aa r X i v : . [ m a t h . C T ] N ov DOUBLES FOR MONOIDAL CATEGORIES
CRAIG PASTRO AND ROSS STREET
Dedicated to Walter Tholen on his 60th birthday
Abstract.
In a recent paper, Daisuke Tambara defined two-sided actions onan endomodule (= endodistributor) of a monoidal V -category A . When A is autonomous (= rigid = compact), he showed that the V -category (that wecall Tamb( A )) of so-equipped endomodules (that we call Tambara modules)is equivalent to the monoidal centre Z [ A , V ] of the convolution monoidal V -category [ A , V ]. Our paper extends these ideas somewhat. For general A ,we construct a promonoidal V -category DA (which we suggest should becalled the double of A ) with an equivalence [ DA , V ] ≃ Tamb( A ). When A is closed, we define strong (respectively, left strong) Tambara modules andshow that these constitute a V -category Tamb s ( A ) (respectively, Tamb ls ( A ))which is equivalent to the centre (respectively, lax centre) of [ A , V ]. Weconstruct localizations D s A and D ls A of DA such that there are equiva-lences Tamb s ( A ) ≃ [ D s A , V ] and Tamb ls ( A ) ≃ [ D ls A , V ]. When A isautonomous, every Tambara module is strong; this implies an equivalence Z [ A , V ] ≃ [ DA , V ]. Introduction
For V -categories A and B , a module T : A (cid:31) / / B (also called “bimodule”,“profunctor”, and “distributor”) is a V -functor T : B op ⊗ A / / V . For amonoidal V -category A , Tambara [Tam06] defined two-sided actions α of A on anendomodule T : A (cid:31) / / A . When A is autonomous (also called “rigid” or “com-pact”) he showed that the V -category Tamb( A ) of Tambara modules ( T, α ) isequivalent to the monoidal centre Z [ A , V ] of the convolution monoidal V -category[ A , V ].Our paper extends these ideas in four ways:(1) our base monoidal category V is quite general (as in [Kel82]) not just vectorspaces;(2) our results are mainly for a closed monoidal V -category A , generalizingthe autonomous case;(3) we show the connection with the lax centre as well as the centre; and,(4) we introduce the double DA of a monoidal V -category A and some local-izations of it, and relate these to Tambara modules. Date : May 29, 2018.
Key words and phrases. monoidal centre, Drinfeld double, monoidal category, Day convolution.The first author gratefully acknowledges support of an international Macquarie UniversityResearch Scholarship while the second gratefully acknowledges support of the Australian ResearchCouncil Discovery Grant DP0771252.
Our principal goal is to give conditions under which the centre and lax centreof a V -valued V -functor monoidal V -category is again such. Some results in thisdirection can be found in [DS07].For general monoidal A , we construct a promonoidal V -category DA withan equivalence [ DA , V ] ≃ Tamb( A ). When A is closed, we define when aTambara module is (left) strong and show that these constitute a V -category(Tamb ls ( A )) Tamb s ( A ) which is equivalent to the (lax) centre of [ A , V ]. Weconstruct localizations D s A and D ls A of DA such that there are equivalencesTamb s ( A ) ≃ [ D s A , V ] and Tamb ls ( A ) ≃ [ D ls A , V ]. When A is autonomous, ev-ery Tambara module is strong, which implies an equivalence Z [ A , V ] ≃ [ DA , V ].These results should be compared with those of [DS07] where the lax centre of[ A , V ] is shown generally to be a full sub- V -category of a functor V -category[ A M , V ] which also becomes an equivalence Z [ A , V ] ≃ [ A M , V ] when A is au-tonomous.As we were completing this paper, Ignacio Lopez Franco sent us his preprint [LF07]which has some results in common with ours. As an example for V -modules of hisgeneral constructions on pseudomonoids, he is also led to what we call the doublemonad. 2. Centres and convolution
