An equivalence between enriched ∞ -categories and ∞ -categories with weak action
aa r X i v : . [ m a t h . A T ] S e p AN EQUIVALENCE BETWEEN ENRICHED ∞ -CATEGORIES AND ∞ -CATEGORIES WITH WEAK ACTION Hadrian Heine
Abstract.
We show that an ∞ -category M left tensored over a monoidal ∞ -category V is completelydetermined by its graph M ≃ × M ≃ → P ( V ) , ( X , Y ) ↦ M ((−) ⊗ X , Y ) , parametrized by the maximal subspace M ≃ in M , equipped with the structure of an enrichmentin the sense of Gepner-Haugseng in the Day-convolution monoidal structure on the ∞ -category P ( V ) of presheaves on V . Precisely, we prove that sending an ∞ -category left tensored over V to itsgraph defines an equivalence between ∞ -categories left tensored over V and a subcategory of all ∞ -categories enriched in presheaves on V . More generally we consider a generalization of ∞ -categoriesleft tensored over V , which Lurie calls ∞ -categories pseudo-enriched in V , and extend the formerequivalence to an equivalence χ between ∞ -categories pseudo-enriched in V and all ∞ -categoriesenriched in presheaves on V . The equivalence χ identifies V -enriched ∞ -categories in the sense ofLurie with V -enriched ∞ -categories in the sense of Gepner-Haugseng. Moreover if V is symmetricmonoidal, we prove that sending an ∞ -category left tensored over V to its graph is lax symmetricmonoidal with respect to the relative tensorproduct on ∞ -categories left tensored over V and thecanonical tensorproduct on P ( V ) -enriched ∞ -categories. Contents
1. Motivation and overview 11.1. Overview 61.2. Notation and terminology 71.3. Acknowledgements 72. Weakly tensored ∞ -categories 83. Weakly enriched ∞ -categories 244. Extracting a weakly enriched ∞ -category 275. A universal property of the extracted weakly enriched ∞ -category 336. A universal property of enriched presheaves 377. The equivalence 428. Generalized Day-convolution 519. Appendix 59References 651. Motivation and overview
In mathematics it ubiquitiously happens that the set of morphisms between two mathematicalstructures of the same kind refines to an object of some monoidal category V as well as the compositionof morphisms refines to a morphism in V . This leads to the notion of V -enriched category, whichsubsumes the following two special cases: ∞ -CATEGORIES AND ∞ -CATEGORIES WITH WEAK ACTION (1) If a V -enriched category has only one object, one may identify it with the enriched endomor-phisms of the unique object, which is an associative algebra in V . So V -enriched categorieswith one object are associative algebras in V .(2) If V is closed monoidal and a V -enriched category C has V -tensors, i.e. for any Y ∈ V and Z ∈ C the functor T ↦ Mor V ( Y , Mor C ( Z , T )) , C → S is corepresentable by some object Y ⊗ Z, the V -tensors endow C with a closed left V -action.Conversely, every category with a closed left V -action has an underlying V -enriched categorythat admits V -tensors, where the V -enrichment comes from the closedness. So V -enrichedcategories with V -tensors are categories with closed left V -action.These two special cases motivate two kind of perspectives on V -enriched categories:(1) V -enriched categories are associative algebras in V with many objects.(2) V -enriched categories are a relaxed version of categories with closed left V -action, in which V -tensors do not need to exist.Goal of this article is to identify both perspectives on enriched categories in the ∞ -categorical world.Here categories enriched in a monoidal category are replaced by ∞ -categories enriched in a monoidal ∞ -category V like the derived ∞ -category of a commutative ring, the ∞ -category of spectra or the ∞ -category of (∞ , n ) -categories for some n ≥ , which leads to the notions of dg- ∞ -categories, spectral ∞ -categories or (∞ , n + ) -categories for some n ≥ . In the ∞ -categorical world perspective 1. wasturned to a definition: Gepner and Haugseng [3] encode homotopy-coherent enrichment in a monoidal ∞ -category V as a many object version of A ∞ -algebra ( = homotopy-coherent associative algebra) in V that has not only one object but a space X of objects. To describe these many object versionsof A ∞ -algebras Gepner and Haugseng build a theory of non-symmetric ∞ -operads parallel to Lurie’stheory of symmetric ∞ -operads [6]. Independently Hinich proposes a different definition of homotopy-coherent enrichment: If V admits the neccessary colimits that are preserved component-wise by thetensorproduct, Hinich [5] endows the ∞ -category of functors X × X → V with a monoidal structure,whose associative algebras were identified by Macpherson [8] with Gepner and Haugseng’s definition.On the other hand Lurie ([6] Definition 4.2.1.25.) takes the second perspective when defining enriched ∞ -categories: A left action of a monoidal ∞ -category V on an ∞ -category is coherently encoded by aLM-monoidal ∞ -category, a cocartesian fibration of ∞ -operads over the operad LM with two colorsgoverning an object with left action, whose restriction to the acting color gives V . Weakening thecondition of cocartesian fibration to a much weaker compatibility condition Lurie relaxes the notionof ∞ -category left tensored over V to the notion of ∞ -category pseudo-enriched in V . By definitionpseudo-enriched ∞ -categories specialize to ∞ -categories left tensored over V iff all V -tensors existand specialize to Lurie’s notion of V -enriched ∞ -categories iff all V -enriched morphism objects exist.Given an arbitrary ∞ -category C pseudo-enriched in V that does not neccessarily admit V -tensors ormorphism objects, one can still consider for any objects Z ∈ V , X , Y ∈ C the space of multi-morphismsMul C ( Z , X; Y ) in the operadic structure on C that as a functor in Z ∈ V defines a presheaf on V . If thispresheaf is representable, it identifies with the V -enriched morphism object of X and Y. Otherwise itmay be seen as a morphism object in presheaves on V . To identify perspectives 1. and 2. we promotefor any ∞ -category C pseudo-enriched in V the graphΓ C ∶ C ≃ × C ≃ → P ( V ) , ( X , Y ) ↦ Mul C (− , X; Y ) to a P ( V ) -enriched ∞ -category in the sense of Gepner-Haugseng ([3] Definition 2.4.4.), where P ( V ) denotes the ∞ -category of presheaves on V endowed with Day-convolution, and for an enriched ∞ -category in the sense of Gepner-Haugseng we always assume that the completion condition ([3] Defi-nition 5.1.7.) is satisfied. For the case that C is an ∞ -category with closed left action over V this wasachieved by Gepner-Haugseng ([3] Theorem 7.4.7.) as well as Hinich ([5] Proposition 6.3.1.) by ex-tending Lurie’s theory of endomorphism objects ([6] 4.7.1) to morphism objects: Given an ∞ -category N EQUIVALENCE BETWEEN ENRICHED ∞ -CATEGORIES AND ∞ -CATEGORIES WITH WEAK ACTION 3 C with closed left action over some monoidal ∞ -category V for any object X ∈ C the V -enriched en-domorphisms of X promote to an associative algebra in V , the endomorphism algebra. Consideringnot only the enriched endomorphisms of one object but all morphism objects between all objects of C the same time one does not get an associative algebra in V anymore but a many object version ofit, a V -enriched ∞ -category in the sense of Gepner-Haugseng with space of objects C ≃ , the maximalsubspace in C . Extending Hinich’s construction we define a functor χ ∶ {∞ -categories pseudo-enriched in V } → Cat P ( V )∞ , C ↦ Γ C , where Cat P ( V )∞ denotes the ∞ -category of small ∞ -categories enriched in the sense of Gepner-Haugsengin the monoidal ∞ -category P ( V ) of presheaves on V . Our main theorem (Theorem 7.3) says that χ is an equivalence: Theorem 1.1.
For any monoidal ∞ -category V the functor χ is an equivalence. Under the equivalence χ the V -enriched ∞ -categories in the sense of Lurie correspond to the V -enriched ∞ -categories in the sense of Gepner-Haugseng: The equivalence χ ∶ {∞ -categories pseudo-enriched in V } ≃ Cat P ( V )∞ restricts to an equivalence {∞ -categories enriched in V } ≃ Cat V ∞ identifying Lurie’s with Gepner-Haugseng’s notion of enriched ∞ -categories.Also under the equivalence χ the ∞ -categories left tensored over V correspond to P ( V ) -enriched ∞ -categories that admit V -tensors in the evident sense.Moreover we improve χ in several respects: ● Source and target of χ carry both closed left actions over Cat ∞ , the ∞ -category of small ∞ -categories, and χ preserves the left actions. In other words χ is a Cat ∞ -enriched equivalence,an equivalence of (∞ , ) -categories. ● If V is symmetric monoidal, the ∞ -category LMod V of small ∞ -categories left tensored over V carries a symmetric monoidal structure, the relative tensorproduct, by identifying ∞ -categories left tensored over V with left modules in Cat ∞ . On the other hand Gepner-Haugseng([3] Proposition 4.3.10.) and Hinich ([5] Corollary 3.5.3.) construct a symmetric monoidalstructure on Cat P ( V )∞ . The restriction of χ to LMod V is lax symmetric monoidal (Theorem7.15). ● We give a version of χ for V a non-symmetric ∞ -operad. If V is only a non-symmetric ∞ -operad, the notion of ∞ -category pseudo-enriched in V is not defined as well as the Day-convolution monoidal structure on P ( V ) . But Lurie gives the notion of ∞ -category weaklyenriched in V ([6] Definition 4.2.1.13.), which we call ∞ -category weakly left tensored over V , and which is a synonym for ∞ -operad equipped with a map to the LM-operad, whoserestriction to Ass is V . On the other hand V embeds into its enveloping monoidal ∞ -categoryEnv ( V ) , the smallest monoidal ∞ -category containing V , and we have the Day-convolutionon P ( Env ( V )) . The functor χ extends to an equivalence {∞ -categories weakly left tensored over V } ≃ Cat P ( Env ( V ))∞ of (∞ , ) -categories (Theorem 7.11). ● By an enriched ∞ -category in the sense of Gepner-Haugseng we always assume that thecompletion condition ([3] Definition 5.1.7.) is satisfied. If this condition is not satisfied, wecall enriched ∞ -categories in the sense of Gepner-Haugseng enriched ∞ -precategories, whichform an ∞ -category PreCat V ∞ . AN EQUIVALENCE BETWEEN ENRICHED ∞ -CATEGORIES AND ∞ -CATEGORIES WITH WEAK ACTION We also have a version of χ for enriched ∞ -precategories: We call an ∞ -category M weaklyleft tensored over V equipped with an essentially surjective map τ ∶ X → M ≃ a flagged ∞ -category weakly left tensored over V , where we take the terminology from [1]. The functor χ extends to an equivalence { flagged ∞ -categories weakly left tensored over V } ≃ PreCat P ( Env ( V ))∞ (Theorem 7.3).In the following we sketch how we construct the functor χ ∶ {∞ -categories weakly left tensored over V } ≃ Cat P ( Env ( V ))∞ and how we prove that χ is an equivalence, where V is a non-symmetric ∞ -operad.By construction the functor Ψ reflects equivalences. So it will be enough to check the following: ● The functor χ has a left adjoint. ● The unit is an equivalence.We start with explaining how we construct χ : Given a small space X and presentably monoidal ∞ -category V Hinich [5] constructs a monoidal structure on the ∞ -category of functors X × X → V , whose associative algebras are canonically identified with V -enriched ∞ -precategories with space ofobjects X. This description of V -enriched ∞ -precategories as associative algebras is very powerful asthe monoidal structure on functors X × X → V acts in various ways: A closed left V -action on an ∞ -category M with colimits leads to a closed left action of the ∞ -category of functors X × X → V on the ∞ -category of functors X → M , where the morphism object of two functors F , G ∶ X → M isthe functor X × X → V sending A , B to Mor M ( F ( A ) , G ( B )) ([5] Proposition 6.3.1.). Especially theendomorphism algebra of any functor X → M is an associative algebra in the monoidal structure onfunctors X × X → V and so canonically identified with a V -enriched ∞ -precategory with space of objectsX , which turns out to be a V -enriched ∞ -category with space of objects X . To give χ we combine this action with the following construction: Any small ∞ -category M weaklyleft tensored over V universally embeds into an ∞ -category left tensored over the enveloping monoidal ∞ -category of V , which we call the enveloping ∞ -category with left action. Embedding the enveloping ∞ -category with left action into presheaves we get an ∞ -category M ′ with closed left action over V ′ that we call the enveloping ∞ -category with closed left action characterized by the following universalproperty: For any ∞ -category N left tensored over V ′ that admits small colimits preserved by the leftaction in both variables there is a canonical equivalence(1) LinFun L V ′ ( M ′ , N ) ≃ LaxLinFun V ( M , N ) between the ∞ -category of left adjoint V ′ -linear functors M ′ → N and the ∞ -category of lax V -linearfunctors M → N , where we restrict the left V ′ -action on N to V . So the graph M ≃ × M ≃ → M ′ , ( A , B ) ↦ Mor M ′ ( A , B ) associated to M is the endomorphism algebra of the inclusion M ≃ ⊂ M ⊂ M ′ and so a V ′ -enriched ∞ -category with space of objects M ≃ , which is the image of M under χ. By the universal property ofthe endomorphism algebra for any V ′ -enriched ∞ -category C with small space of objects X and anymap of spaces τ ∶ X → M ≃ there is a correspondence between V ′ -enriched functors C → χ ( M ) lyingover τ and right C -modules in Fun ( X , M ′ ) lying over τ ∶ X → M ≃ ⊂ M ′≃ . We enhance this by the following theorem:
Theorem 1.2. (Theorem 5.3) Let V be a small non-symmetric ∞ -operad, M a small ∞ -categoryweakly left tensored over V and C a V ′ -enriched ∞ -precategory with small space of objects.We write Fun V ( C , χ ( M )) for the ∞ -category of left C -modules in Fun ( X , M ′ ) lying over functors X → M ⊂ M ′ . N EQUIVALENCE BETWEEN ENRICHED ∞ -CATEGORIES AND ∞ -CATEGORIES WITH WEAK ACTION 5 There is a canonical equivalence
Fun V ( C , χ ( M )) ≃ ≃ Cat P ( Env ( V ))∞ ( C , χ ( M )) over S ( X , M ≃ ) natural in C and M . Note that for any small space X and presentably monoidal ∞ -category V we have a canonicalequivalence Fun ( X × X , V ) rev ≃ Fun ( X × X , V rev ) of monoidal ∞ -categories, where (−) rev indicatesthe reversed monoidal structure, under which a V -enriched ∞ -precategory C with space of objects Xcorresponds to its opposite V rev -enriched ∞ -precategory C op . This allows to dualize the action of the ∞ -category of functors X × X → V : a closed right V -action on an ∞ -category M with colimits leads to aclosed right action of the ∞ -category of functors X × X → V on the ∞ -category of functors X → M thatfor the case M = V is moreover compatible with the left diagonal V -action. We call the ∞ -category P V ( C ) ∶ = RMod C ( Fun ( X , V )) ≃ Fun V rev ( C op , χ ( V rev )) the ∞ -category of V -enriched presheaves on C that is canonically left tensored over V .We prove the following universal property of P V ( C ) : Theorem 1.3. (Theorem 6.1) Let V be a presentably monoidal ∞ -category.For any ∞ -category M left tensored over V that has small colimits that are preserved by the leftaction in both variables and any V -enriched ∞ -precategory C with small space of objects X there is acanonical equivalence LinFun L V ( P V ( C ) , M ) ≃ Fun V ( C , χ ( M )) = LMod C ( Fun ( X , M )) over Fun ( X , M ) , where LinFun L V ( P V ( C ) , M ) denotes the ∞ -category of left adjoint V -linear functors P V ( C ) → M . In [6] Theorem 4.8.4.1. Lurie proves an important universal property of the ∞ -category RMod A ( V ) of right A-modules in a presentably monoidal ∞ -category V : There is a canonical equivalence(2) LinFun L V ( RMod A ( V ) , M ) ≃ LMod A ( M ) , which is Theorem 1.3 for X contractible.For M = P V ( C ) the identity corresponds to a V -enriched functor C → χ ( P V ( C )) , which we call the V -enriched Yoneda-embedding and which by Proposition 6.5 is a V -enriched embedding in the sensethat it induces equivalences on V -enriched morphism objects. In [5] Hinich gives another constructionof V -enriched Yoneda-embedding, which we don’t see how to relate to ours.By Theorem 1.3 we have for any small non-symmetric ∞ -operad V , small ∞ -category M weakly lefttensored over V and V ′ -enriched ∞ -precategory C with small space of objects X a canonical equivalence(3) LinFun L V ′ ( P V ′ ( C ) , M ′ ) ≃ Fun V ′ ( C , χ ( M ′ )) over Fun ( X , M ′ ) , whose pullback to Fun ( X , M ) ⊂ Fun ( X , M ′ ) gives an equivalence(4) ̃ LinFun L V ′ ( P V ′ ( C ) , M ′ ) ≃ Fun V ( C , χ ( M )) , where the left hand side denotes the pullback to Fun ( X , M ) . We combine equivalence 4 with the following theorem, where we writeL ( C ) ⊂ P V ′ ( C ) for the essential image of the V ′ -enriched Yoneda-embedding with its restricted weak left V -action: Theorem 1.4. (Theorem 7.6) Let V be a small non-symmetric ∞ -operad, M a small ∞ -categoryweakly left tensored over V and C a V ′ -enriched ∞ -precategory with small space of objects.The embedding L ( C ) ⊂ P V ′ ( C ) exhibits P V ′ ( C ) as the enveloping ∞ -category with closed left action over V ′ . AN EQUIVALENCE BETWEEN ENRICHED ∞ -CATEGORIES AND ∞ -CATEGORIES WITH WEAK ACTION By Theorem 1.4 and equivalence 1 there is a canonical equivalence(5) ̃ LinFun L V ′ ( P V ′ ( C ) , M ′ ) ≃ ≃ LaxLinFun V ( L ( C ) , M ) . Combining equivalences 4 and 5 we get for any small non-symmetric ∞ -operad V , small ∞ -category M weakly left tensored over V and V ′ -enriched ∞ -precategory C with small space of objects a canonicalequivalence(6) LaxLinFun V ( L ( C ) , M ) ≃ Fun V ( C , χ ( M )) showing that χ admits a left adjoint. Moreover the V ′ -enriched Yoneda-embedding C → χ ( P V ′ ( C )) factors as the unit C → χ ( L ( C )) followed by the canonical embedding χ ( L ( C )) ⊂ χ ( P V ′ ( C )) . So theunit C → χ ( L ( C )) is a V ′ -enriched embedding and by definition induces an essentially surjective map onspaces of objects. By [3] Corollary 5.5.3. a V ′ -enriched embedding between V ′ -enriched ∞ -categoriesis an equivalence if it induces an essentially surjective map on spaces of objects. Thus the unit is anequivalence.Given a small space X and a presentably monoidal ∞ -category V Hinich [5] constructs the monoidalstructure on functors X × X → V and its associated actions via Day-convolution, the universal monoidalstructure on the ∞ -category of functors from one monoidal ∞ -category to another. The space X × Xdoes not carry the structure of a monoidal ∞ -category but at least of a promonoidal ∞ -categoryconstructed by Hinich ([5] Proposition 3.3.6.). Moreover Hinich proves that a promonoidal structureon X × X suffices to guarantee the existence of Day-convolution on the ∞ -category of functors X × X → V ([5] Proposition 2.8.3). Gepner and Haugseng ([3] Definition 2.4.4.) define V -enriched ∞ -categorieswith space of objects X as a many object version of associative algebras in V , precisely as algebras over ageneralized non-symmetric ∞ -operad Ass X with space of colors X × X . Every generalized non-symmetric ∞ -operad can universally be turned into a non-symmetric ∞ -operad, its operadic localization, thathas the same algebras in V . By work of Macpherson [8] the generalized non-symmetric ∞ -operad Ass X localizes to Hinich’s promonoidal structure on X × X . We prefer to work with the generalized non-symmetric ∞ -operad Ass X that seems by far easier to deal with than its operadic localization. Butworking with Ass X makes it neccessary to construct a Day-convolution for generalized non-symmetric ∞ -operads and weakly left, right and bitensored ∞ -categories, which is the content of section 8.1.1. Overview.
Next we give a short overview over the content:Goal of this article is to construct an equivalence χ ∶ {∞ -categories weakly left tensored over V } ≃ Cat P ( Env ( V ))∞ for a non-symmetric ∞ -operad V (Theorem 7.3).Section 2 recalls the notion of (generalized) non-symmetric ∞ -operad (Definition 2.10) as developedby Gepner and Haugseng ([3] Definition 3.1.3.) and introduces weakly left, right and bitensored ∞ -categories (Definition 2.18 and 2.24) that generalize the notions of left, right and bitensored ∞ -categories. Section 3 recalls the notion of enriched ∞ -categories in the sense of Gepner-Haugseng.Section 4 is devoted to construct the functor χ using a theory of Day-convolution for generalizednon-symmetric ∞ -operads and weakly left, right and bitensored ∞ -categories developed in section 8.In section 5 we prove Theorem 1.2 (Theorem 5.3) via a theory of endomorphism objects in families.In section 6 we prove Theorem 1.3 (Theorem 6.1). In section 7 we prove Theorem 1.4 (Theorem 7.6)to finally deduce our main Theorem 7.3. Moreover we refine Theorem 7.3: We show that the functor χ is Cat ∞ -linear (Theorem 7.11) (and so an equivalence of (∞ , ) -categories) and we prove that χ restricted to ∞ -categories left tensored over V is a lax symmetric monoidal functor (Theorem 7.15). N EQUIVALENCE BETWEEN ENRICHED ∞ -CATEGORIES AND ∞ -CATEGORIES WITH WEAK ACTION 7 Notation and terminology.
We write ● Set for the category of small sets, ● ∆ for (a skeleton of) the category of finite, non-empty, partially ordered sets and order pre-serving maps, whose objects we denote by [ n ] = { < ... < n } for n ≥ ● S for the ∞ -category of small spaces, ● Cat ∞ for the ∞ -category of small ∞ -categories.Given an ∞ -category C containing objects A , B we write ● C ( A , B ) for the space of maps A → B in C , ● C / A ∶ = { A } × Fun ({ } , C ) Fun ([ ] , C ) for the ∞ -category of objects over A, ● Ho ( C ) for its homotopy category, ● C ◁ , C ▷ for the ∞ -categories arising from C by adding an initital respectively final object, ● C ≃ for the maximal subspace within C .We often call a fully faithful functor C → D an embedding. We call a functor φ ∶ C → D an inclusion(or subcategory inclusion) or say that φ exhibits C as a subcategory of D if for any ∞ -category B the induced map Cat ∞ ( B , C ) → Cat ∞ ( B , D ) is an embedding (of spaces). A functor φ ∶ C → D is aninclusion if and only if it induces an embedding on maximal subspaces and on all mapping spaces.A functor φ ∶ C → D is an inclusion (embedding) if and only if the induced functor Ho ( φ ) ∶ Ho ( C ) → Ho ( D ) is an inclusion (embedding) and the canonical functor C → Ho ( C ) × Ho ( D ) D is an equivalence.In this case φ is uniquely determined by D and Ho ( φ ) ∶ Ho ( C ) → Ho ( D ) . We call a functor C → S flat if the functor C × S ( − ) ∶ Cat ∞/ S → Cat ∞/ C preserves small colimits.The opposite of a flat functor is flat. By [6] Example B.3.11. every cocartesian fibration is flat andso dually every cartesian fibration is flat. Given functors C → T , T → S such that the composition C → T → S is flat, the canonical functor ( − ) × S C ∶ Cat ∞/ S Ð→ Cat ∞/ C → Cat ∞/ T preserves small colimits and so admits a right adjoint Fun ST ( C , − ) by presentability of Cat ∞/ S .If S , T are contractible, we drop S respectively T from the notation. Note that Fun T ( C , D ) is the ∞ -category of functors C → D over T . For any functors D → T and S ′ → S we have a canonicalequivalence S ′ × S Fun ST ( C , D ) ≃ Fun S ′ S ′ × S T ( S ′ × S C , S ′ × S D ) . [6] Theorem B.4.2. implies the following remark: Remark 1.5.
Let δ ⊂ Fun ([ ] , S ) be a full subcategory such that every equivalence of S belongs to δ and every composition of composable arrows of δ belongs to δ . Let α ∶ T → S be a cocartesianfibration relative to δ . Denote δ T ⊂ Fun ([ ] , T ) the full subcategory spanned by the α -cocartesianlifts of morphisms of E . Let C → T be a cocartesian fibration relative to δ T and D → T a cartesian fibration relative to δ T . Then the functor ψ ∶ Fun ST ( C , D ) → S is a cartesian fibration relative to δ T , where a morphism ofFun ST ( C , D ) lying over a morphism of δ corresponding to a functor γ ∶ [ ] × S C → [ ] × S D over [ ] × S Tis ψ -cartesian if γ sends cocartesian lifts of the morphisms of [ ] to cartesian lifts.1.3. Acknowledgements.
We thank Markus Spitzweck for helpful discussions.
AN EQUIVALENCE BETWEEN ENRICHED ∞ -CATEGORIES AND ∞ -CATEGORIES WITH WEAK ACTION Weakly tensored ∞ -categories In this section we define ∞ -categories weakly left, right and bitensored over a non-symmetric ∞ -operad, which generalize ∞ -categories left, right and bitensored over a monoidal ∞ -category.A monoidal structure on an ∞ -category V may be encoded by a cocartesian fibration V ⊗ → ∆ op , whose fiber over [ n ] is canonically identified with V × n . A left action of a monoidal ∞ -category V on an ∞ -category C may be encoded by a cocartesian fibration O → ∆ op × [ ] such that the fiberover ([ n ] , ) , ([ n ] , ) is canonically identified with V × n × C respectively V × n . Similarly a biaction oftwo monoidal ∞ -categories V , W on C may be encoded by a cocartesian fibration O → ( ∆ /[ ] ) op suchthat the fiber over [ n ] ∗ [ m ] → [ ] ∗ [ ] = [ ] is canonically identified with V × n × C × W × m . On theother hand such a left respectively biaction may also be encoded by functors M → V in case of a left V -action respectively M → V × W in case of a ( V , W ) -biaction such that the fiber of the composition M → V → ∆ op over [ n ] is canonically identified with V × n × C respectively the fiber of the composition M → V × W → ∆ op × ∆ op over [ n ] , [ m ] is canonically identified with V × n × C × W × m . In this section wegeneralize actions of one or two monoidal ∞ -categories to weak actions of one or two non-symmetric ∞ -operads V respectively V , W . We encode these weak actions by certain relative cocartesian fibrations O → ∆ op × [ ] , O → ( ∆ /[ ] ) op allowing the same identification of fibers, and also by certain functors M → V respectively M → V × W allowing the same identification of fibers. We identify both definitionsin Propositions 2.22 and 2.25.We start with defining (generalized) non-symmetric ∞ -operads. For a detailed treatment about(generalized) non-symmetric ∞ -operads we recommend [3] section 3. for details. To define ∞ -operadselegantly we use a simplified variant of Lurie’s notion of categorical pattern based on the concept ofrelative limit: Relative limits 2.1.
Given an ∞ -category K and a functor φ ∶ C → S a functor H ∶ K ⊲ → C isa φ -limit diagram if restriction to K yields for every functor F ∶ K ⊲ → C inverting all morphisms apullback square Fun ( K ⊲ , C )( F , H ) (cid:15) (cid:15) / / Fun ( K , C )( F ∣ K , H ∣ K ) (cid:15) (cid:15) Fun ( K ⊲ , S )( φ ○ F , φ ○ H ) / / Fun ( K , S )( φ ○ F ∣ K , φ ○ H ∣ K ) . Relative cocartesian fibrations 2.2.
Let E ⊂ Fun ([ ] , S ) be a full subcategory. We call a functor φ ∶ C → S a cocartesian fibration relative to E if the pullback [ ] × S C → [ ] along any functor [ ] → Scorresponding to a morphism of E is a cocartesian fibration, whose cocartesian morphisms are φ -cocartesian. Given cocartesian fibrations C → S , D → S relative to E a functor C → D over S is amap of cocartesian fibrations relative to E if it preserves cocartesian lifts of morphisms of E . We write
Cat E ∞/ S ⊂ Cat ∞/ S for the subcategory of cocartesian fibrations relative to E and maps of such. For E = Fun ([ ] , S ) we write Cat cocart ∞/ S for Cat E ∞/ S . The next definition is a variant of Lurie’s notion of fibered object with respect to a categoricalpattern ([6] Definition B.0.19.):
Definition 2.3.
Let E ⊂ Fun ([ ] , S ) be a full subcategory and Λ a collection of functors of the form K ⊲ → S for some ∞ -category K that send any morphism to a morphism of E .We call a cocartesian fibration φ ∶ C → S relative to E fibered with respect to Λ if for any functor α ∶ K ⊲ → S that belongs to Λ the following two conditions hold:(1) the pullback K ⊲ × S C → K ⊲ along α , which is a cocartesian fibration, classifies a limit diagram K ⊲ → Cat ∞ , (2) any φ -cocartesian lift K ⊲ → C of α is a φ -limit diagram. N EQUIVALENCE BETWEEN ENRICHED ∞ -CATEGORIES AND ∞ -CATEGORIES WITH WEAK ACTION 9 We write
Cat E , Λ ∞/ S ⊂ Cat E ∞/ S for the full subcategory spanned by the cocartesian fibrations relative to E fibered with respect to Λ. By [6] Theorem B.0.20. Cat E , Λ ∞/ S ⊂ Cat E ∞/ S is a reflexive full subcategory. Remark 2.4. If φ ∶ C → S is a cocartesian fibration, condition 1. implies condition 2 (Corollary 9.2).
