Equivalences of Graded Categories

TL;DR

This paper classifies G-graded extensions of fusion categories, showing an action of automorphism groups on these extensions and establishing criteria for their equivalence, with applications to understanding their structure.

Contribution

It extends classification techniques for G-graded extensions of fusion categories and introduces an action of automorphism groups on these extensions, linking them via monoidal equivalence.

Findings

Group action of Aut(G)×Aut_⊗(C) on G-graded extensions

Equivalent extensions lie in the same orbit of this group action

Reproves classification of graded extensions using graphical calculus

Abstract

We further the techniques developed by Etingof, Nikshych, and Ostrik in order to classify the $\mathcal{C}$-based equivalences between two $G$-graded extensions of $\mathcal{C}$. The main result of this paper (which follows from this classification) shows that there is an action of the group $\operatorname{Aut}(G)\times \operatorname{Aut}_\otimes(\mathcal{C})$ on the set of all $G$-graded extensions of $\mathcal{C}$, and further, any two extensions in the same orbit of this action are monoidally equivalent. As a warm up for the proof of our classification result we reprove the classification of graded extensions of a fusion category, making extensive use of graphical calculus. Aside from our main result, we provide several other practical applications of our classification of $\mathcal{C}$-based equivalences.