atomic geometric morphism

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.

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.

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).

Atomic morphisms are closed under composition.

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

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

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

This appears as (Johnstone, lemma 3.5.4).

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).

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

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

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

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).

Every etale geometric morphism is atomic.

Section C3.5 of

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