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 $\mathcal{D}$ 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.

中文翻译：

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编织富集融合类别的可拓理论

对于编织融合范畴 $\mathcal{V}$，$\mathcal{V}$-fusion 范畴是配备编织单曲面函子 $\mathcal{F}:\mathcal 的融合范畴 $\mathcal{C}$ {V} \to Z(\mathcal{C})$。给定一个固定的 $\mathcal{V}$-fusion 类别 $(\mathcal{C}, \mathcal{F})$ 和一个固定的 $G$-graded extension $\mathcal{C}\subseteq \mathcal{D} $ 作为一个普通的融合范畴，我们刻画了 $\mathcal{D}$ 的富集 $\widetilde{\mathcal{F}}:\mathcal{V} $\mathcal{C}$ 的丰富。我们表明编织融合类别 $\mathcal{C}$ 的 G-crossed 扩展是 $\mathcal{C}$ 对其自身的规范富集的 G-扩展。作为一个应用程序，我们根据其中心的某些子类别参数化固定的 $G$-graded fusion 类别上的 $G$-crossed braidings 集，