Compact Semisimple Tensor 2-Categories are Morita Connected

TL;DR

This paper extends the concept of Morita connectedness from fusion 2-categories to compact semisimple tensor 2-categories over any field of characteristic zero, broadening the classification framework for these mathematical structures.

Contribution

It generalizes Morita connectedness to compact semisimple tensor 2-categories over arbitrary fields of characteristic zero, including new results on the Picard group of braided fusion 1-categories.

Findings

Proves that the Picard group of any braided fusion 1-category is indfinite.

Establishes Morita connectedness for compact semisimple tensor 2-categories over any characteristic zero field.

Constructs braided fusion 1-categories indexed by Galois cohomology classes.

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.