Example 2.5. ● Denote F in ∗ the category of finite pointed sets. A map ⟨ n ⟩ → ⟨ m ⟩ in F in ∗ is called inert if thepreimage of any element different from the base point consists of precisely one element. Forevery n ≥ ⟨ n ⟩ → ⟨ ⟩ called the standart inert morphisms,where the i-th standart inert morphism τ i ∶ ⟨ n ⟩ → ⟨ ⟩ for 1 ≤ i ≤ n sends only i to 1. For eachn ≥ τ i ∶ ⟨ n ⟩ → ⟨ ⟩ for 1 ≤ i ≤ n determines a functor τ ∶ { , ..., n } ⊲ → F in ∗ . Let Γ be the collection of all of functors τ for n ≥ ∅ ⊲ → F in ∗ selecting ⟨ ⟩ . A cocartesian fibration O → F in ∗ relative to the inert morphisms is fibered with respect toΛ if and only if O → F in ∗ is a symmetric ∞ -operad (in the sense of [6] Definition 2.1.1.10.). ● Denote Ass ∶ = ∆ op the category of finite non-empty totally ordered sets and order preservingmaps. A map [ n ] → [ m ] in Ass is called inert if it corresponds to a map of ∆ of the form [ m ] ≃ { i , i + , ..., i + m } ⊂ [ n ] for some i ≥ . For every n ≥ [ n ] → [ ] called the standart inertmorphisms, where the i-th standart inert morphism τ i ∶ [ n ] → [ ] for 1 ≤ i ≤ n correspondsto the map [ ] ≃ { i − , i } ⊂ [ n ] . For each n ≥ τ i ∶ [ n ] → [ ] for 1 ≤ i ≤ n determines a functor τ ∶ { , ..., n } ⊲ → Ass. Let Λ be the collection ofall τ for n ≥ ∅ ⊲ → Ass selecting [ ] . A cocartesian fibration O → Ass relative to the inert morphisms is fibered with respect toΛ if and only if O → Ass is a non-symmetric ∞ -operad (in the sense of [3] Definition 3.1.3.). Example 2.6.
For every n ≥ [ ] ≃ { i − , i } ⊂ [ n ] in ∆ for 1 ≤ i ≤ nexhibit [ n ] as the colimit in ∆ of the diagram(7) [ ] ← [ ] → [ ] ← [ ] → ... ← [ ] → [ ] . Denote ρ ∶ K ⊲ n → Ass the functor opposite to the functor ( K opn ) ⊳ → ∆ that exhibits [ n ] as the colimitof diagram 7. There is a canonical embedding { , ..., n } ⊂ K n such that the restriction of ρ ∶ K ⊲ n → Assto { , ..., n } ⊲ is τ. Let Λ gen be the collection of all ρ for n ≥ . A cocartesian fibration O → Ass relative to the inert morphisms is fibered with respect to Λ gen ifand only if O → Ass is a generalized non-symmetric ∞ -operad (in the sense of [3] Definition 3.1.12.). Convention 2.7.
When we say (generalized) ∞ -operads, we will always mean (generalized) non-symmetric ∞ -operads. This is as we will mainly deal with (generalized) non-symmetric ∞ -operadsand rarely with (generalized) symmetric ∞ -operads. Our convention is against the convention used in[6], where mainly (generalized) symmetric ∞ -operads are studied and where (generalized) ∞ -operadsalways mean (generalized) symmetric ∞ -operads. Example 2.8.
We will also need the following model of (generalized) ∞ -operads: Denote ∆ ◁ thecategory of finite (evtl. empty) partially ordered sets and set ̃ Ass ∶ = ( ∆ ◁ ) op = Ass ⊳ . We call amorphism in ̃ Ass inert if it is an inert morphism of Ass or its target is the final object of ̃ Ass.In Examples 2.5 and 2.6 we defined collections of functors K ⊲ → Ass denoted Λ respectively Λ gen . We write ̃ Λ respectively ̃ Λ gen for the collections of functors K ⊲ → ̃ Ass that factor as K ⊲ → Ass ⊂ ̃ Ass forsome functor K ⊲ → Ass of Λ respectively Λ gen or are of the form ∅ ⊲ → ̃ Ass selecting the final object.By Corollary 9.4 taking pullback along the embedding Ass ⊂ ̃ Ass defines an equivalence between ̃ Λ- respectively ̃ Λ gen -fibered cocartesian fibrations relative to the inert morphisms and (generalized)non-symmetric ∞ -operads. ∞ -CATEGORIES AND ∞ -CATEGORIES WITH WEAK ACTION Notation 2.9.
Let ψ ∶ O → Ass be a cocartesian fibration relative to the inert morphisms. Wecall a morphism of O inert if it is ψ -cocartesian and its image in Ass is inert. Denote Λ O , Λ gen O thecollection of functors K ⊲ → O that send any morphism to an inert one and such that the compositionK ⊲ → O → Ass belongs to Λ respectively Λ gen . Similarly given a cocartesian fibration ψ ∶ O → ̃ Assrelative to the inert morphisms we call a morphism of O inert if it is ψ -cocartesian and its image in ̃ Ass is inert and write Λ O , Λ gen O for the collection of functors K ⊲ → O that send any morphism to aninert one and such that the composition K ⊲ → O → ̃ Ass belongs to ̃ Λ respectively ̃ Λ gen . Definition 2.10.
Let ψ ∶ O → Ass or ψ ∶ O → ̃ Ass be a cocartesian fibration relative to the inertmorphisms.A O -operad is a cocartesian fibration φ ∶ C → O relative to the inert morphisms that is fibered withrespect to Λ O .A generalized O -operad is a cocartesian fibration φ ∶ C → O relative to the inert morphisms that isfibered with respect to Λ gen O .A (generalized) O -monoidal ∞ -category is a (generalized) O -operad φ ∶ C → O that is a cocartesianfibration. We write Op O ∞ ⊂ Cat inert ∞/ O , Op O , gen ∞ ⊂ Cat inert ∞/ O , Op O , mon ∞ ⊂ Cat cocart ∞/ O for the full subcategories spanned by the O -operads, generalized O -operads respectively O -monoidal ∞ -categories.Let O ′ → O be a map of cocartesian fibrations relative to the inert morphisms of Ass and C → O a (generalized) O -operad respectively (generalized) O -monoidal ∞ -category. Then the pullback O ′ × O C → O ′ is a (generalized) O ′ -operad respectively (generalized) O ′ -monoidal ∞ -category. Remark 2.11.
Let us explain what conditions 1. and 2. say for the case of a cocartesian fibration O → Ass relative to the inert morphisms.Let’s start first with the case of a O -operad: ● Condition 1. says that for any X ∈ O lying over [ n ] ∈ Ass and φ -cocartesian lifts X → X i ofthe standart inert morphisms [ n ] → [ ] for 1 ≤ i ≤ n the induced functor C X → ∏ ni = C X i is anequivalence, and C X is contractible if X ∈ O lies over [ ] . ● Condition 2. says that for any X , Y ∈ C lying over [ n ] respectively [ m ] the induced map C ( Y , X ) → O ( φ ( Y ) , φ ( X )) × ∏ ni = O ( φ ( Y ) ,φ ( X i )) n ∏ i = C ( Y , X i ) is an equivalence.Let’s continue with the case of a generalized O -operad: ● Condition 1.: As φ ∶ C → O is a cocartesian fibration relative to the inert morphisms, for everyn ≥ ∈ O lying over [ n ] the colimit diagram 7 in ∆ uniquely lifts to a diagram(8) X → ( X → X , ← X → X , ← ... → X n − , n ← X n ) in O with all morphisms φ -cocartesian. The induced functor C X → C X × C X1 , C X × C X2 , ... × C Xn − , n C X n is an equivalence. N EQUIVALENCE BETWEEN ENRICHED ∞ -CATEGORIES AND ∞ -CATEGORIES WITH WEAK ACTION 11 ● Condition 2. says that for any X , Y ∈ C lying over [ n ] respectively [ m ] the square C ( Y , X ) (cid:15) (cid:15) / / C ( Y , X ) × C ( Y , X , ) ... × C ( Y , X n − , n ) C ( Y , X n ) (cid:15) (cid:15) O ( φ ( Y ) , φ ( X )) / / O ( φ ( Y ) , φ ( X )) × O ( φ ( Y ) ,φ ( X , )) ... × O ( φ ( Y ) ,φ ( X n − , n )) O ( φ ( Y ) , φ ( X n )) is a pullback square. Definition 2.12.
We call a O -monoidal ∞ -category C → O compatible with small colimits if for any X ∈ O [ ] the fiber C X has small colimits and for any active morphism X → Y in O with Y ∈ O [ ] theinduced functor ∏ ni = C X i ≃ C X → C Y preserves small colimits component-wise.We say that a O -monoidal functor C → D preserves small colimits if for any X ∈ O [ ] the inducedfunctor C X → D X preserves small colimits. We have the following relation between O -operads and generalized O -operads: Proposition 2.13.
Let ψ ∶ O → Ass , φ ∶ C → O be cocartesian fibrations relative to the inert mor-phisms.Then C → O is a O -operad if and only if it is a generalized O -operad and the induced functor C [ ] → O [ ] is an equivalence.Proof. We will first show that the following conditions are equivalent:(1) The functor C [ ] → O [ ] is an equivalence.(2) For any Z ∈ O [ ] the fiber C Z is contractible and for any X ∈ C [ m ] for m ≥ ∈ C [ ] themap C ( X , Y ) → O ( φ ( X ) , φ ( Y )) is an equivalence.As [ ] is the initial object of Ass, the functors O [ ] → O , C [ ] → C are fully faithful. So 2. implies 1.As φ ∶ C → O , ψ ∶ O → Ass are cocartesian fibrations relative to the inert morphisms, φ is a map ofcocartesian fibrations relative to the inert morphisms. For any m ≥ [ m ] → [ ] in Ass areinert. So for any X ∈ C [ m ] and Y ∈ C [ ] the fiber of the map C ( X , Y ) → O ( φ ( X ) , φ ( Y )) over an object f of Ass ([ m ] , [ ]) is equivalent to C [ ] ( X ′ , Y ) → O [ ] ( φ ( X ′ ) , φ ( Y )) for X → X ′ the cocartesian lift of f . So 1. implies 2.Let φ ∶ C → O be a O -operad. Let Z ∈ O [ ] . Then Z determines a functor ∅ ⊲ → O that belongs toΛ O . So the fiber C Z is contractible. Let Y ∈ C [ ] . Then Y determines a functor ∅ ⊲ → C that is a φ -limitdiagram. This means that for any X ∈ C the map C ( X , Y ) → O ( φ ( X ) , φ ( Y )) is an equivalence. So itremains to show that if condition 2. holds, φ ∶ C → O is a O -operad if and only if it is a generalized O -operad. This follows immediately from Remark 2.11. (cid:3) Remark 2.14.
For any cocartesian fibration O → Ass relative to the inert morphisms we set ̃ O ∶ = O ▷ and have an induced functor ̃ O = O ▷ → ̃ Ass = Ass ⊳ that is a cocartesian fibration relative to the inertmorphisms. We write ∞ for the final object of ̃ O , i.e. the cone point. Generalizing Example 2.8 byCorollary 9.4 taking pullback along the embedding O ⊂ ̃ O restricts to equivalencesOp ∞/̃ O ≃ Op ∞/ O , Op gen ∞/̃ O ≃ Op gen ∞/ O . Example 2.15.
Definition 2.10 also includes families of (generalized) O -operads:Let S be an ∞ -category and O → Ass a map of cocartesian fibrations relative to the inert morphisms.The composition O × S → O → Ass is a cocartesian fibration relative to the inert morphisms. ● We call (generalized) O × S-operads S-families of (generalized) O -operads. ∞ -CATEGORIES AND ∞ -CATEGORIES WITH WEAK ACTION ● We call a S-family φ ∶ C → S × O of (generalized) O -operads a S-family of (generalized) O -monoidal ∞ -categories if the functor φ is a map of cocartesian fibrations over O . ● We call a S-family φ ∶ C → S × O of (generalized) O -operads / (generalized) O -monoidal ∞ -categories a cocartesian (cartesian, bicartesian) S-family of (generalized) O -operads if thefunctor φ is a map of cocartesian (cartesian, bicartesian) fibrations over S . ● We call a cocartesian S-family of O -monoidal ∞ -categories compatible with small colimits ifit is compatible with small colimits as a S × O -monoidal ∞ -category.A functor C → S × O is a cocartesian S-family of (generalized) O -monoidal ∞ -categories if and only ifit is a (generalized) S × O -monoidal ∞ -category. Operad structure on sections 2.16.
Given a S-family of (generalized) O -operads the fiber overan object of S is a (generalized) O -operad. Besides the fibers also the sections earn the correspondingstructure: Taking sections turns a S-family of (generalized) O -operads/ O -monoidal ∞ -categories toa (generalized) O -operad/ O -monoidal ∞ -category: By [6] Theorem B.4.2. the adjunction ( − ) × S ∶ Cat ∞/ O ⇄ Cat ∞/ S × O ∶ Fun O S × O ( S × O , − ) restricts to an adjunction ( − ) × S ∶ Op O , gen ∞ ⇄ Op S × O , gen ∞ , whose right adjoint sends S-families of O -operads, O -monoidal ∞ -categories, generalized O -monoidal ∞ -categories to O -operads, O -monoidal ∞ -categories, generalized O -monoidal ∞ -categories. Notation 2.17.
Let S be an ∞ -category, C → O a generalized O -operad and D → S × O a S-family ofgeneralized O -operads. The functor ( − ) × C ∶ Cat ∞/ S → Op S × O , gen ∞ admits a right adjoint Alg S C / O ( − ) . For any functor S ′ → S there is a canonical equivalenceS ′ × S Alg S C / O ( D ) ≃ Alg S ′ C / O ( S ′ × S D ) and a canonical embedding Alg S C / O ( D ) ⊂ Fun SS × C / S × O ( D ) over S . We write Alg S O ( − ) for Alg S O / O ( − ) and Alg S ( − ) for Alg SAss ( − ) and drop S if S is contractible.There is a canonical equivalenceAlg O ( Fun O S × O ( S × O , D )) ≃ Fun S ( S , Alg S O ( D )) . Now we define weak actions by considering the following cocartesian fibrations C → Ass relative tothe inert morphisms: ● Denote BM ∶ = ( ∆ /[ ] ) op . ● Denote LM ⊂ BM , RM ⊂ BM the full subcategories spanned by the maps [ n ] → [ ] that send0 to 0 and at most one object to 1 respectively send n to 1 and at most one object to 0.The right fibration ∆ /[ ] → ∆ is opposite to a left fibration BM → Ass that restricts to cocartesianfibrations LM → Ass , RM → Ass relative to the inert morphisms. A morphism in LM , RM , BM isinert if and only if its image in Ass is. Note that BM → Ass is a generalized ∞ -operad. This impliesthat generalized BM-operads are generalized ∞ -operads over BM . The maps { } ⊂ [ ] , { } ⊂ [ ] yieldtwo embeddings Ass ⊂ BM over Ass, which we call left and right embedding, that factor through LMrespectively RM . We have a canonical equivalence∆ ◁ × ∆ ◁ ≃ ( ∆ ◁ ) /[ ] = ( ∆ /[ ] ) ◁ , ([ n ] , [ m ]) ↦ [ n ] ∗ [ m ] opposite to an equivalence ̃ Ass × ̃ Ass ≃ ̃ BM N EQUIVALENCE BETWEEN ENRICHED ∞ -CATEGORIES AND ∞ -CATEGORIES WITH WEAK ACTION 13 that restricts to equivalences (̃ Ass × { ∅ → ∞ }) ∖ {( ∞ , ∅ )} ≃ ̃ LM , Ass × { ∅ → ∞ } ≃ LM , ({ ∅ → ∞ } × ̃ Ass ) ∖ {( ∅ , ∞ )} ≃ ̃ RM , { ∅ → ∞ } × Ass ≃ RM . We say that a morphism in ̃ Ass preserves the minimum respectively maximum if it is the morphismto the final object or it corresponds to a morphism [ m ] → [ n ] in ∆ sending 0 to 0 respectively m ton . We say that a morphism in Ass preserves the minimum respectively maximum if its image in ̃ Assdoes. A morphism of ̃ BM ≃ ̃ Ass × ̃ Ass is inert if and only if its first component is inert and preservesthe maximum and its second component is inert and preserves the minimum.We have a canonical functor ̃ LM × ̃ RM → ̃ BMthat is the restriction of the functor (̃ Ass × [ ]) × ([ ] × ̃ Ass ) → ̃ BM ≃ ̃ Ass × ̃ Asscorresponding to the commutative square ( − , − ) (cid:15) (cid:15) / / ( − , ∞ ) (cid:15) (cid:15) ( ∞ , − ) / / ( ∞ , ∞ ) . of functors ̃ Ass × ̃ Ass → ̃ Ass × ̃ Ass.By definition 2.10 we have the notion of (generalized) LM , RM , BM-operads and S-families of suchfor an ∞ -category S . We write LMod S ( − ) for Alg SLM ( − ) and drop S from the notation if S is con-tractible. For a S-family of generalized LM-operads D → S × LM we have a functor LMod S ( D ) → Alg S ( Ass × LM D ) over S . In the following we give different models of those, which we call ∞ -categories with weak left action,weak right action respectively weak biaction. Definition 2.18.
Let V → Ass be a generalized ∞ -operad and φ ∶ M → V a map of cocartesianfibrations relative to the inert morphisms of Ass that preserve the maximum.We call φ ∶ M → V an ∞ -category weakly left tensored over V (or an ∞ -category with weak left V -action) if the following conditions hold:(1) for every n ≥ the map [ ] ≃ { n } ⊂ [ n ] in ∆ induces an equivalence θ ∶ M [ n ] → V [ n ] × V [ ] M [ ] . (2) for every X , Y ∈ M lying over [ m ] , [ n ] ∈ Ass the cocartesian lift Y → Y ′ of the map [ ] ≃ { n } ⊂ [ n ] in ∆ induces a pullback square M ( X , Y ) (cid:15) (cid:15) / / V ( φ ( X ) , φ ( Y )) × M ( X , Y ′ ) (cid:15) (cid:15) Ass ([ m ] , [ n ]) / / Ass ([ m ] , [ n ]) × Ass ([ m ] , [ ]) . We call φ ∶ M → V an ∞ -category left tensored over V if V is a monoidal ∞ -category, φ is a mapof cocartesian fibrations over Ass and condition 1. holds (condition 2. is then automatic).
Similarly we define ∞ -categories with weak right V -action. ∞ -CATEGORIES AND ∞ -CATEGORIES WITH WEAK ACTION Remark 2.19.
Condition 2. implies that the functor θ in condition 1. is fully faithful by taking thefiber over the identity of [ n ] in the pullback square of condition 2. So if condition 2. holds, condition1. is equivalent to the condition that θ is essentially surjective.Denote Cat ′∞/
Ass ⊂ Cat ∞/ Ass the subcategory with objects the cocartesian fibrations relative to theinert morphisms that preserve the maximum and with morphisms the maps of such. We write ω LMod gen ⊂ Op ∞ × Fun ({ } , Cat ′∞ / Ass ) Fun ([ ] , Cat ′∞/
Ass ) for the full subcategory spanned by the ∞ -categories with weak left action. We write ω LMod ⊂ ω LMod gen for the full subcategory spanned by the ∞ -categories with weak left action over some ∞ -operad.We write LMod ⊂ Op mon ∞ × Fun ({ } , Cat cocart ∞ / Ass ) Fun ([ ] , Cat cocart ∞/ Ass ) for the full subcategory spanned by the ∞ -categories with left action.We call morphisms of ω LMod functors preserving the weak left actions. We call a functor preservingthe weak left actions a lax V -linear functor if it induces the identity of V on evaluation at the target.We call morphisms of LMod functors preserving the left actions and call a functor preserving the leftactions a V -linear functor if it induces the identity of V on evaluation at the target.We fix the following notation: ● For an ∞ -operad V and ∞ -categories with weak left V -action M → V , N → V denoteLaxLinFun V ( M , N ) ⊂ Fun V ( M , N ) the full subcategory spanned by the lax V -linear functors. ● If V is a monoidal ∞ -category and M , N are ∞ -categories with left V -action, denoteLinFun L V ( M , N ) ⊂ LinFun V ( M , N ) ⊂ LaxLinFun V ( M , N ) the full subcategories spanned by the V -linear functors (preserving small colimits). Proposition 2.20.
There is a canonical equivalence
Cat inert ∞/ LM ≃ Cat inert ∞/ Ass × Fun ({ } , Cat ′∞ / Ass ) Fun ([ ] , Cat ′∞/
Ass ) that restricts to equivalences Op LM , gen ∞ ≃ ω LMod gen , Op LM ∞ ≃ ω LMod , Op LM , mon ∞ ≃ LMod . Proof.
Classifying a functor by a cocartesian fibration gives an equivalence ( Cat cocart ∞/[ ] ) / LM ≃ Fun ([ ] , Cat ∞/ Ass ) . So we have a subcategory inclusion
Cat inert ∞/ Ass × Fun ({ } , Cat ′∞ / Ass ) Fun ([ ] , Cat ′∞/
Ass ) ⊂ Fun ([ ] , Cat ′∞/
Ass ) ⊂ Fun ([ ] , Cat ∞/ Ass ) ≃ ( Cat cocart ∞/[ ] ) / LM ⊂ ( Cat ∞/[ ] ) / LM ≃ Cat ∞/ LM . By Lemma 2.21 1. the subcategory inclusion
Cat inert ∞/ LM ⊂ Cat ∞/ LM factors through ( Cat cocart ∞/[ ] ) / LM and by Lemma 2.21 2. and 3. through Cat inert ∞/ Ass × Fun ({ } , Cat ′∞ / Ass ) Fun ([ ] , Cat ′∞/
Ass ) inducing an equivalence Cat inert ∞/ LM ≃ Cat inert ∞/ Ass × Fun ({ } , Cat ′∞ / Ass ) Fun ([ ] , Cat ′∞/
Ass ) that restricts to the claimed equivalences. N EQUIVALENCE BETWEEN ENRICHED ∞ -CATEGORIES AND ∞ -CATEGORIES WITH WEAK ACTION 15 (cid:3) We used the following Lemma:
Lemma 2.21. (1) Every cocartesian fibration O → LM = Ass × [ ] relative to the inert morphisms is a map ofcocartesian fibrations over [ ] and every map of cocartesian fibrations relative to the inertmorphisms of LM is a map of cocartesian fibrations over [ ] .(2) By 1. every cocartesian fibration O → LM = Ass × [ ] relative to the inert morphisms classifiesa functor φ ∶ M → V ∶ = Ass × LM O over Ass . The functor φ ∶ M → V is a map of cocartesianfibrations relative to the inert morphisms that preserve the maximum.(3) Given cocartesian fibrations O → LM , O ′ → LM relative to the inert morphisms and a functor θ ∶ O → O ′ over LM that is a map of cocartesian fibrations over [ ] classifying a commutativesquare M φ (cid:15) (cid:15) α / / M ′ φ ′ (cid:15) (cid:15) V β / / V ′ , of ∞ -categories over Ass .Then θ ∶ O → O ′ is a map of cocartesian fibrations relative to the inert morphisms of LM if andonly if β is a map of cocartesian fibrations relative to the inert morphisms of Ass and α is a map ofcocartesian fibrations relative to the inert morphisms of Ass that preserve the maximum.Proof.
Every morphism in LM, whose component in Ass is the identity, is inert: a morphism of LMis inert if and only if its image in Ass is inert under the functor LM = Ass × [ ] → Ass given by thenatural transformation ( − ) → ( − ) ∗ [ ] of functors ∆ → ∆ . A morphism in LM, whose componentin [ ] is the identity of 1 (0) is inert if and only if its component in Ass is inert (and additionallypreserves the maximum). (cid:3) For later reference we add the following proposition that guarantees that ω LMod ⊂ Op ∞ × Fun ({ } , Cat ′∞ / Ass ) Fun ([ ] , Cat ′∞/
Ass ) is a reflexive full subcategory (Corollary 2.23). Proposition 2.22.
Let V → Ass be an ∞ -operad.Note that for any Y ∈ V lying over some n ≥ there is a unique morphism Y → Y m (automaticallyinert) lying over the morphism { n } ⊂ [ n ] in ∆ . Denote Λ the collection of functors [ ] ⊲ ≃ [ ] → V corresponding to some morphism Y → Y m in V for some Y ∈ V . Let α ∶ M → V be a map of cocartesian fibrations relative to the inert morphisms that preserve themaximum. α ∶ M → V is a ∞ -category with weak left V -action if and only if α ∶ M → V is fibered withrespect to Λ . Proof.
The functor θ ∶ M [ n ] → V [ n ] × M [ ] induced by the map [ ] ≃ { n } ⊂ [ n ] in ∆ for n ≥ ∈ V [ n ] the functor M Y → M [ n ] → M [ ] ≃ M Y m induced by Y → Y m . ∞ -CATEGORIES AND ∞ -CATEGORIES WITH WEAK ACTION For every X ∈ V [ m ] the commutative square in condition (b) factors as M ( X , Y ) (cid:15) (cid:15) / / V ( φ ( X ) , φ ( Y )) × M ( X , Y m ) (cid:15) (cid:15) V ( φ ( X ) , φ ( Y )) / / (cid:15) (cid:15) V ( φ ( X ) , φ ( Y )) × V ( φ ( X ) , φ ( Y m )) (cid:15) (cid:15) Ass ([ m ] , [ n ]) / / Ass ([ m ] , [ n ]) × Ass ([ m ] , [ ]) . The bottom square is a pullback square as the map V ( φ ( X ) , φ ( Y m )) → Ass ([ m ] , [ ]) is an equivalence.The top square is a pullback square if and only if the square M ( X , Y ) (cid:15) (cid:15) / / M ( X , Y m ) (cid:15) (cid:15) V ( φ ( X ) , φ ( Y )) / / V ( φ ( X ) , φ ( Y m )) is a pullback square. (cid:3) Corollary 2.23.
The ∞ -category ω LMod is a localization relative to Op ∞ of the bicartesian fibration Op ∞ × Fun ({ } , Cat ′∞ / Ass ) Fun ([ ] , Cat ′∞/
Ass ) → Op ∞ . Proof.
Evaluation at the target Fun ([ ] , Cat ′∞/
Ass ) → Cat ′∞/
Ass is a bicartesian fibration as
Cat ′∞/
Ass admits pullbacks. So the pullback Op ∞ × Fun ({ } , Cat ′∞ / Ass ) Fun ([ ] , Cat ′∞/
Ass ) → Op ∞ is a bicartesianfibration. Consequently it is enough to check the following two points: ● The cartesian fibration Op ∞ × Fun ({ } , Cat ′∞ / Ass ) Fun ([ ] , Cat ′∞/
Ass ) → Op ∞ restricts to a cartesianfibration ω LMod → Op ∞ with the same cartesian morphisms. ● The full embedding ω LMod ⊂ Op ∞ × Fun ({ } , Cat ′∞ / Ass ) Fun ([ ] , Cat ′∞/
Ass ) induces on the fiberover every ∞ -operad V a localization.The first point follows immediately from Proposition 2.22. So we check the second point: Let E ⊂ Fun ([ ] , V ) be the full subcategory spanned by the morphisms of V cocartesian over a morphism ofAss that preserves the maximum. Let Λ be the collection of functors of Proposition 2.22. Then byProposition 2.22 the full subcategory ω LMod V ⊂ ( Cat ′∞/
Ass ) / V = Cat E ∞/ V coincides with the localization Cat E , Λ ∞/ V ⊂ Cat E ∞/ V . (cid:3) There is a dual theory of ∞ -categories with weak right action. Next we define ∞ -categories withweak biaction: Definition 2.24.
Let V → Ass , W → Ass be generalized ∞ -operads and φ = ( φ , φ ) ∶ M → V × W a map of cocartesian fibrations relative to the inert morphisms of Ass × Ass , whose first componentpreserves the maximum and whose second component preserves the minimum.We call φ ∶ M → V × W an ∞ -category weakly bitensored over ( V , W ) (or an ∞ -category with weakbiaction over ( V , W ) ) if the following conditions hold:(1) for every n , m ≥ the map [ ] ≃ { n } ⊂ [ n ] in the first component and the map [ ] ≃ { } ⊂ [ m ] in the second component induce an equivalence θ ∶ M [ n ][ m ] → V [ n ] × V [ ] M [ ][ ] × W [ ] W [ m ] , N EQUIVALENCE BETWEEN ENRICHED ∞ -CATEGORIES AND ∞ -CATEGORIES WITH WEAK ACTION 17 (2) for every X , Y ∈ M lying over ([ m ] , [ m ′ ]) , ([ n ] , [ n ′ ]) ∈ Ass × Ass the cocartesian lift Y → Y ′ ofthe map [ ] ≃ { n } ⊂ [ n ] and [ ] ≃ { } ⊂ [ n ′ ] induces a pullback square M ( X , Y ) (cid:15) (cid:15) / / V ( φ ( X ) , φ ( Y )) × W ( φ ( X ) , φ ( Y )) × M ( X , Y ′ ) (cid:15) (cid:15) Ass ([ m ] , [ n ]) × Ass ([ m ′ ] , [ n ′ ]) / / Ass ([ m ] , [ n ]) × Ass ([ m ′ ] , [ n ′ ]) × Ass ([ m ] , [ ]) × Ass ([ m ′ ] , [ ]) . We call φ ∶ M → V an ∞ -category bitensored over ( V , W ) (or an ∞ -category with ( V , W ) -biaction)if V , W are monoidal ∞ -categories, φ is a map of cocartesian fibrations over Ass × Ass and condition1. holds (condition 2. is then automatic).
