An invitation to the theory of locally presentable categories

Talk by Julia Ramos González (UCLouvain) at the Antwerp Algebra Colloquium on March 25, 2022.
The family of locally presentable categories is sufficiently large to contain a myriad of examples of great interest, but small enough to fulfill a range of significant categorical properties. In particular, most of the categories in the toolkit of the working algebraist are locally presentable. For example, the categories of sets, magmas, semigroups, groups, rings, modules, graphs or relational structures are all locally presentable and in general any variety or quasi-variety of algebras is. The understanding of locally presentable categories thus provides us with a general understanding of the structural properties shared by all these categories. The aim of this talk is to present a brief introduction to the theory of locally presentable categories. First, we will provide the basic definitions and analyze several examples. We will then outline the main properties of locally presentable categories. We will conclude by analyzing the different presentations in which locally presentable categories can be found in nature.