✈✈✈✈ Tw op B . We first treat 2. Unwinding the definition, it suffices to solve the following lifting problem ofmarked simplicial sets:(2.3.1) E (cid:31) (cid:127) / / ψ ( ( K / / (cid:127) _ ϕ (cid:15) (cid:15) A f (cid:15) (cid:15) ∆ n +1 / / : : ✉✉✉✉✉ B . Here, E := ∆ { , } ∪ ∆ { n, n − } , the edge ψ (∆ { , } ) (resp. ψ (∆ { n, n +1 } )) is an f -coCartesian(resp. f -Cartesian) edge, and K is the union of the simplicial subsets ∆ I ⊂ ∆ n +1 where I = [2 n + 1] \ { i, n + 1 − i } for 0 < i ≤ n .Let Σ be the simplicial subset ∆ [2 n +1] \{ n } of ∆ n +1 . Put K := K ∪ Σ. It suffices to checkthe following two claims1. The map f has right lifting property with respect to K ֒ → K ;13. The map f has right lifting property with respect to K ֒ → ∆ n +1 .Let us show the first claim. Note that any simplex of Σ which is not in K contains { } as avertex. Thus, we can divide the simplices of K which do not belong to K into the followingtwo classes: • A k is the set of simplices which have N vertices, where N ≤ k + 2, and contain the vertices { , } ; • B k is the set of simplices which have k + 1 vertices and contain { } but do not contain { } . For σ ∈ A k , we denote by σ ′ ∈ B k the simplex obtained by deleting the vertex { } .Let K ( k )1 := K ∪ S σ ∈ A k σ , so K (2 n − = K . Then any element of B k is a simplex of K ( k )1 becausefor any ∆ I ∈ B k , ∆ I ⊔{ } ∈ A k . On the other hand no element of B k +1 belongs to K ( k )1 becauseof the dimension reason. For k < n −
1, we have A k = B k = ∅ because for any I ⊂ [2 , n − I = k , we can find 2 ≤ i ≤ n + 1 such that { i, n + 1 − i } ∩ I = ∅ . This implies that K ( n − = K . On the other hand, for 2 n − ≥ k ≥ n − A k , B k are non-empty.We solve the lifting problem with respect to K ֒ → K ( k )1 inductively. We assume we have amap K ( k )1 → A solving the lifting problem. We choose a total ordering σ < · · · < σ a of thesimplices in A k +1 which do not belong to K ( k )1 , and we have the sequence K ( k )1 =: L ⊂ L ⊂· · · ⊂ L a =: K ( k +1)1 where L i = L i − ∪ σ i . Fix i and put L ′ := L i − , L := L i . Let us solve thelifting problem with respect to L ′ ֒ → L . For this, write σ i = ∆ I with I ⊂ [0 , n + 1]. Recallthat { , } ⊂ I by definition. Now, for j ∈ I \ { , } , ∆ I \{ j } belongs to A k or K , and thus∆ I \{ j } ⊂ L ′ . This holds also for j = 1 since ∆ I \{ } belongs to K . On the other hand, ∆ I \{ } is not a simplex of L ′ because ∆ I \{ } ∈ B k and σ ′ , . . . , σ ′ a are all different to each other. Thisimplies that L ′ ֒ → L ′ ` Λ k +20 ∆ I = L . Consider the following diagram:∆ { , } / / ψ ) ) Λ k +20 / / (cid:15) (cid:15) L ′ / / (cid:15) (cid:15) A f (cid:15) (cid:15) ∆ I / / L / / B . Since ψ (∆ { , } ) is an f -coCartesian edge, we have a map ∆ I → A making the diagram commu-tative. Thus, we have a lifting L → A , and the first claim follows.Let us show the second claim. The idea of the proof is essentially the same as the first claim.If a simplex of ∆ n +1 does not contain the vertex { n } , then it is contained in K . Thus, wecan divide the simplices of ∆ n +1 which do not belong to K into the following two classes: • C k is the set of simplices which have N vertices, where N ≤ k + 2, and contain the vertices { n, n + 1 } ; • D k is the set of simplices which have k + 1 vertices and contain { n } but do not contain { n + 1 } .Let K ( k )2 := K ∪ S σ ∈ C k σ , so K (2 n − = ∆ n +1 . As in the previous case, any element of D k is a simplex of K ( k )2 , and any element of D k +1 does not belong to K ( k )2 . The sets C k , D k arenon-empty if and only if 2 n − ≥ k ≥ n −
1. We solve the lifting problem with respect to K ֒ → K ( k )2 inductively. Assume we have a map K ( k )2 → A solving the problem. We choosea total ordering τ < · · · < τ b of the simplices in C k +1 which do not belong to K ( k )2 , and we14orm a sequence K ( k )2 =: M ⊂ · · · ⊂ M b =: K ( k +1)2 where M i = M i − ∪ τ i . Fix i and put M ′ := M i − , M := M i , τ := τ i = ∆ J . We solve the lifting problem with respect to M ′ ֒ → M .Since { n, n + 1 } ⊂ J , for any j ∈ J \ { n, n + 1 } , ∆ J \{ j } belongs to C k or K , and thuscontained in M ′ . The same holds for j = 2 n because it is contained in K . Finally, ∆ I \{ n +1 } is not contained in M ′ , which implies that M ′ ֒ → M ′ ` Λ k +2 k +2 ∆ J = M . Since ψ (∆ { n, n +1 } ) is a f -Cartesian edge, the lifting problem is solved.Let us prove 3. For this, it suffices to check the lifting problem (2.3.1) where E := ∆ { n − ,n } ∪ ∆ { n +1 ,n +2 } , the edge ψ (∆ { n +1 ,n +2 } ) (resp. ψ (∆ { n − ,n } )) is an f -coCartesian (resp. f -Cartesian)edge, and K is the union of the simplicial subsets ∆ I ⊂ ∆ n +1 where I = [2 n + 1] \ { i, n + 1 − i } for 0 ≤ i < n . The proof is essentially the same as 2, so we only indicate the difference. Put σ := ∆ [2 n +1] \{ n − } . The definition of A k is replaced by the set of simplices which contain vertices { n + 1 , n + 2 } , B k is the one which contain n + 2 but not n + 1. We proceed as before, anddefine L ′ ֒ → L similarly. Small difference from the previous argument is that, in this case, L = L ′ ` Λ k +2 i ∆ I for 0 ≤ i < k + 2 so that ∆ { i,i +1 } in Λ k +2 i is mapped to the edge ∆ { n +1 ,n +2 } in∆ I . The lifting problem can be solved for i = 0 by the assumption that the edge ψ (∆ { n +1 ,n +2 } )is coCartesian, and for 0 < i < k + 2 since f is an inner fibration. The later part also workssimilarly, and we omit the detail. (cid:4) Let C be an ∞ -category, and let (Φ , Θ) : Tw op C → C op × C be the canonical functor. Recallthe notation Tw op K C from 1.12. In particular, for a vertex v : ∆ → C and an edge φ : ∆ → C ,we have the ∞ -categories Tw op v C and Tw op φ C . Let X → Tw op C be a map simplicial sets. Bydefinition, vertices of Θ ∗ ( X ) → C over v correspond to functors Tw op v C → X over Tw op C . Definition. —
Let f : D → C op be a coCartesian fibration. We define D ∨ to be the fullsubcategory of Θ ∗ Φ ∗ ( D ) spanned by the vertices G : Tw op v C → D over C op for some v ∈ C suchthat G sends edges of Tw op v C to f -coCartesian edges in D .Our goal of this section is the following theorem. Let C be an ∞ -category and let f : D → C op be a coCartesian fibration.Then the map f ∨ : D ∨ → C is a Cartesian fibration, and this is equivalent to D ( f ) .Proof. Consider the following diagram: D op (cid:15) (cid:15) Tw op D op (cid:15) (cid:15) Φ D / / Θ D o o D f (cid:15) (cid:15) C Tw op C Φ / / Θ o o C op . Consider the subcategory g D op of Θ ∗ Θ ∗ ( D op ) spanned by the vertices Tw op v C → D op over C such that any edge of Tw op v C is sent to equivalences. We will later show that this category isequivalent to D op . We define M to be the full subcategory of Θ ∗ (Tw op D op ) spanned by thevertices G : Tw op v C → Tw op D op over Tw op C satisfying the following conditions: • The composition Φ D ◦ G sends edges of Tw op v C to f -coCartesian edges; • The composition Θ D ◦ G sends edges of Tw op v C to equivalences.By the first condition, the natural map Tw op D op → Φ ∗ D induces the map M → D ∨ , and bythe second condition, Tw op D op → Θ ∗ D op induces the map M → g D op . Thus, we have the map ν : M → D ∨ × C g D op .Let us show that M , D ∨ , and g D op are Cartesian fibrations over C . Since the verifications aresimilar, and that for M is much more complicated than the other two, we concentrate on this.15et π : M → C be the map. Take a vertex m ∈ M and an edge φ : v → w := π ( m ) in C . We wishto take a right Kan extension as follows:Tw op w C m / / (cid:127) _ (cid:15) (cid:15) Tw op D op p (cid:15) (cid:15) Tw op φ C / / ♦♦♦♦♦♦♦ Tw op C . In order to apply [HTT, 4.3.2.15] to check the existence, take a vertex C := ( v ′ → v ) of Tw op φ C .Then (cid:0) Tw op w C (cid:1) C/ has an initial object C → ( v ′ → w ) which can be depicted as v ′ C (cid:15) (cid:15) v ′ = o o (cid:15) (cid:15) v φ / / w. Choose a following diagram D := Λ ` ∆ { , } ∆ { , ′ , } → D op of the following form: v D (1) “ φ ∗ m ( w → w )” 3 (cid:13) (cid:15) (cid:15) + w D (2) m ( w → w )1 (cid:13) (cid:15) (cid:15) v ′ D (0) ≃ m ( v ′ → w ) 2 (cid:13) ( ( PPPPPPPPPPPP c k PPPPPPPPPPPPPPPPPPPPPP : ♥♥♥♥♥♥♥♥♥♥♥ ♥♥♥♥♥♥♥♥♥♥♥ w w ♥ ♥ ♥ ♥ ♥ ♥ v D (2 ′ ) + w D (3) . Here, “ ⇒ ” are f op -Cartesian edges and the big outer square is a Cartesian pullback square over v → w . The left subscripts indicate the image of the object in C ( e.g. v D (1) is over v ). Theobject m ( w → w ) is a priori an object of Tw op D op , but this determines an edge in D op whichyields the edge (cid:13) . The same procedure yields an edge m ( v ′ → w ), and (cid:13) is an edge equivalentto this edge. We can take such an edge because Θ D ◦ m sends edges of Tw op w C to equivalencesin D op . The edge (cid:13) is an edge that should be equivalent to φ ∗ m ( w → w ) when π is shown tobe Cartesian.Since D ֒ → ∆ ` ∆ { , , } ∆ { , , ′ , } is an inner anodyne and f op is an inner fibration, wecan complete the dashed arrow so that the diagram is commutative and the image in C iscompatible with the map C → ( v ′ → w ). The diagram ∆ { , ′ , } can be considered as a mapfrom ( D (0) → D (2 ′ )) to ( D (0) → D (3)) in Tw op D op over C → ( v ′ → w ). It suffices to show,by [HTT, 4.3.1.4], that this edge in Tw op D op is a p -Cartesian edge. This follows by Lemma 2.3,taking [HTT, 2.4.1.5] into account. Applying [HA, B.4.8], we know that this edge in M is a π -Cartesian edge. Furthermore, by construction, the map ν sends Cartesian edges to Cartesianedges.Now, let us check that ν satisfies the conditions in Lemma 2.2. For this, let us ana-lyze the fibers of M over C . Fix a vertex v ∈ C . Objects of M v correspond to functorsFun Tw op C (cid:0) Tw op v C , Tw op D op (cid:1) satisfying some conditions. The map i : {∗} → Tw op v C sending theunique object to the object v → v yields the map i ∗ : M v → Tw op D op v . We show that this is acategorical equivalence. Consider the following diagram: {∗} F / / (cid:127) _ i (cid:15) (cid:15) Tw op D op p (cid:15) (cid:15) Tw op v C / / ♦♦♦♦♦♦♦ Tw op C . F , there exists a left Kan extension. Take C = ( w → v ) inTw op v C . Then {∗} /C has an initial object {∗} → ( w → v ). By [HTT, 4.3.2.15, 4.3.1.4], it sufficesto check that the map in Tw op D op corresponding to the diagram d F ( ∗ ) (cid:15) (cid:15) d wα o o (cid:15) (cid:15) d ′ ∼ / / d ′′ in D op , where α is an f op -Cartesian edge over v ← w , is a p -coCartesian edge. This followsby Lemma 2.3. By construction, the left Kan extension can be regarded as an object of M v .Invoking [HTT, 4.3.2.17], i ∗ admits a left adjoint i ! : Tw op D op v → M v . By the characterizationof left Kan extension functor [HTT, 4.3.2.16] and the definition of M , i ! is essentially surjective.Since id ∼ −→ i ∗ i ! , i ! is fully faithful, thus, i ! is a categorical equivalence. This implies that i ∗ isalso a categorical equivalence because it is so on the level of homotopy categories.On the other hand, the canonical map D op → Θ ∗ Θ ∗ D op induces a map ι : D op → g D op .This map is in fact an equivalence. Indeed, since g D op → C is a Cartesian fibration and ι sendsCartesian edges to Cartesian edges, it suffices to check that the fibers are equivalence by [HTT,3.3.1.5]. In order to see the equivalence, we may proceed as the proof of the equivalence i ! .Likewise, we have a canonical equivalence ( D ∨ ) v ∼ −→ D v .By construction, we have the following commutative diagram of ∞ -categories: M v i ∗ / / ν v (cid:15) (cid:15) Tw op D op v Φ × Θ (cid:15) (cid:15) ( D ∨ × C g D op ) v / / D v × D op v . Here, the horizontal maps are equivalence. This implies that ν op v is in fact a perfect weak pairing.By the description of Cartesian edges in M , the preservation also holds, and the conditions ofLemma 2.2 are satisfied. Thus, we have D ∨ ∼ = D (( g D op ) op ) ∼ ←− ι D ( D ). (cid:4)
3. Stable R -linear categories We construct the ( ∞ , R -linear categories for an E ∞ -ring R ( e.g. ordinarycommutative ring). This has already been outlined in [GR, Ch.1, 8.3], and the only contributionof ours is to make the construction rigorous. First, we recall the construction of [GH, 4.1]. Let i : [0] → ∆ op be the fully faith-ful inclusion. We may take the right Kan extension functor i ∗ : C at ∞ ≃ Fun([0] , C at ∞ ) → Fun( ∆ op , C at ∞ ). By [HTT, 4.3.2.17], for a functor D • : ∆ op → C at ∞ , we have an equivalenceFun( D • , i ∗ C ) ≃ Fun( D , C ). If we are given a functor E → D , we denote D • × i ∗ D i ∗ E , wherethe fiber product is taken in Fun( ∆ op , C at ∞ ), by D • ∗ E . If E → D is a categorical fibration,then D • ∗ E can be computed termwise by [HTT, 5.1.2.3].We can also have coCartesian fibration version of the above construction. Let Γ ′ be thecategory with objects ([ n ] , i ) where i ∈ [ n ], and a morphism ([ n ] , i ) → ([ n ′ ] , i ′ ) consists of afunction α : [ n ′ ] → [ n ] such that α ( i ′ ) = i . Then the evident functor γ ′ : Γ ′ → ∆ op is a Cartesianfibration. For an ∞ -category C , let C × := γ ′∗ (Γ ′ × C ). Then by 1.2 (or by direct computation), C × is an unstraightening of i ∗ C . For a coCartesian fibration X → ∆ op and a map C → X , weput X ∗ C := X × X × C × in co C art( ∆ op ). We also have a version for Cartesian fibration over ∆ .All of these constructions are compatible via straightening/unstraightening constructions.17inally, by [GH, 4.1.3], C × → ∆ op is a generalized ∞ -operad. If X is a generalized ∞ -operad, X ∗ C is a generalized ∞ -operads as well. Given a map of generalized ∞ -operads C ⊛ → D ⊛ anda functor C → E of ∞ -categories over D , we have the induced map C ⊛ → D ⊛ ∗ E . Let F : C → C at ∞ be a functor. Applying the construction of 1.2 for D = C at ∞ , wehave a functor Y F := Fun( F, C at ∞ ) : C op → d C at ∞ , where d C at ∞ is the ∞ -category of (not nec-essarily small) ∞ -categories (cf. [HTT, 3.0.0.5]). Recall that Y F is the functor sending c ∈ C to Fun( F ( c ) , C at ∞ ). On the other hand, recall from 1.10 that we have the Cartesian fibration C art str ∞ → C at ∞ . This induces the Cartesian fibration F ′ : C art str ∞ × C at ∞ ,F C → C . We define Y ′ F := St( F ′ ) : C op → d C at ∞ (cf. 1.10). The following lemma enables us to identify these twoconstructions: Lemma. —
We have a canonical equivalence Y F ≃ Y ′ F of functors.Proof. For i ∈ { , } , let G i : C op F op −−→ C at op ∞ χ −→ Fun(∆ , d C at ∞ ) { i }→ ∆ −−−−−→ [ C at ∞ , where χ is the map defined in [GHN, A.32]. Informally, χ is the functor sending C ∈ C at ∞ tothe unstraightening equivalence Fun( C op , C at ∞ ) ∼ −→ C art str ( C ). Because unstraightening is anequivalence, we have G ≃ G . By construction, Y F is equivalent to G , thus it remains to showthat G ≃ Y ′ F . It suffices to show the equivalence for C = C at ∞ .For a relative category ( C , W ), we denote by L ( C , W ) the ∞ -localization (cf. [Hn, 1.1.2]).Consider a Cartesian fibration r : ( M , W M ) → ( C , W C ) of relative categories in the sense of [Hn,2.1.1]. We note that this condition is slightly different from the relative Grothendieck fibra-tion compatible with W C in the sense of [GHN, A.28], since [GHN, A.28] requires that all the r -Cartesian morphisms are in W M whereas [Hn] asks only for r -Cartesian morphisms lifting mor-phisms in W C but W C needs to be saturated (cf. [Hn, 1.1.2]). However the construction of [GHN,A.30] can be carried out for Hinich’s one (3) as well. Namely, the functor r corresponds to a nor-mal pseudo-functor St( r ) : C → RelCat (2 , , and yields an ∞ -functor St( r ) ∞ : L ( C , W C ) → C at ∞ by [GHN, A.25]. Since the straightening/unstraightening construction of Lurie is compatiblewith Grothendieck construction, we have the commutative diagram of ∞ -categoriesN M / / (cid:15) (cid:15) Un(St( r ) ∞ ) (cid:15) (cid:15) N C / / L ( C , W C ) . This diagram induces a map L ( M , W M ) → Un(St( r ) ∞ ) over L ( C , W C ). This map is nothing butthe functor θ in [Hn, 2.2.2 (34)], which is proved to be categorical equivalence in [Hn].We let q : X → S et ∆ be the pullback of the Grothendieck fibration E → S et ∆ × ∆ , definedin [GHN, A.31], by the map S et ∆ → S et ∆ × ∆ defined by { } → ∆ . Explicitly, X is thecategory (4) whose fiber over S ∈ S et ∆ is ( S et +∆ ) ◦ /S . Objects of ( S et +∆ ) ◦ /S can be written as A ♮ → S ♯ where A → S is a Cartesian fibration by [HTT, 3.1.4.1]. Given Cartesian fibrations A → S and B → T , a map f from B ♮ → T ♯ to A ♮ → S ♯ in X over T → S in S et ∆ is the map ofmarked simplicial sets B ♮ → A ♮ compatible with T → S . We slightly modify the marking of X from [GHN]: the map f in X is marked if B → A and T → S are categorical equivalences. Note (3) We can also make use of Hinich’s construction [Hn, 2.2.2] instead of [GHN, A.30], which is very similar inspirit. (4)
In the 2nd line of the proof of [GHN, A.31], they say that Y → S ♯ is a fibrant map in S et +∆ . We think this isa typo, and this should be replaced by “a fibrant map in ( S et +∆ ) /S ”. S = T and f is marked, the map B → A is a Cartesian equivalence by [HTT, 3.3.1.5],so the relative category of the fiber is (( S et +∆ ) /S , W S ) where W S is the categorical equivalence.Moreover, given a categorical equivalence X → Y between Cartesian fibrations over S and amap T → S , the base change X × S T → Y × S T is a categorical equivalence by [HTT, 3.3.1.5].Combining with [HTT, 3.3.1.3], all the conditions of [Hn, 2.1.1] are satisfied except for thesaturatedness of the marking.We denote by [ n ] the category whose nerve is ∆ n . Next, we consider the relative category(( S et ∆ ) [1] , W ′ ) where a map ( f, g ) : ( X → Y ) → ( X ′ → Y ′ ) of ( S et ∆ ) [1] is in W ′ preciselyif f, g ∈ W J , where W J is the collection of Joyal equivalent maps of S et ∆ . Then we have amap X → (( S et ∆ ) [1] , W ′ ) of relative categories which induces the map X → L (( S et ∆ ) [1] , W ′ ) ≃ Fun(∆ , C at ∞ ). Here, the equivalence follows by composing the equivalences L (( S et ∆ ) [1] , W ′ ) ≃ L (( S et ∆ , W J ) [1] ) ≃ L (( S et +∆ , W + ) [1] ) ≃ Fun(∆ , C at ∞ ) , where W + denotes the collection of Cartesian equivalences of marked simplicial sets, the middletwo model categories are endowed with projective model structures, the middle equivalencefollows by [Hn, 1.5.1], and the last equivalence follows by [HTT, 4.2.4.4]. This implies that themarking of X is saturated (cf. [Hn, 1.1.2]) because maps in X is marked precisely when itsimage in Fun(∆ , C at ∞ ) are equivalence. Thus, q is a Cartesian fibration of relative categories.Consider the following diagram:Un(St( q ) ∞ ) Un( G ) $ $ ■■■■■■■■■■ L ( X ) / / L ( q ) (cid:15) (cid:15) ∼ o o L ( S et [1]∆ ) (cid:15) (cid:15) ∼ Fun(∆ , C at ∞ ) (cid:15) (cid:15) C art ∞ o o Un( Y ′ F ) y y ttttttttttt L ( S et ∆ ) L ( S et ∆ ) ∼ C at ∞ . In view of the above observation and functoriality, this diagram is commutative, and all thevertical maps are Cartesian fibrations. The map Un(St( q ) ∞ ) → Fun(∆ , C at ∞ ) preserves Carte-sian edges since L ( X ) → L ( S et [1]∆ ) preserves Cartesian edges by the construction in [Hn, 2.2.2].Finally, since Un( G ) and Un( Y ′ F ) are equivalent over each fiber of C at ∞ , we get G ≃ Y ′ F by[HTT, 3.1.3.5] as required. (cid:4) Before constructing the ( ∞ , A -linear categories, we recall the definition ofthe ( ∞ , ∞ -categories Cat ∞ since the construction is a prototype of the construc-tion of LinCat . Recall the functor θ : d C at ∞ → d S pc from 1.4 associating an to ∞ -category C themaximum Kan complex C ≃ . Let ∆ • : ∆ → C at ∞ be the evident functor sending [ n ] to ∆ n . Wehave an equivalence Seq • ( C at ∞ ) ≃ θ ◦ Y ∆ • : ∆ op → d S pc, where Seq • is the functor defined in1.15, by definition. We upgrade this construction by letting Seq • ( Cat ∞ ) := Y ′ ∆ • : ∆ op → d C at ∞ . Proposition ([GR, Ch.10, 2.4.2]) . —
The simplicial ∞ -category Seq • ( Cat ∞ ) defines an ( ∞ , -category such that the underlying ∞ -category is C at ∞ .Proof. The Segal condition holds by 1.11. We need to show the completeness. For this, it sufficesto show that the associated Segal space Seq • ( Cat ∞ ) ≃ is complete. By Lemma 3.2, this Segalspace is naturally equivalent to Seq • ( C at ∞ ), thus complete. (cid:4) Now, we move to the definition of the ( ∞ , A -linear stable categories. Definition. —
Let ∆ + be the augmented simplex category. For a simplicial set S , we defineRM S to be the simplicial subset of S ⊲ × ∆ op+ spanned by all vertex but ( ∞ , [ − ∞ ∈ S ⊲
19s the cone point. For s ∈ S , the vertex ( s, [ n ]) is denoted by (0 s , , . . . , | {z } n +1 ) for n ≥ −
1, and( ∞ , [ n ]) is denoted by (1 , . . . , | {z } n +1 ).If S is an ∞ -category, RM S is an ∞ -category as well. The construction of RM S is functorialwith respect to S .
