An ordinary -enriched category is called atomic if it has a small dense full subcategory of atomic objects, , so that every object of is a small colimit of the functor
More generally, for a cosmos, a -enriched category is atomic if it admits a small -dense full subcategory of atomic objects , such that every object is an enriched coend
Relation to presheaf toposes
A category 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.
Revised on March 10, 2012 23:32:40
by Todd Trimble