nLab
bimodule

Definition

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 objects $\rho (a)$ for $a\in \mathrm{Obj}(C)$ 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

$$C^{\mathrm{op}}\otimes D\to V.$$C^{op} \otimes D \to V \,.

Such a functor is also called a profunctor or distributor.

Some points are in order. Strictly speaking, the construction of $C^{\mathrm{op}}$ from a $V$-category $C$ requires that $V$ be symmetric (or at least braided) monoidal. It’s possible to define $C$-$D$ bimodules without recourse to $C^{\mathrm{op}}$, but then either that should be spelled out, or one should include a symmetry. (If the former is chosen, then closedness one on side might not be the best choice of assumption, in view of the next remark; a more natural choice might be biclosed monoidal.)

Second: bimodules are not that much good unless you can compose them; for that one should add some cocompleteness assumptions to $V$ (with $\otimes$ cocontinuous in both arguments; biclosedness would ensure that), and consider smallness assumptions on the objects $C$, $D$, etc. —Todd.

Examples

• Let $V=\mathrm{Set}$ and let $C=D$. Then the hom functor $C(-,-):C^{\mathrm{op}}\times C\to \mathrm{Set}$ is a bimodule. Bimodules can be thought of as a kind of generalized hom, giving a set of morphisms (or object of $V$) between an object of $C$ and an object of $D$.

• Let $\hat{C}={\mathrm{Set}}^{C^{\mathrm{op}}}$; the objects of $\hat{C}$ are “generating functions” that assign to each object of $C$ a set. Every bimodule $f:D^{\mathrm{op}}\times C\to \mathrm{Set}$ can be curried to give a Kleisli arrow $\tilde{f}:C\to \hat{D}$. Composition of these arrows corresponds to convolution of the generating functions.

  Todd: I am not sure what is trying to be said with regard to “convolution”. I know about Day convolution, but this is not the same thing.

  Also, with regard to “Kleisli arrow”: I understand the intent, but one should proceed with caution since there is no global monad $C\mapsto \hat{C}$ to which Kleisli would refer. Again there are size issues that need attending to.

• Let $V=\mathrm{Vect}$ and let $C=B{A}_1$ and $D=B{A}_2$ be two one-object $\mathrm{Vect}$-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.