nLab Bimod

Definition

For any monoidal category $V$ with coequalisers, one can form a bicategory $Bimod(V)$ of bimodules as follows:

The objects of $Bimod(V)$ are monoids $R$ , $S$ internal to $V$ .
The 1-morphisms are $(R,S)$ -bimodules. That is, objects $A$ in $V$ with a compatible left action of $R$ and right action of $S$ . The composition is given by a generalization of the usual notion of tensor product: For an $(R,S)$ -bimodule $A$ and an $(S,T)$ -bimodule $B$ , a new $(R,T)$ -bimodule $A \otimes_S B$ is computed as the coequaliser of $A$ ‘s right action and $B$ ’s left action of $S$ .

$A \otimes S \otimes B \begin{matrix} \overset{\rho \otimes B}{\rightarrow} \\ \underset{A \otimes \lambda}{\rightarrow} \end{matrix}\,\, A \otimes B \rightarrow A \otimes_S B$
2-morphisms, are bimodule homomorphisms.

In the case where $V =$ Ab, this recovers the usual notion of bimodules and their tensor product.

More generally, profunctors over any monoidal category are referred to as bimodule objects. Composition in this case replaces the colimit above with a coend.

This structure may be extended to that a double category, where the tight morphisms are given by monoid homomorphisms.