The balanced tensor product of module categories
Christopher L. Douglas, Christopher Schommer-Pries, Noah Snyder
TL;DR
The paper addresses the existence and construction of the balanced tensor product for module categories over a finite tensor category. It develops a bimodule-object realization, proving existence for finite module categories and showing that, when the inputs are module categories over $\mathcal{C}$, the balanced tensor product is equivalent to the category of $A$-$B$-bimodule objects in $\mathcal{C}$, i.e. $\mathrm{Mod}(\mathcal{C})$ for suitable algebras $A,B$. It further connects this Deligne-style tensor product with the Kelly tensor product and situates the construction within a monadic framework, enabling explicit descriptions and computations. The results have foundational impact for the 3-category of finite tensor categories and their relation to topological field theories, by giving a concrete, computable model for relative tensor products in the finite setting.
Abstract
The balanced tensor product M (x)_A N of two modules over an algebra A is the vector space corepresenting A-balanced bilinear maps out of the product M x N. The balanced tensor product M [x]_C N of two module categories over a monoidal linear category C is the linear category corepresenting C-balanced right-exact bilinear functors out of the product category M x N. We show that the balanced tensor product can be realized as a category of bimodule objects in C, provided the monoidal linear category is finite and rigid.