The first chapter is devoted to an important class of categories, the locally presentable categories, which is broad enough to encompass a great deal of mathematical life: varieties of algebras, implicational classes of relational structures, interesting cases of posets (domains, lattices), etc., and yet restricted enough to guarantee a number of completeness and smallness properties. Besides, locally presentable categories are closed under a number of categorical constructions (limits, comma-categories), see also Chapter 2. The basic concept, a finitely presentable object, can be regarded as a generalization of the concept of a finite (or compact) element in a Scott domain, i.e., an element a such that for each directed set {di | i ∈ I} with a ≤ ∨ i∈Idi it follows that a ≤ di for some i ∈ I. Now, an object A is finitely presentable if for each directed diagram {Di | i ∈ I} every morphism A → colimi∈IDi factorizes (essentially uniquely) through Di for some i ∈ I.

More generally, an object A is λ-presentable (for a cardinal λ) if every morphism from A to a λ-directed colimit colimi∈IDi factorizes (essentially uniquely) through some Di. A category is locally A-presentable iff it has colimits and is generated (in some strong sense) by a set of λ-presentable objects. We will see that there are many equivalent ways in which locally λ-presentable categories can be introduced: they are precisely

1. the cocomplete categories in which every object is a λ-directed colimit of λ-presentable objects of a certain set (Definition 1.17);

2. the cocomplete categories with a strongly generating set of λ-presentable objects (Theorem 1.20);

3. […]