1. We have a canonical isomorphism RM ∆ ∼ = RM where RM is the cate-gory defined in [GH, 7.1.3] (5) .2. If S is an ∞ -category, then the map RM S → RM is a coCartesian fibration (6) of generalized ∞ -operads.3. Let a be the fiber over [1] ∈ ∆ of the Cartesian fibration RM → ∆ sending ( a , . . . , a n ) to [ a ] . The map of generalized ∞ -operads ( S × RM) ` S × a a → RM S is an equivalence (in Op ns , gen ∞ ). In particular, if T → S is a cofibration of simplicial sets and T → S ′ is a map,the map RM S ` RM T RM S ′ → RM S ` T S ′ is a categorical equivalence.Proof. To see 1, we have the functor RM → ∆ × ∆ op+ by sending ( a , . . . , a n ) ( a i ∈ { , } ) to(0 , [ n − a = 0 and to (1 , [ n ]) if a = 1. It is easy to check that this induces an isomorphismwe need. Via this identification, we see that the notation (0 s , . . . ,
1) is compatible with thatof RM. Note that the map RM → ∆ is a Cartesian fibration. Let us check 2. We haveisomorphisms of simplicial sets( ⋆ ) RM S ∼ = ( S ⊲ × ∆ op+ ) × (∆ × ∆ op+ ) RM ∼ = S ⊲ × ∆ RM . Since S ⊲ × ∆ op+ → (∆ ) ⊲ × ∆ op+ is a coCartesian fibration, and since the map RM S → RMis the base change of this map by the isomorphisms above, it is coCartesian as well. In orderto show that RM S is a generalized ∞ -operad, we only need to check the Segal condition by(non-symmetric analogue of) [HA, 2.1.2.12]. The verification is straightforward.Finally, let us prove 3. We use the theory of categorical patterns [HA, § B]. Let P be thecategorical pattern ( E int , all , { G ∆ [ n ] / → ∆ op } n ) where E int is the set of inert maps and G ∆ [ n ] / isthe simplicial set defined in [GH, 2.3.1]. The associated ∞ -category is Op ns , gen ∞ by [GH, 3.2.9].Since S × a → S × RM is a cofibration in ( S et +∆ ) / P , the pushout is a homotopy pushout. Fora generalized ∞ -operad O ⊛ , let O ⊛ be the marked simplicial set ( O ⊛ , E O ) where E O the set ofinert edges. We have an isomorphism of simplicial sets( ⋆⋆ ) ( S ♭ × RM) a S ♭ × a a ∼ = (cid:0) ( S ♭ × (∆ ) ♯ ) a S ♭ ×{ } ♭ { } ♭ (cid:1) × (∆ ) ♯ RM ∼ = ( S ⋄ ∆ , E ) × (∆ ) ♯ RM , where E is the marking induced by ( S ♭ × (∆ ) ♯ ) ` S ♭ ×{ } ♭ { } ♭ , and the first isomorphism holdssince for any (marked simplicial) sets B, C, D, A ′ over A we have ( B × A A ′ ) ` ( C × A A ′ ) ( D × A A ′ ) ∼ =( B ` C D ) × A A ′ . We wish to apply [HA, B.4.2] to the following diagram of marked simplicialsets (∆ ) ♯ π ←− RM π ′ −→ ∆ op . (5) We think that in [GH, 7.1.1], we should use Simp(∆ ) op instead of Simp(∆ ). (6) In [GH, 7.1.4], it is said that RM is a double ∞ -category, which implies that the map RM → ∆ op is acoCartesian fibration. Unlike BM, which is indeed a double ∞ -category, we think that RM is not. Indeed, sincethere is no map from (0) ∈ RM, the map [0] → [1] in ∆ op cannot be lifted to a map from (0) ∈ RM. However,only the fact that RM is a generalized ∞ -operad is used in [GH]. This can be checked as follows (or directcomputation): The conditions (i), (ii) of [GH, 2.2.6] are easy to check. The condition (iii) follows since BM is ageneralized ∞ -operad and the embedding RM → BM is fully faithful and preserves inert edges.
20e consider the categorical pattern Q := (all , all , ∅ ) on ∆ and the categorical pattern P on ∆ op . Note that ( S et +∆ ) / Q is the coCartesian model structure over ∆ by [HA, B.0.28]. Thenall the conditions of [HA, B.4.2] are satisfied: (5), (6), (8) hold since, in our situation, A = ∅ ,(1), (4) hold since RM → ∆ is a Cartesian fibration, (3) holds since we are taking the set ofall 2-simplices, (2), (7) are easy to check. Thus, the functor π ′ ! ◦ π ∗ is a left Quillen functor.In particular, it preserves weak equivalences because any object is cofibrant. By presentations( ⋆ ) and ( ⋆⋆ ), it remains to show that the map ( S ⋄ ∆ , E ) → ( S ⊲ , E ′ ), where E ′ is the union ofdegenerate edges and the edges lying over the unique non-degenerate edge of ∆ , is a coCartesianequivalence. By a similar argument to [HTT, 4.2.1.2], we are reduced to checking the equivalencein the cases where S = ∆ , ∆ . For S = ∆ , it is in fact an isomorphism, and for S = ∆ , wecan construct a simplicial homotopy. Since ( S et +∆ ) / ∆ is a simplicial model category by [HTT,3.1.4.4], simplicially homotopic objects are weakly equivalent, and the equivalence follows. Thesecond claim of 3 readily follows from the first one. (cid:4) Let C be an ∞ -category.1. We define RMod C to be the full subcategory ofFun(∆ , co C art str (RM)) × { } , co C art str (RM) { RM C → RM } spanned by (homotopy commutative) diagram of ∞ -categories M ⊛ r / / (cid:15) (cid:15) RM C (cid:15) (cid:15) RM RMsuch that r is a base preserving (cf. Definition 1.13) coCartesian fibration of generalized ∞ -operads.2. Let ρ : ∆ ∆ • −−→ C at ∞ → co C art str (RM), where the second functor sends C to RM C → RM.Consider the following diagram RMod ∆ (cid:31) (cid:127) / / X / / (cid:15) (cid:15) (cid:3) Fun(∆ , co C art str (RM)) (cid:15) (cid:15) ∆ ρ / / co C art str (RM) . We define RMod ∆ to be the full subcategory of X spanned by objects in RMod ∆ n over[ n ] ∈ ∆ .3. We put alg : RMod ∆ ∆ { } → ∆ −−−−−−→ co C art str (RM) → co C art str ( a ), where the second functoris induced by the base change by the inclusion a → RM.Let s ∈ S be an object, and i s : { s } ֒ → S be the canonical map. For M ⊛ ∈ RMod S theinduced map M ⊛ × RM S RM s → RM s ≃ RM is a pseudo-enriched ∞ -category in the sense of[GH, 7.2.5] (cf. [GH, 7.2.8]). For a coCartesian fibration M ⊛ → RM of generalized ∞ -operads,we sometimes denote M ⊛ (0 , by M , and call it the underlying ∞ -category . Let A := M ⊛ × RM a .For an object X of M ⊛ (0 , ,..., over [ n + 1] ∈ ∆ op , we have an equivalence M ⊛ (0 , ..., ≃ M × A × n .With this identification, we can write X = ( M , A , . . . , A n ). This object is often denoted by M ⊠ A ⊠ · · · ⊠ A n . 21 emark. —
1. The back-prime ( − ) is put to indicate that the object is Cartesian over ∆ . When we take D of §
2, we erase the back-prime to indicate that it is a coCartesianfibration.2. The reason we employed right module rather than left module is the same as [GH, 7.2.13].However, in our application, we restrict our attention to modules over E ∞ -ring, in whichcase the ∞ -category of right and left modules can be identified (cf. [L2, D.1.2.5]).
1. Let C be an ∞ -category, and consider the diagram ( ⋆ ) M ⊛ p ❋❋❋❋❋❋❋❋❋ r / / N ⊛ q { { ①①①①①①①①① RM C s / / RM where p and q are coCartesian fibrations of generalized ∞ -operads. Furthermore, assumethat for each x ∈ C , the pullback diagram is in co C art str (RM) . Then r sends ( s ◦ p ) -coCartesian edge to ( s ◦ q ) -coCartesian edge.2. The map α : RMod ∆ → ∆ is a Cartesian fibration, and satisfies the Segal condition.Moreover, the map alg sends an α -Cartesian edge to an equivalent edge.Proof. First note that s is a coCartesian fibration by Lemma 3.5. Let e be an ( s ◦ p )-coCartesianedge. We wish to show that r ( e ) is an ( s ◦ q )-coCartesian edge. Note that we are allowed toreplace e by an edge equivalent to it, since being a coCartesian edge is preserved by equivalence.Since s ◦ q is a coCartesian fibration, it suffices to show that r ( e ) is a locally ( s ◦ q )-coCartesianedge by [HTT, 2.4.2.8]. Since e is an ( s ◦ p )-coCartesian edge, p ( e ) is an s -coCartesian edge.This implies that, by replacing e by its equivalent edge, we may assume that there exists x ∈ C such that p ( e ) sits inside RM x in RM C . Thus, it suffices to show that M ⊛ × RM C RM x → N ⊛ × RM C RM x preserves coCartesian edges over RM x . This follows by assumption.Let us show the second claim. We first show that it is a Cartesian fibration. By [HTT,2.3.2.5] and the fact that a fully faithful inclusion is an inner fibration, the map is an innerfibration. Because any base preserving coCartesian fibration of generalized ∞ -operad is stableby base change of generalized ∞ -operad, we get the claim. By construction, the claim for algfollows as well.We are left to show the Segal condition. Let RMod ∼ C be the subcategory of co C art(RM C )spanned by simplices ∆ n → co C art(RM C ) all of whose vertices M ⊛ → RM C are base preservingcoCartesian fibration of generalized ∞ -operads, and all of whose edges are of the form ( ⋆ )such that the base change to RM x for any x ∈ C is in co C art str (RM x ). Then the evidentmap θ : RMod C → RMod ∼ C is a trivial fibration. Indeed, let D := (∆ × ∆ ) ` ∆ ×{ } {∗} , and D ′ := ∆ × { } ∪ { } × ∆ . Then D ′ ♭ → D ♭ is a (Cartesian) marked anodyne. This implies thatthe map θ ′ : Map ♭ ( D ♭ , C at ♮ ∞ ) → Map ♭ ( D ′ ♭ , C at ♮ ∞ ) is a trivial fibration by [HTT, 3.1.2.3]. Thismap is isomorphic to Fun( D, C at ∞ ) → Fun( D ′ , C at ∞ ). We have the inclusionRMod ∼ C ֒ → Fun(∆ , C at ∞ ) × { } , C at ∞ { RM C } ֒ → Fun( D ′ , C at ∞ )where the second map sends F : X → RM C to X F −→ RM C → RM. Similarly, RMod C can beviewed as a subcategory of Fun( D, C at ∞ ). In view of the first claim, θ is a base change of θ ′ ,thus θ is a trivial fibration as well. 22hus, in order to check the Segal condition for RMod, it suffices to show that the canonicalfunctor RMod ∼ ∆ m → RMod ∼ ∆ { ,...,n } × catRMod ∼ ∆ { n } RMod ∼ ∆ { n,...,m } is a categorical equivalence. Since RMod ∼ S is a subcategory of co C art(RM S ), we only need tocheck the conditions of Lemma 1.7. The Segal map is an equivalence for co C art(RM S ) by Lemma3.5.3 and Lemma 1.11. For the rest, it suffices to show the following assertions:1. An object M ⊛ ∈ co C art(RM ∆ n ) is in RMod ∼ ∆ n if and only if the restriction ι ∗ i M ⊛ belongsto RMod ∼ ∆ { i } for any i . Here, ι i : RM ∆ { i } → RM ∆ n is the canonical functor;2. Given f : M → N in co C art(RM ∆ n ) such that M , N ∈ RMod ∼ ∆ n , f is a map in RMod ∼ ∆ n ifand only if ι ∗ i ( f ) is in RMod ∼ ∆ { i } for any i .The second assertion follows by the definition of RMod ∼ C . Let us show the first assertion. ThecoCartesian fibration p : M ⊛ → RM ∆ n is in RMod ∼ ∆ n if and only if, the map M ⊛ → ∆ op exhibits M ⊛ as a generalized ∞ -operad, and the induced map M ⊛ [0] → RM ∆ n , [0] is an equivalence. Wehave the induced coCartesian fibration p [0] : M ⊛ [0] → RM ∆ n , [0] . This is an equivalence if and onlyif it is so after pulling-back by map RM ∆ { i } , [0] → RM ∆ n , [0] for any i by [HTT, 3.3.1.5]. Thusthe equivalence is equivalent to the equivalence of ( ι ∗ i M ⊛ ) [0] → RM ∆ { i } , [0] for any i . Now, wemay assume that p [0] is an equivalence. In view of (an analogue of) [HA, 2.1.2.12], the map M ⊛ → ∆ op is a generalized ∞ -operad if and only if the map π below induced by inert maps M ⊛ [ m ] π −→ M ⊛ { , } × cat M ⊛ { } M ⊛ { , } × cat · · · × cat M ⊛ { m − } M ⊛ { m − ,m } α −→ M ⊛ { , } × catRM ∆ n, { } M ⊛ { , } × cat · · · × catRM ∆ n, { m − } M ⊛ { m − ,m } is an equivalence. Since α is an equivalence, it suffices to show that α ◦ π is an equivalence ifand only it is so after pullback by ι i for any i . Consider the following diagram: M ⊛ [ m ] α / / (cid:15) (cid:15) M ⊛ { , } × catRM ∆ n, { } M ⊛ { , } × cat · · · × catRM ∆ n, { m − } M ⊛ { m − ,m } β (cid:15) (cid:15) RM ∆ n , [ m ] γ / / RM ∆ n , { , } × catRM ∆ n, { } RM ∆ n , { , } × · · · × catRM ∆ n, { m − } RM ∆ n , { m − ,m } Since σ i ! : RM S, [1] → RM S, [0] is a coCartesian fibration for i = 0 ,
1, the fiber product of thetarget of α can be computed by fiber products in the category of simplicial sets. With respectto this model of the fiber product, β is a coCartesian fibration. By direct computation, γ is anisomorphism of simplicial sets. Thus, by [HTT, 3.3.1.5] again, the equivalence of α is equivalentto the equivalence of α over each vertices of RM ∆ n , [ m ] . Thus, M ⊛ is a generalized ∞ -operad ifand only if it is so after pullback by ι i . (cid:4) Let A ⊛ → ∆ op be a monoidal ∞ -category, in other words a coCartesian fibration of ∞ -operads. First, we put RMod C , A := RMod C × catco C art str ( a ) { A } . Since alg : RMod ∆ → co C art str ( a ) × ∆ is a functor in C art str ( ∆ ), we may define RMod ∆ , A := RMod ∆ × catco C art str ( a ) × ∆ ( { A } × ∆ ) . in C art str ( ∆ ). We further put RMod ⊛ := RMod ∆ ∗ (RMod ∆ ) ≃ , RMod ⊛ A := RMod ∆ , A ∗ (RMod ∆ , A ) ≃ . We denote by RMod ⊛ , RMod ⊛ A for the dual coCartesian fibration of RMod ⊛ , RMod ⊛ A .23 emark. — In the definition of RMod C , A , we used the fiber product in C at ∞ , which isdetermined only up to contractible choices. If we need to fix a specific model for the fiber product,we may use RMod S × co C art str ( a ) (co C art str ( a ) A / ) init . Here, C init denotes the full subcategoryspanned by initial objects, which is a contractible Kan complex by [HTT, 1.2.12.9]. The fiberproduct is a fiber product in C at ∞ by [HTT, 2.1.2.2]. In other words, an object of RMod S, A isa pair of an object M ⊛ → RM S in C at RM S ∞ and an equivalence A ⊛ ∼ −→ M ⊛ × RM S a . The map ρ : RMod ⊛ A → ∆ is a complete Segal space, and defines an ( ∞ , -category whose underlying ∞ -category is categorically equivalent to RMod A ∞ A ( C at ∞ ) (cf.[HA, 4.2.2.10] for the notation).Proof. In view of Lemma 3.7, it remains to show the completeness and compute the underlying ∞ -category. We haveRMod ≃ ∆ n ≃ (co C art(RM ∆ n ) bp ) ≃ ≃ (co C art str (RM ∆ n ) bp ) ≃ ≃ Alg RM ∆ n ( C at ∞ ) ≃ . Here co C art − (RM ∆ n ) bp denotes the full subcategory of co C art − (RM ∆ n ) spanned by vertices M ⊛ → RM ∆ n which is base preserving coCartesian fibration of generalized ∞ -operads. On theother hand, we haveAlg RM S ( C at ∞ ) × catAlg( C at ∞ ) { A } ≃ (cid:0) Alg RM ( C at ∞ ) S × catAlg(Cat ∞ ) S Alg( C at ∞ ) (cid:1) × catAlg( C at ∞ ) { A }≃ (cid:0) RMod A ∞ ( C at ∞ ) S × catAlg(Cat ∞ ) S Alg( C at ∞ ) (cid:1) × catAlg( C at ∞ ) { A } (3.9.1) ≃ Fun (cid:0) S, RMod A ∞ A ( C at ∞ ) (cid:1) , where the first equivalence follows by Lemma 3.5.3, and the second by [GH, 7.1.9]. By Lemma3.2, the composition of functors ∆ op ρ ′ −→ C at ∞ κ −→ Spc, where ρ ′ is the functor associated with theCartesian fibration ρ , is equivalent to Seq • (RMod A ∞ A ( C at ∞ )). Thus, the proposition follows. (cid:4) The monoidal ∞ -category A ⊛ is said to be presentable if it comes from an object ofAlg( P r L ). In other words, this is equivalent to saying A is presentable and the tensor product ⊗ : A × A → A preserve small colimits separately in each variable (cf. [HA, 4.8.1.15]). Definition. —
For an ∞ -category C , let L in C at C be the full subcategory of RMod C spannedby coCartesian fibrations p : M ⊛ → RM C which comes from an object of Alg RM C ( P r L ), whichis a subcategory of Alg RM C ( C at ∞ ). We put L in C at C , A := L in C at C × cat C at ∆ op ∞ { A } . Let L in C at ∆ be the full subcategory of RMod ∆ spanned by vertices L in C at ∆ n over [ n ] ∈ ∆ op . We define L in C at ⊛ A := L in C at ∆ ∗ L in C at ≃ A . Remark. —
One may wonder why L in C at C is a full subcategory of RMod C . Indeed, thesubcategory RMod A ∞ A ( P r L ) of RMod A ∞ A ( C at ∞ ) is not full. However, we will use L in C at C toconstruct the ( ∞ , LinCat A , and the underlying ∞ -category of LinCat A does notcoincide with L in C at ∆ , A . Let A be a presentable monoidal ∞ -category. The simplicial ∞ -category L in C at ⊛ A is an ( ∞ , -category whose associated ∞ -category is equivalent to RMod A ∞ A ( P r L ) .Proof. Let us show the Segal condition. For this, it suffices to check the Segal condition for L in C at ∆ . By Lemma 1.7, in view of Proposition 3.9, we only need to show the following claim:An object M ∈ RMod ∆ n belongs to L in C at ∆ n if and only if κ ∗ i M belongs to L in C at ∆ { i,i +1 } forany 0 ≤ i < n . Here κ i : RM ∆ { i,i +1 } → RM ∆ n . The verification is straightforward.24ow, we need to show that it is complete. We have L in C at ≃ ∆ n ≃ Alg RM ∆ n ( P r L ) ≃ . Similarlyto the computation (3.9.1), we haveAlg RM S ( P r L ) × Alg( P r L ) { A } ≃ Fun( S, RMod A ∞ A ( P r L )) . As in Proposition 3.9, use Lemma 3.2 to show that this equivalence induces the equivalencebetween the underlying Segal space of L in C at ⊛ and Seq • (RMod A ∞ A ( P r L )) to conclude. (cid:4) We define
LinCat A to be the ( ∞ , • ( LinCat − op A ) ≃ St( L in C at ⊛ A ), where St denotes the straightening functor, holds. If R is an E -ring, then RMod R can naturally be considered as an object in Alg( P r L ) by [HA, 7.1.2.6]. Inthis case, we denote LinCat
RMod R by LinCat R . The underlying ∞ -category of LinCat A and LinCat R are denoted by L in C at A and L in C at R . We have a conflict of notations for L in C at A with Definition 3.10, but we think that it is clear what it means. Note that, by Proposition 3.11, L in C at R coincides with LinCat St R in [L2, D.1.5.1]. For a 1-morphism F : C → D in LinCat A ,the corresponding monoidal functor of generalized RM-operads is denoted by F ⊛ : C ⊛ → D ⊛ . Remark. —
1. We are taking ( − ) − op in order to have the forgetful functor LinCat A → Cat ∞ . See [GR, Ch.10, 2.4.5]. In [GR, Ch.1, 8.3.1], they used mixture of Cartesian andcoCartesian fibrations to define Seq n ( LinCat A ). We did not employ this approach in orderto avoid too much complications.2. The ( ∞ , Pres appeared in Introduction is by definition
LinCat S , where S is the sphere spectrum. The underlying ∞ -category is P r Lst , the full subcategory of P r L spanned by stable presentable ∞ -categories, by [HA, 4.8.2.18]. The following lemma is useful criterion to detect adjoint maps in
LinCat R . Lemma. —
