nLab
atomic category

Contents

Definition

An ordinary SetSet-enriched category CC is called atomic if it has a small dense full subcategory of atomic objects, Atom(C)Atom(C), so that every object cc of CC is a small colimit of the functor

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

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

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

Properties

Relation to presheaf toposes

Theorem

A category EE 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 (67.80.8.47)