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\begin{document}

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\section*{proper geometric morphism}

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

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

[[!include topos theory - contents]]

\hypertarget{compact_objects}{}\paragraph*{{Compact objects}}\label{compact_objects}

[[!include compact object - 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{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{stability_and_closure_properties}{Stability and closure properties}\dotfill \pageref*{stability_and_closure_properties} \linebreak
\noindent\hyperlink{properness_and_beckchevalley_conditions}{Properness and Beck-Chevalley conditions}\dotfill \pageref*{properness_and_beckchevalley_conditions} \linebreak
\noindent\hyperlink{CompactSites}{Compact sites}\dotfill \pageref*{CompactSites} \linebreak
\noindent\hyperlink{StronglyCompactSites}{Strongly compact sites}\dotfill \pageref*{StronglyCompactSites} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{compact_toposes}{Compact toposes}\dotfill \pageref*{compact_toposes} \linebreak
\noindent\hyperlink{StronglyCompactToposes}{Strongly compact toposes}\dotfill \pageref*{StronglyCompactToposes} \linebreak
\noindent\hyperlink{finite_objects}{Finite objects}\dotfill \pageref*{finite_objects} \linebreak
\noindent\hyperlink{GeometricStacks}{Geometric stacks}\dotfill \pageref*{GeometricStacks} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{References}{References}\dotfill \pageref*{References} \linebreak


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

The notion of \emph{compact topos} is the generalization from [[topology]] to [[topos theory]] of the notion of \emph{[[compact topological space]]}.

More generally, over a general [[base topos]], the notion of \emph{proper geometric morphism} is the generalization to morphisms between toposes of the notion of \emph{[[proper map]]} between [[topological spaces]].

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

\begin{defn}
\label{CompactTopos}\hypertarget{CompactTopos}{}
A [[sheaf topos]] $\mathcal{E}$ is called a \textbf{[[compact topos]]} if the [[direct image]] of the [[global section geometric morphism]] $\Gamma : \mathcal{E} \to Set$ preserves [[directed colimit|directed]] [[joins]] of [[subterminal objects]].

A [[geometric morphism]] $f : \mathcal{F} \to \mathcal{E}$ is called \textbf{proper} if it exhibits $\mathcal{F}$ as a [[compact topos]] over $\mathcal{E}$. (The [[stack semantics]] of $\mathcal{E}$ can be used to formalize this.)

\end{defn}
\begin{defn}
\label{StronglyCompactTopos}\hypertarget{StronglyCompactTopos}{}
A topos is called \textbf{strongly compact} if $\Gamma$ commutes even with all [[filtered colimits]].

A geometric morphism $f : \mathcal{F} \to \mathcal{E}$ is called \textbf{tidy} if it exhibits $\mathcal{F}$ as a strongly compact topos over $\mathcal{E}$.

\end{defn}
(\hyperlink{MoerdijkVermeulen}{MV, p. 53})

This are the first stages of a notion that in [[(∞,1)-topos theory]] continue as follows

\begin{defn}
\label{StronglyCompactTopos}\hypertarget{StronglyCompactTopos}{}
Let $\kappa$ be a [[regular cardinal]] and $-1 \leq n \leq \infty$. Then an [[(∞,1)-topos]] is \emph{$\kappa$-compact of height $n$} if the [[global section geometric morphism]] preserves $\kappa$-[[filtered (∞,1)-category|filtered]] [[(∞,1)-colimits]] of [[n-truncated]] objects.

Accordingly a geometric morphism is \emph{$\kappa$-proper of height $n$} if it exhibits a $\kappa$-compact of height $n$ $(\infty,1)$-topos over a [[base (∞,1)-topos]].

\end{defn}
In this terminology

\begin{itemize}%
\item a topos \emph{compact of height (-1)} is the same as a \emph{compact topos};


\item a topos \emph{compact of height 0} is the same as a \emph{strongly compact topos};



\end{itemize}
\begin{remark}
\label{}\hypertarget{}{}
An [[n-coherent (∞,1)-topos]] is compact of height $n$ in the sense of def. \ref{StronglyCompactTopos}, this is ([[Rational and p-adic Homotopy Theory|Lurie XIII, prop. 2.3.9]]).

