The Picard crossed module of a braided tensor category

Abstract

For a finite braided tensor category $\mathcal{C}$ we introduce its Picard crossed module $\mathfrak{P}\left(\mathcal{C}\right)$ consisting of the group of invertible $\mathcal{C}$-module categories and the group of braided tensor autoequivalences of $\mathcal{C}$. We describe $\mathfrak{P}\left(\mathcal{C}\right)$ in terms of braided autoequivalences of the Drinfeld center of $\mathcal{C}$. As an illustration, we compute the Picard crossed module of a braided pointed fusion category.