Classification of Pointed Fusion Categories of dimension 8 up to weak Morita Equivalence

TL;DR

This paper provides a complete classification of pointed fusion categories of dimension 8 over the complex numbers, including their module categories and braided tensor equivalences, advancing the understanding of their algebraic structures.

Contribution

It offers a comprehensive classification of pointed fusion categories of dimension 8 and their module categories, including braided tensor equivalences of twisted Drinfeld doubles.

Findings

Classified all pointed fusion categories of dimension 8 over .

Determined equivalence classes of module categories for these fusion categories.

Identified braided tensor equivalences of twisted Drinfeld doubles of groups of order 8.

Abstract

In this paper we give a complete classification of pointed fusion categories over $\mathbb{C}$ of global dimension 8. We first classify the equivalence classes of pointed fusion categories of dimension 8, and then we proceed to determine which of these equivalence classes have equivalent categories of modules. This classificaction permits to classify the equivalence classes of braded tensor equivalences of twisted Drinfeld doubles of finite groups of order 8.