Exact factorizations and extensions of fusion categories

TL;DR

This paper introduces the concept of exact factorizations of fusion categories, generalizing group factorizations, and explores their relation to exact sequences and extensions within the framework of fusion categories.

Contribution

It defines and studies exact factorizations of fusion categories, linking them to exact sequences and extensions, and provides new insights and examples in the theory.

Findings

Established the notion of exact factorization of fusion categories.

Connected exact factorizations to exact sequences of fusion categories.

Provided examples and posed open questions in the field.

Abstract

We introduce and study the new notion of an {\em exact factorization} $\mathcal{B}=\mathcal{A}\bullet \mathcal{C}$ of a fusion category $\mathcal{B}$ into a product of two fusion subcategories $\mathcal{A},\mathcal{C}\subseteq \mathcal{B}$ of $\mathcal{B}$. This is a categorical generalization of the well known notion of an exact factorization of a finite group into a product of two subgroups. We then relate exact factorizations of fusion categories to exact sequences of fusion categories with respect to an indecomposable module category, which was introduced and studied by P. Etingof and the author in \cite{EG}. We also apply our results to study extensions of a group-theoretical fusion category by another one, provide some examples, and propose a few natural questions.