Higher Gauge Theory

Abstract

We generalize several aspects of gauge theory to 2-gauge theory. We consider 2-transport on principal 2-bundles with strict 2-group fiber by categorifying a usual definition of principal bundles, following work by Schreiber and Waldorf. We show that locally 2-transport induces a 2-functor from the 2-groupoid of bigons up to thin homotopy to the 2-group. By generalizing the non-Abelian Stokes' Theorem to the 2-group setting, we are able to prove a direct generalization of the Ambrose-Singer Theorem for 2-bundles. Finally we further develop the theory of surface holonomy. In particular we elucidate how such a theory depends on the choice of marking of a surface. We also show that for covering 2-groups the surface holonomy of any connection taking values in a torus computes an invariant of the bundle. Roughly the first half of the thesis is dedicated to a comprehensive introduction to gauge theory and 2-category theory before proceeding to 2-gauge theory.