Let F : C → D be a -morphism in LinCat A . Then the following is equivalent:1. The functor F admits a left (resp. right) adjoint in LinCat A the sense of [GR, Ch.12,1.1.3];2. There exists a monoidal functor G ⊛ : D ⊛ → C ⊛ which is left (resp. right) adjoint to F ⊛ relative to RM in the sense of [HA, 7.3.2.2] and G ⊛ (01) commutes with small colimits.Moreover, if R is an E -ring, and A ⊛ = LMod R , then the above two conditions are equivalentto 3. F ⊛ (01) admits a left adjoint (resp. right adjoint which commutes with small colimits).Proof. Let us show the equivalence of 1 and 2. We only show the non-resp claim, since aproof for right adjoints can be obtained simply by replacing left by right. Let us show 1 to 2.Since F admits a left adjoint, there exists a coCartesian fibration M ⊛ G → RM ∆ , a unit map α : D ⊛ × RM RM ∆ → M ⊛ F ◦ G , a counit map M ⊛ G ◦ F → C ⊛ × RM RM ∆ satisfying some conditions.By taking the base change by the canonical map S × RM → RM S in Lemma 3.5, the datayields a pair of adjoint functors relative to RM. Let us show 2 to 1. Combining Lemma 3.5 and(dual version of) Lemma 1.11, we have an equivalence co C art( S × RM) × catco C art( S × a ) co C art( a ) ≃ co C art(RM S ). First, let us construct a functor G : D → C in LinCat A . Since G ⊛ is assumedmonoidal, the left adjoint G ⊛ yields an object in e G in co C art(∆ × RM). Since e G | ∆ × a is aleft adjoint to the equivalence F ⊛ | a , the restriction of e G is equivalence as well. This impliesthat e G induces an object of co C art( S × RM) × catco C art( S × a ) co C art( a ), and the equivalence aboveyields a an object in co C art(RM ∆ ). Since G ⊛ (01) commutes with small colimits, e G yields a 1-morphism D → C in LinCat A . Let us construct a unit map α similarly. Because F ⊛ admits a25eft adjoint relative to RM, we have a unit map e α : D ⊛ × ∆ → M ⊛ F ◦ G × RM ∆1 (RM × ∆ ) over∆ × RM The restriction of e α to ∆ × a is a unit map of the adjunction of F ⊛ × RM a . Since F ⊛ × RM a is the identity, e α × ∆ × RM (∆ × a ) is an equivalence. Consequently, e α yields a map inco C art( S × RM) × co C art( S × a ) co C art( a ), which induces a desired unit map α using the equivalenceabove. Similarly, we construct a counit map. In order to show that these maps actually givesan adjoint pair ( G, F ), we need to show that certain compositions of maps given by unit andcounit maps are equivalences. The relative adjunction on RM yields corresponding relations inco C art( S × RM), so in order to show the relations in co C art(RM S ), we use the above equivalenceagain.Let us show the equivalence between 2 and 3. The 2 to 3 direction is obvious, so we willshow the other direction. First, consider the case where F ⊛ (01) admits a right adjoint G ⊛ (01) . Inthis case, by [HA, 7.3.2.9], we have a right adjoint G ⊛ relative to RM. This functor is observedto be monoidal when G ⊛ (01) commutes with small colimits in [L2, D.1.5.3], thus the claim follows.Next, assume that F ⊛ (01) admits a left adjoint, denoted by G . We wish to check the conditionsof [HA, 7.3.2.11]. Similarly to the proof of [HA, 7.3.2.7], the condition (1) follows from ourassumption, we only need to check (2). For this, it suffices to show that the induced map φ M,C : G ( M ) ⊗ C C → G ( M ⊗ D C ) for any C ∈ A ≃ LMod R and M ∈ D is an equivalence. Since G admits a right adjoint, G commutes with small colimits. By [HA, 7.2.4.2], C can be writtenas a small filtered colimit of perfect R -modules, and it suffices to show the equivalence when C is a perfect R -module. Thus, it suffices to show the following two assertions: • The map φ M,R n [ m ] is an equivalence for any integers n ≥ m ; • If φ M,C is an equivalence, then φ M,C ′ is an equivalence for any retract C ′ of C .Since the formation of φ commutes with pushouts, we have φ M,C [ m ] ≃ φ M,C [ m ], which implies thefirst assertion. For the second assertion, let I : ∆ → A be a diagram such that I (0) = I (2) = C ′ , I (1) = C and I (∆ { , } ) = id. Then this induces a diagram J := cof( φ M,I ) : ∆ → D such that J (0) = J (2) = cof( φ M,C ′ ), J (1) = cof( φ M,C ) and J (∆ { , } ) = id. Since φ M,C is assumed to bean equivalence, cof( φ M,C ) ≃
0. Since initial objects can be detected in the homotopy category,a retract of 0 is 0. Thus φ M,C ′ ≃ (cid:4) Let R be an E ∞ -ring. Then L in C at R is equipped with canonical symmetric monoidalstructure by [L2, D.2.3.3]. Before concluding this section, we make some construction in termsof CAlg( L in C at R ), which is used to construct a motivic theory associated to an algebra objectin § K be a collection of small simplicial sets. Let Mon prAssoc ( C at ∞ ) ⊗ be the full subcategory ofMon K Assoc ( C at ∞ ) ⊗ (cf. [HA, 4.8.5.14]) spanned by vertices ( C ⊗ , . . . , C ⊗ n ) such that C i is presentablefor any i . Arguing similarly to the proof of [HA, 4.8.5.16 (1)], Mon prAssoc ( C at ∞ ) ⊗ is a symmetric ∞ -operad. In [HA, 4.8.5.10], the full subcategories P r Alg , P r Mod of C at Alg ∞ ( K ) and C at Mod ∞ ( K ) areintroduced. Informally, P r Alg is the ∞ -category of pairs ( C ⊗ , A ) where C ⊗ ∈ Mon prAssoc ( C at ∞ )and A ∈ Alg( C ), and P r Mod is the ∞ -category of pairs ( C ⊗ , M ) where M is an ∞ -categoryleft-tensored over C ⊗ with some suitable presentability. We can promote these ∞ -categoriesto symmetric monoidal ∞ -categories, similarly to [HA, 4.8.5.14] as follows. The ∞ -category P r Alg , ⊗ is simply Mon prAssoc ( C at ∞ ) ⊗ × Mon K Assoc ( C at ∞ ) ⊗ C at Alg ∞ ( K ) ⊗ . We define P r Mod , ⊗ to be the full subcategory of C at Mon ∞ ( K ) ⊗ spanned by the objects of the form(( C ⊗ , M ) , . . . , ( C ⊗ n , M n )) such that C i and M i are presentable for any i . We have the following26iagram P r Alg , ⊗ Θ / / φ ( ( ◗◗◗◗◗◗◗◗◗◗◗◗◗◗ P r Mod , ⊗ ψ v v ❧❧❧❧❧❧❧❧❧❧❧❧❧❧ Mon prAssoc ( C at ∞ ) ⊗ . The maps φ is a coCartesian fibration by [HA, 4.8.5.16]. Now, the symmetric monoidal structureof P r L is closed by [HA, 4.8.1.18]. This implies that the tensor product of P r L commutes withall small colimits separately in each variable. Thus, the argument of [HA, 4.8.5.1] can be appliedto show that the functor P r Mod → Mon prAssoc ( C at ∞ ) is a coCartesian fibration. Similarly to theproof of [HA, 4.8.5.16], the map ψ is also a coCartesian fibration. Since Mon K Assoc ( C at ∞ ) ⊗ canbe identified with Alg( C at ∞ ( K )) ⊗ (cf. proof of [HA, 4.8.5.16]), we can identify Mon prAssoc ( C at ∞ ) ⊗ with Alg( P r L ) ⊗ .The monoidal category of spectra Sp ⊗ =: is an initial object of the ∞ -category CAlg( P r L ).Thus, we have a map → Mod R by [HA, 3.2.1.9]. By [HA, 3.4.3.4], we have a map of symmetric ∞ -operads L in C at ⊗ R := Mod Mod R ( P r L ) ⊗ → Mod ( P r L ) ⊗ ≃ P r L , ⊗ , where the last equivalenceis by [HA, 3.4.2.1]. On the other hand, we have the bifunctor (cf. [HA, 2.2.5.3]) of symmetric ∞ -operads Comm ⊗ × Assoc ⊗ → Comm ⊗ (cf. [HA, 3.2.4.4]). For any symmetric ∞ -operad C ⊗ ,this induces the map CAlg( C ) → Alg
Comm ⊗ Assoc ( C ) ≃ CAlg(Alg( C )) , where the last map follows by definition. Thus, this induces the functorCAlg( L in C at R ) → CAlg( P r L ) → CAlg(Alg( P r L )) . Taking the adjoint, we have Comm ⊗ × CAlg( L in C at R ) → Alg( P r L ) ⊗ ≃ Mon prAssoc ( C at ∞ ) ⊗ . Wetake the base change of P r Alg , ⊗ and P r Mod , ⊗ by this map, which we denote by P r Alg , ⊗ L and P r Mod , ⊗ L .Let us construct a functor P r Mod , ⊗ L → L in C at ⊗ R over Comm ⊗ . Informally, this functor sends( C ⊗ , M ) to M considered as an object of L in C at R via the structural map Mod R → C ⊗ . Let C be an ∞ -category with initial object ∅ ∈ C . The map C ∅ / → C is a trivial fibration by [HTT,4.2.1.6], we way take a quasi-inverse C → C ∅ / . This induces a map I ∅ : ∆ × C → ∆ ⋄ C → C sending (0 , c ) to ∅ and (1 , c ) to c . The object Mod R in CAlg( L in C at R ) is an initial object (cf.[HA, 3.2.1.9, 3.4.4.7]). Thus, applying the above observation, we have the diagram∆ { } × P r Mod , ⊗ (cid:127) _ (cid:15) (cid:15) P r Mod , ⊗ ψ L (cid:15) (cid:15) ∆ × P r Mod , ⊗ I / / e ρ ❤❤❤❤❤❤❤❤❤❤ Comm ⊗ × CAlg( L in C at R )where I is the map induced by I Mod R . We wish to take the right Kan extension of the abovediagram. For the existence, in view of [HTT, 4.3.2.15], it suffices to check the following:Let F : (( C ⊗ , M ) , . . . , ( C ⊗ m , M m )) → (( D ⊗ , N ) , . . . , ( D ⊗ m , N m )) be the map coveringthe identity h m i → h m i in F in ∗ such that the map M i → N i is an equivalence forany i . Then F is a ψ L -Cartesian edge.Since ψ L is a coCartesian fibration since ψ is, it suffices to check that the edge is locally ψ L -Cartesian by [HTT, 5.2.2.4]. This is, indeed, locally Cartesian since it is inner fibration and[HA, 4.2.3.2]. By using this right Kan extension, we have ρ : P r Mod , ⊗ = ∆ { } × P r Mod , ⊗ e ρ −→ ψ − L (Comm ⊗ × { Mod R } ) ≃ L in C at ⊗ R . ∞ -operads. Summing up, we have the following diagram P r Alg , ⊗ L Θ / / ) ) ❙❙❙❙❙❙❙❙❙❙❙❙❙❙❙❙ P r Mod , ⊗ L (cid:15) (cid:15) / / L in C at ⊗ R × CAlg( L in C at R ) s s ❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤ Comm ⊗ × CAlg( L in C at R ) f / / CAlg( L in C at R ) . Let C ⊗ → O ⊗ × S be a coCartesian S -family of O -monoidal ∞ -categories. Let p : O ⊗ × S → S be the projection. We denote by p ⊗∗ ( C ⊗ ) the full subcategory of p ∗ ( C ⊗ ) spanned by verticescorresponding to Alg O ( C s ). Since p ⊗∗ ( C ⊗ ) contains all the coCartesian edges connecting verticesin it, the map p ⊗∗ ( C ⊗ ) → S is a coCartesian fibration as well. Let C ⊗ → C ′⊗ be a map over O ⊗ × S of S -family of O -monoidal ∞ -categories such that for each vertex s ∈ S , the inducedmap C ⊗ s → C ′⊗ s is a map of symmetric ∞ -operads. Then we have p ⊗∗ ( C ⊗ ) → p ⊗∗ ( C ′⊗ ). Thisobservation being applied to the above diagram yields a mapΞ : P r CAlg L := p ⊗∗ ( P r Alg , ⊗ L ) → p ⊗∗ ( L in C at ⊗ R × CAlg( L in C at R )) ≃ CAlg( L in C at R ) × CAlg( L in C at R )over CAlg( L in C at R ), where the last ∞ -category is considered over CAlg( L in C at R ) by the secondprojection. Note that the fiber of P r CAlg L over C ⊗ ∈ CAlg( L in C at R ) is CAlg(Alg( C )). Bythe proof of [HA, 4.8.5.21], Comm ⊗ ⊗ Assoc ⊗ ≃ Comm ⊗ , and in particular, CAlg(Alg( C )) ≃ CAlg( C ). Thus, the fiber Ξ C of Ξ over C ⊗ sends A ∈ CAlg( C ) to (Mod A ( C ) , C ⊗ ). What we havedone so far can be summarized as follows: Lemma. —
Let R be an E ∞ -ring. Then we have the following diagram of coCartesian fibra-tions P r CAlg L φ CAlg ' ' ❖❖❖❖❖❖❖❖❖❖❖ Ξ / / CAlg( L in C at R ) × CAlg( L in C at R ) pr t t ❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤ CAlg( L in C at R ) . Here, the fiber of φ CAlg over C ⊗ ∈ CAlg( L in C at R ) is CAlg( C ) , and Ξ sends A ∈ CAlg( C ) to (Mod A ( C ) , C ⊗ ) .
4. Construction of the bivariant ( ∞ , -functor Assume we are given an ( ∞ , F : C → LinCat A . For each X ∈ C , assume we aregiven an object I X ∈ F ( X ). We do not ask any compatibilities of I X with F . For example, if F ( X ) is a symmetric monoidal, then I X may be taken as a unit object. Then we may considerthe assignment to each 1-morphism f : X → Y in C the object Mor F ( Y ) ( F ( f )( I X ) , I Y ), whereMor M is the morphism object of M ∈ L in C at A . It is natural to ask the functoriality of thisconstruction. This will be encoded in the non-unital right-lax functor C B A ⊛ , which willbe constructed in this section. The construction is a “family version” of [HA, 4.7.1] and [GH,7.3, 7.4]. Let us construct a universal A -module over RMod ∆ , A . As in Remark 3.8, we fix a modelof RMod ∆ , A . We also fix a model for RMod ∆ n , A so that RMod ∆ n , A = RMod ∆ , A × ∆ { [ n ] } . Bydefinition, we have the functor RMod ∆ , A → Fun(∆ , co C art str (RM)) ≃ Fun(∆ , Fun(RM , C at ∞ ))28his induces the functor RM × RMod A → Fun(∆ , C at ∞ ). By taking the unstraightening, wehave the commutative diagram M univ , ⊛ f ' ' PPPPPPPPPPPP h / / RM univ g w w ♥♥♥♥♥♥♥♥♥♥♥♥ RM × RMod A such that f , g are coCartesian fibrations and h preserves coCartesian edges. Furthermore, if wereplace M univ , ⊛ by equivalence, we may assume that h is a categorical fibration. By construction,it is equipped with an equivalence A ⊛ ∼ −→ M univ , ⊛ × RM a . We put M univ , ⊛ [ n ] := M univ , ⊛ × ∆ { [ n ] } .We often abbreviate RMod ∆ , A simply by RMod.By construction, we have the equivalence RM univ[ n ] ≃ RM ∆ n × RMod ∆ n . This induces a mapRM univ[ n ] → RM ∆ n . Note that since all the equivalence in RM ∆ n are degenerate edges, in otherwords, RM ∆ n is gaunt , any functor C → RM ∆ n from an ∞ -category is a categorical fibration by[HTT, 2.3.1.5, 2.4.6.5]. Let φ : [ m ] → [ n ] be a map in ∆ . We have the following pullback diagram in C at ∞ : M univ , ⊛ [ n ] × RM ∆ n RM ∆ m / / (cid:15) (cid:15) M univ , ⊛ [ m ] (cid:15) (cid:15) RMod ∆ n φ ∗ / / RMod ∆ m . Proof.
Let X be the subcategory of F ∈ Fun(∆ × ∆ , C at ∞ ) spanned by the vertices of theform N ⊛ (cid:15) (cid:15) / / N ⊛ (cid:15) (cid:15) RM ∆ m φ ∗ / / RM ∆ n , F (0 , / / (cid:15) (cid:15) F (0 , (cid:15) (cid:15) F (1 , / / F (1 , . such that N ⊛ → RM ∆ m ∈ RMod ∼ ∆ m , using the notation of the proof of Lemma 3.7, and N ⊛ → RM ∆ n ∈ RMod ∼ ∆ n , and the square is a pullback square. Maps are those which induce mapsin RMod ∼ ∆ m and RMod ∼ ∆ n . We have the functor X → RMod ∼ ∆ n . This is a trivial fibration by[HTT, 4.3.2.15]. Let ι i : ∆ × { i } → ∆ × ∆ . We have the following commutative diagram∆ × RMod ∼ ∆ n . . ∆ × X (cid:31) (cid:127) ι × id / / id × ι ∗ ∼ o o (∆ × ∆ ) × X τ (cid:15) (cid:15) ∆ × X ? _ ι × id o o id × ι ∗ / / ∆ × RMod ∼ ∆ m p p C at ∞ . Unstraightening τ , we get the pullback diagram of the form N univ , ⊛ / / α (cid:15) (cid:15) N univ , ⊛ β (cid:15) (cid:15) RM ∆ m × X / / RM ∆ n × X . The commutative diagram above identifies α with M univ , ⊛ [ m ] × RMod ∆ m X → RM ∆ m × X and β with M univ , ⊛ [ n ] → RM ∆ n × RMod ∆ n , and the lemma follows. (cid:4) .3. Recall that Tw op ∆ is the category consisting of maps [ k ] → [ n ] in ∆ , and a morphism([ k ] → [ n ]) → ([ k ′ ] → [ n ′ ]) is a commutative diagram[ k ] (cid:15) (cid:15) [ k ′ ] o o (cid:15) (cid:15) [ n ] / / [ n ′ ]in ∆ . We have the functor Φ × Θ : Tw op ∆ → ∆ op × ∆ sending ([ k ] → [ n ]) to ([ k ] , [ n ]). Thefunctor Θ is a coCartesian fibration. We denote by Tw op[ n ] ∆ the fiber Θ − ([ n ]). We defineTw op ∆ ′ by the full subcategory of Tw op ∆ spanned by the maps σ i : [0] → [ n ]. The categoryTw op ∆ is an analogue of Po op in [HA, 4.7.1], but larger since Lurie considers inert maps whereaswe consider all the maps in ∆ . Recall the map χ : ∆ × ∆ op → RM in [GH, 7.3.6], whose adjoint ∆ → Fun(∆ , RM) is defined by sending [ n ] ∈ ∆ op to (0 , , . . . , | {z } n +1 ) → (1 , . . . , | {z } n +1 ) ∈ Fun(∆ , RM).We have the commutative diagram of functorsTw op ∆ ′{ } (cid:15) (cid:15) Θ ′ x x π ′ ' ' ∆ ∆ × Tw op ∆ Θ ◦ pr =:Θ o o (cid:0) χ ◦ (id × Φ) , Θ ◦ pr (cid:1) =: π / / RM × ∆ Tw op ∆ { }× id =: i O O Θ f f Via the structural map RM × RMod A → RM × ∆ , the map M univ , ⊛ → RM univ can be consideredover RM × ∆ . Thus, the diagram induces a diagram as follows:(4.3.1) Θ ′∗ π ′∗ ( M univ , ⊛ ) (cid:15) (cid:15) Θ ∗ π ∗ ( M univ , ⊛ ) ι o o (cid:15) (cid:15) α / / Θ ∗ ( π ◦ i ) ∗ ( M univ , ⊛ )Θ ′∗ π ′∗ (RM univ ) Θ ∗ π ∗ (RM univ ) o o Let us analyze Θ ∗ ( π ◦ i ) ∗ M univ , ⊛ first. By definition, the composition π ◦ i is equal to thecomposition Tw op ∆ Φ × Θ −−−→ ∆ op × ∆ a × id −−−→ RM × ∆ . We also have the canonical equivalence( a × id) ∗ M univ , ⊛ ≃ A ⊛ × RMod.
Definition. —
A vertex of Θ ∗ ( π ◦ i ) ∗ M univ , ⊛ corresponds to a map f : Tw op[ n ] ∆ → A ⊛ × RModover Tw op ∆ → ∆ op × ∆ . We consider the full subcategory B of Θ ∗ ( π ◦ i ) ∗ M univ , ⊛ spanned bythe vertices satisfying1. The composition pr ◦ f : Tw op[ n ] ∆ → A ⊛ sends an edge of Tw op[ n ] to a coCartesian edge of A ⊛ over ∆ op ;2. The composition pr ◦ f : Tw op[ n ] ∆ → RMod sends any edge of Tw op[ n ] ∆ to an equivalentedge in RMod.
Lemma. —
We have the canonical equivalence ( A ⊛ ) ∨ × ∆ RMod ∼ −→ B over ∆ .Proof. Recall that Θ ∗ is a right Quillen functor. Thus, we have the mapΘ ∗ ( π ◦ i ) ∗ M univ , ⊛ ∼ = Θ ∗ (cid:0) Φ ∗ A ⊛ × Tw op ∆ Θ ∗ RMod (cid:1) ∼ = Θ ∗ Φ ∗ A ⊛ × ∆ Θ ∗ Θ ∗ RMod . A ⊛ ) ∨ × ∆ RMod → Θ ∗ ( π ◦ i ) ∗ M univ , ⊛ . This induces thefunctor in the statement of the lemma. We must show that it is an equivalence. Since thefunctor preserves Cartesian edges over ∆ , it suffices to show the equivalence for each [ n ] ∈ ∆ by [HTT, 3.3.1.5]. For a simplicial set K and an ∞ -category C , let Fun( K, C ) equiv be the fullsubcategory of Fun( K, C ) spanned by functors sending any edge of K to an equivalent edge in C . We only need to show that the constant functor c : RMod ∆ n → Fun(Tw op[ n ] ∆ , RMod ∆ n ) equiv is a categorical equivalence. Consider the following left Kan extension diagram: { [ n ] → [ n ] } / / i (cid:15) (cid:15) RMod ∆ n (cid:15) (cid:15) Tw op[ n ] ∆ / / qqqqqq {∗} . Invoking [HTT, 4.3.2.15], the restriction by i is a trivial fibration, which gives a quasi-inverse to c . (cid:4) Let α : Tw op[ n ] ∆ ′ → RM ∆ n be the functor defined by sending φ : [0] → [ n ]to (0 φ (0) , ′∗ π ′∗ (RM univ ) corresponds to a map f : Tw op[ n ] ∆ ′ → RM univ(0 , , [ n ] ∼ = (RM ∆ n ) (0 , × RMod ∆ n , where (RM ∆ n ) (0 , := RM ∆ n × RM { (0 , } ∼ = ∆ n , for some n . We consider thefull subcategory Pre S tr of Θ ′∗ π ′∗ (RM univ ) spanned by the vertices satisfying the followingconditions:(a) The composition pr ◦ f : Tw op[ n ] ∆ ′ → (RM ∆ n ) (0 , is equal to α ;(b) The composition pr ◦ f : Tw op[ n ] ∆ ′ → RMod ∆ n is constant.2. We put S tr := Θ ′∗ π ′∗ ( M univ , ⊛ ) × Θ ′∗ π ′∗ (RM univ ) Pre S tr. Remark. —
1. We have the following diagram: S tr [ n ] / / ❴❴❴❴❴❴❴❴❴❴❴❴❴❴❴❴❴❴❴❴ (cid:15) (cid:15) ✤✤✤✤✤✤✤✤ ( ( PPPPPPP
RMod ∆ n (cid:15) (cid:15) Fun(Tw op[ n ] ∆ ′ , M univ , ⊛ [ n ] ) b / / (cid:15) (cid:15) Fun(Tw op[ n ] ∆ ′ , RMod ∆ n ) {∗} a / / Fun(Tw op[ n ] ∆ ′ , RM ∆ n ) . Here the dashed functors exhibits S tr as a limit of the diagram in S et ∆ . The map a isa categorical fibration by [HTT, 2.4.6.5] since RM ∆ n is gaunt, and b is also a categoricalfibration since M univ , ⊛ → RM × RMod A is a coCartesian fibration. This implies that thelimit in S et ∆ is actually a limit in C at ∞ . In particular, the category S tr does not depend onthe choice of a “model” of M univ , ⊛ up to equivalences, and the functor S tr [ n ] → RMod ∆ n is a categorical fibration.2. We have the following diagram S tr / / ❴❴❴❴❴❴❴❴❴❴❴❴❴❴❴❴❴❴❴❴❴ (cid:127) _ (cid:15) (cid:15) RMod (cid:15) (cid:15) Θ ′∗ π ′∗ (RM univ ) c / / Θ ′∗ π ′∗ (RM × RMod) ∼ = / / Θ ′∗ Θ ′∗ ( RMod) .