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

\hypertarget{stability_and_closure_properties}{}\subsubsection*{{Stability and closure properties}}\label{stability_and_closure_properties}

\begin{prop}
\label{}\hypertarget{}{}
\begin{enumerate}%
\item Any [[equivalence of categories|equivalence]] is proper and the class of proper maps is closed under composition.


\item If in the [[diagram]]

\begin{displaymath}
\itexarray{
    A&\xrightarrow{p}&B
    \\
    \downarrow^f&\swarrow^g
    \\
    C
 }
\end{displaymath}
$p$ is a [[surjective geometric morphism]] and $f$ is proper then so is $g$.


\item If $h$ is proper and $g$ is a [[geometric embedding]] then $p$ is proper.


\item Any [[hyperconnected geometric morphism]] is proper.


\item $f:F\to G$ is proper iff its [[localic reflection]] $Sh_G(X)\to G$ is, i.e. iff $X$ is a [[compact locale|compact]] [[internal locale]] in $G$.


\item If in a pullback square the bottom morphism is open and surjective and the left morphism is proper then so is the right.



\end{enumerate}
\end{prop}
(\hyperlink{MoerdijkVermeulen}{VM, I.1, I.2})

\begin{prop}
\label{}\hypertarget{}{}
The [[pullback]] of a proper geometric morphism is again proper.

The pullback of a tidy geometric morphism is again tidy.

\end{prop}
(\hyperlink{MoerdijkVermeulen}{VM, theorem 5.8})

\hypertarget{properness_and_beckchevalley_conditions}{}\subsubsection*{{Properness and Beck-Chevalley conditions}}\label{properness_and_beckchevalley_conditions}

A [[geometric morphism]] $f$ of toposes is said to satisfy the \emph{stable} (weak) Beck-Chevalley condition if any pullback of $f$ satisfies the (weak) [[Beck-Chevalley condition]] ((weak)BCC).

\begin{prop}
\label{}\hypertarget{}{}
A map satisfies the stable weak BCC iff it is proper.

\end{prop}
(\hyperlink{MoerdijkVermeulen}{MV, Corollary I.5.9})

\hypertarget{CompactSites}{}\subsubsection*{{Compact sites}}\label{CompactSites}

We discuss classes of [[sites]] such that their [[sheaf topos]] is a compact topos, def. \ref{CompactTopos} (\hyperlink{MoerdijkVermeulen}{VM, I.5}).

(\ldots{})

\hypertarget{StronglyCompactSites}{}\subsubsection*{{Strongly compact sites}}\label{StronglyCompactSites}

We discuss classes of [[sites]] such that their [[sheaf topos]] is a strongly compact topos, def. \ref{CompactTopos} (\hyperlink{MoerdijkVermeulen}{VM, III.4}).

(\ldots{})

\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

\hypertarget{compact_toposes}{}\subsubsection*{{Compact toposes}}\label{compact_toposes}

\begin{prop}
\label{}\hypertarget{}{}
Let $\mathbf{H}$ be a [[topos]] and $X \in \mathbf{H}$ an [[object]]. If

\begin{itemize}%
\item $X$ is a ``[[compact topological space]]-object'' in that:

for every set of morphisms $\{U_i \to X\}_{i \in I}$ such that $\coprod_{i \in I} U_i \to X$ is an [[effective epimorphism]], there is a [[finite set|finite subset]] $J \subset I$ such that $\coprod_{i \in J} U_i \to X$ is still an [[effective epimorphism]];



\end{itemize}
then

\begin{itemize}%
\item The [[slice topos]] $\mathbf{H}_{/X}$ is a compact topos, def. \ref{CompactTopos}.

