Braided Picard groups and graded extensions of braided tensor categories

TL;DR

This paper classifies graded extensions of finite braided tensor categories using their 2-categorical Picard groups, linking them to braided monoidal 2-functors and cohomology, with detailed descriptions for specific categories.

Contribution

It introduces a classification framework for graded extensions of braided tensor categories via 2-categorical Picard groups and relates these to cohomological data.

Findings

Braided extensions correspond to braided monoidal 2-functors into the Picard group.

Extensions are characterized using Eilenberg-Mac Lane cohomology.

Explicit descriptions provided for symmetric and pointed braided fusion categories.

Abstract

We classify various types of graded extensions of a finite braided tensor category $\cal B$ in terms of its $2$-categorical Picard groups. In particular, we prove that braided extensions of $\cal B$ by a finite group $A$ correspond to braided monoidal $2$-functors from $A$ to the braided $2$-categorical Picard group of $\cal B$ (consisting of invertible central $\cal B$-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.