31y [HTT, B.4.5], the map c is a categorical fibration, and the fiber over [ n ] ∈ ∆ coincideswith b above. By definition of S tr, we have the dashed arrows making the diagramcommutative whose fiber over [ n ] is map of 1.3. Informally, an object of S tr consists of M ⊛ → RM ∆ n in RMod ∆ n and objects M , . . . , M n ∈ M (0 , such that M i is an object over (0 i , ∈ RM ∆ n . A vertex of Θ ∗ π ∗ ( M univ , ⊛ ) consists of maps g : ∆ × Tw op[ n ] ∆ → M univ , ⊛ × RMod
RMod ∆ n over RM for some n . We define S tr en , + to be the full subcategory of Θ ∗ π ∗ ( M univ , ⊛ )spanned by the vertices satisfying the following conditions, which are analogous to the conditionsof enriched strings (cf. [GH, 7.3.7]):1. The vertex sits over S tr;2. Given a map φ : ([ k ] → [ n ]) → ([ l ] → [ n ]) in Tw op[ n ] ∆ such that φ (0) = 0, the edge g (0 , φ ) in M univ , ⊛ is a coCartesian edge with respect to the coCartesian fibration s : M univ , ⊛ → RM;3. The map g : ∆ × Tw op[ n ] ∆ → M univ , ⊛ is a s -left Kan extension of g | { }× Tw op[ n ] ;4. Given a map φ : ([ k ] → [ n ]) → ([ l ] → [ n ]) in Tw op[ n ] ∆ , the edge g (1 , φ ) in M univ , ⊛ is acoCartesian edge with respect to the coCartesian fibration s . Remark. —
1. Given a map φ in RM, let Fun φ (∆ , M univ , ⊛ ) coCart be the full subcate-gory of Fun(∆ , M univ , ⊛ ) spanned by coCartesian edges over φ . The canonical inclusionFun φ (∆ , M univ , ⊛ ) coCart → Fun(∆ , M univ , ⊛ ) is a categorical fibration because RM is gauntand coCartesian edges are preserved by equivalence. The conditions except for 1 can berephrased as certain edges in M ⊛ are coCartesian edges over some specified maps φ in RM.Thus, we have the pullback diagrams in S et ∆ : X / / (cid:15) (cid:15) Fun(∆ × Tw op[ n ] ∆ , M univ , ⊛ [ n ] ) a (cid:15) (cid:15) S tr [ n ] / / Fun(Tw op[ n ] ∆ ′ , M univ , ⊛ [ n ] ) , S tr en , +[ n ] / / (cid:15) (cid:15) Q Fun φ (∆ , M univ , ⊛ ) coCart b (cid:15) (cid:15) X / / Q Fun(∆ , M univ , ⊛ ) . The functor a is a categorical fibration by [HTT, 2.2.5.4], and b is also a categorical fibrationby observation above. This implies that the functor S tr en , +[ n ] → S tr [ n ] is a categoricalfibration, and S tr en , + does not depend on the model of M univ , ⊛ .2. Let us have a closer look at objects of S tr en , + . Vertices of S tr en , + correspond to functors g : ∆ × Tw op[ n ] ∆ → M univ , ⊛ × RMod
RMod ∆ n for some n . Because g is over S tr, the mapTw op[ n ] ∆ ′ { } −−→ ∆ × Tw op[ n ] ∆ g −→ M univ , ⊛ × RMod
RMod ∆ n pr −−→ RMod ∆ n is constant, and determines a coCartesian fibration of generalized ∞ -operads M ⊛ → RM ∆ n . Let φ k : [ n − k ] → [ n ] be an inert function considered as in Tw op[ n ] ∆ such that φ (0) = k . Then we can write g (0 , φ k ) ≃ M k ⊠ A k +1 ⊠ A k +2 ⊠ · · · ⊠ A n with M k ∈ M ⊛ (0 , , A i ∈ A . For a careful reader, we note that g (0 , φ k ) may not lie in M ⊛ , but a generalized ∞ -operad equivalent to M ⊛ . Condition 3, 4 implies that A i are the same even if we change k . 32ow, let φ : [ m ] → [ n ] a function considered as an object of Tw op[ n ] ∆ . Put a φ = 0, and for j >
0, define a φj inductively as follows: a φj +1 is the minimum number a φj < k ≤ m suchthat φ ( a φj ) = φ ( k ). This can be depicted as follows:0 = a φ ' ' PPPPPP . . . (cid:15) (cid:15) a φ − w w ♥♥♥♥♥♥ a φ & & ▼▼▼▼▼▼ . . . (cid:15) (cid:15) a φ − w w ♥♥♥♥♥♥ a φ & & ▼▼▼▼▼▼ . . . (cid:15) (cid:15) φ ( a φ ) φ ( a φ ) φ ( a φ )Then condition 2 implies that g (0 , φ ) = M φ (0) ⊠ B ⊠ B · · · ⊠ B m where B a φj = A φ ( a φj − )+1 ⊗· · · ⊗ A φ ( a φj ) for j >
0, and B i = otherwise.As a summary, a vertex consists of a sequence M ⊠ A ⊠ A ⊠ · · · ⊠ A n → M ⊠ A ⊠ · · · ⊠ A n → · · · → M n with lots of “redundant information” determined from the above sequence in an essentiallyunique manner. Let p : M univ , ⊛ → RM × ∆ , q : S tr → ∆ , r : S tr en , + → ∆ , and α : RMod → ∆ be the canonical maps. We recall maps f , g , h from 4.1.1. Let x be a vertex of M univ , ⊛ . Assume we are given a map φ : [ m ] → [ n ] in ∆ , and let e φ be a Cartesian edge in RM × RMod over ∆ with endpoint f ( x ) . Assume that e φ has a g -coCartesian lifting e ′ φ in RM univ with endpoint h ( x ) . Then there exists a h -Cartesianedge y → x in M univ , ⊛ which lifts e ′ φ .2. The maps q , r are Cartesian fibrations.3. The map S tr en , + → S tr sends q -Cartesian edges to r -Cartesian edges, the map S tr → RMod from Remark 4.5 sends q -Cartesian edges to α -Cartesian edges, and S tr en , + → RMod sends r -Cartesian edges to α -Cartesian edges.4. The maps S tr [ n ] → RMod ∆ n , S tr en , +[ n ] → RMod ∆ n are coCartesian fibrations, and themap S tr en , +[ n ] → S tr [ n ] preserves coCartesian edges.Proof. Let us show 1. Since f , g , h are categorical fibrations, we may replace an edge we wishto lift by an edge equivalent to it. In view of Lemma 1.5 applied to the diagram (4.1), we mayreplace the diagram by the diagram of fibers over e φ . Write f ( x ) = ( X, M ⊛ ) where X ∈ RMand M ⊛ → RM ∆ n be in RMod ∆ n . By replacing e φ by an edge equivalent to it, we may assumethat e φ is a morphism of the form (id X , ι ) : ( X, M ⊛ × RM ∆ n RM ∆ m ) → ( X, M ⊛ ), where ι is thecanonical functor. We have the coCartesian fibration M ⊛ X → (RM ∆ n ) X , and let i ∈ ∆ n bethe vertex over which x is lying. The fiber (RM ∆ n ) X is equivalent to ∆ n or {∗} depending onwhether X begins with 0 or 1. We have the g -coCartesian lifting e ′ φ if and only if there exists j ∈ [ m ] such that φ ( j ) = i in the case where X begins with 0 and it always have a lifting when X begins with 1. By replacing e ′ φ by an edge equivalent to it, we may assume that e ′ φ is ofthe form (cid:0) (0 j , . . . , , M ⊛ × RM ∆ n RM ∆ m (cid:1) → (cid:0) (0 i , , . . . , , M ⊛ (cid:1) when X begins with 0. If X begins with 1, erase 0 j , 0 i from the map above. From now on, we only treat the case where X begins with 0 since the other case is similarly and easier to check. Let Γ ∨ φ := Γ ∨ × ∆ ,φ ∆ usingthe notation of 1.3. Then the fiber of h over e ′ φ is Cartesian equivalent to the projection F : (∆ × M ⊛ X ) × ∆ × ∆ n Γ ∨ φ → Γ ∨ φ . M ⊛ X → ∆ n is a coCartesian fibration, this functor is coCartesian fibration as well. Invoking[HTT, 5.2.2.4] (or Lemma 1.5), we may replace F by F × Γ ∨ φ ,e ′ φ ∆ . Because e ′ φ is a coCartesianedge in Γ ∨ φ over ∆ , F × Γ ∨ φ ,e ′ φ ∆ is equivalent to ∆ × ( M ⊛ X × ∆ n ∆ { i } ) → ∆ , and thus, we havea Cartesian lift.Let us show that r is a Cartesian fibration. The argument works similarly, or even simpler, for q , by replacing Tw op ∆ ′ by Tw op ∆ , so we omit. Take a vertex of S tr en , + corresponding to a map g : ∆ × Tw op[ a ] ∆ → M univ , ⊛ , and take a map φ : [ b ] → [ a ]. We define an edge φ ∗ ( g ) → g over φ asfollows. Since Θ is a coCartesian fibration, we have a functor φ ∗ : Tw op[ b ] ∆ → Tw op[ a ] ∆ . We wish totake a p -right Kan extension e : ∆ × Tw op φ ∆ → M univ , ⊛ of g along ∆ × Tw op[ a ] ∆ ֒ → ∆ × Tw op φ ∆ ,and define the corresponding edge in S tr en , + as the desired edge. Let us check the existence ofthe extension. By [HTT, 4.3.2.15], for v := ( i, v ′ : [ k ] → [ b ]) ∈ ∆ × Tw op[ b ] ∆ , we only need tocheck that the diagram on the left below(∆ × Tw op[ a ] ∆ ) v/ g / / (cid:15) (cid:15) M univ , ⊛ h (cid:15) (cid:15) (∆ × Tw op[ a ] ∆ ) ⊳v/ / / ❧❧❧❧❧❧❧❧❧ RM univ ∆ g ( I ) / / (cid:15) (cid:15) M univ , ⊛ h (cid:15) (cid:15) (∆ ) ⊳ / / e ′ ♥♥♥♥♥♥♥ RM univ has a p -limit. The object I := (( i, v ′ ) → ( i, φ ∗ ( v ′ ))) in ∆ × Tw op φ ∆ is an initial object in (∆ × Tw op[ a ] ∆ ) v/ . By [HTT, 4.3.1.7], we need to check that the induced diagram on the right aboveextends to a p -limit diagram. Thus we are reduced to checking the existence of a p -Cartesianedge e ′ ((∆ ) ⊳ ) by [HTT, 4.3.1.4], where g ( I ) is over ((0 φv ′ (0) , , . . . , | {z } k +1 ) , [ a ]) ∈ RM ∆ a × ∆ , andthe cone point is sent to ((0 v ′ (0) , , . . . , | {z } k +1 ) , [ b ]) ∈ RM ∆ b × ∆ . Thus, the existence of a h -Cartesianedge is exactly the content of 1.It remains to check that the edge φ ∗ ( g ) → g is a q -Cartesian edge. In view of [HTT, 2.4.1.4],given any map ∆ n → ∆ , we need to solve the lifting problemΛ nn × ∆ Tw op ∆ ′ f / / (cid:15) (cid:15) M univ , ⊛ (cid:15) (cid:15) ∆ n × ∆ Tw op ∆ ′ / / ✐✐✐✐✐✐✐✐✐✐ RM univ where f | ∆ { n − ,n } × ∆ Tw op ∆ ′ is the edge e . Apply [HA, B.4.8] with C = ∆ n × ∆ Tw op ∆ ′ , C := { n } × ∆ n C . We need to check that e : Tw op φ ∆ ′ → M univ , ⊛ is a h -right Kan extension of g , whichfollows from the construction.The claim 3 follows by concrete description of Cartesian edges. For claim 4, consider thefollowing maps {∗} t ←− ∆ × Tw op[ n ] ∆ u −→ RM × ∆ , where u is the map induced by π . Invoking [HTT, 3.2.2.12], the map t ∗ u ∗ ( M univ , ⊛ ) → t ∗ u ∗ (RM × RMod ∆ n ) is a coCartesian fibration. Let X be the full subcategory of RM univ spanned by functors∆ × Tw op[ n ] ∆ → RMod ∆ n such that all the edges are sent to equivalent edges and the restrictionto Tw op[ n ] ∆ ′ is constant. Then S tr en , +[ n ] is a full subcategory of the pullback of t ∗ u ∗ ( M univ , ⊛ ) by X . The concrete description of coCartesian edges of [HTT, 3.2.2.12] allows us to show that themap S tr en , +[ n ] → X is a coCartesian fibration. By arguing similarly to the last half of the proofof Lemma 4.4, X → RMod ∆ n is a trivial fibration, thus the claim follows for S tr en , +[ n ] . The claim34or S tr [ n ] can be shown similarly. The preservation of coCartesian edges follows by the concretedescription. (cid:4) We apply dualizing construction of § S tr and S tr en , + over ∆ , and induce coCartesian fibrations S tr := D − ∆ ( S tr) → ∆ op , S tr en , + := D − ∆ ( S tr en , + ) → ∆ op . Recall that D − ∆ ( B ) ≃ A ⊛ × RModby Theorem 2.5. We put S tr ∼ := S tr ∗ S tr ≃ [0] . Since S tr → ∆ op is a coCartesian fibration, so is S tr ∼ . We put S tr en , + , ∼ := S tr en , + ∗ ( S tr en , +[0] ) ≃ . By the functoriality of D − , we have the followingcommutative diagram: S tr en , + , ∼ α / / ι (cid:15) (cid:15) A ⊛ × ∆ op RMod pr / / A ⊛ (cid:15) (cid:15) S tr ∼ / / RMod / / ∆ op . Here ι and α are the functors induced by taking dual of the corresponding maps in (4.3.1). Wewish to take the left Kan extension of α along ι . Lemma. —
1. The map S tr en , + , ∼ [0] → S tr ∼ [0] is a categorical equivalence of Kan complexes.2. The coCartesian fibrations S tr ∼ → ∆ op and S tr en , + , ∼ → ∆ op are generalized ∞ -operads.3. The functor S tr ∼ → RMod induces a categorical equivalence S tr ∼ → RMod ∗ S tr ∼ [0] ≃ RMod ⊛ ∗ S tr ∼ [0] (see 3.8 for the notation).Proof. Let us show 1. By definition, S tr en , + , ∼ [0] and S tr ∼ [0] are Kan complexes. By Lemma 1.4, itis enough to show that the map S tr en , +[0] → S tr [0] is a categorical equivalence. For this, it sufficesto show that the map S tr en , +[0] → S tr [0] is a trivial fibration. Since S tr en , +[0] is a full subcategoryof Fun RM (∆ × Tw op[0] ∆ , M ⊛ [0] ) and S tr [0] has a similar description, we invoke [HTT, 4.3.2.15].Let us show 2. Since these are coCartesian fibrations, by [HA, 2.1.2.12], we only need tocheck the Segal condition. We only treat the case S tr en , + , ∼ as the verification is similar, andthis case is more complicated. It suffices to show that S tr en , + is a generalized ∞ -operad. Sincethe Segal condition is stable under taking dual, it suffices to show that the map S tr en , +[ n ] → S tr en , + { , } × cat S tr en , + { } S tr en , + { , } · · · × cat S tr en , + { n − } S tr en , + { n − ,n } is a categorical equivalence. The argument is similar to that of [HA, 4.7.1.13]. Let I ⊂ [ n ]be a subset such that m := I . Let X I be the full subcategory of Fun(∆ × Tw op[ m ] ∆ , M univ , ⊛ [ n ] )spanned by functors which can be lifted to a functor ∆ × Tw op[ m ] ∆ → M univ , ⊛ [ n ] × RM ∆ n RM ∆ I suchthat the composition with M univ , ⊛ [ n ] × RM ∆ n RM ∆ I → M univ , ⊛ [ m ] belongs to S tr en , +[ m ] . By Lemma 4.2,we have X I ≃ S tr en , + I × RMod I RMod ∆ n . Since RMod ∆ satisfies the Segal condition, it suffices toshow that the map S tr en , + → X { , } × cat X { } X { , } × cat · · ·× cat X { n − ,n } is a categorical equivalence.Let Tw op[ n ] ∆ be the full subcategory of Tw op[ n ] ∆ spanned by maps [ m ] → [ n ] which factorsthrough an inert morphism of the form ρ i : [1] → [ n ]. Let p : M univ , ⊛ [ n ] → RM, which is acoCartesian fibration. We let X be the full subcategory of Fun RM (∆ × Tw op[ n ] ∆ , M univ , ⊛ [ n ] )spanned by functors such that 35. We have the inclusion Tw op[ n ] ∆ ′ → Tw op[ n ] ∆ { }× id −−−−→ ∆ × Tw op[ n ] ∆ . Given a vertex, if werestrict the functor along this inclusion, the functor belongs to S tr;2. Let φ : [1] → [ n ] be an inert map. Since Θ is a coCartesian fibration, we have the map φ ∗ : Tw op[1] ∆ → Tw op[ n ] ∆ . By the first condition, given a vertex, the functor ∆ × Tw op[1] ∆ → M univ , ⊛ [ n ] induced by φ ∗ factors through M univ , ⊛ [1] . Then, this functor belongs to S tr en , +[1] .This X is a model for the product and we must show that the map S tr en , + → X is a categoricalequivalence.Now, let Tw op[ n ] Inc (resp. Tw op[ n ] Inc ) be the full subcategory of Tw op[ n ] ∆ spanned by inert maps[ k ] → [ n ] (resp. inert maps [ k ] → [ n ] where k = 0 , i = 0 ,
1, let Y i be the full subcategory ofFun RM (∆ × Tw op[ n ] Inc i , M univ , ⊛ [ n ] ) spanned by functors which satisfies the conditions of Definition4.6 if we replace Tw op[ n ] ∆ by Tw op[ n ] Inc i . We have the following commutative diagram of simplicialsets on the left induced by the commutative diagram on the left: S tr en , + θ / / (cid:15) (cid:15) Y τ (cid:15) (cid:15) X θ / / Y , Tw op[ n ] ∆ Tw op[ n ] Inc o o Tw op[ n ] ∆ O O Tw op[ n ] Inc . O O o o It suffices to show that θ , θ , τ are categorical equivalences. The verification for τ is similar tothe proof of [HA, 4.7.1.13], so we omit. Let us check that θ is a trivial fibration. The verificationof θ is similar, so we omit. The strategy is similar to [HA, 4.7.1.13]. In view of [HTT, 4.3.2.15],it suffices to show the following two assertions:1. For any G ∈ Y , p -left Kan extension of G along the inclusion Tw op[ n ] Inc ֒ → Tw op[ n ] ∆ exists;2. Any F ∈ Fun(∆ × Tw op[ n ] ∆ ) is in X if and only if G := F | ∆ × Tw op[ n ] Inc is in X and F isa p -left Kan extension of G .The verification is standard: Fix an object C := ( a, [ k ] → [ n ]) in ∆ × Tw op[ n ] ∆ and we wish toshow the existence of the p -colimit of the diagram(∆ × Tw op[ n ] Inc ) × ∆ × Tw op[ n ] ∆ (∆ × Tw op[ n ] ∆ ) /C → M univ , ⊛ [ n ] This category has an initial object. More precisely if we write C = ( a, φ : [ k ] → [ n ]), thereexists a unique inert map ψ : [ k ′ ] ֒ → [ n ] such that ψ (0) = φ (0) and ψ ( k ′ ) = φ ( k ). The initialobject is ( a, ψ ) → C . Since p is coCartesian, we get the existence by [HTT, 4.3.1.4]. Thisconstruction also tells us that F is a p -left Kan extension if and only if the induced map F ( a, [ k ] → [ n ]) → F ( a, [ k ′ ] → [ n ]) is a p -coCartesian edge. Thus, we also have the secondassertion.Finally, let us show 3. Since we have S tr ∼ ≃ S tr ∗ S tr ≃ [0] , RMod ∗ S tr ∼ [0] ≃ (RMod ∗ S tr [0] ) ∗ S tr ≃ [0] , it suffices to show the map S tr → RMod ∗ S tr [0] is an equivalence. This is equivalent to showingthe induced functor S tr → RMod ∗ S tr [0] is an equivalence. Since both S tr and RMod ⊛ ∗ S tr [0] are Cartesian fibrations over ∆ and preserves coCartesian edges by Lemma 4.7, it suffices36o check the equivalence for each fiber over ∆ by [HTT, 3.3.1.5]. We choose the followingcommutative diagram (which is possible up to contractible space of choices) M univ[ n ] Q ρ i ! / / (cid:15) (cid:15) Q i ∈ [ n ] M univ[0] (cid:15) (cid:15) RMod [ n ] Q ρ i ! / / Q i ∈ [ n ] RMod [0] . This diagram induces the map S tr [ n ] → RMod ∆ n × cat(RMod ∆0 ) ( × ( n +1)) S tr × ( n +1)[0] . It is reduced toshowing that this is an equivalence. We have an isomorphism Tw op[ n ] ∆ ′ ∼ = ` i ∈ [ n ] { i } sending φ : [0] → [ n ] to φ (0). We consider the maps { i } → RM ∆ { i } → RM ∆ n where the first map sendsto (0 , S tr [ n ] ∼ = Fun RM ∆ n ( a i ∈ [ n ] { i } , M univ , ⊛ [ n ] ) × Fun( ` { i } , RMod ∆ n ) ,α RMod ∆ n ≃ Y i ∈ [ n ] Fun RM ∆ n ( { i } , M univ , ⊛ [ n ] ) × cat Q Fun( { i } , RMod ∆ n ) , ∆ RMod ∆ n , where α is induced by the unique map ` { i } → {∗} , ∆ is the diagonal map. The secondequivalence follows from Remark 4.5. Using this, we may computeRMod ∆ n × cat(RMod ∆0 ) × ( n +1) ( S tr ∼ [0] ) × ( n +1) ≃ RMod ∆ n × cat Q RMod ∆ { i } Y Fun RM ∆ { i } ( { i } , M univ , ⊛ { i } ) . Finally, we haveFun RM ∆ n ( { i } , M univ[ n ] ) ∼ = Fun RM { i } ((0 i , , M univ[ n ] × catRM ∆ n RM { i } ) → Fun RM [0] ((0 , , M univ[0] × catRMod ∆ { i } RMod ∆ n ) ∼ = Fun RM [0] ((0 , , M univ[0] ) × catRMod ∆ { i } RMod ∆ n The middle map is a categorical equivalence by Lemma 4.2. Combining these three equivalences,we have the desired equivalence. (cid:4)
