Braided Picard groups and graded extensions of braided tensor categories
Alexei Davydov, Dmitri Nikshych
TL;DR
The paper develops a comprehensive 2-categorical framework for classifying graded extensions of finite braided tensor categories via Picard-type 2-groups (Brauer-Picard, Picard, Pic_br, Pic_sym). It translates G-extensions into braided/symmetric monoidal 2-functors into these 2-categorical groups and uses Eilenberg–Mac Lane cohomology to describe obstructions and parameter spaces for liftings, along with Whitehead structures. It provides explicit computations for symmetric fusion categories and pointed braided fusion categories, and connects to zesting and Pontryagin–Whitehead invariants, yielding concrete classifications in several important cases. The results extend ENO’s program to refined 2-categorical and braided/symmetric contexts, offering practical tools to determine and construct graded extensions, including central, braided, symmetric, and quasi-trivial cases, with detailed obstruction criteria.
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.