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.