nLab atomic geometric morphism

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 $\mathcal{E}$ is called atomic if its global section geometric morphism is atomic, or in other words, if the constant sheaf functor $\Delta\colon Set\to\mathcal{E}$ is logical.

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. 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).

Proposition

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

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

Proposition

Every étale geometric morphism is atomic.

Related concepts

• atomic topos

• atomic site

References

• Peter Johnstone, Sketches of an Elephant vol. 2 , Oxford UP 2002. (section C3.5, pp.684-695)