The balanced tensor product of module categories

Abstract

The balanced tensor product $M{\otimes }_{A}N$ of two modules over an algebra $A$ is the vector space corepresenting $A$-balanced bilinear maps out of the product $M\times N$. The balanced tensor product $\mathcal{M}{⊠}_{\mathcal{C}}\mathcal{N}$ of two module categories over a monoidal linear category $\mathcal{C}$ is the linear category corepresenting $\mathcal{C}$-balanced right-exact bilinear functors out of the product category $\mathcal{M}\times \mathcal{N}$. We show that the balanced tensor product can be realized as a category of bimodule objects in $\mathcal{C}$, provided the monoidal linear category is finite and rigid.