Doubles for monoidal categories

DOUBLES F OR MONOIDAL CA TEGORIES CRAIG P ASTRO AND ROSS STREET De dic ate d to Walter Tholen on his 60th birthday Abstract. In a recent pap er, Daisuke T amba ra defined tw o-sided actions on an endomo dule (= endodistributor) of a monoidal V -cat egory A . When A is autonomous (= ri gid = compact), he show ed that the V -category (that we call T amb( A )) of so-equipped endomodules (that w e call T ambara mo dules) is equiv alen t to the monoidal cen tre Z [ A , V ] of the conv olution monoidal V - category [ A , V ]. Our paper extends these ideas somewhat. F or general A , we construct a promonoidal V -category D A (which w e s ugge st should be called the double of A ) wi th an equiv alence [ D A , V ] ≃ T amb( A ). When A i s closed, we define str ong (resp ectiv ely , left strong) T ambara modul es and sho w that these constitute a V -category T amb s ( A ) (resp ectiv ely , T amb ls ( A )) which i s equiv alent to the cen tre (respectively , lax cent re) of [ A , V ] . W e construct l ocalizations D s A a nd D ls A o f D A suc h that there are equiv a- lences T amb s ( A ) ≃ [ D s A , V ] and T amb ls ( A ) ≃ [ D ls A , V ]. When A is autonomous, ev ery T ambara mo dule is strong; this impl ies an equiv alence Z [ A , V ] ≃ [ D A , V ]. 1. Introduction F or V -ca tegories A and B , a mo dule T : A  / / B (a lso c alled “bimo dule”, “profunctor” , and “distributor”) is a V -functor T : B op ⊗ A / / V . F or a monoidal V -ca tegory A , T ambara [T am06] defined tw o-sided actions α of A on an endomo dule T : A  / / A . Wh en A is autonomous (also ca lled “r igid” or “com- pact”) he show ed that the V -categ o ry T amb( A ) o f T ambara mo dules ( T , α ) is equiv ale nt to the monoidal c e n tre Z [ A , V ] of the conv olution monoidal V -categor y [ A , V ]. Our pap er extends these ideas in four wa ys: (1) our base monoidal category V is quite general (as in [Ke l82]) not just vector spaces; (2) our results a re ma inly for a close d monoidal V -categ ory A , genera lizing the autono mous case; (3) we show the co nnection with the lax centre as well as the centre; and, (4) we intro duce the double D A o f a mono idal V - category A a nd some loca l- izations of it, and relate these to T a m bara mo dules. Date : M a y 29, 2018. Key wor ds and phr ases. monoidal centre, D r infeld double, monoidal category , Da y conv olution. The fir s t author gratefully ackno wl edge s supp ort of an i n ternationa l M acqua rie Universit y Researc h Sc holarship while the s econ d gratefully ackno wl edges supp ort of the Australian Research Council Di s co v ery Gr an t DP0771252. 1 2 CRAIG P ASTR O AND ROSS STREET Our principa l g o al is to give conditions under which the centre and lax centre of a V -v alued V -functor monoidal V -catego r y is a gain such. Some results in this direction can be found in [DS07]. F or g eneral monoidal A , we construct a pro monoidal V -ca tegory D A with an equiv alence [ D A , V ] ≃ T amb( A ). When A is closed, we define when a T ambara mo dule is (left) strong and show that these cons titute a V -ca teg ory (T amb ls ( A )) T amb s ( A ) which is equiv a len t to the (lax) c en tre of [ A , V ]. W e construct lo calizations D