Colimits in enriched \infty-categories and Day convolution
aa r X i v : . [ m a t h . C T ] J a n COLIMITS IN ENRICHED ∞ -CATEGORIES AND DAYCONVOLUTION VLADIMIR HINICH
Abstract.
Let M be a monoidal ∞ -category with colimits. In this paper westudy colimits of M -functors A → B where B is left-tensored over M and A is an M -enriched category in the sense of [H.EY]. We prove that the enrichedYoneda embedding Y : A → P M ( A ) defined in loc. cit. yields a universal M -functor and, in case when A has a certain monoidal structure, the categoryof enriched presheaves P M ( A ) inherits the same monoidal structure. Introduction M is a monoidal category withcolimits, such that the tensor product in M preserves colimits in both arguments.In this case we prove that the Yoneda embedding Y : A → P M ( A ) of an M -enriched category into the category of enriched presheaves, is universal amongthe functors to a category left tensored over M : one has a natural equivalence(1) Y ∗ : Fun LMod M ( P M ( A ) , B ) → Fun M ( A , B ) . B Fun M ( A , B ). There is a functor in the opposite direction comingfrom the notion of weighted colimit. Given an M -functor f : A → B , where B is a category with colimits left-tensored over M , the weighted colimit colim( f ) : P M ( A ) → B is defined. Thisgives a functor(2) colim : Fun M ( A , B ) → Fun
LMod M ( P M ( A ) , B )quasi-inverse to Y ∗ .1.3. In the case when M is an O -algebra in the category of monoidal categories(that is, M is a O ⊗ Ass -monoidal category), one can define O -monoidal enriched M -categories, as well as O -monoidal left-tensored categories over M . In thiscase, if A is an O -monoidal M -enriched category, P M ( A ) inherits an O -monoidal One can think of colimits for functors of two kinds: functors from one M -enriched categoryto another, and functors from an M -enriched category to a category left tensored over M . Inthis paper we deal with this second kind of functors. structure, Yoneda embedding becomes an O -monoidal M -functor (see 7.2.3), and(1) induces an equivalence(3) Fun O , KM ( P M ( A ) , B ) → Fun OM ( A , B )of the corresponding categories of O -monoidal functors.The O -monoidal structure on P M ( A ) is an enriched version of the Day convo-lution defining a monoidal structure on the presheaves on a monoidal category.1.4. The paper was started with the aim to prove universality of enriched Yonedaembedding constructed in [H.EY]. At first the task seemed very easy: given an M -enriched category A and an M -functor f : A → B , where B is a left M -module with colimits, we define the functor colim f : P M ( A ) → B as a weightedcolimit, using a general Lurie’s machinery [L.HA] of relative tensor product (thisis now explained in the beginning of Section 6). One can easily prove that thecomposition of colim f with the Yoneda embedding gives back f . But we have notfound an easy argument proving that the enriched Yoneda map Y : A → P M ( A ) isuniversal. This is why we had to add Section 3 comparing our working definitionof enriched categories with the one given by Lurie in [L.HA], 4.2.1.28. Nowuniversality of Y follows from the description of colimit preserving left M -modulemaps P M ( A ) → B as operadic left Kan extensions of their restriction to A ,the essential image of the Yoneda embedding. An O -monoidal vertion of theuniversality easily follows from this approach via weighted colimits.Our work has a very considerable overlap with the recent manuscript byHadrien Heine [HH]. In it the category of enriched M -categories is proven tobe equivalent to the one defined by Lurie (we only prove that the functor A A is fully faithful). Heine also proves universality of the Yoneda embedding. Itseems, however, that his methods are insufficient to deduce the O -monoidal ver-sion of the universality.1.5. In Section 2 we provide a digest of the theory of enriched categories andenriched Yoneda lemma. The notion of enriched category used here is the onepresented in [H.EY], Sect. 3. Our definition of enriched categories is practicallyequivalent to the earlier definition of [GH].J. Lurie defines in [L.HA], 4.2.1.28 another notion of M -enriched category, asa category weakly tensored over M , and satisfying some properties.In Section 3 we compare the notion of enriched categories used in this paperwith the one defined by Lurie. We construct a fully faithful functor from thecategory Cat ( M ) of categories enriched over a monoidal category M with colimits,to the category of Lurie enriched M -categories . What is more important to us,we interpret M -functors f : A → B from an enriched category A to a left-tensored category B as functors between the categories weakly tensored over M . H. Heine has recently proven that these two notion of enrichment are equivalent, see [HH].
In Section 4 we review the theory of relative tensor products [L.HA], 3.1 and4.6.In Section 5 we study bar resolutions for enriched presheaves. This is a techni-cal section whose result is only needed in the characterization 6.4.2 of morphismsof M -modules P M ( A ) → B as operadic left Kan extensions.The notion of relative tensor product allows us to define in Section 6 theweighted colimits.In Section 7 we deal with the multiplicative properties of this functor.1.6. Acknowledgement.
We are very grateful to Greg Arone, Ilan Barnea andTomer Schlank for their interest in this work and for sharing their preprint [ABS].We are also grateful to H. Heine for informing us about his work [HH]. The workwas supported by ISF 786/19 grant.2.
Enriched categories and enriched Yoneda: digest
In this section we recall some important constructions of [H.EY]. The notionof operadic left Kan extension is reviewed in 2.4.2.1. The category of operads Op is a subcategory of Cat /F in ∗ , where Fin ∗ is thecategory of finite pointed sets. If O is an operad, we denote Op / O or just Op O thecategory of O -operads, that is operads endowed with a morphism to O .The operad Ass ⊗ governs associative algebras, and Op Ass is the category ofplanar operads. We denote by LM ⊗ , BM ⊗ the operads governing the left modulesand the bimodules respectively. Thus, the operad BM ⊗ has three colors, so thatthe BM ⊗ -algebras are the triples ( A, M, B ) consisting of two associative algebras A and B acting from the left and from the right on M . Similarly, a BM ⊗ -operadhas three components, two planar operads A and B , and a category M , with twocompatible weak actions of A and of B on M .Following [L.HA], 2.3.3 and [H.EY], 2.7.1, we often replace operads with theirstrong approximations. In particular, we use the approximation Ass , LM and BM of Ass ⊗ , LM ⊗ and BM ⊗ as defined in [H.EY], 2.9.2.2. Quivers.
The notion of enriched category, as presented in [H.EY], is basedon a functor(4)
Quiv : Cat op × Op Ass → Op Ass carrying a pair ( X, M ) to a planar operad Quiv X ( M ) whose colors are M -quivers ,that is functors A : X op × X → M .The functor (4) has two variations. The first is a functor(5) Quiv LM : Cat op × Op LM → Op LM , and the second is(6) Quiv BM : Cat op × Op BM → Op BM . VLADIMIR HINICH
The functors are compatible: the
Ass -component of
Quiv LM X ( M , B ) is Quiv X ( M ),and so on.In good cases, the functors Quiv applied to monoidal categories with enoughcolimits, produce a monoidal category.2.2.1.
More details.
In Section 5 we will need a more detailed information aboutthe functor (5) .In what follows ∆ / LM denotes the category of simplices in LM . For a fixed X ∈ Cat , one defines an LM -operad LM X by a presheaf(∆ / LM ) op → S given by the formula LM X ( σ ) = Map( F ( σ ) , X ) , where F : ∆ / LM → Cat is a functor with values in conventional categories combi-natorially defined in [H.EY], 3.2.The LM -operad LM X is always flat, [H.EY], 3.3. This means that the functor Op LM → Op LM given by product with LM X , admits a right adjoint, which is denotedFunop LM ( LM X , ).Finally, given M = ( M a , M m ) ∈ Op LM , one defines the LM -operad Quiv LM X ( M ) asFunop LM ( LM X , M ).Two other variations of the category of quivers, Quiv and
Quiv BM , have a similardescription.Given M = ( M a , M m , M b ) ∈ Op BM , the BM -operad Quiv BM X ( M ) has components( Quiv X ( M a ) , Fun( X, M m ) , M b ).2.2.2. Cat -enrichment.
Let O be an operad or a strong approximation of anoperad. The category of O -operads Op O has a Cat -enrichment that assigns to P , Q ∈ Op O the category Alg P / O ( Q ).The functor Quiv LM X : Op LM → Op LM respects this enrichment. This means that,given P , Q ∈ Op LM , one has a functor(7) Alg P / LM ( Q ) → Alg
Quiv LM X ( P ) / LM ( Quiv LM X ( Q ))extending the mapMap Op LM ( P , Q ) → Map Op LM ( Quiv LM X ( P ) , Quiv LM X ( Q )) . The map (7) is defined as follows. Its target is naturally equivalent, according to[H.EY], 2.8.6, to
Alg
Quiv LM X ( P ) × LM LM X ( Q ). The map (7) can therefore be defined asthe one induced by the canonical evaluation map Quiv LM X ( P ) × LM LM X = Funop LM ( LM X , P ) × LM LM X → P . The functors (4), (6) have a similar description.
Algebras in quivers.
Fixing the second (operadic) argument, we will lookat the functors 4)–(6) as at cartesian families of (planar, LM or BM ) operads. Letus describe our interpretation for the categories of algebras in various operads ofquivers.2.3.1. Enriched precategories.
For a fixed planar operad M with colimits, anassociative algebra in the family Quiv ( M ) is called M -enriched precategory. Wedenote PCat ( M ) = Alg
Ass ( Quiv ( M )) the category of M -enriched precategories.An enriched precategory A has a category X of objects, and an associativemultiplication in the underlying quiver A : X op × X → M .2.3.2. M -functors. Fix an LM -operad, consisting of a planar operad M and a cat-egory B weakly tensored over M . For fixed X ∈ Cat , the LM -operad Quiv LM X ( M , B )consists of the planar operad Quiv X ( M ) and a category Fun( X, B ), weakly ten-sored over Quiv X ( M ).The LM -operads Quiv LM X ( M , B ) form a family Quiv LM ( M , B ).An LM -algebra in it consists of a pair ( A , f ) where A is an M -enriched precat-egory, and f is an A -module in Fun( X, B ).We denote PCat LM ( M , B ) = Alg LM ( Quiv LM ( M , B )) . We interpret A -modules in Fun( X, B ) as M -functors from A to B , whence thenotation(8) Fun M ( A , B ) = LMod A (Fun( X, B )) , the category of left A -modules in Fun( X, B ).2.3.3. Assume now M is a monoidal category with colimits. Applying the aboveto B := M considered as a right M -module (which is the same as left M rev -module), we can define the category of enriched presheaves P M ( A ) = Fun M rev ( A op , M ).It is left-tensored over M and has colimits.Yoneda embedding is an M -functor Y : A → P M ( A ), defined by A -bimodulestructure on A , see details in [H.EY], Section 6.2.3.4. Enriched categories. An M -enriched category is an enriched precategorysatisfying a certain completeness condition. The full embedding Cat ( M ) ⊂ PCat ( M ) is right adjoint to a localization functor L : PCat ( M ) → Cat ( M ) whichcan be described as follows. Given A ∈ Alg
Ass ( Quiv ( M )), we define X as the syb-space of P M ( A ) eq spanned by the representable functors, and define L ( A ) as theendomorphism object in Quiv X ( M ) of the tautological embedding X → P M ( A ),see [H.EY], 7.2.2.3.5. Restriction of scalars.
Given a cartesian family p : Q → B × LM of LM -operads, the embedding Ass → LM induces a functor Alg LM ( Q ) → Alg
Ass ( Q ) VLADIMIR HINICH which is a cartesian fibration. This result can be found in [L.HA], 4.2.3.2 or[H.EY], 2.11.2.3.6. Let ( M , B ) be an LM -operad. The assignment A Fun M ( A , B ) is con-travariant in A . This is a special case of a general setup presented in 2.3.5.Thus, a map f : A → A ′ of M -enriched precategories gives rise to a functor f ∗ : Fun M ( A ′ , B ) → Fun M ( A , B ). The definition of f ∗ allows one to compose amap of M -enriched precategories f : A → A ′ with an M -functor A ′ → B .2.4. Operadic left Kan extensions.
Operadic colimits and operadic left Kanextensions defined in Lurie’s [L.HA], Section 3.1, are a part of the constructionof a free operad algebra. We need a very special case of it, dealing with thealgebras with values in a monoidal category with colimits. In this case operadicleft Kan extension can be defined, similarly to the usual left Kan extensions, asthe functor left adjoint to the forgetful functor.Let O be an operad and let C ∈ Alg O ( Cat L ) be an O -monoidal category withcolimits. Given a morphism f : P → Q of O -operads, one has a forgetful functor f ∗ : Alg Q ( C ) → Alg P ( C ). We define the operadic left Kan extension as the functor f ! : Alg P ( C ) → Alg Q ( C )left adjoint to f ∗ . Operadic left Kan extension, in these circumstances, alwaysexists.2.4.1. Operadic colimit.
Recall [L.T], 4.3.1, the notion of a relative colimit. Let p : C → B be a functor, f ⊲ : K ⊲ → C a diagram and denote f = f ⊲ | K . One saysthat f ⊲ is a p -colimit diagram if the natural map C f ⊲ / → C f/ × D p ◦ f/ D p ◦ f ⊲ / is an equivalence.Operadic colimit is a special kind of relative colimit. Let p : C → O be an O -operad and let f ⊲ : K ⊲ → C act be a functor in the subcategory spanned by theactive arrows.One says that f ⊲ is an operadic colimit diagram if for any C ∈ C the compo-sition K ⊲ f ⊲ → C act ⊕ C → C act is a relative colimit diagram.Let f : K → C act be a diagram in an O -operad C and let g ⊲ : K ⊲ → O be anextension of g = p ◦ f . We say that the diagram f has an operadic colimit over g ⊲ if there exists f ⊲ over g ⊲ that is an operadic colimit diagram.In the case when C is an O -monoidal category with colimits, such that themonoidal structure is compatible with the colimits, any diagram f : K → C hasan operadic colimit. Operadic left Kan extensions.
We present below a definition of operadicleft Kan extension due to Lurie, [L.HA], 3.1.2. We restrict ourselves to a specialcase of what Lurie calls “free algebra”, see [L.HA], 3.1.3.1.Given i : P → Q a morphism of O -operads and an O -operad C , one has anobvious functor i ∗ : Alg Q ( C ) → Alg P ( C ) . For any q ∈ Q we define K q = P × Q Q act /q .Given A ∈ Alg P ( C ) and B ∈ Alg Q ( C ), a morphism f : A → i ∗ ( B ) in Alg P ( C )determines a morphism of functors α → const B ( q ) : K q → C , with α being thecomposition of A with the projection K q → P and const B ( q ) being the constantfunctor with the value B ( q ). Equivalently, this translates into a functor(9) α ⊲q : K ⊲q → C . A morphism f : A → i ∗ ( B ) is called an operadic left Kan extension of A withrespect to i if for any q ∈ Q the functor α ⊲q : K ⊲q → C is an operadic colimitdiagram.In the case when C ∈ Alg O ( Cat L ), the operadic left Kan extension exists anddefines a functor Alg P ( C ) → Alg Q ( C ) left adjoint to i ∗ .3. Lurie’s enriched categories
In this section we compare the notion of M -enriched category and of an M -enriched functor, as presented in 2.3.4 and 2.3.2, with the similar (but simpler)notions of Lurie, [L.HA], 4.2.1.3.1. LM -operads and Lurie’s enriched categories. Probably, the simplestway to define an enriched ∞ -category over a monodial category M is presentedin Lurie’s [L.HA], 4.2.1.28.3.1.1. The map a : Ass → LM of operads induces a base change functor(10) a ∗ : Op LM → Op Ass assigning to each LM -operad O its planar component O a → Ass . The fiber O m at m ∈ LM is a category that is called weakly enriched over O a .We denote by LMod w M the fiber of a ∗ at M ∈ Op Ass . This is the category ofcategories weakly enriched over M .An object of LMod w M is an LM -operad O together with an equivalence M = Ass × LM O . We will sometimes denote it as a pair ( M , A ), where A = { m } × LM O ,or (when M is fixed) as A . VLADIMIR HINICH O , O ′ ∈ LMod w M we define Fun LMod w M ( O , O ′ ) as the fiber of the map Alg O / LM ( O ′ ) → Alg M / Ass ( M )at id M .We will now assume that M is a monoidal category. Here is the definition ofLurie’s M -enriched category.3.1.3. Definition.
