In this thesis we will examine 2-categories and higher categorical structures and formulate 1-categorical theorems in the language of higher categories as well as formulate some internal definitions of these base structures in finitely complete categories. We will begin by defining the relevant 2- categorical structures, such as 2-categories, double categories, bicategories and enriched categories, as well as examples of all. Following this, we will show first how these structures relate to each other (for instance, a 2-category is a special case of a double category) and then demonstrate that the category of V-enriched categories forms a 2-category. Chapter 2 begins with the definition of internal categories in a category C with pullbacks, as well as internal functors and internal natural transformations, after which we will demonstrate that the category of internal categories forms a 2-category. We will then show that in C with pullbacks and terminal object, one can define an internal 2-category and an internal bicategory , and show that these are the same as small 2-categories and small bicategories in the case of C = Set. In the final chapter, we demonstrate that some of the familiar constructions of 1-category theory can actually be defined in a 2-category, and certain theorems about these structures proven using only 2-categorical methods.
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Department of Mathematics and Applied Mathematics