s A and D ls A of D A such tha t there ar e equiv alences T amb s ( A ) ≃ [ D s A , V ] and T amb ls ( A ) ≃ [ D ls A , V ]. When A is autonomo us, ev- ery T ambara mo dule is strong , which implies an equiv alence Z [ A , V ] ≃ [ D A , V ]. These results should b e compar ed with those o f [DS07 ] where the lax centre of [ A , V ] is shown gener ally to b e a full sub- V - category of a functor V -ca tegory [ A M , V ] which also b ecomes a n equiv alence Z [ A , V ] ≃ [ A M , V ] when A is a u- tonomous. As we were completing this pap er, Ig nacio Lop ez F ranco sent us his preprint [LF0 7] which has some results in common with our s. As an example for V -mo dules of his general constructions on pseudomonoids, he is also led to wha t we call the double monad. 2. Centres and convolution W e work with ca tegories enriched in a base monoida l categor y V a s used by Kelly [Kel8 2 ]. It is symmetric, c losed, complete a nd coc o mplete. Let A denote a closed monoidal V -categor y . W e denote the tensor pro duct b y A ⊗ B and the unit b y I in the hop e that this will cause no confusion with the same symbols used for the ba se V itself. W e hav e V -natural isomorphisms A ( A, B C ) ∼ = A ( A ⊗ B , C ) ∼ = A ( B , C A ) defined by ev alua tion and co ev aluation 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 ar e ca nonical isomorphis ms 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 iden tifications just as we do with the a sso c iativit y and unit iso morphisms. W e a lso write B C A for B ( C A ). The Day co n v olution monoidal str ucture [Da y70] on the V -c a tegory [ A , V ] of V -functors fr o m A to V co nsists o f the tensor pro duct F ∗ G and unit J defined by ( 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 J A = A ( I , A ) . DOUBLES FOR MONOIDAL CA TE GORIE S 3 In particular , ( F ∗ A ( A, − )) B ∼ = F ( A B ) and ( A ( A, − ) ∗ G ) B ∼ = G ( B A ) . The centre of a mono ida l categor y was defined in [JS91] and the lax cent re was defined, for example, in [DPS07]. Since the repr esen tables ar e dense in [ A , V ], an ob ject of the lax c ent r e Z l [ A , V ] of [ A , V ] is a pair ( F, θ ) consisting of F ∈ [ A , V ] and a V -natur al family θ of morphisms θ A,B : F ( A B ) / / F ( B A ) such that the diag rams F ( A ⊗ B C ) F ( C A ⊗ B ) θ A ⊗ B,C / / F ( A ( B C )) =   F ( B C A ) θ A, B C " " E E E E E E E E E F (( C A ) B ) θ B,C A < < y y y y y y y y y = O O and F ( I A ) F ( A I ) θ I ,A / / F A =   ? ? ? ? ? ? = ? ?       commute. The hom ob ject Z l [ A , V ](( F , θ ) , ( G, φ )) is defined to b e the equalizer of t wo obvious morphis ms out of [ A , V ]( F, G ). The c ent r e Z [ A , V ] of [ A , V ] is the full s ub- V -categor y of Z l [ A , V ] co ns isting of those ob jects ( F, θ ) with θ inv ertible. 