category theory

# Contents

## Definition

An ordinary $\mathrm{Set}$-enriched category $C$ is called atomic if it has a small dense full subcategory of atomic objects, $\mathrm{Atom}\left(C\right)$, so that every object $c$ of $C$ is a small colimit of the functor

$\mathrm{Atom}\left(C\right)↓c\stackrel{\mathrm{proj}}{\to }\mathrm{Atom}\left(C\right)\stackrel{i}{↪}C.$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 $\mathrm{Atom}\left(C\right)$, such that every object $c$ is an enriched coend

${\int }^{a\in \mathrm{Atom}\left(C\right)}C\left(ia,c\right)\cdot ia.$\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.

Revised on March 10, 2012 23:32:40 by Todd Trimble (67.80.8.47)