Denote
Cat ′∞/
Ass × Ass ⊂ Cat ∞/ Ass × Ass the subcategory of cocartesian fibrations relative to the inertmorphisms, whose first component preserves the maximum and whose second component preserves theminimum, and maps of such. Denote ω BMod gen ⊂ ( Op ∞ × Op ∞ ) × Cat ′∞ / Ass × Ass
Fun ([ ] , Cat ′∞/
Ass × Ass ) the full subcategory spanned by the ∞ -categories with weak biaction. Denote ω BMod ⊂ ω BMod gen the full subcategory spanned by the ∞ -categories weakly bitensored over ∞ -operads. Proposition 2.25.
There is a canonical equivalence
Cat inert ∞/ BM ≃ ( Cat inert ∞/ Ass × Cat inert ∞/ Ass ) × Cat ′∞ / Ass × Ass
Fun ([ ] , Cat ′∞/
Ass × Ass ) that restricts to equivalences Op BM , gen ∞ ≃ ω BMod gen , Op BM ∞ ≃ ω BMod . Proof.
Denote E − , E + ⊂ Fun ([ ] , ̃ Ass ) the full subcategories spanned by the inert morphisms thatpreserve the minimum respectively maximum. Denote Cat ′∞/̃
Ass ×̃ Ass ∶ = Cat E + × E − ∞/̃ Ass ×̃ Ass ⊂ Cat ∞/̃
Ass ×̃ Ass thesubcategory of cocartesian fibrations relative to E + × E − . The canonical equivalence ̃ BM ≃ ̃ Ass × ̃ Assgives an equivalence
Cat inert ∞/̃ BM ≃ Cat ′∞/̃
Ass ×̃ Ass . For a functor M → ̃ Ass × ̃ Ass denote M −∞ , M +∞ the pullbacks of M → ̃ Ass × ̃ Ass along the embedding { ∞ } × ̃ Ass ⊂ ̃ Ass × ̃ Ass respectively ̃ Ass × { ∞ } ⊂ ̃ Ass × ̃ Ass . For any M ∈ Cat ′∞/̃
Ass ×̃ Ass and any V ∈ Cat inert ∞/̃
Ass by Lemma 9.5 the canonical functorsFun E + × E − ̃ Ass ×̃ Ass ( M , V × ̃ Ass ) → Fun E + ̃ Ass ( M +∞ , V ) , Fun E + × E − ̃ Ass ×̃ Ass ( M , ̃ Ass × V ) → Fun E − ̃ Ass ( M −∞ , V ) are equivalences.Denote Γ ⊂ ( Cat inert ∞/̃
Ass × Cat inert ∞/̃
Ass ) × Cat ′∞ /̃ Ass × ̃ Ass
Fun ([ ] , Cat ′∞/̃
Ass ×̃ Ass ) the full subcategory spanned bythe triples ( V → ̃ Ass , W → ̃ Ass , γ ∶ M → V × W ) such that the induced functors M +∞ → V and M −∞ → W are equivalences. By Corollary 9.4 we have canonical equivalences Cat inert ∞/̃
Ass ≃ Cat inert ∞/ Ass , Cat inert ∞/̃ BM ≃ Cat inert ∞/ BM . Taking pullback along Ass ⊂ ̃ Ass gives an equivalence Γ ≃ ( Cat inert ∞/ Ass × Cat inert ∞/ Ass ) × Cat ′∞ / Ass × Ass
Fun ([ ] , Cat ′∞/
Ass × Ass ) . Consequently we need to construct a canonical equivalence Γ ≃ Cat ′∞/̃
Ass ×̃ Ass . Denote Fun ′ ([ ] × [ ] , Cat ′∞/̃
Ass ×̃ Ass ) ⊂ Fun ([ ] × [ ] , Cat ′∞/̃
Ass ×̃ Ass ) the full subcategory spanned bythe commutative squares(9) M (cid:15) (cid:15) / / V (cid:15) (cid:15) W / / T ∞ -CATEGORIES AND ∞ -CATEGORIES WITH WEAK ACTION in Cat ′∞/̃
Ass ×̃ Ass such that T is the final object, V → ̃ Ass × ̃ Ass is a map of cocartesian fibrations over ∗ × ̃ Ass and W → ̃ Ass × ̃ Ass is a map of cocartesian fibrations over ̃ Ass × ∗ such that the correspondingclassified functors ̃ Ass → Cat ∞/̃
Ass are constant with value a cocartesian fibration relative to the inertmorphisms.Sending square 11 to the triple ( V , W , M → V × W ) defines a canonical equivalenceFun ′ ([ ] × [ ] , Cat ′∞/̃
Ass ×̃ Ass ) ≃ ( Cat inert ∞/̃
Ass × Cat inert ∞/̃
Ass ) × Cat ′∞ /̃ Ass × ̃ Ass
Fun ([ ] , Cat ′∞/̃
Ass ×̃ Ass ) , via which we may view Γ as a full subcategory of Fun ′ ([ ] × [ ] , Cat ′∞/̃
Ass ×̃ Ass ) . We have a canonical functor θ ∶ ̃ Ass × ̃ Ass × [ ] × [ ] → ̃ BM ≃ ̃ Ass × ̃ Ass and call a morphism in ̃ Ass × ̃ Ass × [ ] × [ ] inert if its image under θ in ̃ BM is. The subcategory
Cat inert ∞/̃
Ass ×̃ Ass ×[ ]×[ ] ⊂ Cat ∞/̃
Ass ×̃ Ass ×[ ]×[ ] is contained in the subcategory ( Cat cocart ∞/[ ]×[ ] ) /̃ Ass ×̃ Ass ×[ ]×[ ] ≃ Fun ([ ] × [ ] , Cat ∞/̃
Ass ×̃ Ass ) . The sub-category inclusion
Cat inert ∞/̃
Ass ×̃ Ass ×[ ]×[ ] ⊂ Fun ([ ] × [ ] , Cat ∞/̃
Ass ×̃ Ass ) yields an equivalence(10) Cat inert ∞/̃
Ass ×̃ Ass ×[ ]×[ ] ≃ Fun ([ ] × [ ] , Cat ′∞/̃
Ass ×̃ Ass ) . The functor θ yields a functor α ∶ Cat ′∞/̃
Ass ×̃ Ass → Cat inert ∞/̃
Ass ×̃ Ass ×[ ]×[ ] ≃ Fun ([ ] × [ ] , Cat ′∞/̃
Ass ×̃ Ass ) that sends O to the commutative square(11) O (cid:15) (cid:15) / / O +∞ × ̃ Ass (cid:15) (cid:15) ̃ Ass × O −∞ / / ̃ Ass × ̃ Assand so factors through Γ . The functor ̃ BM ≃ ̃ Ass × ̃ Ass × {( , )} → ̃ Ass × ̃ Ass × [ ] × [ ] is a section of θ and so yields a functor β ∶ Γ ⊂ Fun ([ ] × [ ] , Cat ′∞/̃
Ass ×̃ Ass ) ≃ Cat inert ∞/̃
Ass ×̃ Ass ×[ ]×[ ] → Cat ′∞/̃
Ass ×̃ Ass left inverse to α. The equivalence id ≃ β ○ α exhibits α as left adjoint to β : For any Q ∶ = ( V , W , M → V × W ) ∈ Γ and O ∈ Cat ′∞/̃
Ass ×̃ Ass the projectionFun E − ̃ Ass ( O −∞ , V ) × Fun E +× E − ̃ Ass × ̃ Ass ( O , ̃ Ass × V ) Fun E + × E − ̃ Ass ×̃ Ass ( O , M ) × Fun E +× E − ̃ Ass × ̃ Ass ( O , V ×̃ Ass ) Fun E + ̃ Ass ( O +∞ , V ) → Fun E + × E − ̃ Ass ×̃ Ass ( O , M ) is an equivalence and so induces on maximal subspaces a natural equivalenceΓ ( α ( O ) , Q ) ≃ Cat ′∞/̃
Ass ×̃ Ass ( O , β ( Q )) that is induced by β. As the functor β is conservative, the functor β ∶ Γ → Cat ′∞/̃
Ass ×̃ Ass is anequivalence. So we obtain the desired equivalence
Cat inert ∞/ BM ≃ ( Cat inert ∞/ Ass × Cat inert ∞/ Ass ) × Cat ′∞ / Ass × Ass
Fun ([ ] , Cat ′∞/
Ass × Ass ) that restricts to the claimed equivalences. (cid:3) The involution of BM We have a canonical involution on ∆ sending [ n ] to [ n ] and a map α ∶ [ n ] → [ m ] in ∆ to the map [ n ] ≃ [ n ] α Ð→ [ m ] ≃ [ m ] , where the bijection [ n ] ≃ [ n ] sends i to n − i and similar for m . The involution on ∆ yields involutionson Ass = ∆ op and BM = ( ∆ /[ ] ) op . The involution on BM restricts to an equivalenceLM ≃ Ass × [ ] ≃ RM ≃ Ass × [ ] N EQUIVALENCE BETWEEN ENRICHED ∞ -CATEGORIES AND ∞ -CATEGORIES WITH WEAK ACTION 19 that is the product of the involution on Ass with the identity of [ ] . The involution on Ass yields an involution on Op ∞ , Op gen ∞ denoted by ( − ) rev , the involution on BMyields an involution on Op BM ∞ , Op BM , gen ∞ that restricts to equivalences ω LMod gen ≃ Op LM , gen ∞ ≃ Op RM , gen ∞ ≃ ω RMod gen , ω
LMod ≃ Op LM ∞ ≃ Op RM ∞ ≃ ω RMod , where ω RMod gen , ω
RMod are defined similarly, under which an ∞ -category with weak left V -actioncorresponds to an ∞ -category with weak right V rev -action.When working with ∞ -operads and ∞ -categories weakly left tensored over such we often use thefollowing abusive notation and terminology: Notation 2.27.
Let θ ∶ W → Ass be an ∞ -operad. Set V ∶ = W [ ] . We say that θ endows V with thestructure of an ∞ -operad. Abusively we call V an ∞ -operad leaving the structure given by θ implicite.If we need to talk about θ , we write V ⊗ for W .Let φ ∶ M → W be an ∞ -category weakly left tensored over W . Set D ∶ = M [ ] . We say that φ exhibits D as weakly left tensored over V or φ endows D with a weak left V -action. Abusively we call D an ∞ -category weakly left tensored over V (or an ∞ -category with weak left V -action) leaving thestructure given by φ implicite. If we need to talk about φ (and not only D ), we write D ⊛ → V ⊗ for φ ∶ M → W .Similarly let γ ∶ M → V ⊗ × W ⊗ be an ∞ -category weakly bitensored over ( V ⊗ , W ⊗ ) . Set D ∶ = M [ ][ ] . We say that γ exhibits D as weakly bitensored over ( V , W ) . Abusively we say that D is weaklybitensored over ( V , W ) and call D an ∞ -category weakly bitensored over ( V , W ) . Enveloping O -monoidal ∞ -category 2.28. Let O → Ass be a cocartesian fibration relative to theinert morphisms. Denote Act ( O ) ⊂ Fun ([ ] , O ) the full subcategory spanned by the active morphismsin O . For every generalized O -operad C we setEnv O ( C ) ∶ = Act ( O ) × Fun ({ } , O ) C . The diagonal embedding O ⊂ Fun ([ ] , O ) factors as O ⊂ Act ( O ) . The pullback of the embedding O ⊂ Act ( O ) (seen over O via evaluation at the source) along the functor C → O is an embedding C ⊂ Env O ( C ) . The functor Env O ( C ) → Act ( O ) → Fun ({ } , O ) is a generalized O -monoidal ∞ -category and theembedding C ⊂ Env O ( C ) is a map of generalized O -operads satisfying the following universal propertyby Proposition 9.12: For every generalized O -monoidal ∞ -category D → O (in fact for any cocartesianfibration D → O ) composition with the embedding C ⊂ Env O ( C ) yields an equivalenceFun ⊗ O ( Env O ( C ) , D ) ≃ Alg C / O ( D ) between maps Env O ( C ) → D of cocartesian fibrations over O and maps C → D of cocartesian fibrationsrelative to the inert morphisms of O .For any Z ∈ O [ ] we have a canonical equivalenceEnv O ( C ) Z ≃ ( O [ ] ) / Z × O [ ] C [ ] . Especially if O [ ] is a space, which is satisfied for O ∈ { Ass , LM , RM , BM } , the embedding C ⊂ Env O ( C ) over O yields an equivalence C Z ≃ Env O ( C ) Z . So if C is a O -operad, Env O ( C ) is a O -monoidal ∞ -category due to Proposition 2.13.We call Env O ( C ) → O the enveloping (generalized) O -monoidal ∞ -category of C → O and drop O for O = Ass . We have the following important lemma:
Lemma 2.29.
Let θ ∶ O ′ → O be a map of cocartesian fibrations relative to the inert morphisms of Ass . If θ is a right fibration relative to the active morphisms, the canonical map Env O ′ ( O ′ × O C ) → O ′ × O Env O ( C ) ∞ -CATEGORIES AND ∞ -CATEGORIES WITH WEAK ACTION of O ′ -monoidal ∞ -categories is an equivalence.Proof. The functor θ ∶ O ′ → O yields an equivalence Act ( O ′ ) ≃ Fun ({ } , O ′ ) × Fun ({ } , O ) Act ( O ) thatgives rise to an equivalence O ′ × O Env O ( C ) = O ′ × Fun ({ } , O ) ( Act ( O ) × Fun ({ } , O ) C ) ≃ Act ( O ′ ) × Fun ({ } , O ′ ) ( O ′ × O C ) = Env O ′ ( O ′ × O C ) inverse to the canonical map. (cid:3) The assumptions of Lemma 2.29 are satisfied for the canonical embeddingsAss ⊂ LM , Ass ⊂ RM , Ass ⊂ BM . Now we consider the enveloping monoidal and LM-monoidal ∞ -category and presheaves on that: Enveloping ∞ -category with closed left action 2.30.Proposition 2.31. Let V be a small ∞ -operad.(1) There is an embedding of ∞ -operads V ⊂ V ′ with V ′ a presentable monoidal ∞ -category suchthat for any presentable monoidal ∞ -category W restriction along the embedding V ⊂ V ′ yieldsan equivalence Fun ⊗ , L ( V ′ , W ) ≃ Alg V ( W ) , where the left hand side denotes the full subcategory of all left adjoint monoidal functors.(2) Let M be a small ∞ -category weakly left tensored over V . There is an embedding of ∞ -categories M ⊂ M ′ with weak left action with M ′ a presentable ∞ -category with left action over V ′ such that for any presentable ∞ -category N with left V ′ -action restriction along the embedding M ⊂ M ′ yields an equivalence LinFun L V ′ ( M ′ , N ) ≃ LaxLinFun V ( M , N ) , where we restrict the left V ′ -action on N to a weak left V -action, and the left hand side denotesthe full subcategory of all left adjoint V ′ -linear functors.Proof. Denote
Cat cc ∞ ⊂ ̂ Cat ∞ the subcategory of ∞ -categories with small colimits and functors pre-serving small colimits. By [6] Proposition 4.8.1.3. and Remark 4.8.1.8. we have the following facts:the subcategory inclusion Cat cc ∞ ⊂ ̂ Cat ∞ admits a left adjoint ρ that by its universal property sendsa small ∞ -category C to P ( C ) . The ∞ -category Cat cc ∞ carries a closed symmetric monoidal structuresuch that ρ ∶ ̂ Cat ∞ → Cat cc ∞ is symmetric monoidal when ̂ Cat ∞ carries the cartesian structure.So for O ∈ { Ass , LM } we get an induced adjunction Alg O ( ̂ Cat ∞ ) ⇄ Alg O ( Cat cc ∞ ) . We have a canonical equivalenceAlg O ( ̂ Cat ∞ ) ≃ Mon O ( ̂ Cat ∞ ) ≃ ̂ Op O , mon ∞ and the right adjoint identifies Alg O ( Cat cc ∞ ) with the subcategory of Alg O ( ̂ Cat ∞ ) ≃ ̂ Op O , mon ∞ of O -monoidal ∞ -categories compatible with small colimits.The composition Alg O ( ̂ Cat ∞ ) → Alg O ( Cat cc ∞ ) ⊂ Alg O ( ̂ Cat ∞ ) sends a small O -monoidal ∞ -category C to a presentable O -monoidal ∞ -category, whose fiber over any X ∈ O [ ] is the presentable ∞ -category P ( C X ) . Especially it sends the enveloping monoidal ∞ -category Env ( V ) of a small ∞ -operad V to apresentable monoidal ∞ -category V ′ , which comes with an embedding of ∞ -operads V ⊂ Env ( V ) ⊂ V ′ satisfying universal property 1.2.: The ∞ -category M weakly left tensored over V corresponds to a LM-operad O that embedsinto its enveloping LM-monoidal ∞ -category Env LM ( O ) , whose restriction to Ass is Env ( V ) . The LM-monoidal ∞ -category Env LM ( O ) is sent to a presentable LM-monoidal ∞ -category O ′ corresponding N EQUIVALENCE BETWEEN ENRICHED ∞ -CATEGORIES AND ∞ -CATEGORIES WITH WEAK ACTION 21 to a presentable ∞ -category M ′ with left action over V ′ . The embedding O ⊂ Env LM ( O ) ⊂ O ′ of LM-operads corresponds to an embedding M ⊂ M ′ of ∞ -categories with weak left action that satisfiesuniversal property 2. (cid:3) Next we define multi-morphism spaces in ∞ -operads and ∞ -categories with weak left action: Multi-morphism spaces in ∞ -categories with weak left action 2.32. We fix the followingnotation: ● Let V be an ∞ -operad and X , ..., X n , Y ∈ V for some n ≥
0. We writeMul V ( X , ..., X n ; Y ) for the full subspace of V ⊗ ( X , Y ) spanned by the morphisms X → Y in V lying over the activemorphism [ ] → [ n ] in ∆ , where X ∈ V ⊗[ n ] ≃ ( V [ ] ) × n corresponds to ( X , ..., X n ) . ● Let M be an ∞ -category weakly left tensored over some ∞ -operad V and X , ..., X n − ∈ V , X n , Y ∈ M for some n ≥
0. We writeMul M ( X , ..., X n ; Y ) for the full subspace of M ⊛ ( X , Y ) spanned by the morphisms X → Y in M ⊛ lying over [ ] ≃ { } ⊂ [ n ] in ∆ , where X ∈ M ⊛[ n ] ≃ V × n × M corresponds to ( X , ..., X n ) .Now we define morphism objects in an ∞ -category weakly left tensored over some ∞ -operad V . Warning 2.33.
The following definition of morphism object in case that V is a monoidal ∞ -categoryis not the definition ([6] Definition 4.2.1.28.) given by Lurie. See Lemma 2.37 for the relation betweenboth definitions. Morphism objects 2.34.Definition 2.35.
Let M be an ∞ -category weakly left tensored over some ∞ -operad V . A morphism object of X , Y ∈ M is an object Mor M ( X , Y ) ∈ V together with a multi-morphism α ∈ Mul M ( Mor M ( X , Y ) , X; Y ) that induces for every objects Z , ..., Z n ∈ V an equivalence Mul V ( Z , ..., Z n ; Mor M ( X , Y )) ≃ Mul M ( Z , ..., Z n , X; Y ) . We call an ∞ -category M weakly left tensored over V closed if for every X , Y ∈ M there is amorphism object Mor M ( X , Y ) ∈ V . We will often use the following lemma:
Lemma 2.36.
Let M be an ∞ -category weakly left tensored over an ∞ -operad V . The embedding M ⊂ M ′ of ∞ -categories with weak left action preserves morphism objects.Proof. Note that an object of Env ( V ) is a tensorproduct of objects in V ⊂ Env ( V ) . So V ′ is generatedby V under small colimits and tensor products.For any X , Y ∈ M and Z , ..., Z k ∈ V we have a canonical equivalence V ′ ( Z ⊗ ... ⊗ Z n ; Mor M ( X , Y )) ≃ Mul V ′ ( Z , ..., Z n ; Mor M ( X , Y )) ≃ Mul V ( Z , ..., Z n ; Mor M ( X , Y )) ≃ Mul M ( Z , ..., Z n , X; Y ) ≃ Mul M ′ ( Z , ..., Z n , X; Y ) ≃ Mul V ′ ( Z , ..., Z n ; Mor M ′ ( X , Y )) ≃ V ′ ( Z ⊗ ... ⊗ Z n ; Mor M ′ ( X , Y )) . (cid:3) If V is a monoidal ∞ -category, the definition of morphism object gets easier: ∞ -CATEGORIES AND ∞ -CATEGORIES WITH WEAK ACTION Lemma 2.37.
Let V be a monoidal ∞ -category and M an ∞ -category with weak left V -action. Let X , Y ∈ M , T ∈ V and α ∈ Mul M ( T , X; Y ) a multi-morphism.Then α exhibits T as the morphism object of X and Y , i.e. for every Z , ..., Z n ∈ V the multi-morphism α induces an equivalence ψ ∶ V ( ⊗ ( Z , ..., Z n ) , T ) ≃ Mul M ( Z , ..., Z n , X; Y ) if and only if the following two conditions are satisfied:(1) For every Z , ..., Z n ∈ V the canonical map γ ∶ Mul M ( ⊗ ( Z , ..., Z n ) , X; Y ) → Mul M ( Z , ..., Z n , X; Y ) is an equivalence.(2) For every Z ∈ V the canonical map ρ Z ∶ V ( Z , T ) → Mul M ( Z , X; Y ) is an equivalence.Note that the morphism γ is an equivalence if M is left tensored over V .Proof. If α exhibits T as the morphism object of X and Y, then condition 2. holds.Condition 2. is equivalent to the condition that for every Z , ..., Z n ∈ V the canonical map ρ ⊗( Z ,..., Z n ) ∶ V ( ⊗ ( Z , ..., Z n ) , T ) → Mul M ( ⊗ ( Z , ..., Z n ) , X; Y ) is an equivalence. We have ψ = γ ○ ρ ⊗( Z ,..., Z n ) . (cid:3) Lurie makes the following definition ([6] Definition 4.2.1.25. and 4.2.1.28.):
Definition 2.38.
We say that an ∞ -category M weakly left tensored over a monoidal ∞ -category V is pseudo-enriched in V if condition 1. of Lemma 2.37 holds for any X , Y ∈ M . We say that an ∞ -category M weakly left tensored over a monoidal ∞ -category V is enriched in V (in the sense of Lurie) if condition 1. and 2. of Lemma 2.37 holds for any X , Y ∈ M . So an ∞ -category M weakly left tensored over a monoidal ∞ -category V is closed if and only if M is a V -enriched ∞ -category in the sense of Lurie. Graph of an ∞ -category with weak left action 2.39. Given an ∞ -category M with closed weakleft V -action the functor V op × M op × M → S , Mul M ( − , − ; − ) is adjoint to a functor Mor M ( − , − ) ∶ M op × M → V ⊂ Fun ( V op , S ) . If the weak left V -action on M is not closed, we may embed M into its enveloping ∞ -category M ′ with closed left V ′ -action to get a functorMor M ′ ( − , − ) ∶ M ′ op × M ′ → V ′ that we restrict to a functor M op × M → V ′ . For any X , Y ∈ M , Z , ..., Z n ∈ V we have a canonicalequivalence Mor M ′ ( X , Y )( Z ⊗ ... ⊗ Z n ) ≃ Mul M ( Z , ..., Z n , X; Y ) . Now we specialize morphism objects to endomorphism objects, which carry the structure of anassociative algebra:
N EQUIVALENCE BETWEEN ENRICHED ∞ -CATEGORIES AND ∞ -CATEGORIES WITH WEAK ACTION 23 Endomorphism objects 2.40.
Let M ⊛ → V ⊗ be an ∞ -category with weak left action and X ∈ M . The functor σ ∶ [ ] → Ass corresponding to the morphism { } ⊂ [ ] in ∆ gives rise to an ∞ -categoryFun Ass ([ ] , M ⊛ ) . We have a canonical functorFun Ass ([ ] , M ⊛ ) → M ⊛[ ] × M ⊛[ ] ≃ V × M × M . The pullback M [ X ] ∶ = {( X , X )} × M × M Fun
Ass ([ ] , M ⊛ ) → V is a right fibration classifying the presheaf Mul M ( − , X; X ) ∶ V op → S . So we may identify objects of M [ X ] with pairs ( A , α ) consisting of an object A ∈ V and a multi-morphism α ∈ Mul M ( A , X; X ) .Evaluation at σ ∶ [ ] → Ass defines a functorLMod ( M ) ⊂ Fun
Ass ( Ass , M ⊛ ) → Fun
Ass ([ ] , M ⊛ ) that gives rise to a commutative square { X } × M LMod ( M ) (cid:15) (cid:15) / / M [ X ] (cid:15) (cid:15) Alg ( V ) / / V . The forgetful functor LMod ( M ) → M × Alg ( V ) is a map of cartesian fibrations over Alg ( V ) thatis fiberwise conservative and so reflects cartesian morphisms over Alg ( V ) . Hence the functor { X } × M LMod ( M ) → Alg ( V ) is a right fibration. If a multi-morphism α ∈ Mul M ( A , X; X ) exhibits A asthe endomorphism object of X, then by Proposition 2.41, a modification of [6] Theorem 4.7.1.34.,the ∞ -category { X } × M LMod ( M ) admits a final object lying over ( A , α ) ∈ M [ X ] . By sending thisfinal object to Alg ( V ) we get a canonical associative algebra structure E on A , which we call theendomorphism algebra structure on A and usually do notationally identify with A . The endomorphismalgebra structure on A satisfies the following universal property:Note that a final object of the ∞ -category { X } × M LMod ( M ) corresponds to a representation ofthe right fibration { X } × M LMod ( M ) → Alg ( V ) , i.e. an object E ∈ Alg ( V ) and an equivalence { X } × M LMod ( M ) ≃ Alg ( V ) / E over Alg ( V ) classifying an equivalence of spaces { X } × M LMod B ( M ) ≃ Alg ( V )( B , E ) natural in B ∈ Alg ( V ) . Proposition 2.41.
Let M → V be a ∞ -category with weak left action and X ∈ M . Let α ∈ Mul M ( A , X; X ) be a multi-morphism.If α exhibits A as the endomorphism object of X , the ∞ -category { X } × M LMod ( M ) admits a finalobject lying over ( A , α ) ∈ M [ X ] .Proof. The ∞ -category M → V with weak left action embeds into its enveloping closed monoidal ∞ -category M ′ → V ′ . We get an induced embedding { X } × M LMod ( M ) ⊂ { X } × M ′ LMod ( M ′ ) coveringthe embedding M [ X ] ⊂ M ′ [ X ] . By Lemma 2.36 the embedding M ⊂ M ′ preserves morphism objectsso that α exhibits A as the endomorphism object of X in M ′ . Thus ( A , α ) ∈ M [ X ] represents the rightfibration M ′ [ X ] → V ′ or equivalently is the final object of M ′ [ X ] . So by [6] Theorem 4.7.1.34. the ∞ -category { X } × M ′ LMod ( M ′ ) admits a final object lying over ( A , α ) ∈ M [ X ] ⊂ M ′ [ X ] so that the ∞ -category { X } × M LMod ( M ) admits a final object, too. (cid:3) In the next section we will use the following proposition: ∞ -CATEGORIES AND ∞ -CATEGORIES WITH WEAK ACTION Proposition 2.42.
Let S be an ∞ -category and M ⊛ → V a S -family of ∞ -categories with weak leftaction, which for any functor B → S gives rise to a weak left action of Fun S ( B , V ) on Fun S ( B , M ) .Let α ∈ Fun S ( B , M ) be a functor over S that admits an endomorphism object in Fun S ( B , V ) that isuniversal in the following sense:For any functor B ′ → B over S the induced functor Fun S ( B , M ) → Fun S ( B ′ , M ) of ∞ -categorieswith weak left action preserves the endomorphism object.Then there is a canonical equivalence B × M LMod S ( M ) ≃ B × Alg S ( V ) { } Alg S ( V ) [ ] over B × S Alg S ( V ) { } , where the pullbacks are formed via α respectively the endomorphism algebra β ∶ = Fun S ( B , M )( α, α ) ∈ Alg ( Fun S ( B , V )) ≃ Fun S ( B , Alg S ( V )) , which is stable under pullback alongany functor B ′ → B . Proof.
