1.1. Introduction

This is a short introduction on basic category theory with emphasis on proofs of results and examples to help illustrate the categorical perspective. We include definitions and examples of categories, functors and natural transformations we then take a deeper look into the idea of universal properties such as adjoints, limits, initial and terminal objects, and monads. We look at how these universal properties are linked as well as how they show up in categories we are used to working with.

Category theory is a branch of maths developed in the 1940's by Saunders Maclane and Samuel Eilenberg as an offshoot of algebraic topology (\cite{wiki:history}). Category theory aims to generalise the construction of mathematical objects such as: mappings, products, quotient spaces algebras and modules.

Category theory is a vast area of mathematics and this report only states the very beginnings, interesting areas to look at after reading this Include: Representables and The Yoneda Lemma, Higher category theory and enriched category theory. This report has a ground up approach where we aim to prove every result stated. We also include lots of worked examples. The main references for this report are from the books: Adámek - Herrlick - Strecker \cite{ACC}, Leinster \cite{Leinster} and Riehl \cite{Riehl} some supporting theory is also from nCatLab \cite{nLab} other references used will be cited in the report.

\subsection{Ethics}

It is important to uphold the academic integrity and ethical principals when writing a paper to ensure its credibility. In this paper we have, to our best ability, cited every author of books, webpages and other sources where we have adapted or used their ideas and material. We state where we have added our own proofs of theorems or examples. Additionally, we expect the publication of this report will cause no harm to human kind, animals or nature. The purpose of the paper is to compile ideas and research and add our own proofs and perspective on the examples to promote the advancement of knowledge in the field.

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