We work with categories enriched in a base monoidal category V as used byKelly [Kel82]. It is symmetric, closed, complete and cocomplete.Let A denote a closed monoidal V -category. We denote the tensor product by A ⊗ B and the unit by I in the hope that this will cause no confusion with the samesymbols used for the base V itself. We have V -natural isomorphisms A ( A, B C ) ∼ = A ( A ⊗ B, C ) ∼ = A ( B, C A )defined by evaluation and coevaluation morphisms e l : B C ⊗ B / / C, d l : A / / B ( A ⊗ B ) ,e r : A ⊗ C A / / C, d r : B / / ( A ⊗ B ) A . Consequently, there are canonical isomorphisms A ⊗ B C ∼ = A ( B C ) , C A ⊗ B ∼ = ( C A ) B , ( B C ) A ∼ = B ( C A ) and I C ∼ = C ∼ = C I which we write as if they were identifications just as we do with the associativityand unit isomorphisms. We also write B C A for B ( C A ).The Day convolution monoidal structure [Day70] on the V -category [ A , V ] of V -functors from A to V consists of the tensor product F ∗ G and unit J definedby ( F ∗ G ) A = Z U,V A ( U ⊗ V, A ) ⊗ F U ⊗ GV ∼ = Z V F ( V A ) ⊗ GV ∼ = Z U F U ⊗ G ( A U )and JA = A ( I, A ) . OUBLES FOR MONOIDAL CATEGORIES 3
In particular,( F ∗ A ( A, − )) B ∼ = F ( A B ) and ( A ( A, − ) ∗ G ) B ∼ = G ( B A ) . The centre of a monoidal category was defined in [JS91] and the lax centre wasdefined, for example, in [DPS07]. Since the representables are dense in [ A , V ], anobject of the lax centre Z l [ A , V ] of [ A , V ] is a pair ( F, θ ) consisting of F ∈ [ A , V ]and a V -natural family θ of morphisms θ A,B : F ( A B ) / / F ( B A )such that the diagrams F ( A ⊗ B C ) F ( C A ⊗ B ) θ A ⊗ B,C / / F ( A ( B C )) = (cid:15) (cid:15) F ( B C A ) θ A,BC " " EEEEEEEEE F (( C A ) B ) θ B,CA < < yyyyyyyyy = O O and F ( I A ) F ( A I ) θ I,A / / F A = (cid:31) (cid:31) ?????? = ? ? (cid:127)(cid:127)(cid:127)(cid:127)(cid:127)(cid:127) commute. The hom object Z l [ A , V ](( F, θ ) , ( G, φ )) is defined to be the equalizer oftwo obvious morphisms out of [ A , V ]( F, G ). The centre Z [ A , V ] of [ A , V ] is thefull sub- V -category of Z l [ A , V ] consisting of those objects ( F, θ ) with θ invertible.3. Tambara modules
Let A denote a monoidal V -category. We do not need A to be closed for thedefinition of Tambara module although we will require this restriction again later.A left Tambara module on A is a V -functor T : A op ⊗ A / / V together witha family of morphisms α l ( A, X, Y ) : T ( X, Y ) / / T ( A ⊗ X, A ⊗ Y )which are V -natural in each of the objects A , X and Y , satisfying the two equations α l ( I, X, Y ) = 1 T ( X,Y ) and T ( X, Y ) T ( A ⊗ A ′ ⊗ X, A ⊗ A ′ ⊗ Y ) . α l ( A ⊗ A ′ ,X,Y ) $ $ IIIIIIIIIIII T ( A ′ ⊗ X, A ′ ⊗ Y ) α l ( A ′ ,X,Y ) / / α l ( A,A ′ ⊗ X,A ′ ⊗ Y ) z z uuuuuuuuuuuu Similarly, a right Tambara module on A is a V -functor T : A op ⊗ A / / V together with a family of morphisms α r ( B, X, Y ) : T ( X, Y ) / / T ( X ⊗ B, Y ⊗ B ) CRAIG PASTRO AND ROSS STREET which are V -natural in each of the objects B , X and Y , satisfying the two equations α r ( I, X, Y ) = 1 T ( X,Y ) and T ( X, Y ) T ( X ⊗ B ⊗ B ′ , Y ⊗ B ⊗ B ′ ) . α r ( B ⊗ B ′ ,X,Y ) $ $ IIIIIIIIIIII T ( X ⊗ B, Y ⊗ B ) α r ( B,X,Y ) / / α r ( B ′ ,B ⊗ X,B ⊗ Y ) z z uuuuuuuuuuuu A Tambara module ( T, α ) on A is a V -functor T : A op ⊗ A / / V togetherwith both left and right Tambara module structures satisfying the “bimodule”compatibility condition T ( X, Y ) T ( A ⊗ X, A ⊗ Y ) α l ( A,X,Y ) / / T ( A ⊗ X ⊗ B, A ⊗ Y ⊗ B ) . α r ( B,A ⊗ X,A ⊗ Y ) (cid:15) (cid:15) T ( X ⊗ B, Y ⊗ B ) α r ( B,X,Y ) (cid:15) (cid:15) α l ( A,X ⊗ B,Y ⊗ B ) / / The morphism defined to be the diagonal of the last square is denoted by α ( A, B, X, Y ) : T ( X, Y ) / / T ( A ⊗ X ⊗ B, A ⊗ Y ⊗ B )and we can express a Tambara module structure purely in terms of this, however,we need to refer to the left and right structures below. Proposition 3.1.
