Concept
The theory of accessible categories is a part of mathematics, specifically of . 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 Grothendieck continued the development of the theory for homotopy-theoretic purposes in his (still partly unpublished) 1991 manuscript Les dérivateurs.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. 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 (aleph-0), the first infinite cardinal number, and , the first uncountable cardinal). 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 . A direct limit (also known as a directed colimit) 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. In the category of all sets, the finitely presentable objects coincide with the finite sets. The -presentable objects are the sets of cardinality smaller than . In the , 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.