Murray, MichaelVarghese, MathaiRoberts, David Michael2011-03-182011-03-182010http://hdl.handle.net/2440/62680This thesis introduces two main concepts: a fundamental bigroupoid of a topological groupoid and 2-covering spaces, a categorification of covering spaces. The first is applied to the second to show, among other things, that the fundamental 2-group of the 2-covering space is a sub-2-group of the fundamental 2-group of the base. Along the way we derive general results about localisations of the 2-categories of categories and groupoids internal to a site at classes of weak equivalences, construct a topological fundamental bigroupoid of locally well-behaved spaces, and finish by providing a rich source of examples of 2-covering spaces, including a functorial 2-connected 2-covering space.category theory; groupoids; algebraic topology; homotopy theoryThesis20110105144102