
Accessible category

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 Grothendieck continued the development of the theory for homotopy-theoretic purposes in his (still partly unpublished) 1991 manuscript "Les dérivateurs".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.
formula_1-directed colimits and formula_1-presentable objects.
Let formula_1 be an infinite regular cardinal, i.e. a cardinal number that is not the sum of a smaller number of smaller cardinals; examples are formula_4 (aleph-0), the first infinite cardinal number, and formula_5, the first uncountable cardinal). A partially ordered set formula_6 is called formula_1-directed if every subset formula_8 of formula_9 of cardinality less than formula_1 has an upper bound in formula_9. In particular, the ordinary directed sets are precisely the formula_12-directed sets.
Now let formula_13 be a category. A direct limit (also known as a directed colimit) over a formula_1-directed set formula_6 is called a formula_1-directed colimit. An object formula_17 of formula_13 is called formula_1-presentable if the Hom functor formula_20 preserves all formula_1-directed colimits in formula_13. It is clear that every formula_1-presentable object is also formula_24-presentable whenever formula_25, since every formula_24-directed colimit is also a formula_1-directed colimit in that case. A formula_12-presentable object is called finitely presentable.
formula_1-accessible and locally presentable categories.
The category formula_13 is called formula_1-accessible provided that:
An formula_12-accessible category is called finitely accessible.
A category is called accessible if it is formula_1-accessible for some infinite regular cardinal formula_1. 
When an accessible category is also cocomplete, it is called locally presentable.
A functor formula_50 between formula_1-accessible categories is called formula_1-accessible provided that formula_53 preserves formula_1-directed colimits.
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 formula_50 between locally presentable categories: