Braided Picard groups and graded extensions of braided tensor categories

Abstract

We classify various types of graded extensions of a finite braided tensor category $\mathcal{B}$ in terms of its $2$-categorical Picard groups. In particular, we prove that braided extensions of $\mathcal{B}$ by a finite group $A$ correspond to braided monoidal $2$-functors from $A$ to the braided $2$-categorical Picard group of $\mathcal{B}$ (consisting of invertible central $\mathcal{B}$-module categories). Such functors can be expressed in terms of the Eilnberg-Mac~Lane cohomology. We describe in detail braided $2$-categorical Picard groups of symmetric fusion categories and of pointed braided fusion categories.