3. T ambara modules Let A denote a monoidal V -catego ry . W e do no t need A to b e clos ed for the definition of T ambara mo dule although we will require this r estriction again later . A left T amb ar a mo dule o n A is a V -functor T : A op ⊗ A / / V together with a family of morphisms α l ( A, X , Y ) : T ( X, Y ) / / T ( A ⊗ X , A ⊗ Y ) which are V -natur al in each of the o b jects A , X and Y , sa tis fying the tw o eq uations 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 ) $ $ I I I I I I I I I I I I T ( A ′ ⊗ X, A ′ ⊗ Y ) α l ( A ′ ,X,Y ) / / α l ( A,A ′ ⊗ X,A ′ ⊗ Y ) z z u u u u u u u u u u u u Similarly , a right T amb ar a mo dule on A is a V -functor T : A op ⊗ A / / V together with a family o f morphisms α r ( B , X , Y ) : T ( X , Y ) / / T ( X ⊗ B , Y ⊗ B ) 4 CRAIG P ASTR O AND ROSS STREET which a re V -natura l in ea c h of the ob jects B , X a nd Y , satisfying the tw o 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 ) $ $ I I I I I I I I I I I I T ( X ⊗ B , Y ⊗ B ) α r ( B ,X,Y ) / / α r ( B ′ ,B ⊗ X,B ⊗ Y ) z z u u u u u u u u u u u u A T amb ar a mo dule ( T , α ) o n A is a V -functor T : A op ⊗ A / / V together with b oth left and rig h t T a m bara mo dule structures satisfying the “bimo dule” compatibility co ndition 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 )   T ( X ⊗ B , Y ⊗ B ) α r ( B ,X,Y )   α l ( A,X ⊗ B ,Y ⊗ B ) / / The morphism defined to be the diag onal 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 expr ess a T ambara mo dule struc tur e purely in terms of this, how ever, we need to refer to the left and r igh t structures b elow. Prop osition 3.1 . Su pp ose A is a monoidal V - c ate gory and T : A op ⊗ A / / V is a V - fu n ctor. (a) If A is right close d, ther e is a bije ction b et we en V -natura l families of mor- phisms α l ( A, X , Y ) : T ( X, Y ) / / T ( A ⊗ X , A ⊗ Y ) and V -natur al families of morphisms β l ( A, X , Y ) : T ( X, Y A ) / / T ( A ⊗ X , Y ) . (b) Under t he bije ction of (a), the family α l is a left T amb ar a structu re if and only if the family β l satisfies the two e quations 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 ′ ) =   T ( A ′ ⊗ X, Y A ) . β l ( A ′ ,X,Y A ) / / β l ( A,A ′ ⊗ X,Y ) O O (c) If A is left close d, t her e is a bije ction b etwe en V -natur al families of morphisms α r ( B , X , Y ) : T ( X , Y ) / / T ( X ⊗ B , Y ⊗ B ) and V -natur al families of morphisms β r ( B , X , Y ) : T ( X , B Y ) / / T ( X ⊗ B , Y ) . DOUBLES FOR MONOIDAL CA TE GORIE S 5 (d) Under the bije ction of (c), the family α r is a right T amb ar a structure if and only if the family β r satisfies the two e quations 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 )) =   T ( X ⊗ B , B ′ Y ) . β r ( B ,X, B ′ Y ) / / β r ( B ′ ,X ⊗ B ,Y ) O O (e) If A is close d, the families α l and α r form a T amb ar a mo dule structu re if and only if the families β l and β r , c orr esp onding under (a) and (c), satisfy the c ondition 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 )   T ( X ⊗ B , Y A ) β r ( B ,X,Y A )   β l ( A,X ⊗ B ,Y ) / / Pr o of. The bijection of (a) is defined b y the formulas β l ( A, X , Y ) =  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 ) / /  and α l ( A, X , Y ) =  T ( X, Y ) T ( X, ( A ⊗ Y ) A ) T ( X, d r ) / / T ( A ⊗ X , A ⊗ Y ) β l ( A,X,A ⊗ Y ) / /  . That the pro cesses ar e mutually inv erse uses the adjunction iden tities o n the mor- phisms e a nd d . The bijection of (c) is o btained dually by reversing the tensor pro duct. T ransla tio n of the conditio ns from the α to the β as r equired for (b), (d) and (e) is straig h tforward.  