Monoidal 2-structure of Bimodule Categories

TL;DR

This paper develops a tensor product for bimodule categories, establishing a monoidal 2-category structure, and explores its implications for module categories over tensor categories, including group representations.

Contribution

It introduces a tensor product for bimodule categories, proving the 2-category forms a monoidal 2-category, and relates bimodule categories to center modules, extending previous results.

Findings

Bimodule categories form a monoidal 2-category under the new tensor product.

De-equivariantization is equivalent to tensoring over Rep(G).

Determined the group of invertible irreducible Rep(G)-module categories.

Abstract

We define a notion of tensor product of bimodule categories and prove that with this product the 2-category of C-bimodule categories for fixed tensor C is a monoidal 2-category in the sense of Kapranov and Voevodsky. We then provide a monoidal-structure preserving 2-equivalence between the 2-category of C-bimodule categories and Z(C)-module categories (module categories over the center). For finite group G we show that de-equivariantization is equivalent to tensor product over category Rep(G) of finite dimensional representations. We derive Rep(G)-module fusion rules and determine the group of invertible irreducible Rep(G)-module categories extending earlier results for abelian groups.