By taking pullback we may reduce to the case that the functor B → S is the identity. For anyfunctor θ ∶ T → S we have a canonical equivalenceFun S ( T , S × M LMod S ( M )) ≃ { α ○ θ } × Fun S ( T , M ) LMod ( Fun S ( T , M )) ≃ Alg ( Fun S ( T , V )) / β ○ θ ≃ Fun S ( T , S × Alg S ( V ) { } Alg S ( V ) [ ] ) over Fun S ( T , Alg S ( V )) ≃ Alg ( Fun S ( T , V )) that represents the desired equivalence over Alg S ( V ) . (cid:3) Weakly enriched ∞ -categories Next we give the definition of enriched ∞ -categories in the sense of Gepner-Haugseng and a variationof it (Definition 3.5), which we call weakly enriched ∞ -categories.We start with defining many object versions of the ∞ -categories LM , RM , BM of 2 parametrizingleft, right and biactions:
Many object versions of BM 3.1.
The forgetful functor ν ∶ ∆ → Set , [ n ] ↦ { , ..., n } gives riseto a functor ρ ∶ ∆ /[ ] (−) op ÐÐÐ→ ∆ /[ ] ν Ð→ Set /{ , } → Set , where the last functor takes the fiber over 1. For any space X the compositionBM ρ op ÐÐ→
Set op Fun (− , X ) ÐÐÐÐÐ→ S classifies a left fibration BM X → BM that is a generalized BM-monoidal ∞ -category. If X is con-tractible, the functor BM X → BM is an equivalence. Note that the pullback of BM X → BM along theright embedding Ass ⊂ RM ⊂ BM is the identity of Ass and the fiber over id [ ] ∈ BM is X . We write LM X → LM , Ass X → Assfor the pullbacks of BM X → BM along the embeddings LM ⊂ BM , Ass ⊂ LM ⊂ BM . Note that the left fibration Ass X → Ass classifies the functorAss (−) op ÐÐÐ→
Ass ν op ÐÐ→
Set op Fun (− , X ) ÐÐÐÐÐ→ S . We write RM X ∶ = RM × LM LM X → RM for the pullback of LM X → LM along the equivalenceRM ≃ LM . Moreover we have a relative versions of BM X : Given a left fibration X → S classifying a functor α ∶ S → S the composition BM × S BM × α ÐÐÐ→ BM × S ρ op × S ÐÐÐ→
Set op × S Fun (− , −) ÐÐÐÐÐ→ S N EQUIVALENCE BETWEEN ENRICHED ∞ -CATEGORIES AND ∞ -CATEGORIES WITH WEAK ACTION 25 classifies a cocartesian S-family of generalized BM-operads BM SX → S × BM, whose pullback along theright embedding Ass ⊂ BM is the identity and whose fiber over id [ ] ∈ BM is X → S . We writeAss SX → S × Ass , LM SX → S × LM , RM SX → S × RMfor the pullbacks of BM SX → S × BM along the left embedding Ass ⊂ BM, the embedding LM ⊂ BMrespectively the equivalence RM ≃ LM ⊂ BM . Now we are ready to make the following definition of enriched ∞ -precategory given by Gepner-Haugseng in [3] Definition 2.4.4.: Definition 3.2.
Gepner-HaugsengGiven a space X and an ∞ -operad V an ∞ -precategory enriched in V (or V -precategory) with spaceof objects X is a map Ass X → V of generalized ∞ -operads. Remark 3.3.
Definition 3.2 is a slight variant from the original definition of Gepner and Haugseng:Gepner and Haugseng define homotopy-coherent enrichment in an ∞ -operad V under the namecategorical algebra in V or V -enriched ∞ -category with space of objects X, which in our language isan ∞ -precategory enriched in V rev with space of objects X, where V rev denotes the reversed ∞ -operadstructure. Given objects A , B , C ∈ X the composition maps of a V -precategory C with space of objectsX are multi-morphisms Mor C ( B , C ) , Mor C ( A , B ) → Mor C ( A , C ) in our definition but maps Mor C ( A , B ) , Mor C ( B , C ) → Mor C ( A , C ) in the definition of Gepner and Haugseng. Remark 3.4.
The full subcategory inclusion Op ∞ ⊂ Op gen ∞ of ∞ -operads into generalized ∞ -operadsadmits a left adjoint denoted by L and called operadic localization. So a V -precategory with space ofobjects X is likewise a map of ∞ -operads L ( Ass X ) → V . The abstract ∞ -operad L ( Ass X ) can be explicitely constructed: Gepner-Haugseng construct apresentation of L ( Ass X ) as a simplicial operad ([3] Definition 4.2.2. and Corollary 4.2.8.) that isidentified by Macpherson [8] with a construction of Hinich ([5] 3.2.11.). We decided to work with Ass X instead of L ( Ass X ) as Ass X seems a simpler object, is much easier to define and has the technicallyconvenient advantage that the functor Ass X → Ass is a left fibration.Next we define weakly enriched ∞ -categories: For any small ∞ -operad V denote V ′ the Day-convolution monoidal structure on presheaves on the enveloping monoidal ∞ -category of V . Definition 3.5.
Given a space X and a small ∞ -operad V an ∞ -precategory weakly enriched in V with space of objects X is an ∞ -precategory enriched in V ′ with space of objects X . The canonical embedding V ⊂ V ′ of ∞ -operads yields an embedding Alg Ass X ( V ) ⊂ Alg
Ass X ( V ′ ) of ∞ -categories enriched in V into ∞ -categories weakly enriched in V . For later applications we need the opposite weakly enriched ∞ -precategory: Opposite weakly enriched ∞ -precategory 3.6. For any [ n ] ∈ ∆ we have a bijection [ n ] ≃ [ n ] sending i to n − i.Denote τ the non-identity involution on ∆ and ν ∶ ∆ → Set the functor forgetting the order. Wehave a canonical equivalence ν ○ τ ≃ ν that yields for any space X a canonical equivalenceAss revX = τ ∗ ( Ass X ) ≃ Ass X over Ass that yields on the fiber over [ ] ∈ Ass the equivalence X × X ≃ X × X switching the factors. ∞ -CATEGORIES AND ∞ -CATEGORIES WITH WEAK ACTION So for any ∞ -operad V we get a canonical equivalence(12) Alg Ass X ( V ) ≃ Alg
Ass revX ( V ) ≃ Alg
Ass X ( V rev ) that sends a V -precategory with space of objects X to its opposite V rev -precategory with the samespace of objects but reversed graph. So we get an induced equivalence(13) Alg Ass X ( V ′ ) ≃ Alg
Ass X ( V ′ rev ) ≃ Alg
Ass X ( V rev ′ ) , using the canonical equivalence V ′ rev ≃ V rev ′ , that sends an ∞ -precategory weakly enriched in V withspace of objects X to its opposite ∞ -precategory weakly enriched in V rev . N EQUIVALENCE BETWEEN ENRICHED ∞ -CATEGORIES AND ∞ -CATEGORIES WITH WEAK ACTION 27 Extracting a weakly enriched ∞ -category In this section we construct the functor χ ∶ ω LMod → { weakly enriched ∞ -precategories } (4.5) via a generalized version of Day-convolution developed in section 8.Following Hinich ([5]) we construct for any space X and ∞ -operad V an ∞ -operad structure onthe ∞ -category Fun ( X × X , V ) that weakly acts on Fun ( X , V ) from the right, and whose associativealgebras are V -precategories with space of objects X. In case of Hinich’s construction the identificationof Alg ( Fun ( X × X , V )) with V -precategories with space of objects X follows from work of Macpherson([8]), in our construction this follows tautologically. We define V -enriched presheaves on C as right C -modules in Fun ( X , V ) for this weak action. To construct the ∞ -operad structure on Fun ( X × X , V ) and its weak right action on Fun ( X , V ) we use a Day-convolution for generalized ∞ -operads, whichwe construct in section 8:Let O → Ass be a cocartesian fibration relative to the inert morphisms, C → O a generalized O -monoidal ∞ -category and D → O a generalized O -operad.By Theorem 8.7 and Proposition 8.9 the functor ( − ) × O C ∶ Op O , gen ∞ → Op O , gen ∞ admits a right adjoint Fun O ( C , − ) that preserves O -operads, and preserves O -monoidal ∞ -categoriescompatible with small colimits in case that the fibers of C → O are small. For any map O ′ → O ofcocartesian fibrations relative to the inert morphisms of Ass we have a canonical equivalence O ′ × O Fun O ( C , D ) ≃ Fun O ′ ( O ′ × O C , O ′ × O D ) . If D is an O -operad, for any Z ∈ O [ ] we have a canonical equivalenceFun O ( C , D ) Z ≃ Fun ( C Z , D Z ) . For O ∈ { Ass , LM , RM , BM } and any space X by 3.1 we have a generalized O -monoidal ∞ -category O X → O and set Quiv O X ( − ) ∶ = Fun O ( O X , − ) . More generally by 3.1 we have for any left fibration X → S a generalized O -monoidal ∞ -category O SX → S × O and set Quiv S , O X ( − ) ∶ = Fun S × O ( O SX , − ) , and Quiv O X , S ( − ) ∶ = Fun O S × O ( S × O , Quiv S , O X ( − )) . We consider the following cases:(1) O = Ass: For any ∞ -operad V the ∞ -operad Quiv AssX ( V ) has underlying ∞ -category Fun ( X × X , V ) and its associative algebras are V -precategories with space of objects X . (2) O = BM: For any ∞ -category M weakly bitensored over ∞ -operads ( V , W ) the BM-operadQuiv BMX ( M ) exhibits Fun ( X , M ) as weakly bitensored over Fun ( X × X , V ) from the left and W from the right, where the weak right action of W on Fun ( X , M ) is the diagonal action.Dualizing 2. the ∞ -category Fun ( X , M ) is canonically weakly bitensored over V from the leftand Fun ( X × X , W ) from the right, where the weak left action of V is the diagonal action. ∞ -CATEGORIES AND ∞ -CATEGORIES WITH WEAK ACTION (3) We apply this especially to V seen as weakly bitensored over itself. In this case the ∞ -categoryFun ( X , V ) is canonically weakly bitensored over V from the left and Fun ( X × X , V ) from theright and for any associative algebra C in Fun ( X × X , V ) we call P V ( C ) ∶ = RMod C ( Fun ( X , V )) the ∞ -category of V -enriched presheaves on C . So P V ( C ) carries an induced weak left V -action.We extend this via the enveloping closed monoidal ∞ -category V ′ ∶ The ∞ -category Fun ( X , V ) is canonically weakly bitensored over V ′ from the left and Fun ( X × X , V ′ ) from the right andfor any associative algebra C in Fun ( X × X , V ′ ) we set P V ( C ) ∶ = RMod C ( Fun ( X , V )) . (4) O = LM , RM: For any ∞ -category M weakly left (right) tensored over an ∞ -operad V the ∞ -category Fun ( X , M ) is canonically weakly left (right) tensored over Fun ( X × X , V ) . ByProposition 4.7 the weak left action of Fun ( X × X , V ) on Fun ( X , M ) is closed if the weak leftaction of V on M is. Especially the weak left action of Fun ( M ≃ × M ≃ , V ′ ) on Fun ( M ≃ , M ′ ) isclosed and we call the endomorphism algebra of the canonical embedding M ≃ ⊂ M ⊂ M ′ the ∞ -precategory weakly enriched in V associated to M (with space of objects M ≃ ), denoted by χ ( M ) . More generally given a map τ ∶ X → M ≃ the pullback τ ∗ ( χ ( M )) ∈ Alg ( Fun ( X × X , V )) isthe endomorphism algebra of τ with respect to the left action of Fun ( X × X , V ) on Fun ( X , M ) . Remark 4.1. If D → S × O is a bicartesian S-family of O -monoidal ∞ -categories compatible with smallcolimits, by Proposition 8.9 the functor Quiv O , SX ( D ) → S × O is a bicartesian S-family of O -monoidal ∞ -categories compatible with small colimits and the functor Quiv O X , S ( D ) → O is a O -monoidal ∞ -category compatible with small colimits.Especially for S contractible: if an O -operad D is a O -monoidal ∞ -category compatible with smallcolimits, by Proposition 8.9 2. the O -operad Quiv O X ( D ) is a O -monoidal ∞ -category compatible withsmall colimits, whose tensorproduct admits the following description: ● For O = Ass: Given functors F , G ∶ X × X → V their tensorproduct F ⊗ G ∶ X × X → V is the leftkan-extension of the functorX × X × X ( α,β ) ÐÐÐ→ ( X × X ) × ( X × X ) F × G ÐÐ→ V × V ⊗ Ð→ V along the functor γ ∶ X × X × X → X × X, where α sends ( A , B , C ) to ( B , C ) , β sends ( A , B , C ) to ( A , B ) and γ sends ( A , B , C ) to ( A , C ) . So concretely we have ( F ⊗ G )( A , C ) ≃ colim B ∈ X F ( B , C ) ⊗ G ( A , B ) for A , C ∈ X . ● For O = LM: Given functors F ∶ X × X → V , G ∶ X → D the left action F ⊗ G ∶ X → D is the leftkan-extension of the functorX × X ( id ,β ) ÐÐÐ→ ( X × X ) × X F × G ÐÐ→ V × D ⊗ Ð→ D along the functor γ ∶ X × X → X, where β sends ( B , C ) to B and γ sends ( B , C ) to C. Soconcretely we have ( F ⊗ G )( C ) ≃ colim B ∈ X F ( B , C ) ⊗ G ( B ) for C ∈ X . Remark 4.2.
Let O ∈ { Ass , LM , RM , BM } . The embedding Op O ∞ ⊂ Op O , gen ∞ of O -operads into gener-alized O -operads admits a left adjoint L that sends O X to a O -operad L ( O X ) . By [8] Hinich constructsexplicite presentations of L ( O X ) ([5] 3.2.11.) that allow him to deduce that the functor ( − ) × O L ( O X ) ∶ Op O ∞ → Op O ∞ admits a right adjoint Quiv O X ( − ) : Hinich proves that the functor L ( O X ) → O is flat ([5]Proposition 3.3.6.) and deduces from flatness that the functor ( − ) × O L ( O X ) ∶ Op O ∞ → Op O ∞ admits N EQUIVALENCE BETWEEN ENRICHED ∞ -CATEGORIES AND ∞ -CATEGORIES WITH WEAK ACTION 29 a right adjoint Quiv O X ( − ) ([5] Proposition 2.8.3). Contrary to that we work with the generalized O -operad O X and need to construct a Day-convolution for generalized O -operads.Using Hinich’s explicite description of L ( O X ) one can check with some effort that the local equiva-lence O X → L ( O X ) is a strong approximation in the sense of [6] Definition 2.3.3.6., which implies thatany base-change B × O O X → B × O L ( O X ) of it along a map B → O of generalized O -operads is again alocal equivalence. This guarantees that Hinich’s and our Quiv O X ( D ) are equivalent. Notation 4.3.
For O ∈ { Ass , LM , RM , BM } we setQuiv O ∶ = Quiv O , S × Op O ∞ S ∗ × Op O ∞ ( S × U O ) ,ω Quiv O ∶ = Quiv O , S × Op O ∞ S ∗ × Op O ∞ ( S × ω U O ) , where U O → Op O ∞ , ω U O → Op O ∞ denote the cocartesian Op O ∞ -families of O -operads classifying the iden-tity respectively the functor sending an O -operad to presheaves with Day-convolution on its enveloping O -monoidal ∞ -category. For O = Ass we drop O from the notation.By Remark 4.1 the functor ω Quiv O → S × Op O ∞ × O is a bicartesian S × Op O ∞ -family of O -monoidal ∞ -categories compatible with small colimits. We have a canonical embedding U O ⊂ ω U O of cocartesianOp O ∞ -families of O -operads that yields an embedding Quiv O ⊂ ω Quiv O of S × Op O ∞ -families of O -operads.For functors S → S , α ∶ S → Op O ∞ classifying a left fibration X → S respectively a cocartesian S-familyof O -operads D → S × O we have canonical equivalences(14) Quiv S × O X ( D ) ≃ S × ( S × Op O ∞ ) Quiv O (15) Quiv S × O X ( D ′ ) ≃ S × ( S × Op O ∞ ) ω Quiv O , where D ′ → S × O classifies the composition of α ∶ S → Op O ∞ with the functor sending an O -operad topresheaves with Day-convolution on its enveloping O -monoidal ∞ -category. Notation 4.4.
We set PreCat ∞ ∶ = Alg S × Op ∞ ( Quiv ) ,ω PreCat ∞ ∶ = Alg S × Op ∞ ( ω Quiv ) and for any ∞ -operad V we write PreCat V ∞ , ω PreCat V ∞ for the fiber over V .By the equivalences 14, 15 we have a canonical equivalenceAlg ( Quiv S , X ( D )) ≃ Fun S ( S , Alg S ( Quiv S × AssX ( D ))) ≃ Fun S × Op ∞ ( S , PreCat ∞ ) , (16) Alg ( Quiv S , X ( D ′ )) ≃ Fun S ( S , Alg S ( Quiv S × AssX ( D ′ ))) ≃ Fun S × Op ∞ ( S , ω PreCat ∞ ) . The comparison functor 4.5.
Now we are ready to construct the functor χ ∶ ω LMod → ω PreCat ∞ that endows the graph Mor C ( − , − ) ∶ C ≃ × C ≃ → V ′ of an ∞ -category C weakly left tensored over V withthe structure of a V ′ -enriched ∞ -precategory.Denote Fun ([ ] , S ) × Fun ({ } , S ) ω LModthe pullback of evaluation at the target along the functor ω LMod → S sending a ∞ -category withweak left action to its maximal subspace. So objects of Fun ([ ] , S ) × Fun ({ } , S ) ω LMod are pairs ( M , ϕ ∶ X → M ≃ ) . The diagonal embedding S ⊂ Fun ([ ] , S ) yields an embedding ω LMod ⊂ Fun ([ ] , S ) × Fun ({ } , S ) ω LMod . ∞ -CATEGORIES AND ∞ -CATEGORIES WITH WEAK ACTION The functor χ will be constructed as the restriction of a functorΘ ∶ Fun ([ ] , S ) × Fun ({ } , S ) ω LMod → PreCat ∞ that sends a pair ( M , ϕ ∶ X → M ≃ ) to the graph X × X ϕ × ϕ ÐÐ→ M ≃ × M ≃ Mor M ′ (− , −) ÐÐÐÐÐÐ→ V ′ with the structureof a V ′ -enriched ∞ -precategory with space of objects X . A functor ψ ∶ S → Fun ([ ] , S ) × Fun ({ } , S ) ω LModover Fun ({ } , S ) × Op ∞ precisely corresponds to a cocartesian S-family M ⊛ → V ⊗ of ∞ -categories withweak left action equipped with a map F ∶ X → M of cocartesian fibrations over S starting at a leftfibration X → S. To construct Θ we will extract from ψ a functorS → ω PreCat ∞ over S × Op ∞ . By functoriality the cocartesian S-family M ⊛ → V ⊗ of ∞ -categories with weak left actiongives rise to an enveloping cocartesian S-family M ′⊛ → V ′⊗ of ∞ -categories with closed left action andan embedding M ⊂ M ′ of cocartesian S-families of ∞ -categories with weak left action. So we get aleft action of the ∞ -category Fun S ( X × S X , V ′ ) on the ∞ -category Fun S ( X , M ′ ) . By Proposition 4.7the functor F ∶ X → M ⊂ M ′ admits an endomorphism object Mor Fun S ( X , M ′ ) ( F , F ) ∈ Fun S ( X × S X , V ′ ) with respect to this left action. Specializing 16 we have a canonical equivalenceAlg ( Fun S ( X × S X , V ′ )) ≃ Fun S × Op ∞ ( S , ω PreCat ∞ ) . The endomorphism algebra Mor
Fun S ( X , M ′ ) ( F , F ) ∈ Alg ( Fun S ( X × S X , V ′ )) corresponds to the desiredfunctor S → ω PreCat ∞ over S × Op ∞ . So for ψ the identity we obtain a functorΘ ∶ Fun ([ ] , S ) × Fun ({ } , S ) ω LMod → ω PreCat ∞ over S × Op ∞ sending an ∞ -category with weak left action M equipped with a map ϕ ∶ X → M ≃ to theendomorphism algebra Mor Fun ( X , M ′ ) ( ϕ, ϕ ) ∈ Alg ( Fun ( X × X , V ′ )) . We restrict Θ to the following fullsubcategory: We write ω LMod fl ⊂ Fun ([ ] , S ) × Fun ({ } , S ) ω LModfor the full subcategory of pairs ( M , ϕ ∶ X → M ≃ ) with ϕ essentially surjective. We call ω LMod fl the ∞ -category of flagged ∞ -categories with weak left action and have the following remark: Remark 4.6.
Ayala-Francis ([2] Definition 0.12.) introduce the notion of flagged ∞ -category, whichis an ∞ -category C equipped with an essentially surjective map of spaces X → C ≃ , and shows thatflagged ∞ -categories are a model for Segal spaces, precisely that the restricted Yoneda-embedding Cat ∞ fl ⊂ P ( Cat ∞ fl ) → P ( ∆ ) along the restricted diagonal embedding ∆ ⊂ Cat ∞ ⊂ Cat ∞ fl is fully faithful with essential image theSegal spaces. So every flagged ∞ -category ( M , ϕ ∶ X → M ≃ ) with weak left action has an underlyingflagged ∞ -category aka Segal space. Moreover we can identify flagged ∞ -categories with flagged ∞ -categories with weak left action over S as the forgetful functor ω LMod S → Cat ∞ is an equivalence.We write χ for the restriction ω LMod ⊂ ω LMod fl ⊂ Fun ([ ] , S ) × Fun ({ } , S ) ω LMod Θ Ð→ ω PreCat ∞ . The next proposition for the case that S is contractible is due to Hinich ([5] Proposition 6.3.1.)with a very different proof:
N EQUIVALENCE BETWEEN ENRICHED ∞ -CATEGORIES AND ∞ -CATEGORIES WITH WEAK ACTION 31 Proposition 4.7.
Let S be an ∞ -category and M ⊛ → V ⊗ a cocartesian S -family of ∞ -categories withweak left action. Let X → S be a left fibration and F , G ∶ X → M be functors over S , where F is a mapof cocartesian fibrations over S .Assume that for every Z ∈ X lying over some s ∈ S the images F ( Z ) , G ( Z ) ∈ M s admit a morphismobject.Then the functors F , G admit a morphism object Mor
Fun S ( X , M ) ( F , G ) ∈ Fun S ( X × S X , V ) that isuniversal in the following sense:For every functor T → S , left fibration Y → T and functor Y → T × S X over T the induced functor Fun S ( X , M ) → Fun T ( Y , T × S M ) preserves the morphism object Mor
Fun S ( X , M ) ( F , G ) . Proof.
We first reduce to the case that M ⊛ → V ⊗ is a bicartesian S-family of presentable ∞ -categorieswith left action: The cocartesian S-family M ⊛ → V ⊗ of ∞ -categories with weak left action embedsinto its enveloping cocartesian S-family M ′⊛ → V ′⊗ of ∞ -categories with closed left action. For every s ∈ S the induced embedding M s ⊂ M ′ s preserves morphism objects. The embedding M ⊂ M ′ yields anembedding Fun S ( X , M ) ⊂ Fun S ( X , M ′ ) of ∞ -categories with weak left action.Having reduced to this case by Proposition 9.7 the map F extends to a map ¯F ∶ Fun S ( X , V ) → M of cocartesian fibrations over S that admits a right adjoint R ∶ M → Fun S ( X , V ) relative to S . We will show that the compositionT ∶ = R ○ G ∈ Fun S ( X , Fun S ( X , V )) ≃ Fun S ( X × S X , V ) is the morphism object of F , G . We will first construct a map β ∶ T ⊗ F → G in Fun S ( X , M ) . Denote E the tensorunit of Fun S ( X × S X , V ) ≃ Fun S ( X , Fun S ( X , V )) . We have ¯F ○ E ≃ F . The unit id → R ○ ¯F gives rise to a mapE → R ○ ¯F ○ E ≃ R ○ F . The functor R ∶ M → Fun S ( X , V ) over S yields a functorR ∗ ∶ Fun S ( X , M ) → Fun S ( X , Fun S ( X , V )) ≃ Fun S ( X × S X , V ) laxly compatible with the weak left actions.So for any functor H ∶ X × S X → V over S we get a mapH ≃ H ⊗ E → H ⊗ R ∗ ( F ) → R ∗ ( H ⊗ F ) in Fun S ( X , V ) adjoint to a map θ ∶ ¯F ○ H → H ⊗ Fin Fun S ( X , M ) natural in H. For any s ∈ S and Z ∈ X s we have that θ ( Z ) ∶ ¯F ( H ( Z )) ≃ colim Y ∈ X H ( Y , Z ) ⊗ F ( Y ) → ( H ⊗ F )( Z ) ≃ colim Y ∈ X H ( Y , Z ) ⊗ F ( Y ) is the canonical equivalence so that θ is an equivalence.We define β ∶ T ⊗ F → G as the counitT ⊗ F ≃ ¯F ○ T = ( ¯F ○ R ) ○ G → Gand have to see that β yields for any functor H ∶ X → V over S an equivalence ρ ∶ Fun S ( X , V )( H , T ) ≃ Fun S ( X , M )( H ⊗ F , T ⊗ F ) → Fun S ( X , M )( H ⊗ F , G ) . The map ρ factors asFun S ( X , V )( H , T ) → Fun S ( X , M )( ¯F ○ H , ¯F ○ T ) Ð→ Fun S ( X , M )( ¯F ○ H , G ) ∞ -CATEGORIES AND ∞ -CATEGORIES WITH WEAK ACTION ≃ Fun S ( X , M )( H ⊗ F , G ) . Consequently we need to check that the map ρ ′ ∶ Fun S ( X , V )( H , T ) → Fun S ( X , M )( ¯F ○ H , ¯F ○ T ) Ð→ Fun S ( X , M )( ¯F ○ H , G ) is an equivalence. The map ρ ′ is inverse to the mapFun S ( X , M )( ¯F ○ H , G ) → Fun S ( X , V )( R ○ ¯F ○ H , T ) → Fun S ( X , V )( H , T ) induced by the unit id → R ○ ¯F using the triangular identities. (cid:3) N EQUIVALENCE BETWEEN ENRICHED ∞ -CATEGORIES AND ∞ -CATEGORIES WITH WEAK ACTION 33 A universal property of the extracted weakly enriched ∞ -category In this section we prove that for any ∞ -category M weakly left tensored over an ∞ -operad V andany ∞ -precategory C weakly enriched in V with space of objects X we have a canonical equivalence(17) ω PreCat V ∞ ( C , χ ( M )) ≃ LMod C ( Fun ( X , M )) ≃ over S ( X , M ≃ ) ≃ Fun ( X , M ) ≃ natural in C , M and V (Theorem 5.3), where the ∞ -category of leftmodules is formed with respect to the restricted left action of Fun ( X × X , V ′ ) on Fun ( X , M ) . Notation 5.1.
This motivates to setFun V ( C , χ ( M )) ∶ = LMod C ( Fun ( X , M )) and call Fun V ( C , χ ( M )) the ∞ -category of V ′ -enriched functors C → χ ( M ) . Given an ∞ -operad V we construct a canonical functor ( ω PreCat V ∞ ) op × ω LMod V → S , C , M ↦ LMod C ( Fun ( X , M )) ≃ (Proposition 5.2) adjoint to a functor ω LMod V → Fun (( ω PreCat V ∞ ) op , S ) that we canonically factorthrough χ ∶ ω LMod V → ω PreCat V ∞ (Theorem 5.3). But technically more challenging we also considerfunctoriality in V . We start with making precise how the assignment C , M ↦ LMod C ( Fun ( X , M )) on the left hand sideof equivalence 17 is functorial. Given a cocartesian fibration X → S denote X rev → S the cocartesianfibration arising from X → S by taking fiberwise the opposite ∞ -category. Proposition 5.2.
There is a canonical functor ρ ∶ ω PreCat rev ∞ × Op ∞ ω LMod → Cat ∞ , C , M ↦ LMod C ( Fun ( X , M )) over Op ∞ that induces on the fiber over an ∞ -operad V a functor ( ω PreCat V ∞ ) op × ω LMod V → Cat ∞ . Proof.