\end{itemize}
\end{prop}
\begin{remark}
\label{}\hypertarget{}{}
Beware that $X$ being a ``compact topological space-object'' is different from it being a [[compact object]] (the difference being that between compactness of height (-1) and height 0). For the latter case see prop. \ref{SliceOverCompactIsStronglyCompact} below.

\end{remark}
\begin{proof}
The [[terminal object]] of $\mathbf{H}_{/X}$ is the [[identity]] $id_X : X \to X$ in $\mathbf{H}$. A [[subterminal object]] of $\mathbf{H}_{/X}$ is a [[monomorphism]] $U \hookrightarrow X$ in $\mathbf{H}$.

The [[global section geometric morphism]] $\Gamma_X : \mathbf{H}_{/X} \to Set$ sends an object $[E \to X]$ to its set of [[sections]]

\begin{displaymath}
\Gamma_X([E \to X]) = \mathbf{H}(X, E) \times_{\mathbf{H}(X,X)} \{id_X\}
  \,.
\end{displaymath}
Therefore it sends all subterminal object in $\mathbf{H}_{/X}$ to the [[empty set]] except the terminal object $X$ itself, which is sent to the singleton set.

So let $X$ now be a compact-topological-space-object and $U_\bullet : I \to \mathbf{H}_{/X}$ is directed system of subterminals.

If their union $\vee_i U_i$ does not cover $X$, then $\Gamma_X(\vee_i U_i) = \emptyset$. But then also none of the $U_i$ can be $X$ itself, and hence also $\Gamma_X(U_i) = \emptyset$ for all $i \in I$ and so $\vee_i \Gamma_X(U_i) = \emptyset$. On the other hand, if $\vee_i U_i  = X$ then the $\{U_i \to X\}_{i \in I}$ form a cover, hence then by assumption there is a finite subset $\{U_i \to X\}_{i \in J}$ which still covers. By the assumption that the system $U_\bullet$ is a [[directed set]] it also contains the union $X = \vee_{i \in J} U_i$. Therefore $\vee_{i \in I} \Gamma_X(U_i) = \Gamma_X(X) = *$ is the singleton, as is $\Gamma_X(\vee_{i \in I} U_i) = \Gamma_X(X)$. So $\Gamma_X$ preserves directed unions of subterminals and hence $\mathbf{H}_{/X}$ is a compact topos.

\end{proof}
\hypertarget{StronglyCompactToposes}{}\subsubsection*{{Strongly compact toposes}}\label{StronglyCompactToposes}

The following propositions say in summary that

\begin{enumerate}%
\item the \emph{[[petit topos]]} over a [[compact topological space]] that is also [[Hausdorff topological space|Hausdorff]] is strongly compact.


\item the \emph{[[gros topos]]} over a [[compact object]] is strongly compact.



\end{enumerate}
See also (\hyperlink{MoerdijkVermeulen}{VM, III.1}).

\begin{prop}
\label{}\hypertarget{}{}
Examples of strongly compact toposes $\mathcal{E}$, def. \ref{StronglyCompactTopos}, include the following.

\begin{enumerate}%
\item Every [[coherent topos]] is strongly compact.


\item The [[sheaf topos]] over a [[compact topological space|compact]] [[Hausdorff space|Hausdorff]] [[topological space]] is strongly compact.



\end{enumerate}
\end{prop}
(\hyperlink{MoerdijkVermeulen}{MV, Examples III.1.1})

\begin{prop}
\label{SliceOverCompactIsStronglyCompact}\hypertarget{SliceOverCompactIsStronglyCompact}{}
Let $\mathbf{H}$ be a topos over [[Set]] and $X \in \mathbf{H}$ an object. Then the following are equivalent

\begin{enumerate}%
\item $X$ is a [[compact object]] (in the sense that the [[hom functor]] $\mathbf{H}(X,-)$ preserves [[filtered colimits]])


\item the [[slice topos]] $\mathbf{H}_{/X}$ is strongly compact, def. \ref{StronglyCompactTopos}.



