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,[1] and Gabriel and Ulmer (1971).
[2] It has been further developed in 1989 by Michael Makkai and Robert Paré, with motivation coming from model theory, a branch of mathematical logic.
[3] A standard text book by Adámek and Rosický appeared in 1994.
[4] Accessible categories also have applications in homotopy theory.
[5][6] Grothendieck continued the development of the theory for homotopy-theoretic purposes in his (still partly unpublished) 1991 manuscript Les dérivateurs.
[7] Some properties of accessible categories depend on the set universe in use, particularly on the cardinal properties and Vopěnka's principle.
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
has an upper bound in
In particular, the ordinary directed sets are precisely the
-presentable if the Hom functor
-directed colimit in that case.
-presentable object is called finitely presentable.
-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.
-accessible categories is called
One can show that every locally presentable category is also complete.
[9] Furthermore, a category is locally presentable if and only if it is equivalent to the category of models of a limit sketch.
[10] Adjoint functors between locally presentable categories have a particularly simple characterization.
between locally presentable categories: Adámek, J.; Rosický, J.
Locally Presentable and Accessible Categories.
Cambridge University Press.