Accessible category

The theory of accessible categories originates from the work of Grothendieck completed by 1969 (Grothendieck (1972)) and Gabriel-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.^[1] Accessible categories have also applications in homotopy theory.^[1]^[2] Grothendieck also continued the development of the theory for homotopy-theoretic purposes in his (still partly unpublished) 1991 manuscript Les dérivateurs (Grothendieck (1991)). Some properties of accessible categories depend on the set universe in use, particularly on the cardinal properties.^[3]

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 $\operatorname {Hom}(X,-)$ 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, and an $\aleph _{0}$ -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 precisely the finitely presented modules (which are not necessarily the same as the finitely generated 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 an accessible category $C$ is also cocomplete, $C$ is called locally presentable. Locally presentable categories are also complete.

References

1 2 J. Rosicky "On combinatorial model categories", Arxiv, 16 August 2007. Retrieved on 19 January 2008.
↑ J. Rosicky, Injectivity and accessible categories
↑ J. Adamek and J. Rosicky, Locally Presentable and Accessible Categories, Cambridge University Press 1994

Accessible category

Definition

Examples

Further notions

References

Further reading