Extension Theory for Braided-Enriched Fusion Categories. (2nd July 2021)

• Record Type: Journal Article
• Authors: Jones, Corey; Morrison, Scott; Penneys, David; Plavnik, Julia
• Is Part Of: International mathematics research notices. Volume 2022:Number 20(2022)
• Page Start: 15632
• Page End: 15683
• Publication Date: 2021-07-02
• Subjects: Mathematics -- Periodicals; 510
• DOI: 10.1093/imrn/rnab133
• Languages: English
• ISSNs: 1073-7928

Abstract

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.