Let M ⊛ → RM ∆ be an object of L in C at ∆ . Let M ⊛ i → RM be the pullback by RM ∆ { i } → RM ∆ for i = 0 , . Assume we are given M i ∈ M i . Then there exists Mor( M , M ) equipped with a map M ⊠ Mor( M , M ) → M over the active map in M ⊛ having the universalproperty that for any A ∈ A , the induced map Map A ( A, Mor( M , M )) → Map M ( M ⊗ M A, M ) is a homotopy equivalence. If F ⊛ : M ⊛ → M ⊛ is the monoidal functor of generalized ∞ -operadsassociated with M ⊛ , then Mor M ( M , M ) ≃ Mor M ( F ( M ) , M ) , where Mor M is the morphismobject (cf. [HA, 4.2.1.33]).Proof. Consider the functorMap( M ⊗ M ( − ) , M ) : A op ( M ⊗ ,M ) −−−−−−→ M op × M Map −−−→ S pc . It suffices to show that this functor is equivalent to Map M ( F ( M ) ⊗ M ( − ) , M ). Using [HTT,5.2.1.4], choose a functor G : M × ∆ → M associated to M . Let ι i : M i → M be the canonical37unctor, and F ′ := ι ◦ F . Then G determines a map of functors ι → F ′ . This induces the mapof functors Map M (cid:0) F ′ ( − ) , ( − ) (cid:1) → Map M (cid:0) ι ( − ) , ( − ) (cid:1) : M op0 × M → S pc . This induces an equivalence Map M (cid:0) F ( − ) , ( − ) (cid:1) ∼ −→ Map M (cid:0) ι ( − ) , ( − ) (cid:1) . Thus, we haveMap M (cid:0) M ⊗ M ( − ) , M (cid:1) ∼ ←− Map M (cid:0) F ( M ⊗ M ( − )) , M (cid:1) ≃ Map M (cid:0) F ( M ) ⊗ M ( − ) , M (cid:1) , and we get the desired equivalence. (cid:4) To proceed, we need to restrict our attention to L in C at in RMod. Let A be a pre-sentable monoidal ∞ -category (cf. 3.10). We put L S tr ∼ := S tr ∼ × RMod ⊛ L in C at ⊛ A , L S tr en , + , ∼ := S tr en , + , ∼ × RMod ⊛ L in C at ⊛ A where the fiber products are taken in Op ns , gen ∞ . Assume we are givena diagram A / / f (cid:15) (cid:15) O p (cid:15) (cid:15) B / / < < ②②②②② ∆ op . Assume that f is a map of generalized ∞ -operads. An operadic p -left Kan extension of thisdiagram consists of a factorization ( A × ∆ ) ` A ×{ } B h ′ −→ M h ′′ −→ ∆ op × ∆ where h ′ is an inneranodyne and h ′′ is a ∆ -family of generalized ∞ -operads (cf. [GH, A.3.1]), and an operadic p -leftKan extension of M × ∆ { } ≃ A → O along the inclusion M × ∆ { } ֒ → M (cf. [GH, A.3.3]). Proposition. —
Let A be a presentable monoidal ∞ -category, and consider the followingdiagram: L S tr en , + , ∼ / / (cid:15) (cid:15) A ⊛ p (cid:15) (cid:15) L S tr ∼ / / H ssssss ∆ op . Then the diagram admits an operadic p -left Kan extension H . For M ⊛ → RM ∆ in L in C at ∆ and an object ( M , M ) of L S tr ∼ over M ⊛ , we have H ( M , M ) ≃ Mor M ( M , M ) .Proof. By small object argument, we may take a factorization of the map( L S tr en , + , ∼ × ∆ ) a L S tr en , + , ∼ ×{ } L S tr ∼ → ∆ op × ∆ into inner anodyne followed by a ∆ -family of generalized ∞ -operads. We abbreviate L ( − ) as( − ) to ease the notation. For a generalized ∞ -operad O ⊛ , we denote by O ⊛ act the subcategoryof O ⊛ consisting of active maps. Let M ∈ S tr ∼ [1] . Invoking [GH, A.3.4], it suffices to show thatthe diagram S tr en , + , ∼ × S tr ∼ ( S tr ∼ act ) /M / / (cid:15) (cid:15) A ⊛ π (cid:15) (cid:15) ( S tr en , + , ∼ × S tr ∼ ( S tr ∼ act ) /M ) ⊲ / / ✐✐✐✐✐✐✐✐✐✐ ∆ op extends to an operadic π -colimit diagram. Since S tr en , + , ∼ → S tr ∼ preserves coCartesian edgesover ∆ op by Lemma 4.7.3, the map S tr en , + , ∼ × S tr ∼ ( S tr ∼ act ) /M → ( ∆ opact ) / [1] is a coCartesianfibration using [HTT, 2.4.3.2]. This implies that the inclusion S tr en , + , ∼ [1] × S tr ∼ [1] ( S tr ∼ [1] ) /M → S tr en , + , ∼ × S tr ∼ ( S tr ∼ act ) /M is left cofinal by [HTT, 4.1.2.15]. Thus, by (non-symmetric analogue of)[HA, 3.1.1.4], it suffices to show the existence of the operadic colimit for S tr en , + , ∼ [1] × S tr ∼ [1] ( S tr ∼ [1] ) /M .38et M ⊛ → RM ∆ be the generalized ∞ -operad over which M is defined. We put S tr (en , + , ) ∼ M := S tr (en , + , ) ∼ [1] × RMod ∆1 { M } . By Lemma 4.7, S tr en , + , ∼ [1] → S tr ∼ [1] is a map between coCartesian fi-bration over RMod ⊛ [1] which preserves coCartesian edges. Thus, by the same argument asabove, we are reduced to showing the existence of operadic colimit of S tr en , + , ∼ M × S tr ∼ M ( S tr ∼ M ) /M .Let F M : Tw op[1] ∆ ′ → M ⊛ be the functor corresponding to M . Put M i := F M ( { i } → [1]) in M ⊛ [0] × RM ∆1 RM ∆ { i } . Unwinding the definition, existence of Mor M ( M , M ) is equivalent to theexistence of an initial object of S tr en , + , ∼ M × S tr ∼ M ( S tr ∼ M ) /M . Thus an initial object exist by Lemma4.9. Since A ⊛ is compatible with small colimits, [GH, A.2.7] implies the existence operadiccolimit, whose value at the cone point is nothing but Mor M ( M , M ). (cid:4) Let us carry out one of the main constructions of this paper. Let C be an ( ∞ , D : C → LinCat − op A of ( ∞ , D : C → L in C at A be the associated functor of ∞ -categories to D , and assume we are given thefollowing commutative diagram C ≃ M / / (cid:15) (cid:15) L S tr ∼ [0] (cid:15) (cid:15) C D / / L in C at A . Recall that giving the 2-functor is equivalent to giving a monoidal functor D ⊛ : C ⊛ → L in C at ⊛ A of the generalized ∞ -operads. We have the functor C ⊛ [0] ≃ C ≃ M −→ L S tr ∼ [0] , also denoted by M .Using this, we have the map of generalized ∞ -operads H M : C ⊛ D ⊛ ∗ M −−−−→ L in C at ⊛ ∗ L S tr ∼ [0] ∼ ←− L S tr ∼ H −→ A ⊛ , where the equivalence follows by Lemma 4.8. Let us describe this informally. For a 1-morphism f : X → Y in C , we have the map D ( f ) : D ( X ) → D ( Y ). The functor M defines M ( X ) ∈ D ( X ), M ( Y ) ∈ D ( Y ). Then H M sends f to Mor D ( f ) ( M ( X ) , M ( Y )).In our application, it is not hard to construct M . Assume that the inclusion C ≃ → C factorsas C ≃ → C ′ F −→ C , where C ′ is an ∞ -category. We moreover assume C ′ has an initial object ∅ .Fix an object I ∈ DF ( ∅ ). Consider the diagram {∅} I / / (cid:15) (cid:15) L S tr [0] p (cid:15) (cid:15) C ′ D ◦ F / / M I ♦♦♦♦♦♦♦ L in C at A , ∆ . The functor p is a categorical fibration by Remark 4.5. Since p is equivalent to the base changeof M univ , ⊛ [0] , (0 , ( ≃ S tr [0] ) → RMod ∆ which is a coCartesian fibration, p is a coCartesian fibrationby [HTT, 2.4.4.3]. Thus we may take a p -left Kan extension of I . This extension is denotedby M I . By the above construction, we have the map of generalized ∞ -operads H M I : C ⊛ → A ⊛ associated to I . Remark. —
1. The map H M of generalized ∞ -operads induces a non-unital right-lax func-tor of ( ∞ , H M : C B A ⊛ . Let f : X → Y be a 1-morphism in C , and let D ( f ) : D ( X ) → D ( Y ) be the associated1-morphism in LinCat A . Then the functor M defines M X ∈ D ( X ) for each X , and H M ( f ) ≃ Mor D ( Y ) ( D ( f )( M X ) , M Y )39y viewing 1-morphisms in B A ⊛ as objects of A . This interpretation of H M is moreconceptual, but a priori discards some information from H M when we take the localizationto pass from A ⊛ to B A ⊛ . We believe that H M is more essential than H M , and theconstruction in the next section, for which we use H M rather than H M crucially, shouldbe able to be carried out within the realm of ( ∞ , §
6, we apply this construction to Gaitsgory-Rozenblyum’s 6-functor formalism. Then H M I ( f ) becomes the corresponding bivariant homology theory in the sense of Fulton-MacPherson [FM, § H M I is supposed to encode all the axioms of thetheory. However, it is still not satisfactory because treating ( ∞ , ∞ -categories. In §
5, we will extract a functor between ∞ -categories which ismuch easier to handle, yet retains some important features of bivariant homology theory. Assume that the functor M : C ≃ → S tr [0] can be lifted to a functor f M : C ′ → S tr [0] com-patible with D : C → L in C at A . The construction above yields a functor H M | C ′ sending a sequence C f −→ C → . . . f n −→ C n in C to Mor D ( f ) ( M ( C ) , M ( C )) ⊠ · · · ⊠ Mor D ( f n ) ( M ( C n − ) , M ( C n ))in A ⊛ [ n ] . On the other hand, we also have a functor f M sending the sequence to A ⊠ · · · ⊠ A ,where we take n -times product and A is a unit-object of A . On the other hand, we have amap f M ( f i ) : M ( C i − ) → M ( C i ). This yields a map A → Mor D ( f i ) ( M ( C i − ) , M ( C i )). Thus itis natural to expect for a map of functors f M → H M | C ′ , which we will construct in the rest ofthis section. This map will be used in the next section.Recall the notation of 1.3. Let α : Γ ∨ → Tw op ∆ be the unique functor over ∆ sending ([ n ] , i )to a i : [ n − i ] → [ n ] in Tw op ∆ such that a i (0) = i . Let z : Γ ∨ → RM × ∆ be the map sending([ n ] , i ) to (0 , ∈ RM over ∆ . Put D := (∆ [1] × Γ ∨ ) a { }× Γ ∨ (∆ [1] × Tw op ∆ ) . The maps Γ ∨ → ∆ and Θ : Tw op ∆ → ∆ induce the map Θ D : D → ∆ . We may check easilythat D is (nerve of) a category, and Θ D is a coCartesian fibration. We also have a map ofsimplicial sets π D : D → RM × ∆ such that the restriction to ∆ [1] × Tw op ∆ is π , and restrictionto { } × Γ ∨ is z and ∆ × ([ n ] , i ) is the unique active map. Definition. —
1. Let S tr en , ++ be the full subcategory of Θ D, ∗ π ∗ D ( M univ , ⊛ ) spanned byfunctors F : D [ n ] := D × ∆ , Θ D { [ n ] } → M univ , ⊛ satisfying the following conditions:(a) The restriction F | ∆ × Tw op[ n ] ∆ belongs to S tr en , +[ n ] ;(b) The functor F is an p -left Kan extension of F | ∆ × Tw op[ n ] ∆ where p : M univ , ⊛ → RM × ∆ .2. Let γ ( M univ , ⊛ ) → ∆ be the full subcategory of γ ∨∗ z ∗ ( M univ , ⊛ ) ∼ = γ ∨∗ γ ∨∗ ( M univ , ⊛ (0 , ) spannedby the functors Γ ∨ [ n ] ∼ = ∆ n → M univ , ⊛ [ n ] , (0 , such that the composition ∆ n → M univ , ⊛ [ n ] , (0 , → RM univ[ n ] , (0 , ≃ ∆ n × RMod ∆ n is of the form id × m where m : ∆ n → RMod ∆ n factorsthrough RMod ≃ ∆ n .The functors { } × Γ ∨ → D ← ∆ × Tw op ∆ induce the diagram γ ( M univ , ⊛ ) ← S tr en , ++ α −→ S tr en , + .
1. The map S tr en , ++ → ∆ is a Cartesian fibration, α preserves Cartesianedges, and α [ n ] is a trivial fibration. In particular, α is a Cartesian equivalence. As usual,we put S tr en , ++ := D − ∆ ( S tr en , ++ ) . . The map γ ( M univ , ⊛ ) → ∆ is a Cartesian fibration. Moreover, we have a canonical equiv-alence γ ∨∗ ( M univ[0] , (0 , × Γ ∨ ) ≃ / ∆ ≃ γ ( M univ , ⊛ ) ≃ / ∆ of right fibrations over ∆ (recall 1.4 for thenotation). We put γ ( M univ , ⊛ ) := D − ∆ ( γ ( M univ , ⊛ )) as usual.Proof. Let us show the first claim. We can check that S tr en , ++ is a Cartesian fibration byexactly the same argument as Lemma 4.7. By description of coCartesian edges, we see that α preserves Cartesian edges. The fiber α [ n ] is trivial fibration by [HTT, 4.3.2.15].Let us show the second assertion. The first claim is a straightforward application of [HTT,3.2.2.12], so let us check the second claim. To ease the notations, we abbreviate γ ∨∗ , γ ∨∗ by γ ∗ , γ ∗ . Let F : C → D be a coCartesian fibration of ∞ -categories. We may consider the followingdiagram: C FF ′ (cid:15) (cid:15) / / (cid:3) C F (cid:15) (cid:15) γ ∗ γ ∗ ( D × Γ ∨ ) / / (cid:15) (cid:15) D γ ∗ ( D × Γ ∨ ) . All the vertical arrows are coCartesian fibrations. We define γ ( C F ) to be the full subcategoryof γ ∗ γ ∗ C F spanned by vertices ∆ n → C F, [ n ] such that the composition ∆ n → C F, [ n ] → γ ∗ γ ∗ ( D × Γ ∨ ) [ n ] ∼ = ∆ n × Fun(∆ n , D ) is of the form id × m where m factors through Fun(∆ n , D ) ≃ . Weclaim that the composition γ ( C F ) ⊂ γ ∗ γ ∗ C F → γ ∗ γ ∗ ( C × ∆ ) ∼ = γ ∗ ( C × Γ ∨ )is a categorical equivalence between Cartesian fibrations. Indeed, using [HTT, 3.2.2.12], wecan check that this is a map between Cartesian fibrations that preserves Cartesian edges. Theinduced map between fibers over [ n ] ∈ ∆ can be computed explicitly, and the equivalence follows.Moreover, the equivalence induces the equivalence γ (cid:0) C F × γ ∗ ( D × Γ ∨ ) γ ∗ ( D × Γ ∨ ) ≃ / ∆ (cid:1) ∼ −→ γ ∗ ( C × Γ ∨ ) ≃ / ∆ . Now let RMod ⊛ , str ∆ ⊂ RMod ⊛ ∆ be the subcategory spanned by all objects of RMod ⊛ A andmorphisms M ⊛ → N ⊛ over [ n ] → [ m ] which sends coCartesian edge over RM ∆ n to coCartesianedge over RM ∆ m . Let RMod str∆ n be the fiber over [ n ]. We apply the observation above to thecoCartesian fibration F : M univ , ⊛ [0] , (0 , × RMod ∆0 RMod str∆ → RMod str∆ . The unstraightening of F ′ isthe compositionΓ ∨ × ∆ Φ Cart ( γ, RMod str∆ ) ≃ γ ∗ γ ∗ (RMod str∆ × Γ ∨ ) → RMod str∆ St( F ) −−−→ C at ∞ . By Proposition 3.9, we have Γ ∨ × ∆ RMod ⊛ , str ∆ ≃ Γ ∨ × ∆ Φ Cart ( γ, RMod str∆ ). Unwinding thedefinition, the unstraightening of the composition Γ ∨ × ∆ RMod ⊛ , str ∆ → C at ∞ can be identifiedwith M univ , ⊛ (0 , → RM univ(0 , base changed to RMod ⊛ , str ∆ . Thus, the claim follows by the observationabove. (cid:4) Let p : A ⊛ → ∆ op be a monoidal ∞ -category.1. An edge e : ∆ → A ⊛ is said to exhibit e (1) as a unit object if e is p -coCartesian edge and p (0) = [0].2. Let A ⊛ be the full subcategory of Fun(∆ , A ⊛ ) spanned by unit objects.41he inclusion A ⊛ → A ⊛ is a categorical fibration. Moreover, the map A ⊛ → ∆ op is a trivialfibration by a similar argument to [HA, 3.2.1.4]. We put S tr en , + := S tr en , ++ × A ⊛ A ⊛ . Remark. —
Informally objects of S tr en , ++ consists of the data of S tr en , + , which contains asequence in M univ , ⊛ [ n ] of the form M ⊠ A ⊠ · · · ⊠ A n → M ⊠ A ⊠ · · · ⊠ A n → · · · → M n , together with a sequence M ⊗ A ⊗ · · · ⊗ A n → M ⊗ A ⊗ · · · ⊗ A n → · · · → M n in M univ[0] , (0 , . Since ⊗ is defined essentially uniquely, it is not surprising that α : S tr en , ++ → S tr en , + is a trivial fibration. Furthermore, S tr en , + consists of data as above such that any A i is a unit object of A ⊛ for any i . The main feature of this construction can be seen from the following lemma:
Lemma. —
The composition S tr en , + → S tr en , ++ → γ ( M univ , ⊛ ) is a categorical equivalence.Proof. Using the description of coCartesian edges in Lemma 4.7, we can check that the mapis a functor between coCartesian fibrations that preserves coCartesian edges, so it suffices toshow that the fiber over [ n ] ∈ ∆ op is an equivalence. In this case, Γ ∨ × ∆ { [ n ] } ∼ = ∆ n . Weconsider the following category D + : Let E be the category of two objects −
1, 0 such thatHom( − ,
0) = { a } , Hom(0 , −
1) = { b } , Hom( − , −
1) = { id } , Hom(0 ,
0) = { id , a ◦ b } . Weconsider D ′ := ( E × ∆ n ) ` { }× ∆ n (∆ × Tw op[ n ] ∆ ), where the coproduct is taken in the categoryof small (ordinary) categories. There is a unique map f : ( − , → (1 , [ n ] → [ n ]) where 0 ∈ ∆ n .Using this morphism, we define D + := D ′ ` f, ∆ { , } ∆ where the coproduct is taken in thecategory of small categories. The functor b : ∆ → E induces the faithful functor D [ n ] → D ′ .There is a unique extension of D → RM to D ′ . Note that the morphism a × id i of E × ∆ n issent to the map (0 , → (0 , , . . . , D + by putting ∆ { } to (1) in RM. Let C ⊂ D + be the full subcategory consisting of objects in ∆ × Tw op[ n ] ∆ and∆ .Now, let S D be the full subcategory of Fun RM ( D + , M univ , ⊛ [ n ] ) which is spanned by functors F such that F | {− }× ∆ n belongs to γ ( M univ , ⊛ ) and which is a p -left Kan extension along theinclusion {− } × ∆ n ֒ → D + , where p : M univ , ⊛ [ n ] → RM. We put S tr en , := S tr en , + × A ⊛ A ⊛ . Wehave the following diagram S D / / τ & & ▼▼▼▼▼▼▼▼▼▼▼▼▼ φ , , S tr en , +[ n ] ψ / / (cid:15) (cid:15) S tr en , [ n ] γ ( M univ , ⊛ )It suffices to show that φ , ψ , τ are categorical equivalences. Consider the following two left Kanextension diagrams: ∆ n × {− } / / (cid:15) (cid:15) M univ , ⊛ [ n ] p (cid:15) (cid:15) D + / / rrrrrr RM , C / / (cid:15) (cid:15) M univ , ⊛ [ n ] p (cid:15) (cid:15) D + / / ; ; ✇✇✇✇ RM . Invoking [HTT, 4.3.2.15], these diagrams yields the trivial fibrations τ , φ . The map ψ is a trivialfibration by Lemma 4.13. (cid:4) .16. This lemma yields a functor γ ( M univ , ⊛ ) ∼ ←− S tr en , + → S tr en , ++ α −→ S tr en , + which is defined up to a contractible space of choices. Let us apply this construction to thesituation in 4.11. Assume that we are given an ( ∞ , D , and assume that the functor C ≃ → C factors as C ≃ → C ′ → C . We assume further that the map M : C ≃ → S tr ∼ [0] is lifted to f M : C ′ → S tr [0] which is compatible with D . If C ′ admits an initial object, the functor M I in4.11 satisfies this condition. Since S tr [0] ≃ M univ , ⊛ [0] , (0 , , we have the composition s M : γ ∗ ( C ′ × Γ) ≃ / ∆ op f M −→ γ ∗ ( L S tr [0] × Γ) ≃ / ∆ op → L γ ( M univ , ⊛ ) → L S tr en , + , where the second functor follows from Lemma 4.13. Note that the last two functors are basechanged from RMod ⊛ to L in C at ⊛ . Thus, we have a diagram γ ∗ ( C ′ × Γ) ≃ / ∆ op s M / / L S tr en , + , ∼ A / / ι (cid:15) (cid:15) A ⊛ (cid:15) (cid:15) L S tr ∼ / / H ttttttttttt ∆ op . Recall that the operadic left Kan extension is equipped with a map A → H ◦ ι . Putting f M := A ◦ s M , H M | C ′ := H ◦ ι ◦ s M . Then we have the map of functors f M → H M | C ′ : Φ co ( γ, C ′ ) ≃ / ∆ op = γ ∗ ( C ′ × Γ) ≃ / ∆ op → A ⊛ as desired.
