Foundations of Categorified Representation Theory
This thesis develops the foundations of the program of groupoidification and presents an application of this program --- the Fundamental Theorem of Hecke Operators. In stating this theorem, we develop a theory of enriched bicategories and construct the Hecke bicategory --- a categorification of the intertwining operators between permutation representations of a finite group. As an immediate corollary, we obtain a categorification of the Iwahori-Hecke algebra, which leads to solutions of the Zamolodchikov tetrahedron equation. Such solutions are a positive step towards invariants of 2-tangles in 4-dimensional space and constructions of higher-categories with braided structures.