A left (res pectively , r igh t) T ambara mo dule T on A will b e called str ong when the mor phis ms β l ( A, X , Y ) : T ( X , Y A ) / / T ( A ⊗ X , Y ) (resp ectiv ely , β r ( B , X , Y ) : T ( X , B Y ) / / T ( X ⊗ B , Y )) corres p onding via Pr opo s ition 3.1 to the left (resp ectively , r igh t) T ambara structur e, are in vertible. A T ambara module is ca lled left (resp e ctiv ely , right ) str ong when it is strong as a le ft (resp ectively , right) mo dule and stro ng when it is b oth left and right strong. In particular, notice that the ho m V -functor (= identit y mo dule) of A is a strong T ambara mo dule. 6 CRAIG P ASTR O AND ROSS STREET Prop osition 3.2 . Su pp ose A is a monoidal V - c ate gory and T : A op ⊗ A / / V is a V -functor. If A is right (left) autonomous then every left (right) T amb ar a mo dule is str ong. Pr o of. If A ∗ denotes a right dual for A with unit η : I / / A ∗ ⊗ A then an inv erse for β l is defined by the comp osite T ( A ⊗ X , Y ) T ( A ∗ ⊗ A ⊗ X , A ∗ ⊗ Y ) α l ( A ∗ ,A ⊗ X,Y ) / / T ( X, A ∗ ⊗ Y ) T ( η , 1) / / .  W rite L T amb( A ) for the V -categ ory who se ob jects are left T ambara mo dules T = ( T , α l ) and whose hom L T amb( A )( T , T ′ ) in V is defined to b e the intersection ov er all A , X a nd Y o f the equa lizers of the pairs of mo r phisms: [ A op ⊗ A , V ]( T , T ′ ) V ( T ( X , Y ) , T ′ ( A ⊗ X, A ⊗ Y )) V ( α l , 1) ◦ pr A ⊗ X,A ⊗ Y / / V (1 ,α l ) ◦ pr X,Y / / . Equiv a len tly , we can define the hom a s an intersection of equa lizers of pairs of morphisms: [ A op ⊗ A , V ]( T , T ′ ) V ( T ( X , Y A ) , T ′ ( A ⊗ X, Y )) V ( β l , 1) ◦ pr A ⊗ X,Y / / V (1 ,β l ) ◦ pr X,Y A / / . Comp osition is defined so that we hav e a V -functor ι : L T amb( A ) / / [ A op ⊗ A , V ] which forgets the left mo dule s tructure on T . In fact, L T a m b( A ) beco mes a monoidal V -ca tegory in such a wa y that the forgetful V - functor ι b ecomes str ong monoidal. F or this, the monoida l structure on [ A op ⊗ A , V ] is the usual tensor pro duct (= co mp osition) o f endomo dules: ( T ⊗ A T ′ )( X, Y ) = Z Z T ( X, Z ) ⊗ T ′ ( Z, Y ) . When T and T ′ are left T ambara mo dules, the left T ambara structure ( T ⊗ A T ′ )( X, Y ) / / ( T ⊗ A T ′ )( A ⊗ X, A ⊗ Y ) on T ⊗ A T ′ is defined by tak ing its compos ite with the copro jection copr Z int o the ab o ve co end to be the co mposite 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 w e obtain monoidal V -categor ies R T a m b( A ) a nd T amb( A ) of right T am- bara and all T a m bara modules on A . W e write L T amb s ( A ) for the full sub- V -catego ry of L T amb( A ) consisting of the strong left T a m bara mo dules. W e wr ite T amb ls ( A ), T amb r s ( A ) and T amb s ( A ) for the full sub- V -ca teg ories of T amb( A ) consisting of the left s trong, rig h t s tr ong and strong T ambara mo dules resp ectively . If A is a utonomous then T a m b( A ) = T amb ls ( A ) = T a mb r s ( A ) = T a mb s ( A ) by Pr opo s ition 3.2. DOUBLES FOR MONOIDAL CA TE GORIE S 7 4. The Ca yley functor Consider a right c lo sed monoidal V -catego r y A . Ther e is a Cayley V -fu nctor Υ : [ A , V ] / / [ A op ⊗ A , V ] defined as follows. T o each ob ject 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 defined by taking its co mposite with the pro jectio n pr X,Y : [ A op ⊗ A , V ]( T F , T G ) / / V ( F ( Y X ) , G ( Y X )) to b e the pr o jection pr Y X : [ A , V ]( F , G ) / / V ( F ( Y X ) , G ( Y X )) . Prop osition 4.1. The Cayley V - functor Υ is str ong monoidal; it takes Day c on- volution to c omp osition of endomo dules. Pr o of. W e hav e 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 ).  