Extension Theory for Braided-Enriched Fusion Categories

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