Accessible category

The theory of accessible categories was introduced in 1989 by mathematicians Michael Makkai and Robert Paré in the setting of category theory. Their motivation was model theoretic, a branch of mathematical logic.J. Rosicky [http://arxiv.org/abs/0708.2185 "On combinatorial model categories"] , "Arxiv", 16 August,2007. Retrieved on 19 January, 2008.] . Some properties of accessible categories depends on the set universe in use, particularly on the cardinal properties J. Adamek and J. Rosicky, Locally Presentable and Accessible Categories, Cambridge University Press 1994.] . It turned out that accessible categories have applications in homotopy theory J. Rosicky, Injectivity and accessible categories ]

Definition

Let $K$ be an infinite regular cardinal and let $C$ be a category.An object $X$ of $C$ is called $K$-presentable if the Hom functor $Hom(X,-)$ preserves $K$ - directed colimits.The category $C$ is called $K$ - accessible provided that :
* $C$ has $K$-directed colimits
* $C$ has a set $P$ of $K$ - presentable objects such that every object of $C$ is a $K$ - directed colimit of objects of $P$

A category $C$ is called accessible if $C$ is $K$ - accessible for some infinite regular cardinal $K$.

A $aleph\_0$-presentable object is usually called finitely presentable, andan $aleph\_0$-accessible category is often called finitely accessible.

Examples

*The category $R$-Mod of (left) $R$-modules is finitely accessible for any ring $R$. The objects that are finitely presentable in the above sense are finitely generated modules (which are not necessarily finitely presented modules unless $R$ is noetherian).
*The category of simplicial sets is finitely-accessible.
*The category Mod(T) of models of some first-order theory T with countable signature is $aleph\_1$ -accessible. $aleph\_1$ -presentable objects are models with a countable number of elements.

Further notions

When the category $C$ is cocomplete, $C$ is called a locally presentable category.Locally presentable categories are also complete.

References

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