Fusion 2-categories with no line operators are grouplike

TL;DR

This paper proves that certain fusion 2-categories with specific endomorphism categories have their indecomposable objects forming a finite group, revealing a grouplike structure.

Contribution

It establishes a new structural result linking fusion 2-categories with trivial endomorphism categories to finite groups.

Findings

Indecomposable objects form a finite group

Fusion 2-categories with Vec or SVec endomorphisms are grouplike

No line operators in these fusion 2-categories

Abstract

We show that if $\mathcal{C}$ is a fusion $2$-category in which the endomorphism category of the unit object is $\rm{Vec}$ or $\rm{SVec}$, then the indecomposable objects of $\mathcal{C}$ form a finite group.