Theory and Applications of Categories, Vol. 21, No. 4, 2008, pp. 61–75.

DOUBLES FOR MONOIDAL CATEGORIES

Dedicated to Walter Tholen on his 60th birthday

CRAIG PASTRO AND ROSS STREET

Abstract. In a recent paper, Daisuke Tambara defined two-sided actions on an endomodule (= endodistributor) of a monoidal 𝒱 -category 𝒜. When 𝒜 is autonomous (= rigid = compact), he showed that the 𝒱 -category (that we call Tamb(𝒜)) of soequipped 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 be called the double of 𝒜) with an equivalence [𝒟𝒜 , 𝒱 ] ≃ Tamb(𝒜). When 𝒜 is closed, we define strong (respectively, left strong) Tambara modules and show 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 ]. We construct localizations 𝒟_{s}𝒜 and 𝒟_{ls}𝒜 of 𝒟𝒜 such that there are equivalences Tamb_{s}(𝒜) ≃ [𝒟_{s}𝒜 , 𝒱 ] and Tamb_{ls}(𝒜) ≃ [𝒟_{ls}𝒜 , 𝒱 ]. When 𝒜 is autonomous, every Tambara module is strong; this implies an equivalence Z[𝒜 , 𝒱 ] ≃ [𝒟𝒜 , 𝒱 ].

1. Introduction

For 𝒱 -categories 𝒜 and ℬ, a module T : 𝒜 [figure] ℬ (also called “bimodule”, “profunctor”, and “distributor”) is a 𝒱 -functor T : ℬ^{op} ⊗ 𝒜 [figure] 𝒱 . For a monoidal 𝒱 -category 𝒜, Tambara [Tam06] defined two-sided actions α of 𝒜 on an endomodule T : 𝒜 [figure] 𝒜. When 𝒜 is autonomous (also called “rigid” or “compact”) he showed that the 𝒱 -category Tamb(𝒜) of Tambara modules (T, α) is equivalent 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 vector spaces;

2. our results are mainly for a closed monoidal 𝒱 -category 𝒜, generalizing the autonomous case;

3. we show the connection with the lax centre as well as the centre; and,

The first author gratefully acknowledges support of an international Macquarie University Research Scholarship while the second gratefully acknowledges support of the Australian Research Council Discovery Grant DP0771252.

Received by the editors 2007-09-14 and, in revised form, 2008-05-30. Published on 2008-06-06 in the Tholen Festschrift. 2000 Mathematics Subject Classification: 18D10. Key words and phrases: monoidal centre, Drinfeld double, monoidal category, Day convolution. \begin{matrix}c \\ ◯ \end{matrix} Craig Pastro and Ross Street, 2008. Permission to copy for private use granted.

4. we introduce the double 𝒟𝒜 of a monoidal 𝒱 -category 𝒜 and some localizations of it, and relate these to Tambara modules.

Our principal goal is to give conditions under which the centre and lax centre of a 𝒱 -valued 𝒱 -functor monoidal 𝒱 -category is again such. Some results in this direction can be found in [DS07].

For general monoidal 𝒜, we construct a promonoidal 𝒱 -category 𝒟𝒜 with an equivalence [𝒟𝒜 , 𝒱 ] ≃ Tamb(𝒜). When 𝒜 is closed, we define when a Tambara 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 ]. We construct localizations 𝒟_{s}𝒜 and 𝒟_{ls}𝒜 of 𝒟𝒜 such that there are equivalences Tamb_{s}(𝒜) ≃ [𝒟_{s}𝒜 , 𝒱 ] and Tamb_{ls}(𝒜) ≃ [𝒟_{ls}𝒜 , 𝒱 ]. When 𝒜 is autonomous, every 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 autonomous.

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 his general constructions on pseudomonoids, he is also led to what we call the double monad.

2. Centres and convolution

We work with categories enriched in a base monoidal category 𝒱 as used by Kelly [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 same symbols 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 [figure] C, d_{l} : A [figure]^{B}(A⊗B), e_{r} : A⊗C^{A} [figure] C and d_{r} : B [figure] (A⊗B)^{A}.