Suppose A is a monoidal V -category and T : A op ⊗ A / / V is a V -functor. (a) If A is right closed, there is a bijection between V -natural families of mor-phisms α l ( A, X, Y ) : T ( X, Y ) / / T ( A ⊗ X, A ⊗ Y ) and V -natural families of morphisms β l ( A, X, Y ) : T ( X, Y A ) / / T ( A ⊗ X, Y ) . (b) Under the bijection of (a), the family α l is a left Tambara structure if and onlyif the family β l satisfies the two equations β l ( I, X, Y ) = 1 T ( X,Y ) and T ( X, Y A ⊗ A ′ ) T ( A ⊗ A ′ ⊗ X, Y ) β l ( A ⊗ A ′ ,X,Y ) / / T ( X, ( Y A ) A ′ ) = (cid:15) (cid:15) T ( A ′ ⊗ X, Y A ) . β l ( A ′ ,X,Y A ) / / β l ( A,A ′ ⊗ X,Y ) O O (c) If A is left closed, there is a bijection between V -natural families of morphisms α r ( B, X, Y ) : T ( X, Y ) / / T ( X ⊗ B, Y ⊗ B ) and V -natural families of morphisms β r ( B, X, Y ) : T ( X, B Y ) / / T ( X ⊗ B, Y ) . OUBLES FOR MONOIDAL CATEGORIES 5 (d)
Under the bijection of (c), the family α r is a right Tambara structure if andonly if the family β r satisfies the two equations β r ( I, X, Y ) = 1 T ( X, Y ) and T ( X, B ⊗ B ′ Y ) T ( X ⊗ B ⊗ B ′ , Y ) β r ( B ⊗ B ′ ,X,Y ) / / T ( X, B ( B ′ Y )) = (cid:15) (cid:15) T ( X ⊗ B, B ′ Y ) . β r ( B,X, B ′ Y ) / / β r ( B ′ ,X ⊗ B,Y ) O O (e) If A is closed, the families α l and α r form a Tambara module structure ifand only if the families β l and β r , corresponding under (a) and (c), satisfy thecondition T ( X, B Y A ) T ( A ⊗ X, B Y ) β l ( A,X, B Y ) / / T ( A ⊗ X ⊗ B, Y ) . β r ( B,A ⊗ X,Y ) (cid:15) (cid:15) T ( X ⊗ B, Y A ) β r ( B,X,Y A ) (cid:15) (cid:15) β l ( A,X ⊗ B,Y ) / / Proof.
The bijection of (a) is defined by the formulas β l ( A, X, Y ) = (cid:16) T ( X, Y A ) T ( A ⊗ X, A ⊗ Y A ) α l ( A,X,Y A ) / / T ( A ⊗ X, Y ) T ( A ⊗ X,e r ) / / (cid:17) and α l ( A, X, Y ) = (cid:16) T ( X, Y ) T ( X, ( A ⊗ Y ) A ) T ( X,d r ) / / T ( A ⊗ X, A ⊗ Y ) β l ( A,X,A ⊗ Y ) / / (cid:17) . That the processes are mutually inverse uses the adjunction identities on the mor-phisms e and d . The bijection of (c) is obtained dually by reversing the tensorproduct. Translation of the conditions from the α to the β as required for (b), (d)and (e) is straightforward. (cid:3) A left (respectively, right) Tambara module T on A will be called strong whenthe morphisms β l ( A, X, Y ) : T ( X, Y A ) / / T ( A ⊗ X, Y )(respectively, β r ( B, X, Y ) : T ( X, B Y ) / / T ( X ⊗ B, Y ))corresponding via Proposition 3.1 to the left (respectively, right) Tambara structure,are invertible. A Tambara module is called left (respectively, right ) strong when itis strong as a left (respectively, right) module and strong when it is both left andright strong. In particular, notice that the hom V -functor (= identity module) of A is a strong Tambara module. CRAIG PASTRO AND ROSS STREET
Proposition 3.2.
Suppose A is a monoidal V -category and T : A op ⊗ A / / V is a V -functor. If A is right (left) autonomous then every left (right) Tambaramodule is strong.Proof. If A ∗ denotes a right dual for A with unit η : I / / A ∗ ⊗ A then an inversefor β l is defined by the composite T ( A ⊗ X, Y ) T ( A ∗ ⊗ A ⊗ X, A ∗ ⊗ Y ) α l ( A ∗ ,A ⊗ X,Y ) / / T ( X, A ∗ ⊗ Y ) T ( η, / / . (cid:3) Write LTamb( A ) for the V -category whose objects are left Tambara modules T = ( T, α l ) and whose hom LTamb( A )( T, T ′ ) in V is defined to be the intersectionover all A , X and Y of the equalizers of the pairs of morphisms:[ A op ⊗ A , V ]( T, T ′ ) V ( T ( X, Y ) , T ′ ( A ⊗ X, A ⊗ Y )) V ( α l , ◦ pr A ⊗ X,A ⊗ Y / / V (1 ,α l ) ◦ pr X,Y / / . Equivalently, we can define the hom as an intersection of equalizers of pairs ofmorphisms:[ A op ⊗ A , V ]( T, T ′ ) V ( T ( X, Y A ) , T ′ ( A ⊗ X, Y )) V ( β l , ◦ pr A ⊗ X,Y / / V (1 ,β l ) ◦ pr X,Y A / / . Composition is defined so that we have a V -functor ι : LTamb( A ) / / [ A op ⊗ A , V ] which forgets the left module structure on T . In fact, LTamb( A ) becomes amonoidal V -category in such a way that the forgetful V -functor ι becomes strongmonoidal. For this, the monoidal structure on [ A op ⊗ A , V ] is the usual tensorproduct (= composition) of endomodules:( T ⊗ A T ′ )( X, Y ) = Z Z T ( X, Z ) ⊗ T ′ ( Z, Y ) . When T and T ′ are left Tambara modules, the left Tambara structure( T ⊗ A T ′ )( X, Y ) / / ( T ⊗ A T ′ )( A ⊗ X, A ⊗ Y )on T ⊗ A T ′ is defined by taking its composite with the coprojection copr Z into theabove coend to be the composite T ( X, Z ) ⊗ T ′ ( Z, Y ) T ( A ⊗ X, A ⊗ Z ) ⊗ T ′ ( A ⊗ Z, A ⊗ Y ) α l ⊗ α l / / ( T ⊗ A T ′ )( A ⊗ X, A ⊗ Y ) copr A ⊗ Z / / . Similarly we obtain monoidal V -categories RTamb( A ) and Tamb( A ) of right Tam-bara and all Tambara modules on A .We write LTamb s ( A ) for the full sub- V -category of LTamb( A ) consisting of thestrong left Tambara modules. We write Tamb ls ( A ), Tamb rs ( A ) and Tamb s ( A )for the full sub- V -categories of Tamb( A ) consisting of the left strong, right strongand strong Tambara modules respectively.If A is autonomous then Tamb( A ) = Tamb ls ( A ) = Tamb rs ( A ) = Tamb s ( A )by Proposition 3.2. OUBLES FOR MONOIDAL CATEGORIES 7 The Cayley functor
Consider a right closed monoidal V -category A . There is a Cayley V -functor Υ : [ A , V ] / / [ A op ⊗ A , V ]defined as follows. To each object F ∈ [ A , V ], define Υ( F ) = T F by T F ( X, Y ) = F ( Y X ) . The effect Υ
F,G : [ A , V ]( F, G ) / / [ A op ⊗ A , V ]( T F , T G ) of Υ on homs is definedby taking its composite with the projectionpr X,Y : [ A op ⊗ A , V ]( T F , T G ) / / V ( F ( Y X ) , G ( Y X ))to be the projectionpr Y X : [ A , V ]( F, G ) / / V ( F ( Y X ) , G ( Y X )) . Proposition 4.1.
