Pivotal Brauer-Picard groupoids and graded extensions

TL;DR

This paper extends graded extension theory to pivotal and spherical tensor categories, defining new categorical groups and classifying their extensions with an obstruction theory for structure extension.

Contribution

It introduces pivotal and spherical Brauer-Picard 2-categorical groups and classifies graded extensions of tensor categories using these structures.

Findings

Classification of graded extensions via monoidal 2-functors.

Realization of categorical groups as fixed points of 2-categorical actions.

Development of an obstruction theory for extending pivotal and spherical structures.

Abstract

We develop pivotal and spherical versions of graded extension theory. We define the corresponding analogues of Brauer-Picard $2$-categorical groups and realize them as fixed points of natural $\mathbb{Z}$ and $\mathbb{Z}/2\mathbb{Z}$ $2$-categorical actions. We classify graded extensions of a pivotal tensor category by monoidal $2$-functors into the pivotal Brauer-Picard $2$-categorical group. A similar statement is proven for spherical (unimodular) tensor categories. We also develop an obstruction theory for determining when pivotal and spherical structures can be extended.