In fact, Υ la nds in the left T ambara mo dules by defining, for each F ∈ [ A , V ], the structur e α l ( A, X , Y ) =  F ( Y X ) F (( d r ) X ) / / F (( A ⊗ Y ) A ⊗ X )  on T F . It is helpful to observe that the β l corres p onding to this α l (via Prop osi- tion 3.1) is given b y the iden tity β l ( A, X , Y ) =  F ( Y A ⊗ X ) 1 / / F ( Y A ⊗ X )  , showing that T F bec omes a strong left mo dule. T o see that there is a V -functor ˆ Υ : [ A , V ] / / L T amb s ( A ) sa tisfying ι ◦ ˆ Υ = Υ , w e merely o bserve that pr A ⊗ X,Y ◦ Υ F, G = pr Y A ⊗ X = pr ( Y A ) X = pr X,Y A ◦ Υ F, G . 8 CRAIG P ASTR O AND ROSS STREET Prop osition 4 .2. If A is a right close d monoidal V -c ate gory then the V -functor ˆ Υ : [ A , V ] / / L T amb s ( A ) is an e quivalenc e. Pr o of. Define ζ : L T a m b( 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 ◦ ˆ Υ F, G = pr I ,Y ◦ ι T F ,T G ◦ ˆ Υ F, G = pr I ,Y ◦ Υ F, G = pr Y and pr X,Y ◦ ι T F ,T G ◦ ˆ Υ 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 inv erse o f ˆ Υ F, G , so that ˆ Υ is fully faithful. T o see that ˆ Υ is essentially surjective on o b jects, take a strong left mo dule T . P ut F Y = T ( I , Y ) as a V -functor in Y . Then the isomo rphism β l ( X, I , Y ) y ie lds T F ( X, Y ) = F ( Y X ) = T ( I , Y X ) ∼ = T ( X, Y ); so ˆ Υ( F ) ∼ = T .  Now supp ose we hav e an ob ject ( F , θ ) of the lax centre Z l [ A , V ] of [ A , V ]. Then T F bec omes a right T ambara mo dule by defining α r ( B , X , Y ) =  F ( Y X ) F ( B ( Y ⊗ B ) X ) F (( d l ) X ) / / F ( Y ⊗ B ) X ⊗ B θ B, ( Y ⊗ B ) X / /  . If A is left closed, the β r corres p onding to this α r (via P rop o sition 3 .1) is defined by β r ( B , X , Y ) =  F ( B Y X ) θ B,Y X / / F ( Y X ⊗ B )  . It is easy to see tha t, in this wa y , T F = ˆ Υ( F ) actually b ecomes a (tw o-sided) T ambara module which w e write a s ˆ Υ( F, θ ), and we ha ve a V -functor ˆ Υ : Z l [ A , V ] / / T amb ls ( A ) . Prop osition 4 .3. If A is a close d monoida l V -c ate gory then the V -functor ˆ Υ : Z l [ A , V ] / / T amb ls ( A ) is an e quivalenc e which r est ricts to an e qu ival enc e ˆ Υ : Z [ A , V ] / / T amb s ( A ) . Pr o of. The pro of of full faithfulness pro ceeds alo ng the lines of the b eginning of the pro of of P ropo sition 4.2. F or essential sur jectivit y on o b jects, take a left strong T ambara mo dule ( T , α ). Then β l ( A, X , Y ) : T ( X , Y A ) / / T ( A ⊗ X, Y ) is inv ert- ible. Define the V -functor F : A / / V b y F X = T ( I , X ) as in the pro of of Prop osition 4.2, and define θ A,Y : F ( A Y ) / / F ( Y A ) to b e the compo site T ( I , A Y ) T ( A, Y ) β r ( A,I ,Y ) / / T ( I , Y A ) β l ( A,I ,Y ) − 1 / / . This is ea s ily seen to yield an ob ject ( F , θ ) of the la x centre Z l [ A , V ] with ˆ Υ( F, θ ) ∼ = T F . Th us w e have the first equiv alence . Clearly θ is invertible if and only if β r is; the seco nd equiv alence fo llows.  