Compact Semisimple Tensor 2-Categories are Morita Connected
Thibault D. Décoppet, Sean Sanford
TL;DR
The paper extends Morita theory from fusion 2-categories over algebraically closed fields to compact semisimple tensor 2-categories over arbitrary characteristic-zero fields, proving that every such category is Morita equivalent to a connected one. Central to the argument is a two-step reduction: (1) collapsing the symmetric center to a nondegenerate or slightly degenerate braided fusion 1-category, and (2) inflating along finite field extensions to ensure invertible objects appear in every connected component, followed by a descent argument using Picard-group torsion. The work establishes a deep link between Morita invertibility and Witt groups of braided fusion 1-categories, showing injective maps from Galois cohomology $H^4(\mathrm{Gal}(\overline{\mathds{k}}/\mathds{k});\overline{\mathds{k}}^{\times})$ into Witt groups, and it lays groundwork for classifying compact semisimple tensor 2-categories via Morita theory. It also discusses symmetric center collapsing, inflation, and Galois grading as mechanisms to achieve Morita connectedness, with applications to Witt groups, Witt spaces, and conjectural spectral sequences guiding future computations. Overall, the results reveal a robust higher-categorical analogue of Noether-type cohomological classifications, tying together centers, centralizers, and Galois actions in a unified Morita/Witt framework.
Abstract
In arXiv:2211.04917, it was shown that, over an algebraically closed field of characteristic zero, every fusion 2-category is Morita equivalent to a connected fusion 2-category, that is, one arising from a braided fusion 1-category. This result has recently allowed for a complete classification of fusion 2-categories. Here we establish that compact semisimple tensor 2-categories, which generalize fusion 2-categories to an arbitrary field of characteristic zero, also enjoy this ``Morita connectedness'' property. In order to do so, we generalize to an arbitrary field of characteristic zero many well-known results about braided fusion 1-categories over an algebraically closed field. Most notably, we prove that the Picard group of any braided fusion 1-category is indfinite, generalizing the classical fact that the Brauer group of a field is torsion. As an application of our main result, we derive the existence of braided fusion 1-categories indexed by the fourth Galois cohomology group of the absolute Galois group that represent interesting classes in the appropriate Witt groups.