5. Bivariant homology functor
In the last section, we “extracted” a functor which encodes axioms of bivariant theory. However,the description is still very inexplicit. Assume we are given a “6-functor formalism” for thecategory of schemes Sch. Then there should be an associated cohomology theory H ∗ : Sch op → Mod R . There should also be an associated Borel-Moore homology H BM : Sch prop → Mod R . Here,Sch prop is the category of schemes and consider only proper morphisms as morphisms. The goalof this section is to construct a unified functor from which we can retrieve these two theorieseasily, as well as explaining the relations of these theories. Let Sch be a category which admits finite limits. In particular, it admits a final object.We fix a final object denoted by ∗ . Let prop, sep be a class of morphisms of Sch which satisfyconditions [GR, Ch.7, 1.1.1] if we put adm = prop, vert = sep, horiz = all. We put Corr := Corr (Sch) propsep;all . Let A ⊛ be a presentable monoidal ∞ -category, and A : A ⊛ [1] . We assume weare given the following diagram Sch op D / / (cid:15) (cid:15) L in C at A (cid:15) (cid:15) Corr D / / LinCat - op A . We fix two objects
I, J ∈ D ( ∗ ). By 4.11, we have the functors M I , M J : Sch op → L S tr [0] . Theimage of X ∈ Sch op is denoted by I X , J X . For f : X → Y in Sch, we put f ∗ := D ( f ). Similarly,43f f ∈ sep, let V f be the 1-morphism X → YX f (cid:15) (cid:15) XY. in Corr . We put f ! := D ( V f ) : D ( X ) → D ( Y ). Example. —
As the notation suggests, the main example we have in mind is the case whereSch is the category of schemes of finite type over a base scheme S , and prop is the class of propermorphisms, and sep is the class of separated morphisms. We denote by f Ar propsep (Sch) the category whose objects consists of morphisms X → Y insep. Assume we are given two objects f : X → Y and f : X → Y . Let S ( f , f ) be the setof diagrams of the form below on the left(5.2.1) X f (cid:15) (cid:15) W f ′ (cid:15) (cid:15) g ′ / / α o o (cid:3) X f (cid:15) (cid:15) Y Y g / / Y , X W / / o o β (cid:15) (cid:15) X X W ′ / / o o X , where α belongs to prop. An element of S ( f , f ) defined by W and W ′ are equivalent,denoted by W ∼ W ′ if there exists a diagram of the form above on the right. We define theset of morphisms from f to f by S ( f , f ) / ∼ . Note that β is automatically an equivalence.The composition of ( X → Y ) → ( X → Y ) → ( X → Y ) is defined by the following diagram W × Y Y / / (cid:15) (cid:15) (cid:3) W / / (cid:15) (cid:15) (cid:3) X (cid:15) (cid:15) W / / (cid:15) (cid:15) (cid:3) X (cid:15) (cid:15) X (cid:15) (cid:15) Y / / Y / / Y . The functor f Ar propsep (Sch) → Sch sending X → Y to Y is a Grothendieck fibration. Remark. —
For Y ∈ Sch, let Sch propsep /Y be the subcategory of of Sch /Y consisting of objects f : X → Y in Sch /Y such that f is in sep, and morphisms X (cid:15) (cid:15) α / / X ′ (cid:15) (cid:15) Y Y such that α is in prop. Given a morphism Y → Y , we can choose a base change functorSch prop /Y → Sch prop Y . This yields a pseudo-functor Sch op → C at (2 , . The Grothendieck fibration f Ar propsep (Sch) → Sch is associated to this pseudo-functor.Now, the goal of this section is to show the following theorem:
Under the setting of 5.1, there exists a functor
H : f Ar propsep (Sch) op → A suchthat for f ∈ sep , the object H( f ) ∈ A is equivalent to Mor D ( V f ) ( I X , J Y ) ≃ Mor( f ! ( I X ) , J Y ) using he notation of 5.1. Assume we are given a morphism m : f → f given by the diagram (5.2.1)on the left. Then H( m ) is equivalent to the composition of the following morphisms Mor( f ( I X ) , J Y ) → Mor( g ∗ f ( I X ) , g ∗ J Y ) ≃ Mor( f ′ ! g ′∗ ( I X ) , g ∗ J Y ) ≃ Mor( f α ! α ∗ ( I X ) , J Y ) → Mor( f ( I X ) , J Y ) . Remark. —
1. Consider the functor i : Sch → f Ar propsep (Sch) sending X to X → X . ThenH ◦ i op0 : Sch op → A is called the cohomology theory . On the other hand, let Sch propsep bethe subcategory of Sch consisting of objects X such that the map X → ∗ is in sep andmorphisms X → Y which are in prop. Consider the functor i : Sch propsep → f Ar propsep (Sch) op sending X to X → ∗ . Then H ◦ i is called the Borel-Moore homology theory . Then functorH describes the relations between these theories.2. Even though we can unify cohomology and Borel-Moore homology theories, it is not com-pletely satisfactory because we are not able to retrieve all the features of bivariant ho-mology theory. In fact, bivariant homology theory has 3 operations: contravariant func-toriality with respect to all the morphisms, covariant functoriality with respect to propermorphisms, and product structure. In our treatment, product structure is missing. Wewonder if there is an upgraded version of the functor H so that all the axioms [FM, § Let ∞ be the cone point of Sch ⊲ . We have the functor c : Sch ⊲ → Sch sending ∞ to ∗ suchthat the composition Sch → Sch ⊲ → Sch is isomorphic to the identity. A morphism in Sch ⊲ isdefined to be in sep, prop if and only if the image by c is in sep, prop. We have the functor Corr (Sch ⊲ ) propsep;all → Corr (Sch) propsep;all D −→ LinCat - op A . Now, we have the functor M : (Sch ⊲ ) ≃ → S tr ∼ [0] such that M ( ∞ ) = I and M | Sch = J − . Wedenote by Corr ⊛ the generalized ∞ -operad defining Corr (Sch ⊲ ) propsep;all .Till 5.11, we will focus on constructing the following sequence of functors between ∞ -categories over ∆ op :(5.4.1) H ⊤ : Φ co (Γ × ∆ , Sch op ) prop α −→ ←− s ∗ Corr ⊛ β ←− ∼ Ψ ⋆ ←− s ∗ Corr ⊛ Ψ H M −−−→ Ψ( ←− s ∗ A ⊛ ) . The undefined notations will be introduced later. Since β is a categorical equivalence, H ⊤ isdefined canonically up to contractible space of choices.If we are given a Cartesian fibration f : C → ∆ op and an ∞ -category D , we introduced thenotation Φ co ( f, D ) in 1.2. In this section, this ∞ -category is also denoted by Φ co ( C , D ) especiallywhen the structural map f is clear. In particular, when we use this notation, the ∞ -category C is always considered over ∆ op . Remark. —
A vertex of Φ co (Γ × ∆ , Sch op ) prop over [ n ] ∈ ∆ op is a diagram in Sch of the form(5.8.1). The functor H ⊤ sends this diagram to a sequenceMor( I X , J Y ) ⊠ Mor( J Y , J Y ) ⊠ Mor( J Y , J Y ) ⊠ · · · ⊠ Mod( J Y n − , J Y n ) → Mor( I X , J Y ) ⊠ Mor( J Y , J Y ) ⊠ · · · ⊠ Mod( J Y n − , J Y n ) → · · · → Mor( I X n , J Y n )in A ⊛ . Here, Mor( J Y i , J Y i +1 ) is taken over the functor D ( f : Y i +1 → Y i ) =: f ∗ : D ( Y i ) → D ( Y i +1 ).Since we have the morphism A → Mor( J Y i , J Y i +1 ) corresponding to f ∗ J Y i ≃ J Y i +1 id −→ J Y i +1 theabove sequence yields a sequence in A ⊛ Mor( I X , J Y ) ⊠ ⊠ · · · ⊠ → Mor( I X , J Y ) ⊠ ⊠ · · · ⊠ → · · · → Mor( I X n , J Y n ) . H ⊤ , will be constructed in 5.11. Taking the tensor product in A , this sequenceyields Mor( I X , J Y ) → Mor( I X , J Y ) → · · · → Mor( I X n , J Y n ) . This is the functor H • which will be constructed in 5.12. Let ←− s : ∆ op → ∆ op be the functor ( − ) ⊳ . First, let us prepare some result on ←− s . Let F : ∆ op × ∆ → ∆ op be a functor sending ([ n ] , i ) to [ n +1 − i ], F | ∆ op × ∆ { } = ←− s , F | ∆ op × ∆ { } = id,and ([ n ] , → ([ n ] ,
1) to the map d : [ n ] ⊳ → [ n ] such that d | [ n ] = id. This defines a naturaltransform of functors ←− S : ←− s → id.Let A : ∆ op → S et +∆ be a functor such that(*) For any vertex [ n ] ∈ ∆ op , A ([ n ]) is an ∞ -category, and for any inert map f : [ n ] → [ m ] such that m ∈ [ m ] is sent to n ∈ [ n ], the functor between ∞ -categories A ( f ) isa categorical fibration.Since A is assumed to be fibrant with respect to the projective model structure, A := N + A ∆ op → ∆ op (cf. [HTT, § ←− S induces a functor A ◦←− s → A in ( S et +∆ ) ∆ op . By assumption (*), this morphism is a fibration withrespect to the projective model structure. Thus, invoking [HTT, 3.2.5.18], we have a fibrationof coCartesian fibrations(5.5.1) q A : ←− s ∗ A ≃ N + A ◦←− s ( ∆ op ) → N + A ( ∆ op ) =: A where the first isomorphism of simplicial sets by (adjoint of) [HTT, 3.2.5.14]. We often abbre-viate q A by q . This map q A is, in particular, a categorical fibration by [HA, B.2.7]. Remark. —
Let A → ∆ op be a coCartesian fibration. Then A is coCartesian equivalent toN + A ( ∆ op ) such that A satisfies (*). Indeed, let ∆ op → S et +∆ be a functor, and put the Reedymodel structure on ( S et +∆ ) ∆ op associated to the Cartesian model structure on S et +∆ . Recall thatfor a simplicial set A and X ∈ ( S et +∆ ) ∆ op , an object hom( A, X ) in S et +∆ is defined in [Du, 4.1].Now, for any cofibration A → B of simplicial sets and a Reedy fibrant object X , the inducedmap hom( B, X ) → hom( A, X ) is a fibration in S et +∆ by [Du, 4.5]. In particular, in view of[Du, 4.2], the the map X ([ n ]) → X ([ m ]) is a fibration for any injective map [ m ] → [ n ]. Thisimplies that any Reedy fibrant object satisfies (*). Now, for F X ( ∆ op ), where X is the object( A → ∆ op ) ♮ in ( S et +∆ ) / N( ∆ op ) , there exists a objectwise weak equivalence, thus Reedy weakequivalence, F X ( ∆ op ) ≃ A • such that A • is a Reedy fibrant, thus the claim follows Let further assume that A is an ∞ -operad. Let e : x → y be an edge in ←− s ∗ A over [ n ] such that σ x → σ y is an equivalence. Then e is a q -Cartesian edge. If t ∈ ←− s ∗ A isan object such that σ ( t ) is a final object in A [1] , then t is q -final.Proof. Let C → ∆ op be a map, φ : [ a ] → [ b ] in ∆ op and c ∈ C [ a ] , d ∈ C [ b ] . Then Map φ ( c, d )denotes the union of connected components of Map( c, d ) lying over φ . Let us show the first claim.Since q is an inner fibration, it suffices to show by [HTT, 2.4.4.3] that for any φ : [ m ] → [ n ] in ∆ op and z ∈ ( ←− s ∗ A ) [ m ] , the diagramMap φ ←− s ∗ A ( z, x ) / / (cid:15) (cid:15) Map φ ←− s ∗ A ( z, y ) (cid:15) (cid:15) Map φ A ( q ( z ) , q ( x )) / / Map φ A ( q ( z ) , q ( y ))46nduced by e is a homotopy Cartesian diagram. Since A is an ∞ -operad, for any w ∈ A [ n ] , wehave homotopy equivalencesMap φ ←− s ∗ A ( z, w ) ≃ Map ( ←− s ∗ A ) [0] (( σ ◦ φ ) ! ( z ) , σ ( w )) × Y
The edge e is not q -coCartesian. Recall the notations from 1.3. Let φ : [ n ] → [ m ] in ∆ op , and put D φ := Γ × γ, ∆ op ,φ ∆ .We put D i := D φ × ∆ { i } for i = 0 ,
1. Let us describe the category D φ more explicitly. It canbe depicted as the following diagram:[ n ] φ (cid:15) (cid:15) / / / / . . . / / φ (0) / / α w w ♣♣♣♣♣♣♣♣♣♣♣♣♣♣ / / . . . / / φ ( k ) / / α k y y sssssssssss . . . / / φ ( m ) / / α m y y rrrrrrrrrrr . . . / / m ] 0 / / / / . . . / / k / / . . . / / m. Here α k for k ∈ [ m ] is the unique map from ([ n ] , φ ( k )) to ([ m ] , k ). We define a function φ ′ : [ φ ( m )] → [ m ] by φ ′ ( i ) := min (cid:8) k ∈ [ m ] | i ≤ φ ( k ) (cid:9) for i ∈ [ n ]. The unique map from ([ n ] , i )to ([ m ] , φ ′ ( i )) is denoted by β i . By construction, α k = β φ ( k ) .Let us define another functor δ : Γ → ∆ op . We put δ ([ n ] , i ) := [ n − i ]. For a map f : ([ n ] , i ) → ([ m ] , j ) given by a map φ : [ m ] → [ n ], we define δ ( f ) : [ n − i ] → [ m − j ] in ∆ op to be mapcorresponding to the function sending k ∈ [ m − j ] to φ ( k + j ) − i . Note that if φ is an inert map,then so is δ ( f ) for any f over φ . Let Φ co (Γ × ∆ , Sch op ) prop be the simplicial subset of Φ co (Γ × ∆ , Sch op )consisting of simplices ∆ k → Φ co (Γ × ∆ , Sch op ) satisfying the following two conditions: • For any vertex corresponding to a functor f : Γ [ n ] × ∆ → Sch op , the square f | ∆ { i,i +1 } × ∆ ,considered as a square in Sch, is a pullback square for any 0 ≤ i < n ; • for any edge corresponding to a functor f : D φ × ∆ → Sch op over φ : [ n ] → [ m ] in ∆ op ,the morphism f ( α k , { } ) is proper and f ( α k , { } ) is an equivalence for any k ∈ [ m ].Now, let us construct a functor α : Φ co (Γ × ∆ , Sch op ) prop → ←− s ∗ Corr ⊛ of categories. A vertexof Φ co (Γ × ∆ , Sch op ) prop corresponds to a diagram F : (∆ n × ∆ ) op → Sch as follows:(5.8.1) X n / / (cid:15) (cid:15) (cid:3) X n − / / (cid:15) (cid:15) (cid:3) . . . / / (cid:3) X (cid:15) (cid:15) / / (cid:3) X (cid:15) (cid:15) Y n / / Y n − / / . . . / / Y / / Y . Here, we put X i := F ( i,
1) and Y i := F ( i, F is contravariant, whichis why ( i,
1) corresponds to X i . A morphism from ( X i → Y i ) to ( X ′ i → Y ′ i ) is a morphism ofdiagrams such that the morphisms X i → X ′ i are proper and morphisms Y i → Y ′ i are equivalences.47he functor α in (5.4.1) is defined to be the functor sending the diagram above to the followingobject in Seq n +1 Corr(Sch ⊲ ) X n / / (cid:15) (cid:15) (cid:3) . . . / / (cid:3) X / / (cid:3) (cid:15) (cid:15) X (cid:15) (cid:15) / / (cid:3) X (cid:15) (cid:15) / / ∞ .Y n = (cid:15) (cid:15) / / . . . / / Y / / = (cid:15) (cid:15) Y / / = (cid:15) (cid:15) Y Y n / / = (cid:15) (cid:15) . . . / / Y / / = (cid:15) (cid:15) Y Y n / / = (cid:15) (cid:15) . . . / / Y ... = (cid:15) (cid:15) . . .Y n By definition, we have the following commutative diagram of functorsΦ co (Γ × ∆ , Sch op ) prop / / ∆ { } → ∆ (cid:15) (cid:15) ←− s ∗ Corr ⊛ q Corr (cid:15) (cid:15) Φ co (Γ , Sch op ) ≃ / ∆ op / / (Corr(Sch) propsep;all ) ⊛ / / Corr ⊛ . Let C → ∆ op be a coCartesian fibration. Let Ψ C := γ ∗ δ ∗ ( C ). By [HTT, 3.2.2.12], thefunctor Ψ C → ∆ op is a coCartesian fibration. A vertex of Ψ C over [ n ] ∈ ∆ op correspondsto a functor ∆ n → C over ∆ op where ∆ n → ∆ op sends i ∈ ∆ n to [ n − i ] and i → i + 1 to d : [ n − i ] → [ n − i − ⋆ C to be the full subcategory of Ψ C spanned by thefunctors φ : ∆ n → C such that the following condition holds: • For each 0 ≤ i < n , the edge φ ( i → ( i + 1)) in C is coCartesian over the map [ n − i ] → [ n − i −
1] in ∆ op defined by the inert map d : [ n − i − → [ n − i ].Let C → D be a map of coCartesian fibrations over ∆ op which preserves coCartesian edges overthe inert map d . Then the induced functor Ψ C → Ψ D induces Ψ ⋆ C → Ψ ⋆ D .Now, let us describe coCartesian edges explicitly. Recall the notations of 5.7. Let φ : [ m ] → [ n ] be a map in ∆ op . Then the diagram δ : D φ → ∆ op can be depicted as[ n ] φ (cid:15) (cid:15) [ n ] / / [ n − / / . . . / / [ n − φ (0)] / / δ ( α ) t t ❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥ / / . . . / / [ n − φ ( k )] / / δ ( α k ) v v ♥♥♥♥♥♥♥♥♥♥♥♥ . . . / / [ n − φ ( m )] / / δ ( α m ) v v ♥♥♥♥♥♥♥♥♥♥♥♥♥♥ . . . / / [0][ m ] [ m ] / / [ m − / / . . . / / [ m − k ] / / . . . / / [0] . In the case φ is an inert map, the map δ ( α k ) is the unique inert map which sends 0 to 0. Avertex of Ψ C over [ n ] (resp. [ m ]) is a functor D → C (resp. D → C ) over ∆ op , and an edgebetween these vertices is a functor F : D φ → C . The description of coCartesian edges of [HTT,3.2.2.12], coCartesian edges over φ are exactly the functors F such that F ( α k ) are coCartesianedges in C over ∆ op . This description, in particular, implies that the induced map Ψ ⋆ C → ∆ op is a coCartesian fibration as well. 48ow, consider the following diagram ∆ op z (cid:15) (cid:15) id id | | ∆ op Γ γ / / δ o o ∆ op where z is the functor sending [ n ] to ([ n ] , γ ∗ δ ∗ → γ ∗ z ∗ z ∗ δ ∗ ≃ idinduces the functor Ψ C → C over ∆ op . Thus, we have a functor G : Ψ ⋆ C → C . Our desiredfunctor β in (5.4.1) is the functor G in the case where C = ←− s ∗ Corr ⊛ . Remark. —
Let us informally describe objects of Ψ( ←− s ∗ A ⊛ ). An object of A ⊛ [ n ] is denoted by M ⊠ M ⊠ · · · ⊠ M n where M i ∈ A by identifying A ⊛ [ n ] and A n . Then objects of Ψ( ←− s ∗ A ⊛ ) over[ n ] ∈ ∆ op are diagrams of the form( M −∞ ⊠ M ⊠ M ⊠ · · · ⊠ M n ) → ( M −∞ ⊠ M ⊠ · · · ⊠ M n ) → · · · → ( M n −∞ ) , where the map ( M i −∞ ⊠ M ii +1 ⊠ · · · ⊠ M in ) → ( M i +1 −∞ ⊠ M i +1 i +2 ⊠ · · · ⊠ M i +1 n ) consists of data M i −∞ ⊗ M ii +1 → M i +1 −∞ and M ij → M i +1 j for j ≥ i + 2. Objects of Ψ A ⊛ over [ n ] are diagramswithout the M i −∞ -factors. In order for it to belong to Ψ ⋆ A ⊛ , the map M ij → M i +1 j should beequivalences. Finally, H ⊤ sends the diagram (5.8.1) to the diagram above such that M i −∞ ≃ Mor( I X i , J Y i ), M ij ≃ Mor( J Y j , J Y j ).
1. Let p : C → D be a categorical fibration between coCartesian fibrationsover ∆ op . Then Ψ p : Ψ C → Ψ D is a categorical fibration. Moreover, an edge e : D φ → C of Ψ C is Ψ p -Cartesian if e ( β i ) is p -Cartesian for i ≤ φ ( m ) and e ([ n ] , i ) for i > φ ( m ) is p -final.2. The functor G is a categorical equivalence.Proof. Let us check the first assertion. The functor Ψ p is a categorical fibration by [HA, B.4.5].Consider the following diagram D v / / i (cid:15) (cid:15) C p (cid:15) (cid:15) D φ w / / e > > ⑥⑥⑥⑥⑥⑥⑥⑥ D . In order to show that the edge e is Cartesian, it suffices to show that the diagram above is a p -right Kan extension diagram by invoking [HA, B.4.8], as usual. For this, we must show thatthe diagram ( D φ ) ⊳ ([ n ] ,i ) / → C is a p -limit diagram. Let i ≤ φ ( m ). Then we have the map β i in D φ . This is an initial object of ( D φ ) ([ n ] ,i ) / . Thus, the assumption that e ( β i ) is a p -Cartesianedge implies that the diagram is p -limit. If i > φ ( m ), then ( D φ ) ([ n ] ,i ) / is empty. Then e ([ n ] , i )must be a p -final object, which follows by assumption.Let us show the second claim. The functor G is a functor between coCartesian fibrations over ∆ op , and moreover, it sends coCartesian edges to coCartesian edges. Thus, by [HTT, 3.3.1.5],it suffices to show that the fibers are trivial fibrations. This follows from [HTT, 4.3.2.15]. (cid:4) .11. We currently have the following diagramΦ co (Γ × ∆ , Sch op ) prop β − ◦ α / / ι = { ∆ { } ֒ → ∆ } (cid:15) (cid:15) H ′⊤ + + Ψ ⋆ ←− s ∗ Corr ⊛ / / Ψ q Corr (cid:15) (cid:15) X ρ / / F ′ (cid:15) (cid:15) (cid:3) Ψ( ←− s ∗ A ⊛ ) F =Ψ q A (cid:15) (cid:15) Φ co (Γ , Sch op ) ≃ / ∆ op / / H Ψ ⋆ Corr ⊛ / / Ψ ⋆ A ⊛ G ∼ (cid:15) (cid:15) / / Ψ A ⊛ A ⊛ . Let E be the collection of the edges D φ → A ⊛ of Ψ A ⊛ such that φ is the identity. Let E ′ be the collection of the F -Cartesian edges in Ψ( ←− s ∗ A ⊛ ) which sits over edges in E . Now, thecondition (A) of [HTT, 3.1.1.6] follows by Lemma 5.10, (B) follows by definition, and (C)follows by combining Lemma 5.6 and Lemma 5.10. Thus, invoking [HTT, 3.1.1.6], the map(Ψ( ←− s ∗ A ⊛ ) , E ′ ) → (Ψ A ⊛ , E ) in S et +∆ has the right lifting property with respect to any markedanodyne. We put the induced marking on Ψ ⋆ A ⊛ from Ψ A ⊛ , and to X by the pullback diagram.The marked simplicial sets are denoted by Ψ ⋆ A ⊛ and X respectively. The map F ′ : X → Ψ ⋆ A ⊛ also has the right lifting property with respect to any marked anodyne.For X, Y ∈ S et +∆ , the marked simplicial set X Y is denoted by Fun + ( Y, X ). Now, considerthe following sequence of functors:Fun + (Φ co (Γ × ∆ , Sch op ) prop ,♭ , X ) a −→ Fun + (Φ co (Γ × ∆ , Sch op ) prop ,♭ , Ψ ⋆ A ⊛ )Fun(Φ co (Γ × ∆ , Sch op ) prop , Ψ ⋆ A ⊛ ) b −→ ∼ Fun(Φ co (Γ × ∆ , Sch op ) prop , A ⊛ ) . By [HTT, 3.1.2.3], a has the right lifting property with respect to any marked anodyne. On theother hand, by [HTT, 1.2.7.3], b is a categorical equivalence.Now, by 4.16, we have the map → H in the ∞ -category Fun(Φ co (Γ , Sch op ) ≃ , A ⊛ ). By com-posing with ι , this induces the map f : ◦ ι → ( G ◦ F ′ ) ◦ H ′⊤ in Fun(Φ co (Γ × ∆ , Sch op ) prop , A ⊛ ).Because b is a categorical equivalence, we can take a map f ′ such that b ( f ′ ) ≃ f . For each object S ∈ Φ co (Γ × ∆ , Sch op ) prop , we have the edge f ( S ) of Φ co (Γ × ∆ , Sch op ) prop . The image of thisedge f ( S ) in ∆ op is constant. Since G is a functor over ∆ op , the image of f ′ ( S ) in ∆ op is con-stant. Thus, by definition of E , f ′ defines a marked edge of Fun + (Φ co (Γ × ∆ , Sch op ) prop ,♭ , Ψ ⋆ A ⊛ ).Since a has the right lifting property with respect to any marked anodyne, we may lift the markededge f ′ along a , and we get a functor H ′⊤ , : Φ co (Γ × ∆ , Sch op ) prop → X and a map H ′⊤ , → H ′⊤ whose composition with G ◦ F ′ is equivalent to f . Finally, put H ⊤ , := ρ ◦ H ′⊤ , . By construction,the edge H ⊤ , ( S ) → H ⊤ ( S ) is F -Cartesian for any S ∈ Φ co (Γ × ∆ , Sch op ) prop . Let δ + : Γ × ∆ → ∆ op be the functor whose restriction to Γ ×{ } is δ and δ + (([ n ] , i ) ,
1) :=[0] such that δ + (cid:0) (([ n ] , i ) , → (([ n ] , i ) , (cid:1) is equal to the map [ n − i ] → [0] in ∆ op correspondingto the function [0] → [ n − i ] sending 0 to n − i . We have the following diagramΓ ι (cid:15) (cid:15) δ (cid:1) (cid:1) γ (cid:29) (cid:29) ∆ op ∆ op ←− s o o Γ × ∆ γ + / / δ + o o ∆ op Γ ι O O [1] f f γ B B where ι i is the inclusion into ∆ { i } ⊂ ∆ and γ + := γ ◦ pr . Let C → ∆ op be a coCartesianfibration. We put Ψ + C := γ + ∗ ◦ ( δ + ◦ ←− s ) ∗ ( C ). We have the map θ : Ψ + → Ψ ◦ ←− s ∗ by using the50djunction id → ι , ∗ ι ∗ , and θ : Ψ + → Φ co (Γ , ( − ) [1] ) by using the adjunction id → ι , ∗ ι ∗ . Wedefine a category Ψ + ,⋆ C by the full subcategory of Ψ + C spanned by vertices corresponding to thefunctors φ : ∆ n × ∆ ≃ Γ [ n ] × ∆ → C × ∆ op , ←− s ∆ op over ∆ op satisfying the following condition: • for each i ∈ ∆ n , the edge φ ( { i } × ∆ ) is a coCartesian edge over the unique active map[ n ] ⊳ → [0] ⊳ in ∆ op .Similarly to the proof of Lemma 5.10.2, the map θ : Ψ + ,⋆ C → Ψ( ←− s ∗ C ) is a categorical equiva-lence. We now define H • as in the following diagram:Φ co (Γ × ∆ , Sch op ) prop H ⊤ , / / H • . . Ψ( ←− s ∗ A ⊛ ) Ψ + ,⋆ ( A ⊛ ) ∼ θ o o θ (cid:15) (cid:15) Φ co (Γ , A ) . Lemma. —
The functor H • preserves coCartesian edges.Proof. For a coCartesian fibration p : C → ∆ op , an edge in C is said to be an inert edge if itis p -coCartesian over an inert map in ∆ op . First, let us check that H • preserves inert edges.Preservation for α is easy to check, that for β follows because it is a Cartesian equivalence. Let[ n ] → [ m ] be the inert map sending 0 ∈ [ m ] to 0 ∈ [ n ]. A coCartesian edge over such a map iscalled a 0 -inert map . Let C → D be a functor between coCartesian fibrations over ∆ op such that0-inert maps are preserved. Then, by the description of coCartesian edges in 5.9, the inducedfunctor Ψ C → Ψ D preserves inert edges. In particular, for a map of coCartesian fibrations C ′ → D ′ over ∆ op which preserves inert edges, the induced functor Ψ ←− s ∗ C ′ → Ψ ←− s ∗ D ′ preservesinert edges because ←− s ∗ C ′ → ←− s ∗ D ′ preserves 0-inert edges. Thus, H ⊤ preserves inert edges.Now, for any inert map [ n ] → [ m ] in ∆ op sending 0 ∈ [ m ] to 0 ∈ [ n ], a map X → Y in ←− s ∗ A ⊛ is coCartesian over ∆ op if and only if σ X → σ Y is an equivalence and q A ( X ) → q A ( Y ) is acoCartesian edge in A ⊛ . The description of coCartesian edges in 5.9 implies that H ⊤ , preservesinert edge as well. Finally, in order to check that H • preserves inert edge, we may describecoCartesian edges of Ψ + ,⋆ A ⊛ similarly to 5.9, and using the fact that F ◦ H ⊤ , is a lifting of along G .We have shown that H • preserves inert edges. Let us treat the general case. Let φ : [ m ] → [ n ]be a map in ∆ op , and e : v → v be a coCartesian edge in Φ co ( . . . ) prop over φ . We wish toshow that H • ( e ) is a coCartesian edge. Let ξ : H • ( v ) → w be a coCartesian edge over φ .Then we have a map D : ∆ → Φ co (∆ , A ) such that D (∆ { , } ) = ξ , D (∆ { , } ) = H • ( e ). Put A := D (∆ { , } ). We mush show that A is an equivalence. Let f : v → v be a coCartesian edgeover σ i . We have a diagram (∆ × ∆ ) ⊳ → Φ co (Γ , A ) depicted as D w / / /o/o/o/o/o/o/o/o/o/o A (cid:15) (cid:15) w ′ B (cid:15) (cid:15) H • ( v ) H • ( e ) / / ξ =} <| :z 9y 8x 7w 6v 5u 4t 3s 3s 2r 1q H • ( v ) H • ( f ) / / /o/o/o/o/o/o H • ( v )[ m ] φ / / [ n ] σ i / / [0]The arrows mean that the edges are coCartesian. Note that H • ( f ) is coCartesian since σ i isan inert map. Let a : X → Y be a map in Fun(∆ n , A ). Then it is an equivalence if and only if σ i ! ( a ) ∈ A is an equivalence for any i . Thus, it suffices to show that B is an equivalence. Now, φ ◦ σ i is an inert map, so a composition of H • ( f ) and H • ( e ) is coCartesian. Thus, B is anequivalence by [HTT, 2.4.1.7, 2.4.1.5]. (cid:4) .13. We have constructed the functor H • . By taking the straightening functor, this is afunctor between certain simplicial objects in C at ∞ . We need to extract a functor of ∞ -categories“associated to” H • . In fact, a simplicial ∞ -category C • has two directions of morphisms. Amorphism of C , namely an object of Fun(∆ , C ), and an object of C , namely an object ofFun(∆ , C ). We wish to “integrate” these two types of morphisms. The functor Int we willconstruct in the rest of this section enables us to do this.Let C be an ∞ -category. By 1.2, the simplicial object ∆ • : ∆ → C at ∞ induces the functor M • C := Fun(∆ • , C ) : ∆ op → C at ∞ . Now, let p : D → C be a Cartesian fibration. An edge∆ → D is said to be p -equivalent if the edge ∆ → D → C is an equivalence. We define f M n p to be the subcategory of M n D spanned by functors ∆ n → D such that any induced edge∆ → ∆ n → D is p -Cartesian, and morphisms ∆ n × ∆ → D such that for any vertex k of∆ n , the induced edge ∆ { k }× id −−−−→ ∆ n × ∆ → D is p -equivalent. Then M • D induces the functor f M • p : ∆ op → C at ∞ . Proposition. —
There exists a functor
Int : Fun(∆ • , C at ∞ ) → C at ∞ such that the followingholds.1. Let C be an ∞ -category. Then we have a canonical functor D : Int( M • C ) → C ;2. If we are given a Cartesian fibration p : D → C of ∞ -categories, the induced functor Int( f M • p ) → Int( M • D ) D −→ D is an equivalence.3. For C • ∈ Fun(∆ • , C at ∞ ) , we have the functor α : C ≃ → Int( C • ) ≃ and for x, y ∈ C , afunctor α x,y : { x } × cat C ≃ , { } Fun(∆ , C ) ≃ × cat { } , C ≃ ,s C ≃ × cat t, C ≃ { y } → Map
Int( C • ) ( α ( x ) , α ( y )) . The maps α and α x,y are functorial with respect to C • . If C • = M • C with an ∞ -category C , we have C ≃ C , C ≃ Fun(∆ , C ) by definition. Under this identification, we have D ≃ ◦ α ≃ id and the following diagram commutes: { x } × cat C ≃ , { } Fun(∆ , C ) ≃ × cat { } , C ≃ , { } Fun(∆ , C ) ≃ × cat { } , C ≃ { y } α x,y / / , , ❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨ Map
Int( C • ) ( α ( x ) , α ( y )) D ( α ( x ) ,α ( y )) (cid:15) (cid:15) Map C ( x, y ) . where the diagonal map is the composition map. Remark. —
Consider C • = f M • p for a Cartesian fibration p : D → C . In this case, the fiberproduct of assertion 3 is the space of maps of the form x a (cid:15) (cid:15) ⋆ b / / y where a is p -equivalent and b is p -Cartesian.For now, assume we have proven the proposition, and complete the proof of the theorem. Proof of Theorem 5.3.
