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Drinfeld centers and Morita equivalence classes of fusion 2-categories

Part of: Modules, bimodules and ideals

Published online by Cambridge University Press:  17 June 2025

Thibault D. Décoppet Open the ORCID record for Thibault D. Décoppet [Opens in a new window]
Thibault D. Décoppet*
Affiliation:
Mathematical Institute, University of Oxford, Oxford OX2 6GG, UK Current address: Department of Mathematics, Harvard University, Cambridge, MA 02138, USA decoppet@math.harvard.edu
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Abstract

We prove that the Drinfeld center of a fusion 2-category is invariant under Morita equivalence. We go on to show that the concept of Morita equivalence between connected fusion 2-categories corresponds to a notion of Witt equivalence between braided fusion 1-categories. A strongly fusion 2-category is a fusion 2-category whose braided fusion 1-category of endomorphisms of the monoidal unit is $\mathbf{Vect}$ or $\mathbf{SVect}$. We prove 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. We proceed to show that every fusion 2-category is Morita equivalent to a connected fusion 2-category. As a consequence, we find that every rigid algebra in a fusion 2-category is separable. This implies in particular that every fusion 2-category is separable. Conjecturally, separability ensures that a fusion 2-category is 4-dualizable. We define the dimension of a fusion 2-category, and prove that it is always non-zero. Finally, we show that the Drinfeld center of any fusion 2-category is a finite semisimple 2-category.

Keywords

Drinfeld centerfusion 2-categoryMorita equivalenceseparable algebraWitt equivalence

MSC classification

Primary: 16D90: Module categories ; module theory in a category-theoretic context; Morita equivalence and duality
Type
Research Article
Information
Compositio Mathematica , Volume 161 , Issue 2 , June 2025 , pp. 305 - 340
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Copyright
© The Author(s), 2025. The publishing rights in this article are licensed to Foundation Compositio Mathematica under an exclusive licence

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