\end{enumerate}
\end{prop}
\begin{proof}
The [[direct image]] $\Gamma_X$ of the [[global section geometric morphism]]

\begin{displaymath}
((-) \times X \dashv  \Gamma_X) : \mathbf{H}_{/X} 
    \stackrel{\overset{(-) \times X}{\leftarrow}}{\underset{\mathbf{\Gamma}_X}{\to}} 
  \mathbf{H}
   \stackrel{\overset{\Delta}{\leftarrow}}{\underset{\mathbf{H}(*,-)}{\to}}
  Set
\end{displaymath}
is given by the [[hom functor]] out of the [[terminal object]]. The terminal object in $\mathbf{H}_{/X}$ is the [[identity]] [[morphism]] $id_X : X \to X$. So the terminal geometric morphism takes any $[E \to X]$ in $\mathbf{H}_{/X}$ to the set of [[sections]], given by the [[pullback]] of the [[hom set]] along the inclusion of the [[identity]]

\begin{displaymath}
\Gamma_X([E \to X]) = \mathbf{H}(X,E) \times_{\mathbf{H}(X,X)} \{id\}
  \,.
\end{displaymath}
By the discussion at \emph{\href{overcategory#LimitsAndColimits}{overcategory -- limits and colimits}} we have that [[colimits]] in $\mathbf{H}_{/X}$ are computed in $\mathbf{H}$. So if $[E \to X] \simeq \underset{\longrightarrow_i}{\lim}{[E_i \to X]}$ is a [[filtered colimit]] in $\mathbf{H}_{/X}$, then $E \simeq \underset{\longrightarrow_i}{\lim}{E_i }$ is a filtered colimit in $\mathbf{H}$.

If now $X \in \mathbf{H}$ is a [[compact object]], then this commutes over this colimit and hence

\begin{displaymath}
\begin{aligned}
    \Gamma_X([E \to X]) 
    &= 
    \mathbf{H}(X,\underset{\longrightarrow_i}{\lim} E_i) \times_{\mathbf{H}(X,X)} \{id\}
    \\
    & \simeq
    (\underset{\longrightarrow_i}{\lim}\mathbf{H}(X, E_i)) \times_{\mathbf{H}(X,X)} \{id\}
    \\
    &\simeq
    \underset{\longrightarrow_i}{\lim}
    (\mathbf{H}(X, E_i) \times_{\mathbf{H}(X,X)} \{id\})
    \\
    & \simeq 
    \underset{\longrightarrow_i}{\lim}
    \Gamma_X([E_i \to X])
   \end{aligned}
  \,,
\end{displaymath}
where in the second but last step we used that in the [[topos]] [[Set]] [[universal colimits|colimits are preserved by pullback]].

This shows that $\Gamma_X(-) : \mathbf{H}_{/X} \to Set$ commutes over filtered colimits if $X$ is a [[compact object]].

Conversely, assume that $\Gamma_X(-)$ commutes over all filtered colimits. For every ([[filtered category|filtered]]) [[diagram]] $F_\bullet : I \to \mathbf{H}$ there is the corresponding filtered diagram $X \times F_\bullet : I \to \mathbf{H}_{/X}$, where $[X \times F_i \to X]$ is the projection. As before, the product with $X$ preserves forming colimits

\begin{displaymath}
\underset{\longrightarrow_i}{\lim} ([X \times F_i \to X])
  \simeq
  [X \times  (\underset{\longrightarrow_i}{\lim} F_i) \to X]
  \,.
\end{displaymath}
Moreover, sections of a trivial [[bundle]] are maps into the fiber

\begin{displaymath}
\Gamma_X([X \times F_i \to X]) \simeq \mathbf{H}(X,F_i)
  \,.
\end{displaymath}
So it follows that $X$ is a compact object:

\begin{displaymath}
\begin{aligned}
    \mathbf{H}(X, \underset{\longrightarrow_i}{\lim} F_i)
    & 
   \simeq 
   \Gamma_X( [X \times (\underset{\longrightarrow_i}{\lim} F_i) \to X])
    \\
    & \simeq 
   \Gamma_X(\underset{\longrightarrow_i}{\lim} [X \times  F_i \to X])
   \\
   & \simeq
   \underset{\longrightarrow_i}{\lim}
   \Gamma_X( [X \times  F_i \to X])   
   \\
   & \simeq
   \underset{\longrightarrow_i}{\lim}
   \mathbf{H}(X,F_i)     
  \end{aligned}
  \,.
\end{displaymath}
\end{proof}
\hypertarget{finite_objects}{}\subsubsection*{{Finite objects}}\label{finite_objects}

\begin{prop}
\label{}\hypertarget{}{}
An object $X \in \mathcal{T}$ in a topos $\mathcal{T}$ is a [[Kuratowski finite object]] precisely if the [[étale geometric morphism]]

\begin{displaymath}
\mathcal{T}_{/X} \to \mathcal{T}
\end{displaymath}
out of the [[slice topos]] is a proper geometric morphism. And precisely if $X$ is even \emph{decidable} is this a tidy geometric morphism.

\end{prop}
(\hyperlink{MoerdijkVermeulen}{Moerdijk-Vermeulen, examples III 1.4})

\hypertarget{GeometricStacks}{}\subsubsection*{{Geometric stacks}}\label{GeometricStacks}

A typical condition on a [[geometric stack]] to qualify as an [[orbifold]]/[[Deligne-Mumford stack]] is that its [[diagonal]] be proper. This is equivalent to the corresponding map of toposes being a proper geometric morphism (e.g. \hyperlink{Carchedi12}{Carchedi 12, section 2}, \hyperlink{LurieSpectral}{Lurie Spectral, section 3}).

\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[separated geometric morphism]], [[Hausdorff topos]]

\end{itemize}
\hypertarget{References}{}\subsection*{{References}}\label{References}

The theory of proper geometric morphisms is largly due to

\begin{itemize}%
\item [[Ieke Moerdijk]], Jacob Vermeulen, \emph{Relative compactness conditions for toposes} (\href{https://dspace.library.uu.nl/handle/1874/2374}{pdf})


\item [[Ieke Moerdijk]], Jacob Vermeulen, \emph{Proper maps of toposes} , Memoirs of the American Mathematical Society, no. 705 (2000)



\end{itemize}
based on the [[locale|localic]] case discussed in

\begin{itemize}%
\item Jacob Vermeulen, \emph{Proper maps of locales}, J. Pure Applied Alg. 92 (1994)

\end{itemize}
A textbook account is in section C3.2 of

\begin{itemize}%
\item [[Peter Johnstone]], \emph{[[Sketches of an Elephant]]}

\end{itemize}
Discussion with relation to properness of [[geometric stacks]] includes

\begin{itemize}%
\item [[David Carchedi]], section 2 of \emph{\'E{}tale Stacks as Prolongations} (\href{http://arxiv.org/abs/1212.2282}{arXiv:1212.2282})

\end{itemize}
Discussion of higher compactness conditions in [[(∞,1)-topos theory]] is in section 3 of

\begin{itemize}%
\item [[Jacob Lurie]], \emph{Spectral Schemes} (\href{http://www.math.harvard.edu/~lurie/papers/DAG-VII.pdf}{pdf})

\end{itemize}
and in section 2.3 of

\begin{itemize}%
\item [[Jacob Lurie]], \emph{[[Rational and p-adic Homotopy Theory]]}

\end{itemize}
and for the special case of [[spectral Deligne-Mumford stacks]] in section 1.4 of

\begin{itemize}%
\item [[Jacob Lurie]], \emph{[[Quasi-Coherent Sheaves and Tannaka Duality Theorems]]}

\end{itemize}
[[!redirects proper map of toposes]] [[!redirects proper maps of toposes]]

[[!redirects proper geometric morphisms]]

[[!redirects compact topos]] [[!redirects strongly compact topos]]

[[!redirects compact toposes]] [[!redirects strongly compact toposes]]

[[!redirects compact topoi]] [[!redirects strongly compact topoi]]



\end{document}