Crossed pointed categories and their equivariantizations

TL;DR

This paper introduces a new cohomology theory for crossed modules, classifies crossed pointed categories, and explores their equivariantizations to produce braided fusion categories, including criteria for modularity and underlying algebraic structures.

Contribution

It generalizes existing cohomology theories, classifies a broad class of categories, and links them to quasi-Hopf algebras, expanding understanding of braided fusion categories.

Findings

Classification of crossed pointed categories via quasi-abelian third cohomology

Construction of braided fusion categories through equivariantization

Criteria for modularity of the resulting categories

Abstract

We propose the notion of quasi-abelian third cohomology of crossed modules, generalizing Eilenberg and MacLane's abelian cohomology and Ospel's quasi-abelian cohomology, and classify crossed pointed categories in terms of it. We apply the process of equivariantization to the latter to obtain braided fusion categories which may be viewed as generalizations of the categories of modules over twisted Drinfeld doubles of finite groups. As a consequence, we obtain a description of all braided group-theoretical categories. A criterion for these categories to be modular is given. We also describe the quasi-triangular quasi-Hopf algebras underlying these categories.