Module categories over equivariantized tensor categories

TL;DR

This paper characterizes indecomposable module categories over equivariantized tensor categories using original categories and additional data, providing a 2-categorical perspective on equivariantization.

Contribution

It offers a classification of module categories over equivariantized tensor categories and introduces a 2-categorical interpretation of equivariantization.

Findings

Classification of indecomposable module categories over $ ext{Rep}(G)$-equivariant tensor categories.

A 2-categorical framework for understanding equivariantization of module categories.

Explicit description of module categories via original categories and extra data.

Abstract

For a finite tensor category $\mathcal C$ and a Hopf monad $T:\mathcal C\to \mathcal C$ satisfying certain conditions we describe exact indecomposable left $\mathcal C^T$-module categories in terms of left $\mathcal C$-module categories and some extra data. We also give a 2-categorical interpretation of the process of equivariantization of module categories.