Tensor functors between Morita duals of fusion categories

Abstract

Given a fusion category $\mathcal{C}$ and an indecomposable $\mathcal{C}$-module category $\mathcal{M}$, the fusion category ${\mathcal{C}}_{\mathcal{M}}^{\ast }$ 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 ${\mathcal{C}}_{\mathcal{M}}^{\ast }$ and ${\mathcal{D}}_{\mathcal{N}}^{\ast }$ 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.

Final version. Accepted for publication in Letters in Mathematical Physics