Connected and Lagrangian étale algebras in Z_1(2Vect^π_G) are classified by subgroups, braided fusion categories with group actions, and 4-group morphism data; this yields a parametrization of bosonic fusion 2-categories.
Higher Witt Groups for 2-Categories I: Centralizers
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abstract
In this article, we investigate monoidal, braided, sylleptic centralizers of monoidal, braided, sylleptic 2-functors. We specifically focus on multifusion 2-categories and show that monoidal, braided, sylleptic centralizers are multifusion again, via studying the corresponding enveloping algebras. We provide a characterization of the non-degeneracy condition for monoidal, braided, and sylleptic fusion 2-categories, via vanishing of their centers. Applying Double Centralizer Theorems, we establish the relationship between monoidal, braided, symmetric local modules and free modules. In particular, we obtain factorization properties of non-degenerate monoidal, braided, and sylleptic fusion 2-categories. Main results in this article will be used to study higher Witt equivalences of non-degenerate monoidal, braided, sylleptic 2-categories in the sequential articles.
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On \'Etale Algebras and Bosonic Fusion 2-Categories
Connected and Lagrangian étale algebras in Z_1(2Vect^π_G) are classified by subgroups, braided fusion categories with group actions, and 4-group morphism data; this yields a parametrization of bosonic fusion 2-categories.