The theory of accessible categories is a part of mathematics, specifically of category theory. It attempts to describe categories in terms of the "size" (a cardinal number) of the operations needed to generate their objects. The theory originates in the work of Grothendieck completed by 1969, and Gabriel and Ulmer (1971). It has been further developed in 1989 by Michael Makkai and Robert Paré, with motivation coming from model theory, a branch of mathematical logic. A standard text book by Adámek and Rosický appeared in 1994. Accessible categories also have applications in homotopy theory. Grothendieck continued the development of the theory for homotopy-theoretic purposes in his (still partly unpublished) 1991 manuscript Les dérivateurs. Some properties of accessible categories depend on the set universe in use, particularly on the cardinal properties and Vopěnka's principle.

κ-directed colimits and κ-presentable objects

Let \kappa be an infinite regular cardinal, i.e. a cardinal number that is not the sum of a smaller number of smaller cardinals; examples are \aleph {0} (aleph-0), the first infinite cardinal number, and \aleph{1}, the first uncountable cardinal). A partially ordered set (I, \leq) is called \kappa-directed if every subset J of I of cardinality less than \kappa has an upper bound in I. In particular, the ordinary directed sets are precisely the \aleph_0-directed sets. Now let C be a category. A direct limit (also known as a directed colimit) over a \kappa-directed set (I, \leq) is called a \kappa-directed colimit. An object X of C is called \kappa**-presentable** if the Hom functor preserves all \kappa-directed colimits in C. It is clear that every \kappa**-presentable** object is also \kappa'-presentable whenever , since every \kappa'-directed colimit is also a \kappa-directed colimit in that case. A \aleph_0-presentable object is called finitely presentable.

Examples

κ-accessible and locally presentable categories

The category C is called \kappa-accessible provided that: An \aleph_0-accessible category is called finitely accessible. A category is called accessible if it is \kappa-accessible for some infinite regular cardinal \kappa. When an accessible category is also cocomplete, it is called locally presentable. A functor F : C \to D between \kappa-accessible categories is called \kappa-accessible provided that F preserves \kappa-directed colimits.

Examples

Theorems

One can show that every locally presentable category is also complete. Furthermore, a category is locally presentable if and only if it is equivalent to the category of models of a limit sketch. Adjoint functors between locally presentable categories have a particularly simple characterization. A functor F : C \to D between locally presentable categories: