The theory ofaccessible categories is a part of mathematics, specifically of category theory. It attempts to describe categories in terms of the "size" of the operations needed to generate their objects. The theory originates in the work of Grothendieck completed by 1969, and Gabriel and Ulmer. It has been further developed in 1989 by Michael Makkai and RobertParé, 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 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.
-directed colimits and -presentable objects
Let be an infinite regular cardinal, i.e. a cardinal number that is not the sum of a smaller number of smaller cardinals; examples are . A partially ordered set is called -directed if every subset of of cardinality less than has an upper bound in. In particular, the ordinary directed sets are precisely the -directed sets. Now let be a category. A direct limit over a -directed set is called a -directed colimit. An object of is called -presentable if the Hom functor preserves all -directed colimits in. It is clear that every -presentable object is also -presentable whenever , since every -directed colimit is also a -directed colimit in that case. A -presentable object is called finitely presentable.
Examples
In the category Set of all sets, the finitely presentable objects coincide with the finite sets. The -presentable objects are the sets of cardinality smaller than.
In the category of all groups, an object is finitely presentable if and only if it is a finitely presented group, i.e. if it has a presentation with finitely many generators and finitely many relations. For uncountable regular, the -presentable objects are precisely the groups with cardinality smaller than.
contains a set of -presentable objects such that every object of is a -directed colimit of objects of.
An -accessible category is called finitely accessible. A category is called accessible if it is -accessible for some infinite regular cardinal. When an accessible category is also cocomplete, it is called locally presentable. A functor between -accessible categories is called -accessible provided that preserves -directed colimits.
Examples
The category Set of all sets and functions is locally finitely presentable, since every set is the direct limit of its finite subsets, and finite sets are finitely presentable.
The category -Mod of -modules is locally finitely presentable for any ring.
The category Mod of models of some first-order theory T with countable signature is -accessible. -presentable objects are models with a countable number of elements.
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 between locally presentable categories:
is a left adjoint if and only if it preserves small colimits,
is a right adjoint if and only if it preserves small limits and is accessible.