The Cayley V -functor Υ is strong monoidal; it takes Day con-volution to composition of endomodules.Proof. We have the calculation:(Υ( F ) ⊗ A Υ( G ))( X, Y ) = Z Z Υ( F )( X, Z ) ⊗ Υ( G )( Z, Y )= Z Z F ( Z X ) ⊗ G ( Y Z ) ∼ = Z Z,U,V A ( U, Z X ) ⊗ F U ⊗ A ( V, Y Z ) ⊗ GV ∼ = Z Z,U,V A ( X ⊗ U, Z ) ⊗ F U ⊗ A ( Z ⊗ V, Y ) ⊗ GV ∼ = Z U,V A ( X ⊗ U ⊗ V, Y ) ⊗ F U ⊗ GV ∼ = Z U,V A ( U ⊗ V, Y X ) ⊗ F U ⊗ GV ∼ = Υ( F ∗ G )( X, Y ) , and of course Υ( A ( I, − ))( X, Y ) = A ( I, Y X ) ∼ = A ( X, Y ). (cid:3) In fact, Υ lands in the left Tambara modules by defining, for each F ∈ [ A , V ],the structure α l ( A, X, Y ) = (cid:16) F ( Y X ) F (( d r ) X ) / / F (( A ⊗ Y ) A ⊗ X ) (cid:17) on T F . It is helpful to observe that the β l corresponding to this α l (via Proposi-tion 3.1) is given by the identity β l ( A, X, Y ) = (cid:16) F ( Y A ⊗ X ) / / F ( Y A ⊗ X ) (cid:17) , showing that T F becomes a strong left module. To see that there is a V -functorˆΥ : [ A , V ] / / LTamb s ( A ) satisfying ι ◦ ˆΥ = Υ, we merely observe thatpr A ⊗ X,Y ◦ Υ F,G = pr Y A ⊗ X = pr ( Y A ) X = pr X,Y A ◦ Υ F,G . CRAIG PASTRO AND ROSS STREET
Proposition 4.2. If A is a right closed monoidal V -category then the V -functor ˆΥ : [ A , V ] / / LTamb s ( A ) is an equivalence.Proof. Define ζ : LTamb( A )( T F , T G ) / / [ A , V ]( F, G ) by pr Y ◦ ζ = pr I,Y ◦ ι T F ,T G .Then pr Y ◦ ζ ◦ ˆΥ F,G = pr
I,Y ◦ ι T F ,T G ◦ ˆΥ F,G = pr
I,Y ◦ Υ F,G = pr Y and pr X,Y ◦ ι T F ,T G ◦ ˆΥ F,G ◦ ζ = pr X,Y ◦ Υ F,G ◦ ζ = pr Y X ◦ ζ = pr I,Y X ◦ ι T F ,T G = pr X,Y ◦ ι T F ,T G . It follows that ζ is the inverse of ˆΥ F,G , so that ˆΥ is fully faithful. To see that ˆΥ isessentially surjective on objects, take a strong left module T . Put F Y = T ( I, Y )as a V -functor in Y . Then the isomorphism β l ( X, I, Y ) yields T F ( X, Y ) = F ( Y X ) = T ( I, Y X ) ∼ = T ( X, Y );so ˆΥ( F ) ∼ = T . (cid:3) Now suppose we have an object (
F, θ ) of the lax centre Z l [ A , V ] of [ A , V ]. Then T F becomes a right Tambara module by defining α r ( B, X, Y ) = (cid:16) F ( Y X ) F ( B ( Y ⊗ B ) X ) F (( d l ) X ) / / F ( Y ⊗ B ) X ⊗ Bθ B, ( Y ⊗ B ) X / / (cid:17) . If A is left closed, the β r corresponding to this α r (via Proposition 3.1) is definedby β r ( B, X, Y ) = (cid:16) F ( B Y X ) θ B,Y X / / F ( Y X ⊗ B ) (cid:17) . It is easy to see that, in this way, T F = ˆΥ( F ) actually becomes a (two-sided)Tambara module which we write as ˆΥ( F, θ ), and we have a V -functorˆΥ : Z l [ A , V ] / / Tamb ls ( A ) . Proposition 4.3. If A is a closed monoidal V -category then the V -functor ˆΥ : Z l [ A , V ] / / Tamb ls ( A ) is an equivalence which restricts to an equivalence ˆΥ : Z [ A , V ] / / Tamb s ( A ) . Proof.
