Extension theory for braided-enriched fusion categories

TL;DR

This paper develops a framework for understanding how braided-enriched fusion categories can be extended and classified, especially focusing on G-crossed extensions and braidings, generalizing existing classification results.

Contribution

It introduces a characterization of compatible enrichments for G-graded extensions and extends Nikshych's classification of braidings to broader contexts.

Findings

Characterized enrichments compatible with G-graded extensions.

Showed G-crossed extensions are G-extensions of canonical enrichments.

Parameterized G-crossed braidings via subcategories of the center.

Abstract

For a braided fusion category $\mathcal{V}$, a $\mathcal{V}$-fusion category is a fusion category $\mathcal{C}$ equipped with a braided monoidal functor $\mathcal{F}:\mathcal{V} \to Z(\mathcal{C})$. Given a fixed $\mathcal{V}$-fusion category $(\mathcal{C}, \mathcal{F})$ and a fixed $G$-graded extension $\mathcal{C}\subseteq \mathcal{D}$ as an ordinary fusion category, we characterize the enrichments $\widetilde{\mathcal{F}}:\mathcal{V} \to Z(\mathcal{D})$ of $\mathcal{D}$ which are compatible with the enrichment of $\mathcal{C}$. We show that G-crossed extensions of a braided fusion category $\mathcal{C}$ are G-extensions of the canonical enrichment of $\mathcal{C}$ over itself. As an application, we parameterize the set of $G$-crossed braidings on a fixed $G$-graded fusion category in terms of certain subcategories of its center, extending Nikshych's classification of the braidings on a fusion category.