Denote ̂ Cart ⊂ Fun ([ ] , ̂ Cat ∞ ) the subcategory with objects the cartesian fibrations and mor-phisms the commutative squares, whose top functor preserves cartesian morphisms.By Proposition 9.8 there is a canonical equivalenceOp ∞ × Fun ({ } , ̂ Cat ∞ ) ̂ Cart ≃ Fun Op ∞ ( ω PreCat rev ∞ , Op ∞ × ̂ Cat ∞ ) over Op ∞ . So the functor ρ is adjoint to a functor ρ ′ ∶ ω LMod → Op ∞ × Fun ({ } , ̂ Cat ∞ ) ̂ Cartover Op ∞ . The functor ρ ′ is classified by a map θ ∶ X → ω LMod × Op ∞ ω PreCat ∞ of cocartesianfibrations over ω LMod that yields on every fiber a cartesian fibration, whose cartesian morphisms arepreserved by the fiber transports. In the following we will construct θ ∶ The cocartesian S × ω LMod-family ω Quiv LM of left tensored ∞ -categories gives rise to a mapLMod S × ω LMod ( ω Quiv LM ) → Alg S × ω LMod ( Ass × LM ω Quiv LM ´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶ ω LMod × Op ∞ ω Quiv ) ≃ ω PreCat ∞ × Op ∞ ω LModof cocartesian fibrations over S × ω LMod.We define θ as the pullbackQuiv LM m × ω Quiv LM m LMod S × ω LMod ( ω Quiv LM ) → ω PreCat ∞ × Op ∞ ω LMod . ∞ -CATEGORIES AND ∞ -CATEGORIES WITH WEAK ACTION Thus θ is a map of cocartesian fibrations over ω LMod that induces on the fiber over any ∞ -category M with weak left action over some ∞ -operad V the canonical functor θ M ∶ ( Quiv LM M ) m × ( ω Quiv LM M ) m LMod S ( ω Quiv LM M ) → Alg S ( ω Quiv V ) ≃ ω PreCat V ∞ . As ω Quiv LM M → S × LM is a map of cartesian fibrations over S , the functor θ M is a cartesian fibration,whose cartesian morphisms are those, whose image in ( Quiv LM M ) m is cartesian over S . The functor θ M classifies a functor ( ω PreCat V ∞ ) op → Cat ∞ sending an ∞ -precategory C weakly enriched in V withspace of objects X toFun ( X , M ) × Fun ( X , M ′ ) LMod C ( Fun ( X , M ′ )) ≃ LMod C ( Fun ( X , M )) . (cid:3) To state Theorem 5.3 we use that for any cocartesian fibration X → S there is a canonical embedding X ⊂ Fun S ( X rev , S × S ) of cocartesian fibrations over S that induces fiberwise the Yoneda-embedding(Lemma 9.10). Theorem 5.3.
The functor ω LMod χ Ð→ ω PreCat ∞ ⊂ Fun Op ∞ ( ω PreCat rev ∞ , Op ∞ × ̂ S ) over Op ∞ is adjoint to the composition ω PreCat rev ∞ × Op ∞ ω LMod ρ Ð→ Cat ∞ (−) ≃ ÐÐ→ S ⊂ ̂ S . Proof.
Denote ̂ Cart ⊂ Fun ([ ] , ̂ Cat ∞ ) the subcategory with objects the cartesian fibrations and mor-phisms the commutative squares, whose top functor preserves cartesian morphisms.By Proposition 9.8 there is a canonical equivalenceOp ∞ × Fun ({ } , ̂ Cat ∞ ) ̂ Cart ≃ Fun Op ∞ ( ω PreCat rev ∞ , Op ∞ × ̂ Cat ∞ ) over Op ∞ . So the functor ρ is adjoint to a functor ρ ′ ∶ ω LMod → Op ∞ × Fun ({ } , ̂ Cat ∞ ) ̂ Cartover Op ∞ . By the proof of 5.2 the functor ρ ′ is classified by the map θ ∶ Quiv LM m × ω Quiv LM m LMod S × ω LMod ( ω Quiv LM ) → ω PreCat ∞ × Op ∞ ω LModof cocartesian fibrations over ω LMod that induces on the fiber over any ∞ -category M with weak leftaction a right fibration. Denote ̂ U ⊂ ̂ R ⊂ Fun ([ ] , ̂ Cat ∞ ) the full subcategories spanned by the not neccessarily small right fibrations respectively representableright fibrations, i.e. the right fibrations C → D such that C has a final object. By Proposition 9.9evaluation at the target ̂ U → ̂ Cat ∞ is a cocartesian fibration classifying the identity. By Proposition9.8 there is a canonical equivalence ̂ R ≃ Fun ̂ Cat ∞ (̂ U rev , ̂ Cat ∞ × ̂ S ) over ̂ Cat ∞ , whose pullback gives an equivalenceOp ∞ × Fun ({ } , ̂ Cat ∞ ) ̂ R ≃ Fun Op ∞ ( ω PreCat rev ∞ , Op ∞ × ̂ S ) . The Yoneda-embedding of ω PreCat ∞ relative to Op ∞ factors as ω PreCat ∞ ≃ Op ∞ × Fun ({ } , ̂ Cat ∞ ) ̂ U ⊂ Op ∞ × Fun ({ } , ̂ Cat ∞ ) ̂ R ≃ Fun Op ∞ ( ω PreCat rev ∞ , Op ∞ × ̂ S ) . The functor ω LMod χ Ð→ ω PreCat ∞ ≃ Op ∞ × Fun ({ } , ̂ Cat ∞ ) ̂ U ⊂ Op ∞ × Fun ({ } , ̂ Cat ∞ ) ̂ R N EQUIVALENCE BETWEEN ENRICHED ∞ -CATEGORIES AND ∞ -CATEGORIES WITH WEAK ACTION 35 over Op ∞ is classified by the map ω LMod × ω PreCat { } ∞ Fun ([ ] , ω PreCat ∞ ) ′ → ω PreCat { }∞ × Op ∞ ω LModof cocartesian fibrations over ω LMod that induces on the fiber over any ∞ -category M with weak leftaction a right fibration, whereFun ([ ] , ω PreCat ∞ ) ′ ⊂ Fun ([ ] , ω PreCat ∞ ) denotes the full subcategory spanned by the morphisms lying over an equivalence in Op ∞ . Conse-quently we need to construct a canonical equivalence ( Fun ([ ] , S ) × Fun ({ } , S ) ω LMod ) × ω Quiv LM m LMod S × ω LMod ( ω Quiv LM ) ≃ ω LMod × ω PreCat { } ∞ Fun ([ ] , ω PreCat ∞ ) ′ over Alg S × ω LMod ( Ass × LM ω Quiv LM ´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶ ω LMod × Op ∞ ω Quiv ) ≃ ω PreCat { }∞ × Op ∞ ω LMod.By Corollary 2.42 there is a canonical equivalenceLMod S × ω LMod ( Quiv LM ) ≃ Quiv LM m × ω PreCat { } ∞ ω PreCat [ ]∞ over ω PreCat { }∞ × ( S × Op ∞ ) Quiv LM m , where the pullback is formed via χ ∶ Quiv LM m → ω PreCat ∞ . The compositionQuiv LM m ≃ Fun ([ ] , S ) × Fun ({ } , S ) ω LMod ⊂ Fun ([ ] , S ) × Fun ({ } , S ) Quiv LM m ≃ Fun ([ ] , Quiv LM m ) cart → Fun ({ } , Quiv LM m ) is the identity. As χ ∶ Quiv LM m → ω PreCat ∞ is a map of cartesian fibrations over S , the functor χ ∶ Quiv LM m → ω PreCat ∞ factors asQuiv LM m ≃ Fun ([ ] , S ) × Fun ({ } , S ) ω LMod
Fun ([ ] , S )× S χ ÐÐÐÐÐÐÐÐ→
Fun ([ ] , S ) × Fun ({ } , S ) ω PreCat ∞ ≃ Fun ([ ] , ω PreCat ∞ ) cart → Fun ({ } , ω PreCat ∞ ) where Fun ([ ] , ω PreCat ∞ ) cart ⊂ Fun ([ ] , ω PreCat ∞ ) denotes the full subcategory spanned by themorphisms of ω PreCat ∞ that are cartesian over S (and so lie over an equivalence of ∞ -operads).Hence we have a canonical equivalenceQuiv LM m × ω PreCat { } ∞ ω PreCat [ ]∞ ≃ Quiv LM m × Fun ([ ] ,ω PreCat ∞ ) cart ( ω PreCat [ ]∞ × ω PreCat ∞ Fun ([ ] , ω PreCat ∞ ) cart ) over ω PreCat { }∞ × ( S × Op ∞ ) Quiv LM m , where the pullback is along the functorQuiv LM m ≃ Fun ([ ] , S ) × Fun ({ } , S ) ω LMod
Fun ([ ] , S )× S χ ÐÐÐÐÐÐÐÐ→
Fun ([ ] , S ) × Fun ({ } , S ) ω PreCat ∞ ≃ Fun ([ ] , ω PreCat ∞ ) cart . As ω PreCat ∞ → S is a cartesian fibration, we have a factorization system on ω PreCat ∞ with leftclass the morphisms lying over equivalences of spaces and right class the cartesian morphisms. Thusby [7] Proposition 5.2.8.17. evaluation at { → } ⊂ [ ] Fun ([ ] , ω PreCat ∞ ) × ω PreCat ∞ Fun ([ ] , ω PreCat ∞ ) ≃ Fun ([ ] , ω PreCat ∞ ) → Fun ([ ] , ω PreCat ∞ ) restricts to an equivalence ω PreCat [ ]∞ × ω PreCat ∞ Fun ([ ] , ω PreCat ∞ ) cart ≃ Fun ([ ] , ω PreCat ∞ ) ′ . So we get a canonical equivalenceQuiv LM m × Fun ([ ] ,ω PreCat ∞ ) cart ( ω PreCat [ ]∞ × ω PreCat ∞ Fun ([ ] , ω PreCat ∞ ) cart ) ≃ ∞ -CATEGORIES AND ∞ -CATEGORIES WITH WEAK ACTION ( Fun ([ ] , S ) × Fun ({ } , S ) ω LMod ) × ( Fun ([ ] , S )× Fun ({ } , S ) ω PreCat { } ∞ ) Fun ([ ] , ω PreCat ∞ ) ′ ≃ ω LMod × ω PreCat { } ∞ Fun ([ ] , ω PreCat ∞ ) ′ over ω PreCat { }∞ × Op ∞ ω LMod . (cid:3) N EQUIVALENCE BETWEEN ENRICHED ∞ -CATEGORIES AND ∞ -CATEGORIES WITH WEAK ACTION 37 A universal property of enriched presheaves
Goal of this section is to prove the following theorem:
Theorem 6.1.
Let V be a monoidal ∞ -category compatible with small colimits, C a V -precategory withsmall space of objects X and M an ∞ -category left tensored over V compatible with small colimits.There is a canonical equivalence Ψ ∶ LinFun L V ( P V ( C ) , M ) ≃ Fun V ( C , χ ( M )) = LMod C ( Fun ( X , M )) over LinFun L V ( Fun ( X , V ) , M ) ≃ Fun ( X , M ) . If V is presentable, any left V -action compatible with small colimits is closed. So by Theorem 5.3we have a canonical equivalenceFun V ( C , χ ( M )) ≃ ≃ ̂ PreCat V ∞ ( C , χ ( M )) . So the equivalence Ψ of Theorem 6.1 induces on maximal subspaces an equivalenceLinFun L V ( P V ( C ) , M ) ≃ ≃ ̂ PreCat V ∞ ( C , χ ( M )) that sends for M = P V ( C ) the identity to a V -enriched functor C → χ ( P V ( C )) , which we call the V -enriched Yoneda-embedding, and which we prove to be an embedding (Proposition 6.5).Naturality in M and the Yoneda-lemma imply the following corollary: Corollary 6.2.
Restriction along the V -enriched Yoneda-embedding C → χ ( P V ( C )) defines an equiv-alence LinFun L V ( P V ( C ) , M ) ≃ ≃ ̂ PreCat V ∞ ( C , χ ( M )) . To construct the functor Ψ we start with the following Lemmas 6.3 and 6.4 : Let X be a spaceand V an ∞ -operad. The ∞ -category Fun ( X , V ) is weakly bitensored over Fun ( X × X , V ) from theright and V from the left, where the weak left action is the diagonal weak left action. Thus the ∞ -category Fun ( X , Fun ( X , V )) ≃ Fun ( X × X , V ) is weakly bitensored over Fun ( X × X , V ) from the leftand Fun ( X × X , V ) from the right, where the weak right action is the diagonal action. The next lemmaidentifies this weak biaction: Lemma 6.3.
Let X be a space and V an ∞ -operad.The weak biaction on Fun ( X × X , V ) from above is induced by the canonical ∞ -operad structure on Fun ( X × X , V ) .Proof. First note that we have a canonical equivalenceBM X × BM BM revX ≃ BM × Ass
Ass X classified by the equivalence that sends a functor φ ∶ [ n ] → [ ] to the equivalenceFun ( φ − ( ) , X ) × Fun ( φ − ( ) , X ) ≃ Fun ( φ − ( ) ∐ φ − ( ) , X ) ≃ Fun ([ n ] , X )) . This equivalence yields for any BM-operad O → BM a natural equivalenceAlg O / BM ( Quiv
BMX ( Quiv
BMX ( BM × Ass V rev ) rev )) ≃ Alg O × BM BM X / BM ( Quiv
BMX ( BM × Ass V rev ) rev ) ≃ Alg O × BM BM X × BM BM revX / BM ( BM × Ass V ) ≃ Alg O × BM BM × Ass
Ass X / BM ( BM × Ass V ) ≃ Alg O / BM ( BM × Ass
Quiv X ( V )) representing an equivalenceQuiv BMX ( Quiv
BMX ( BM × Ass V rev ) rev ) ≃ BM × Ass
Quiv X ( V ) . (cid:3) ∞ -CATEGORIES AND ∞ -CATEGORIES WITH WEAK ACTION Let X be a space and M an ∞ -category weakly bitensored over ( V , W ) . For any A ∈ Alg ( W ) wehave a canonical equivalenceFun ( X , RMod A ( M )) ≃ RMod A ( Fun ( X , M )) over Fun ( X , M ) as the weak right action of W on Fun ( X , M ) is the diagonal weak action. ButFun ( X , RMod A ( M )) , RMod A ( Fun ( X , M )) carry canonical weak left actions over Fun ( X × X , V ) , whichare identified by the next lemma: Lemma 6.4.
Let X be a space and M an ∞ -category weakly bitensored over ( V , W ) . For any A ∈ Alg ( W ) the canonical equivalence Fun ( X , RMod A ( M )) ≃ RMod A ( Fun ( X , M )) over Fun ( X , M ) refines to a Fun ( X × X , V ) -linear equivalence.More coherently, there is a canonical equivalence ψ ∶ RMod LM (( LM × RM ) × BM Quiv
BMX ( M )) ≃ Quiv
LMX ( RMod LM (( LM × RM ) × BM M )) over Alg ( W ) and LM × BM Quiv
BMX ( M ) ≃ Quiv
LMX ( LM × BM M ) that induces on underlying ∞ -categories the canonical equivalence (18) RMod ( Fun ( X , M )) ≃ Fun ( X , RMod ( M )) . Proof.
Via the enveloping closed BM-monoidal ∞ -category we can reduce to the case that M is a BM-monoidal ∞ -category compatible with small colimits. Let M ′ → ̃ BM the ̃ BM-monoidal ∞ -categorycorresponding to M . We will construct a map ψ ′ ∶ RMod ̃ LM ((̃ LM × ̃ RM ) × ̃ BM Quiv ̃ BMX ( M ′ )) → Quiv ̃ LMX ( RMod ̃ LM ((̃ LM × ̃ RM ) × ̃ BM M ′ )) of cocartesian fibrations over ̃ LM that restricts on ̃ Ass to the identity, induces on underlying ∞ -categories the equivalence 18, and restricts to an equivalence ψ with the desired properties.For any ̃ LM-operad O we have a canonical equivalenceAlg O /̃ LM ( RMod ̃ LM ((̃ LM × ̃ RM ) × ̃ BM M ′ )) ≃ Alg O ×̃ RM /̃ LM ×̃ RM ((̃ LM × ̃ RM ) × ̃ BM M ′ ) . So we have canonical equivalencesAlg O /̃ LM ( RMod ̃ LM ((̃ LM × ̃ RM ) × ̃ BM Quiv ̃ BMX ( M ′ ))) ≃ Alg O ×̃ RM /̃ LM ×̃ RM (̃ LM × ̃ RM ) × ̃ BM Quiv ̃ BMX ( M ′ )) ≃ Alg ( O ×̃ RM )× (̃ LM × ̃ RM ) (̃ LM ×̃ RM )× ̃ BM ̃ BM X /̃ LM ×̃ RM ((̃ LM × ̃ RM ) × ̃ BM M ′ ) , Alg O /̃ LM ( Quiv ̃ LMX ( RMod ̃ LM ((̃ LM × ̃ RM ) × ̃ BM M ′ ))) ≃ Alg O × ̃ LM ̃ LM X /̃ LM ( RMod ̃ LM ((̃ LM × ̃ RM ) × ̃ BM M ′ )) ≃ Alg ( O ×̃ RM )× (̃ LM × ̃ RM ) ̃ LM X ×̃ RM /̃ LM ×̃ RM ((̃ LM × ̃ RM ) × ̃ BM M ′ ) . The functor ψ ′ is induced by the map ̃ LM X × ̃ RM → (̃ LM × ̃ RM ) × ̃ BM ̃ BM X of left fibrations over ̃ LM × ̃ RM classifying the natural transformation of functors ̃ LM × ̃ RM → S thatsends ([ n ] , i ) , ([ m ] , j ) to the identity of Fun ([ n ] , X ) respectively the map Fun ([ n ] , X ) → Fun ([ − ] , X ) depending if i = (cid:3) N EQUIVALENCE BETWEEN ENRICHED ∞ -CATEGORIES AND ∞ -CATEGORIES WITH WEAK ACTION 39 Let X be a space and V an ∞ -operad. Combining Lemmas 6.3 and 6.4 for any C ∈ Alg ( Fun ( X × X , V )) there is a canonical equivalenceFun ( X , RMod C ( Fun ( X , V ))) ≃ RMod C ( Fun ( X × X , V )) over Fun ( X , Fun ( X , V )) ≃ Fun ( X × X , V ) of ∞ -categories with left Fun ( X × X , V ) -action that yields forany B ∈ Alg ( Fun ( X × X , V )) an equivalenceLMod B ( Fun ( X , RMod C ( Fun ( X , V )))) ≃ LMod B ( RMod C ( Fun ( X × X , V ))) ≃ BMod B , C ( Fun ( X × X , V )) . over LMod B ( Fun ( X × X , V )) . Let X be a small space, V a monoidal ∞ -category compatible with small colimits and M an ∞ -category left tensored over V compatible with small colimits.Denote Ψ the compositionLinFun L V ( RMod C ( Fun ( X , V )) , M ) → LinFun
LFun ( X × X , V ) ( Fun ( X , RMod C ( Fun ( X , V ))) , Fun ( X , M )) → Fun L ( LMod C ( Fun ( X , RMod C ( Fun ( X , V )))) , LMod C ( Fun ( X , M ))) → LMod C ( Fun ( X , M )) , where the last functor evaluates at C ∈ BMod C , C ( Fun ( X × X , V )) ≃ LMod C ( Fun ( X , RMod C ( Fun ( X , V )))) . The functor Ψ is natural in M ∈ LMod V ( Cat cc ∞ ) and for M = RMod C ( Fun ( X , V )) the functor Ψsends the identity of M to C ∈ BMod C , C ( Fun ( X × X , V )) ≃ LMod C ( Fun ( X , RMod C ( Fun ( X , V )))) . Denote φ ∶ Fun ( X , V ) → RMod C ( Fun ( X , V )) the free functor. The functor Ψ fits into a commutativesquare(19) LinFun L V ( RMod C ( Fun ( X , V )) , M ) φ ∗ (cid:15) (cid:15) Ψ / / LMod C ( Fun ( X , M )) ν (cid:15) (cid:15) LinFun L V ( Fun ( X , V ) , M ) ≃ / / Fun ( X , M ) , where ν denotes the forgetful functor.Next we show that Ψ is an equivalence proving theorem 6.1: Theorem 6.1.
Denote φ ∶ Fun ( X , V ) ⇄ RMod C ( Fun ( X , V )) ∶ γ the free-forgetful adjunction. The right adjoint γ is V -linear and makes the left adjoint φ canonically V -linear as the composition γ ○ φ ≃ ( − ) ⊗ C is V -linear.So this adjunction yields an adjunction γ ∗ ∶ Fun ( X , M ) ≃ LinFun L V ( Fun ( X , V ) , M ) ⇄ LinFun L V ( RMod C ( Fun ( X , V )) , M ) ∶ φ ∗ . By the commutativity of square 19 it will be enough to prove the following two conditions:(1) The left vertical functor φ ∗ in the square is monadic.(2) For any functor H ∶ X → M the canonical map θ ∶ C ⊗ H → C ⊗ φ ∗ ( γ ∗ ( H )) ≃ C ⊗ ν ( Ψ ( γ ∗ ( H ))) → ν ( Ψ ( γ ∗ ( H ))) in Fun ( X , M ) is an equivalence. ∞ -CATEGORIES AND ∞ -CATEGORIES WITH WEAK ACTION We start with the first condition: The right adjoint φ ∗ is conservative as RMod C ( Fun ( X , V )) isgenerated by the free right C -modules under geometric realizations. Moreover φ ∗ preserves smallcolimits as by Lemma 9.6 small colimits exist in ∞ -categories of V -linear functors and are formedobjectwise.To check condition 2. it is enough to prove the case M = Fun ( X , V ) and H ∶ X → Fun ( X , V ) corresponding to the identity or equivalently H adjoint to the tensorunit E ∶ X × X → V of Fun ( X × X , V ) as both vertical adjunctions in the commtative square are natural in M with respect to V -linear functorspreserving small colimits. In this case the left Fun ( X × X , V ) -action on Fun ( X , M ) ≃ Fun ( X × X , V ) arises from the canonical monoidal structure on Fun ( X × X , V ) .The unit E → φ ∗ ( γ ∗ ( E )) is the mapE ≃ (( − ) ⊗ E ) ○ E → (( − ) ⊗ C ) ○ Einduced by the unit E → C , where ( − ) ⊗ C , ( − ) ⊗ E ≃ id ∶ Fun ( X , V ) → Fun ( X , V ) are induced by thecanonical right Fun ( X × X , V ) -action on Fun ( X , V ) . The canonical right Fun ( X × X , V ) -action on Fun ( X , V ) is compatible with the diagonal left V -action.The diagonal left V -action on Fun ( X , V ) induces a left Fun ( X × X , V ) -action on Fun ( X , Fun ( X , V )) ≃ Fun ( X × X , V ) compatible with the diagonal right Fun ( X × X , V ) -action. By Lemma 6.3 this biaction isinduced from the canonical monoidal structure on Fun ( X × X , V ) . Consequently for any B ∈ Fun ( X × X , V ) the induced functorFun ( X , ( − ) ⊗ B ) ∶ Fun ( X , Fun ( X , V )) → Fun ( X , Fun ( X , V )) is canonically equivalent to the functor ( − ) ⊗ B ∶ Fun ( X × X , V ) → Fun ( X × X , V ) induced by thecanonical monoidal structure on Fun ( X × X , V ) . Hence the unit E → φ ∗ ( γ ∗ ( E )) is the unit E → C . By definition the left C -module structure on ν ( Ψ ( γ ∗ ( H ))) ∶ X → M is the left C -module structureon C coming from the associative algebra structure. Therefore the map C ⊗ ν ( Ψ ( γ ∗ ( H ))) → ν ( Ψ ( γ ∗ ( H ))) is the multiplication C ⊗ C → C . (cid:3) By the next proposition the V -enriched Yoneda-embedding C → χ ( P V ( C )) is in fact a V -enrichedembedding: Proposition 6.5.
Let V be a presentable monoidal ∞ -category compatible with small colimits. The V -enriched Yoneda-embedding C → χ ( P V ( C )) is a V -enriched embedding.Proof. The associative algebra structure on C in Fun ( X × X , V ) gives rise to a ( C , C ) -bimodule structureon C corresponding to a left C -module structure on the free right C -module C ′ on the tensorunit ofFun ( X × X , V ) with respect to the canonical left action of Fun ( X × X , V ) on RMod C ( Fun ( X × X , V )) . We have a canonical equivalence(20) RMod C ( Fun ( X × X , V )) ≃ Fun ( X , RMod C ( Fun ( X , V ))) , under which C ′ corresponds to the composition ι ∶ X ⊂ Fun ( X , S ) ⊗(−) ÐÐÐ→
Fun ( X , V ) free ÐÐ→
RMod C ( Fun ( X , V )) = P V ( C ) . By Lemma 6.4 the equivalence 20 identifies the canonical left Fun ( X × X , V ) -actions. By definitionof Ψ the canonical left action of C on C ′ corresponds to that left action of C on ι , which correspondsto the V -enriched Yoneda-embedding C → χ ( P V ( C )) . By the description of endomorphism objects ofProposition 4.7 to see that the V -enriched Yoneda-embedding is an embedding, we need to check thatthe left action of C on ι exhibits C as the endomorphism object of ι . This follows via equivalence 20: By N EQUIVALENCE BETWEEN ENRICHED ∞ -CATEGORIES AND ∞ -CATEGORIES WITH WEAK ACTION 41 Lemma 6.6 the canonical left action of C on C ′ exhibits C ∈ Alg ( Fun ( X × X , V )) as the endomorphismalgebra of C ′ ∈ RMod C ( Fun ( X × X , V )) . (cid:3) For completeness we add the following well-known lemma:
Lemma 6.6.
Let V be a monoidal ∞ -category and A an associative algebra in V .The associative algebra A in V gives rise to a ( A , A ) -bimodule structure on A corresponding toa left A -module structure on the free right A -module A ′ on the tensorunit of V with respect to thecanonical left V -action on RMod A ( V ) .This left A -module structure on A ′ exhibits A as the endomorphism algebra of A ′ ∈ RMod A ( V ) .Proof. We may assume that V is a presentable monoidal ∞ -category compatible with small colimitsby embedding V into its ∞ -category of presheaves with Day-convolution. As V is presentable, the left V -action on RMod A ( V ) is closed. Thus A ′ admits an endomorphism object Mor RMod A ( V ) ( A ′ , A ′ ) ∈ V with respect to the left V -action on RMod A ( V ) . By its universal property the canonical left A-modulestructure on A ′ corresponds to a map of associative algebras A → Mor
RMod A ( V ) ( A ′ , A ′ ) in V , whichwe want to see as an equivalence. For any X ∈ V the induced map V ( X , A ) → V ( X , Mor
RMod A ( V ) ( A ′ , A ′ )) ≃ RMod A ( V )( X ⊗ A ′ , A ′ ) is a section of the map ξ ∶ RMod A ( V )( X ⊗ A ′ , A ′ ) → V ( X ⊗ A , A ) → V ( X , A ) that forgets and composes with the morphism θ ∶ X ≃ X ⊗ → X ⊗ A . But ξ is an equivalence as θ exhibits X ⊗ A ′ as the free right A-module on X as the composition X ⊗ A θ ⊗ A ÐÐ→ X ⊗ A ⊗ A → X ⊗ A isthe identity. (cid:3) ∞ -CATEGORIES AND ∞ -CATEGORIES WITH WEAK ACTION The equivalence
In section 4 we constructed a functorΘ ∶ Fun ([ ] , S ) × Fun ({ } , S ) ω LMod → ω PreCat ∞ over S × Op ∞ sending an ∞ -category M weakly left tensored over an ∞ -operad V equipped with amap ϕ ∶ X → M ≃ to the endomorphism algebraMor Fun ( X , M ′ ) ( ϕ, ϕ ) ∈ Alg ( Fun ( X × X , V ′ )) ≃ ( ω PreCat V ∞ ) X . Moreover we restricted the functor Θ to functorsΘ ∶ ω LMod fl → ω PreCat ∞ , χ ∶ ω LMod → ω PreCat ∞ . Now we come to our main goal: we prove that the functor Θ ∶ ω LMod fl → ω PreCat ∞ is an equivalencethat restricts to a fully faithful functor χ with essential image the weakly enriched ∞ -categories, whichwe define in the following: For an ∞ -precategory C weakly enriched in an ∞ -operad V with small spaceof objects X denote L ( C ) ⊂ P V ′ ( C ) the essential image of the functorX ⊂ Fun ( X , S ) ⊗(−) ÐÐÐ→
Fun ( X , V ′ ) free ÐÐ→ P V ′ ( C ) endowed with the restricted weak left action over V . Denote ι ∶ X → L ( C ) ≃ the induced map.The following definition is non-standart but equivalent to the usual definition due to [5] 7.2.: Definition 7.1.
Let C be an ∞ -precategory (weakly) enriched in an ∞ -operad V with small space ofobjects X . We call C an ∞ -category (weakly) enriched in V if ι ∶ X → L ( C ) ≃ is an equivalence. We may think of L ( C ) ≃ as the genuine space of objects of C and as ι as the comparison map betweenthe space of objects and the genuine space of objects.One may interprete definition 7.1 the following way: Remark 7.2.
