nLab atomic topos

Idea

An atomic topos is a topos $\mathcal{E}$ where the global sections functor $\Gamma: \mathcal{E} \to Set$ 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 $f$ is called atomic if its inverse image functor $f^*$ is logical.

Definition

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

Definition

A non-zero object $A$ of a topos $\mathcal{E}$ is an atom if its only subobjects are $A$ and $0$.

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.

Proof

See Johnstone, C3.5.8 and Barr-Diaconescu, Theorem A.

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

Proposition

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

Proof

See (Johnstone, cor. C3.5.2)

Proposition

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

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

• A category of presheaves $Set^{\mathcal{C}^{op}}$ is atomic precisely iff $\mathcal{C}$ is a groupoid (cf. Barr-Diaconescu (1980)).

• Another example of an atomic Grothendieck topos is the Schanuel topos. More generally, any category of G-sets is an atomic Grothendieck topos.

Related concepts

• atomic geometric morphism

• atomic site

• connected topos

References

• Michael Barr, Radu Diaconescu, Atomic Toposes, J. Pure Appl. Algebra 17 (1980) 1-24 [doi:10.1016/0022-4049(80)90020-1, pdf, pdf]

• Olivia Caramello, Atomic toposes and countable categoricity , Appl. Cat. Struc. 20 no. 4 (2012) pp.379-391. (arXiv:0811.3547)

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

• Jérémie Marquès?, Atomic Toposes with Co-Well-Founded Categories of Atoms , arXiv:2406.14346 (2024). (abstract)