Tensor functors between Morita duals of fusion categories

TL;DR

This paper characterizes tensor functors between Morita duals of fusion categories, providing a framework to understand their structure and applications to equivariantizations, group-theoretical categories, and categorification of known sequences.

Contribution

It offers a detailed description of tensor functors between dual fusion categories using data from the original categories, advancing the understanding of their interrelations.

Findings

Describes tensor functors between dual fusion categories.

Analyzes G-equivariantizations and group-theoretical fusion categories.

Proposes a categorification of the Rosenberg-Zelinsky sequence.

Abstract

Given a fusion category $\mathcal{C}$ and an indecomposable $\mathcal{C}$-module category $\mathcal{M}$, the fusion category $\mathcal{C}^*_\mathcal{M}$ of $\mathcal{C}$-module endofunctors of $\mathcal{M}$ is called the (Morita) dual fusion category of $\mathcal{C}$ with respect to $\mathcal{M}$. We describe tensor functors between two arbitrary duals $\mathcal{C}^*_\mathcal{M}$ and $\mathcal{D}^*_\mathcal{N}$ in terms of data associated to $\mathcal{C}$ and $\mathcal{D}$. We apply the results to $G$-equivariantizations of fusion categories and group-theoretical fusion categories. We describe the orbits of the action of the Brauer-Picard group on the set of module categories and we propose a categorification of the Rosenberg-Zelinsky sequence for fusion categories.