Consequently, there are canonical isomorphisms

A⊗BC ≅ ^{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 associativity and 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 defined by

(F ∗ G)A = ∫ ^{U,V} 𝒜 (U ⊗ V, A) ⊗ F U ⊗ GV
≅ ∫ ^{V} F (^{V} A) ⊗ GV
≅ ∫ ^{U} F U ⊗ G(A^{U})

and

JA = 𝒜 (I, A).

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 was defined, for example, in [DPS07]. Since the representables are dense in [𝒜 , 𝒱 ], an object 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) [figure] F (B^{A})

such that the diagrams

[figure: F (^{A⊗B}C); F (C^{A⊗B}); θ_{A⊗B,C}; F (^{A}(^{B}C)); =; F (^{B}C^{A}); θ_{A,^{B} C}; F ((C^{A})^{B}); θ_{B,C^{A}}; =; and; F (^{I}A); F (A^{I}); θ_{I,A}; F A; =; =]

commute. The hom object Z_{l}[𝒜 , 𝒱 ]((F, θ), (G, ϕ)) is defined to be the equalizer of two obvious morphisms out of [𝒜 , 𝒱 ](F, G). The centre Z[𝒜 , 𝒱 ] of [𝒜 , 𝒱 ] is the full 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 the definition 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 [figure] 𝒱 together with a family of morphisms

α_{l}(A, X, Y) : T (X, Y) [figure] 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

[figure: T (X, Y); T (A ⊗ A′ ⊗ X, A ⊗ A′ ⊗ Y).; α_{l}(A⊗A′,X,Y); T (A′ ⊗ X, A′ ⊗ Y); α_{l}(A′,X,Y); α_{l}(A,A′⊗X,A′⊗Y)]

Similarly, a right Tambara module on 𝒜 is a 𝒱 -functor T : 𝒜^{op} ⊗ 𝒜 [figure] 𝒱 together with a family of morphisms

α_{r}(B, X, Y) : T (X, Y) [figure] T (X ⊗ B, Y ⊗ B)

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

[figure: T (X, Y); T (X ⊗ B ⊗ B′, Y ⊗ B ⊗ B′).; α_{r}(B⊗B′,X,Y); T (X ⊗ B, Y ⊗ B); α_{r}(B,X,Y); α_{r}(B′,B⊗X,B⊗Y)]

A Tambara module(T, α) on 𝒜 is a 𝒱 -functor T : 𝒜^{op} ⊗ 𝒜 [figure] 𝒱 together with both left and right Tambara module structures satisfying the “bimodule” compatibility condition

[figure: 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 diagonal of the last square is denoted by

α(A, B, X, Y) : T (X, Y) [figure] 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.

3.1. Proposition. Suppose 𝒜 is a monoidal 𝒱 -category and T : 𝒜^{op} ⊗ 𝒜 [figure] 𝒱 is a 𝒱 -functor.

(a) If 𝒜 is right closed, there is a bijection between 𝒱 -natural families of morphisms

α_{l}(A, X, Y) : T (X, Y) [figure] T (A ⊗ X, A ⊗ Y)

and 𝒱 -natural families of morphisms

β_{l}(A, X, Y) : T (X, Y^{A}) [figure] T (A ⊗ X, Y).

(b) Under the bijection of (a), the family α_{l} is a left Tambara structure if and only if the family β_{l} satisfies the two equations β_{l}(I, X, Y) = 1_{T (X,Y)} and

[figure: 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)]

(c) If 𝒜 is left closed, there is a bijection between 𝒱 -natural families of morphisms

α_{r}(B, X, Y) : T (X, Y) [figure] T (X ⊗ B, Y ⊗ B)

and 𝒱 -natural families of morphisms

β_{r}(B, X, Y) : T (X,^{B}Y) [figure] T (X ⊗ B, Y).

(d) Under the bijection of (c), the family α_{r} is a right Tambara structure if and only if the family β_{r} satisfies the two equations β_{r}(I, X, Y) = 1_{T} (X, Y) and

[figure: 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)]

(e) If 𝒜 is closed, the families α_{l} and α_{r} form a Tambara module structure if and only if the families β_{l} and β_{r}, corresponding under (a) and (c), satisfy the condition

[figure: 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)]

Proof. The bijection of (a) is defined by the formulas

β_{l}(A, X, Y) = ( [figure: 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) = ( [figure: 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 processes are mutually inverse uses the adjunction identities on the morphisms e and d. The bijection of (c) is obtained dually by reversing the tensor product. Translation of the conditions from the α to the β as required for (b), (d) and (e) is straightforward.

A left (respectively, right) Tambara module T on 𝒜 will be called strong when the morphisms

β_{l}(A, X, Y) : T (X, Y^{A}) [figure] T (A ⊗ X, Y)
(respectively, β_{r}(B, X, Y) : T (X,^{B}Y) [figure] 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 it is strong as a left (respectively, right) module and strong when it is both left and right strong. In particular, notice that the hom 𝒱 -functor (= identity module) of 𝒜 is a strong Tambara module.

3.2. Proposition. Suppose 𝒜 is a monoidal 𝒱 -category and T : 𝒜^{op} ⊗ 𝒜 [figure] 𝒱 is a 𝒱 -functor. If 𝒜 is right (left) autonomous then every left (right) Tambara module is strong.

Proof. If A^{∗} denotes a right dual for A with unit η : I [figure] A^{∗} ⊗ A then an inverse for β_{l} is defined by the composite

[figure: T (A ⊗ X, Y); T (A^{∗} ⊗ A ⊗ X, A^{∗} ⊗ Y); α_{l}(A^{∗},A⊗X,Y); T (X, A^{∗} ⊗ Y); T (η,1)] .

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 intersection over all A, X and Y of the equalizers of the pairs of morphisms:

[figure: [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}] .

Equivalently, we can define the hom as an intersection of equalizers of pairs of morphisms:

[figure: [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}}] .

Composition is defined so that we have a 𝒱 -functor ι : LTamb(𝒜) [figure] [𝒜^{op} ⊗ 𝒜 , 𝒱 ] which forgets the left module structure on T. In fact, LTamb(𝒜) becomes a monoidal 𝒱 -category in such a way that the forgetful 𝒱 -functor ι becomes strong monoidal. For this, the monoidal structure on [𝒜^{op} ⊗𝒜 , 𝒱 ] is the usual tensor product (= composition) of endomodules:

(T ⊗_{𝒜} T′)(X, Y) = ∫ ^{Z} T (X, Z) ⊗ T′(Z, Y).

When T and T′ are left Tambara modules, the left Tambara structure

(T ⊗_{𝒜} T′)(X, Y) [figure] (T ⊗_{𝒜} T′)(A ⊗ X, A ⊗ Y)

on T ⊗_{𝒜} T′ is defined by taking its composite with the coprojection copr_{Z} into the above coend to be the composite

[figure: T (X, Z) ⊗ T′(Z, Y); T (A ⊗ X, A ⊗ Z) ⊗ T′(A ⊗ Z, A ⊗ Y); α_{l}⊗α_{l}]
[figure: (T ⊗_{𝒜} T′)(A ⊗ X, A ⊗ Y); ^{copr^{A⊗Z}}] .

Similarly we obtain monoidal 𝒱 -categories RTamb(𝒜) and Tamb(𝒜) of right Tambara and all Tambara modules on 𝒜.

We write LTamb_{s}(𝒜) for the full sub-𝒱 -category of LTamb(𝒜) consisting of the strong 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 strong and strong Tambara modules respectively.

If 𝒜 is autonomous then Tamb(𝒜) = Tamb_{ls}(𝒜) = Tamb_{rs}(𝒜) = Tamb_{s}(𝒜) by Proposition 3.2.

4. The Cayley functor

Consider a right closed monoidal 𝒱 -category 𝒜. There is a Cayley 𝒱 -functor

Υ : [𝒜 , 𝒱 ] [figure] [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) [figure] [𝒜^{op} ⊗ 𝒜 , 𝒱 ](T_{F} , T_{G}) of Υ on homs is defined by taking its composite with the projection

pr_{X,Y} : [𝒜^{op} ⊗ 𝒜 , 𝒱 ](T_{F} , T_{G}) [figure] 𝒱 (F (Y^{X}), G(Y^{X}))

to be the projection

pr_{Y^{X}} : [𝒜 , 𝒱 ](F, G) [figure] 𝒱 (F (Y^{X}), G(Y^{X})).