Let C be an ∞ -precategory weakly enriched in an ∞ -operad V with small space ofobjects X . We say that two points in the space of objects X are equivalent in C if their images under ι areequivalent in L ( C ) . For any Z ∈ X let’s write Mor C ( Z , Z ) ∈ V ′ for the V ′ -enriched endomorphism algebraof Z provided by C . Note that the functor V ′ ( V ′ , − ) ∶ V ′ → S is lax monoidal and so sends associative algebras in V ′ toassociative monoids in S giving rise to grouplike monoids of units.The map ι ∶ X → L ( C ) ≃ is an equivalence if and only if the following two conditions hold: ● Two points in X are equivalent if and only if they are equivalent in C . ● The V ′ -enriched endomorphism algebra Mor C ( Z , Z ) ∈ V ′ of any Z ∈ X has monoid of units theloop space of Z in X . The V ′ -enriched Yoneda-embedding C → χ ( P V ′ ( C )) induces a V ′ -enriched embedding C → χ ( L ( C )) lying over the map ι corresponding to an equivalence C ≃ ι ∗ ( χ ( L ( C ))) ≃ Θ ( L ( C ) , ι ) of ∞ -precategories weakly enriched in V . So the functor Θ ∶ ω LMod fl → ω PreCat ∞ is essentiallysurjective. Especially an ∞ -precategory C weakly enriched in an ∞ -operad V is an ∞ -category weaklyenriched in V if and only if the V ′ -enriched embedding C → χ ( L ( C )) is an equivalence.We seek to prove the following theorem: N EQUIVALENCE BETWEEN ENRICHED ∞ -CATEGORIES AND ∞ -CATEGORIES WITH WEAK ACTION 43 Theorem 7.3.
The functor Θ ∶ ω LMod fl → ω PreCat ∞ is an equivalence and restricts to an equivalence χ ∶ ω LMod → ω Cat ∞ . The next Lemma 7.4 reduces Theorem 7.3 to show that for any ∞ -operad V the induced functorΘ V ∶ ( ω LMod fl ) V → ω PreCat V ∞ is an equivalence and restricts to an equivalence χ V ∶ ω LMod V → ω Cat V ∞ . Lemma 7.4.
The functor Θ ∶ ω LMod fl → ω PreCat ∞ preserves cartesian morphisms over S × Op ∞ .Proof. The functor Fun ([ ] , S ) → Fun ({ } , S ) × Fun ({ } , S ) is a map of cartesian fibrations overFun ({ } , S ) . So the functor Fun ([ ] , S ) × Fun ({ } , S ) ω LMod → Fun ({ } , S ) is a cartesian fibration. By Corollary 2.23 the forgetful functor ω LMod → Op ∞ × Cat ∞ is a map ofcartesian fibrations over Op ∞ so that the composition ω LMod → Op ∞ × Cat ∞ → Op ∞ × S is, too.So the functor Fun ([ ] , S ) × Fun ({ } , S ) ω LMod → Op ∞ is a cartesian fibration. Hence the functor σ ∶ Fun ([ ] , S ) × Fun ({ } , S ) ω LMod → Fun ({ } , S ) × Op ∞ is a cartesian fibration. By Lemma 7.5 the full subcategory ω LMod fl ⊂ Fun ([ ] , S ) × Fun ({ } , S ) ω LModis a colocalization relative to S × Op ∞ . So the restriction ω LMod fl → S × Op ∞ is a cartesian fibration,whose cartesian morphisms are the composites of a colocal equivalence followed by a σ -cartesianmorphism.The functor ̂ PreCat ∞ → S × ̂ Op ∞ is a map of cartesian fibrations over S and so also its pullback ω PreCat ∞ = Op ∞ × ̂ Op ∞ ̂ PreCat ∞ → S × Op ∞ is a map of cartesian fibrations over S , where the pullbackis taken along the functor θ ∶ Op ∞ → ̂ Op ∞ sending an ∞ -operad to its enveloping closed monoidal ∞ -category.The functor θ factors through the subcategory ̂ Op L ∞ ⊂ ̂ Op ∞ with the same objects but with mor-phisms the maps of ∞ -operads that admit a right adjoint relative to Ass . The canonical functor ̂ PreCat ∞ → ̂ Op ∞ is a cocartesian fibration and so also its pullback ρ ∶ ̂ Op L ∞ × ̂ Op ∞ ̂ PreCat ∞ → ̂ Op L . But ρ is also a locally cartesian fibration and so a bicartesian fibration. Thus also its pullback ω PreCat ∞ = Op ∞ × ̂ Op ∞ ̂ PreCat ∞ → Op ∞ is a bicartesian fibration and so the functor ω PreCat ∞ → S × Op ∞ is acartesian fibration.As next we show that the functor Θ ∶ Fun ([ ] , S ) × S ω LMod → ω PreCat ∞ is a map of cartesianfibrations over S × Op ∞ . For this it is enough to check that the composition κ ∶ Fun ([ ] , S ) × S ω LMod Θ Ð→ ω PreCat ∞ → ω Quivis a map of cartesian fibrations over S × Op ∞ as the forgetful functor ω PreCat ∞ → ω Quiv is a map ofcartesian fibrations over S × Op ∞ that reflects cartesian morphisms (as it induces fiberwise conservativefunctors). But κ is a map of cartesian fibrations over S × Op ∞ by the description of endomorphismobjects used in the definition of Θ . ∞ -CATEGORIES AND ∞ -CATEGORIES WITH WEAK ACTION By Lemma 7.5 the functor Θ inverts the colocal equivalences of the colocalization ω LMod fl ⊂ Fun ([ ] , S ) × S ω LMod. So by the description of cartesian morphisms of the cartesian fibration ω LMod fl → S × Op ∞ also the functorΘ ∶ ω LMod fl ⊂ Fun ([ ] , S ) × S ω LMod Θ Ð→ ω PreCat ∞ is a map of cartesian fibrations over S × Op ∞ . (cid:3) For Lemma 7.4 we used the following lemma:
Lemma 7.5.
The full subcategory ω LMod fl ⊂ Fun ([ ] , S ) × Fun ({ } , S ) ω LMod is coreflexive, where a morphism ( M , ϕ ) → ( N , ψ ) is a colocal equivalence if the functor M → N inducesequivalences on morphism objects between objects in the essential image of ϕ .A morphism in Fun ([ ] , S ) × Fun ({ } , S ) ω LMod is a colocal equivalence if and only if its image underthe functor Θ ∶ Fun ([ ] , S ) × Fun ({ } , S ) ω LMod → ω PreCat ∞ is an equivalence.Proof. Given an ∞ -category with weak left action M → V and a map ϕ ∶ X → M ≃ denote M ∣ ϕ ⊂ M thefull subcategory with weak left V -action spanned by the essential image of ϕ. Then we have a canonicalmap ( M ∣ ϕ , ϕ ) → ( M , ϕ ) in Fun ([ ] , S ) × Fun ({ } , S ) ω LMod (inducing the identity under evaluation atthe source) that yields for any ∞ -category with weak left action N → W and map ρ ∶ Y → N ≃ anequivalence ( Fun ([ ] , S ) × Fun ({ } , S ) ω LMod )(( N , ρ ) , ( M ∣ ϕ , ϕ )) ≃ ( Fun ([ ] , S ) × Fun ({ } , S ) ω LMod )(( N , ρ ) , ( M , ϕ )) provided that ρ ∶ Y → N ≃ is essentially surjective. (cid:3) To see that Θ V , χ V are fully faithful, we use the following theorem: Theorem 7.6.
Let C be an ∞ -precategory weakly enriched in an ∞ -operad V with small space ofobjects X .The canonical embedding L ( C ) ⊂ P V ′ ( C ) exhibits P V ′ ( C ) as the enveloping ∞ -category with closed left V ′ -action.Proof. Denote L ( C ) ′ the enveloping ∞ -category with closed left action over V ′ associated to L ( C ) . We want to see that the unique V ′ -linear left adjoint extension L ( C ) ′ → P V ′ ( C ) of the embeddingL ( C ) ⊂ P V ′ ( C ) is an equivalence.Let M , N be ∞ -categories with left V ′ -action compatible with small colimits, τ ∶ Y → M ≃ a mapand θ ∶ M → N a left adjoint V ′ -linear functor. By Theorem 6.1 the induced functorFun ( X , Y ) × Fun ( X , M ) LinFun L V ′ ( P V ′ ( C ) , M ) → Fun ( X , Y ) × Fun ( X , N ) LinFun L V ′ ( P V ′ ( C ) , N ) is equivalent to the functorFun ( X , Y ) × Fun ( X , M ) Fun V ( C , χ ( M )) → Fun ( X , Y ) × Fun ( X , N ) Fun V ( C , χ ( N )) that is canonically equivalent to the functorFun V ′ ( C , χ ( M , τ )) → Fun V ′ ( C , χ ( N , θ ○ τ )) that is an equivalence if the V ′ -enriched functor χ ( M , τ ) → χ ( N , θ ○ τ ) is an equivalence.Denote M ∣ τ ⊂ M , N ∣ θ ○ τ ⊂ N the full subcategories with weak left V -action spanned by the essentialimages of τ respectively θ ○ τ. The V ′ -linear functor θ ∶ M → N restricts to a lax V -linear functor M ∣ τ → N ∣ θ ○ τ that is essentially surjective. N EQUIVALENCE BETWEEN ENRICHED ∞ -CATEGORIES AND ∞ -CATEGORIES WITH WEAK ACTION 45 The induced functorFun ( X , Y ) × Fun ( X , M ) LinFun L V ′ ( L ( C ) ′ , M ) → Fun ( X , Y ) × Fun ( X , N ) LinFun L V ′ ( L ( C ) ′ , N ) is canonically equivalent to the functorFun ( X , Y ) × Fun ( X , M ∣ τ ) LaxLinFun V ( L ( C ) , M ∣ τ ) → Fun ( X , Y ) × Fun ( X , N ∣ θ ○ τ ) LaxLinFun V ( L ( C ) , N ∣ θ ○ τ ) that is an equivalence if θ ∶ M → N induces equivalences on morphism objects between objects in theessential image of τ. We apply this first to θ the canonical V ′ -linear functor L ( C ) ′ → P V ′ ( C ) and τ themap X → L ( C ) ≃ ⊂ L ( C ) ′ ≃ using that the V ′ -enriched functor χ ( L ( C ) ′ , τ ) ≃ C → χ ( P V ′ ( C ) , θ ○ τ ) ≃ C is the identity. So we get a V ′ -linear left adjoint functor α ∶ P V ′ ( C ) → L ( C ) ′ such that the composition P V ′ ( C ) α Ð→ L ( C ) ′ → P V ′ ( C ) is the identity and the composition X → P V ′ ( C ) ≃ α ≃ Ð→ L ( C ) ′ ≃ is the canonical map X → L ( C ) ≃ ⊂ L ( C ) ′ ≃ . Hence the composition L ( C ) ⊂ P V ′ ( C ) α Ð→ L ( C ) ′ factorsas L ( C ) ρ Ð→ L ( C ) ⊂ L ( C ) ′ for some lax V -linear functor ρ . So the embedding L ( C ) ⊂ P V ′ ( C ) factorsas L ( C ) ρ Ð→ L ( C ) ⊂ P V ′ ( C ) . Thus α induces equivalences on morphism objects between objects in theessential image of the map X → P V ′ ( C ) ≃ . Taking θ to be α ∶ P V ′ ( C ) → L ( C ) ′ and τ to be the map X → P V ′ ( C ) ≃ we get a V ′ -linear left adjointfunctor β ∶ L ( C ) ′ → P V ′ ( C ) such that the compositionL ( C ) ′ β Ð→ P V ′ ( C ) α Ð→ L ( C ) ′ is the identity. (cid:3) Theorems 6.1 and 7.6 give the following theorem:
Theorem 7.7.
Let V be an ∞ -operad, C an ∞ -precategory weakly enriched in V with small space ofobjects X and M an ∞ -category weakly left tensored over V .There is a canonical equivalence LaxLinFun V ( L ( C ) , M ) ≃ Fun V ( C , χ ( M )) over Fun ( X , M ) that induces on maximal subspaces the map ω LMod V ( L ( C ) , M ) ≃ ω PreCat V ∞ ( C , χ ( M )) over S ( X , M ≃ ) given by composition with the V -enriched embedding C → χ ( L ( C )) . For any essentially surjective map τ ∶ Y → M ≃ the map ( ω LMod fl ) V (( L ( C ) , ι ) , ( M , τ )) → ω PreCat V ∞ ( C , Θ ( M , τ )) over S ( X , Y ) given by composition with the canonical equivalence C ≃ Θ ( L ( C ) , ι ) is an equivalence.Proof. By Theorem 6.1 there is a canonical equivalenceLinFun L V ′ ( P V ′ ( C ) , N ′ ) ≃ Fun V ′ ( C , χ ( N ′ )) over LinFun L V ′ ( Fun ( X , V ′ ) , N ′ ) ≃ Fun ( X , N ′ ) . By Theorem 7.6 the canonical functorLinFun L V ′ ( P V ′ ( C ) , N ′ ) → LaxLinFun V ( L ( C ) , N ′ ) ∞ -CATEGORIES AND ∞ -CATEGORIES WITH WEAK ACTION over Fun ( X , N ′ ) is an equivalence. So we get a canonical equivalenceLaxLinFun V ( L ( C ) , N ′ ) ≃ Fun V ′ ( C , χ ( N ′ )) over Fun ( X , N ′ ) , whose pullback to Fun ( X , N ) ⊂ Fun ( X , N ′ ) is an equivalence ρ ∶ LaxLinFun V ( L ( C ) , N ) ≃ Fun V ( C , χ ( N )) over Fun ( X , N ) that for N = L ( C ) sends the identity to the V -enriched embedding C → χ ( L ( C )) .So by naturality shown in Theorem 5.3 and the Yoneda-lemma the equivalence ρ induces on maximalsubspaces the map ω LMod V ( L ( C ) , N ) ≃ ω PreCat V ∞ ( C , χ ( N )) over S ( X , N ≃ ) given by composition with the V -enriched embedding C → χ ( L ( C )) . The canonical map ( ω LMod fl ) V (( L ( C ) , ι ) , ( M , τ )) → ω PreCat V ∞ ( C , Θ ( M , τ )) over S ( X , Y ) factorsas the pullback ( ω LMod fl ) V (( L ( C ) , ι ) , ( M , τ )) ≃ S ( X , Y ) × S ( X , M ≃ ) ω LMod V ( L ( C ) , M ) ≃ S ( X , Y ) × S ( X , M ≃ ) ω PreCat V ∞ ( C , χ ( M )) ≃ ω PreCat V ∞ ( C , Θ ( M , τ )) , where the equivalence in the middle is shown in the first part. (cid:3) Theorem 7.3.
By Lemma 7.4 it is enough to check that for any ∞ -operad V the induced functorΘ V ∶ ( ω LMod fl ) V → ω PreCat V ∞ is an equivalence and restricts to an equivalence χ V ∶ ω LMod V → ω Cat V ∞ . By Theorem 7.7 the functor Θ V admits a left adjoint that sends a V ′ -precategory C to ( L ( C ) , τ ) such that the unit C → Θ ( L ( C ) , τ ) is the canonical equivalence. By Lemma 7.5 the functor Θ V isconservative and so an equivalence by the triangular identities. By definition 7.1 a V ′ -precategory C is a V ′ -category if and only if ( L ( C ) , τ ) belongs to ω LMod, i.e. τ ∶ C → χ ( L ( C )) is an equivalence. (cid:3) In the following we discuss some corollaries of Theorem 7.3:
Proposition 7.8.
Under the equivalence ω LMod ≃ ω Cat ∞ of Theorem 7.3 the ∞ -categories with closed weak left action over some ∞ -operad V correspond tothe V -enriched ∞ -categories and the ∞ -categories pseudo-enriched in some monoidal ∞ -category V correspond to the ∞ -categories enriched in P ( V ) ⊂ V ′ .Proof. Let V be a monoidal ∞ -category. The lax monoidal embedding V ⊂ Env ( V ) admits a monoidalleft adjoint L (a left adjoint relative to Ass) so that the embedding P ( V ) ⊂ V ′ = P ( Env ( V )) coincideswith L ∗ . (cid:3) Corollary 7.9.
The embedding ω Cat ∞ ⊂ ω PreCat ∞ admits a left adjoint that restricts to a localization between the full subcategories of enriched ∞ -categories and enriched ∞ -precategories. N EQUIVALENCE BETWEEN ENRICHED ∞ -CATEGORIES AND ∞ -CATEGORIES WITH WEAK ACTION 47 A V -enriched functor C → D for some ∞ -operad V and V -category D is a local equivalence ifand only if it induces an essentially surjective map on spaces of objects and induces equivalences onmorphism objects.More concretely, the composition ω PreCat ∞ ≃ ω LMod fl → ω LMod ≃ ω Cat ∞ , C ↦ χ ( L ( C )) is left adjoint to the embedding ω Cat ∞ ⊂ ω PreCat ∞ . Proof.
The diagonal embedding S ⊂ Fun ([ ] , S ) is right adjoint to evaluation at the target relativeto S . Thus the embedding ω LMod ⊂ Fun ([ ] , S ) × Fun ({ } , S ) ω LMod and so the embedding ω LMod ⊂ ω LMod fl is right adjoint to the canonical projection (relative to ω LMod). We conclude by invokingthe equivalence Θ ∶ ω LMod fl ≃ ω PreCat ∞ that restricts to the equivalence χ ∶ ω LMod ≃ ω Cat ∞ .Let V be an ∞ -operad. If C is a V -precategory, then the V ′ -category χ ( L ( C )) is a V -category asthe canonical morphism C → χ ( L ( C )) induces an essentially surjective map on spaces of objects andinduces equivalences on morphism objects. A V -enriched functor C → D is a local equivalence if andonly if L ( C ) → L ( D ) is an equivalence. (cid:3) An ( ∞ , ) -categorical equivalence 7.10. Let V be an ∞ -operad. In the following (Proposition7.11) we will enhance the equivalence of ∞ -categories χ ∶ ω LMod V ≃ ω Cat V ∞ of Theorem 7.3 to a Cat ∞ -linear equivalence (an equivalence of ( ∞ , ) -categories), where the leftaction of Cat ∞ on ω LMod V sends an ∞ -category K and an ∞ -category M ⊛ → V ⊗ weakly left tensoredover V to the ∞ -category K × M ⊛ → V ⊗ weakly left tensored over V , and the left action of Cat ∞ on ω Cat V ∞ = Cat V ′ ∞ sends an ∞ -category K and a V ′ -category C to the image under the functor ⊗ ∶ Cat S ∞ × Cat V ′ ∞ → Cat S × V ′ ∞ → Cat V ′ ∞ using that V ′ is canonically left tensored over S . So given an ∞ -category K and ∞ -categories M , N weakly left tensored over V we have a canonical equivalence ω Cat V ∞ ( K ⊗ χ ( M ) , χ ( N )) ≃ LaxLinFun V ( K × M , N ) ≃ ≃ Fun ( K , LaxLinFun V ( M , N )) ≃ . So the left action of
Cat ∞ on ω Cat V ∞ is closed with morphism ∞ -category of two weakly V -enriched ∞ -categories C , D the ∞ -category LaxLinFun V ( L ( C ) , L ( D )) ≃ Fun V ( C , D ) . Especially the full subcat-egory
Cat V ∞ of ω Cat V ∞ is enriched in Cat ∞ with the same morphism ∞ -categories. Theorem 7.11.
Let V be an ∞ -operad. The equivalence ω LMod V ≃ ω Cat V ∞ of Theorem 7.3 is Cat ∞ -linear.Proof. We use that an ∞ -category with finite products carries a unique structure of a cartesiansymmetric monoidal ∞ -category and a finite products preserving functor yields a unique symmetricmonoidal functor on cartesian structures. The ∞ -category ω LMod ≃ ω Cat ∞ admits finite productspreserved by the forgetful functor ψ ∶ ω Cat ∞ → Op ∞ , where the product of an ∞ -category C weaklyenriched in V and an ∞ -category D weakly enriched in W is the image under the functor Cat V ′ ∞ × Cat W ′ ∞ → Cat V ′ × W ′ ∞ → Cat ( V × W ) ′ ∞ induced by the canonical lax monoidal functor V ′ × W ′ → P ( Env ( V ) × Env ( W )) → P ( Env ( V × W )) = ( V × W ) ′ . So the cocartesian fibration ψ uniquely promotes to a symmetric monoidal functor ω LMod × → Op ×∞ on cartesian structures that is a locally cocartesian fibration. The ∞ -operad V is canonically a leftmodule over Ass, the final ∞ -operad, encoded by a map of ∞ -operads LM → Op ×∞ . So taking the ∞ -CATEGORIES AND ∞ -CATEGORIES WITH WEAK ACTION pullback we get a locally cocartesian fibration of symmetric ∞ -operads θ ∶ LM × Op ×∞ ω LMod × → LMthat is cocartesian fibration as its locally cocartesian morphisms are closed under composition.Thus θ encodes a left action of ω LMod
Ass on ω LMod V that sends an ∞ -category B → Ass weaklyleft tensored over Ass and an ∞ -category M → V weakly left tensored over V to the ∞ -category B × Ass M → V weakly left tensored over V . Restricting along the finite products preserving embedding Cat ∞ ≃ { Ass } × Mon ( Cat ∞ ) LMod ⊂ { Ass } × Op ∞ ω LMod , K ↦ K × Ass → Ass we get a left action of
Cat ∞ on ω LMod V that sends an ∞ -category K and an ∞ -category M → V weakly left tensored over V tothe ∞ -category K × M → V weakly left tensored over V . The canonical equivalenceLM × Op ×∞ ω LMod × ≃ LM × Op ×∞ ω Cat ×∞ over LM exhibits the equivalence ω LMod V ≃ ω Cat V ∞ as { Ass } × Op ∞ ω LMod ≃ Cat P ( Env ( Ass ))∞ -linear. The equivalence { Ass } × Op ∞ ω LMod ≃ Cat P ( Env ( Ass ))∞ restricts to the canonical equivalence { Ass } × Mon ( Cat ∞ ) LMod ≃ Cat ∞ ≃ Cat S ∞ . So restricting along thefinite products preserving embedding { Ass } × Mon ( Cat ∞ ) LMod ⊂ { Ass } × Op ∞ ω LMod the equivalence ω LMod V ≃ ω Cat V ∞ is Cat ∞ -linear. The left action of Cat ∞ on ω Cat V ∞ sends an ∞ -category K and a V -category C to theimage under the functor Cat S ∞ × Cat V ′ ∞ → Cat S × V ′ ∞ → Cat V ′ ∞ . (cid:3) Let V be an ∞ -operad and C an ∞ -precategory weakly enriched in V with space of objects X . Set P V ( C ) ∶ = RMod C ( Fun ( X , V )) , where we take right modules with respect to the restricted weak biaction on Fun ( X , V ) over V ′ fromthe left and Fun ( X × X , V ′ ) from the right. Corollary 7.12.
Let V be an ∞ -operad and C an ∞ -category weakly enriched in V . The following ∞ -categories are canonically equivalent: ● P V ( C ) ● Fun V rev ( C op , χ ( V rev )) ● LaxLinFun V rev ( L ( C op ) , V rev ) ● The ∞ -category of morphisms from C op → χ ( V rev ) in Cat V ′ rev ∞ .Proof. Denote X the space of objects of C . There is a canonical equivalence P V ( C ) = RMod C ( Fun ( X , V )) ≃ LMod C op ( Fun ( X , V rev )) = Fun V rev ( C op , χ ( V rev )) ≃ LaxLinFun V rev ( L ( C op ) , V rev ) . (cid:3) Compatibility with the symmetric monoidal structure 7.13.
Given a E k -monoid V in Op ∞ (for example a E k -monoid in Mon ( Cat ∞ ) corresponding to a E k + -monoidal ∞ -category) for 0 ≤ k ≤ ∞ Gepner-Haugseng ([3] Proposition 4.3.10.) and Hinich ([5] Corollary 3.5.3.) construct a E k -monoidalstructure on Cat V ∞ . We construct a locally cocartesian fibration of symmetric ∞ -operads with targetE k , whose fiber over the unique color is ω LMod V ≃ ω Cat V ∞ , that restricts to the E k -monoidal structureon Cat V ∞ . N EQUIVALENCE BETWEEN ENRICHED ∞ -CATEGORIES AND ∞ -CATEGORIES WITH WEAK ACTION 49 In the following we use that an ∞ -category with finite products carries a unique structure of acartesian symmetric monoidal ∞ -category and a finite products preserving functor yields a uniquesymmetric monoidal functor on cartesian structures ([6] Corollary 2.4.1.9.). The forgetful functor ρ ∶ ω LMod → S × Op ∞ preserves finite products (in fact all small limits) and so promotes to asymmetric monoidal functor ρ × ∶ ω LMod × → ( S × Op ∞ ) × on cartesian structures. The functor ρ is amap of cocartesian fibrations over Op ∞ so that ρ × is a map of locally cocartesian fibrations over Op ×∞ . Next we restrict the locally cocartesian fibration ω LMod × → Op ×∞ to cocartesian fibrations (Lemma7.14).We write ω LMod cl ⊂ ω LMod , PLMod ⊂ ω LMod for the full subcategories spanned by the ∞ -categories with closed weak left action respectively pseudo-enriched ∞ -categories. Note that LMod ⊂ PLMod and that LMod , PLMod , ω
LMod cl are closed in ω LMod under finite products. Moreover thecocartesian fibration ω LMod → Op ∞ restricts to cocartesian fibrations ω LMod cl → Op ∞ , PLMod → Mon ( Cat ∞ ) . We start with the following Lemma:
Lemma 7.14.
The locally cocartesian fibration ω LMod × → Op ×∞ restricts to cocartesian fibrations ( ω LMod cl ) × → Op ×∞ , PLMod × → Mon ( Cat ∞ ) × with the same cocartesian morphisms.Proof. We need to check that the cocartesian morphisms of the cocartesian fibrations ω LMod cl → Op ∞ , PLMod → Mon ( Cat ∞ ) are closed under finite products.A morphism f ∶ M → N in ω LMod cl is cocartesian over its image F ∶ V → W in Op ∞ if and only iffor all A , B ∈ M the induced morphism F ( Mor M ( A , B )) → Mor N ( f ( A ) , f ( B )) in W is an equivalence.Let f ∶ M → N , ̃ f ∶ ̃ M → ̃ N be morphisms in ω LMod cl lying over morphisms F ∶ V → W , ̃ F ∶ ̃ V → ̃ W inOp ∞ and let A , B ∈ M , A ′ , B ′ ∈ ̃ M . Then the morphism f × ̃ f ∶ M × ̃ M → N × ̃ N lying over F × ̃ F inducesthe morphism ( F × ̃ F )( Mor M ×̃ M (( A , A ′ ) , ( B , B ′ ))) → Mor N ×̃ N (( f ( A ) , ̃ f ( A ′ )) , ( f ( B ) , ̃ f ( B ′ ))) that is the morphism ( F ( Mor M ( A , B )) → Mor N ( f ( A ) , f ( B )) , ̃ F ( Mor ̃ M ( A ′ , B ′ )) → Mor ̃ N (̃ f ( A ′ ) , ̃ f ( B ′ ))) . A morphism f ∶ M → N in PLMod is cocartesian over its image F ∶ V → W in Mon ( Cat ∞ ) if andonly if for all A , B ∈ M the induced morphism F ∗ ( Mul M ( − , A; B )) → Mul N ( − , f ( A ) ; f ( B )) in P ( W ) isan equivalence.Let f ∶ M → N , ̃ f ∶ ̃ M → ̃ N be morphisms in PLMod lying over morphisms F ∶ V → W , ̃ F ∶ ̃ V → ̃ W inOp ∞ and let A , B ∈ M , A ′ , B ′ ∈ ̃ M . Then the morphism f × ̃ f ∶ M × ̃ M → N × ̃ N lying over F × ̃ F inducesthe morphism ( F × ̃ F ) ∗ ( Mul M ×̃ M ( − , ( A , A ′ ) ; ( B , B ′ ))) → Mor N ×̃ N ( − , ( f ( A ) , ̃ f ( A ′ )) ; ( f ( B ) , ̃ f ( B ′ ))) that is the image of the morphism ( F ∗ ( Mul M ( − , A; B )) → Mul N ( − , f ( A ) ; f ( B )) , ̃ F ∗ ( Mul ̃ M ( − , A ′ ; B ′ )) → Mul ̃ N ( − , ̃ f ( A ′ ) ; ̃ f ( B ′ ))) under the functor P ( W ) × P ( ̃ W ) → P ( W × ̃ W ) . (cid:3) Given a E k + -monoidal ∞ -category V for 0 ≤ k ≤ ∞ the ∞ -categoryLMod V = LMod V ( Cat ∞ ) ∞ -CATEGORIES AND ∞ -CATEGORIES WITH WEAK ACTION carries a canonical E k -monoidal structure, the relative tensorproduct: The forgetful functor LMod = LMod ( Cat ∞ ) → Mon ( Cat ∞ ) preserves finite products and so yields a symmetric monoidal functor θ ∶ LMod × → Mon ( Cat ∞ ) × on cartesian structures. As the θ -cocartesian morphisms are closed under finite products, the symmet-ric monoidal functor θ is a cocartesian fibration, whose pullback along a map of symmetric ∞ -operadsE k → Mon ( Cat ∞ ) × corresponding to V is the E k -monoidal structure on LMod V encoding the relativetensorproduct. Theorem 7.15.
