Elements of Abelian Categories

Abstract

In this thesis, we construct the Baer sum of extensions in the sense of Yoneda and prove that it endows the
Ext-classes with an Abelian group structure. Extensions are defined by means of equivalence classes of short
exact sequences in Abelian categories. Now, in order to do that, we carefully lay a solid foundation for the
theory of Abelian categories by starting with elementary concepts from category theory and reviewing limits.
We then study pointed, semiadditive, preadditive, additive, Puppe-exact and finally, Abelian categories.
We prove that every Abelian category is additive and give several characterizations for additive, exact and
Abelian categories. Lastly, we prove several technical results about exact sequences that will help us achieve
our initial purpose.

Popular Abstract

In high school, linear algebra is understood as the study of systems of linear equations. Later, in university, one thinks about linear algebra differently, from a higher level viewpoint. Now, although matrices may play an important role, the focus is on vector spaces and linear maps. Even though this is not typically mentioned in a first course on linear algebra in university, we can think of all vector spaces and all linear maps like an infinite graph, where nodes are vector spaces and linear maps are edges with direction. We can compose linear maps that follow one after another, so we get another edge g ◦ f. With this realization, we have discovered one of the most important concepts in mathematics, that of a category. We call this system of nodes and directed edges the category of vector spaces and linear maps, and we denote it by Vect.

However, once we start learning about category theory, we notice that there are some properties about Vect that the original definition of a category does not distinguish. For example, if f : V → W is a linear map, then we are very used to taking its kernel and its image, and we have the renowned First Isomorphism Theorem, or we might observe that the set of all linear maps V → W is more than a set, it is also a vector space. Also, observe that the vector space {0} of dimension zero has the property that for any vector space V, there is only one map into it from V (the constant map to 0) and only one map to V from it (the inclusion). This is what we will call a zero object.

We will abstract properties like these in great generality to axiomatically study categories that behave like Vect. We will see many kinds of new categories, building up until we arrive to the concept of an Abelian
category, which is the central object of study. We will be very comfortable working with them, as they have the properties that we liked about Vect.The second half of the thesis is rather technical, but just like in Vect, we can define in an Abelian category the concept of an exact sequence, and then we can find a surprising way to sum them, just like we can sum numbers. This sum is what we call the Baer sum.

• author: González Pérez, Diego
• supervisor: Gustav Jasso
• organization: Mathematics (Faculty of Engineering); Mathematics (Faculty of Sciences); Centre for Mathematical Sciences
• alternative title: The Yoneda Ext and the Baer Sum
• course: MATK11 20231
• year: 2024
• type: M2 - Bachelor Degree
• subject: Mathematics and Statistics
• keywords: category theory, abelian category, functor, exact category, additive category, pointed category, Ext functor, extension, preadditive category, zero object, constant morphism
• publication/series: Bachelor’s Theses in Mathematical Sciences
• report number: LUNFMA-4167-2024
• ISSN: 1654-6229
• other publication id: 2024:K11
• language: English
• id: 9170311
• date added to LUP: 2024-12-05 16:08:16
• date last changed: 2024-12-05 16:08:16