4.1. Proposition. The Cayley 𝒱 -functor Υ is strong monoidal; it takes Day convolution to composition of endomodules.

Proof. We have the calculation:

(Υ(F) ⊗_{𝒜} Υ(G))(X, Y) = ∫ ^{Z} Υ(F)(X, Z) ⊗ Υ(G)(Z, Y)
= ∫ ^{Z} F (Z^{X}) ⊗ G(Y^{Z})
≅ ∫ ^{Z,U,V} 𝒜 (U, Z^{X}) ⊗ F U ⊗ 𝒜 (V, Y^{Z}) ⊗ GV
≅ ∫ ^{Z,U,V} 𝒜 (X ⊗ U, Z) ⊗ F U ⊗ 𝒜 (Z ⊗ V, Y) ⊗ GV
≅ ∫ ^{U,V} 𝒜 (X ⊗ U ⊗ V, Y) ⊗ F U ⊗ GV
≅ ∫ ^{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 lands in the left Tambara modules by defining, for each F ∈ [𝒜 , 𝒱 ], the structure

α_{l}(A, X, Y) = ( \xymatrix{F (Y^{X}) \ar[r]^{F ((d_{r})^{X})} & F ((A ⊗ Y)^{A⊗X})} )

on T_{F}. It is helpful to observe that the β_{l} corresponding to this α_{l} (via Proposition 3.1) is given by the identity

β_{l}(A, X, Y) = ( \xymatrix{F (Y^{A⊗X}) \ar[r]^{1} & F (Y^{A⊗X})} ),

showing that T_{F} becomes a strong left module. To see that there is a 𝒱 -functor Υ̂ : [𝒜 , 𝒱 ] [figure] LTamb_{s}(𝒜) satisfying ι ∘ Υ̂ = Υ, we merely observe that

pr_{A⊗X,Y} ∘ Υ_{F,G} = pr_{Y^{A⊗X}} = pr_{(Y^{A})^{X}} = pr_{X,Y^{A}} ∘ Υ_{F,G}.

4.2. Proposition. If 𝒜 is a right closed monoidal 𝒱 -category then the 𝒱 -functor Υ̂ : [𝒜 , 𝒱 ] [figure] LTamb_{s}(𝒜) is an equivalence.

Proof. Define ζ : LTamb(𝒜)(T_{F} , T_{G}) [figure] [𝒜 , 𝒱 ](F, G) by pr_{Y} ∘ ζ = pr_{I,Y} ∘ ι_{T_{F} ,T_{G}}. Then

pr_{Y} ∘ ζ ∘ Υ̂_{F,G} = pr_{I,Y} ∘ ι_{T_{F} ,T_{G}} ∘ Υ̂_{F,G} = pr_{I,Y} ∘ Υ_{F,G} = pr_{Y}

and

pr_{X,Y} ∘ ι_{T_{F} ,T_{G}} ∘ Υ̂_{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 Υ̂_{F,G}, so that Υ̂ is fully faithful. To see that Υ̂ is essentially 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 Υ̂(F) ≅ T.

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) = ( [figure: 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} corresponding to this α_{r} (via Proposition 3.1) is defined by

β_{r}(B, X, Y) = ( \xymatrix{F (^{B}Y^{X}) \ar[r]^{θ_{B,Y^{X}}} & F (Y^{X⊗B})} ).

It is easy to see that, in this way, T_{F} = Υ̂(F) actually becomes a (two-sided) Tambara module which we write as Υ̂(F, θ), and we have a 𝒱 -functor

Υ̂ : Z_{l}[𝒜 , 𝒱 ] [figure] Tamb_{ls}(𝒜).

4.3. Proposition. If 𝒜 is a closed monoidal 𝒱 -category then the 𝒱 -functor

Υ̂ : Z_{l}[𝒜 , 𝒱 ] [figure] Tamb_{ls}(𝒜)

is an equivalence which restricts to an equivalence

Υ̂ : Z[𝒜 , 𝒱 ] [figure] Tamb_{s}(𝒜).

Proof. The proof of full faithfulness proceeds along the lines of the beginning of the proof of Proposition 4.2. For essential surjectivity on objects, take a left strong Tambara module (T, α). Then β_{l}(A, X, Y) : T (X, Y^{A}) [figure] T (A ⊗ X, Y) is invertible. Define the 𝒱 -functor F : 𝒜 [figure] 𝒱 by F X = T (I, X) as in the proof of Proposition 4.2, and define θ_{A,Y} : F (^{A}Y) [figure] F (Y^{A}) to be the composite

[figure: T (I,^{A}Y); T (A, Y); β_{r}(A,I,Y); T (I, Y^{A}); β_{l}(A,I,Y)^{−1}] .