Let V be a E k + -monoidal ∞ -category for ≤ k ≤ ∞ .The subcategory inclusion LMod V ⊂ PLMod V ≃ Cat P ( V )∞ is lax E k -monoidal.Proof. The subcategory inclusion LMod ⊂ PLMod yields an inclusion LMod × ⊂ PLMod × of symmetricmonoidal ∞ -categories over Mon ( Cat ∞ ) × . (cid:3) Denote
Cat cc ∞ ⊂ ̂ Cat ∞ the ∞ -category of ∞ -categories with small colimits and small colimitspreserving functors. By [6] Proposition 4.8.1.3. the ∞ -category Cat cc ∞ carries a closed symmetricmonoidal structure such that the subcategory inclusion Cat cc ∞ ⊂ ̂ Cat ∞ is lax symmetric monoidal.Given a E k + -monoidal ∞ -category V compatible with small colimits for 0 ≤ k ≤ ∞ the lax sym-metric monoidal subcategory inclusion Cat cc ∞ ⊂ ̂ Cat ∞ yields a lax E k -monoidal subcategory inclusionLMod V ( Cat cc ∞ ) ⊂ LMod V ( ̂ Cat ∞ ) . Corollary 7.16.
Let V be a presentably E k + -monoidal ∞ -category for ≤ k ≤ ∞ .The subcategory inclusion LMod V ( Cat cc ∞ ) ⊂ ω ̂ LMod cl V ≃ ̂ Cat V ∞ is lax E k -monoidal. Corollary 7.17.
Let V be a presentably E k + -monoidal ∞ -category for ≤ k ≤ ∞ .Then V carries a canonical structure of an E k -algebra in ̂ Cat V ∞ , i.e. is a E k -monoidal V -category. N EQUIVALENCE BETWEEN ENRICHED ∞ -CATEGORIES AND ∞ -CATEGORIES WITH WEAK ACTION 51 Generalized Day-convolution
In this section we construct a Day-convolution for generalized O -operads, where O → Ass isa cocartesian fibration relative to the inert morphisms. Our Day-convolution specializes to theDay-convolution for generalized non-symmetric ∞ -operads constructed by Haugseng ([4]) and non-symmetric ∞ -operads constructed by Lurie ([6] Theorem 2.2.6.22.).More precisely, we show that for any generalized O -monoidal ∞ -category C → O the induced functor ( − ) × O C ∶ Op O , gen ∞ → Op O , gen ∞ admits a right adjoint, which by definition assigns the Day-convolution to a generalized O -operad.To prove this we use the following strategy: Denote Cat inert ∞/ O the ∞ -category of cocartesian fibrationsrelative to the inert morphisms of O and functors over O preserving cocartesian lifts of inert morphisms.We first show that for any cocartesian fibration C → O relative to the inert morphisms, which is aflat functor, the induced functor ( − ) × O C ∶ Cat inert ∞/ O → Cat inert ∞/ O admits a right adjoint Fun O ( C , − ) ∶ Cat inert ∞/ O → Cat inert ∞/ O (Corollary 8.6 following from Proposition 8.1). Ina second step (Theorem 8.7) we show that the functor Fun O ( C , − ) ∶ Cat inert ∞/ O → Cat inert ∞/ O sends generalized O -operads to generalized O -operads, and sends O -operads to O -operads. In the third and last step(Proposition 8.9 ) we we show that if C → O has small fibers, the functor Fun O ( C , − ) ∶ Cat inert ∞/ O → Cat inert ∞/ O preserves O -monoidal ∞ -categories compatible with small colimits and O -monoidal functors preservingsmall colimits between such.Let S be an ∞ -category and E ⊂ Fun ([ ] , S ) a full subcategory. Denote Cat E ∞/ S ⊂ Cat ∞/ S thesubcategory with objects the cocartesian fibrations relative to E and with morphisms the functorsover S preserving cocartesian lifts of morphisms of E . Denote ρ ∶ E ⊂ Fun ([ ] , S ) → S the functor thatevaluates at the source.We start with the following proposition:
Proposition 8.1.
Let S , T be ∞ -categories and E ⊂ Fun ([ ] , S ) , δ ⊂ Fun ([ ] , T ) full subcategories suchthat every equivalence of S belongs to E and every composition of composable arrows of E belongs to E and the same for δ. Let C → T be a cocartesian fibration relative to δ and C → S a flat functor thatsends cocartesian lifts of morphisms of δ to E . This guarantees that the functor ( − ) × S C ∶ Cat ∞/ S → Cat ∞/ C → Cat ∞/ T induces a functor Cat E ∞/ S → Cat δ ∞/ T . If evaluation at the source ρ ∶ E → S is flat, the functor ( − ) × S C ∶ Cat E ∞/ S → Cat δ ∞/ T admits a rightadjoint.Proof. As the functors ρ ∶ E → S and C → S are flat, the composition C ′ ∶ = E × Fun ({ } , S ) C → E ρ Ð→ S isflat. So we have a canonical functor C ′ = E × Fun ({ } , S ) C → E × C → S × T , whose projection to the firstfactor is flat. Hence given a functor D → T we may form Fun S ×∗ S × T ( C ′ , S × D ) and define Fun S , E ( C , D ) as the full subcategory of Fun S ×∗ S × T ( C ′ , S × D ) spanned by the functors E X × S C → D over T for someX ∈ S that send morphisms of E X × S C , whose image in C is a cocartesian lift of a morphism of δ , tococartesian lifts of a morphism of δ .Evaluation at the source Fun ([ ] , S ) → S is a cartesian fibration, whose cartesian morphisms aresent to equivalences under evaluation at the target. The restriction E ⊂ Fun ([ ] , S ) → S is still acartesian fibration relative to E , whose cartesian morphisms lying over E are sent to equivalencesunder evaluation at the target. Hence the pullback C ′ → E → S is a cartesian fibration relative to E , whose cartesian lifts of morphisms of E get equivalences in C and so also in T . So the functor ∞ -CATEGORIES AND ∞ -CATEGORIES WITH WEAK ACTION Fun S ×∗ S × T ( C ′ , S × D ) → S is a cocartesian fibration relative to E that restricts to a cocartesian fibrationFun S , E ,δ ( C , D ) → S relative to E with the same cocartesian morphisms.We want to prove that for any cocartesian fibrations φ ∶ B → S , D → T relative to E respectively δ there is a canonical equivalenceFun δ T ( B × S C , D ) ≃ Fun E S ( B , Fun S , E ,δ ( C , D )) . We first reduce to the case that B = S, i.e. that there is a canonical equivalence(21) Fun δ T ( C , D ) ≃ Fun E S ( S , Fun S , E ,δ ( C , D )) ∶ Denote E B ⊂ Fun ([ ] , B ) the full subcategory spanned by the φ -cocartesian morphisms lying overmorphisms of E . There is a canonical equivalence E B ≃ B × S E over B , where we view E , E B over Srespectively B via evaluation at the source. There is a canonical equivalence ( B × S C ) ′ ≃ B × S C ′ over B , where we view C ′ , ( B × S C ) ′ over S respectively B via evaluation at the source. This equivalenceyields a canonical equivalence B × S Fun S ×∗ S × T ( C ′ , S × D ) ≃ Fun B ×∗ B × T (( B × S C ) ′ , B × D ) over B that restricts to an equivalence(22) B × S Fun S , E ,δ ( C , D ) ≃ Fun B , E B ,δ ( B × S C , D ) . Note that there is a canonical equivalence Fun E S ( B , Fun S , E ,δ ( C , D )) ≃ Fun E B B ( B , B × S Fun S , E ,δ ( C , D )) . Thus via equivalence 22 the general case follows from equivalence 21, which we will prove in the fol-lowing:The diagonal embedding S ⊂ Fun ([ ] , S ) and so the diagonal embedding S ⊂ E admit a left adjointrelative to S, where we now view E over S via evaluation at the target. This localization relative to Sgives rise to a localization C ⊂ C ′ relative to C and so also T with left adjoint the projection C ′ → C .This localization gives rise to a localizationFun T ( C , D ) ⊂ Fun T ( C ′ , D ) . Note that the following three conditions on a functor α ∶ C ′ → D over T are equivalent, where θ ∶ C ′ → E → S denotes evaluation at the source: ● α inverts local equivalences, i.e. those morphisms of C ′ , whose image in C is an equivalence. ● α inverts morphisms of C ′ that are θ -cartesian lying over morphisms of E via θ , which areexactly the morphsms of C ′ , whose image in C is an equivalence and whose image under θ belongs to E . ● α inverts local equivalences with local target (whose image under θ has to belong to E ).Given a functor C ′ → D over T corresponding to a functor C ′ → S × D over S × T corresponding to asection β of κ ∶ Fun S ×∗ S × T ( C ′ , S × D ) → S the functor C ′ → D over T satisfies the conditions above if andonly if the section β sends morphisms of E to κ -cocartesian morphisms.In other words the full subcategories Fun T ( C , D ) ⊂ Fun T ( C ′ , D ) andFun E S ( S , Fun S ×∗ S × T ( C ′ , S × D )) ⊂ Fun S ( S , Fun S ×∗ S × T ( C ′ , S × D )) ≃ Fun S × T ( C ′ , S × D ) ≃ Fun T ( C ′ , D ) coincide so that we obtain a canonical equivalenceFun T ( C , D ) ≃ Fun E S ( S , Fun S ×∗ S × T ( C ′ , S × D )) . This equivalence restricts to the desired equivalenceFun δ T ( C , D ) ≃ Fun E S ( S , Fun S , E ,δ ( C , D )) for the following reason: Given a functor C → D over T corresponding to a section β of κ ∶ Fun S ×∗ S × T ( C ′ , S × D ) → S. Then β factors through Fun S , E ( C , D ) if and only if for every X ∈ S the composition E X × S C → C → D sends morphisms, whose image in C is a cocartesian lift of a morphism of δ , to cocartesian lifts N EQUIVALENCE BETWEEN ENRICHED ∞ -CATEGORIES AND ∞ -CATEGORIES WITH WEAK ACTION 53 of morphisms of δ . This is equivalent to the condition that the given functor C → D over T preservescocartesian lifts of morphisms of δ . (cid:3) Corollary 8.2.
Let S , T be ∞ -categories and E ⊂ Fun ([ ] , S ) , δ ⊂ Fun ([ ] , T ) full subcategories suchthat every equivalence of S belongs to E and every composition of composable arrows of E belongs to E and the same for δ. Let T → S be a functor that sends δ to E and C → T a cocartesian fibrationrelative to δ .If the composition C → T → S and evaluation at the source ρ ∶ E → S are flat, the functor ( − ) × S C ≃ ( − ) × S T × T C ∶ Cat E ∞/ S → Cat δ ∞/ T admits a right adjoint, denoted by Fun S , E T ,δ ( C , − ) . Remark 8.3.
Let the assumptions be as in Corollary 8.2 and φ ∶ S ′ → S a cocartesian fibration relativeto E . Set T ′ ∶ = S ′ × S T . Denote E ′ ⊂ Fun ([ ] , S ′ ) , δ ′ ⊂ Fun ([ ] , T ′ ) the full subcategories spanned bythe cocartesian lifts of morphisms of E respectively δ. Evaluation at the source induces equivalences E ′ ≃ S ′ × S E over S ′ and δ ′ ≃ T ′ × T δ over T ′ byassumption on φ. Let D → T be a cocartesian fibration relative to δ . By Proposition 8.1 we havefunctors Fun S , E ,δ ( C , D ) → S and Fun S ′ , E ′ ,δ ′ ( T ′ × T C , T ′ × T D ) → S ′ . By their universal property wehave a canonical equivalenceFun S ′ , E ′ ,δ ′ ( T ′ × T C , T ′ × T D ) ≃ S ′ × S Fun S , E ,δ ( C , D ) over S ′ . Especially important for us is the case S = T , E = δ ∶ Corollary 8.4.
Let S be an ∞ -category and E ⊂ Fun ([ ] , S ) a full subcategory such that every equiv-alence of S belongs to E and every composition of composable arrows of E belongs to E . Let C → S bea cocartesian fibration relative to E .If the funtors C → S and evaluation at the source ρ ∶ E → S are flat, the functor ( − ) × S C ∶ Cat E ∞/ S → Cat E ∞/ S admits a right adjoint, denoted by Fun S , E ( C , − ) . Example 8.5.
Let O → Ass be a cocartesian fibration relative to the inert morphisms. Denote E ⊂ Fun ([ ] , O ) the full subcategory spanned by the inert morphisms. Evaluation at the source E → O is a cartesian fibration: Proof.
The inert-active factorization system on Ass lifts to an inert-active factorization system on O . Thus by [7] Lemma 5.2.8.19. E ⊂ Fun ([ ] , O ) is a colocalization relative to O (via evaluation atthe source). For any ∞ -category B evaluation at the source Fun ([ ] , B ) → B is a cartesian fibration.So evaluation at the source Fun ([ ] , O ) → O is a cartesian fibration and so also its colocalization E ⊂ Fun ([ ] , O ) → O relative to O . (cid:3) Corollary 8.6 and Example 8.5 give the following corollary:
Corollary 8.6.
Let O → Ass be a cocartesian fibration relative to the inert morphisms and C → O acocartesian fibration relative to the inert morphisms, which is a flat functor.The induced functor ( − ) × O C ∶ Cat inert ∞/ O → Cat inert ∞/ O admits a right adjoint, which we abusively denote by Fun O ( C , − ) . Let φ ∶ D → O be a cocartesian fibration relative to the inert morphisms and Λ gen O the collectionof functors defined before definition 2.10. We say that φ ∶ D → O satisfies the Segal condition if thepullback K ⊲ × O C → K ⊲ along any functor α ∈ Λ gen O classifies a limit diagram K ⊲ → Cat ∞ . We continue with the following proposition: ∞ -CATEGORIES AND ∞ -CATEGORIES WITH WEAK ACTION Theorem 8.7.
Let O → Ass , C → O , D → O be cocartesian fibrations relative to the inert morphismssuch that C → O is a flat functor.(1) If D → O satisfies the Segal condition, also Fun O ( C , D ) does.(2) If C → O is a cocartesian fibration and D → O is a generalized O -operad, then Fun O ( C , D ) isa generalized O -operad.(3) If C → O is a cocartesian fibration and D → O is a O -operad, then Fun O ( C , D ) is a O -operad. Remark 8.8.
We expect that in 2. and 3. it is not neccessary to assume that C → O is a cocartesianfibration. But we don’t know a proof of showing this with our methods. Proof.
Denote E ⊂ Fun ([ ] , O ) the full sucategory spanned by the inert morphisms.1.: By definition for any X ∈ O we have a canonical equivalenceFun O ( C , D ) X ≃ Fun cocart E X ( E X × O C , E X × O D ) . Moreover if X lies over [ ] ∈ Ass we have Fun cocart E X ( E X × O C , E X × O D ) ≃ Fun ( C X , D X ) so thatFun O ( C , D ) X is contractible if D X is. Let X lie over [ n ] ∈ Ass with n ≥ . We want to see that the inert maps X → X i → X i , j induce an equivalenceFun cocart E X ( E X × O C , E X × O D ) ≃ Fun cocart E X1 ( E X × O C , E X × O D ) × Fun cocart E X1 , ( E X1 , × O C , E X1 , × O D ) ...... × Fun cocart E Xn − , n ( E Xn − , n × O C , E Xn − , n × O D ) Fun cocart E Xn ( E X n × O C , E X n × O D ) . Denote E ′ X ⊂ E X the full subcategory spanned by the inert morphisms X → Y in O with Y lying over [ ] , [ ] ∈ Ass . Note that a functor K ⊲ → O of sending the cone point to X and sending any morphismto an inert one canonically lifts to E X . For any ∞ -category A with finite limits restriction Fun ( E X , A ) → Fun ( E ′ X , A ) admits a fully faithfulright adjoint with local objects the functors φ ∶ E X → A that send any canonical lift K ⊲ → E X of afunctor K ⊲ → O of Λ O , sending the cone point to X, to a limit diagram. So for A = Cat ∞ pullbackalong E ′ X ⊂ E X gives an equivalenceFun cocart E X ( E X × O C , E X × O D ) ≃ Fun cocart E ′ X ( E ′ X × O C , E ′ X × O D ) . Consequently it is enough to show that the canonical functorFun cocart E ′ X ( E ′ X × O C , E ′ X × O D ) ≃ Fun cocart E ′ X1 ( E ′ X × O C , E ′ X × O D ) × Fun cocart E ′ X1 , ( E ′ X1 , × O C , E ′ X1 , × O D ) ...... × Fun cocart E ′ Xn − , n ( E ′ Xn − , n × O C , E ′ Xn − , n × O D ) Fun cocart E Xn ( E X n × O C , E X n × O D ) is an equivalence.Note that if X lies over [ ] ∈ Ass we have E ′ X = E X contractible, and if X lies over [ ] ∈ Ass we have E ′ X = Λ . More generally we have that the inert maps X → X i → X i , j induce an equivalence ρ ∶ ( E ′ X ∐ E ′ X1 , E ′ X ∐ E ′ X2 , ... ∐ E ′ Xn − , n − E ′ X n − ∐ E ′ Xn − , n E ′ X n ) ◁ ≃ E ′ X . Especially the cocartesian fibration E ′ X × O D → E ′ X classifies a limit diagram.Note that for any ∞ -category K and cocartesian fibrations A → K , B → K we have canonical mapsK × A ∞ → K × K ◁ A of cocartesian fibrations over K. The induced canonical functorFun cocartK ◁ ( A , B ) → Fun ( A ∞ , B ∞ ) × Fun cocartK ( K × A ∞ , K × K ◁ B ) Fun cocartK ( K × K ◁ A , K × K ◁ B ) is an equivalence. If the cocartesian fibration B → K ◁ classifies a limit diagram, the functor B ∞ → Fun cocartK ( K , K × K ◁ B ) N EQUIVALENCE BETWEEN ENRICHED ∞ -CATEGORIES AND ∞ -CATEGORIES WITH WEAK ACTION 55 is an equivalence so that the functorFun ( A ∞ , B ∞ ) → Fun ( A ∞ , Fun cocartK ( K , K × K ◁ B )) ≃ Fun cocartK ( K × A ∞ , K × K ◁ B ) is an equivalence. In this case the functorFun cocartK ◁ ( A , B ) → Fun cocartK ( K × K ◁ A , K × K ◁ B ) is an equivalence. So the functor ρ is an equivalence.2.: Let D → O be a generalized O -operad. Let α ∶ X → Y be a morphism in O . Denote E α ∶ = [ ] × O E the pullback of evaluation at the source E → O along the functor [ ] → O corresponding to α. Bydefinition we have a canonical equivalence [ ] × O Fun O ( C , D ) ≃ Fun ′ E α ( E α × O C , E α × O D ) , where Fun ′ E α ( E α × O C , E α × O D ) ⊂ Fun E α ( E α × O C , E α × O D ) denotes the full subcategory spanned bythe functors E α × O C → E α × O D over E α that induce on the fiber over 0 and 1 a map E X × O C → E X × O D of cocartesian fibrations over E X respectively a map E Y × O C → E Y × O D of cocartesian fibrations over E Y . Denote α i ∶ X α Ð→ Y → Y i and α i , j ∶ X α Ð→ Y → Y i , j the composition of α ∶ X → Y with the canonicalinert morphisms in O , whose target lies over [ ] respectively [ ] . Note that a morphism α → β inFun ([ ] , O ) yields a functor E β → E α , which we apply to β = α i , β = α i , j . By 1. we know that Fun O ( C , D ) satisfies the Segal condition. It remains to check the following:The inert maps Y → Y i → Y i , j in O induce an equivalence σ D ∶ Fun ′ E α ( E α × O C , E α × O D ) ≃ Fun ′ E α ( E α × O C , E α × O D ) × Fun ′ E α , ( E α , × O C , E α , × O D ) ...... × Fun ′ E α n − , n ( E α n − , n × O C , E α n − , n × O D ) Fun ′ E α n ( E α n × O C , E α n × O D ) . Every generalized O -operad D → O embeds into its enveloping generalized O -monoidal ∞ -category D ′ → O via an embedding D ⊂ D ′ of cocartesian fibrations relative to the inert morphisms of O . So forany Z ∈ O the embedding E Z × O D ⊂ E Z × O D ′ is a map of cocartesian fibrations over E Z . Hence for anymorphism β in O we get an induced embedding Fun ′ E α ( E α × O C , E α × O D ) ⊂ Fun ′ E α ( E α × O C , E α × O D ′ ) . So σ D is the restriction of σ D ′ . Moreover using that D → O satisfies the Segal condition, the functor σ D is the pullback of the functor σ D ′ along the evident embedding. Consequently we can reduce tothe case that D → O is a cocartesian fibration.Denote E ′ α ⊂ E α the full subcategory spanned by the inert morphisms Z → Z ′ such that Z ′ lies over [ ] , [ ] ∈ Ass if Z = Y.As D satisfies the Segal condition, the functor θ ∶ Fun ′ E α ( E α × O C , E α × O D ) → Fun ′ E α ( E ′ α × O C , E α × O D ) ≃ Fun ′ E ′ α ( E ′ α × O C , E ′ α × O D ) that restricts along the embedding E ′ α × O C ⊂ E α × O C over E α is conservative and admits a fullyfaithful right adjoint that sends a functor over E α to its right kan-extension. So θ is an equivalence.Consequently it is enough to show that the canonical functorFun ′ E ′ α ( E ′ α × O C , E ′ α × O D ) ≃ Fun ′ E ′ α ( E ′ α × O C , E ′ α × O D ) × Fun ′ E ′ α , ( E ′ α , × O C , E ′ α , × O D ) ...... × Fun ′ E ′ α n − , n ( E ′ α n − , n × O C , E ′ α n − , n × O D ) Fun ′ E ′ α n ( E ′ α n × O C , E ′ α n × O D ) is an equivalence.For any cartesian fibration M → [ ] the functor M ∐ ( M ×{ }) ( M × [ ]) → M ∞ -CATEGORIES AND ∞ -CATEGORIES WITH WEAK ACTION is an equivalence. So we get canonical equivalences E ′ α ≃ E X ∐ ( E ′ Y ×{ }) ( E ′ Y × [ ]) , E ′ α i ≃ E X ∐ ( E ′ Yi ×{ }) ( E ′ Y i × [ ]) , E ′ α i , j ≃ E X ∐ ( E ′ Yi , j ×{ }) ( E ′ Y i , j × [ ]) . Set K ∶ = E ′ Y ∐ E ′ Y1 , E ′ Y ∐ E ′ Y2 , ... ∐ E ′ Yn − , n − E ′ Y n − ∐ E ′ Yn − , n E ′ Y n . Then the canonical functor(23) E ′′ α ∶ = E X ∐ ( K ×{ }) ( K × [ ]) → E ′ α ∐ E ′ α , E ′ α ∐ E ′ α , ... ∐ E ′ α n − , n E ′ α n is an equivalence. By the first part of the proof the canonical functor ρ ∶ K ◁ ≃ E ′ Y is an equivalence.The embedding K ⊂ E ′ Y yields an embedding E ′′ α = E X ∐ ( K ×{ }) ( K × [ ]) → E X ∐ ( E ′ Y ×{ }) ( E ′ Y × [ ]) ≃ E ′ α . As D → O satisfies the Segal condition, this functor induces an equivalenceFun ′ E ′ α ( E ′ α × O C , E ′ α × O D ) ≃ Fun ′ E ′′ α ( E ′′ α × O C , E ′′ α × O D ) . So it is enough to show that the canonical functorFun ′ E ′′ α ( E ′′ α × O C , E ′′ α × O D ) → Fun ′ E ′ α ( E ′ α × O C , E ′ α × O D ) × Fun ′ E ′ α , ( E ′ α , × O C , E ′ α , × O D ) ...... × Fun ′ E ′ α n − , n ( E ′ α n − , n × O C , E ′ α n − , n × O D ) Fun ′ E ′ α n ( E ′ α n × O C , E ′ α n × O D ) is an equivalence. This follows from the equivalence 23.3. follows immediately from 2. (cid:3) Proposition 8.9.
Let O → Ass be a cocartesian fibration relative to the inert morphisms, C → O acocartesian fibration with small fibers and D → O a O -operad, whose fibers have small colimits.(1) If D → O is a locally cocartesian fibration, Fun O ( C , D ) → O is a locally cocartesian fibration.(2) If D → D ′ is a map of O -operads and locally cocartesian fibrations over O that induces onevery fiber a small colimits preserving functor between ∞ -categories with small colimits, theinduced map Fun O ( C , D ) → Fun O ( C , D ′ ) is a map of locally cocartesian fibrations over O .(3) If C → O is a generalized O -monoidal ∞ -category and D → O is a O -monoidal ∞ -categorycompatible with small colimits, Fun O ( C , D ) → O is a O -monoidal ∞ -category compatible withsmall colimits.Proof.
1. As Fun O ( C , D ) → O is a O -operad, it is enough to show that for any active morphism α ∶ X → Y in O with Y lying over [ ] ∈ Ass the induced pullback [ ] × O Fun O ( C , D ) → [ ] is acocartesian fibration.By Lemma 8.10 we have for any F ∈ Fun O ( C , D ) X lying over F ′ ∶ Fun ( C X , D X ) and G ∈ Fun O ( C , D ) Y ≃ Fun ( C Y , D Y ) a canonical equivalence { α } × O ( X , Y ) Fun O ( C , D )( F , G ) ≃ Fun ( C X , D Y )( α D ∗ ○ F ′ , G ○ α C ∗ ) ≃ Fun ( C Y , D Y )( lan α C ∗ ( α D ∗ ○ F ′ ) , G ) , where lan α C ∗ ∶ Fun ( C X , D Y ) → Fun ( C Y , D Y ) denotes the left adjoint to pre-composition along α C ∗ . α ∶ X → Y in O with Y lying over [ ] yields the composition α ∗ ∶ m ∏ i = Fun O ( C , D ) X i ≃ Fun O ( C , D ) X → Fun ( C X , D X ) Fun ( C X ,α D ∗ ) ÐÐÐÐÐÐÐ→
Fun ( C X , D Y ) lan α C ∗ ÐÐÐ→
Fun ( C Y , D Y ) , N EQUIVALENCE BETWEEN ENRICHED ∞ -CATEGORIES AND ∞ -CATEGORIES WITH WEAK ACTION 57 which proves 2.3. The functor α ∗ factors as m ∏ i = Fun O ( C , D ) X i ≃ m ∏ i = Fun ( C X i , D X i ) → Fun ( m ∏ i = C X i , m ∏ i = D X i ) → Fun ( C X , m ∏ i = D X i ) Fun ( C X ,α D ∗ ) ÐÐÐÐÐÐÐ→
Fun ( C X , D Y ) lan α C ∗ ÐÐÐ→
Fun ( C Y , D Y ) and thus preserves small colimits component-wise if D → O is compatible with small colimits.Let D → O be compatible with small colimits. To verify that the functor Fun O ( C , D ) → O isa cocartesian fibration, we need to show that its locally cocartesian morphisms are closed undercomposition. As Fun O ( C , D ) → O is a O -operad, it is enough to check that for any active morphisms β ∶ Z → X , α ∶ X → Y in O with Y lying over [ ] the canonical natural transformation ( α ○ β ) ∗ → α ∗ ○ β ∗ is an equivalence.Denote β i ∶ Z β Ð→ X → X i the composition of β with the evident inert morphism X → X i with X i lying over [ ] . The induced functor β ∗ ∶ Fun O ( C , D ) Z Ð→ Fun O ( C , D ) X ≃ ∏ mi = Fun O ( C , D ) X i induceson the i-th component the functor ( β i ) ∗ ∶ Fun O ( C , D ) Z Ð→ Fun O ( C , D ) X i ≃ Fun ( C X i , D X i ) . Hence thefunctor β ∗ factors asFun O ( C , D ) Z → Fun ( C Z , D Z ) ( Fun ( C Z , ( β i ) D ∗ )) i ÐÐÐÐÐÐÐÐÐÐ→ m ∏ i = Fun ( C Z , D X i ) ∏ mi = lan ( β i ) C ∗ ÐÐÐÐÐÐÐ→ m ∏ i = Fun ( C X i , D X i ) . So we conclude by observing that the functorFun ( C Z , D X ) ≃ m ∏ i = Fun ( C Z , D X i ) ∏ mi = lan ( β i ) C ∗ ÐÐÐÐÐÐÐ→ m ∏ i = Fun ( C X i , D X i ) → Fun ( m ∏ i = C X i , m ∏ i = D X i ) → Fun ( C X , m ∏ i = D X i ) ≃ Fun ( C X , D X ) is canonically equivalent to the functor lan β C ∗ ∶ Fun ( C Z , D X ) → Fun ( C X , D X ) ∶ given a functor F ∶ C Z → D X i the composition C X → C X i lan ( β i ) C ∗ ( F ) ÐÐÐÐÐÐ→ D X i is canonically lan β C ∗ ( F ) . (cid:3) We used the following lemma, which is an adaption of [6] Proposition 2.2.6.6. to the non-symmetricsetting:
Lemma 8.10.
Let O → Ass be a cocartesian fibration relative to the inert morphisms, C → O a flatfunctor that is a cocartesian fibration relative to the inert morphisms of O and D → O a O -operad.For any active morphism α ∶ X → Y in O with Y lying over [ ] ∈ Ass and any F , G ∈ Fun O ( C , D ) lying over X , Y ∈ O with images F ′ ∈ Fun ( C X , D X ) , G ′ ∈ Fun ( C Y , D Y ) there is a canonical equivalence { α } × O ( X , Y ) Fun O ( C , D )( F , G ) ≃ { F ′ , G ′ } × ( Fun ( C X , D X )× Fun ( C Y , D Y )) Fun [ ] ([ ] × O C , [ ] × O D ) ≃ Fun ( C X , D Y )( α D ∗ ○ F ′ , G ′ ○ α C ∗ ) . Proof.
