#### Date of Award

6-2019

#### Document Type

Open Access

#### Degree Name

Bachelor of Science

#### Department

Mathematics

#### First Advisor

George Todd

#### Keywords

Category Theory, Universal Property, Topology, Abstract Algebra, Arrow, Functor, Natural Transformation

#### Abstract

Category theory unifies and formalizes the mathematical structure and concepts in a way that various areas of interest can be connected. For example, many have learned about the sets and its functions, the vector spaces and its linear transformation, and the group theories and its group homomorphism. Not to mention the similarity of structure in topological spaces, as the continuous function is its mapping. In sum, category theory represents the abstractions of other mathematical concepts. Hence, one could use category theory as a new language to define and simplify the existing mathematical concepts as the universal properties. The goal of this thesis is to provide an understanding of the basic category theory and to derive the universal property of certain mathematical concepts, such as the direct product, the quotient group, and the discrete topology. We start with the basic definitions of Category Theory, namely defining category, functor, natural transformation, and the adjoint. After establishing the basic definition, we will study some notable examples, as well as to propose some interesting examples of our own. Built upon the understanding of the definitions and examples, we will discuss some related questions and come to an application of category theory, the universal property.

#### Recommended Citation

Zhang, Niuniu, "Category Theory and Universal Property" (2019). *Honors Theses*. 2364.

https://digitalworks.union.edu/theses/2364

#### Included in

Algebra Commons, Geometry and Topology Commons, Logic and Foundations Commons, Set Theory Commons