nLab atomic site

Idea

Atomic sites are sites $(\mathcal{C}, J_{at})$ equipped with the atomic topology $J_{at}$. The corresponding sheaf toposes $Sh(\mathcal{C}, J_{at})$ are precisely the atomic Grothendieck toposes.

Definition

Definition

A site $(\mathcal{C}, J_{at})$ is called atomic if the covering sieves $S$ of $J_{at}$ are exactly the inhabited sieves $S\neq\emptyset$. A Grothendieck topology $J_{at}$ of this form is called atomic.

Example

Let $FinSet^{op}_{mono}$ be the opposite of the category $FinSet_{mono}$ with objects finite sets and monomorphisms. Then $(FinSet^{op}_{mono}, J_{at})$ is an atomic site and the corresponding sheaf topos $Sh(FinSet^{op}_{mono}, J_{at})$ is the Schanuel topos. That $J_{at}$ is indeed a Grothendieck topology is ensured by prop. .

Properties

Proposition

Let $\mathcal{C}$ be a category. Then $\mathcal{C}$ can be made into an atomic site if and only if for any diagram

$$\array{ & & A \\ & & \downarrow\\ B & \to & C }$$

there is an object $D$ and arrows $D \to A, B$ such that the following diagram commutes:

$$\array{ D & \to & A \\ \downarrow & & \downarrow\\ B & \to & C }$$

Proof

This is exactly what is needed for the pullback stability axiom to hold, and the other axioms are immediate.

The condition occurring in the proposition is called the (right) Ore condition. It is a result by P. T. Johnstone (1979) that $Set^{\mathcal{C}^{op}}$ is a De Morgan topos precisely if $\mathcal{C}$ satisfies the Ore condition. Whence we see that every atomic Grothendieck toposes is a (Boolean) subtopos of a De Morgan presheaf topos.

Recall that the dense topology $J_d$ on a category $\mathcal{C}$ consists of all sieves $S\in J_d(C)$ with the property that given $f:D\to C$ there exists $g:E\to D$ such that $f\cdot g\in S$. The atomic topology is a special case of this:

Proposition

Let $\mathcal{C}$ be a category satisfying the Ore condition. Then the atomic topology $J_{at}$ coincides with the dense topology $J_d$.

Proof

For $\mathcal{C}=\emptyset$ the claim is trivial. So let $C\in\mathcal{C}$ be an object and $S$ a sieve on $C$.

Assume $S\in J_d(C)$, then for $id\colon C\to C$ there exists $g:E\to C$ with $id\cdot g\in S$ whence $S\in J_{at}(C)$.

Conversely, assume $S\in J_{at}(C)$ and let $f:D\to C$ be a morphism. Then there exists $g\in S$ by assumption and the diagram $D\overset{f}{\rightarrow} C \overset{g}{\leftarrow} E$ can be completed to a commutative square $f\cdot i = g\cdot h$ but $g\cdot h\in S$ since $g\in S$ and $S$ is a sieve. Whence $f \cdot i\in S$ and, accordingly, $S\in J_d(C)$.

In other words, the atomic topology is just the dense topology on categories satisfying the Ore condition. Since the corresponding sheaf toposes of the dense topology are just the double negation subtoposes of the corresponding presheaf topos we finally get:

Proposition

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

The sheaves of atomic sheaf toposes $Sh(\mathcal{C}, J_{at})$ are easy to describe:

Proposition

Let $(\mathcal{C}, J_{at})$ be an atomic site. A presheaf $P\in Set^{\mathcal{C}^{op}}$ is a sheaf for $J_{at}$ iff for any morphism $f:D\to C$ and any $y\in P(D)$ , if $P(g)(y)=P(h)(y)$ for all diagrams

$$E\overset{g}{\underset{h}{\rightrightarrows}} D\overset{f}{\to} C$$

with $f\cdot g=f\cdot h$ , then $y=P(f)(x)$ for a unique $x\in P(C)$.

For the proof see Mac Lane-Moerdijk (1994, pp.126f).

The more general definition

Thus far we have presented the classical approach as presented in Mac Lane-Moerdijk (1994) going back to Barr-Diaconescu (1980) but it was observed by O. Caramello (2012) that the atomic topology can in fact be defined on arbitrary categories not only on those satisfying the Ore condition.

