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\begin{document}

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\section*{open morphism}

\begin{quote}%
This page is about the general concept. For open [[continuous functions]] see at \emph{[[open map]]}.


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

\hypertarget{topology}{}\paragraph*{{Topology}}\label{topology}

[[!include topology - contents]]

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

[[!include topos theory - contents]]

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

\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{for_maps_between_topological_spaces}{For maps between topological spaces}\dotfill \pageref*{for_maps_between_topological_spaces} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{for_morphisms_between_locales}{For morphisms between locales}\dotfill \pageref*{for_morphisms_between_locales} \linebreak
\noindent\hyperlink{for_geometric_morphisms_of_toposes}{For geometric morphisms of toposes}\dotfill \pageref*{for_geometric_morphisms_of_toposes} \linebreak
\noindent\hyperlink{for_morphisms_in_a_topos}{For morphisms in a topos}\dotfill \pageref*{for_morphisms_in_a_topos} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

\hypertarget{for_maps_between_topological_spaces}{}\subsubsection*{{For maps between topological spaces}}\label{for_maps_between_topological_spaces}

A [[function]] $f : X \to Y$ between [[topological space]]s is called an \textbf{[[open map]]} if the [[image]] of every [[open set]] in $X$ is also open in $Y$.

Recall that $f$ is a \textbf{[[continuous map]]} if the [[preimage]] of every [[open subspace|open set]] in $Y$ is open in $X$. For defining open maps typically one restricts attention to open [[continuous map]]s, although it also makes sense to speak of open functions that are not continuous.

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

\begin{itemize}%
\item For any two topological spaces $X$, $Y$, the projection map $\pi \colon X \times Y \to Y$ out of the [[product topological space]] is open.


\item If $G$ is a [[topological group]] and $H$ is a subgroup, then the projection to the coset space $p \colon G \to G/H$, where $G/H$ is provided with the quotient topology (making $p$ a quotient map), is open. This follows easily from the observation that if $U$ is open in $G$, then so is

\begin{displaymath}
p^{-1}(p(U)) = U H = \bigcup_{h \in H} U h
\end{displaymath}

\item If $p \colon A \to B$ and $q \colon C \to D$ are open maps, then their product $p \times q \colon A \times C \to B \times D$ is also an open map.



\end{itemize}
\hypertarget{for_morphisms_between_locales}{}\subsubsection*{{For morphisms between locales}}\label{for_morphisms_between_locales}

A continuous map $f\colon X \to Y$ of topological spaces defines a homomorphism $f^*\colon Op(Y) \to Op(X)$ between the [[frames]] of open sets of $X$ and $Y$. If $f$ is open, then this frame homomorphism is also a [[complete lattice|complete]] [[Heyting algebra]] homomorphism; the converse holds if $Y$ is a [[T-D space]]. Accordingly, we define a map $f\colon X \to Y$ of [[locales]] to be \textbf{open} if it is, as a frame homomorphism $f^*\colon Op(Y) \to Op(X)$, a complete Heyting algebra homomorphism, i.e. it preserves arbitrary [[meets]] and the Heyting implication.

This is equivalent to saying that $f^*\colon Op(Y) \to Op(X)$ has a left adjoint $f_!$ (by the [[adjoint functor theorem]] for posets) which satisfies the [[Frobenius reciprocity]] condition that $f_!(U \cap f^* V) = f_!(U) \cap V$.

In the special case that $Y$ is the one-point locale, the Frobenius reciprocity condition is automatically satisfied.

\hypertarget{for_geometric_morphisms_of_toposes}{}\subsubsection*{{For geometric morphisms of toposes}}\label{for_geometric_morphisms_of_toposes}

[[categorification|Categorifying]], a [[geometric morphism]] $f\colon X \to Y$ of [[toposes]] is an [[open geometric morphism]] if its [[inverse image functor]] $f^*\colon Y \to X$ is a [[Heyting functor]].

\hypertarget{for_morphisms_in_a_topos}{}\subsubsection*{{For morphisms in a topos}}\label{for_morphisms_in_a_topos}

A [[class]] $R \subset Mor(\mathcal{E})$ of [[morphism]]s in a [[topos]] $\mathcal{E}$ is called a \textbf{class of open maps} if it satisfies the following axioms.

\begin{enumerate}%
\item Every [[isomorphism]] belongs to $R$;


\item The [[pullback]] of a morphism in $R$ belongs to $R$.


\item If the pullback of a morphism $f$ along an [[epimorphism]] lands in $R$, then $f$ is also in $R$.


\item For every [[set]] $S$ the canonical morphism $(\coprod_{s \in S} *) \to *$ from the $S$-fold [[coproduct]] of the [[terminal object]] to the terminal object is in $R$.


\item For $\{X_i \stackrel{f_i}{\to} Y_i\}_{i \in I} \subset R$ then also the [[coproduct]] $\coprod_i X_i \to \coprod_i Y_i$ is in $R$.


\item If in a diagram of the form

\begin{displaymath}
\itexarray{
     Y &&\stackrel{p}{\to}&& X
     \\
     & {}_{\mathllap{g}}\searrow && \swarrow_{\mathrlap{f}}
     \\
     && B
  }
\end{displaymath}
we have that $p$ is an [[epimorphism]] and $g$ is in $R$, then $f$ is in $R$.



\end{enumerate}
The class $R$ is called a class of \textbf{\'e{}tale maps} if in addition to the axioms 1-5 above it satisfies

\begin{enumerate}%
\item for $f : X \to Y$ in $R$ also the [[diagonal]] $Y \to Y \times_X Y$ is in $R$.


\item If in

\begin{displaymath}
\itexarray{
     Y &&\stackrel{p}{\to}&& X
     \\
     & {}_{\mathllap{g}}\searrow && \swarrow_{\mathrlap{f}}
     \\
     && B
  }
\end{displaymath}
we have that $p$ is an epimorphism, and $p, g \in R$, then $f\in R$.



\end{enumerate}
For instance (\hyperlink{JoyalMoerdijk}{JoyalMoerdijk, section 1}).

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

\begin{itemize}%
\item [[open geometric morphism]]


\item [[essential sublocale]]


\item [[closed morphism]], [[proper morphism]]


\item [[étale map]]


\item [[separated morphism]]



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

\begin{itemize}%
\item [[Peter Johnstone]], \emph{Complemented sublocales and open maps} , Annals of Pure and Applied Logic \textbf{137} (2006) pp.240--255.


\item [[Peter Johnstone]], \emph{[[Sketches of an Elephant]]}, volume II, Oxford University Press 2002. (Section C1.5, pp.520-522)


\item [[André Joyal]], [[Ieke Moerdijk]], \emph{A completeness theorem for open maps}, Annals of Pure and Applied Logic \textbf{70} no.1 (1994) pp.51-86. \href{http://www.ams.org/mathscinet-getitem?mr=1303663}{MR95j:03104}, 



\end{itemize}
An application:

\begin{itemize}%
\item [[André Joyal]], [[Mogens Nielsen]] and [[Glynn Winskel]], \emph{Bisimulation from open maps}, BRICS report series RS-94-7 (1994) (\href{http://ojs.statsbiblioteket.dk/index.php/brics/article/view/21663}{abstract page}, \href{http://www.brics.dk/RS/94/7/BRICS-RS-94-7.pdf}{pdf}).

\end{itemize}


\end{document}