Showing changes from revision #3 to #4: Added | Removed | Changed

Let $V$ be a closed monoidal category. Recall that for $C$ a category enriched over $V$, a $V$-module is a $V$-functor $\rho : C \to V$. We think of the object $\rho(a)$ for $a \in Obj(C)$ as the objects on which $C$ acts, and of $\rho(C(a,b))$ as the action of $C$ on these objects.

In this language a $C$-$D$ bimodule for $V$-categories $C$ and $D$ is a $V$-functor

Such a functor is also called a profunctor or distributor.

Examples

• Let $V = Vect$ and let $C=A_{1}B{A}_1$ C = \mathbf{A}_1 \mathbf{B}A_1 and $D = \mathbf{A}_2$ be two one-object $Vect$-enriched categories, whose endomorphism vector spaces are hence algebras. Then a $C$-$D$ bimodule is a vector space $V$ with an action of $A_1$ on the left and and action of $A_2$ on the right.