Let M be a monoidal category. A Lurie M -enriched category A is an LM -operad O with the equivalences M = Ass × LM O , A = { m } × LM O ,satisfying the following properties.1. The natural map ⊕ m i → ⊗ m i induces an equivalence Map(( ⊗ m i ) ⊕ a, b ) → Map( ⊕ m i ⊕ a, b ) for any m i ∈ M and a, b ∈ A . Here we use thesign ⊕ as in [L.HA], 2.1.1.15 .2. For any a, b ∈ A the weak enrichment functor hom A ( a, b ) : M op → S ,defined by the formulahom A ( a, b )( m ) = Map( m ⊕ a, b ) , is representable.Lurie M -enriched categories form a category denoted Cat
Lur ( M ). This is a fullsubcategory of LMod w M spanned by the Lurie M -enriched categories.In this section we assign, assuming that the base monoidal category M hascolimits, to an M -enriched category A a Lurie M -enriched category ¯ A . We provethat the category Cat ( M ) defined in 2.3.4 is equivalent to a full subcategory of Cat
Lur ( M ), see 3.4.1. Note that H. Heine has recently proven [HH] that the twonotions are equivalent.3.1.4. Let M be a monoidal category with colimits.For any M -enriched category A we define ¯ A ⊂ P M ( A ), the full subcategory of P M ( A ) spanned by the representable presheaves. Obviously, ¯ A is an M -enrichedcategory in the sense of Lurie. By [H.EY], 6.1.4, this defines a functor λ : Cat ( M ) → Cat
Lur ( M ) . Corollary 3.4.1 asserts that this functor is fully faithful. As H. Heine shows in[HH] that λ is actually an equivalence, see Remark 3.4.2. The first property is a pseudo-enrichment in Lurie’s terminology. see [L.HA], 4.2.1.25. Thecondition makes sense for the number of factors n ≥ Baby Yoneda functor.
In what follows we denote by ( M , ¯ A ) ⊗ the LM -operad formed by the category ¯ A weakly tensored over a monoidal category M .Let M be a monoidal category with colimits, A be an M -enriched categoryand B be a category with colimits left-tensored over M .Let ( M , ¯ A ) be the corresponding Lurie enriched category.Our aim is to construct an equivalence(11) Fun LMod w M ( ¯ A , B ) → Fun M ( A , B ) . B Fun M ( A , B ) , as defined by the formula (8), and the preservation of Cat -enrichment by
Quiv LM ,see 2.2.2, yields a canonical functor(12) Fun M ( A , B ′ ) × Fun
LMod w M ( B ′ , B ) → Fun M ( A , B ) . In particular, for B ′ := ¯ A , we get(13) Fun M ( A , ¯ A ) × Fun
LMod w M ( ¯ A , B ) → Fun M ( A , B ) . We claim that the Yoneda embedding Y : A → P M ( A ) factors uniquely throughthe full embedding ¯ A → P M ( A ); the natural M -functor y : A → ¯ A so defined willbe denoted y and called the baby Yoneda functor . The existence (and uniqueness)of the baby Yoneda immediately follows from the lemma below.3.2.2. Lemma.
Let B be a full subcategory in C that is weakly enriched over amonoidal category M . Then, for an associative algebra A in M , one has LMod A ( B ) = LMod A ( C ) × C B . (cid:3) Note the following obvious property of the M -functor y : A → ¯ A .3.2.3. Lemma.
The forgetful functor
Fun M ( A , ¯ A ) → Fun( X, ¯ A ) carries y to amap i : X → ¯ A identifying X with the maximal subspace of ¯ A . (cid:3) The functor (11) is an equivalence. M , B ) is an LM -monoidal category with colimits. Let A ∈ Alg ( Quiv X ( M )). Let Φ : X → B be an A -module in Fun( X, B ) and let a : φ → Φbe an arrow in Fun( X, B ).We will now show how to verify whether a presents Φ as a free A -modulegenerated by φ .The A -module structure on Φ defines an active arrow ( A , Φ) → Φ in
Quiv LM X ( M , B )that is explicitly described in [H.EY] 4.2.1 and 4.3.1, case (w2) with n = 2, k = 1. Here is the result. The active arrow above is given by a map A := [1] × LM LM X → ( M , B ) act where A = A ⊔ C ( C × [1]) ⊔ C A , with A = X × X op × X , A = X , C = Tw ( X ) op × X , the map C → A is givenby the projection Tw ( X ) op → X × X op , whereas C → A is the projection to thelast factor.This yields, for any x ∈ X , a functor(14) θ Φ ⊲x : ( Tw ( X ) op ) ⊲ → ( M , B ) act carrying the terminal object of the left-hand side to Φ( x ) and the arrow α : z → y from Tw ( X ) to the pair ( A ( y, x ) , Φ( z )).We denote by θ Φ x the restriction of θ Φ ⊲x to Tw ( X ) op . Note that the functors θ Φ x make sense for any Φ ∈ Fun( X, B ) (no A -module structure is needed).3.3.2. Here is a small observation. A functor f : K ⊲ → C can be uniquely pre-sented by a map f | K → const f ( ∗ ) in Fun( K, C ), with const f ( ∗ ) being the constantfunctor having the value f ( ∗ ) ∈ C .In particular, given f as above and an arrow α : f → f | K in C , we get acanonocally defined functor f ′ : K ⊲ → C with f ′ | K = f and f ′ ( ∗ ) = f ( ∗ ).3.3.3. Here is another observation. Let p : C → B be a cocartesian fibrationand let, as above, f : K ⊲ → C be a functor.As above, f gives rise to a map f | K → const f ( ∗ ) in Fun( K, C ), as well as toits image p ◦ f | K → const p ◦ f ( ∗ ) in Fun( K, B ). Since the map Fun(
K, p ) is acocartesian fibration, we get a map of functors f → −→ f : K ⊲ → C , such that p ◦ −→ f = const p ◦ f ( ∗ ) and for such that for any x ∈ K the arrow f ( x ) →−→ f ( x ) is p -cocartesian. In this case we will say that −→ f is obtained from f by acocartesian shift.3.3.4. We will denote by θ a⊲x : ( Tw ( X ) op ) ⊲ → ( M , B ) act the functor induced by a : φ → Φ as explained in 3.3.2, whose restriction to Tw ( X ) op is θ φx and the value at the terminal object is Φ( x ).One has the following.3.3.5. Lemma.
A map a : φ → Φ in Fun( X, B ) presents Φ as a free A -module ifand only if for any x ∈ X the diagram θ a⊲x is an operadic colimit diagram.Proof. The map a : φ → Φ presents Φ as a free A -module generated by φ if a induces a cocartesian arrow ( A , φ ) → Φ in
Quiv LM X ( M , B ). This easily translatesto our condition. (cid:3) M , B ) is an LM -monoidal category with colimits, thecriterion 3.3.5 can be formulated in terms of colimits in B .Let ( M , B ) be an LM -monoidal category with colimits and let Φ : X → B bean A -module.For a fixed x ∈ X we denote by −→ θ φx : Tw ( X ) op → B the functor obtained from θ φx by the cocartesian shift, as explained in 3.3.3. Wenow have3.3.7. Corollary.
Let ( M , B ) be an LM -monoidal category with colimits. A map a : φ → Φ in Fun( X, B ) presents Φ as a free A -module if and only if for any x ∈ X the diagram −→ θ a⊲x induces an equivalence colim( −→ θ φx ) → Φ( x ) . (cid:3) The evaluation of (13) at y defines the canonical functor y ∗ : Fun LMod w M ( ¯ A , B ) → Fun M ( A , B ) . Lemma 3.2.3 asserts that G ◦ y ∗ = G ′ where G : Fun M ( A , B ) → Fun( X, B )is the forgetful functor and G ′ : Fun LMod w M ( ¯ A , B ) → Fun( X, B ) is given by thecomposition with X → ¯ A .3.3.8. We will now prove that y ∗ is an equivalence when B has colimits. Ourproof will use the description of the source and the target of y ∗ by monads onFun( X, B ).Let us consider the following diagram.(15) Fun LMod w M ( ¯ A , B ) y ∗ / / G ′ ( ( PPPPPPPPPPPP
Fun M ( A , B ) G w w ♦♦♦♦♦♦♦♦♦♦♦♦ Fun( X, B ) F F ′ h h Here the functor F , left adjoint to G , is the free A -module functor. The functor F ′ , left adjoint to G ′ , is defined by the operadic left Kan extension with respectto the map of LM -operads ǫ : M ⊔ X → ( M , ¯ A ) . The equivalence G ′ = G ◦ y ∗ gives rise to a morphism of functors η : F → y ∗ ◦ F ′ .Here is the main result of this section.3.3.9. Proposition.
The functor y ∗ defined above is an equivalence. Proof.
According to [L.HA], 4.7.3.16, we have to verify the following conditions.1. The functors G ′ and G preserve geometric realizations.2. The functors G and G ′ are conservative.3. η ( φ ) is an equivalence for any φ ∈ Fun( X, B ).The functor G is conservative by [L.HA], 3.2.2.6 and preserves colimits by[L.HA], 4.2.3.5. The functor G ′ is conservative (by [L.HA], 3.2.2.6) and preservesgeometric realizations by [L.HA], 3.2.3.1.It remains to verify that η ( φ ) : F ( φ ) → y ∗ ◦ F ′ ( φ ) is an equivalence for any φ : X → B . We will do so by verifying that the unit of the adjunction(16) a : φ → G ′ ◦ F ′ ( φ )satisfies the condition of 3.3.7 with Φ = G ′ ◦ F ′ ( φ ).The map (16) defines, for any x ∈ X , Φ( x ) as an operadic colimit, see [L.HA],3.1.1.20, 3.1.3.5, which we are now going to describe.To shorten the notation, we denote X = M ⊔ X and A = ( M , ¯ A ), both con-sidered as LM -operads. Similarly, P will denote the LM -operad ( M , P M ( A )). Wedenote F x = X × A A act /x and define the functor¯ φ x : F x → ( M , B )as the composition of the projection F x → X and the map X → ( M , B ) inducedby φ .The map (16) defines Φ( x ) as the operadic colimit of ¯ φ x .In 3.3.11 and 3.3.12 below we define a functor τ x : Tw ( X ) op → F x and provethat θ φx : Tw ( X ) op → ( M , B ) factors as θ φx = ¯ φ x ◦ τ x .Then we prove (see 3.3.13) that τ x is cofinal. This implies that the diagram θ a⊲x is an operadic colimit diagram. This, by 3.3.5, proves the assertion. (cid:3) Construction of τ x , a general idea. Here is how τ x looks like. An object f ∈ F x is given by a collection ( m , . . . , m n , z, β ) where m i ∈ M , z ∈ X and β : ( m , . . . , m n , z ) → x is an arrow in Q over an active arrow ( a n m ) → m in LM . Note that an arrow β can be equivalently described (by the Yoneda lemma,see [H.EY], Sect. 6) by an arrow ⊗ m i → A ( z, x ) in M .The functor τ x : Tw ( X ) op → F x will carry an arrow α : z → y to τ x ( α ) = ( A ( y, x ) , z, α ∗ : A ( y, x ) → A ( z, x )) . We present below a more accurate description of τ x .3.3.11. Construction of τ x . The functor τ x is defined by its components τ X : Tw ( X ) op → X and τ A : Tw ( X ) op → A act /x and an equivalence of their compositions Tw ( X ) op → A . The functor τ X is the composition(17) Tw ( X ) op → X × X op → M × X, where the second map carries ( z, y ) to ( A ( y, x ) , z ). Since ¯ A is a full subcategory of P := P M ( A ), A act /x is a full subcategory of P act /Y ( x ) .The functor τ A is therefore uniquely defined by a functor Tw ( X ) op → P act /Y ( x ) whose composition with the forgetful functor to P is given by (17).Since P is LM -monoidal, τ A is determined by a functor Tw ( X ) op → P /Y ( x ) assigning to α : z → y an arrow A ( y, x ) ⊗ Y ( z ) → Y ( x ) in P .The right fibration p : Tw ( X ) op → X op × X carrying α : z → y to the pair( y, z ), is classified by the functor h : X × X op → S given by the formula h ( z, y ) =Map X ( z, y ).One has a functor X op × X → P carrying ( y, z ) to A ( y, x ) ⊗ Y ( z ). In order to liftit to a functor Tw ( X ) op → P /Y ( x ) , it is enough to present a morphism of functorsMap X ( z, y ) → Map M ( A ( y, x ) , A ( z, x )). This latter comes from functoriality of A .We have the following.3.3.12. Lemma. θ φx = ¯ φ x ◦ τ x .Proof. The functor θ φx : Tw ( X ) op → ( M , B ) factors as θ φx = ¯ φ x ◦ τ x as ¯ φ x factorsthrough X , so that the composition ¯ φ x ◦ τ x can be expressed through τ X given bythe formula (17). (cid:3) Lemma.
The functor τ x : Tw ( X ) op → F x is cofinal.Proof. We use Quillen’s Theorem A, see [L.T], 4.1.3.1.We claim that the comma category(18) Tw ( X ) op × F x ( F x ) f/ has a terminal object for any f ∈ F x . In fact, let f = ( m , . . . , m n , z, u ) where u : ⊗ m i → A ( z, x ) is an arrow in M . The terminal object of (18) is given by τ x (id z ) = ( A ( z, x ) , z, id A ( z,x ) ). (cid:3) Enriched categories and Lurie enriched categories.
As an easy con-sequence of the above, we have the following.3.4.1.
Corollary.
The functor λ : Cat ( M ) → Cat
Lur ( M ) is fully faithful.Proof. Let A , A ′ ∈ Cat ( M ). The mapMap Cat ( M ) ( A , A ′ ) → Fun M ( A , P M ( A ′ )) eq is embedding, identifying the left-hand side with the subspace of the right-handside consisting of f : A → P M ( A ′ ) with representable images. In other words, itinduces an equivalenceMap Cat ( M ) ( A , A ′ ) → Fun M ( A , ¯ A ′ ) eq . According to the theorem, one has an equivalenceFun M ( A , P M ( A ′ )) → Fun
LMod w M ( A , P M ( A ′ )) which identifies Map Cat ( M ) ( A , A ′ ) with Map LMod w M ( ¯ A , ¯ A ′ ). (cid:3) Note that H. Heine has recently proven [HH] that λ is an equivalence.3.4.2. Remark.