This is easily seen to yield an object (F, θ) of the lax centre Z_{l}[𝒜 , 𝒱 ] with Υ̂(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.

5. The double monad

Tambara modules are actually Eilenberg-Moore coalgebras for a fairly obvious comonad on [𝒜^{op} ⊗ 𝒜 , 𝒱 ]. We begin by looking at the case of left modules.

Let Θ_{l} : [𝒜^{op} ⊗ 𝒜 , 𝒱 ] [figure] [𝒜^{op} ⊗ 𝒜 , 𝒱 ] be the 𝒱 -functor defined by the end

Θ_{l}(T)(X, Y) = ∫_{A} T (A ⊗ X, A ⊗ Y).

There is a 𝒱 -natural family ϵ_{T} : Θ_{l}(T) [figure] T defined by the projections

pr_{I} : ∫_{A} T (A ⊗ X, A ⊗ Y) [figure] T (X, Y).

There is a 𝒱 -natural family δ_{T} : Θ_{l}(T) [figure] Θ_{l}(Θ_{l}(T)) defined by taking its composite with the projection

pr_{B,C} : ∫_{B,C} T (B ⊗ C ⊗ X, B ⊗ C ⊗ Y) [figure] T (B ⊗ C ⊗ X, B ⊗ C ⊗ Y)

to be the projection

pr_{B⊗C} : ∫_{A} T (A ⊗ X, A ⊗ Y) [figure] T (B ⊗ C ⊗ X, B ⊗ C ⊗ Y).

It is now easily checked that Θ_{l} = (Θ_{l}, δ, ϵ) is a comonad on [𝒜^{op} ⊗ 𝒜 , 𝒱 ].

There is 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) = ∫_{B} T (X ⊗ B, Y ⊗ B)

and

Θ(T)(X, Y) = ∫_{A,B} T (A ⊗ X ⊗ B, A ⊗ Y ⊗ B).

We can easily identify the 𝒱 -categories of Eilenberg-Moore coalgebras for these three comonads.

5.1. Proposition. 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′) [figure] Θ_{l}(T ⊗_{𝒜} T′) and 𝒜 (−, −) [figure] Θ_{l}(𝒜 (−, −)) with components

∫ ^{Z}∫_{A} T (A⊗X, A⊗Z)⊗∫_{B} T′(B⊗X, B⊗Z) [figure] ∫_{C} ∫ ^{U} T (C ⊗X, U)⊗T′(U, C ⊗Y) (1)

and

𝒜 (X, Y) [figure] ∫_{A} 𝒜 (A ⊗ X, A ⊗ Y)  (2)

defined as follows. The morphism (1) is determined by its precomposite with the coprojection copr_{Z} and postcomposite with the projection pr_{C}; the result is defined to be the composite

∫_{A} T (A ⊗ X, A ⊗ Z)⊗ ∫_{B} T′(B ⊗ X, B ⊗ Z)

T (C ⊗ X, C ⊗ Z) ⊗ T′(C ⊗ Z, C ⊗ Y)^{[figure: pr_{C} ⊗ pr_{C}]}

[figure: ^{copr^{C⊗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:

5.2. Proposition. 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 Φ which therefore become opmonoidal monads whose 𝒱 -categories of Eilenberg-Moore algebras are monoidally isomorphic to LTamb(𝒜), RTamb(𝒜) and Tamb(𝒜), respectively. Straightforward applications of the Yoneda Lemma, show that the formulas for these adjoints are

Φ_{l}(S)(U, V) = ∫ ^{A,X,Y} 𝒜 (U, A ⊗ X) ⊗ 𝒜 (A ⊗ Y, V) ⊗ S(X, Y),
Φ_{r}(S)(U, V) = ∫ ^{B,X,Y} 𝒜 (U, X ⊗ B) ⊗ 𝒜 (Y ⊗ B, V) ⊗ S(X, Y), and
Φ(S)(U, V) = ∫ ^{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 ] [figure] [𝒴 ^{op}, 𝒱 ] are equivalent to 𝒱 - functors Ψ̌ : 𝒴 ^{op} ⊗ 𝒳 [figure] 𝒱 , which are also called modules Ψ̌ : 𝒳 [figure] 𝒴 from 𝒳 to 𝒴 . The equivalence is defined by:

Ψ̌(Y, X) = Ψ(𝒳 (−, X))(Y)

and

Ψ(M)(Y) = ∫ ^{X} Ψ̌(Y, X) ⊗ M(X).

