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\begin{document}

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\section*{proper morphism}

\begin{quote}%
For the concept in [[topology]] see at \emph{[[proper map]]}.


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

\hypertarget{geometry}{}\paragraph*{{Geometry}}\label{geometry}

[[!include higher geometry - contents]]

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

\noindent\hyperlink{definitions}{Definitions}\dotfill \pageref*{definitions} \linebreak
\noindent\hyperlink{between_topological_spaces}{Between topological spaces}\dotfill \pageref*{between_topological_spaces} \linebreak
\noindent\hyperlink{between_schemes}{Between schemes}\dotfill \pageref*{between_schemes} \linebreak
\noindent\hyperlink{between_locales}{Between locales}\dotfill \pageref*{between_locales} \linebreak
\noindent\hyperlink{between_toposes}{Between toposes}\dotfill \pageref*{between_toposes} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{definitions}{}\subsection*{{Definitions}}\label{definitions}

\hypertarget{between_topological_spaces}{}\subsubsection*{{Between topological spaces}}\label{between_topological_spaces}

While every [[continuous map]] sends [[compact subsets]] to [[compact subsets]] (see at \emph{[[continuous images of compact spaces are compact]]}), it is not true in general that the [[preimage]] of a compact set along a continuous map is compact.

A [[continuous function]] $f : X \to Y$ between [[topological space]]s is \textbf{[[proper map]]} if the [[inverse image]] of any [[compact space|compact]] subset is itself compact.

A notion of proper homotopy between proper maps leads to [[proper homotopy theory]].

Similarly, one can consider the conditions on morphisms in other geometric situations, like [[algebraic geometry]], and properness often either reflects or implies good behaviour with respect to the compact objects (cf. also proper push-forward).

\hypertarget{between_schemes}{}\subsubsection*{{Between schemes}}\label{between_schemes}

A \textbf{proper morphism of [[schemes]]} is by definition a morphism $f:X\to Y$ which is

\begin{enumerate}%
\item [[separated morphism|separated]],


\item of [[finite type morphism|finite type]]


\item [[universally closed morphism|universally closed]] (the latter means that for every $h: Z\to Y$ the pullback $h^*(f): Z\times_Y X\to Z$ is closed).



\end{enumerate}
There is a classical and very practical [[valuative criterion of properness]] due Chevalley.

We say that a scheme $X$ is \textbf{proper} if the canonical map $X \to \operatorname{Spec} \mathbb{Z}$ to the [[terminal object]] is proper.

Proper schemes are analogous to [[compact topological spaces]]. This is one reason why one uses the terminology ``quasi-compact'' when referring to schemes whose underlying topological space is compact.

The [[base change]] formulas for [[cohomology]] for proper and for smooth morphisms of schemes motivated [[Alexander Grothendieck|Grothendieck]] (in [[Pursuing Stacks]]) to define abstract proper and smooth functors in the setting of [[fibered categories]]; this is further expanded on in (\hyperlink{Maltsiniotis}{Maltsiniotis}).

\hypertarget{between_locales}{}\subsubsection*{{Between locales}}\label{between_locales}

Recall that a [[locale]] $L$ is given by a [[frame]] $O(L)$, its [[frame of opens]], and that a [[continuous map]] $f$ from $K$ to $L$ is given by an [[adjunction]] $f^* \dashv f_* \colon O(K) \rightleftarrows O(L)$ such that the [[inverse image]] function $f^*$ preserves finitary [[meets]] (or equivalently is a frame [[homomorphism]], since it must preserve all [[joins]]).

Such a map $f$ is \textbf{proper} iff the [[direct image]] function $f_*$ preserves [[directed joins]] (or equivalently is [[Scott-continuous function|Scott-continuous]], or equivalently is a morphism of [[preframes]]), and also satisfies the [[Frobenius reciprocity]]-like condition that $f_*(U\cup f^*(V)) = f_*(U) \cup V$ (which by itself states that the map is [[closed map|closed]]).

In particular, the map $L\to 1$ is proper iff $L$ is both [[compact locale|compact]] and [[covert space|covert]]. But in this case the second condition is redundant, since every compact locale is automatically covert; see [[covert space]] for a proof.

Proper maps of locales can also be characterized as those that are universally closed, i.e. every pullback of them (along any map of locales) is closed.

\hypertarget{between_toposes}{}\subsubsection*{{Between toposes}}\label{between_toposes}

Proper maps of locales can be generalized to [[geometric morphisms]] of [[Grothendieck toposes]]; see [[proper geometric morphism]].

The topos-theoretic condition refers only to directed unions of subterminal objects, suggesting a stronger condition that it preserve all [[filtered colimits]]. This is a strictly stronger condition even for locales (i.e. [[localic toposes]]), called being \emph{tidy}. In fact properness and tidiness are the first two rungs on an infinite ladder of higher properness for [[higher toposes]].

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

\begin{itemize}%
\item [[proper map]], [[open map]], [[closed map]]


\item [[open morphism]], [[closed morphism]]


\item [[proper base change theorem]]


\item [[proper homotopy theory]]


\item [[maps from compact spaces to Hausdorff spaces are closed and proper]]



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

\begin{itemize}%
\item [[Georges Maltsiniotis]], \emph{Structures d'asph\'e{}ricit\'e{}, foncteurs lisses, et fibrations}, Ann. Math. Blaise Pascal \textbf{12}, pp. 1-39 (2005) (\href{http://people.math.jussieu.fr/~maltsin/ps/asphbl.ps}{ps}).

\end{itemize}
(TO ADD: The definition of a proper dg algebra, proper dg category, proper A-inf cat ???)

\begin{itemize}%
\item wikipedia \href{http://en.wikipedia.org/wiki/Proper_morphism}{proper morphism}
\item Daniel Halpern-Leistner, Anatoly Preygel, \emph{Mapping stacks and categorical notions of properness}, \href{http://arxiv.org/abs/1402.3204}{arxiv/1402.3204}

\end{itemize}
[[!redirects proper morphism]] [[!redirects proper morphisms]] [[!redirects proper scheme]] [[!redirects proper schemes]] [[!redirects proper morphism of schemes]] [[!redirects proper morphisms of schemes]]

[[!redirects proper functor]] [[!redirects proper functors]]



\end{document}