DOUBLES FOR MONOIDAL CA TE GORIE S 9 5. The double monad T ambara mo dules are actually E ile n berg -Moo re coalgebra s for a fairly o b vious comonad on [ A op ⊗ A , V ]. W e beg in b y lo oking at the cas e of left mo dules. Let Θ l : [ A op ⊗ A , V ] / / [ A op ⊗ A , V ] b e the V -functor defined by the end Θ 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 pr o jections pr I : Z A T ( A ⊗ X , A ⊗ Y ) / / T ( X, Y ) . There is a V -na tur al family δ T : Θ l ( T ) / / Θ l (Θ l ( T )) defined b y taking its com- po site with the pro jectio n pr B ,C : Z B ,C T ( B ⊗ C ⊗ X, B ⊗ C ⊗ Y ) / / T ( B ⊗ C ⊗ X, B ⊗ C ⊗ Y ) to b e the pr o jection pr B ⊗ C : Z A T ( A ⊗ X , A ⊗ Y ) / / T ( B ⊗ C ⊗ X , B ⊗ C ⊗ Y ) . It is now easily c heck ed tha t Θ l = (Θ l , δ, ǫ ) is a comonad on [ A op ⊗ A , V ]. There ar e als o a comona d Θ r on [ A op ⊗ A , V ], a distributive law Θ r Θ l ∼ = Θ l Θ r , and a co monad 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 ) . W e can easily identify the V - categories of Eile n berg -Moo re coalgebra s for these three comona ds. Prop osition 5 .1. The r e ar e isomorph isms of V -c ate gories • [ A op ⊗ A , V ] Θ l ∼ = L T amb( A ) , • [ A op ⊗ A , V ] Θ r ∼ = R T a m b( A ) , and • [ A op ⊗ A , V ] Θ ∼ = T amb( A ) . In fact, Θ l , Θ r and Θ are all monoidal comonads on [ A op ⊗ A , V ]. F or exa mple, the structur e on Θ l is provided by the V -natural tr ansformations Θ l ( T ) ⊗ A Θ l ( T ′ ) / / Θ l ( T ⊗ A T ′ ) and A ( − , − ) / / Θ l ( A ( − , − )) with comp onents (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 a s follows. The morphism (1) is determined by its pr ecompo s ite with the copro jection copr Z and p ostcomp osite with the pro jection pr C ; the r esult is defined 10 CRAIG P ASTR O AND ROSS STREET to b e the c omposite 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 copro jection copr I . It follo ws that [ A op ⊗ A , V ] Θ l bec omes mono idal with the underlying functor b ecoming strong monoidal; see [Mo e02] and [McC02]. Clear ly w e hav e: Prop osition 5 .2. The isomorph isms of Pr op osition 5.1 ar e monoidal. The next thing to o bserve is that Θ l , Θ r and Θ all have left adjoints Φ l , Φ r and Φ which therefore beco me opmonoidal mo nads who s e V -categor ies o f Eilenberg -Moo re algebras are monoida lly isomo r phic to L T a m b( A ), R T amb ( A ) and T amb( A ), re- sp ectiv ely . Stra igh tforward a pplications of the Y one da Lemma, show that the for- m ulas for these adjoints are Φ 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 ) , and Φ( 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 equiv alen t to V -functors ˇ Ψ : Y op ⊗ X / / V , which are also c alled mo dules ˇ Ψ : X  / / Y from X to Y . The equiv a lence is defined by: ˇ Ψ( Y , X ) = Ψ( X ( − , X ))( Y ) and Ψ( M )( Y ) = Z X ˇ Ψ( Y , X ) ⊗ M ( X ) . It follows that Φ l , Φ r and Φ determine mona ds ˇ Φ l , ˇ Φ r and ˇ Φ on A op ⊗ A in the bicateg ory V - Mo d . The for m ulas are: ˇ Φ l ( X, Y , U , V ) = Z A A ( U, A ⊗ X ) ⊗ A ( A ⊗ Y , V ) , ˇ Φ r ( X, Y , U , V ) = Z B A ( U, X ⊗ B ) ⊗ A ( Y ⊗ B , V ) , and ˇ Φ( X , Y , U , V ) = Z A,B A ( U, A ⊗ X ⊗ B ) ⊗ A ( A ⊗ Y ⊗ B , V ) . 