It follows that Φ_{l}, Φ_{r} and Φ determine monads Φ̌_{l}, Φ̌_{r} and Φ̌ on 𝒜^{op} ⊗ 𝒜 in the bicategory 𝒱 -Mod. The formulas are:

Φ̌_{l}(X, Y, U, V) = ∫ ^{A} 𝒜 (U, A ⊗ X) ⊗ 𝒜 (A ⊗ Y, V),
Φ̌_{r}(X, Y, U, V) = ∫ ^{B} 𝒜 (U, X ⊗ B) ⊗ 𝒜 (Y ⊗ B, V), and
Φ̌(X, Y, U, V) = ∫ ^{A,B} 𝒜 (U, A ⊗ X ⊗ B) ⊗ 𝒜 (A ⊗ Y ⊗ B, V).

6. 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 Φ̌_{l}, Φ̌_{r} and Φ̌ on 𝒜^{op} ⊗ 𝒜 in the bicategory 𝒱 -Mod. We call them the left double, right double and double of the monoidal 𝒱 -category 𝒜. They all have the same objects as 𝒜^{op} ⊗ 𝒜. The homs are defined by

𝒟_{l}𝒜 ((X, Y), (U, V)) = ∫ ^{A} 𝒜 (U, A ⊗ X) ⊗ 𝒜 (A ⊗ Y, V),
𝒟_{r}𝒜 ((X, Y), (U, V)) = ∫ ^{B} 𝒜 (U, X ⊗ B) ⊗ 𝒜 (Y ⊗ B, V), and
𝒟𝒜 ((X, Y), (U, V)) = ∫ ^{A,B} 𝒜 (U, A ⊗ X ⊗ B) ⊗ 𝒜 (A ⊗ Y ⊗ B, V).

6.1. Proposition. 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 which the equivalences in Proposition 6.1 become monoidal when the right-hand sides are given 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,A,B} 𝒜 (H, A ⊗ X) ⊗ 𝒜 (A ⊗ Y, Z) ⊗ 𝒜 (Z, B ⊗ U) ⊗ 𝒜 (B ⊗ V, K)
= ∫ ^{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 α̃_{l} : (X, Y) [figure] (A ⊗ X, A ⊗ Y) denote the morphism in 𝒟_{l}𝒜 defined by the composite

[figure: I; 𝒜 (A ⊗ X, A ⊗ X) ⊗ 𝒜 (A ⊗ Y, A ⊗ Y); j_{A⊗X}⊗j_{A⊗Y}]
[figure: 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})(α̃_{l}) = α_{l}. When 𝒜 is right closed, we let β̃_{l} : (X, Y^{A}) [figure] (A ⊗ X, Y) denote the morphism in 𝒟_{l}𝒜 defined by the composite

[figure: 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})(β̃_{l}) = β_{l}.

Similarly, we have the morphism α̃_{r} : (X, Y) [figure] (X ⊗ B, Y ⊗ B) in 𝒟_{r}𝒜, and also, when 𝒜 is left closed, the morphism β̃_{r} : (X,^{B}Y) [figure] (X ⊗ B, Y).

There are 𝒱 -functors 𝒟_{l}𝒜 [figure] 𝒟𝒜 [figure] 𝒟_{r}𝒜 which are the identity functions on 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 α̃_{l} and α̃_{r} can be regarded also as morphisms of 𝒟𝒜. Under closedness assumptions, the morphisms β̃_{l} and β̃_{r} can also be regarded as morphisms of 𝒟𝒜.