Let p : f Ar(Sch) propsep → Sch be the Cartesian fibration (cf. Remark 5.2).By Definition 5.8, we have f M • p ≃ St(Φ co (Γ × ∆ , Sch op ) prop ). On the other hand, we have52t(Φ co (Γ , A )) ≃ M • A . By Lemma 5.12, we can take the straightening of the functor H • . Thus,we have the diagramInt(St(Φ co (Γ × ∆ , Sch op ) prop )) ∼ (cid:15) (cid:15) IntSt( H • ) / / Int(St(Φ co (Γ , A ))) ∼ Int( M • A ) (cid:15) (cid:15) f Ar(Sch) propsep A . The composition of these functors is the desired functor. The required properties follows byunwinding the construction and the description 5.13.3. (cid:4)
The rest of this section is devoted to proving the proposition. We need some preparations.
Let F : C → D be a Cartesian fibration of (ordinary) categories. Let us construct acategory T as follows: The objects consists of pairs ( c → c ′ , d ) where d ∈ D and c → c ′ is a mapin the fiber C d . A morphism ( c → c ′ , d ) → ( c → c ′ , d ) is a pair of a morphism d → d in D and a diagram c (cid:15) (cid:15) c α / / /o/o/o β o o c (cid:15) (cid:15) c ′ / / c ′ where α is a Cartesian edge over d → d and β is a morphism in C d . We apply this constructionto the case where F = γ : Γ → ∆ op , and get a Cartesian fibration p T : T → ∆ op whose fiber over[ n ] ∈ ∆ op is Tw op ∆ n . Remark. —
This is nothing but the unfurling construction of Barwick (cf. [BGN, 3.2]), andthe construction above can be generalized to a Cartesian fibration between ∞ -categories. Werestricted our attention to ordinary categories just to avoid too much complications.Now, let us construct a functor F : T → ∆ op × ∆ op . Integers 0 ≤ a ≤ b ≤ n determine amap a → b in ∆ n . This map is denoted by ab . The object of T over [ n ] ∈ ∆ op defined by ab is denoted by ( n, ab ). We put F ( n, ab ) := ([ a ] , [ n − b ]). Now, let f : ( n, ab ) → ( m, a ′ b ′ ) be amorphism. Let φ f : [ m ] → [ n ] be a function corresponding to the morphism [ n ] → [ m ] in ∆ op ,and by definition of morphisms in T , f can be written as a diagram a (cid:15) (cid:15) e a / / /o/o/o o o (cid:15) (cid:15) a ′ (cid:15) (cid:15) b / / e b / / /o/o/o b ′ , in Γ where are Cartesian edges over ∆ op . By construction, we have φ f ( a ′ ) = e a , φ f ( b ′ ) = e b , e a ≤ a , and b ≤ e b . Let ψ a := φ f | [ a ′ ] , which yields a function [ a ′ ] → [ e a ] since φ f ( a ′ ) = e a . We alsodefine a function ψ b : [ m − b ′ ] → [ n − e b ] by ψ b ( i ) := φ f ( i + b ′ ) − e b , which is well-defined since φ f ( b ′ ) = e b . For c ≤ d , let d c,d : [ c ] → [ d ] be the function such that d ( i ) = i + d − c , namelythe inert map such that d c,d ( c ) = d . We define the map F ( f ) : ([ a ] , [ n − b ]) → ([ a ′ ] , [ m − b ′ ]) by( ψ a , d n − e b,n − b ◦ ψ b ).Now, assume we are given a functor ∆ op × ∆ op → C . Assume that C admits finite limits.Invoking [HTT, 4.3.3.7], we have the right Kan extension functor p T , ∗ : Fun( T , C ) → Fun( ∆ op , C ),which is a right adjoint to the restriction functor p ∗ T : Fun( ∆ op , C ) → Fun( T , C ). Note that for53 : T → C , p T , ∗ ( f )([ n ]) ≃ lim ←− T [ n ] ( f ) by [HTT, 4.3.1.9] since p T is a Cartesian fibration. Usingthis functor, we define L : Fun( ∆ op × ∆ op , C ) ◦ F −−→ Fun( T , C ) p T , ∗ −−→ Fun( ∆ op , C ) . Finally, let us construct the functor by the compositionInt : Fun( ∆ op , C at ∞ ) ∼ ←− Fun( ∆ op , CSS ) → Fun( ∆ op , Fun( ∆ op , S pc))(5.14.1) ≃ Fun( ∆ op × ∆ op , S pc) L −→ Fun( ∆ op , S pc) JT −−→ C at ∞ . Here JT is the localization functor in 1.15. We wish to show that this functor satisfies theconditions of Proposition 5.13.
An object i → j of Ar(∆ n ) is denoted by ( i ; j ). For example the simplicial set Ar(∆ )can be depicted as (0; 0) (cid:15) (cid:15) (0; 1) (cid:15) (cid:15) / / (1; 1) (cid:15) (cid:15) (0; 2) / / (1; 2) / / (2; 2) . A functor σ : ∆ k × ∆ l → Ar(∆ n ) is said to be a square if σ ( i, j ) = ( a + i ; b + j ) for some integers a , b . Small squares are the squares such that k, l ≤
1. The map σ is a monomorphism ofsimplicial sets, so squares can be viewed as simplicial subsets of Ar(∆ n ). Let φ : ∆ k → Ar(∆ n )be a k -simplex. The ordered pair ( φ (0) , φ ( k )) of vertices of Ar(∆ n ) is called the terminal pair of φ . Let σ be a square. A simplicial subset X of Ar(∆ n ) is said to be saturated if a simplex φ with terminal pair T is contained in X , then any simplex with terminal pair T belongs to X . Example. —
1. Let X be the union of all the squares in Ar(∆ n ). Then X is saturated.Indeed, any square is saturated and the union of saturated simplicial sets is saturated, weget the claim. Let φ : ∆ k → Ar(∆ n ) be a k -simplex, and put φ (0) = ( i ; j ), φ ( k ) = ( i k ; j k ).We can check that φ belongs to X if and only if i k ≤ j .Let us give an alternative description of X which is useful for us. We also denote elementsof Tw(∆ n ) by ( a ; b ). For ( a ; b ) ∈ Tw(∆ n ), consider the map φ : ∆ a × ∆ n − b → Ar(∆ n )such that φ ( i, j ) = ( i, b + j ), which is in fact a square. This gives us a functor ∆ • × ∆ n −• : Tw(∆ n ) → S et ∆ and the morphism of functors ∆ • × ∆ n −• → X where X denotesthe constant functor. Thus, we get a map lim −→ Tw(∆ n ) ∆ • × ∆ n −• → X . This map is anisomorphism of simplicial sets.2. Let X be the union of the small squares in Ar(∆ n ). Then X is saturated. Indeed, let φ : ∆ k → Ar(∆ n ) be a k -simplex, and put φ (0) = ( i ; j ), φ ( k ) = ( i k ; j k ). Then φ belongsto X if and only if i k ≤ j , i k − i ≤
1, and j k − j ≤ Let X be a simplicial subset of Ar(∆ n ) which is saturated and any smallsquares σ belongs to X . Then the inclusion X → Ar(∆ n ) is an inner anodyne.Proof. Given two vertices a = ( i ; j ), b = ( i ′ ; j ′ ) of Ar(∆ n ), the distance of these points, denotedby Dist( a, b ), is defined to be | ( i ′ − i ) + ( j ′ − j ) | if either i ≤ i ′ and j ≤ j ′ or i ≥ i ′ and j ≥ j ′ ,and ∞ otherwise. If Dist( a, b ) = ∞ , then there is no morphism in Ar(∆ n ) from a to b or b a . We denote the distance by D ( a, b ). Given a k -simplex φ : ∆ k → Ar(∆ n ), the length isdefined to be the distance of the terminals, namely Dist( φ (0) , φ ( k )). Let X k be the simplicialsubset of Ar(∆ n ) which is the union of X and simplices of length ≤ k . Since any small squarebelongs to X , we have X = X and X n = Ar(∆ n ). It suffices to check that X k − → X k isan inner anodyne. When k = 2, this follows by the assumption that any small square belongsto X , so we may assume that k >
2. Let S k be the union of the empty set ∅ and the set ofpairs ( a, b ) of objects of Ar(∆ n ) such that D ( a, b ) = k . Put a total order of S k such that ∅ isthe minimum. Take P ∈ S k . Let Y P be the union of X k − and simplices whose terminals are P ′ ∈ S k for P ′ ≤ P , especially, P ∅ = X k − . Let P + be the successor of P . It remains to showthat Y P → Y P + is an inner anodyne. If there exists a simplex of Y P with terminal pair in P + ,then Y P = Y P + since X is assumed saturated. Thus, we may assume that no simplex of Y P has terminal pair P + . Let T be the finite set of simplices φ : ∆ k → Ar(∆ n ) with terminals P + such that φ ( l ) = φ ( l + 1) for any l . The last condition is equivalent to φ being non-degenerate.For any φ ∈ T , the simplices φ | ∆ [ k ] \{ k } , φ | ∆ [ k ] \{ } belong to X k − . Now, for any subset I ⊂ [ k ], φ | ∆ [ k ] \ I belongs to Y P if and only if I contains either 0 or k . Indeed, “if” direction is clear byinduction hypothesis. If I does not contain both 0, k , then the terminals of φ | ∆ [ k ] \ I is P + andthis is not contained in Y P , thus the claim. Let P + = (( i, j ) , ( i ′ , j ′ )). Let φ be the uniqueelement of T such that φ ( l ) = ( i ; j + l ) for l ≤ j ′ − j . φ φ ψ foldunfold upper cornerAn upper corner of φ ∈ T is l ∈ [ k ] \ { , k } such that φ ( l −
1) = ( w − z ), φ ( l ) = ( w ; z ), and φ ( l + 1) = ( w ; z + 1). For φ, ψ ∈ T , we say that ψ is obtained by folding φ , or φ is obtained by unfolding ψ , if there exists 0 < l < k such that φ ( a ) = ψ ( a ) for any a = l and ψ ( l ) = ( w +1; z − φ ( l ) = ( w ; z ). Note that the l -vertex is an upper corner of ψ . We also note that the set ofupper corners U φ of φ ∈ T completely determines φ . Considering the number of foldings from φ , we can put a total ordering on T so that if ψ is obtained by folding φ then φ < ψ . Then φ is the minimum element in T . For φ ∈ T , let Z φ be the union of Y P and ψ for ψ ≤ φ . Let φ ′ isthe successor of φ . It suffices to show that Y P → Z φ and Z φ → Z φ ′ are inner anodynes. Sinceit is similar, we only check the latter case. Put Λ U := S l ∈ U φ ′ ∪{ ,k } ∆ [ k ] \{ l } . We have the mapΛ U → ∆ k φ ′ −→ Ar(∆ n ). Since k >
2, the inclusion Λ U → ∆ k is an inner anodyne by [J, 2.12 (iv)].Thus, it remains to show that Z φ ′ = Z φ ⊔ Λ U ∆ k . For l ∈ U φ ′ , it is clear that φ ′ | ∆ [ k ] \{ l } is in Z φ because the simplicial set ψ ∈ T obtained from φ ′ by unfolding the corner l satisfies ψ ≤ φ bythe choice of the ordering of T and φ ′ | ∆ [ k ] \{ l } is also a simplex of ψ . Let σ be a simplicial subsetof φ ′ which contains U φ ′ ∪ { , k } as vertices. Assume σ belongs to Z φ . Then there would exist ψ ≤ φ such that σ is a simplex of ψ . However, since φ ′ is the minimum element in T whichcontains all the vertices in U φ ′ , we should have ψ ≥ φ ′ . This is a contradiction, and we have σ Z φ , which completes the proof. (cid:4) Let ∆ • , • : ∆ × ∆ → C at ∞ be the functor sending ([ m ] , [ n ]) to ∆ m × ∆ n .Let Ar(∆ • ) : ∆ → C at ∞ be the functor sending [ n ] ∈ ∆ to Ar(∆ n ) . Then we have a canonical quivalence of functors L op (∆ • , • ) ≃ Ar(∆ • ) .Proof. First, let us construct a functor L op (∆ • , • ) → Ar(∆ • ). Let A be the composition T op F op −−→ ∆ × ∆ ∆ • , • −−−→ C at ∞ and B be the composition T op → ∆ Ar −→ C at ∞ . We must define a morphism θ : A → B of functors because we have the adjunction ( p op T , ! , p op , ∗ T ). Let ( m, a ′ b ′ ) → ( n, ab ) be amap in T op corresponding to a map f in T op using the notation of 5.14. Let θ ( n, ab ) : ∆ a × ∆ n − b → Ar(∆ n ) be the functor sending ( i, j ) to ( i ; j + b ). bn a n Ar(∆ n )∆ a × ∆ n − b θ ( n, ab )Note that since Hom( x, y ) is either singleton or empty set for any x, y ∈ Ar(∆ n ), θ ( n, ab ) isdetermined uniquely. The following diagram commutes, thus we have the morphism θ :∆ a ′ × ∆ m − b ′ ψ a × ψ b / / θ ( m,a ′ b ′ ) (cid:15) (cid:15) ∆ a × ∆ n − bθ ( n,ab ) (cid:15) (cid:15) Ar(∆ m ) φ f / / Ar(∆ n ) . Now, let us show that this functor is an equivalence. It suffices to check this for each [ n ] ∈ ∆ .Then L op (∆ • , • )([ n ]) ≃ lim −→ ( a ; b ) ∈ Tw∆ n ∆ a × ∆ n − b where the colimit is taken in C at ∞ . EndowTw∆ n with the Reedy category structure by declaring deg( ab ) = a + b , the direct subcategoryas the one spanned by maps of the form ( a ; b ) → ( a ′ ; b ) for a ≤ b , the inverse subcategoryas the one spanned by maps of the form ( a ; b ) → ( a ; b ′ ) for b ≥ b ′ (cf. [Hr, 15.1.2]). Thenone can check that for any functor of (ordinary) categories F : Tw∆ n → C , the latching objectL ( a ; b ) F ≃ F ( a − b ). Thus, the map L ( a ; b ) ∆ • ,n −• → ∆ a,n − b is a cofibration, and the functor∆ • ,n −• : Tw∆ n → S et +∆ is a Reedy cofibrant diagram. Moreover, by [Hr, 15.10.2], it has fibrantconstant. Thus, by [Hr, 19.9.1], the (ordinary) colimit of ∆ • ,n −• as a simplicial set is equivalentto homotopy colimit by [HTT, 4.2.4.1]. Now the desired claim follows from Lemma 5.16 as wellas Example 5.15.1. (cid:4) Let C be an ∞ -category, and ( S L , S R ) be a factorization system (cf. [HTT,5.2.8.8]). Let Fun(Ar(∆ n ) , C ) ′ be the full subcategory of Fun(Ar(∆ n ) , C ) spanned by the functors F such that F (( i ; j ) → ( i ; j + 1)) is in S L and F (( i ; j ) → ( i + 1; j )) is in S R . We have thefunctor ∆ n → Ar(∆ n ) sending i to ( i ; i ) . The induced functor Fun(Ar(∆ n ) , C ) ′ → Fun(∆ n , C ) is a trivial fibration.Proof. Let X be the union of small squares in Ar(∆ n ). Then the inclusion X → Ar(∆ n ) isan inner anodyne by Lemma 5.16 and 5.15.2. We can similarly define Fun( X, C ) ′ in Fun(X , C )56imilarly to Fun(Ar(∆ n ) , C ) ′ . Let us check that the functors Fun(Ar(∆ n ) , C ) ′ α −→ Fun( X, C ) ′ β −→ Fun(∆ n , C ) are trivial fibrations. The functor α is the pullback of the map Fun(Ar(∆ n ) , C ) → Fun( X, C ). This functor is a trivial fibration by [HTT, 2.3.2.5]. It remains to show that β is atrivial fibration.For a vertex v = ( a ; b ) ∈ Ar(∆ n ), let D v be the 2-simplex of Ar(∆ n ) such such that the 0-vertices are ( a ; b ), ( a ; b + 1), ( a + 1; b + 1). If some of the vertices are not defined, we put D v = ∅ .Similarly, let U v be the 2-simplex such that the 0-vertices are ( a ; b ), ( a + 1; b ), ( a + 1; b + 1), and ∅ if some of the vertices are not defined. We put Y := ∆ n ֒ → Ar(∆ n ), and for i >
0, we define X i := Y i − ∪ S v ∈ Y i − D v , Y i := X i ∪ S v ∈ X i U v inductively. Y X Y We have X n = X . We define Fun( X i , C ) ′ and Fun( Y i , C ) ′ likewise. Since the inclusion X i → Y i is an inner anodyne, the map Fun( Y i , C ) ′ → Fun( X i , C ) ′ is a trivial fibration. The mapFun( X i +1 , C ) ′ → Fun( Y i , C ) ′ is a trivial fibration by [HTT, 5.2.8.17]. Thus, β is a trivial fibrationas required. (cid:4) Proof of Proposition 5.13.
