Extension Theory for Braided-Enriched Fusion Categories. (2nd July 2021) - Jones, Corey; Morrison, Scott; Penneys, David; Plavnik, Julia
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 $\mathcal{F}:\mathcal{V} → Z(\mathcal{C})$ . 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 $\widetilde{\mathcal{F}}:\mathcal{V} → Z(\mathcal{D})$ of $\mathcal{D}$ that 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.