The balanced tensor product of module categories

TL;DR

This paper explores the construction of the balanced tensor product of module categories over a monoidal linear category, showing it can be realized as a category of bimodule objects under certain finiteness and rigidity conditions.

Contribution

It extends the concept of balanced tensor products from modules over algebras to module categories over monoidal linear categories, providing a categorical realization as bimodule objects.

Findings

Balanced tensor product characterized as bimodule category

Realization requires finite and rigid monoidal categories

Generalizes tensor product concepts to higher categorical structures

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.