nLab
atomic geometric morphism

Context

Topos Theory

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Contents

Definition

Definition

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

Note

As shown in prop. 2 below, every atomic morphism f:𝒮f : \mathcal{E} \to \mathcal{S} is also a locally connected geometric morphism. The connected objects A,f !A*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 Γ *\Gamma^* logical then it preserves the isomorphism **Ω* \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)