Let B be weakly enriched over a monoidal category M . Here isa reasonable way to assign to B an M -enriched category A . Let X = B eq andlet i : X → B be the natural embedding. The LM -operad Quiv LM X ( M , B ) defines aweak enrichment of Fun( X, B ) over Quiv X ( M ).The M -enriched category A can now be defined as the endomorphism objectof i ∈ Fun( X, B ) (if it exists). Apparently, this is precisely how [HH] proves that λ is an equivalence.3.4.3. It is well-known that the forgetful functor p : Alg LM ( Cat ) → Alg
Ass ( Cat )is a cartesian fibration. In particular, given a monoidal functor f : A → B and a category X left-tensored over B , we have an LM -monoidal cartesian lifting f ! : ( A , X ) → ( B , X ) in Alg LM ( Cat ). Lemma.
The arrow f ! : ( A , X ) → ( B , X ) is also p ′ -cartesian, where p ′ : Op LM → Op Ass is the forgetful functor.Proof.
The embedding
Alg LM ( Cat ) → Op LM has a left adjoint functor denoted P LM (monoidal envelope functor). Similarly, P Ass : Op Ass → Alg
Ass ( Cat ) is left adjointto the embedding. The lemma immediately follows from the equivalence P LM ( O ) a = P Ass ( O a )valid for any O ∈ Op LM . (cid:3) Relative tensor product and duality
Introduction.
This section is mostly an exposition of (parts of) Lurie’s[L.HA], 4.6 and 3.1. In it we do the following.1. Starting with a monoidal category C with geometric realizations, we con-struct a 2-category BMOD ( C ) called the Morita 2-category of C , whoseobjects are associative algebras in C , so that the category of morphismsFun( A, B ) is the category of A - B -bimodules. Composition of arrows in BMOD ( C ) is given by the relative tensor product of bimodules. The de-scription of BMOD ( C ) is based on a study of an operad Tens over
Ass ⊗ ∆ op ,see [H.EY], 2.10.5 (3) describing collections of bimodules and multilinearmaps between them. More precisely, a category object in
Cat , that is, a simplicial object in
Cat satisfying theSegal condition. Lurie [L.HA] describes it as a family of operads based on ∆ op .
2. Duality for bimodules describes adjunction between morphisms in
BMOD ( C ).The special case, when the unit and the counit of the adjunction areequivalences, describes a Morita equivalence between the correspondingassociative algebras in C .3. A more general type of the relative tensor product of bimodules, withdifferent bimodules belonging to different categories, is described usingthe same operad Tens and the ones obtained from it by a base change.4.2.
Morita -category and the operad Tens . We now present a construc-tion of the Morita 2-category
BMOD ( C ) for a monoidal category C with geometricrealizations. We describe BMOD ( C ) as a Segal simplicial object in Cat , carrying [ n ]to a category BMOD n ( C ). The category BMOD n ( C ) can be described as the categoryof algebras in C over a certain planar operad (in sets).We will denote this operad Tens n . Algebras over it are collections ( A , . . . , A n )of associative algebras in C ,together with a collection of A i − - A i -bimodules M i for i = 1 , . . . , n . So, BMOD n ( C ) = Alg
Tens n ( C ). To define the simplicial object BMOD • ( C ), we have to provide a compatible collection of functors s ∗ : BMOD n ( C ) → BMOD m ( C ) defined for each s : [ m ] → [ n ] in ∆, together with the coherence data.For a map s : [ m ] → [ n ] the functor s ∗ : BMOD n ( C ) → BMOD m ( C ) comes from acorrespondence between the operads Tens m and Tens n .We will present an operad Tens s over Ass ⊗ [1] with fibers Tens m and Tens n over0 and 1 respectively. The functors i ∗ : Alg
Tens s ( C ) → BMOD m ( C ) and i ∗ : Alg
Tens s ( C ) → BMOD n ( C ) aredefined by the embeddings i : Tens m → Tens s and i : Tens n → Tens s .The map s ∗ : BMOD n ( C ) → BMOD m ( C ) will be defined as the composition i ∗ ◦ i where i is left adjoint to i ∗ .In order to describe the compatibility of s ∗ with respect to composition, wewill define an operad Tens over
Ass ⊗ ∆ op = Ass ⊗ × Com ∆ op . We will have Tens s = Tens × Ass ⊗ ∆op Ass ⊗ [1] where s : [1] → ∆ op defines the map Ass ⊗ [1] → Ass ⊗ ∆ op .4.2.1. The operad
Tens . Tens is the operad in sets governing the following col-lection of data.1. For each n ≥ A ,n , . . . , A n,n and A i − - A i bi-modules M i,n for i = 1 , . . . , n .2. For each map s : [ m ] → [ n ] in ∆ the collection of arrows:a. Morphism of algebras A s ( i ) ,n → A i,m , for i = 0 , . . . , m .b. Multilinear morphisms (see remark below) M s ( i − ,n × . . . × M s ( i ) ,n → M i,m of A s ( i − ,n - A s ( i ) ,n -bimodules. Ass ⊗ denotes the operad governing associative algebras.
3. The collections of arrows defined in (2) for each s : [ m ] → [ n ] compose inan obvious way. Remark.
Multilinearity in the last sentence means that, in case s ( i − < s ( i ),the map is compatible with the actions of all intermidiate A j,n , j = s ( i −
1) +2 , . . . , s ( i ) −
1; it means nothing if s ( i ) = s ( i −
1) + 1; and it means an A s ( i ) ,n -bimodule map A s ( i ) ,n → M i,m if s ( i −
1) = s ( i ).4.2.2. The map to
Ass ⊗ ∆ op . An Ass ⊗ ∆ op -algebra in an Ass ⊗ -operad C is givenby a functor A : ∆ op → Alg
Ass ( C ). This functor defines a canonical Tens -algebra defined by the formulas A i,n = A ([ n ]) = M i,n . This gives a functor π ∗ : Alg
Ass ∆op ( C ) → Alg
Tens ( C ) realized as the inverse image with respect to themap π : Tens → Ass ⊗ ∆ op . φ : S → ∆ op one defines Tens S (or Tens φ ) as Com S × Com ∆op
Tens .One defines p : BMOD ( C ) → ∆ op as a category over ∆ op representing the functor(19) Fun ∆ op ( B, BMOD ( C )) = Alg
Tens B ( C ) . In the case when C has geometric realizations and a monoidal structure pre-serving geometric realizations, BMOD ( C ) is a cocartesian fibration over ∆ op , so itdefines a simplicial object BMOD • ( C ) in Cat , see [L.HA], 4.4.3.12. It satisfies theSegal condition by [L.HA], 4.4.3.11.4.2.4.
Remark.
Note that
BMOD ( C ) is not complete. The zero component BMOD ( C )is the category of algebras in C which is not a space. An equivalence defined by BMOD ( C ) is Morita equivalence which is not equivalence in BMOD ( C ).4.3. Duality.
In this subsection we apply the general notion of adjunction in a2-category to the Morita 2-category described in the previous subsection.4.3.1.
Definition. (see [L.HA], 4.6.2.3) Let C be a monoidal category with geo-metric realizations. Let A, B be two associative algebras in C , M ∈ A BMod B ( C )and N ∈ B BMod A ( C ). A map c : B → N ⊗ A M is said to exhibit N as left dualof M (or M as a right dual of N ) if there exists e : M ⊗ B N → A in A BMod A ( C )such that the compositions M = M ⊗ B B id M ⊗ c → M ⊗ B N ⊗ A M e ⊗ id M → M and N = B ⊗ B N c ⊗ id N → N ⊗ A M ⊗ B N id N ⊗ e → N are equivalent to id M and id N , respectively.A dual pair of bimodules M ∈ A BMod B ( C ) and N ∈ B BMod A ( C ) determines anadjunction (see [L.HA], 4.6.2.1)(20) F : LMod B ( C ) −→←− LMod A ( C ) : G given by the formulas F ( X ) = M ⊗ B X and G ( Y ) = N ⊗ A Y , for any left C -tensored category M with geometric realizations. This adjunction deserves thename Morita adjunction .A Morita adjunction is called a Morita equivalence if the arrows c and e areequivalences.Two properties of Morita adjunction are listed below. The first one, Propo-sition 4.3.2, describes a good behavior of Morita adjunctions under composi-tion. The second one, Proposition 4.3.3, claims that the left dualizability of M ∈ A BMod B ( C ) is independent of the algebra B .4.3.2. Proposition. (see [L.HA] , 4.6.2.6) let C be a monoidal category with geo-metric realizations, A, B, C three associative algebras in C . If c : B → N ⊗ A M exhibits N as a left dual to M ∈ A BMod B and c ′ : C → N ′ ⊗ B M ′ exhibits N ′ as aleft dual to M ′ ∈ B BMod C then the composition C c ′ → N ′ ⊗ B M ′ = N ′ ⊗ B B ⊗ B M ′ c → N ′ ⊗ B N ⊗ A M ⊗ B M ′ exhibits N ′ ⊗ B N as a left dual to M ⊗ B M ′ . Proposition. (see [L.HA] , 4.6.2.12, 4.6.2.13) Let C be as above. A bimod-ule M ∈ A BMod B ( C ) is left dualizable if and only if its image M ′ in LMod A ( C ) = A BMod ( C ) is left dualizable. Moreover, if N ∈ B BMod A ( C ) is left dualto M , its image in RMod A ( C ) is a left dual of M ′ ∈ LMod A ( C ) . (cid:3) Remark.
In the classical context of associative rings, an (
A, B )-bimodule N is right-dualizable iff it is finitely generated projective as a right A -module.This property is independent of B and right dualizability of N is sufficient to havean adjunction between the categories of left A and B - modules. This adjunctionis an equivalence, for B = End A ( N ), if N is a generator in RMod A . It wouldbe very nice to describe in our general context a condition on a right dualizablemodule N ∈ RMod A leading to Morita equivalence.4.3.5. We can fix a right-dualizable A -module N ∈ RMod A ( C ) and try to recon-struct a would-be Morita equivalence.Let M ∈ LMod A ( C ) be the right dual of N .The category RMod A ( C ) is left-tensored over C . So, given N ∈ RMod A ( C ), onecan define an endomorphism object End A ( N ) which, if it exists, acquires anassociative algebra structure. Since N is right dualizable, this object does exist,as one has a canonical equivalence(21) Map C ( X, N ⊗ A M ) = Map RMod A ( C ) ( X ⊗ N, N )by [L.HA], 4.6.2.1 (3), so that End A ( N ) = N ⊗ A M as an object of C . Corollary.
Let C be a monoidal category with geometric realizations, A an associative algebra in C , N ∈ RMod A ( C ) a right dualizable A -module. Then M ∈ LMod A ( C ) , the right dual of N , has a canonical structure of A - End A ( N ) -bimodule and the pair ( M, N ) defines a Morita adjunction F : LMod
End A ( N ) ( C ) −→←− LMod A ( C ) , with F ( X ) = M ⊗ End A ( N ) X and G ( Y ) = N ⊗ A Y , for which the coevaluation c : End A ( N ) → N ⊗ A M is an equivalence.Proof. See [L.HA], 4.6.2.1 (2). (cid:3)
Note that this construction produces the A - A -bimodule evaluation map(22) e : M ⊗ End A ( N ) N → A. Remark.
The algebra B = End A ( N ) can be also described in terms ofthe left A -module M . In fact, the category LMod A ( C ) is right-tensored over C , soit is left-tensored over the reversed monoidal category C rev . The endomorphismobject of M ∈ LMod A ( C ) in C rev exists, and it coincides with the algebra B op .4.4. Relative tensor product.
The relative tensor product of bimodules withvalues in a monoidal category C is encoded in the composition of arrows of BMOD ( C ). There exists a slightly more general relative tensor product, for thebimodules having values in different categories.Let now C ∈ Alg
Tens S ( Cat K ) where K is a collection of categories containing∆ op .We wish to study tensor product of bimodules with values in C .We define, slightly generalizing 4.2.3, p : BMOD φ ( C ) → S as a category over S representing the functorFun S ( B, BMOD φ ( C )) = Alg
Tens B ( C ) . One has4.4.1.
Proposition. The map p : BMOD φ ( C ) → S is a cocartesian fibra-tion. An arrow ˜ α in BMOD φ ( C ) over α : x → y in S is p -cocartesian iff thecorresponding F α ∈ Alg
Tens φ ( α ) / Tens S ( C ) is an operadic left Kan extensionof its restriction F φ ( x ) : Tens φ ( x ) → C . Let f : C → D be a Tens φ -monoidal functor preserving geometric realiza-tions. Then the induced map BMOD φ ( f ) : BMOD φ ( C ) → BMOD φ ( D ) preservescocartesian arrows.Proof. The first two claims are just [L.HA], Corollary 4.4.3.2, with O = Tens S .The condition (*) is fulfilled as C is Tens S -monoidal category with geometric realizations, commuting with the tensor product. Claim 3 follows from Claim2. (cid:3) ≻ : [1] → ∆ op be defined by the arrow ∂ : [1] → [2] in ∆. We have then Tens ≻ = Com [1] × Com ∆op
Tens .One has natural embeddings i : Tens → Tens ≻ and i : Tens → Tens ≻ induced by the embedding of the ends { } → [1] and { } → [1].Note that Tens = BM ⊗ and Tens = BM ⊗ ⊔ Ass ⊗ BM ⊗ .4.4.3. Let C be a Tens ≻ -monoidal category. Up to equivalence, C is uniquelydescribed by a collection of five monoidal categories C a , C b , C c , C a ′ , C c ′ , three bi-module categories C m ∈ C a BMod C b ( Cat ), C n ∈ C b BMod C c ( Cat ), C k ∈ C a ′ BMod C c ′ ( Cat ),monoidal functors φ a : C a → C a ′ and φ c : C c → C c ′ , and a C b -bilinear functor C m × C n → C k of C a - C c -bimodule categories.The embedding i : Tens → Tens ≻ induces i ∗ : Alg
Tens ≻ ( C ) → Alg
Tens / Tens ≻ ( C ) . The relative tensor product functor(23) RT : Alg
Tens / Tens ≻ ( C ) → Alg
Tens ≻ ( C )is defined as the functor left adjoint to i ∗ .The functor RT exists if C ∈ Alg
Tens ≻ ( Cat K ) where K contains ∆ op .It makes sense to fix associative algebras A ∈ Alg
Ass ( C a ), B ∈ Alg
Ass ( C b ), C ∈ Alg
Ass ( C c ), and restrict (23) to Tens -algebras in C having algebra-components A, B and C . If A ′ = φ a ( A ) ∈ Alg
Ass ( C a ′ ) and C ′ = φ c ( C ) ∈ Alg
Ass ( C c ′ ), thisgives(24) RT A,B,C : A BMod B ( C m ) × B BMod C ( C n ) → A ′ BMod C ′ ( C k ) . Two-sided bar construction.
