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Grothendieck also continued the development of the theory for homotopy-theoretic purposes in his (still partly unpublished) 1991 manuscript Les dérivateurs (Grothendieck (1991)).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 $Hom(X,-)$ 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 the category $C$ is cocomplete, $C$ is called a locally presentable category. Locally presentable categories are also complete.

References

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

Further reading

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

• Gabriel, P; Ulmer, F (1971), Lokal Präsentierbare Kategorien, Lecture Notes in Mathematics 221, Springer

• Grothendieck, Alexander; et al. (1972), Théorie des Topos et Cohomologie Étale des Schémas, Lecture Notes in Mathematics 269, Springer

• Grothendieck, Alexander (1991), Les dérivateurs, Contemporary Mathematics, manuscript (Les Dérivateurs: Texte d'Alexandre Grothendieck. Édité par M. Künzer, J. Malgoire, G. Maltsiniotis)

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