Doubles for monoidal categories

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, (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.

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

defined by evaluation and coevaluation morphisms

Consequently, there are canonical isomorphisms

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

In particular,

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

such that the diagrams

] consisting of those objects (F, θ) with θ invertible.

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 / / V together with a family of morphisms

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

Similarly, a right Tambara module on A is a V -functor T : A op ⊗ A / / V together with a family of morphisms

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 z z u u u u u u u u u u u u A Tambara module(T, α) on A is a V -functor T : A op ⊗ A / / V together with both left and right Tambara module structures satisfying the “bimodule” compatibility condition

The morphism defined to be the diagonal of the last square is denoted by

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 :

and V -natural families of morphisms

(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

and V -natural families of morphisms

(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

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

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

That the processes are mutually inverse uses the adjunction identities on the morphisms e and

…(Full text truncated)…