Topics in 2-categorical Algebra
type: CreativeWork
description: 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.
abstract: 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.
author: Sutton, Matthew
keywords: Mathematics and Applied Mathematics
date published: 2023
date modified: 2026-07-22T22:23:46Z
provider: University of Cape Town
publisher: Networked Digital Library of Theses and Dissertations
type: WebPage
source organization: Department of Mathematics and Applied Mathematics

PropertyValue
property ID: Handle
value: http://hdl.handle.net/11427/39858