# Some non-braided fusion categories of rank 3

## TL;DR

This paper classifies fusion categories with three simple objects based on specific fusion rules, providing a concrete graphical calculus framework, and discusses pivotality and sphericity, potentially completing the classification under Ostrik's conjecture.

## Contribution

It offers a classification of all fusion categories with three simple objects under certain fusion rules, assuming Ostrik's conjecture, and introduces a practical graphical calculus framework.

## Key findings

- Classification of fusion categories with three simple objects
- Development of a concrete graphical calculus for fusion categories
- Elementary proof that the quadruple dual functor is naturally isomorphic to the identity

## Abstract

We classify all fusion categories for a given set of fusion rules with three simple object types. If a conjecture of Ostrik is true, our classification completes the classification of fusion categories with three simple object types. To facilitate the discussion we describe a convenient, concrete and useful variation of graphical calculus for fusion categories, discuss pivotality and sphericity in this framework, and give a short and elementary re-proof of the fact that the quadruple dual functor is naturally isomorphic to the identity.