topos theory

# Contents

## Definition

###### Definition

A geometric morphism $f:ℰ\stackrel{\stackrel{{f}^{*}}{←}}{\underset{{f}_{*}}{\to }}ℱ$ is called atomic if its inverse image ${f}^{*}$ is a logical functor.

###### Definition

A sheaf topos is called atomic if its global section geometric morphism is atomic.

Generally, a topos over a base topos $\Gamma :ℰ\to 𝒮$ is called an atomic topos if $\Gamma$ is atomic.

###### Note

As shown in prop. 2 below, every atomic morphism $f:ℰ\to 𝒮$ is also a locally connected geometric morphism. The connected objects $A\in ℰ,{f}_{!}A\simeq *$ are called the atoms of $ℰ$.

See (Johnstone, p. 689).

## Properties

###### Proposition

Atomic morphisms are closed under composition.

###### Proposition

An atomic geometric morphism is also a locally connected geometric morphism.

###### Proof

By this proposition a logical morphism with a right adjoint has also a left adjoint.

###### Proposition

If an atomic morphism is also a connected, then it is even hyperconnected.

This appears as (Johnstone, lemma 3.5.4).

###### Corollary

Let $ℰ$ be a Grothendieck topos with enough points. Then $ℰ$ is a Boolean topos precisely if it is an atomic topos.

This appears as (Johnstone, cor. 3.5.2).

###### Proof

If ${\Gamma }^{*}$ logical then it preserves the isomorphism $*\coprod *\simeq \Omega$ characterizing a Boolean topos and hence $ℰ$ is Boolean if it is atomic.

For the converse…

###### Proposition

A localic geometric morphism is atomic precisely if it is an etale geometric morphism.

This appears as (Johnstone, lemma 3.5.4 (iii)).

## Examples

###### Proposition

Every etale geometric morphism is atomic.

## References

Section C3.5 of

Revised on April 20, 2011 15:38:19 by Urs Schreiber (131.211.233.58)