The following explicit formula for the calcu-lation of relative tensor product explains why does it exist for categories withgeometric realizations.Recall that
Tens governs 5-tuples of objects, ( A, M, B, N, C ), where
A, B, C are associative monoids, M is an A - B -bimodule and N is a B - C -bimodule. Wedenote the colors of Tens by a, m, b, n, c . The operad Tens has colors a ′ , k, c ′ .Define a functor u : ∆ op → Tens carrying [ n ] to mb n c ∈ Tens , where theaction of u on the arrows is defined as follows. • Faces correspond to the action maps mb → m , bb → b or bn → n . • Degeneracies correspond to the unit maps 1 → b . We extend the map u : ∆ op → Tens to u + : ∆ op+ → Tens ≻ carrying theterminal object of ∆ op+ to k ∈ Tens .Let q : C → Tens ≻ present a Tens ≻ -monoidal category. The map Fun(∆ op , q ) :Fun(∆ op , C ) → Fun(∆ op , Tens ≻ ) is a cocartesian fibration. The functor u + definesan arrow β : u → u ∗ in Fun(∆ op , Tens ≻ ), u ∗ being the constant functor with thevalue k ∈ Tens ≻ . Therefore, any φ ∈ Alg
Tens / Tens ≻ ( C ) gives rise to a unique lift β ! : φ ◦ u → X , where X is a simplicial object in C k . We will denote X = −−→ φ ◦ u and call it the two-sided bar construction, −−→ φ ◦ u = Bar ( φ ).The following explicit description of relative tensor product is a reformulationof [L.HA], 4.4.2.8.4.4.5. Proposition.
Let C be a Tens ≻ -monoidal category with geometric realiza-tions and the tensor structure commuting with the geometric realizations, and let q : C → Tens ≻ be the corresponding cocartesian fibration. Given a commutativediagram (25) Tens φ / / (cid:15) (cid:15) C q (cid:15) (cid:15) Tens ≻ Φ tttttt Tens ≻ of marked categories, with φ corresponding to a pair of bimodules M ∈ A BMod B and N ∈ B BMod C with values in C . Then there exists Φ presenting a relativetensor product of M and N . Vice versa, any extension Φ of φ presents a relativetensor product of M with N if and only if the following conditions are fulfilled. • Φ carries the maps A → A and A → A to q -cocartesian arrows in C . • The functor Φ induces an equivalence | Bar ( φ ) | → Φ( k ) . Associativity.
To formulate associativity, we need to use Proposition 4.4.1applied to the family φ : S → ∆ op defined by the commutative square(26) [1] ∂ (cid:15) (cid:15) ∂ / / [2] ∂ (cid:15) (cid:15) [2] ∂ / / [3]in ∆, and a Tens φ -monoidal category C with geometric realizations, see [L.HA],4.4.3.14.4.5. Variants.
The tensor product of bimodules (24) commutes with the functorforgetting the left A -module structure and the right C -module structure. We would like to formulate this observation as follows. Let
Ten ≻ be the fullsuboperad of Tens ≻ spanned by the colors a, b, a ′ , m, n, k ∈ [ Tens ≻ ]. There is anobvious embedding i : Ten ≻ → Tens ≻ and the functor i ∗ : Alg
Tens ≻ → Alg
Ten ≻ forgets the right module structure on the bimodules described by the colors n and k .Similarly, it makes sense to describe a yet smaller suboperad En ≻ spannedby the colors b, m, n, k ∈ [ Tens ≻ ]. We denote j : En ≻ → Tens ≻ the obviousembedding that forgets both the right module structure on bimodules describedby n, k and the left module structure on bimodules described by m, k .We define Ten and En as for Tens ≻ ; this yields the functors i ∗ and the leftadjoints RT exactly as for Tens ≻ -monoidal categories. One has4.5.1. Proposition.
The forgetful functors i ∗ and j ∗ commute with the relativetensor product.Proof. The commutative square(27)
Alg
Tens ≻ ( C ) i ∗ (cid:15) (cid:15) i ∗ / / Alg
Tens / Tens ≻ ( C ) i ∗ (cid:15) (cid:15) Alg
Ten ≻ / Tens ≻ ( C ) i ∗ / / Alg
Ten / Tens ≻ ( C )defines a morphism of functors RT ◦ i ∗ → i ∗ ◦ RT . To prove that this functor is an equivalence, we use the description of RT in termsof the two-sided bar construction. The functor u + : ∆ op → Tens ≻ factors through i : Ten ≻ → Tens ≻ , so the bar construction used to calculate RT as a colimit, isthe same for both setups.The version for j : En ≻ → Tens ≻ is proven in the same way. (cid:3) Reduction. An A - B -bimodule in C can be equivalently described as a left A -module in the category RMod B ( C ).We present below a similar transformation of Tens ≻ -monoidal categories com-patible with the formation of the weighted colimit.The construction is based on the notion of bilinear map of operads and theirtensor product as presented in [H.EY], 2.10.4.6.1. We define a map p : Tens ≻ → BM ⊗ as the obvious map carrying thecolors a, a ′ , b, m to a ∈ [ BM ], n, k to m ∈ [ BM ] and c, c ′ to b ∈ [ BM ]. We have Ten ≻ = LM ⊗ × BM ⊗ Tens ≻ .One has a standard bilinear map Pr : LM ⊗ × RM ⊗ → BM ⊗ defined in [L.HA],4.3.2.1 and [H.EY], 2.10.7. There is a lifting of Pr to a bilinear map(28) µ : Ten ≻ × RM ⊗ → Tens ≻ uniquely defined by its action on the colors. • µ ( ∗ , m ) = ∗ where ∗ is any color of Ten ≻ . • µ ( n, b ) = c, µ ( k, b ) = c ′ .4.6.2. Let C be a Tens ≻ -operad. Following a general pattern [H.EY], 2.10.1, wedefine a Ten ≻ -operad C red := Alg µ RM / Tens ≻ ( C ) as the one representing the functor K ∈ Cat / Ten ≻ Map
Cat + / Tens ♮ ≻ ( K ♭ × RM ♮ , C ♮ ) . We call C red the reduction of C .Here is a more convenient description of C red in the case when C is a Tens ≻ -monoidal category. In this case C is classifined by a lax cartesian structure e C : Tens ≻ → Cat . Composing it with µ , we get a functor e C ◦ µ : Ten ≻ × RM ⊗ → Cat , defining(29) C : Ten ≻ → Fun lax ( RM ⊗ , Cat ) , that is, a functor with the values in RM -monoidal categories. Composing it withthe functor Alg RM , we get a functor e C red : Ten ≻ → Cat classifying C red .Here is a more detailed information about the functor C . For x = a, a ′ , m, b , C ( x ) is the RM -monoidal category ( C x , [0]) describing the trivial action of the triv-ial monoidal category on C x . Thus, one has C red x = C x for these values of x . Fur-thermore, C ( n ) = ( C n , C c ) ⊗ and C ( k ) = ( C n , C c ′ ) ⊗ , so that C red n = Alg RM ( C n , C c )and C red k = Alg RM ( C k , C c ′ ).The standard embedding i : Ten ≻ → Tens ≻ identifies the m -component of C with i ∗ ( C ).This defines a functor G : C red → i ∗ ( C ) forgetting the right module structureon the components C red n , C red k .The restriction with respect to µ (28) defines a natural map(30) θ : Alg
Tens ≻ ( C ) → Alg
Ten ≻ ( C red ) , whose composition with the map induced by G is the obvious restriction(31) Alg
Tens ≻ ( C ) → Alg
Ten ≻ / Tens ≻ ( C ) . We believe that the map (30) is an equivalence, that is that µ presents Tens ≻ asa tensor product.We will actually verify a somewhat weaker statement Proposition 4.6.4 thatwill be used in Section 6. Lemma.
The bilinear map µ : Ten × RM ⊗ → Tens obtained by restriction of µ , presents Tens as a tensor product of Ten with BM ⊗ .Proof. We compose µ with the standard strong approximations RM → RM ⊗ , ten → Ten as described in [H.EY], 2.9, with ten = BM ⊔ Ass LM . We get abilinear map ten × RM → Tens that is easily seen to be a strong approxima-tion. (cid:3) Proposition.
Let C be a Tens ≻ -monoidal category with colimits. Then µ induces a commutative diagram (32) Alg
Tens / Tens ≻ ( C ) RT / / θ ∼ (cid:15) (cid:15) Alg
Tens ≻ ( C ) θ (cid:15) (cid:15) Alg
Ten / Ten ≻ ( C red ) RT red / / Alg
Ten ≻ ( C red ) , where RT red is the relative tensor product defined for the Ten ≻ -monoidal category C red .Proof. If C is a Tens ≻ -monoidal category with colimits, C red is a Ten ≻ -monoidalcategory with colimits. This implies that RT red is defined as the functor left ad-joint to the restriction i red ∗ : Alg
Ten ≻ ( C red ) → Alg
Ten / Ten ≻ ( C red ). The equivalence θ is defined by the universal property of C red = Alg µ RM / Tens ≻ ( C ) and Lemma 4.6.3.The equivalence θ ◦ i ∗ = i red ∗ ◦ θ induces a morphism of functors(33) RT red ◦ θ → θ ◦ RT . We claim that this morphism is an equivalence.Let φ ∈ Alg
Tens / Tens ≻ ( C ) be given by a pair of bimodules M ∈ A BMod B ( C m ) , N ∈ B BMod C ( C n ) and let Φ = RT ( φ ). By 4.4.5, the composition Φ ◦ u + : ∆ op+ → C is arelative colimit diagram. This is equivalent to saying that the cocartesian shift −−−−→ Φ ◦ u + : ∆ op+ → C k is a colimit diagram.The map u + : ∆ op+ → Tens ≻ factors through i : Ten ≻ → Tens ≻ . By 4.4.5,the claim of Proposition 4.6.4 will be proven once we verify that θ (Φ) ◦ u + isa relative colimit diagram in C red , or, equivalently, that the cocartesian shift −−−−−−→ θ (Φ) ◦ u + : ∆ op+ → C red k is a colimit diagram.The composition G ◦ −−−−−−→ θ (Φ) ◦ u + : ∆ op+ → C k is a colimit diagram as the com-position G ◦ θ is the restriction (31). According to [L.HA], 3.2.3.1, G createscolimits. This proves the proposition. (cid:3) Remark.
Let C ∈ Alg
Ass ( C c ) be the c -component of φ . We define a Ten ≻ -monoidal subcategory C red C of C red as follows. The restriction Alg RM → Alg
Ass / RM applied to the functor C (29) yields a morphism of functors e C red → Alg
Ass / RM ◦ C . The
Ten ≻ -monoidal subcategory C red C is defined as the fiber of this functor at theobject of Alg
Ass / RM ◦ C determined by the cocartesian arrow C → φ c ( C ), in thenotation of 4.4.3. The functor θ (Φ) ◦ u + : ∆ op+ → C red canonically factors through C red C . By [L.HA], 3.2.3.1, the functor ∆ op+ → C red C so defined is also a relativecolimit diagram.5. Bar resolutions for enriched presheaves
Let C = ( C a , C m ) be an LM -monoidal category with colimits, let A be an asso-ciative algebra in C a and M be an A -module in C m .A bar resolution of M is an augmented simplicial object Bar • ( A, M ) → M in LMod A ( C m ), which is a colimit diagram, such that Bar n ( A, M ) are free A -modules.The pair ( A, M ) is given by a map of operads γ : LM → C , whose compositionwith a certain functor u + : ∆ op+ → LM gives an operadic colimit diagram, whosecocartesian shift 3.3.3 is equivalent to G ( Bar ( A, M )), where G : LMod A ( C m ) → C m is the forgetful functor.In Subsection 5.1 we explain how to reconstruct Bar ( A, M ) from −−−→ γ ◦ u + , interms an action of the monad corresponding to A .We use a similar reasoning to describe the bar resolution for enriched presheavesin Subsection 5.2. Here we are able to say more than for general modules. Ingeneral, we have no chance to introduce the monad action on γ ◦ u + instead of −−−→ γ ◦ u + , as the category LMod A ( C m ) is not left-tensored over C a .As for the enriched presheaves that are defined as LMod A op (Fun( X op , M )), theyhave a left M -module structure. This allows us to encode the bar resolution fora presheaf f ∈ P M ( A ) into an operadic colimit diagram(34) K ⊲ → ( M , P M ( A )) ⊗ , with an appropriate choice of a category K , see 5.2.6 for the precise formulation.This is the main result of this section formulated in 5.3.1. We later use it toprove an important result 6.4.2.5.1. Bar resolution of a module.
Let O be an LM -operad and let γ : LM → O be an LM -algebra in O defined by a pair ( A, M ) where A is an associative algebrain the planar operad O a and M ∈ O m is a left A -module.A very special case of the two-sided bar construction 4.4.4 gives the followingsimplicial resolution of a module.We define the functor u + : ∆ op+ → LM by the formula u + ([ n ]) = a n +1 m, n ≥ − , with the face maps defined by the multiplication in a and by its action on m .Note that the image of u + belongs to LM act , the active part of LM .5.1.1. Lemma.
The composition γ ◦ u + : ∆ op+ → O is an operadic colimit diagram. In the case when O is a monoidal LM -categorywith geometric realizations, it induces an equivalence | Bar ( A, M ) | → M in O m ,where Bar ( A, M ) = −−→ γ ◦ u is the simplicial object in O m .Proof. This is a direct consequence of [L.HA], 4.4.2.5, 4.4.2.8, applied to thetensor product A ⊗ A M = M . (cid:3) Bar ( A, M ) = −−→ γ ◦ u has a canonicallifting to an augmented simplicial object in LMod A ( O m ). In 5.1.2—5.1.5 we showhow this resolution can be described in terms of our construction.Let us note a few easy facts about monadic adjunction. Recall that any adjointpair of functors F : C −→←− D : G endows the composition A := G ◦ F with a structure of a monad (that is, anassociative algebra object in the endomorphism category) on C and a naturalfunctor(35) D → Mod A ( C )is defined. An adjoint pair is called monadic if (35) is an equivalence. The Barr-Beck theorem [L.HA], 4.7.3.5, provides a neccesary and sufficient condition foran adjunction to be monadic.5.1.3. Lemma.
Let F : C → D : G be a monadic adjunction. Then for any K ∈ Cat the induced adjunction F ′ : Fun( K, C ) −→←− Fun( K, D ) : G ′ is also monadic.Proof. The conditions of the Barr-Beck theorem are obviously verified for G ′ asthey are verified for G . (cid:3) Corollary.
