Locally presentable categories

The concept of a locally presentable category is one of the most fruitful concepts of category theory. The definition, generalizing the concept of an algebraic lattice, is natural and simple. The scope is very broad: varieties and quasivarieties of (many-sorted) algebras, Horn classes of relational structures, and functor-categories are all locally presentable. Furthermore, locally presentable categories enjoy a number of important properties: they are complete and cocomplete, wellpowered and co-wellpowered, and they have a strong generator.

The definition of a locally presentable category is due to P. Gabriel and F. Ulmer. Their lecture notes [Gabriel, Ulmer 1971], by now classical, are a profound, but by no means easily readable, treatise on the topic. One of the aims of our monograph is to make the fundamentals of the theory of locally presentable categories more accessible to readers who work in category theory, computer science, and related areas. We have collected these fundamentals in Chapter 1, where the basic properties of locally presentable categories are proved and several equivalent ways of introducing these categories are exhibited. For example, we show that locally presentable categories are precisely the categories sketchable by a limit sketch (i.e., the categories of all set-valued functors preserving specified limits).

Accessible categories

These generalize locally presentable categories by weakening cocompleteness to the existence of some directed colimits. The collection of all categories obtained by this generalization is much broader than that of all locally presentable categories, and it includes categories such as

An important special case: for each sketch in the sense of C.