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2012-11-26T19:44:00Z

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The theory of '''accessible categories''' was introduced in 1989 by mathematicians [[Michael Makkai]] and Robert Paré in the setting of [[category theory]]. While the original motivation came from [[model theory]], a branch of [[mathematical logic]],<ref name=ref1>J. Rosicky [http://arxiv.org/abs/0708.2185 "On combinatorial model categories"], ''[[Arxiv]]'', 16 August 2007. Retrieved on 19 January 2008.</ref> it turned out that accessible categories have applications in [[homotopy theory]].<ref name=ref1/><ref name=ref3>J. Rosicky, Injectivity and accessible categories</ref> Some properties of accessible categories depend on the [[Axiomatic set theory|set universe]] in use, particularly on the [[cardinal number|cardinal]] properties.<ref name=ref2>J. Adamek and J. Rosicky, Locally Presentable and Accessible Categories, Cambridge University Press 1994</ref>

==Definition==
Let <math>K</math> be an infinite [[regular cardinal]] and let <math>C</math> be a [[category (mathematics)|category]].
An object <math>X</math> of <math>C</math> is called '''<math>K</math>-presentable''' if the [[Hom functor]] <math>Hom(X,-)</math> preserves <math>K</math>-directed [[colimit]]s.
The category <math>C</math> is called '''<math>K</math>-accessible''' provided that :
* <math>C</math> has <math>K</math>-directed colimits
* <math>C</math> has a set <math>P</math> of <math>K</math>-presentable objects such that every object of <math>C</math> is a <math>K</math>-directed colimit of objects of <math>P</math>

A category <math>C</math> is called '''accessible''' if <math>C</math> is <math>K</math>-accessible for some infinite regular cardinal <math>K</math>.

A <math>\aleph_0</math>-presentable object is usually called '''finitely presentable''', and
an <math>\aleph_0</math>-accessible category is often called '''finitely accessible'''.

==Examples==

*The category [[Category of modules|<math>R</math>-Mod]] of (left) <math>R</math>-modules is finitely accessible for any ring <math>R</math>. The objects that are finitely presentable in the above sense are precisely the [[finitely presented module]]s (which are not necessarily the same class of objects than [[finitely generated module]]s unless <math>R</math> is [[noetherian ring|noetherian]]).
*The category of [[simplicial set]]s is finitely-accessible.
*The category Mod(T) of models of some [[first-order theory]] T with countable signature is <math>\aleph_1</math> -accessible. <math>\aleph_1</math> -presentable objects are models with a countable number of elements.

==Further notions==

When the category <math>C</math> is [[cocomplete]], <math>C</math> is called a locally presentable category.
Locally presentable categories are also [[Complete category|complete]].

==References==
{{reflist}}

==Further reading==
* {{Citation
| last = Makkai | first = Michael
| last2 = Paré | first2 = Robert
| title = Accessible categories: The foundation of Categorical Model Theory
| publisher = AMS
| series = Contemporary Mathematics
| year = 1989
| isbn = 0-8218-5111-X }}
* {{Citation
| last = Adámek | first = Jiří
| last2 = Rosicky | first2 = Jiří
| title = Locally presentable and accessible categories
| publisher = CUP
| series = LNM Lecture Notes
| year = 1994
| isbn = 0-521-42261-2 }}

[[Category:Category theory]]

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