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Abstract For a finite group , a ‐crossed braided fusion category is a ‐graded fusion category with additional structures, namely, a ‐action and a ‐braiding. We develop the notion of ‐crossed braided zesting: an explicit method for constructing new ‐crossed braided fusion categories from a given one by means of cohomological data associated with the invertible objects in the category and grading group . This is achieved by adapting a similar construction for (braided) fusion categories recently described by the authors. All ‐crossed braided zestings of a given category are ‐extensions of their trivial component and can be interpreted in terms of the homotopy‐based description of Etingof, Nikshych, and Ostrik. In particular, we explicitly describe which ‐extensions correspond to ‐crossed braided zestings.
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Extension Theory for Braided-Enriched Fusion Categories
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 $$\mathcal{D}$$ 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.
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- PAR ID:
- 10283116
- Date Published:
- Journal Name:
- International Mathematics Research Notices
- ISSN:
- 1073-7928
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1093/imrn/rnab133
