Generalised Orbifolds and G-equivariantisation

TL;DR

This paper explores how orbifold data in ribbon categories can be used to construct new categories and establishes an equivalence between orbifold categories and G-equivariantizations in the context of topological field theory.

Contribution

It provides a constructive proof showing that orbifold categories derived from a ribbon category are equivalent to G-equivariantizations of a G-crossed ribbon category.

Findings

Orbifold data in ribbon categories can define new ribbon categories.

The constructed orbifold category is equivalent to G-equivariantization of a G-crossed ribbon category.

The proof offers a concrete method to realize this equivalence.

Abstract

In a construction motivated by topological field theory, a so-called orbifold datum $\mathbb{A}$ in a ribbon category $C$ allows one to define a new ribbon category $C_{\mathbb{A}}$. If $C$ is the neutral component of a $G$-crossed ribbon category $B$, and $\mathbb{A}$ is an orbifold datum in $C$ defined in terms of $B$, one finds that $C_{\mathbb{A}}$ is equivalent to the equivariantisation $B^G$ of $B$ as a ribbon category. We give a constructive proof of this equivalence.