Let F : C −→←− D : G be a monadic pair with A = G ◦ F therespective monad, and let f : K → C be a functor. There is an equivalencebetween decompositions f = G ◦ f ′ , f ′ : K → D , and A -module structures on f .Proof. This is just a reformulation of 5.1.3. (cid:3) −−−→ γ ◦ u + has a canonical lifting to anaugmented simplicial object in LMod A ( O m ).We have a monadic pair F : O m → LMod A ( O m ) : G , so, in order to lift thefunctor −−−→ γ ◦ u + , we have to present an action of A = G ◦ F on it.The left O a -module structure on Fun(∆ op+ , O m ) is detailed in 5.1.6 below. Theaction of A = G ◦ F on the functor −−−→ γ ◦ u + is determined by a map of operadsΓ : LM → ( O a , Fun(∆ op+ , O m )) whose restriction to Ass ⊂ LM coincides with A andsatisfying Γ( m ) = −−−→ γ ◦ u + .5.1.6. K LM . An LM -monoidal category O = ( O a , O m ) encodes an action of amonoidal category O a on a category O m , or, in other words, a monoidal functor O a → End
Cat ( O m ).Fix K ∈ Cat . The functor C Fun(
K, C ) defines a monoidal functorEnd( C ) → End(Fun(
K, C )). Thus, any LM -monoidal category O defines a monoidalfunctor O a → End( O m ) → End(Fun( K, O m )), that is an LM -monoidal category( O a , Fun( K, O m )).We wish to present two more constructions of the LM -monoidal category ( O a , Fun( K, O m )).1. Applying the functor Fun LM ( K, ) defined in [H.EY], 6.1.6, we get an LM -monoidal category with the a -component Fun LM ( K, O ) a = Fun( K, O a ) andFun LM ( K, O ) m = Fun( K, O m ). The forgetful functor Alg LM ( Cat ) → Alg
Ass ( Cat )being a cartesian fibration, the LM -monoidal category ( O a , Fun( K, O m )) describedabove, is equivalent to i ∗ (Fun LM ( K, O )), where i : O a → Fun( K, O a ) is induced bythe map K → [0].2. Let K ∈ Cat . Denote by K LM the LM -monoidal category describing the actionof the terminal monoidal category [0] on K . We claim that Funop LM ( K LM , O ) givesyet another presentation of ( O a , Fun( K, O m )).We start with the map of cocartesian fibrations over LM (36) q : K × LM → K LM constructed as an obvious natural transformation of the classifying functors LM → Cat . The map q induces(37) K × Fun LM ( K LM , O ) → K LM × LM Fun LM ( K LM , O ) → O , and, therefore, an LM -operad map Q : Fun LM ( K LM , O ) → Fun LM ( K, O ) . The monoidal component of Q is the monoidal functor i : O a → Fun( K, O a )mentioned above. Therefore, the map Q factors through Q ′ : Fun LM ( K LM , O ) → i ∗ (Fun LM ( K, O )) . One can easily see that Q ′ is an equivalence. LM → Funop LM ((∆ op+ ) LM , O ), that is, an LM -operad map (∆ op+ ) LM → O , whose m -component is γ ◦ u + : ∆ op+ → O m .Let P LM be the monoidal envelope of LM . The monoidal part of P LM has objects a n , n ≥
0, and the arrows are generated by the unit → a , the product aa → a ,subject to the standard identities. The module part of P LM consists of the objects a n m with the obvious action of the monoidal part.One has an LM -monoidal functor i : (∆ op+ ) LM → P LM that is the unit on themonoidal part, and u + : ∆ op+ → P LM carrying [ n −
1] to a n m .Since O is LM -monoidal, the map γ : LM → O uniquely extends to Γ : P LM → O .The composition with i yields the required LM -operad map (∆ op+ ) LM → O .5.2. Bar resolution for presheaves.
We apply the reasoning of the previoussubsection to enriched presheaves. Given a monoidal category M and an M -enriched category A with the space of objects X , we want to describe a barresolution for f ∈ P M ( A ) = Fun M rev ( A op , M ).The pair γ = ( A op , f ) is given by the functor(38) γ : LM → Quiv LM X op ( M rev , M ) . The information we need about the bar resolution of f is contained in the com-position γ ◦ u + : ∆ op+ → LM → Quiv LM X op ( M rev , M ) . Since
Quiv LM X op ( M rev , M ) = Funop LM ( LM X op , ( M rev , M )) , the functor γ defines (and is uniquely defined by)(39) γ ′ : LM X op → ( M rev , M ) , where the LM -operad LM X op is the one discussed in 2.2.1.5.2.1. We will need to know more about the LM -operad LM X and its base change LM ◦ X := ∆ op+ × LM LM X .The explicit description of LM X is given in [H.EY], 3.2. According to thisdescription, LM X is presented by a functor(∆ / LM ) op → S carrying σ : [ n ] → LM to Map( F ( σ ) , X ) where F : ∆ / LM → Cat has values inconventional categories described by certain diagrams, see [H.EY], 3.2, especiallythe diagrams (51), (55), (60).The base change LM ◦ X = ∆ op+ × LM LM X is described by the collection of F ( σ ) for σ : [ n ] → LM that factor through u + : ∆ op+ → LM .The categories F ( σ ) for these values of σ canonically decompose F ( σ ) = F − ( σ ) ⊔ [ n ], where [ n ] appears as the rightmost component of F ( σ ) in the graphic presentation [H.EY], (55), (60), the component containing the vertex y , see op.cit. , diagram (51).This means that LM ◦ X = LM ◦ , − X × X , so that the canonical projection LM ◦ X → ∆ op+ factors through the projection to LM ◦ , − X .5.2.2. The restriction of (39) to LM ◦ X op gives therefore(40) γ ◦ : LM ◦ , − X op × X op → u ∗ + ( M rev , M ) , where u ∗ + ( M rev , M ) is the base change of ( M rev , M ) considered as a category over LM .We compose γ ◦ with the equivalence op : u ∗ + ( M rev , M ) → u ∗ + ( M , M ) .We get a functor γ ◦ , − : LM ◦ , − X op → Fun( X op , u ∗ + ( M , M )) . Since the projection LM ◦ X op → ∆ op+ factors through LM ◦ , − X op → LM , the map γ ◦ , − defines a map(41) γ ◦ , − : LM ◦ , − X op → u ∗ + (Fun LM ( X op , ( M , M ))) , where we use the notation of [H.EY], 6.1.6 to define the target of the map.5.2.3. The right-hand side of (41) has, as Ass -component, the monoidal categoryFun( X op , M ). There is a monoidal functor c : M → Fun( X op , M )carrying m ∈ M to the corresponding constant functor. The arrow(42) c ! : ( M , Fun( X op , M )) → Fun LM ( X op , ( M , M )) . induced by c is cartesian in Op LM by Lemma 3.4.3. Since the Ass -component of γ ◦ , − factors through the map Ass X op → M defining A op , the map (41) factorsthrough c ! giving the map that we denote by the same letter(43) γ ◦ , − : LM ◦ , − X op → u ∗ + ( M , Fun( X op , M )) . LM ◦ , − X op has one object over the terminal object [ −
1] of ∆ op+ .We will denote this object by ∗ (note that it is not a terminal object). The functor γ ◦ , − applied to ∗ gives G ( f ) ∈ Fun( X op , M ) (once more, G is the forgetful functor P M ( A ) → Fun( X op , M )).An object of LM ◦ , − X op over [ n − n ≥
1, is given by a collection of objects( y, x n , y n , . . . , x ) of X .The functor γ ◦ , − carries ( y, x n , y n , . . . , x ) to( f ( y ) , A ( y n , x n ) , . . . , A ( y , x ) , Y ( x )) ∈ M n × Fun( X op , M ) , where Y is the Yoneda embedding. Note that this is an equivalence over op : ∆ op+ → ∆ op+ . γ ◦ , − do with the arrows.An arrow α in LM ◦ , − X op from ( y, x n , y n , . . . , x ) to ∗ is given by a collection ofmaps α i : x i → y i +1 (or α n : x n → y ).The functor γ ◦ , − carries α to the arrow( f ( y ) , A ( y n , x n ) , . . . , A ( y , x ) , Y ( x )) → f defined by the map f ( y ) ⊗ A ( y n , x n ) ⊗ . . . ⊗ A ( y , x ) → f ( x ) , defined by the A op -module structure on P M ( A ) ∋ f .We are now ready to formulate the enriched presheaf analog of Lemma 5.1.1.5.2.6. Lemma.
The functor (44) ( LM ◦ , − X op ) / ∗ → ( M , Fun( X op , M )) induced by γ ◦ , − , is an operadic colimit diagram.Proof. Note that the source of the functor (44) has form K ⊲ , where K = ∆ op × ∆ op+ ( LM ◦ , − X op ) / ∗ . The evaluation map e x : Fun( X op , M ) → M , x ∈ X op , commutes withthe left M -module structure, so, in order to prove the lemma, it is sufficient toverify that for any x ∈ X op the composition of (44) with the evaluation map isan operadic colimit diagram K ⊲ → ( M , M ).We know that the composition γ ◦ u + : ∆ op+ → Quiv LM X op ( M rev , M ) is an operadiccolimit diagram. Since M is a monoidal category with colimits, the restrictionfunctor(45) Fun LM (∆ op+ , Quiv LM X op ( M rev , M )) → Fun LM (∆ op , Quiv LM X op ( M rev , M ))has a left adjoint carrying γ ◦ u to γ ◦ u + . Since, by adjunction, the arrow (45)can be rewritten as(46) Fun LM ( LM ◦ X op , ( M , M )) → Fun LM (∆ op × ∆ op+ LM ◦ X op , ( M , M )) , this implies that the functor γ ◦ is an operadic left Kan extension of its restrictionto ∆ op × ∆ op+ LM ◦ X op . This easily implies the claim. (cid:3) M , P M ( A )) G → ( M , Fun( X op , M )) . This implies the main result of this section, Proposition 5.3.1.To deduce the factorization, we need, according to 5.1.4, to present the actionof the monad G ◦ F defined by A op on the functor (43). As a first step, wewill describe a left Quiv X op ( M rev )-tensored structure on the target of (43), thecategory u ∗ + ( M , Fun( X op M )). Here is a general construction in the context of BM -monoidal categories.5.2.8. Condensation.
Let O = ( O a , O m , O b ) be a BM -monoidal category. We canlook at O m as an object of RMod O b ( LMod O a ( Cat )). Its bar construction gives anaugmented simplicial object in
LMod O a ( Cat ) that classifies a cocartesian fibrationover ∆ op+ . The total category of this cocartesian fibration in v ∗ + ( O m , O b ) wherethe functor v + : ∆ op → RM is defined as the composition of u + with op : LM → RM .The resulting LM -monoidal category will be called the condensation of O anddenoted by cond ( O ).Applying the condensation functor to O = Quiv BM X op ( M rev ), we get an LM -monoidal category whose monoidal part is Quiv X op ( M rev ) and whose left-tensoredpart is v ∗ + (Fun( X op , M ) , M rev ) = u ∗ + ( M, Fun( X op , M )).5.2.9. Therefore, in order to show that (43) canonically factors through (47),we have to extend γ ◦ , − to a map of operads(48) Γ : ( LM ◦ , − X op ) LM → cond ( Quiv BM X op ( M rev )) . LM -operads by simplicialspaces over LM , or, equivalently, by presheaves on ∆ / LM . We will describe thefunctors corresponding to the source and the target of Γ, and we will show that(38) allows one to construct a map of these presheaves.We will now describe the source of (48). First of all, LM ◦ , − X op is a category over∆ op+ , so the space of its k -simplices decomposes( LM ◦ , − X op ) k = a τ :[ k ] → ∆ op+ LM ◦ , − X op ( u + ◦ τ ) , where LM ◦ , − X op ( u + ◦ τ ) = Map( F − ( u + ◦ τ ) , X op ), see the notation of 5.2.1.Let σ : [ n ] → LM be an object of ∆ / LM presented by a sequence σ : a d m → . . . a d k m → a d k +1 → . . . a d n . We will assume k = − σ factors through Ass → LM .Then the source of (48) is described by the following formula.(49) ( LM ◦ , − X op ) LM ( σ ) = ( [0] , if k = − , ( LM ◦ , − X op ) k = ` τ :[ k ] → ∆ op+ LM ◦ , − X op ( u + ◦ τ ) otherwise . The description of the target of (48) will be presented in 5.2.14, after a certaindigression. Internal mapping operad, reformulated.
We need some detail on internaloperad objects, [H.EY], 2.8.The direct product in the category P ( C ) of presheaves has a right adjointassigning to a pair F, G ∈ P ( C ) a presheaf Fun P ( C ) ( F, G ) whose value at c ∈ C is calculated as the limit(50) Fun P ( C ) ( F, G )( c ) = lim a → b → c Map( F ( b ) , G ( a )) . Fix a category B and let us look for a similar description of the internal Homin Cat /B . The latter is a full subcategory of P (∆ /B ), so we can try to use theformula (50) with C = ∆ /B . Given F, G ∈ Cat /B , we are looking for an objectFun Cat /B ( F, G ) furnishing an equivalence(51) Map
Cat /B ( H, Fun
Cat /B ( F, G )) = Map
Cat /B ( H × F, G ) . Since
Cat /B is a full subcategory of P (∆ /B ), and since the representable presheavesbelong to Cat /B , the object Fun Cat /B ( F, G ), if it exists, is equivalent to Fun P (∆ /B ) ( F, G ).This provides a very easy criterion for the existence of Fun
Cat /B ( F, G ): it existsif and only if Fun P (∆ /B ) ( F, G ) is a category over B . Note that, by definition, for b ∈ B , the fiber Fun Cat /B ( F, G ) b identifies with Fun( F b , G b ).Let us now apply the above reasoning to operads. If P is a flat O -operad, onedefines a marked category π : P ′ → P over P by the formulas(52) P ′ = Fun in ([1] , O ) × O P , where Fun in ([1] , O ) denotes the category of inert arrows in O . Let s, t : Fun in ([1] , O ) → O be the standard projections. An arrow in P ′ is marked iff its projections to P and to O (via s ) are inert. P ′ considered as a category over O is flat. One de-fines Fun ♯ O ( P ′ , Q ) as the full subcategory of Fun Cat / O ( P ′ , Q ) spanned by the arrows α : P ′ o → Q o , for some o ∈ O , carrying the marked arrows in P ′ o to equivalences.Proposition 2.8.3 of [H.EY] claims that, for any O -operad Q , Fun ♯ O ( P ′ , Q ) is an O -operad representing Funop O ( P , Q ).In particular, for s : [ n ] → O ,Funop O ( P , Q )( s ) ⊂ lim u → v → s Map( P ′ ( v ) , Q ( u )) , consists of collections whose each component corresponding to u = v : [1] → [0] k → [ n ] s → O carries the marked arrows of P ′ s ( k ) to equivalences in Q s ( k ) .5.2.12. Remark.