6. Doubles The bica tegory V - M o d admits the Kle isli construction fo r monads. W rite D l A , D r A and D A for the Kleisli V -c a tegories for the monads ˇ Φ l , ˇ Φ r and ˇ Φ on A op ⊗ A in the bicateg ory V - Mo d . W e call them the left double , right double and double DOUBLES FOR MONOIDAL CA TE GORIE S 11 of the mo no idal V - category A . They all hav e the same o b jects as A op ⊗ A . The homs 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 D A (( X , Y ) , ( U, V )) = Z A,B A ( U, A ⊗ X ⊗ B ) ⊗ A ( A ⊗ Y ⊗ B , V ) . Prop osition 6 .1. The r e ar e c anonic al e qu ival enc es of V -c ate gorie s: • Ξ l : L T a m b( A ) ≃ [ D l A , V ] , • Ξ r : R T a mb( A ) ≃ [ D r A , V ] , and • Ξ : T amb( A ) ≃ [ D A , V ] . It follows from the main result of Day [Day70] that these doubles D l A , D r A and D A all admit promonoidal s tructures ( P l , J l ), ( P r , J r ) and ( P , J ) for which the equiv a le nces in Prop osition 6.1 beco me mono idal when the right-hand sides are given the corr e sponding convolution s tructures. F or example, w e ca lculate 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 ) . F urthermo r e, there are some sp ecial morphisms that ex ist in these do ubles D l A , D r A and D A . Let ˜ α l : ( X , Y ) / / ( A ⊗ X , A ⊗ Y ) denote the morphism in D l A defined by the comp osite 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 pr oper ty that Ξ l ( T , α l )( X, Y ) = T ( X , Y ) and Ξ l ( T , α l )( ˜ α l ) = α l . When A is rig h t clo sed, we let ˜ β l : ( X , Y A ) / / ( A ⊗ X , Y ) denote the mor- phism in D l A defined b y the comp osite 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 hav e the mor phism ˜ α r : ( X , Y ) / / ( X ⊗ B , Y ⊗ B ) in D r A , a nd also, when A is le ft closed, the morphism ˜ β r : ( X , B Y ) / / ( X ⊗ B , Y ). There are V -functor s D l A / / D A o o D r A w hich are the ident ity functions on ob jects and a re defined o n homs using pro jections with B = I for the left leg 12 CRAIG P ASTR O AND ROSS STREET and the pro jections A = I for the seco nd leg. In this way , the mor phisms ˜ α l and ˜ α r can b e r egarded also a s morphisms of D A . Under clos e dness assumptions, the morphisms ˜ β l and ˜ β r can a lso be regar ded as morphisms of D A . Let Σ l denote the set of morphisms ˜ β l : ( X , Y A ) / / ( A ⊗ X, Y ), let Σ r denote the set o f mor phis ms ˜ β r : ( X , B Y ) / / ( X ⊗ B , Y ), and let Σ deno te the set o f morphisms Σ = Σ l ∪ Σ r . Under a ppropriate closedness as s umptions on A , w e can form v arious V - categories of fractions suc h as: • L D A = D l A [Σ − 1 l ] and R D A = D r A [Σ − 1 r ], • D ls A = D A [Σ − 1 l ] and D r s A = D A [Σ − 1 r ], and • D s A = D A [Σ − 1 ]. The following result is now a utomatic. Theorem 6. 