nLab atomic topos

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Topos Theory

topos theory

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Toposes

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Extra stuff, structure, properties

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In higher category theory

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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

Recall that a geometric morphism ff is called atomic if its inverse image functor f *f^* is logical.

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 June 21, 2024 at 09:00:41. See the history of this page for a list of all contributions to it.