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Register a new userExtension theory for braided-enriched fusion categories
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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 $mathcal{D}$ which 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 Nikshychs classification of the braidings on a fusion category.
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We show any slightly degenerate weakly group-theoretical fusion category admits a minimal extension. Let $d$ be a positive square-free integer, given a weakly group-theoretical non-degenerate fusion category $mathcal{C}$, assume that $text{FPdim}(mathcal{C})=nd$ and $(n,d)=1$. If $(text{FPdim}(X)^2,d)=1$ for all simple objects $X$ of $mathcal{C}$, then we show that $mathcal{C}$ contains a non-degenerate fusion subcategory $mathcal{C}(mathbb{Z}_d,q)$. In particular, we obtain that integral fusion categories of FP-dimensions $p^md$ such that $mathcal{C}subseteq text{sVec}$ are nilpotent and group-theoretical, where $p$ is a prime and $(p,d)=1$.
We introduce a finiteness property for braided fusion categories, describe a conjecture that would characterize categories possessing this, and verify the conjecture in a number of important cases. In particular we say a category has F if the associated braid group representations factor over a finite group, and suggest that categories of integral Frobenius-Perron dimension are precisely those with property F.
We classify various types of graded extensions of a finite braided tensor category $cal B$ in terms of its $2$-categorical Picard groups. In particular, we prove that braided extensions of $cal B$ by a finite group $A$ correspond to braided monoidal $2$-functors from $A$ to the braided $2$-categorical Picard group of $cal B$ (consisting of invertible central $cal B$-module categories). Such functors can be expressed in terms of the Eilnberg-Mac~Lane cohomology. We describe in detail braided $2$-categorical Picard groups of symmetric fusion categories and of pointed braided fusion categories.
We establish rank-finiteness for the class of $G$-crossed braided fusion categories, generalizing the recent result for modular categories and including the important case of braided fusion categories. This necessitates a study of slightly degenerate braided fusion categories and their centers, which are interesting for their own sake.
This is a study of weakly integral braided fusion categories with elementary fusion rules to determine which possess nondegenerately braided extensions of theoretically minimal dimension, or equivalently in this case, which satisfy the minimal modular extension conjecture. We classify near-group braided fusion categories satisfying the minimal modular extension conjecture; the remaining Tambara-Yamagami braided fusion categories provide arbitrarily large families of braided fusion categories with identical fusion rules violating the minimal modular extension conjecture. These examples generalize to braided fusion categories with the fusion rules of the representation categories of extraspecial $p$-groups for any prime $p$, which possess a minimal modular extension only if they arise as the adjoint subcategory of a twisted double of an extraspecial $p$-group.
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