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 SBA \otimes_S B is computed as the coequaliser of A’s right action and B’s left action of S.

ASBρB AλABA SB 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. Composition in this case replaces the colimit above with a coend.

Last revised on February 16, 2021 at 00:05:52. See the history of this page for a list of all contributions to it.