Note that a map P ′ x → Q x factoring through π : P ′ x → P x automatically carries the marked arrows to equivalences.5.2.13. Condensation, for BM -operads. We need more explicit formulas for thecondensation of a BM -monoidal category. This operation can, actually, be definedin the greater generality of BM -operads. Given a BM -operad p : O → BM , we will define its “condensation” q : O ′ = cond ( O ) → LM so that • O ′ × LM Ass = O × BM Ass − . • O ′ m = v ∗ + ( O × BM RM ),where v + : ∆ op+ → RM is defined as the composition v + = op ◦ u + .We will be using a presentation of operads by presheaves on ∆ / BM and ∆ / LM .Recall that BM = (∆ / [1] ) op ; its objects are arrows s : [ n ] → [1] and an arrowfrom s : [ n ] → [1] to t : [ m ] → [1] is given by f : [ m ] → [ n ] such that s ◦ f = t .An object s : [ n ] → [1] of BM defined by the formulas s ( i ) = 0 , i = 0 , . . . , k, s ( i ) =1 , i > k , is otherwise denoted a k mb n − k − , see [H.EY], 2.9.2.We denote by BM the full subcategory of BM spanned by s : [ n ] → [1] with s (0) = 0. The category LM is the full subcategory of BM spanned by the arrows s : [ n ] → [1] having at most one value of 1. The subcategories BM and RM aredefined as images of BM and LM under op : BM → BM .The full embeddings LM → BM and RM → BM have left adjoint functors ℓ : BM → LM and r : BM → RM erasing superfluous values of 1 and 0 respectively.The functors ℓ and r induce functors ∆ / BM → ∆ / LM and ∆ / BM → ∆ / RM . We willdenote by ∆ act / RM the category of simplices in RM whose all arrows are active.For τ : [ n ] → BM given by τ : a c mb d → . . . → a c k mb d k → a c k +1 → . . . → a c n or τ : a c mb d → . . . → a c k mb d k → b d k +1 → . . . → b d n , we denote by τ − the k -simplex a c mb d → . . . → a c k mb d k (if τ is a simplex in Ass − ⊂ BM , we put k = − τ − is empty in this case).For σ ∈ ∆ / LM we define(53) Π( σ ) = { τ ∈ ∆ / BM | ℓτ = σ, r ( τ − ) ∈ ∆ act / RM } . Note that for σ ∈ ∆ / Ass k = − σ ) = { σ } .The assignment σ Π( σ ) is obviously functorial, with a map α : [ m ] → [ n ]defining Π( σ ) → Π( σ ◦ α ) carrying τ to τ ◦ α .Let a BM -operad O be described by a presheaf F ∈ P (∆ / BM ). We define apresheaf F ′ ∈ P (∆ / LM ) describing cond ( O ) by the formula(54) F ′ ( σ ) = a τ ∈ Π( σ ) F ( τ ) . Lemma.
The presheaf F ′ defined above represents an LM -operad. Proof.
1. Segal condition follows from the definition of Π( σ ). To verify complete-ness, we can fix w = a k m ∈ LM and study the simplicial space n ` τ ∈ Π( w n ) F ( τ ),where w n is the degenerate n -simplex determined by w ∈ LM . This simplicialspace is equivalent to the product of the fibers { a k } × BM O and v ∗ + ( O ), which is,of course, complete as a simplicial space. Thus, F ′ represents a category over LM which we will denote by cond ( O ).2. It remains to prove that cond ( O ) is fibrous.Let us describe cocartesian liftings of the inerts in LM . These are of two types • Erasing m . Such inert has form α : a n m → a l . • Not erasing m : either α : a n → a l or α : a n m → a l m .In the first case the cocartesian lifting of α in F ′ ( α ) having a source in F ( t ) for t = a n mb k is an inert arrow in F ( τ ), where τ ∈ Π( α ) is (the only) inert arrowfrom a n mb k to a l such that ℓ ( τ ) = α .In the second case, for α : a n m → a l m , with the source t = a n mb k , is the inertarrow in F ( τ ) where τ is defined by the conditions ℓ ( τ ) = α , r ( τ ) = id mb k .The rest of the fibrousness conditions [L.HA], 2.3.3.28 or [H.EY], 2.6.3 easilyfollow from the above description. (cid:3) / LM .For σ : [ n ] → LM we have cond ( Quiv BM X op ( M rev ))( σ ) = a ˆ σ ∈ Π( σ ) Quiv BM X op ( M rev )(ˆ σ )(55) ⊂ a ˆ σ ∈ Π( σ ) lim u → v → ˆ σ Map( BM ′ X op ( v ) , M (op ◦ u )) , where the inclusion means that we have to choose the connected componentspreserving the inerts, and BM ′ X op is defined by the formula (52).5.2.15. We are now ready to construct Γ. It consists of a compatible collectionof maps(56) Γ σ,τ,s,t : LM ◦ , − X op ( u + ◦ τ ) × BM ′ X op ( t ) → M ( π ◦ op ◦ s )for each σ ∈ ∆ LM , ˆ σ = σ ∗ ( v + ◦ op ◦ τ ) ∈ Π( σ ) and s → t → ˆ σ in ∆ / BM .Equivalently, this can be described by a compatible collection(57) Γ σ,τ : LM ◦ , − X op ( u + ◦ τ ) × BM ′ X op (ˆ σ ) → M ( π ◦ op ◦ ˆ σ )for each σ ∈ ∆ LM and ˆ σ = σ ∗ ( v + ◦ op ◦ τ ) ∈ Π( σ ). BM ′ X op → BM X op , so, by Remark 5.2.12, it will induces amap to Fun ♯ ( BM ′ X op , M rev ) as needed.The explicit formulas for BM X show that BM X op (ˆ σ ) = BM X op ( σ ), so we will rewritethe source of (57) asMap( F − ( u + ◦ τ ) , X op ) × Map( F ( σ ) , X op ) = Map( F − ( u + ◦ τ ) ⊔ F ( σ ) , X op ) =Map( F ( u + ◦ τ ) ⊔ F ( σ k ) F ( σ ) , X op ) = Map( F ( σ ) , X op ) × Map( F ( σ k ) ,X op ) Map( F ( τ ) , X op ) , where σ k denotes the k -simplex in LM with the constant value m ∈ LM . Similarly,one has a canonical presentation(58) M ( π ◦ op ◦ ˆ σ ) = M ( π ◦ op ◦ σ ) × M ( π ◦ σ n ) M ( π ◦ op ◦ τ ) . γ ′ (39) is described by a map of presheaves, given, for each σ : [ n ] → LM , by a map(59) γ σ : Map( F ( σ ) , X op ) → M ( π ◦ op ◦ σ ) . The collection of maps Γ σ,τ is defined as the fiber product of γ σ and γ τ .5.3. Conclusion.
We have just constructed the functor Γ (48) defined by thecollection of maps Γ σ,τ (57) obtained as the fiber product of γ σ and γ u + τ (59).This implies the main result of this section.5.3.1. Proposition.
The functor Γ (48) defines an operadic colimit diagram (60) ( LM ◦ , − X op ) / ∗ → ( M , P M ( A )) . Weighted colimits
In this paper we study weighted colimits of M -functors.Given an M -functor f : A → B where A is M -enriched category and B is aleft M -module, and an enriched presheaf W ∈ P M ( A ), we will define a weightedcolimit colim W ( f ) ∈ B . The construction is functorial in W , so that colim W ( f ) isthe evaluation at W of a certain colimit preserving functor colim( f ) : P M ( A ) → B preserving the left M -module structure. The composition colim( f ) ◦ Y with theenriched Yoneda embedding yields f : A → B .The construction of weighted colimit is also functorial in f : A → B . This willimply Theorem 6.4.7 claiming that the Yoneda embedding Y : A → P M ( A ) is auniversal M -functor.6.1. Internal Hom.
We keep the notation of 4.4.3. Let C = Cat K .Given A, B, C associative algebras in C , we have a functor RT A,B,C : A BMod B ( C ) × B BMod C ( C ) → A BMod C ( C )defined by the relative tensor product. This functor has a right adjointFun K A : A BMod B ( C ) op × A BMod C ( C ) → B BMod C ( C ) which we are now going to describe.6.1.1. The formulaMap B BMod C ( N, Fun K A,B,C ( M, K )) = Map A BMod C ( M ⊗ B N, K )determines Fun K A,B,C as presheaf. In the case when M is a free A − B -bimodule M = A ⊗ X ⊗ B , this presheaf is represented by the B − C -bimodule Fun K ( X ⊗ B, K ); since any bimodule is a colimit of free bimodules and since the Yonedaembedding commutes with the limits, this proves the existence of Fun K A,B,C ( M, K )in general .6.1.2. The functor Fun K A,B,C , as defined above, depends on three algebras
A, B, C .We will omit B and C from the notation for the following reason.Let b : B → B ′ and c : C → C ′ be algebra maps. Then the B ′ − C ′ -bimoduleFun K A,B ′ ,C ′ ( M, K ) identifies with the restriction of scalars of the B − C -bimoduleFun K A,B,C ( M, K ).6.1.3. Thus, we now assume that B = C = . For M, K ∈ LMod A ( C ), we candefine Fun K A ( M, K ) as the full subcategory of Fun
LMod wA ( M, K ), see 3.1.2, spannedby the lax LM -monoidal functors f = (id A , f m ) : ( A, M ) → ( A, K ) satisfying twoextra properties: • f is LM -monoidal. • f m preserves K -indexed colimits.As we explained above, in the case when M is an A − B -bimodule and K isan A − C -bimodule, Fun K A ( M, K ) acquires a natural B − C -bimodule structure.6.2. Tensor product over
Quiv . We apply the notion of tensor product de-scribed in 4.4 to the following context.Let K be a collection of categories containing ∆ op . Fix a monoidal category M ∈ Alg
Ass ( Cat K ) and a K -strongly small category X ∈ cat K as in [H.EY], 4.4.2.Let B be a left M -module. Let N = Fun( X, M ). This is a right M -module.6.2.1. Lemma. N is right dualizable. Its right dual is M = Fun( X op , M ) con-sidered as a left M -module.Proof. We deduce the duality between M and N from the special case M = S K .In this case N = P K ( X op ), M = P K ( X ) and the duality is given by the maps c : S K → P K ( X op ) ⊗ P K ( X ) = P K ( X op × X )defined as the colimit-preserving map preserving the terminal objects, and e : P K ( X ) ⊗ P K ( X op ) → S K This is a temporary notation. The forgetful functor
Cat K → Cat creates the limits. extending the Yoneda map X op × X → S to preserve colimits.To get the duality for arbitrary M , we use Proposition 4.3.2. We have threeassociative algebras in Cat K , A = B = S K and C = M , the adjoint pair( P K ( X op ) , P K ( X )) that we just constructed, and the one defined by ( M , M ).The composition of these gives an adjunction for ( N, M ). (cid:3) Let us now apply Corollary 4.3.6 to A = M and N = Fun( X, M ). We getthe right dual module M = Fun( X op , M ) and the endomorphism ring B = Quiv X ( M ), see [H.EY], 4.5.3. We also get a canonical M - M -bimodule map(61) e : Fun( X op , M ) ⊗ Quiv X ( M ) Fun( X, M ) → M . We would like to comment about the right
Quiv X ( M )-module structure onFun( X op , M ).According to Remark 4.3.7, the reverse monoidal category Quiv X ( M ) rev iden-tifies with the endomorphism object of the left M -module Fun( X op , M ), that is,of the right M rev -module Fun( X op , M ). This means that the right Quiv X ( M )-action on Fun( X op , M ) is defined by the same construction as the left action onFun( X, M ), but with M rev replacing M and X op replacing X .Tensoring (61) with B over M and using associativity of the relative tensorproduct 4.4.6, we get a map of left M -modules(62) e B : Fun( X op , M ) ⊗ Quiv X ( M ) Fun( X, B ) → B . This gives a
Ten ≻ -algebra Q = Q X, M , B in Cat K consisting of the followingcategories and operations between them described above. • Monoidal categories Q a = Q a ′ = M , Q b = Quiv X ( M ), • A bimodule category Q m = Fun( X op , M ), a left Quiv X ( M )-module Q n =Fun( X, B ) and a left M -module Q k = B .6.2.2. We will present the collection of Ten ≻ -monoidal categories Q X, M , B as acartesian family based on B := cat K × Alg LM ( Cat K ) op , see [H.EY], 2.11.2.Fix a category B . The category Cat cart /B is the full subcategory of Cat /B spannedby cartesian fibrations.We define the category of families C := Cat K , cart /B as the subcategory of Cat cart /B spanned by the families X → B classified by the functors B op → Cat K , withmorphisms X → X ′ over B inducing colimit preserving functors X X → X ′ X foreach X ∈ B . The category C has a symmetric monoidal structure induced fromthe cartesian structure on Cat cart /B : if the families X and Y over B are classifiedby the functors ˜ X , ˜ Y : B op → Cat K , the tensor product of X and Y is classified bythe functor X ˜ X ( X ) ⊗ ˜ Y ( X ). It seems very plausible that e is an equivalence. We do not need this fact. We can now follow the construction of Q X, M , B as a Ten ≻ -algebra object in C := Cat K , cart /B with B = cat K × Alg LM ( Cat K ) op , thinking of P ( X ), P ( X op ), M and B as corresponding cartesian families in C .For an operad O , we denote by Mon lax O the full subcategory of Op O spanned bythe O -monoidal categories.The described above construction defines a functor(63) Q : cat op K × Alg LM ( Cat K ) → Mon lax
Ten ≻ carrying the triple ( X, M , B ) to the Ten ≻ -monoidal category Q X, M , B consideredas a Ten ≻ -operad.We now have a relative tensor product functor(64) RT : Alg
Ten / Ten ≻ ( Q ) → Alg
Ten ≻ ( Q ) . Let A ∈ Alg
Ass ( Quiv X ( M )) be an M -enriched precategory with the space ofobjects X .The restriction RT , A of (64) is a functor(65) RT , A : RMod A (Fun( X op , M )) × LMod A (Fun( X, B )) → B . Taking into account that
RMod A (Fun( X op , M )) = P M ( A ) and LMod A (Fun( X, B )) =Fun M ( A , B ), we finally get the functor called weighted colimit ,(66) colim : P M ( A ) × Fun M ( A , B ) → B , carrying a pair ( W ∈ P M ( A ) , f : A → B ) to colim W ( f ) := f ⊗ W ∈ B . Thisfunctor preserves colimits separately in both arguments, as well as left M -tensoredstructure in the first argument.In particular, for a fixed f : A → B the functor colim( f ) : P M ( A ) → B preserving the colimits and the left M -tensored structure, is defined.Since the bifunctor (66) is a special case of the relative tensor product, itdefines, using the notation (6.1), a canonical functor (that we also denote ascolim)(67) colim : Fun M ( A , B ) → Fun KM ( P M ( A ) , B )6.3. Properties of the weighted colimit.
Lemma.
Let Y : A → P M ( A ) be the Yoneda embedding. Then colim( Y ) =id P M ( A ) .Proof. Look at the
Tens ≻ -monoidal category F in Cat K having the followingcomponents. • Monoidal categories F a = F a ′ = M , F b = F c = F c ′ = Quiv X ( M ), • F m = F k = Fun( X op , M ), F n = Quiv X ( M ),with the standard M - Quiv X ( M )-bimodule structure on Fun( X op , M ) and the unit Quiv X ( M )- Quiv X ( M )-bimodule structure on Quiv X ( M ).We will study the relative tensor product defined by F . Let A be an associativealgebra in Quiv X ( M ). The relative tensor product with A - A -bimodule A definesthe identity functor on RMod A (Fun( X op , M )) = P M ( A ).We will now show that the calculation of colim( Y ) has the same answer.We apply the reduction procedure described in 4.6. Let F ′ = F red A be a Ten ≻ -monoidal category obtained from F by reduction, see 4.6.2 and 4.6.5. It has thefollowing components. • Monoidal categories F ′ a = F ′ a ′ = M , F ′ b = Quiv X ( M ). • F ′ m = Fun( X op , M ), F ′ n = RMod A ( Quiv X ( M )), F ′ k = RMod A (Fun( X op , M )).The category Fun( X, Fun( X op , M )) has a structure of Quiv X ( M )– Quiv X ( M )-bimodule described by the BM -monoidal category Quiv BM X ( Quiv BM X op ( M rev ) rev ) , as in [H.EY], 6.1.7. This bimodule identifies with Quiv X ( M ). This implies that F ′ n as a left Quiv X ( M )-module identifies with Fun( X, P M ( A )), so that F ′ is equiv-alent to the category Q X, M , B with B = P M ( A ). (cid:3) The colimit functor defined by formula(66) is functorial in B .6.3.2. Lemma.
