Tittel: An introduction to abelian categories
Veileder: Steffen Oppermann
Sammendrag: There are many ways to think about category theory. In some sense, category theory is a game with dots and arrows, in addition to a couple of rules on how to use them. Then you explore. From a mathematical point of view, categories may be a way of generalizing many different mathematical structures (such as sets, groups and vector spaces) in a very abstract setting. It strips them of details and seeks to find the common pattern that they share. The main theorem of the thesis is an example of this.
Most people come across the second isomorphism theorem long before seeing the definition of a category, including myself. When seeing it in the categorical setting, I decided to understand its proof properly and the result is this thesis. Every definition, remark, observation and lemma is used to prove the theorem. Also, as you will notice when reading, all proofs are included. My goal is that anyone who knows the definition of an abelian group will be able to understand every step in every proof written. The amount of detail is mostly a consequence of how I learn mathematics myself, namely without much intuition. (If I do have intuition on something, I don’t really trust it.)
Hopefully this thesis can be a gentle introduction to basic category theory and abelian categories for those who have not seen it before. However, it should be mentioned that some very central terms in category theory have been left out, for example functors and natural transformations. Also, the foundations for category theory will not be discussed. For those interested, I recommend Foundations for Category Theory by Daniel Murfet as a place to start.
The main sources which were used during the writing process are Introduction to Categories and Categorical Logic by Samson Abramsky and Nikos Tzevelekos, and Abelian Categories by Daniel Murfet. Also, I would like to mention my supervisor Steffen Oppermann who has been very helpful and generous with his time.