Definition’

Let $\mathcal{C}$ be a category. The atomic topology $J_{at}$ on $\mathcal{C}$ is the smallest Grothendieck topology containing all the nonempty sieves. A site of the form $(\mathcal{C}, J_{at})$ is called atomic.

Note that $J_{at}$ is well defined as the intersection of all Grothendieck topologies with the property that all nonempty sieves cover. The following proposition justifies the terminology:

Proposition

Let $(\mathcal{C}, J_{at})$ be an atomic site. Then $Sh(\mathcal{C}, J_{at})$ is an atomic Grothendieck topos.

Proof

The main idea is to consider the full subcategory $\mathcal{C}'$ on those objects $U$ with $\emptyset\notin J_{at}(U)$ together with the induced topology $J'_{at}=J_{at}|_{\mathcal{C}'}$. Then one shows that $\mathcal{C}'$ satisfies the Ore condition and concludes by the comparison lemma that $Sh(\mathcal{C}', J'_{at})\simeq Sh(\mathcal{C}, J_{at})$. For the details see Caramello (2012, prop.1.4).

Example

Consider the category $\mathcal{C}$ on the ‘walking co-span’ $A\overset{f}{\rightarrow} C\overset{g}{\leftarrow} B$. $\mathcal{C}$ does not satisfy the Ore condition. The atomic topology $J_{at}$ is given by

$$J_{at}(A)=\{\{id_A\},\emptyset\} \qquad J_{at}(B)=\{\{id_B\},\emptyset\}$$

$$J_{at}(C)=\{\{ id_C, f,g\},\{f \}, \{ g \},\{f,g\},\emptyset\} \quad .$$

Here $\emptyset\in J_{at}(A)$ , respectively $\emptyset\in J_{at}(B)$, due to the stability axiom applied to $f^\ast(\{g\})=\emptyset$ , respectively to $g^\ast(\{f\})=\emptyset$ . Whereas $\emptyset\in J_{at}({C})$ by the transitivity axiom applied to $\{f\}\in J_{at}(C)$ and the sieve $\emptyset$ since $f^{\ast}(\emptyset)=\emptyset\in J_{at}(A)$.

Accordingly the subcategory $\mathcal{C}'$ is empty and $Sh(\mathcal{C},J_{at})\simeq 1$ is degenerate. In particular, $Sh(\mathcal{C},J_{at})$ is not equivalent to $Sh(\mathcal{C},J_d)\simeq Sh_{\neg\neg}(Set^{\mathcal{C}^{op}})\simeq Set\times Set$. So we see that the atomic topology on $\mathcal{C}$ is distinct from the dense topology. For completeness we describe the latter:

$$J_{d}(A)=\{\{id_A\}\} \qquad J_{d}(B)=\{\{id_B\}\}$$

$$J_{d}(C)=\{\{id_C, f,g\},\{f,g\}\} \quad .$$

Further details on $Set^{\mathcal{C}^{op}}$, the topos of hypergraphs, may be found at hypergraph.

In the example, we observed that dense and the atomic topology need not coincide for categories not satisfying the Ore condition. In fact more can be said here:

Proposition

Let $\mathcal{C}$ be a category. Then $J_{d}\subseteq J_{at}$ in general, but $J_d=J_{at}$ precisely if $\mathcal{C}$ satisfies the Ore condition.

Proof

The proof of prop. already showed that the sieves of the dense topology $J_d$ are never empty regardless of the Ore condition. From prop. follows that the atomic topology $J_{at}$ will additionally contain empty sieves precisely if $\mathcal{C}$ does not satisfy the Ore condition.

In particular, $Sh(\mathcal{C},J_{at})\subseteq Sh(\mathcal{C},J_{d})\simeq Sh_{\neg\neg}(Set^{\mathcal{C}^{op}})$ .

Related entries

• dense topology

• atomic topos

• atomic geometric morphism

• Ore condition

• De Morgan topos

Reference

• Michael Barr, Radu Diaconescu, Atomic Toposes , JPAA 17 (1980) pp.1-24. (pdf)

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

• Saunders Mac Lane, Ieke Moerdijk, Sheaves in Geometry and Logic , Springer Heidelberg 1994. (pp.115, 126)