For g : B → B ′ an arrow in LMod M ( Cat K ) , the following diagram (68) P M ( A ) × Fun M ( A , B ) colim / / (cid:15) (cid:15) B (cid:15) (cid:15) P M ( A ) × Fun M ( A , B ′ ) colim / / B ′ , with the vertical arrows defined by g , is commutative.Proof. The functor g : B → B ′ induces a Ten ≻ -monoidal colimit-preserving func-tor Q X, M , B → Q X, M , B ′ . By Proposition 4.4.5, this induced a map preservingweighted colimits. (cid:3) Corollary.
Let A be an M -enriched category and let B be a left M -moduleswith colimits. For any colimit-preserving map F : P M ( A ) → B of left M -modulesthere is a natural equivalence F ∼ → colim( F ◦ Y ) . Universality.
We define the map(69) Y ∗ : Fun KM ( P M ( A ) , B ) → Fun M ( A , B )as the composition with the Yoneda embedding Y : A → P M ( A ).In this subsection we will show that Y ∗ is an equivalence.6.4.1. The weighted colimit defines a map (67) in the opposite direction. Ac-cording to Corollary 6.3.3, the composition Y ∗ ◦ colim is equivalent to identity.We will now prove that the other composition is also equivalent to identity.Recall 3.2 that ¯ A ⊂ P M ( A ) is the full subcategory spanned by the representablefunctors.In what follows we denote P = P M ( A ) , P = ( M , P ) ∈ Alg LM ( Cat K ) , A = ( M , ¯ A ) ⊂ P . Thus, A is an LM -suboperad of P .6.4.2. Proposition.
Any F ∈ Fun
LMod M ( Cat K ) ( P, B ) is an operadic left Kan exten-sion of ¯ F = F | ¯ A .Proof. Let f ∈ P . Denote F f = A × P P act /f .We have to verify that the composition F f → A ¯ F → B extends to an operadiccolimit diagram F ⊲f → B carrying the terminal object of F ⊲f to F ( f ).The general case is immediately reduced to the case B = P and F = id P .Thus, we have to verify that any f ∈ P is an operadic colimit of the functor¯ y : F f → P defined as the composition of the projection F f → A with theembedding to P .The plan of our proof is as follows.Following to 5.3.1 , the pair γ = ( A op , f ) gives rise to a factorization of thefunctor γ ◦ , − , see (43),(70) LM ◦ , − X op Γ → P G → ( M , Fun( X op , M ))through the forgetful functor G , so that Γ( ∗ ) = f where ∗ is a (unique) objectof LM ◦ , − X op over [ − ∈ ∆ op+ . The restriction of Γ to ∆ op × ∆ op+ LM ◦ , − X op factors throughthe full embedding A → P .Recall that K = ∆ op × ∆ op+ ( LM ◦ , − X op ) / ∗ induces an equivalence K ⊲ = ( LM ◦ , − X op ) / ∗ . This yields a functor (71) τ : K → ( LM ◦ , − X op ) / ∗ → F f and an operadic colimit diagram (by Proposition 5.3.1)(72) Γ : K ⊲ → P extending the composition of (71) with the projection F f → P .It remains to verify that τ is cofinal.6.4.3. Cofinality of τ . To prove that τ is cofinal, we use Quillen’s Theorem A inthe form of [L.T], 4.1.3.1. We have to verify that, for any φ ∈ F f , the commacategory K φ = K × F f ( F f ) φ/ is weakly contractible.The proof goes as follows. In 6.4.5 below we present an object t φ of K φ anda functor F : K φ → Fun(Λ , K φ ) whose evaluation at 1 ∈ Λ is id K φ and at2 ∈ Λ is the composition K φ → { t φ } ֒ → K φ . The functor F provides a null-homotopy for the identity map on K φ . This proves cofinality of τ and, finally,Proposition 6.4.2. (cid:3) p : K → ∆ op the obvious projection. We also denote p : K φ → ∆ op the composition K φ → K → ∆ op . The functor F : K φ → Fun(Λ , K φ )will be defined as the one assigning to t ∈ K φ a p -product diagram t ← t ′ → t φ in the sense of [L.T], 4.3.1.1. The latter means that for any x ∈ K φ the diagramMap K φ ( x, t ′ ) / / (cid:15) (cid:15) Map K φ ( x, t ) × Map K φ ( x, t φ ) (cid:15) (cid:15) Map ∆ op ( p ( x ) , p ( t ′ )) / / Map ∆ op ( p ( x ) , p ( t )) × Map ∆ op ( p ( x ) , p ( t φ ))is cartesian. To construct F , we first construct a functor ¯ F : ∆ op → Fun(Λ , ∆ op )(this is very easy), and then prove that for any t there exists a p -product diagram t ← t ′ → t φ whose image under p is ¯ F ( p ( t )). By the uniqueness of relative productdiagrams, the functor ¯ F lifts to a functor F : K φ → Fun(Λ , K φ ).6.4.5. We define q : F f → ∆ op+ so that the composition u + ◦ op ◦ q is the projectionto F f → P act /f → LM . This condition uniquely determines q as any object in P act /f has image in u + (∆ op+ ). The definition of q is chosen so that for t ∈ K the equality p ( t ) = q ( τ ( k )) holds.Let φ ∈ F f be defined by the collection ( m , . . . , m n , z, β ) with m i ∈ M , z ∈ X op and β : ( m , . . . , m n , z ) → f defined by a map ⊗ m i → f ( z ) (we will alsodenote it by β ). Given d ∈ K defined by a sequence ( y, x k , y k , . . . , x , y , x ) together with ar-rows α i : x i → y i +1 ( α k : x k → y ), an object t : φ → τ ( d ) of K φ is defined by acollection of maps ⊗ r k i − m i → f ( y ); ⊗ r j − i = r j +1 m i → A ( y j , x j ) , j = k, . . . , ⊗ ni = r +1 m i → A ( z, x ) , for a certain sequence of numbers 1 ≤ r k ≤ . . . ≤ r ≤ n defining the arrow a n m → a k m in LM that is the image of t .We define an object t φ of K φ by the arrow t φ : φ → τ ( d φ ) where d φ =( z, z, id z ) ∈ K , so that τ ( d φ ) is given by ( f ( z ) , z, id f ( z ) ), and t φ : φ → τ ( d φ ),an arrow in F f over the map a n m → am induced by a n → a in LM , that is theidentity on z and β : ⊗ m i → f ( z ) on the a -component.Note that q ( t φ ) is the map h n − i → h i corresponding to { } ∈ [ n − F : ∆ op → Fun(Λ , ∆ op ). Its opposite carries [ k − ∈ ∆to the diagram [ k − → [ k ] ← [0] in ∆ with the arrows ∂ : [ k − → [ k ] and[0] → { } ∈ [ k ].We claim that for any t ∈ K φ with p ( t ) = h k − i , there is a p -product diagram t ← t ′ → t φ over ¯ F ( h k i ).We will now define an arrow d ′ → d in K with a decomposition of t : φ → τ ( d )via τ ( d ′ ) → τ ( d ), as well as a map d ′ → d φ decomposing t φ .We put d ′ = ( y, x k , y k , . . . , x , z, z ) together with α i : x i → y i +1 and id z : z → z .We have τ ( d ′ ) = ( f ( y ) , A ( y k , x k ) , . . . , A ( y , x ) , A ( z, x ) , z ), so that the collectionof maps (73), together with the unit → A ( z, z ), yields a map that we denote t ′ : φ → τ ( d ′ ).The arrow d ′ → d in K is given by the commutative diagram(74) y • x k ◦ α k o o (cid:15) (cid:15) y k • . . . α k − o o x ◦ α o o (cid:15) (cid:15) z • z ◦ o o y • O O x k ◦ α k o o y k • O O . . . α k − o o x ◦ α o o where all unnamed arrows appearing in the diagram are the identity maps. Thearrow d ′ → d φ is given by the commutative diagram(75) y • x k ◦ α k o o y k • . . . α k − o o x ◦ α o o z • z ◦ o o (cid:15) (cid:15) z • O O z ◦ o o where, once more, all unnamed arrows are the identity maps.Thus, for a fixed map p ( d ′′ ) → p ( d ′ ) in ∆ op , its lifting in d ′′ → d ′ in K isdescribed by the same collection of data as a pair of maps d ′′ ← d and d ′′ → d φ . Therefore, the diagram d ← d ′ → d φ is a p -product diagram in K . For the samereason the diagram t ← t ′ → t φ is a p -product diagram.This proves the cofinality of τ : K → F f , and, therefore, Proposition 6.4.2.We are now ready to deduce the main result of this section.6.4.6. Proposition.
The composition colim ◦ Y ∗ is equivalent to identity.Proof. Let F : P M ( A ) → B be a colimit-preserving functor preserving the left M -tensored structure. Put f = Y ∗ ( F ) and F ′ = colim( f ). Both F and F ′ areoperadic left Kan extensions of f , so they are canonically equivalent. (cid:3) This proves our main universality result.6.4.7.
Theorem.
The map (69) is an equivalence. (cid:3) Multiplicative structures
We will now assume that M ∈ Alg O ⊗ Ass ( Cat K ). In this case many objectsmentioned above acquire a structure of O -algebra . We will see in this sectionthat7.1. From associative algebras to left modules. µ : P × Q → R ,and an R -operad X , the P -operad p : Alg µ Q / R ( X ) → P is defined as the objectrepresenting the functor K ∈ Cat / P Map
Cat + / R ♮ ( K ♭ × Q ♮ , X ♮ ) , see [H.EY], 2.10. In the case X is R -monoidal, Alg µ Q / R ( X ) is P -monoidal. In thecase µ represents R as a tensor product of P and Q , one has an equivalence Alg P ( Alg µ Q / R ( X )) = Alg R ( X ) . We will suppress the letter µ from the notation if it is clear from the context.We will need the following general claim about cocartesian fibrations.7.1.2. Lemma.
Let P f / / p (cid:31) (cid:31) ❄❄❄❄❄❄❄❄ Q q (cid:127) (cid:127) ⑧⑧⑧⑧⑧⑧⑧⑧ B be a commutative diagram with p, q cocartesian fibrations. Assume that This sentence becomes slightly imprecise if O is not monochrome: an O -algebra consistsof more than one object. (1) For any b ∈ B f b : P b → Q b is a cocartesian fibration. (2) For any α : b → b ′ in B the functor α ! : P b → P b ′ carries f b -cocartesian arrows to f b ′ -cocartesian arrows.Then f is a cocartesian fibration.Proof. By [L.T], 2.4.2.11, f is a locally cocartesian fibration, with locally co-cartesian arrows of the form u = u ′′ ◦ u ′ where u ′ is p -cocartesian and u ′′ ∈ P b ′ f b ′ -cocartesian, with p ( u ) : b → b ′ . Condition (2) ensures that the compositionof locally cocartesian arrows is locally cocartesian. This implies the claim. (cid:3) Let C ∈ Alg O ⊗ LM ( Cat K ) where K is a collection of categories containing ∆ op .7.1.3. Proposition.
The forgetful functor (76)
Alg LM / O ⊗ LM ( C ) → Alg
Ass / O ⊗ LM ( C ) is an O -monoidal cocartesian fibration.Proof. This result is very close to [L.HA], 4.5.3. We will apply Lemma 7.1.2 tothe forgetful functor f : Alg LM / O ⊗ LM ( C ) → Alg
Ass / O ⊗ LM ( C ) over B := O .For o ∈ O we denote C o the LM -monoidal category obtained from C by the basechange along LM o → O ⊗ LM .The fiber f o of f at o ∈ O is a cartesian fibration as it is a forgetful functor Alg LM ( C o ) → Alg
Ass ( C o ). Since these are categories with geometric realizations, f o is also a cocartesian fibration. The same is true for f o at any o ∈ O ascocartesian fibrations are closed under products. This proves the condition (1) ofLemma 7.1.2. Let us verify the condition (2). Given α : o → o ′ in O , the functor α ! : Alg LM ( C o ) → Alg LM ( C o ′ )is induced by the colimit preserving LM -monoidal functor C α : C o → C o ′ inducedby α . An arrow ( A, M ) → ( B, N ) in
Alg LM ( C o ) is f o -cocartesian if it induces anequivalence B ⊗ A M → N . Thus, α ! preserves this property. (cid:3) The forgetful functor (76) is classified by a lax O -monoidal functor LMod : Alg
Ass / O ⊗ LM ( C ) → Cat , see Prop. A.2.1. of [H.R].In particular, we have the following.7.1.4. Corollary.
The functor
LMod defined as above assigns an O -monoidal cat-egory of left modules LMod A ( C ) to an O -algebra object A in Alg
Ass / O ⊗ LM ( C ) . (cid:3) Remark.
It is worthwhile, in order to keep track of what we are doing, torepeat the above construction in down-to-earth terms.An O -algebra object in Alg
Ass / O ⊗ LM ( C ) consists of a collection of assiciativealgebra objects A o in LM -monoidal categories C o , with operations ⊗ A o i → A o corresponding to each operation α : ( o , . . . , o n ) → o in O . The category LMod A ( C )has components LMod A ( C ) o = LMod A o ( C o ), and the operation α : ( o , . . . , o n ) → o in O assigns to a collection M i ∈ LMod A ai ( C a i ) the pushforward of ⊗ A o i -module ⊗ M i along ⊗ A o i → A o .We apply Corollary 7.1.4 a number of times.7.2. O -monoidal enriched categories. O -monoidal left-tensored categories. Let C = Cat K . This is a symmetricmonoidal category, so it has a structure of O ⊗ LM -monoidal category. For M ∈ Alg O ⊗ Ass ( Cat K ) the category LMod M ( Cat K ) is O -monoidal. An O -monoidal left M -module B is defined as an O -algebra in LMod M ( Cat K ).7.2.2. O -monoidal enriched categories. In [H.EY], 3.5.2, a lax SM functor quiv : Op Ass → Fam cart Op Ass , as well as its relatives quiv LM : Op LM → Fam cart Op LM and quiv BM : Op LM → Fam cart Op BM ,are constructed.Since the functor Alg
Ass : Fam cart Op Ass → Cat preserves the limits, the compo-sition
Alg
Ass ◦ quiv : Op Ass → Cat carrying M ∈ Op Ass to the category
PCat ( M )of M -enriched precategories, is lax symmetric monoidal. The same holds for thecompositions Alg LM ◦ quiv LM : Op LM → Cat and
Alg BM ◦ quiv BM : Op BM → Cat .Fix an operad O . If M is an O -algebra in Op Ass (for instance, M ∈ Alg O ⊗ Ass ( Cat K )),then the category PCat ( M ) is O -monoidal.Similarly, if M is an O -algebra in Op LM (for instance, M ∈ Alg O ⊗ LM ( Cat K )), thenthe category PCat LM ( M ) is O -monoidal. Definition.
