nLab atomic category

Definition

Recall that an object $c$ in an ordinary $Set$-enriched category $C$ is atomic if $hom(c,-)$ preserves small colimits. The category $C$ itself is called atomic if it has a small dense full subcategory of atomic objects, $Atom(C)$, so that every object $c$ of $C$ is a small colimit of the functor

$$Atom(C) \downarrow c \stackrel{proj}{\to} Atom(C) \stackrel{i}{\hookrightarrow} C.$$

More generally, for $V$ a cosmos, a $V$-enriched category $C$ is atomic if it admits a small $V$-dense full subcategory of atomic objects $Atom(C)$, such that every object $c$ is an enriched coend

$$\int^{a \in Atom(C)} C(i a, c) \cdot i a.$$

Properties

Relation to presheaf toposes

Theorem

A category is equivalent to a category of presheaves, hence of the form $[C^{op}, Set]$, if and only if it is cocomplete and atomic.

Indeed, it suffices that the atomic objects form a strong generator, rather than a dense one. See Centazzo, Rosický & Vitale 2004 for a proof.

Related concepts

• atom

• tiny object

• compact object

• Cauchy completion

References

• Claudia Centazzo, Jiří Rosický, Enrico M. Vitale: A characterization of locally $D$-presentable categories, Cahiers de topologie et géométrie différentielle catégoriques 45 2 (2004) 141-146 [numdam:CTGDC_2004__45_2_141_0, pdf]