topos theory

# Contents

## Definition

###### Definition

A geometric morphism $f : \mathcal{E} \stackrel{\overset{f^*}{\leftarrow}}{\underset{f_*}{\to}} \mathcal{F}$ 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 $\Gamma : \mathcal{E} \to \mathcal{S}$ is called an atomic topos if $\Gamma$ is atomic.

###### Note

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

See (Johnstone, p. 689).

## Properties

### General

###### 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 $\mathcal{E}$ be a Grothendieck topos with enough points. Then $\mathcal{E}$ 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 $\mathcal{E}$ 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)).

### Decomposition of atomic toposes

Atomic toposes decompose as disjoint unions of connected atomic toposes. Connected atomic toposes with a point of a topos are the classifying toposes of localic groups.

An example of a connected atomic topos without a point of a topos is given in (Johnstone, example D3.4.14).

## Examples

###### Proposition

Every etale geometric morphism is atomic.

## References

Section C3.5 of

Revised on June 19, 2013 03:11:27 by Urs Schreiber (82.169.65.155)