Gauging Generalised Symmetries : Group-Theoretic Higher Fusion Categories and Higher Representation Theory

In this thesis we will construct novel non-invertible symmetries by gauging finite invertible symmetries, discuss the various formalisms for performing this construction using one and two dimensions as motivating examples, and invest- igate what the most general notion of gauging is in three dimensions. While unitary fusion categorical symmetries are now well under control for describing symmetries of oriented unitary quantum field theories in two dimensions, the same cannot be said for unitary fusion 2-categorical symmetries in three dimensions. It is our hope that by conjecturing that all such underlying fusion 2-categories are group-theoretic, that we can systematically construct all examples and describe how they should lift to unitary fusion 2-categories.