The proof of full faithfulness proceeds along the lines of the beginning ofthe proof of Proposition 4.2. For essential surjectivity on objects, take a left strongTambara module (
T, α ). Then β l ( A, X, Y ) : T ( X, Y A ) / / T ( A ⊗ X, Y ) is invert-ible. Define the V -functor F : A / / V by F X = T ( I, X ) as in the proof ofProposition 4.2, and define θ A,Y : F ( A Y ) / / F ( Y A ) to be the composite T ( I, A Y ) T ( A, Y ) β r ( A,I,Y ) / / T ( I, Y A ) β l ( A,I,Y ) − / / . This is easily seen to yield an object (
F, θ ) of the lax centre Z l [ A , V ] with ˆΥ( F, θ ) ∼ = T F . Thus we have the first equivalence. Clearly θ is invertible if and only if β r is;the second equivalence follows. (cid:3) OUBLES FOR MONOIDAL CATEGORIES 9 The double monad
Tambara modules are actually Eilenberg-Moore coalgebras for a fairly obviouscomonad on [ A op ⊗ A , V ]. We begin by looking at the case of left modules.Let Θ l : [ A op ⊗ A , V ] / / [ A op ⊗ A , V ] be the V -functor defined by the endΘ l ( T )( X, Y ) = Z A T ( A ⊗ X, A ⊗ Y ) . There is a V -natural family ǫ T : Θ l ( T ) / / T defined by the projectionspr I : Z A T ( A ⊗ X, A ⊗ Y ) / / T ( X, Y ) . There is a V -natural family δ T : Θ l ( T ) / / Θ l (Θ l ( T )) defined by taking its com-posite with the projectionpr B,C : Z B,C T ( B ⊗ C ⊗ X, B ⊗ C ⊗ Y ) / / T ( B ⊗ C ⊗ X, B ⊗ C ⊗ Y )to be the projectionpr B ⊗ C : Z A T ( A ⊗ X, A ⊗ Y ) / / T ( B ⊗ C ⊗ X, B ⊗ C ⊗ Y ) . It is now easily checked that Θ l = (Θ l , δ, ǫ ) is a comonad on [ A op ⊗ A , V ].There are also a comonad Θ r on [ A op ⊗ A , V ], a distributive law Θ r Θ l ∼ = Θ l Θ r ,and a comonad Θ = Θ r Θ l :Θ r ( T )( X, Y ) = Z B T ( X ⊗ B, Y ⊗ B )and Θ( T )( X, Y ) = Z A,B T ( A ⊗ X ⊗ B, A ⊗ Y ⊗ B ) . We can easily identify the V -categories of Eilenberg-Moore coalgebras for thesethree comonads. Proposition 5.1.
There are isomorphisms of V -categories • [ A op ⊗ A , V ] Θ l ∼ = LTamb( A ) , • [ A op ⊗ A , V ] Θ r ∼ = RTamb( A ) , and • [ A op ⊗ A , V ] Θ ∼ = Tamb( A ) . In fact, Θ l , Θ r and Θ are all monoidal comonads on [ A op ⊗ A , V ]. For example,the structure on Θ l is provided by the V -natural transformations Θ l ( T ) ⊗ A Θ l ( T ′ ) / / Θ l ( T ⊗ A T ′ ) and A ( − , − ) / / Θ l ( A ( − , − )) with components(1) Z Z Z A T ( A ⊗ X, A ⊗ Z ) ⊗ Z B T ′ ( B ⊗ X, B ⊗ Z ) / / Z C Z U T ( C ⊗ X, U ) ⊗ T ′ ( U, C ⊗ Y )and(2) A ( X, Y ) / / Z A A ( A ⊗ X, A ⊗ Y )defined as follows. The morphism (1) is determined by its precomposite with thecoprojection copr Z and postcomposite with the projection pr C ; the result is defined to be the composite Z A T ( A ⊗ X, A ⊗ Z ) ⊗ Z B T ′ ( B ⊗ X, B ⊗ Z ) pr C ⊗ pr C / / T ( C ⊗ X, C ⊗ Z ) ⊗ T ′ ( C ⊗ Z, C ⊗ Y ) copr C ⊗ Z / / Z U T ( C ⊗ X, U ) ⊗ T ′ ( U, C ⊗ Y ) . The morphism (2) is simply the coprojection copr I . It follows that [ A op ⊗ A , V ] Θ l becomes monoidal with the underlying functor becoming strong monoidal; see [Moe02]and [McC02]. Clearly we have: Proposition 5.2.
The isomorphisms of Proposition 5.1 are monoidal.
