The Picard crossed module of a braided tensor category

TL;DR

This paper introduces the Picard crossed module for finite braided tensor categories, linking invertible module categories and braided autoequivalences, with explicit computations for braided pointed fusion categories.

Contribution

It defines the Picard crossed module in terms of the Drinfeld center and provides explicit calculations for specific categories, advancing the understanding of their automorphism structures.

Findings

The Picard crossed module is described via braided autoequivalences of the Drinfeld center.

Explicit computation of the Picard crossed module for braided pointed fusion categories.

Establishes a new framework connecting invertible module categories and autoequivalences.

Abstract

For a finite braided tensor category we introduce its Picard crossed module consisting of the group of invertible module categories and the group of braided tensor autoequivalences. We describe the Picard crossed module in terms of braided autoequivalences of the Drinfeld center of the braided tensor category. As an illustration, we compute the Picard crossed module of a braided pointed fusion category.