2. F or a close d monoidal V - c ate gory A , ther e ar e e quivalenc es of V -c ate gories: • [L D A , V ] ≃ L T a m b s ( A ) ≃ [ A , V ] , • [ D ls A , V ] ≃ T amb ls ( A ) ≃ Z l [ A , V ] , and • [ D s A , V ] ≃ T amb s ( A ) ≃ Z [ A , V ] . The first equiv alence of Theorem 6.2 implies that L D A a nd A are Morita eq uiv- alent. This b egs the question of w he ther there is a V - functor relating them mor e directly . Indeed there is . W e hav e a V -functor Π : D l A / / A defined on ob jects by Π( X , Y ) = Y X and by defining the effect Π : D l A (( X , Y ) , ( U, V )) / / A ( Y X , V U ) on hom ob jects to hav e its comp osite with the A -copr o jection equal to the comp osite 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 ) comp osition / / 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 ) to isomorphisms. So Π induces a V -functor ˆ Π : L D l A / / A ; this induces the first equiv alence o f Theorem 6.2. F or clos ed monoidal A , the seco nd and third equiv alences of Theorem 6.2 show that bo th the lax cent re and the cen tre of the c o n volution monoida l V -categ o ry [ A , V ] are again functor V -catego ries [ D ls A , V ] and [ D s A , V ]. Since Z l [ A , V ] and Z [ A , V ] ar e monoidal with the tenso r pro ducts colimit pr eserving in ea c h v ariable, using the cor resp ondence in [Day70], there are la x bra ided and br aided promo noidal structures on D ls A and D s A which are such that [ D ls A , V ] and [ D s A , V ] b ecome closed monoida l under convolution, and the equiv alences of Theorem 6 .2 b ecome lax braide d and braided mo no idal equiv alenc e s. DOUBLES FOR MONOIDAL CA TE GORIE S 13 R emark. W e are g rateful to Bria n Day for p oin ting out that the V -c a tegory A M app earing in [DS07] is equiv alen t to the full s ub- V -categor y of D A consisting o f the ob jects o f the form ( I , Y ). He also p oin ted out that a consequence of Theor em 6.2 is that the centre of V as a V -categor y is equiv a len t to V itse lf. This also can b e seen dir ectly by using the V -natur alit y in X of the cent re structure u X : A ⊗ X / / X ⊗ A on an ob ject A o f 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 -categor y is not equiv a le n t to V . References [Day7 0] Brian Day . On closed categories of functors, in R e p orts of the M idwest Cate gory Seminar IV , Lecture N ote s in Mathematics 137 (1970), pp. 1–38. [DPS07] B. Da y , E. P anc hadc haram, and R. Street. Lax braidings and the lax cent re, Con temporary Mathematics 441 (2007). See h ttp://www.ams.org/bo okstore-getitem/item=co nm-441 . [DS07] Brian Da y and Ross Street. Cen tres of monoidal categories of functors, Con temporary Mathematics 431 (2007 ), pp. 187–202. [JS91] Andr ´ e Jo ya l and Ross Street. T ortile Y ang-Baxter operators in tensor categories, J. Pure Appl. Algebra 71 (1991), pp. 43–51. [Kel82] G. M. Kelly . Basic concepts of enriche d category theory , London Mathematical So- ciet y Lecture Note Series 64. Cambridge University Press, Cambridge, 1982. Also at h ttp://www.tac.m ta.ca/tac /reprints/articles/10 /tr10.pdf . [LF07] I. L. Lop ez F ranco. Hopf modul es for autonomous pseudomonoids and the monoidal cen tre, preprint (2007). [McC02] Paddy McCrudden. Opmonoidal monads, Theory Appl. Categories 10 (2002), pp. 469– 485. [Mo e02] I. Moer dijk, Monads on tensor categories, J. Pure Appl. Algebra 168 no. 1-2 (2002), pp. 189–208. [T am06] D. T ambara. Dis tr i butors on a tensor categ ory , Hokk aido Math. Journal 35 (2006), pp. 379–425. E-mail addr ess : { craig, street } @maths .mq.edu.au Centre of A ustralian Ca tegor y Theor y , Dep a r tment of Ma thema tics, Macquarie University, NSW 2109 Australia