nLab Bimod

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-cells are (R,S)-bimodules. That is, objects A in V with a compatible left action of R and right action of S. 1-cell composition is then 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-cells, as expected are just module homomorphisms. In the case where V = Ab, this recovers the usual notion of bimodules and 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.

See also

• bimodule
• bimodule object
• Bimod