nLab atomic category

Definition

An ordinary $Set$-enriched category $C$ 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 $E$ is equivalent to a presheaf topos (functors with values in the 1-category Set of 0-groupoids) if and only if it is cocomplete and atomic.

This is due to Marta Bunge, who showed it is enough to have a regular cocomplete category with a generating set of atomic objects.

Related concepts

• atom

• compact object

• Cauchy completion