Clifford theory for tensor categories

TL;DR

This paper extends Clifford theory to tensor categories, describing module categories over strongly graded tensor categories in terms of subcategories linked to group subgroups.

Contribution

It introduces a framework for understanding module categories over strongly graded tensor categories via subgroup-associated subcategories.

Findings

Module categories over strongly graded tensor categories can be described as induced from subcategories.

The structure of module categories is linked to subgroups of the grading group.

Provides a new perspective on tensor category gradings and their module categories.

Abstract

A graded tensor category over a group $G$ will be called a strongly $G$-graded tensor category if every homogeneous component has at least one multiplicativily invertible object. Our main result is a description of the module categories over a strongly $G$-graded tensor category as induced from module categories over tensor subcategories associated with the subgroups of $G$.