We saw in Chapter 1 that locally presentable categories encompass a considerable collection of “everyday” categories. There are, however, other categories which are well-related to λ-directed colimits, but are not (co)complete, e.g., the categories of fields, Hilbert spaces, linearly ordered sets, etc. These are the accessible categories we investigate in the present chapter.

Accessible categories can be introduced in several equivalent ways: they are precisely

1. the categories with λ-directed colimits in which every object is a λ-directed colimit of λ-presentable objects of a certain set (Definition 2.1);

2. the full sub categories of functor categories SetA closed under Adirected colimits and λ-pure subobjects (Corollary 2.36);

3. the free cocompletions of small categories w.r.t. λ-directed colimits (Theorem 2.26);

4. the categories sketchable by a small sketch (Corollary 2.61);

5. the categories axiomatizable by basic theories in first-order logic (Theorem 5.35).

The relationship between locally presentable and accessible categories is that (a) a category is locally presentable iff it is accessible and complete (Corollary 2.47) and (b) a category is accessible if it is a full, cone-reflective subcategory of a locally presentable category, closed under λ-directed colimits (Theorem 2.53). Moreover, every accessible category has a full embedding into the category of graphs preserving λ-directed colimits (Theorem 2.65).

The natural choice of morphisms between λ-accessible categories is the λ-accessible functors, i.e., functors preserving λ-directed colimits. For example, each left or right adjoint between accessible categories is accessible (Proposition 2.23). In the last section we investigate properties of the 2–category of λ-accessible categories and A-accessible functors. We show, for example, that the Eilenberg-Moore category of a λ-accessible monad is accessible (Theorem 2.78).