Compact Semisimple 2-Categories

TL;DR

This paper introduces the concept of compact semisimple 2-categories over arbitrary fields, characterizes their structure, and extends key results from finite semisimple 2-category theory to this broader setting.

Contribution

It defines compact semisimple 2-categories, proves their equivalence to module categories over tensor 1-categories, and generalizes important finite semisimple results.

Findings

Every compact semisimple 2-category is equivalent to a category of separable module 1-categories.

Over algebraically closed or real closed fields, these categories are finite.

Key results from finite semisimple 2-category theory extend to compact semisimple 2-categories.

Abstract

Working over an arbitrary field, we define compact semisimple 2-categories, and show that every compact semisimple 2-category is equivalent to the 2-category of separable module 1-categories over a finite semisimple tensor 1-category. Then, we prove that, over an algebraically closed field or a real closed field, compact semisimple 2-categories are finite. Finally, we explain how a number of key results in the theory of finite semisimple 2-categories over an algebraically closed field of characteristic zero can be generalized to compact semisimple 2-categories.