The next thing to observe is that Θ l , Θ r and Θ all have left adjoints Φ l , Φ r and Φwhich therefore become opmonoidal monads whose V -categories of Eilenberg-Moorealgebras are monoidally isomorphic to LTamb( A ), RTamb( A ) and Tamb( A ), re-spectively. Straightforward applications of the Yoneda Lemma, show that the for-mulas for these adjoints areΦ l ( S )( U, V ) = Z A,X,Y A ( U, A ⊗ X ) ⊗ A ( A ⊗ Y, V ) ⊗ S ( X, Y ) , Φ r ( S )( U, V ) = Z B,X,Y A ( U, X ⊗ B ) ⊗ A ( Y ⊗ B, V ) ⊗ S ( X, Y ) , andΦ( S )( U, V ) = Z A,B,X,Y A ( U, A ⊗ X ⊗ B ) ⊗ A ( A ⊗ Y ⊗ B, V ) ⊗ S ( X, Y ) . Recall that left adjoint V -functors Ψ : [ X op , V ] / / [ Y op , V ] are equivalent to V -functors ˇΨ : Y op ⊗ X / / V , which are also called modules ˇΨ : X (cid:31) / / Y from X to Y . The equivalence is defined by:ˇΨ( Y, X ) = Ψ( X ( − , X ))( Y )and Ψ( M )( Y ) = Z X ˇΨ( Y, X ) ⊗ M ( X ) . It follows that Φ l , Φ r and Φ determine monads ˇΦ l , ˇΦ r and ˇΦ on A op ⊗ A inthe bicategory V - Mod . The formulas are:ˇΦ l ( X, Y, U, V ) = Z A A ( U, A ⊗ X ) ⊗ A ( A ⊗ Y, V ) , ˇΦ r ( X, Y, U, V ) = Z B A ( U, X ⊗ B ) ⊗ A ( Y ⊗ B, V ) , andˇΦ( X, Y, U, V ) = Z A,B A ( U, A ⊗ X ⊗ B ) ⊗ A ( A ⊗ Y ⊗ B, V ) . Doubles
The bicategory V - Mod admits the Kleisli construction for monads. Write D l A , D r A and DA for the Kleisli V -categories for the monads ˇΦ l , ˇΦ r and ˇΦ on A op ⊗ A in the bicategory V - Mod . We call them the left double , right double and double OUBLES FOR MONOIDAL CATEGORIES 11 of the monoidal V -category A . They all have the same objects as A op ⊗ A . Thehoms are defined by D l A (( X, Y ) , ( U, V )) = Z A A ( U, A ⊗ X ) ⊗ A ( A ⊗ Y, V ) , D r A (( X, Y ) , ( U, V )) = Z B A ( U, X ⊗ B ) ⊗ A ( Y ⊗ B, V ) , and DA (( X, Y ) , ( U, V )) = Z A,B A ( U, A ⊗ X ⊗ B ) ⊗ A ( A ⊗ Y ⊗ B, V ) . Proposition 6.1.
There are canonical equivalences of V -categories: • Ξ l : LTamb( A ) ≃ [ D l A , V ] , • Ξ r : RTamb( A ) ≃ [ D r A , V ] , and • Ξ : Tamb( A ) ≃ [ DA , V ] . It follows from the main result of Day [Day70] that these doubles D l A , D r A and DA all admit promonoidal structures ( P l , J l ), ( P r , J r ) and ( P, J ) for whichthe equivalences in Proposition 6.1 become monoidal when the right-hand sides aregiven the corresponding convolution structures. For example, we calculate that P l and J l are as follows: P l (( X, Y ) , ( U, V ); (
H, K )) = ( D l A (( X, Y ) , − ) ⊗ A D l A (( U, V ) , − ))( H, K )= Z Z,A,B A ( H, A ⊗ X ) ⊗ A ( A ⊗ Y, Z ) ⊗ A ( Z, B ⊗ U ) ⊗ A ( B ⊗ V, K )= Z A,B A ( H, A ⊗ X ) ⊗ A ( A ⊗ Y, B ⊗ U ) ⊗ A ( B ⊗ V, K )and J l ( H, K ) = A ( H, K ) . Furthermore, there are some special morphisms that exist in these doubles D l A , D r A and DA . Let ˜ α l : ( X, Y ) / / ( A ⊗ X, A ⊗ Y ) denote the morphism in D l A defined by the composite I A ( A ⊗ X, A ⊗ X ) ⊗ A ( A ⊗ Y, A ⊗ Y ) j A ⊗ X ⊗ j A ⊗ Y / / D l A (( X, Y ) , ( A ⊗ X, A ⊗ Y )) copr A / / . The V -functor Ξ l has the property that Ξ l ( T, α l )( X, Y ) = T ( X, Y ) and Ξ l ( T, α l )(˜ α l ) = α l . When A is right closed, we let ˜ β l : ( X, Y A ) / / ( A ⊗ X, Y ) denote the mor-phism in D l A defined by the composite I A ( A ⊗ X, A ⊗ X ) ⊗ A ( A ⊗ Y A , Y ) j A ⊗ X ⊗ e r / / D l A (( X, Y A ) , ( A ⊗ X, Y )) copr A / / . Then Ξ l ( T, α l )( ˜ β l ) = β l .Similarly, we have the morphism ˜ α r : ( X, Y ) / / ( X ⊗ B, Y ⊗ B ) in D r A , andalso, when A is left