Computing the group of minimal non-degenerate extensions of a super-Tannakian category

Nikshych, Dmitri

doi:10.48550/arxiv.2201.04076

Cited by 1 publication

(2 citation statements)

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“…Let G be a finite group. It follows from the above corollary and example 2.1 of [Nik22] that there is a bijection of sets BrPic(Mod(Rep(G))) H 3 (G; k × ). Namely, in the Tannakian case, every minimal non-degenerate extension of Rep(G) is Witt-trivial.…”

Section: Connected Fusion 2-categoriesmentioning

confidence: 80%

“…On the other hand, the above corollary shows that BrPic(2SVect) Z/2Z. Namely, the group of minimal non-degenerate extensions of SVect is identified with Z/16Z by example 2.2 of [Nik22], and only two out of the sixteen minimal non-degenerate extensions of SVect are Witt-trivial (see appendix A.3.2 of [DGNO10]). But, it follows from corollary 2.3.8 that every minimal nondegenerate extension of SVect induces a braided autoequivalence of Z (2SVect).…”

Section: Connected Fusion 2-categoriesmentioning

confidence: 99%

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On the Drinfeld Centers of Fusion 2-Categories

Décoppet¹

2022

Preprint

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We prove that the Drinfeld center of a fusion 2-category is invariant under Morita equivalence and under taking the 2-Deligne tensor product with an invertible fusion 2-category. We go on to show that the concept of Morita equivalence between connected fusion 2-categories recovers exactly the notion of Witt equivalence between braided fusion 1-categories. Then, we introduce the notion of separable fusion 2-category. Conjecturally, separability ensures that a fusion 2-category is 4-dualizable. We define the dimension of a fusion 2-category, which is a scalar whose non-vanishing is equivalent to separability. In addition, we prove that a fusion 2-category is separable if and only if its Drinfeld center is finite semisimple. We then establish the separability of every strongly fusion 2-category, that is fusion 2-category whose braided fusion 1-category of endomorphisms of the monoidal unit is Vect or SVect. We proceed to show that every fusion 2-category is Morita equivalent to the 2-Deligne tensor product of a strongly fusion 2-category and an invertible fusion 2-category. Finally, we prove that every fusion 2-category is separable.

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Section: Connected Fusion 2-categoriesmentioning

confidence: 80%

Section: Connected Fusion 2-categoriesmentioning

confidence: 99%