nLab
atomic topos

Contents

Context

Topos Theory

topos theory

Background

Toposes

Internal Logic

Topos morphisms

Extra stuff, structure, properties

Cohomology and homotopy

In higher category theory

Theorems

Contents

Idea

An atomic topos is a topos \mathcal{E} where the global sections functor Γ:Set\Gamma: \mathcal{E} \to Set is atomic. In the case where \mathcal{E} is a Grothendieck topos, there are some (arguably more explicit) alternative characterizations of atomic toposes, as in Theorem , which make it clearer why they are called “atomic”.

Definition

Definition

A topos over a base topos Γ:𝒮\Gamma : \mathcal{E} \to \mathcal{S} is called an atomic topos if Γ\Gamma is atomic. Unless otherwise specified, the base topos will be taken to be SetSet.

Definition

A non-zero object AA of a topos \mathcal{E} is an atom if its only subobjects are AA and 00.

Theorem

Let \mathcal{E} be a Grothendieck topos. Then the following are equivalent:

  1. \mathcal{E} is an atomic topos.

  2. \mathcal{E} is the category of sheaves on an atomic site.

  3. The subobject lattice of every object of \mathcal{E} is a complete atomic Boolean algebra.

  4. \mathcal{E} has a small generating set of atoms.

  5. Every object of \mathcal{E} can be written as a disjoint union of atoms.

Properties

Proposition

Let \mathcal{E} be an atomic topos. Then \mathcal{E} is Boolean.

This appears as one direction of (Johnstone, cor. C3.5.2).

Proof

If Γ *\Gamma^* is logical then it preserves the isomorphism **Ω* \coprod * \simeq \Omega characterizing a Boolean topos.

Proposition

Let \mathcal{E} be a Boolean Grothendieck topos with enough points. Then \mathcal{E} is an atomic topos.

Proposition

Atomic Grothendieck toposes Sh(𝒞,J at)Sh(\mathcal{C}, J_{at}) for (𝒞,J at)(\mathcal{C}, J_{at}) an atomic site are precisely the double negation subtoposes Sh ¬¬(Set 𝒞 op)Sh_{\neg\neg}(Set^{\mathcal{C}^{op}}) for a De Morgan presheaf topos Set 𝒞 opSet^{\mathcal{C}^{op}}.

Proof

For the argument see at atomic site.

Decomposition of atomic toposes

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

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

Examples

References

Last revised on August 6, 2016 at 07:47:44. See the history of this page for a list of all contributions to it.