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 $\mathcal{E}$ 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. 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).

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 etale geometric morphism is atomic.

References

Last revised on August 6, 2016 at 07:37:56. See the history of this page for a list of all contributions to it.