Abstract

A locally presentable category can be thought of as a category which is quite large (complete and cocomplete) but which is also quite small (controlled by a mere set of data). The conditions are mild enough that there are many common categories which lie in this optimal range. However, they are also strong enough to admit a concrete representation theorem: in one form, this says that the locally presentable categories are precisely the categories of models of essentially algebraic theories.

I will give a basic introduction to locally presentable categories in the spirit of the book by Adámek and Rosický, covering some motivation, some key properties and various forms of the representation theorem. Those interested could go on to read about accessible categories, an even more general notion enjoying similar properties and an interesting representation theorem.