Denote Fun ′ E X ( E X × O C , E X × O D ) ⊂ Fun E X ( E X × O C , E X × O D ) the full subcategory spanned by the functors E X × O C → E X × O D over E X that send morphisms, whoseimage in C is cocartesian over O , to morphisms, whose image in D is cocartesian over O . DenoteFun ′[ ]× O E ([ ] × O E × O C , [ ] × O E × O D ) ⊂ Fun [ ]× O E ([ ] × O E × O C , [ ] × O E × O D ) the full subcategory spanned by the functors over [ ] × O E , whose images in Fun E X ( E X × O C , E X × O D ) and Fun E Y ( E Y × O C , E Y × O D ) belong to Fun ′ E X ( E X × O C , E X × O D ) , Fun ′ E Y ( E Y × O C , E Y × O D ) . ∞ -CATEGORIES AND ∞ -CATEGORIES WITH WEAK ACTION There is a canonical equivalence { α } × O ( X , Y ) Fun O , E ( C , D )( F , G ) ≃ {( F , G )} × ( Fun O , E ( C , D ) X × Fun O , E ( C , D ) Y ) Fun O ([ ] , Fun O , E ( C , D )) ≃ {( F , G )} × ( Fun ′ E X ( E X × O C , E X × O D )× Fun ′ E Y ( E Y × O C , E Y × O D )) Fun ′[ ]× O E ([ ] × O E × O C , [ ] × O E × O D ) . Consequently it is enough to see that the canonical functor β ∶ Fun ′[ ]× O E ([ ] × O E × O C , [ ] × O E × O D ) → Fun ′ E X ( E X × O C , E X × O D ) × Fun ( C X , D X ) Fun [ ] ([ ] × O C , [ ] × O D ) is an equivalence, which is the restriction of the functor γ ∶ Fun [ ]× O E ([ ] × O E × O C , [ ] × O E × O D ) → Fun E X ( E X × O C , E X × O D ) × Fun ( C X , D X ) Fun [ ] ([ ] × O C , [ ] × O D ) . The diagonal embedding gives an embedding [ ] × O C ⊂ [ ] × O E × O C that restricts to an embedding C X ⊂ E X × O C . We have a canonical equivalenceFun E X ( E X × O C , E X × O D ) × Fun ( C X , D X ) Fun [ ] ([ ] × O C , [ ] × O D ) ≃ Fun [ ]× O E ( E X × O C , [ ] × O E × O D ) × Fun [ ] × OE ( C X , [ ]× O E × O D ) Fun [ ]× O E ([ ] × O C , [ ] × O E × O D ) ≃ Fun [ ]× O E (( E X × O C ) ∐ C X ([ ] × O C ) , [ ] × O E × O D ) , under which γ identifies with the functor ψ ∗ ∶ Fun [ ]× O E ([ ] × O E × O C , [ ] × O E × O D ) → Fun [ ]× O E (( E X × O C ) ∐ C X ([ ] × O C ) , [ ] × O E × O D ) induced by the functor ψ ∶ ( E X × O C ) ∐ C X ([ ] × O C ) → [ ] × O E × O C . So we need to see that ψ ∗ restrictsto an equivalenceFun ′[ ]× O E ([ ] × O E × O C , [ ] × O E × O D ) → Fun ′[ ]× O E (( E X × O C ) ∐ C X ([ ] × O C ) , [ ] × O E × O D ) , where the last full subcategory is spanned by the functors over [ ] × O E , whose restriction to E X × O C sends morphisms, whose image in C is cocartesian over O , to morphisms, whose image in D iscocartesian over O . We first observe that the functor ψ ∗ is conservative as the canonical functorFun O , E ( C , D ) Y ≃ Fun ′ E Y ( E Y × O C , E Y × O D ) → Fun ( C Y , D Y ) is an equivalence.Secondly we observe that the functor ψ is fully faithful and ψ ∗ admits a fully faithful left adjointthat sends a functor ( E X × O C ) ∐ C X ([ ] × O C ) → [ ] × O E × O D over [ ] × O E that belongs to Fun ′[ ]× O E (( E X × O C ) ∐ C X ([ ] × O C ) , [ ] × O E × O D ) to its right kan-extension along ψ relative to [ ] × O E , which belongs to Fun ′[ ]× O E ([ ] × O E × O C , [ ] × O E × O D ) , i.e.whose restriction to Fun E Y ( E Y × O C , E Y × O D ) belongs to Fun ′ E Y ( E Y × O C , E Y × O D ) ≃ Fun ( C Y , D Y ) . (cid:3) We also use Day-convolution in families of (generalized) ∞ -operads: Let S be an ∞ -category and O → Ass a cocartesian fibration relative to the inert morphisms. The composition O × S → O → Ass isa cocartesian fibration relative to the inert morphisms. We use the definitions of example 2.15.Replacing O → Ass by O × S → O → Ass we get the following corollary:
Corollary 8.11.
Let S be an ∞ -category, O → Ass a cocartesian fibration relative to the inert mor-phisms and C → S × O a cocartesian fibration.If D → S × O is a S -family of (generalized) O -operads, the functor Fun S × O ( C , D ) → S × O is a S -family of (generalized) O -operads. N EQUIVALENCE BETWEEN ENRICHED ∞ -CATEGORIES AND ∞ -CATEGORIES WITH WEAK ACTION 59 If D → S × O is a cocartesian S -family of (generalized) O -monoidal ∞ -categories compatible withsmall colimits, the functor Fun S × O ( C , D ) → S × O is a cocartesian S -family of (generalized) O -monoidal ∞ -categories. Appendix
In this appendix we give some needed proofs.
Lemma 9.1.
Let K be a ∞ -category and φ ∶ C → S a functor.(1) If a functor H ∶ K ⊲ → C is a φ -limit diagram, for every functor G ∶ K ⊲ → C over S the canonicalfunctor (24) Fun S ( K ⊲ , C )( G , H ) → Fun S ( K , C )( G ∣ K , H ∣ K ) is an equivalence.(2) If C → S is a cocartesian fibration, a functor H ∶ K ⊲ → C is a φ -limit diagram if and only if forevery functor G ∶ K ⊲ → C over S sending all morphisms to φ -cocartesian morphisms the map24 is an equivalence.Proof.
1. follows from .... If C → S is a cocartesian fibration, the commutative square in the definitionof φ -limit induces on the fiber over every object of Fun ( K ⊲ , S )( φ ○ F , φ ○ H ) the map 24 for G theobjectwise φ -cocartesian lift of the natural transformation φ ○ F → φ ○ H starting at F . (cid:3) Corollary 9.2.
Let K be a ∞ -category and φ ∶ C → S a functor.Consider the following conditions:(1) The pullback K ⊲ × S C → K ⊲ along α , which is a cocartesian fibration, classifies a limit diagram K ⊲ → Cat ∞ , (2) any φ -cocartesian lift K ⊲ → C of α is a φ -limit diagram.Denote ( − ) ∣ K the functor C α (−∞) ≃ Fun cocartK ⊲ ( K ⊲ , K ⊲ × S C ) → Fun cocartK ( K , K × S C ) Condition 2. implies that condition 1. is equivalent to the condition that ( − ) ∣ K is essentiallysurjective.If φ ∶ C → S is a cocartesian fibration, condition 1. implies condition 2.Proof. Condition 1. says that ( − ) ∣ K is an equivalence. By Lemma 9.1 1. condition 2. implies thatfor any cocartesian section H of the pullback K ⊲ × S C → K ⊲ along α and any section F of the samepullback the induced mapFun K ⊲ ( K ⊲ , K ⊲ × S C )( F , H ) → Fun K ( K , K × S C )( F ∣ K , H ∣ K ) is an equivalence. So condition 2. implies that ( − ) ∣ K is fully faithful. So if condition 2. holds, condition1. is equivalent to the condition that ( − ) ∣ K is essentially surjective.If φ ∶ C → S is a cocartesian fibration, by Lemma 9.1 2. condition 2. is equivalent to the conditionthat ( − ) ∣ K is fully faithful. So in this case condition 1. implies condition 2. (cid:3) ∞ -CATEGORIES AND ∞ -CATEGORIES WITH WEAK ACTION Lemma 9.3.
Let S be an ∞ -category. Denote Θ ⊂ Cat ∞/ S ▷ the full subcategory spanned by thefunctors X → S ▷ such that X has a final object lying over the final object of S ▷ .Taking pullback along the embedding S ⊂ S ▷ defines a functor κ ∶ Θ ⊂ Cat ∞/ S ▷ → Cat ∞/ S that admits a fully faithful left adjoint with colocal objects the functors X → S ▷ , whose fiber over thefinal object of S ▷ is contractible.Proof. Sending X to X ▷ defines a section β ∶ Cat ∞/ S → Θ ⊂ Cat ∞/ S ▷ of κ. For any functor Y → S and any X ∈ Θ the induced map
Cat ∞/ S ▷ ( Y ▷ , X ) ≃ Cat ∞/ S ( Y , κ ( X )) is inverse to the map Cat ∞/ S ( Y , κ ( X )) → Cat ∞/ S ( Y ▷ , κ ( X ) ▷ ) → Cat ∞/ S ▷ ( Y ▷ , X ) induced by the functor κ ( X ) ▷ → X over S ▷ , whose pullback to S is the identity of κ ( X ) and whoseimage of the final object of κ ( X ) ▷ is the final object of X . The functor κ ( X ) ▷ → X over S ▷ is anequivalence if and only if the fiber of the functor X → S ▷ over the final object of S ▷ is contractible. (cid:3) Note that a functor X → S ▷ , whose fiber over the final object of S ▷ is contractible, is a cocartesianfibration relative to the morphisms of the form A → ∞ for some A ∈ S if and only if the ∞ -categoryX has a final object lying over the final object of S ▷ . Corollary 9.4.
Let S be an ∞ -category. Denote Γ ⊂ Cat ∞/ S ▷ the full subcategory spanned by thecocartesian fibration relative to the morphisms of the form A → ∞ for some A ∈ S , whose fiber overthe final object of S ▷ is contractible.Taking pullback along the embedding S ⊂ S ▷ defines an equivalence Γ ≃ Cat ∞/ S . Lemma 9.5.
Let S , T be ∞ -categories and E ⊂ Fun ([ ] , S ) , E ′ ⊂ Fun ([ ] , T ) full subcategories con-taining all equivalences. Assume that T has a final object ∞ and that E ′ contains all morphisms ofthe form A → ∞ for some A ∈ T . Let X → S × T be a cocartesian fibration relative to E × E ′ and Y → S a cocartesian fibration relative to E . The canonical functor
Fun E × E ′ S × T ( X , Y × T ) → Fun E S ( X ∞ , Y ) that takes the fiber over ∞ ∈ T is an equivalence.Proof. Note that the embedding X ∞ ⊂ X sends cocartesian lifts of morphisms of E to cocartesian liftsof morphisms of E × E ′ . We first prove the case that E is the full subcategory of Fun ([ ] , S ) spannedby the equivalences. Denote Θ ⊂ Fun S ( X , Y ) be the full subcategory spanned by the functors X → Yover S that invert cocartesian lifts of morphisms of E ′ or equivalently of morphism of the form A → ∞ for some A ∈ T. In this case we want to see that the canonical functor(25) Θ ⊂ Fun S ( X , Y ) → Fun S ( X ∞ , Y ) is an equivalence.As T has a final object, the full subcategory spanned by the final objects of T is a reflexive fullsubcategory with any morphism a local equivalence. As X → T is a cocartesian fibration relative tothe morphisms of the form A → ∞ for some A ∈ T, the full subcategory X ∞ ⊂ X is a reflexive fullsubcategory with local equivalences to a local object the cocartesian lifts of a morphism of the formA → ∞ for some A ∈ T. As any cocartesian lift of a morphism of E ′ lies over an equivalence in S, N EQUIVALENCE BETWEEN ENRICHED ∞ -CATEGORIES AND ∞ -CATEGORIES WITH WEAK ACTION 61 the localization X ∞ ⊂ X is a localization relative to S , which guarantees that the functor 25 is anequivalence.Now we treat the general case: A functor X → Y × T over S × T is a map of cocartesian fibrationsrelative to E × E ′ if and only if it is a map of cocartesian fibrations relative to E and E ′ . A functorX → Y × T over S × T is a map of cocartesian fibrations relative to E ′ if and only if its correspondingfunctor X → Y over S belongs to Θ . A map X → Y × T of cocartesian fibrations relative to E ′ is amap of cocartesian fibrations relative to E if and only if for any t ∈ T the induced functor X t → Yis a map of cocartesian fibrations relative to E if and only if the induced functor X ∞ → Y is a mapof cocartesian fibrations relative to E using that the functor X t → Y factors as X t → X ∞ → Y andthe functor X t → X ∞ over S is a map of cocartesian fibrations relative to E . Thus the equivalence 25restricts to the desired equivalenceFun E × E ′ S × T ( X , Y × T ) → Fun E S ( X ∞ , Y ) . (cid:3) Lemma 9.6.
Let V be a monoidal ∞ -category and M , N categories with a left V -action.If N admits small colimits and for every Z ∈ V the functor Z ⊗ ( − ) ∶ N → N preserves small colimits,the ∞ -categories LinFun lax V ( M , N ) , LinFun V ( M , N ) admit small colimits and the forgetful functors LinFun lax V ( M , N ) → Fun ( M , N ) , LinFun V ( M , N ) → Fun ( M , N ) preserve small colimits.So in this case the full subcategory LinFun cc V ( M , N ) ⊂ LinFun V ( M , N ) spanned by the small colimitspreserving V -linear functors is closed in LinFun lax V ( M , N ) under small colimits.Proof. By definition LinFun V ( M , N ) is the full subcategory of Fun V ( M , N ) spanned by the functors M → N over V that preserve cocartesian morphisms over Ass . The functor N → V is a locally cocartesian fibration that is compatible with small colimits by ourassumption on N . So by ... the ∞ -category LinFun V ( M , N ) admits small colimits which are preservedby the forgetful functor LinFun V ( M , N ) → Fun ( M , N ) . As V is a monoidal ∞ -category, the full subcategory V ⊂ Env ( V ) is a monoidal localization. We havea canonical equivalence LinFun lax V ( M , N ) ≃ LinFun
Env ( V ) ( Env LM ( M ) , N ) , under which the forgetful functor LinFun lax V ( M , N ) → Fun ( M , N ) corresponds to the compositionLinFun Env ( V ) ( Env LM ( M ) , N ) → Fun ( Env LM ( M ) , N ) → Fun ( M , N ) . (cid:3) Proposition 9.7.
Let B → T be a cocartesian fibration with small fibers, V → T a monoid in ̂ Cat cocart ∞/ T classifying a functor T → Alg ( Cat cc ∞ ) and W → T a right module over V in ̂ Cat cocart ∞/ T classifying afunctor T → RMod ( Cat cc ∞ ) . There is a canonical equivalence
LinFun T , cc V ( Fun T ( B rev , V ) , W ) ≃ Fun T ( B , W ) over T . Especially there is a canonical equivalence LinFun T , cc V ( Fun T ( B rev , V ) , Fun T ( B rev , V )) ≃ Fun T ( B × T B rev , V ) over T . ∞ -CATEGORIES AND ∞ -CATEGORIES WITH WEAK ACTION Proof.
Let’s first treat the case that T is contractible.In this case we have to show that for every small ∞ -category B , every monoidal ∞ -category V compatible with small colimits and every ∞ -category M right tensored over V compatible with smallcolimits composition with the functor B ⊂ Fun ( B op , S ) ⊗(−) ÐÐÐ→
Fun ( B op , V ) defines an equivalence ρ ∶ LinFun cc V ( Fun ( B op , V ) , M ) ≃ Fun ( B , M ) . The functor Fun ( B op , S ) × V → Fun ( B op , V ) adjoint to the functor B op × Fun ( B op , S ) × V → S × V ⊗ Ð→ V preserves small colimits in each component and thus yields a functor α ∶ Fun ( B op , S ) ⊗ V → Fun ( B op , V ) that is an equivalence by [6] Proposition 4.8.1.17.The unique small colimits preserving monoidal functor ⊗ ( − ) ∶ S → V yields a small colimits preservingfunctor θ ∶ Fun ( B op , S ) → Fun ( B op , V ) . θ exhibits the diagonal right V -action on Fun ( B op , V ) asthe free right V -module on Fun ( B op , S ) in Cat cc ∞ as the equivalence α factors as Fun ( B op , S ) ⊗ V → Fun ( B op , V ) ⊗ V → Fun ( B op , V ) . Thus composition with θ defines an equivalenceLinFun cc V ( Fun ( B op , V ) , M ) ≃ Fun cc ( Fun ( B op , S ) , M ) . By the universal property of the ∞ -category of presheaves Fun ( B op , S ) on B we have a canonicalequivalence Fun cc ( Fun ( B op , S ) , M ) ≃ Fun ( B , M ) . So ρ is an equivalence.Now let’s turn to the general case: Denote U ⊂ R ⊂ Fun ([ ] , Cat ∞ ) the full subcategories spannedby the (representable) right fibrations. By Proposition 9.8 and 9.9 there is a canonical equivalence R ≃ Fun
Cat ∞ ( U rev , Cat ∞ × S ) over Cat ∞ , where U , R → Cat ∞ are evaluation at the target and U rev → Cat ∞ denotes the pullback of U → Cat ∞ along the opposite ∞ -category involution. So we get a Yoneda embedding U ⊂ R ≃ Fun
Cat ∞ ( U rev , Cat ∞ × S ) over Cat ∞ , whose pullback along the functor T → Cat ∞ classified by B → T is an embedding B ⊂ Fun T ( B rev , T × S ) over T. As S is the initial object of the ∞ -category Cat cc ∞ , there is a canonical mapT × S → V of cocartesian fibrations over T . The functor B ⊂ Fun T ( B rev , T × S ) → Fun T ( B rev , V ) over T yields a functor ΨLinFun T , cc V ( Fun T ( B rev , V ) , W ) → Fun T ( Fun T ( B rev , V ) , W ) → Fun T ( B , W ) over T . We will show that Ψ is an equivalence.To prove this, it is enough to check that for every functor [ ] → T the pullback [ ] × T Ψ is anequivalence. So we can reduce to the case that T = [ ] .Ψ induces on the fiber over every i ∈ [ ] the functorLinFun cc V i ( Fun ( B opi , V i ) , W i ) → Fun ( B i , W i ) that is an equivalence by the first part of the proof. Hence Ψ is essentially surjective.Let F i ∶ B i → W i be functors for i ∈ [ ] corresponding to V i -linear small colimits preserving functors¯F i ∶ Fun ( B opi , V i ) → W i .It remains to show that Ψ induces an equivalence ϕ ∶ LinFun [ ] , cc V ( Fun [ ] ( B rev , V ) , W )( ¯F , ¯F ) → Fun [ ] ( B , W )( F , F ) . N EQUIVALENCE BETWEEN ENRICHED ∞ -CATEGORIES AND ∞ -CATEGORIES WITH WEAK ACTION 63 Denote ψ the functor B → B , κ the monoidal functor V → V , τ the monoidal functor W → W and φ the monoidal functor Fun ( B op0 , V ) → Fun ( B op1 , V ) .ϕ is canonically equivalent to the functorLinFun cc V ( Fun ( B op0 , V ) , κ ∗ ( W ))( τ ○ ¯F , ¯F ○ φ ) ≃ Fun ( B , W )( τ ○ F , F ○ ψ ) . (cid:3) Proposition 9.8.
Denote
Cocart ⊂ Fun ([ ] , Cat ∞ ) the subcategory with objects the cocartesian fibra-tions and morphisms the maps that yield on evaluation at the source a functor preserving the evidentcocartesian morphisms. Denote L ⊂ Fun ([ ] , Cat ∞ ) the full subcategory spanned by the left fibrationsand U → Cat ∞ the cocartesian fibration classifying the identity.There is a canonical equivalence Cocart ≃ Fun
Cat ∞ ( U , Cat ∞ × Cat ∞ ) over Cat ∞ , where Cocart → Cat ∞ is evaluation at the target, that restricts to an equivalence L ≃ Fun
Cat ∞ ( U , Cat ∞ × S ) over Cat ∞ .Proof. We show that for every functor ψ ∶ S → Cat ∞ classified by a cocartesian fibration X → S functorsS → Cocart over
Cat ∞ naturally correspond to functors S → Fun
Cat ∞ ( U , Cat ∞ × Cat ∞ ) over Cat ∞ :Functors α ∶ S → Fun ([ ] , Cat ∞ ) over Cat ∞ correspond to natural transformations of functorsS → Cat ∞ with target ψ and thus are classified by a map β of cocartesian fibrations over S with target X.The functor α ∶ S → Fun ([ ] , Cat ∞ ) factors through Cocart if and only if β is a cocartesian fibration. Soa functor S → Cocart over
Cat ∞ corresponds to a cocartesian fibration over X. A cocartesian fibrationover X is classified by a functor γ ∶ X → Cat ∞ adjoint to a functor S → Fun
Cat ∞ ( U , Cat ∞ × Cat ∞ ) over Cat ∞ .The functor α ∶ S → Fun ([ ] , Cat ∞ ) factors through S if and only if β is a left fibration if and onlyif γ ∶ X → Cat ∞ factors through S . (cid:3) Proposition 9.9.
Denote U ⊂ R the full subcategory spanned by the representable right fibrations, i.e.the right fibrations C → D such that C has a final object.Evaluation at the target U → Cat ∞ is a cocartesian fibration classifying the identity.Proof. Denote V → Cat ∞ the cocartesian fibration classifying the identity. We will construct anequivalence V ≃ U over Cat ∞ by naturally identifying for any ∞ -category S the set of equivalenceclasses of functors S → V over Cat ∞ and S → U over Cat ∞ .A functor S → V over Cat ∞ is classified by a cocartesian fibration C → S equipped with a sectionof C → S. A functor S → U ⊂ Fun ([ ] , Cat ∞ ) over Cat ∞ is classified by a map of cocartesian fibrations D → C over S that induces on the fiber over any s ∈ S a representable right fibration.A section α of C → S gives rise to the map S × C { } C [ ] → C { } of cocartesian fibrations over Sthat induces on the fiber over any s ∈ S the representable right fibration ( C s ) / α ( s ) → C s and induceson sections the functor Fun S ( S , C ) / α → Fun S ( S , C ) that is an equivalence iff α is a final object inFun S ( S , C ) , which is the case if for any s ∈ S the image α ( s ) is final in C s . Given a map of cocartesian fibrations ψ ∶ D → C over S such that for any s ∈ S the fiber D s has afinal object the ∞ -category Fun S ( S , D ) has a final object β such that for any s ∈ S the image β ( s ) isfinal in D s . So we get a a section ψ ○ β of C → S.For D = S × C { } C [ ] we have Fun S ( S , D ) ≃ Fun S ( S , C ) / α over Fun S ( S , C ) so that ψ ○ β = α. In general the induced map S × D { } D [ ] → S × C { } C [ ] of cocartesian fibrations over S induces onthe fiber over any s ∈ S the equivalence ( D s ) / β ( s ) ≃ ( C s ) / α ( s ) (coming from the fact that D s → C s is a ∞ -CATEGORIES AND ∞ -CATEGORIES WITH WEAK ACTION right fibration). Hence the induced map D ≃ S × D { } D [ ] → S × C { } C [ ] of cocartesian fibrations overS is an equivalence. So D → C is equivalent over C to S × C { } C [ ] → C { } . (cid:3) Corollary 9.10.
Given a cocartesian fibration X → S there is a canonical embedding X ⊂ Fun S ( X rev , S × S ) of cocartesian fibrations over S that induces fiberwise the Yoneda-embedding.Proof. Denote by U ⊂ R ⊂ Fun ([ ] , Cat ∞ ) the full subcategories spanned by the right fibrations respectively representable right fibrations, i.e.the right fibrations C → D such that C has a final object. By Proposition 9.9 evaluation at the target U → Cat ∞ is a cocartesian fibration classifying the identity. By Proposition 9.8 there is a canonicalequivalence R ≃ Fun
Cat ∞ ( U rev , Cat ∞ × S ) over Cat ∞ , whose pullback along the functor S → Cat ∞ classified by the cocartesian fibration X → Sgives an equivalence S × Fun ({ } , Cat ∞ ) R ≃ Fun S ( X rev , S × S ) over S and so a full embedding X ≃ S × Fun ({ } , Cat ∞ ) U ⊂ S × Fun ({ } , Cat ∞ ) R ≃ Fun S ( X rev , S × S ) over S. (cid:3) Enveloping cocartesian fibrations 9.11.
Let S be a ∞ -category and L , R ⊂ Fun ([ ] , S ) full sub-categories containing all equivalences.We call ( L , R ) a factorization system on S if the embedding R ⊂ Fun ([ ] , S ) admits a left adjointand a morphism in Fun ([ ] , S ) is a local equivalence if and only if its image under evaluation at thesource belongs to L and its image under evaluation at the target is an equivalence.Given a functor C → S the pullback C ′ ∶ = Fun ([ ] , S ) × Fun ({ } , S ) C → Fun ({ } , S ) is a cocartesian fibration, whose cocartesian morphisms are those that get equivalences in C . Thediagonal embedding S ⊂ Fun ([ ] , S ) yields an embedding C ⊂ Fun ([ ] , S ) × Fun ({ } , S ) C over Fun ({ } , S ) .Moreover C ′ → S is the universal cocartesian fibration, i.e. for any cocartesian fibration D → S theinduced functor Fun cocartS ( C ′ , D ) → Fun S ( C , D ) is an equivalence.Denote Env ( C ) ⊂ C ′ the pullbackR × Fun ({ } , S ) C → R → Fun ({ } , S ) . Proposition 9.12.
Let γ ∶ C → S be a cocartesian fibration relative to L .(1) The embedding Env ( C ) ⊂ C ′ admits a left adjoint, where a morphism of C ′ is a local equivalenceif and only if its image in C is cocartesian over S and lies over a morphism of L and its imagein S under evaluation at the target is an equivalence.So Env ( C ) → S is a cocartesian fibration, whose cocartesian morphisms are precisely those,whose image in C is cocartesian over S and lies over a morphism of L . N EQUIVALENCE BETWEEN ENRICHED ∞ -CATEGORIES AND ∞ -CATEGORIES WITH WEAK ACTION 65 (2) For any cocartesian fibration ρ ∶ D → S the embedding C ⊂ Env ( C ) induces an equivalence Fun cocartS ( Env ( C ) , D ) ≃ Fun LS ( C , D ) , where the right hand side denotes the full subcategory of functors preserving cocartesian liftsof morphisms of L . Proof.
1. follows immediately from the fact that R ⊂ Fun ([ ] , S ) is a localization with the describedlocal equivalences and C → S is a cocartesian fibration relative to L.1. implies that the embedding C ⊂ Env ( C ) is a map of cocartesian fibrations relative to L . So theembedding C ⊂ Env ( C ) induces a functorFun cocartS ( Env ( C ) , D ) → Fun LS ( C , D ) . Moreover 1. implies that restrictionFun S ( C ′ , D ) → Fun S ( Env ( C ) , D ) along the functor Env ( C ) ⊂ C ′ restricts to an equivalence on the full subcategory of Fun S ( C ′ , D ) spanned by the functors inverting local equivalences, and a functor C ′ → D over S inverting localequivalences is a map of cocartesian fibrations over S if and only if its restriction Env ( C ) ⊂ C ′ → D is. As we have a canonical equivalence Fun cocartS ( C ′ , D ) ≃ Fun S ( C , D ) , it is enough to see that a map ψ ∶ C ′ → D of cocartesian fibrations over S inverts local equivalences if its restriction τ ∶ C ⊂ C ′ → D isa map of cocartesian fibrations relative to L . Given a morphism h in C ′ corresponding to a morphism f ∶ X → Y in C and a commutative square γ ( X ) (cid:15) (cid:15) γ ( f ) / / γ ( Y ) (cid:15) (cid:15) X ′ g / / Y ′ in S we have a unique commutative square τ ( X ) (cid:15) (cid:15) τ ( f ) / / τ ( Y ) (cid:15) (cid:15) ψ ( X ) ψ ( f ) / / ψ ( Y ) in D , where both vertical morphisms are ρ -cocartesian, that lifts the former commutative square.If h is a local equivalence, f ∶ X → Y is γ -cocartesian and lies over a morphism of L and g is anequivalence. So if τ ∶ C ⊂ C ′ → D is a map of cocartesian fibrations relative to L, the image τ ( f ) is ρ -cocartesian. Therefore ψ ( f ) is also ρ -cocartesian as both vertical maps in the diagram are. As ψ ( f ) lies over the equivalence g, it is itself an equivalence. (cid:3) References [1] David Ayala and John Francis. Fibrations of ∞ -categories, 2017.[2] David Ayala and John Francis. Flagged higher categories , pages 137–173. 01 2018.[3] David Gepner and Rune Haugseng. Enriched -categories via non-symmetric -operads.
Advances in Mathematics ,279:575 – 716, 2015.[4] Rune Haugseng. ∞ -operads via symmetric sequences, 2017.[5] Vladimir Hinich. Yoneda lemma for enriched -categories. Advances in Mathematics
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