Accessible category

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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. Their motivation was model theoretic, a branch of mathematical logic.^[1]. Some properties of accessible categories depends on the set universe in use, particularly on the cardinal properties^[2]. It turned out that accessible categories have applications in homotopy theory^[1]^[3]

1 Definition
2 Examples
3 Further notions
4 References
5 Further reading

[edit] 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 $H o m (X, - )$ 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, and an $\aleph_0$ -accessible category is often called finitely accessible.

[edit] 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.

[edit] Further notions

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

[edit] References

^ ^a ^b J. Rosicky "On combinatorial model categories", Arxiv, 16 August 2007. Retrieved on 19 January 2008.
^ J. Adamek and J. Rosicky, Locally Presentable and Accessible Categories, Cambridge University Press 1994.
^ J. Rosicky, Injectivity and accessible categories

[edit] Further reading

Makkai, Michael & Paré, Robert (1989), Accessible categories: The foundation of Categorical Model Theory, Contemporary Mathematics, AMS, ISBN 0-8218-5111-X

Adámek, Jiří & Rosicky, Jiří (1994), Locally presentable and accessible categories, LNM Lecture Notes, CUP, ISBN 0-521-42261-2