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\begin{document}

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\section*{atomic site}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{topos_theory}{}\paragraph*{{Topos Theory}}\label{topos_theory}

[[!include topos theory - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{example}{Example}\dotfill \pageref*{example} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{the_more_general_definition}{The more general definition}\dotfill \pageref*{the_more_general_definition} \linebreak
\noindent\hyperlink{related_entries}{Related entries}\dotfill \pageref*{related_entries} \linebreak
\noindent\hyperlink{reference}{Reference}\dotfill \pageref*{reference} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

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

\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

\begin{defn}
\label{}\hypertarget{}{}
A [[site]] $(\mathcal{C}, J_{at})$ is called \textbf{atomic} if the [[covering sieve|covering]] [[sieves]] $S$ of $J_{at}$ are exactly the [[inhabited set|inhabited]] sieves $S\neq\emptyset$. A [[Grothendieck topology]] $J_{at}$ of this form is called \emph{atomic}.

\end{defn}
\hypertarget{example}{}\subsection*{{Example}}\label{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. \ref{atomic_ore}.

\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\begin{prop}
\label{atomic_ore}\hypertarget{atomic_ore}{}
Let $\mathcal{C}$ be a [[category]]. Then $\mathcal{C}$ can be made into an atomic site if and only if for any diagram

\begin{displaymath}
\itexarray{
  & & A \\
  & & \downarrow\\
  B & \to & C
}
\end{displaymath}
there is an object $D$ and arrows $D \to A, B$ such that the following diagram commutes:

\begin{displaymath}
\itexarray{
  D & \to & A \\
  \downarrow & & \downarrow\\
  B & \to & C
}
\end{displaymath}
\end{prop}
\begin{proof}
This is exactly what is needed for the pullback stability axiom to hold, and the other axioms are immediate.

\end{proof}
The condition occurring in the proposition is called the (right) [[Ore condition]]. It is a result by [[Peter Johnstone|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 topos|atomic Grothendieck toposes]] is a ([[Boolean topos|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 $f\cdot g\in J_d(C)$. The atomic topology is a special case of this:

\begin{prop}
\label{atomic_dense}\hypertarget{atomic_dense}{}
Let $\mathcal{C}$ be a [[category]] satisfying the [[Ore condition]]. Then the atomic topology $J_{at}$ coincides with the dense topology $J_d$.

\end{prop}
\begin{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, $S\in J_d(C)$.

\end{proof}
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:

\begin{prop}
\label{}\hypertarget{}{}
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|double negation subtoposes]] $Sh_{\neg\neg}(Set^{\mathcal{C}^{op}})$ for a De Morgan presheaf topos $Set^{\mathcal{C}^{op}}$. $\qed$

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

\begin{prop}
\label{atomic_sheaf}\hypertarget{atomic_sheaf}{}
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

\begin{displaymath}
E\overset{g}{\underset{h}{\rightrightarrows}} D\overset{f}{\to} C
\end{displaymath}
with $f\cdot g=f\cdot h$ , then $y=P(f)(x)$ for a unique $x\in P(C)$.

\end{prop}
For the proof see Mac Lane-Moerdijk (\hyperlink{MM94}{1994}, pp.126f).

\hypertarget{the_more_general_definition}{}\subsection*{{The more general definition}}\label{the_more_general_definition}

Thus far we have presented the classical approach as presented in Mac Lane-Moerdijk (\hyperlink{MM94}{1994}) going back to Barr-Diaconescu (\hyperlink{BD80}{1980}) but it was observed by [[Olivia Caramello|O. Caramello]] (\hyperlink{Ca12}{2012}) that the atomic topology can in fact be defined on arbitrary categories not only on those satisfying the [[Ore condition]].

\begin{defn}
\label{}\hypertarget{}{}
Let $\mathcal{C}$ be a category. The \emph{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 \emph{atomic}.

\end{defn}
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:

\begin{prop}
\label{}\hypertarget{}{}
Let $(\mathcal{C}, J_{at})$ be an atomic site. Then $Sh(\mathcal{C}, J_{at})$ is an [[atomic topos|atomic Grothendieck topos]].

\end{prop}
\begin{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 [[dense sub-site|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 (\hyperlink{Ca12}{2012}, prop.1.4).

\end{proof}
\begin{example}
\label{}\hypertarget{}{}
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

\begin{displaymath}
J_{at}(A)=\{\{id_A\},\emptyset\} \qquad J_{at}(B)=\{\{id_B\},\emptyset\}
\end{displaymath}
\begin{displaymath}
J_{at}(C)=\{\{ id_C, f,g\},\{f \}, \{ g \},\{f,g\},\emptyset\} \quad .
\end{displaymath}
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:

\begin{displaymath}
J_{d}(A)=\{\{id_A\}\} \qquad J_{d}(B)=\{\{id_B\}\}
\end{displaymath}
\begin{displaymath}
J_{d}(C)=\{\{id_C, f,g\},\{f,g\}\} \quad .
\end{displaymath}
Further details on $Set^{\mathcal{C}^{op}}$, the topos of hypergraphs, may be found at [[hypergraph]].

\end{example}
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:

\begin{prop}
\label{}\hypertarget{}{}
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]].

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

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

\hypertarget{related_entries}{}\subsection*{{Related entries}}\label{related_entries}

\begin{itemize}%
\item [[dense topology]]


\item [[atomic topos]]


\item [[atomic geometric morphism]]


\item [[Ore condition]]


\item [[De Morgan topos]]



\end{itemize}
\hypertarget{reference}{}\subsection*{{Reference}}\label{reference}

\begin{itemize}%
\item [[Michael Barr]], [[Radu Diaconescu]], \emph{Atomic Toposes} , JPAA \textbf{17} (1980) pp.1-24. (\href{http://www.math.mcgill.ca/barr/papers/atom.top.pdf}{pdf})


\item [[Olivia Caramello]], \emph{Atomic toposes and countable categoricity} , Appl. Cat. Struc. \textbf{20} no. 4 (2012) pp.379-391. (\href{http://arxiv.org/abs/0811.3547}{arXiv:0811.3547})


\item [[Saunders Mac Lane]], [[Ieke Moerdijk]], \emph{[[Sheaves in Geometry and Logic]]} , Springer Heidelberg 1994. (pp.115, 126)



\end{itemize}
[[!redirects atomic topology]]



\end{document}