Let Σ_{l} denote the set of morphisms β̃_{l} : (X, Y^{A}) [figure] (A ⊗ X, Y), let Σ_{r} denote the set of morphisms β̃_{r} : (X,^{B}Y) [figure] (X ⊗ B, Y), and let Σ denote the set of morphisms Σ = Σ_{l} ∪ Σ_{r}. Under appropriate closedness assumptions on 𝒜, we can form various 𝒱 -categories of fractions such as:

• L𝒟𝒜 = 𝒟_{l}𝒜 [Σ_{l}^{−1}] and R𝒟𝒜 = 𝒟_{r}𝒜 [Σ_{r}^{−1}],

• 𝒟_{ls}𝒜 = 𝒟𝒜 [Σ_{l}^{−1}] and 𝒟_{rs}𝒜 = 𝒟𝒜 [Σ_{r}^{−1}], and

• 𝒟_{s}𝒜 = 𝒟𝒜 [Σ^{−1}].

The following result is now automatic.

6.2. Theorem. For a closed monoidal 𝒱 -category 𝒜, there are equivalences of 𝒱 - categories:

• [L𝒟𝒜 , 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 L𝒟𝒜 and 𝒜 are Morita equivalent. This begs the question of whether there is a 𝒱 -functor relating them more directly. Indeed there is. We have a 𝒱 -functor

Π : 𝒟_{l}𝒜 [figure] 𝒜

defined on objects by Π(X, Y) = Y^{X} and by defining the effect

Π : 𝒟_{l}𝒜 ((X, Y), (U, V)) [figure] 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)
[figure: V^{(−)}⊗(−)^{A⊗X}; 𝒜 (V^{A⊗X}, V^{U}) ⊗ 𝒜 ((A ⊗ Y)^{A⊗X}, V^{A⊗X})]
[figure: composition; 𝒜 ((A ⊗ Y)^{A⊗X}, V^{U})]
[figure: 𝒜 ((d_{r})^{X},V^{U}); 𝒜 (Y^{X}, V^{U})] .

It is easy to see that Π takes the morphisms β̃_{l} : (X, Y^{A}) [figure] (A⊗X, Y) to isomorphisms. So Π induces a 𝒱 -functor

Π̂ : L𝒟_{l}𝒜 [figure] 𝒜;

this induces the first equivalence of Theorem 6.2.

For closed monoidal 𝒜, the second and third equivalences of Theorem 6.2 show that 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 promonoidal structures on 𝒟_{ls}𝒜 and 𝒟_{s}𝒜 which are such that [𝒟_{ls}𝒜 , 𝒱 ] and [𝒟_{s}𝒜 , 𝒱 ] become closed monoidal under convolution, and the equivalences of Theorem 6.2 become lax braided and braided monoidal equivalences.

6.3. 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 of the 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 using the 𝒱 -naturality in X of the centre structure u_{X} : A ⊗ X [figure] 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 Seminar IV, Lecture Notes in Mathematics 137 (1970): 1–38.

[DPS07] B. Day, E. Panchadcharam, and R. Street. Lax braidings and the lax centre, in Hopf Algebras and Generalizations, Contemporary Mathematics 441 (2007): 1–17.

[DS07] Brian Day and Ross Street. Centres of monoidal categories of functors, Contemporary Mathematics 431 (2007): 187–202.

[JS91] André Joyal and Ross Street. Tortile Yang-Baxter operators in tensor categories, J. Pure Appl. Algebra 71 (1991): 43–51.

[Kel82] G. M. Kelly. Basic concepts of enriched category theory, London Mathematical Society Lecture Note Series 64. Cambridge University Press, Cambridge, 1982.

        Also at http://www.tac.mta.ca/tac/reprints/articles/10/tr10abs.html.

[LF07] Ignacio L. Lopez Franco. Hopf modules for autonomous pseudomonoids and the monoidal centre, (2007), arXiv:0710.3853v2.

[McC02] Paddy McCrudden. Opmonoidal monads, Theory Appl. Categories 10 (2002): 469–485.

[Moe02] I. Moerdijk, Monads on tensor categories, J. Pure Appl. Algebra 168 no. 1-2 (2002): 189–208.

[Tam06] D. Tambara. Distributors on a tensor category, Hokkaido Math. Journal 35 (2006): 379–425.

Centre of Australian Category Theory Department of Mathematics Macquarie University NSW 2109 Australia

Email: craig@maths.mq.edu.au
      street@maths.mq.edu.au

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