Let M ∈ Alg O ⊗ Ass ( Cat K ). An O -monoidal M -enriched precategoryis an O -algebra object in PCat ( M ).7.2.3. O -monoidal M -functors. Let ( M , B ) ∈ Alg O ⊗ LM ( Cat K ). This means that M is an O ⊗ Ass -monoidal category with colimits and B is an O -monoidal categorywith colimits left-tensored over M .By 7.2.2, the category PCat LM ( M , B ) = Alg LM ( Quiv LM ( M , B )) is O -monoidal.By 7.1.3, the forgetful functor PCat LM ( M , B ) → PCat ( M )is an O -monoidal cocartesian fibration. Therefore, it is classified by the lax O -monoidal functor PCat ( M ) → Cat carrying A ∈ PCat ( M ) to Fun M ( A , B ). In particular, for any O -monoidal M -enriched category A the category Fun M ( A , B )is O -monoidal. This is an enriched form of the Day convolution [L.HA], 2.6.Let us repeat that if O is not monochrome, M is actually a collection ofmonoidal categories, PCat ( M ) is actually a collection of categories, etc. Definition.
Given ( M , B ) ∈ Alg O ⊗ LM ( Cat K ) and an O -monoidal M -enriched cat-egory A , an O -monoidal M -functor f : A → B is an O -algebra in Fun M ( A , B ).We denote Fun OM ( A , B ) = Alg O (Fun M ( A , B ) the category of O -monoidal M -functors from A to B .7.3. O -monoidal Yoneda embedding. M is O ⊗ Ass -monoidal category in
Cat K and let A be an O -monoidal M -enriched precategory, that is, an O -algebra in PCat ( M ).We will now repeat the construction of Yoneda embedding for A .Denote π : BM → Ass the canonical map of (planar) operads. For any planaroperad C (or a family of of planar operads) the functor π ∗ : Alg
Ass ( C ) → Alg BM ( C )carries an associative algebra A to the A - A -bimodule A .The folding functor φ : Op BM → Op LM defined in [H.EY], 3.6, preserves limits,so the functor φ ◦ π ∗ : Alg
Ass ( C ) → Alg LM ( φ ◦ π ∗ ( C )) also preserves limits, and,therefore, carries O -algebras to O -algebras.We apply this to C := Quiv ( M ) and an O -algebra A in PCat ( M ). We get an O -algebra in PCat LM ( M ⊗ M rev , M ).According to 7.2.3, this defines an O -monoidal M ⊗ M rev -functor ˜ Y : A ⊠ A op → M .7.3.2. One uses the adjoint associativity equivalence to deduce Yoneda embed-ding from the functor A ⊠ A op → M .Recall [H.EY], 6.1.7. Below B is a left M ⊗ M ′ -module, A ∈ PCat ( M ) and A ′ ∈ PCat ( M ′ ). There is a canonical equivalence(77) Fun M ⊗ M ′ ( A ⊠ A ′ , B ) = Fun M ( A , Fun M ′ ( A ′ , B )) . Let now O be an operad, M and M ′ be in Alg O ⊗ Ass ( Cat K ) and B be an O -monoidal category left-tensored over M ⊗ M ′ . Then the left and the right side of(77) is an O -monoidal category by (7.2.3). Proposition.
Under these assumptions the equivalence (77) canonically lifts toan equivalence of O -monoidal categories.Proof. Let T = ( M , B , M ′ rev ) be the BM -monoidal category defined by the left M ⊗ M ′ -module B . The equivalence (77) is constructed in [H.EY], 6.1.7, froman equivalence(78) Funop BM ( BM X × BM rev X ′ , T ) = Quiv BM X ( Quiv BM X ′ ( T rev ) rev ) . Both left and right hand sides, cosidered as functors of T , preserve limits. There-fore, in case T is O -monoidal, (78) is an equivalence of O ⊗ BM -monoidal cate-gories. (cid:3) Y : A ⊠ A op → M is O -monoidal, immediately imply that the Yoneda embedding Y : A → P M ( A ) is O -monoidal.7.4. Universality of Yoneda embedding in the monoidal setting, 1.
Re-call that the composition with the Yoneda embedding defines an equivalence (69) Y ∗ : Fun KM ( P M ( A ) , B ) → Fun M ( A , B )for any left M -tensored category with colimits B . In this, final, subsection wepresent an O -monoidal version of this equivalence.7.4.1. Assume that M is O ⊗ Ass -monoidal and let B , B ′ ∈ Alg O ( LMod M ( Cat K )). Definition.
The category of O -monoidal functors Fun O , KM ( B , B ′ ) is the full sub-category of the fiber of the map Alg ( M , B ) / O ⊗ LM ( M , B ′ ) → Alg M / O ⊗ Ass ( M )at id M , spanned by the lax O ⊗ LM -monoidal functors f = (id M , f m ) : ( M , B ) → ( M , B ′ ) satisfying two extra properties: • f is O ⊗ LM -monoidal. • f m preserves K -indexed colimits.7.4.2. Let now A be an O -monoidal M -enriched category.In this case P M ( A ) is also O -monoidal (as a left M -module) and the Yonedaembedding is a monoidal M -functor, see 7.3.3. In Theorem 7.4.4 below we showthat the equivalence (69) induces an equivalence(79) Y O ∗ : Fun O , KM ( P M ( A ) , B ) → Fun OM ( A , B ) . B Fun M ( A , B ) , as defined by the formula (8), defines a functor Alg O ( LMod M ( Cat K )) → Alg O ( Cat )which yields a canonical functor(80) Fun OM ( A , P M ( A )) × Fun O , KM ( P M ( A ) , B ) → Fun OM ( A , B ) . Composing it with the O -monoidal Yoneda embedding Y : A → P M ( A ), we getthe map (79).7.4.4. Theorem.
Let O be an operad, M ∈ Alg O ⊗ Ass ( Cat K ) be an O ⊗ Ass -monoidal category with K -imdexed colimits. Let, furthermore, A be an O -monoidal M -enriched category and let B ∈ Alg O ( LMod M ( Cat K )) be an O -monoidal categorywith colimits left-tensored over M . Then the functor (79) is an equivalence. We will deduce Theorem 7.4.4 applying the general tensor product constructionof 4.4 to the O -monoidal setup. This is based on the monoidal properties of thefunctor (63) that we will now study.7.5. Monoidal properties of the functor Q (63). In this subsection we provethe following result.7.5.1.
Proposition.
The functor Q (63) has a natural lax symmetric monoidalstructure, with the SM structure on the source and on the target of Q defined asfollows. • The cartesian SM structure on
Mon lax
Ten ≻ . • The cocartesian structure on cat op K (induced by the product of categories)and the SM structure on Alg LM ( Cat K ) induced by the SM structure on Cat K . The proof of the proposition is given in 7.5.6–7.5.8.7.5.2.
Families of monoidal categories.
Recall that, for an operad O , we denoteby Mon lax O the full subcategory of Op O spanned by the O -monoidal categories.A functor X : B → Mon lax O has two equivalent descriptions presented below.(X1) As a lax bifibration p : X → B op × O , such that for any b ∈ B the fiber p : X b → O is an O -monoidal category. Recall [H.D], 3.1.2, that p is alax bifibration if p B : X → B op is a cartesian fibration, p O : X → O is acocartesian fibration, and p is both a map of cartesian fibrations over B op and a map of cocartesian fibrations over O .(X2) As an O -algebra object in Cat cart /B op , the full subcategory of Cat /B op spannedby the cartesian fibrations, see [H.EY], 2.11.3.We wish to express in these terms what makes a functor X : B → Mon lax O laxsymmetric monoidal.Let Cat cart be the full subcategory of Fun([1] , Cat ) spanned by the cartesianfibrations.The composition b : Cat cart → Fun([1] , Cat ) → Fun( { } , Cat ) =
Cat is acartesian fibration. We will consider it as a family of (cartesian) SM cate-gories. The cartesian SM structure om
Cat cart induces a cartesian SM structureon
Alg O ( Cat cart ). We have7.5.3.
Proposition.
The equivalence between
Alg O ( Cat cart /B ) and Fun( B, Mon lax O ) extends to an equivalence between lax SM functors X : B → Mon lax O and the fiberof the functor b : Alg
Com ( Alg O ( Cat cart )) → Alg
Com ( Cat ) at B . The proof of the proposition is given in 7.5.5. The following lemma settles thecase of trivial operad.
Lemma.
The Grothendieck construction establishes an equivalence betweenlax SM functors X : B → Cat and the fiber of the functor b : Alg
Com ( Cat cart ) → Alg
Com ( Cat ) at B .Proof. The claim is well-known for cocartesian fibrations, see [H.R], A.2.1. Givena functor X : B op → Cat , the corresponding cartesian fibration p : Y → B canbe presented as a result of the following three steps. • We compose X with op : Cat → Cat . Since op is an equivalence, thisoperation preserves lax symmetric monoidal structure on X . • We apply to op ◦ X the cocartesian version of Grothendieck construction.We get a cocartesian fibration p : X → B op that is a commutative algebraobject in Coc . • We apply another functor op : Coc → Cart carrying p : X → B op to q : Y := X op → B . This functor is also an equivalence and thereforepreserves the commutative algebra structure. (cid:3) Proof of Proposition 7.5.3.
We denote by
Bifib lax O the category of laxbifibrations p : X → B × O endowed with the cartesian SM structure.By Lemma 7.5.4, lax SM monoidal functors X : B → Mon lax O correspond tocommutative algebras p : X → B × O in Bifib lax O . On the other hand, the equiv-alence of two descriptions (X1) and (X2) for families of monoidal categories allowsone to identify commutative algebras in Bifib lax O with commutative algebras in Alg O ( Cat cart )). (cid:3)
Quiv BM : cat op K × Alg
Ass ( Cat K ) → Mon lax BM . This yields, by Proposition 7.5.3, a commutative algebra structure on thecorresponding BM -algebra Q in Cat cart , lying over B op0 where B = cat op K × Alg
Ass ( Cat K ).Note that Q defines a bimodule in Cat cart , K /B op that is right-dualizable.7.5.7. We will now extend Q to a commutative algebra Q in Alg
Tens ≻ ( Cat cart , K ).This is done as follows. For any B and any SM category C with colimits wedenote by Alg rd BM ( C ) the category of right-dualizable bimodules in C . We denoteby Alg du Tens ≻ ( C ) the category of dual pairs of bimodules defined as follows.We denote by Tens ′≻ the operad governing the following data. • Associative algebras
A, B . • M ∈ A BMod B , N ∈ B BMod A . • A B -bilinear map M × N → A of A - A -bimodules. Any
Tens ′≻ -algebra defines a Tens ≻ -algebra. This defines a map of operads j : Tens ≻ → Tens ′≻ . We define
Alg du Tens ≻ ( C ) as the full subcategory of Alg
Tens ′≻ ( C ) spanned by thebiliner maps M × N → A presenting M as right dual of N . The functor j • : Alg du Tens ≻ ( C ) → Alg
Tens ≻ ( C ) is defined as the composition of the full embeddingwith j ∗ . The forgetful functor G : Alg du Tens ≻ ( C ) → Alg rd BM ( C )is an equivalence, whence the functor j ♯ = j • ◦ G − : Alg rd BM ( C ) → Alg
Tens ≻ ( C ) . The functor j ♯ makes equally sense for families of categories. We will apply itto the family Cat cart , K . The functor(81) j ♯ : Alg rd BM ( Cat cart , K ) → Alg
Tens ≻ ( Cat cart , K )preserves products, and, therefore, carries Q , a commutative algebra in Alg BM ( Cat cart , K ),to a commutative algebra in Alg
Tens ≻ ( Cat cart , K ) over B op0 . We will call it Q .7.5.8. In the last step of the construction we will incorporate the left M -module B into the picture.For a monoidal category C with geometric realizations there is a functor(82) Alg
Tens ≻ / Ass ( C ) × Alg
Ass ( C ) Alg LM / Ass ( C ) → Alg
Ten ≻ / Ass ( C )defined by the tensor product ( B N C , C L ) N ⊗ C L , ( A ′ K C ′ , C L ) K ⊗ C L .(find better words)We will apply the functor (82) to the family of categories C = Cat cart , K over Cat .The functor (82) in this case preserves products, and, therefore, commutativealgebra objects. We define B = cat op K × Alg LM ( Cat K ) = B × Alg
Ass ( Cat K ) Alg LM ( Cat K ) . We denote by p and p the natural projections of B to B and Alg LM ( Cat K ).These are SM functors. Therefore, p ∗ ( Q ) is an commutative algebra objectof Alg
Tens ≻ ( Cat cart , K ). Furthermore, applying p ∗ to the universal family of LM -algebras over Alg LM ( Cat K ), we get a commutative algebra object in Alg LM / Ass ( Cat cart , K ).The image under (82) of this pair yields a commutative algebra object in Alg
Ten ≻ / Ass ( Cat cart , K ).This is the family Q defined in (63). Proposition 7.5.1 is proven. Proof of Theorem 7.4.4.
Let us recall the setup. M is a O ⊗ Ass -monodalcategory with colimits, A ∈ Cat ( M ) is an O -monoidal M -enriched category and B is an O -monoidal category left-tensored over M . By Proposition 7.5.1, Q X, M , B is a Ten ≻ -algebra object in Alg O ( Cat K ).This allows one to define an O -monoidal version of the relative tensor productfunctor that yields, similarly to (66), an O -monoidal functor(83) colim : P M ( A ) × Fun M ( A , B ) → B . Let [ O ] = O eq be the space of colors of the operad O . We have a forgetful functor G : Alg O ( Cat K ) → ( Cat K ) [ O ] that commutes with sifted colimits, see [L.HA],3.2.3.1.By Proposition 4.4.1(3), the forgetful functor G commutes with the weightedcolimits.Choose o ∈ [ O ] and look at the following diagram(84) Fun O , KM ( P M ( A ) , B ) Y O ∗ / / G a (cid:15) (cid:15) Fun OM ( A , B ) colim o o G a (cid:15) (cid:15) Fun KM ( P M o ( A o ) , B o ) Y ∗ o / / Fun M o ( A o , B o ) , colim o o with Y ∗ o induced by the Yoneda embedding for A o and G a denoting the a -component of the functor forgetting the O -algebra structure. Both squares ofthe diagram are homotopy commutative.Since for all o ∈ [ O ] the arrows Y ∗ o and colim are homotopy inverse, and sincethe forgetful functor G is conservative, Y O ∗ and colim are also homotopy inverse.This proves that (79) is an equivalence. References [ABS] Arone, Barnea, Schlank, Noncommutative CW spectra, manuscript in preparation.[GH] D. Gepner, R. Haugseng, Enriched ∞ -categories via non-symmetric operads, Adv.Math., 279(2015), 575–716.[H.D] V. Hinich, So, what is the derived functor?, arXiv:1811.12255, Homology, homotopy,applications, 22(2020), no. 2, 279–293.[H.EY] V. Hinich, Yoneda lemma for enriched infinity categories, arXiv:1805.0507635, Adv.Math., 367(2020), 107129.[H.lec] V. Hinich, Lectures on ∞ -categories, preprint arXiv:1709.06271.[H.L] V. Hinich, Dwyer-Kan localization, revisited, preprint arXiv:1311.4128, Homology,homotopy, applications, 18(2016), 27–48.[H.R] V. Hinich, Rectification of algebras and modules, preprint arXiv:1311.4130, Doc.Math. 20 (2015), 879—926.[HH] H. Heine, An equivalence between enriched ∞ -categories and ∞ -categories with aweak action, arXiv:2009.02428. Department of Mathematics, University of Haifa, Mount Carmel, Haifa 3498838,Israel
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