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Keywords:
Fusion Categories; Braided Categories; Symmetric Fusion Categories
Fusion Categories; Braided Categories; Symmetric Fusion Categories
Summary:
We introduce, for a symmetric fusion category $\Cal A$ with Drinfeld centre $\Cal Z(\Cal A)$, the notion of $\Cal Z(\Cal A)$-crossed braided tensor category. These are categories that are enriched over $\Cal Z(\Cal A)$ equipped with a symmetric tensor product, while being braided monoidal with respect to the usual tensor product on $\Cal Z(\Cal A)$. In the Tannakian case where $\Cal A={\bf Rep}(G)$ for a finite group $G$, the 2-category of $\Cal Z(\Cal A)$-crossed braided categories is shown to be equivalent to the 2-category of $G$-crossed braided tensor categories. A similar result is established for the super Tannakian case where $\Cal A$ is the representation category of a finite super group.
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