On Some Notions of Cohomology for Fusion Categories

Abstract

In this dissertation, we study two main topics: superfusion categories, and embeddings of symmetric fusion categories into modular fusion categories. Using a construction of Brundan and Ellis, we give a formula relating the fermionic 6$j$-symbols of a superfusion category to the 6$j$-symbols of the corresponding underlying fusion category, and prove a version of Ocneanu rigidity for superfusion categories. Inspired by the work of Lan, Kong, and Wen on the group of modular extensions of a symmetric fusion category, we also give definitions for the low cohomology groups of a finite supergroup and show these definitions are functorial. This dissertation includes previously published material.