We have already constructed the functor Int. We will show thatthe functor satisfies the required properties. Let C n,m := Fun(∆ n × ∆ m , C ) ≃ . Then we havethe functor C • , • : ∆ op × ∆ op → S pc. Let G : Fun( ∆ op , C at ∞ ) → Fun( ∆ op × ∆ op , S pc) be thecomposition of functors in (5.14.1). By construction, G ( M • C ) ≃ C • , • . In the situation of5.13.2, we denote by e D n,m be the full subcategory (thus space) of D n,m spanned by of maps∆ n × ∆ m → D such that for each vertex i ∈ ∆ n , the edges { i } × ∆ { j,j +1 } are p -equivalent andfor each j ∈ ∆ m the edges ∆ { i,i +1 } × { j } are p -Cartesian.We have the map ∆ n → Ar(∆ n ) sending i to ( i ; i ). This induces the functor of cosimplicial ob-jects ι : ∆ • → Ar(∆ • ). For an ∞ -category C , let A ( C ) be the simplicial spaces Fun(Ar(∆ • ) , C ) ≃ ,namely Fun(Ar(∆ n ) , C ) ≃ is assigned to [ n ]. The functor ι induces the functor A ( C ) → Seq • ( C ).Moreover, we have A ( C ) := Fun(Ar(∆ • ) , C ) ≃ ≃ Fun( L op (∆ • , • ) , C ) ≃ ≃ (cid:0) L (Fun(∆ • , • , C )) (cid:1) ≃ ≃ L (Fun(∆ • , • , C ) ≃ ) =: L ( C • , • ) . where the 1st equivalence follows by Corollary 5.17, and the 3rd equivalence by Lemma 1.4.For the 2nd equivalence, the functor can be constructed using the adjointness of ( p T , ! , p ∗ T ) and( p ∗ T , p T , ∗ ). Then the equivalence is reduced to the equivalence for each term. In this situation, p T , ! and p T , ∗ can be computed by colimits and limits. Thus, we have the functorInt( M • C ) := JT( LG ( M • C )) ≃ JT( A ( C )) → JT(Seq • C ) ∼ −→ C . This is the required functor of 5.13.1.Now, let e A ( p ) be the simplicial subspaces of A ( D ) such that for each [ n ], we consider thesubspace spanned by the functors φ : Ar(∆ n ) → D such that “vertical edges” φ (( i ; j ) → ( i ; j +1))are p -equivalent and “horizontal edges” φ (( i ; j ) → ( i + 1; j )) are p -Cartesian. Let Fun(∆ , D ) cart be the full subcategory of Fun(∆ , D ) spanned by p -Cartesian edges. Then the inclusion57un(∆ , D ) cart → Fun(∆ , D ) is a categorical fibration by [HTT, 2.4.6.5]. Similarly, we de-fine Fun(∆ , D ) cons to be the full subcategory spanned by p -equivalent edges. Then we have e A ( p ) ≃ A ( D ) × cat Q E h Fun(∆ , D ) Y E h Fun(∆ , D ) cart × cat Q E v Fun(∆ , D ) Y E v Fun(∆ , D ) cons , where E h (resp. E v ) is the set of horizontal (res. vertical) edges in Ar(∆ n ). Thus, may write e A ( p ) using limits. We also have similar presentation for e D • , • using limits. Thus, we have L ( e D • , • ) ≃ e A ( p ). It remains to check that the composition e A ( p )([ n ]) → Fun(Ar(∆ n ) , D ) ≃ → Fun(∆ n , D ) ≃ is a homotopy equivalence of spaces. This follows from Corollary 5.18.For C • in Fun( ∆ • , C at ∞ ), unwinding the definition, the space LG ( C • )([0]) is equivalentto C ≃ and LG ( C • )([1]) is equivalent to Fun(∆ , C ) ≃ × cat { } , C ≃ ,s C ≃ . Now, the adjunction map LG ( C • ) → Seq • JT( LG ( C • )) =: Seq • Int( C • ) induces the desired maps of 5.13.3. (cid:4) Remark. —
Let C be an ∞ -category. The proof shows, in fact, Fun(∆ n , Int( M • C )) ≃ ≃ Fun(Ar(∆ n ) , C ) ≃ .
6. Examples
In this section, we exhibit some concrete examples of ( ∞ , Corr (Sch) → LinCat R toapply the results of previous sections. We fix a noetherian scheme S , and let Sch( S ) be a full subcategory of the categoryof noetherian S -schemes which is stable under pullbacks. We denote by T ri be the (2 , T ri ⊗ the (2 , Definition. — A category of coefficients is a functor D : Sch( S ) op → T ri ⊗ . For a morphism f : X → Y in Sch( S ), we denote D ( f ) : D ( Y ) → D ( X ) by f ∗ . The category of coefficients issaid to be premotivic (cf. [CD2, 1.4.2]) if the following conditions are satisfied: • For any smooth separated morphism of finite type f in Sch( S ), the 1-morphism f ∗ , con-sidered as a morphism in T ri, admits a left adjoint f ♯ , and f ∗ and f ♯ satisfies some basechange property (cf. [CD2, 1.1.10]); • For any morphism f , f ∗ , considered as a morphism in T ri, admits a right adjoint, denotedby f ∗ (cf. [CD2, 1.1.12]); • For any smooth separated morphism of finite type f : X → Y , the canonical morphism f ♯ (( − ) ⊗ f ∗ ( − )) → f ♯ ( − ) ⊗ ( − ) of functors D ( X ) × D ( Y ) → D ( Y ) is an equivalence (cf.[CD2, 1.1.27]); • For any X ∈ Sch( S ), the category D ( X ) is closed ( i.e. admits an internal hom).The category Sch( S ) is assumed to be adequate in the sense of [CD2, 2.0.1]: • It is closed under finite sums and pullbacks along morphisms of finite type; • Any quasi-projective S -scheme belongs to Sch( S );58 Any separated morphism of finite type (7) in Sch( S ) admits a compactification (cf. [CD2,2.0.1 (c)] for more precise statement); • Chow’s lemma holds (cf. [CD2, 2.0.1 (d)] for more precise statement).The category of coefficients is said to be motivic if it is premotivic, and moreover, satisfiesthe following conditions (see [CD2, 2.4.45] for more details): • For p : A T → T , the unit map p ♯ p ∗ → id is an equivalence (homotopy property); • For a smooth separated morphism of finite type f : X → T and a section s : T → X , thefunctor f ♯ s ∗ induces a categorical equivalence (stability property); • We have D ( ∅ ) = 0, i ∗ i ∗ → id is an equivalence for any closed immersion Z ֒ → T , and( j ∗ , i ∗ ) is conservative where j is the open immersion T \ Z ֒ → T (localization property); • For any proper morphism f , f ∗ admits a right adjoint (adjoint property).One of the main results of the theory is the following theorem, which roughly says that theproper base change theorem holds for motivic category of coefficients. (Voevodsky, Ayoub, Cisinski-D´eglise [CD2, 2.4.26, 2.4.28]) . — Let D : Sch( S ) op → T ri ⊗ be a motivic category of coefficients. Then the support property and the proper base changeproperty holds: Consider a Cartesian diagram in Sch( S ) X ′ g ′ / / f ′ (cid:15) (cid:15) (cid:3) X f (cid:15) (cid:15) Y ′ g / / Y.
1. If f is proper and g is an open immersion, the canonical map g ! f ′∗ → f ∗ g ′ ! of functors D ( X ′ ) → D ( Y ) , constructed using the equivalence g ′ ! f ′∗ ∼ −→ f ∗ g ! , is an equivalence (namely,the support property holds).2. If f is proper, then the adjunction map g ∗ f ∗ → f ′∗ g ′∗ of functors D ( X ) → D ( Y ′ ) is anequivalence (namely, the proper base change property holds). The following theorem is essentially a consequence of Gaitsgory and Rozenblyum’s exten-sion theorem as well as the theorem above. The theorem roughly says as follows: Assume weare given a motivic category of coefficients D , and assume we wish to upgrade this to an ( ∞ , LinCat . Then all we need to construct is onlyan ∞ -enhancement of D , which is often easy to carry out. The author learned the techniquefrom the thesis of A. Khan [K]. Theorem. —
Let R be an E ∞ -ring, and let D ∗ : Sch( S ) op → CAlg( L in C at R ) be a functor.Assume that the composition Sch( S ) op → CAlg( L in C at R ) → T ri ⊗ (7) In [CD2], they do not impose the morphism to be of finite type. We think this is a typo, otherwise, all theseparated morphisms in Sch( S ) need to be of finite type. s a motivic category of coefficients. Here, the second functor is defined by [HA, 4.8.2.18] and[HA, 1.1.2.14]. Then we have the following commutative diagram Sch( S ) op D ∗ / / (cid:15) (cid:15) L in C at R (cid:15) (cid:15) Corr ( S ) propsep;all D ∗ ! / / LinCat - op R . Here, prop , sep , all denote the classes of proper morphisms, separated morphisms, and all mor-phisms. Remark. —
In fact, we may relax the condition if we do not need ∞ -enhancement of ⊗ and H om . Indeed, in order to get D ∗ ! , it suffices to assume that we are given a functor Sch( S ) op → L in C at R such that the induced functor Sch( S ) op → T ri can be promoted to a motivic categoryof coefficients Sch( S ) op → T ri ⊗ . Proof.
Consider the functor Sch( S ) op D ∗ −−→ L in C at R → LinCat R . Let us show that this functorsatisfies the right Beck-Chevalley condition ([GR, Ch.7, 3.1.5]) with respect to open immersions,namely satisfies the following two conditions: • For any open immersion j : U → X in Sch( S ), the 1-morphism j ∗ : D ( X ) → D ( U ) in LinCat R admits a left adjoint, denoted by j ! ; • For a Cartesian diagram in Sch( S )(6.3.1) X ′ f ′ (cid:15) (cid:15) g ′ / / (cid:3) X f (cid:15) (cid:15) Y ′ g / / Y such that g is an open immersion, the canonical 2-morphism of functors g ′ ! ◦ f ′∗ → f ∗ ◦ g ! is an equivalence.For the existence of left adjoint, it suffices to check that the 1-morphism j ∗ considered as afunctor between underlying ∞ -category admits a left adjoint by Lemma 3.13. An exact functor F between stable ∞ -categories admits left or right adjoint if and only if so does the functorbetween its homotopy categories h F by [NRS, 3.3.1]. Since an open immersion is separatedsmooth of finite type, this follows from the fact that the induced category of coefficients ispremotivic. In order to show that the adjunction map is an equivalence, it suffices to show thisfor the associated homotopy category as well. Thus, the equivalence follows by the base changeproperty of f ♯ in the axiom of premotivic category. Invoking [GR, Ch.7, 3.2.2 (b)], we get afunctor D : Corr ( S ) openopen;all → LinCat R , where open denotes the class of open immersions.Restricting D to Corr ( S ) isoopen;all , where iso is the class of isomorphisms, and take ( − ) - op toboth sides, we get the 2-functor ( Corr ( S ) isoall;open ) - op → LinCat - op R . Since the 2-morphismsare equivalences in the category Corr ( S ) isoall;open , in other words it is an ( ∞ , Corr ( S ) isoall;open ≃ ( Corr ( S ) isoall;open ) - op by inverting the 2-morphisms. Thus, weobtain D : Corr ( S ) isoall;open → LinCat - op R .Now, we wish to show that the composition Sch( S ) → Corr isoall;open D −−→ LinCat - op R sat-isfies the left Beck-Chevalley condition ([GR, Ch.7, 3.1.2]) with respect to “prop”. An adjointpair of 1-morphisms ( f, g ) in a 2-category C is equivalent to giving an adjoint pair of 1-maps( f ′ , g ′ ) in the 2-category C - op , where f ′ , g ′ are corresponding 1-maps in C - op to f , g . Thusverifying the left Beck-Chevalley condition amounts to checking the following two conditions:60 For any proper morphism f : X → Y , the 1-morphism f ∗ : D ( Y ) → D ( X ) in LinCat R admits a right adjoint, denoted by f ∗ ; • For a Cartesian diagram (6.3.1) in Sch( S ) such that f is proper and g is any morphism,the adjunction map g ∗ f ∗ → f ′∗ g ′∗ is an equivalence;In order to check that the 1-morphism f ∗ admits a right adjoint in LinCat R , we need to showthat the underlying functor, denoted by ( f ∗ ) ◦ , between ∞ -category (without linear ∞ -categorystructure) admits right adjoint which commutes with small colimits by Lemma 3.13. Since ( f ∗ ) ◦ is a morphism in P r L by definition of L in C at, ( f ∗ ) ◦ admits a right adjoint f ◦∗ . We need to checkthat this functor commutes with small colimits. Since f ◦∗ is an exact functor by [HA, 1.1.4.1], itsuffices to check that it commutes with small coproducts by [HTT, 4.4.2.7]. This commutation isequivalent to the commutation of small coproducts of the functor associated homotopy categoriesh( f ◦∗ ). By [HTT, 5.2.2.9], h( f ◦∗ ) is right adjoint to h(( f ∗ ) ◦ ), and h( f ◦∗ ) admits a right adjoint h f ! ,because f is proper, by the adjointness axiom of motivic category of coefficients. Thus the claimfollows. The second condition can be checked in the homotopy category, which is nothing butTheorem 6.2. In addition to the left Beck-Chevalley condition, the condition [GR, Ch.7, 5.2.2]holds since the support property holds by Theorem 6.2. This enables us to invoke [GR, Ch.7,5.2.4] for adm = prop, co - adm = open, and get a functor D : Corr ( S ) propall;sep → LinCat - op R .Finally, we take ( − ) - op to get D ∗ ! . (cid:4) Motivic theory of modules6.4.
Assume we are in the situation of Theorem 6.3. Let R ′ be an E ∞ -algebra over R . Thenwe have the scalar extension functor L in C at R → L in C at R ′ (cf. [HA, D.2.4]). Thus, we have D R ′ : Sch( S ) D −→ L in C at R → L in C at R ′ . Now, recall the notations of 3.14. Consider the followingdiagram: P r CAlg L φ CAlg (cid:15) (cid:15) pr ◦ Ξ / / CAlg( L in C at R ′ ) , Sch( S ) op D R ′ / / A CAlg( L in C at R ′ )Assume we are given a dotted arrow in the diagram so that the diagram commutes. Then bycomposing with pr ◦ Ξ, we get a new functor Mod A ( D R ′ ) : Sch( S ) op → L in C at R ′ . Assume thatfor any morphism f in Sch( S ) op , the edge A ( f ) is φ CAlg -coCartesian. In this case, by [CD2,7.2.13, 7.2.18], the underlying theory of coefficients of Mod A ( D R ′ ) is in fact a motivic theory ofcoefficients. Indeed, the underlying category is compatible with [CD2] by [HA, 4.3.3.17]. Thus,we can apply Theorem 6.3.Finally, the construction of A is essentially the same as choosing a commutative algebraobject A ( S ) of D R ′ ( S ). Because we assume that A ( f ) is a coCartesian edge for any morphism f in Sch( S ) op , the following is a φ CAlg -left Kan extension diagram: { S } (cid:15) (cid:15) A ( S ) / / P r CAlg L φ CAlg (cid:15) (cid:15)
Sch( S ) op D R ′ / / A ♥♥♥♥♥♥♥♥♥♥♥♥ CAlg( L in C at R ′ ) . Thus by [HTT, 4.3.2.15, 4.3.2.16], we have the claim. Summing up, if we fix A S ∈ CAlg( D R ′ ( S )),there exists an ∞ -enhancement of the motivic theory associating f : X → S to Mod f ∗ A S ( D R ′ ( X )). ´Etale cohomology theory .5. Let O ⊗ be a symmetric ∞ -operad. Let Mon pres O ( C at ∞ ) be the subcategory of Mon O ( C at ∞ )(cf. [HA, 2.4.2.1]) spanned by O -monoidal ∞ -categories which are compatible with small colimitsand each fiber over X ∈ O is presentable, and those O -monoidal functors which preserve smallcolimits. Lemma. —
Let D be a coCartesian symmetric monoidal ∞ -category. Mon pres D ( C at ∞ ) ≃ Fun( D , CAlg( P r L )) . Proof.
The proof is similar to [GL, 3.3.4.11]. Let K be the set of small simplicial sets. Thenthe inclusion P r L → C at ∞ ( K ) is fully faithful. By [HA, 4.8.1.9] and [HTT, 5.5.3.5], we haveMon pres D ( C at ∞ ) ≃ Alg D ( P r L ). We invoke [HA, 2.4.3.18] to conclude. (cid:4) Let S be a noetherian scheme, and Sch( S ) be the category of noetherian S -schemes. Let´Et be the full subcategory of Fun(∆ , Sch( S )) spanned by ´etale morphisms X → Y over S .We have the functor ´Et → Sch( S ) sending X → Y to Y , which is a Cartesian fibration. Bystraightening, we have the functor ´Et : Sch( S ) op → C at ∞ sending T ∈ Sch( S ) to ´Et( T ), thecategory of ´etale schemes over T . We fix a ring Λ, and consider Mod ⊗ Λ ∈ CAlg( P r LSt ). Theconstruction 1.14 induces a functor PS hv Λ : Sch( S ) op ´Et −→ C at ∞ op −→ C at ∞ Fun( − , Mod ⊗ Λ ) −−−−−−−−−→ CAlg( P r Lst )sending T ∈ Sch( S ) to PS hv Λ ( T ) := Fun(´Et( T ) op , Mod ⊗ Λ ), the ∞ -category of Mod Λ -valuedpresheaves with pointwise symmetric monoidal structure. Now, let F be a presheaf in PS hv Λ ( T ).We say that F is a sheaf if for any ´etale hypercovering U • → V where V ∈ ´Et( T ), the inducedmap F ( V ) → lim ←− F ( U • )is an equivalence. We define S hv Λ ( T ) to be the full subcategory of PS hv Λ ( T ) spanned by sheaves.By [L2, 1.3.4.3], the fully faithful inclusion S hv Λ ( T ) ֒ → PS hv Λ ( T ) admits a left adjoint. By [L2,2.1.2.2], we have an equivalence h S hv Λ ( T ) ≃ D ( T ´et , Λ), where the last category is the (ordinary)derived category. By Lemma 6.5, the functor PS hv Λ gives a coCartesian fibration of symmetric ∞ -operads PS hv ⊗ Λ → Sch( S ) op , × with compatibility conditions. In view of [L2, 1.3.4.4] (or [GL,3.2.2.6]), we may invoke [HA, 2.2.1.9] to get a coCartesian fibration S hv ⊗ Λ → Sch( S ) op , × whichis the fiberwise localization. Since presentability is preserved by localizations, S hv ⊗ Λ yields thefunctor S hv Λ : Sch( S ) op → CAlg( P r Lst ) . By construction, this is an ∞ -enhancement of the functor Sch( S ) op → T ri ⊗ sending T to D ( T ´et , Λ) with pullback functors. When Λ is torsion and there exists an integer n invertiblein S such that n Λ = 0, this functor forms a motivic category of coefficients. This is a conse-quence of marvelous works in SGA, but we need slightly to be careful since we are dealing with unbounded derived categories. To check that it is premotivic, non-trivial points are to check theexistence of f ♯ and the projection formula. The existence of f ♯ follows from [SGA 4, Exp. XVIII,Thm 3.2.5]. For the projection formula, since all the functors involved commute with colimits,we are reduced to checking the formula for compact objects, in which case it is well-known. Toshow that it is motivic, the stability property follows from [CD2, 2.4.19]. The other propertiesare standard. Remark. —
We have treated the torsion cohomology theory, but we can further upgrade thisto ℓ -adic cohomology theory. However, as in the classical theory, the adic formalism is morecomplicated than the torsion theory. In order to avoid too much complications, we decided notto go into adic formalism. 62 table motivic A -homotopy theory6.7. Let S be a noetherian scheme of finite Krull dimension. We put Sch( S ) to be the categoryof noetherian S -schemes of finite Krull dimension. In Robalo’s thesis [R1, 9.3.1], the functor SH ⊗ : Sch( S ) op → CAlg( P r LSt ) ≃ CAlg( L in C at S )is constructed, where S denotes the sphere spectrum and the last equivalence is from [HA,4.8.2.18]. Let us recall his construction for the sake of completeness. The first half of theconstruction is parallel to that of ´etale theory except that we use Nisnevich topology ratherthan ´etale topology and take Λ := S . Then we get a sheaf S hv Nis S : Sch( S ) op → CAlg( P r Lst ) . We need two more operations to acquire SH : localize A and invert P . For each T ∈ Sch( S ),let S T be the collection of morphisms T → p ∗ p ∗ T of S hv S ( T ) where T is a unit object, and p : A T → T . We localize S hv S ( T ) by S T (cf. [HTT, 5.5.4.15]). Invoking Lemma 6.5 similarlyto the construction of S hv out of PS hv, we obtain a functor S hv Nis , A S : Sch( S ) op → CAlg( P r Lst ).Finally, we need to invert P . In fact, this is the crucial part of Robalo’s article [R2]. Heconstructed a map (cf. [R2, 2.6])Loc : P ( free ⊗ (∆ )) ⊗ → P ( L ⊗ ( free ⊗ (∆ ) , ∗ ) ( free ⊗ (∆ ))) ⊗ in CAlg( P r L ). Giving an object of CAlg( P r L ) P ( free ⊗ (∆ )) ⊗ / is equivalent to giving a presentablesymmetric monoidal category C ⊗ and an object X ∈ C . Assume given X ∈ S hv Nis , A S ( S ).The corresponding object of CAlg( P r L ) P ( free ⊗ (∆ )) ⊗ / is denoted by X ′ . Consider the followingdiagram { S } X ′ / / (cid:15) (cid:15) CAlg( P r L ) P ( free ⊗ (∆ )) ⊗ /p (cid:15) (cid:15) Sch( S ) op S hv Nis , A S / / X ′′ ✐✐✐✐✐✐✐✐ CAlg( P r L ) . By [HTT, 2.1.2.2], p is a left fibration, and since S is a final object of Sch( S ), we may take the p -left Kan extension. For C ⊗ ∈ CAlg( P r L ), let Cons( C ⊗ ) be the constant functor Sch( S ) op → CAlg( P r L ) at { C ⊗ } . The Kan extension determines a diagram in Fun(Sch( S ) op , CAlg( P r L )) S hv Nis , A S X ′′ ←−− Cons( P ( free ⊗ (∆ )) ⊗ ) Cons(Loc) −−−−−−→
Cons( P ( L ⊗ ( free ⊗ (∆ ) , ∗ ) ( free ⊗ (∆ ))) ⊗ ) . The pushout of this diagram is denoted by S hv Nis , A S [ X − ]. Finally, let p : P S → S . Let p ∗ := S hv Nis , A S ( p ), and let p ∗ be a right adjoint. Then we define SH ⊗ := S hv Nis , A S [( p ∗ p ∗ S ) − ] . By [R2, 2.23], this functor, in fact, lands in CAlg( P r Lst ). By [HTT, 5.1.2.3], the pushout can becomputed object-wise. Namely, we have an equivalence SH ⊗ ( X ) ≃ S hv Nis , A S ( X ) a P ( free ⊗ (∆ )) ⊗ P ( L ⊗ ( free ⊗ (∆ ) , ∗ ) ( free ⊗ (∆ ))) ⊗ . Thus by [R2, 2.4.4, 2.37], this coincides with the classical stable A -homotopy category of [CD2,1.4.3]. The underlying triangulated category forms a motivic category of coefficients by [CD2,2.4.48], and we may apply the previous theorem to get a 6-functor formalism.63 .8 Example. — For the future record, we summarize what we have constructed. Let k be a perfect field, and take S = Spec( k ). Let R be a (discrete) ring. Recall that Voevodskyintroduced the Eilenberg-MacLane spectrum HR S in CAlg( SH ( S )), which is a Z -module (cf.[CD1, 2.12] for more detail). A principal application of the construction in 6.4 is when we take A ( S ) to be HR S ⊗ Z R . This spectrum yields a motivic theory of coefficients as in [CD1, 4.3].Our construction above gives an ∞ -enhancement of this theory. In particular, by Theorem 5.3for I = Spec( k ) , J = Spec( k ) ( d ), where d is an integer and ( d ) denotes the Tate twist, we havea functor H( d ) : f Ar propsep (Sch( k )) op → Mod R ≃ D ( R ) , such that p : X → Y over k is sent to Mor Mod HR Y ( p ! p ∗ Y , Y ( d )) in D ( R ). For exampleH(id : X → X ) coincides with the motivic cohomology H ∗ M ( X, R ( d )), at least when X is smooth(cf. [CD2, 11.2.3, 11.2.c]), and H( X → Spec( k )) is nothing but the motivic Borel-Moore theory.The functor H unifies these two theories, and gives an ∞ -enhancement. Arithmetic D -module theory6.9. Let k be a perfect field of characteristic p >
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