closed, the morphism ˜ β r : ( X, B Y ) / / ( X ⊗ B, Y ).There are V -functors D l A / / DA o o D r A which are the identity functionson objects and are defined on homs using projections with B = I for the left leg and the projections A = I for the second leg. In this way, the morphisms ˜ α l and˜ α r can be regarded also as morphisms of DA . Under closedness assumptions, themorphisms ˜ β l and ˜ β r can also be regarded as morphisms of DA .Let Σ l denote the set of morphisms ˜ β l : ( X, Y A ) / / ( A ⊗ X, Y ), let Σ r denotethe set of morphisms ˜ β r : ( X, B Y ) / / ( X ⊗ B, Y ), and let Σ denote the set ofmorphisms Σ = Σ l ∪ Σ r . Under appropriate closedness assumptions on A , we canform various V -categories of fractions such as: • L DA = D l A [Σ − l ] and R DA = D r A [Σ − r ], • D ls A = DA [Σ − l ] and D rs A = DA [Σ − r ], and • D s A = DA [Σ − ].The following result is now automatic. Theorem 6.2.
For a closed monoidal V -category A , there are equivalences of V -categories: • [L DA , V ] ≃ LTamb s ( A ) ≃ [ A , V ] , • [ D ls A , V ] ≃ Tamb ls ( A ) ≃ Z l [ A , V ] , and • [ D s A , V ] ≃ Tamb s ( A ) ≃ Z [ A , V ] . The first equivalence of Theorem 6.2 implies that L DA and A are Morita equiv-alent. This begs the question of whether there is a V -functor relating them moredirectly. Indeed there is. We have a V -functorΠ : D l A / / A defined on objects by Π( X, Y ) = Y X and by defining the effectΠ : D l A (( X, Y ) , ( U, V )) / / A ( Y X , V U )on hom objects to have its composite with the A -coprojection equal to the composite A ( U, A ⊗ X ) ⊗ A ( A ⊗ Y, V ) V ( − ) ⊗ ( − ) A ⊗ X / / A ( V A ⊗ X , V U ) ⊗ A (( A ⊗ Y ) A ⊗ X , V A ⊗ X ) composition / / A (( A ⊗ Y ) A ⊗ X , V U ) A (( d r ) X ,V U ) / / A ( Y X , V U ) . It is easy to see that Π takes the morphisms ˜ β l : ( X, Y A ) / / ( A ⊗ X, Y ) toisomorphisms. So Π induces a V -functorˆΠ : L D l A / / A ;this induces the first equivalence of Theorem 6.2.For closed monoidal A , the second and third equivalences of Theorem 6.2 showthat both the lax centre and the centre of the convolution monoidal V -category[ A , V ] are again functor V -categories [ D ls A , V ] and [ D s A , V ]. Since Z l [ A , V ] and Z [ A , V ] are monoidal with the tensor products colimit preserving in each variable,using the correspondence in [Day70], there are lax braided and braided promonoidalstructures on D ls A and D s A which are such that [ D ls A , V ] and [ D s A , V ] becomeclosed monoidal under convolution, and the equivalences of Theorem 6.2 becomelax braided and braided monoidal equivalences. OUBLES FOR MONOIDAL CATEGORIES 13
Remark.
We are grateful to Brian Day for pointing out that the V -category A M appearing in [DS07] is equivalent to the full sub- V -category of DA consisting ofthe objects of the form ( I, Y ).He also pointed out that a consequence of Theorem 6.2 is that the centre of V as a V -category is equivalent to V itself. This also can be seen directly by usingthe V -naturality in X of the centre structure u X : A ⊗ X / / X ⊗ A on an object A of V , and the fact that u I = 1, to deduce that u X = c A,X (the symmetry of V ).Generally, the centre of V as a monoidal Set -category is not equivalent to V . References [Day70] Brian Day. On closed categories of functors, in
Reports of the Midwest Category SeminarIV
E-mail address : { craig, street } @[email protected]