Pivotal Brauer-Picard groupoids and graded extensions
Agustina Czenky, David Jaklitsch, Dmitri Nikshych, Julia Plavnik, David Reutter, Sean Sanford, Harshit Yadav
TL;DR
This work develops a comprehensive framework for pivotal and spherical graded extensions of finite tensor categories by introducing pivotal and spherical Brauer-Picard 2-groupoids. It realizes these 2-groupoids as fixed points of natural $B\underline{\mathbb{Z}}$ and $B\underline{\mathbb{Z}/2\mathbb{Z}}$-actions, and classifies $G$-graded extensions via monoidal 2-functors into these fixed-point 2-groupoids. An obstruction theory is developed to determine when pivotal structures extend across graded components, with both algebraic and homotopical formulations. The paper also extends sphericalization to unimodular contexts, showing compatibility with the extension classification and providing a unified approach to spherical and pivotal graded extensions, including detailed examples. Overall, it generalizes prior semisimple results beyond semisimplicity and clarifies how pivotal and spherical data interact with graded extension theory in a 2-categorical setting.
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 structures can be extended.