\documentclass[12pt,titlepage]{article}

\usepackage{amsmath}
\usepackage{mathrsfs}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsthm}
\usepackage{mathtools}
\usepackage{graphicx}
\usepackage{color}
\usepackage{ucs}
\usepackage[utf8x]{inputenc}
\usepackage{xparse}
\usepackage{hyperref}

%----Macros----------
%
% Unresolved issues:
%
%  \righttoleftarrow
%  \lefttorightarrow
%
%  \color{} with HTML colorspec
%  \bgcolor
%  \array with options (without options, it's equivalent to the matrix environment)

% Of the standard HTML named colors, white, black, red, green, blue and yellow
% are predefined in the color package. Here are the rest.
\definecolor{aqua}{rgb}{0, 1.0, 1.0}
\definecolor{fuschia}{rgb}{1.0, 0, 1.0}
\definecolor{gray}{rgb}{0.502, 0.502, 0.502}
\definecolor{lime}{rgb}{0, 1.0, 0}
\definecolor{maroon}{rgb}{0.502, 0, 0}
\definecolor{navy}{rgb}{0, 0, 0.502}
\definecolor{olive}{rgb}{0.502, 0.502, 0}
\definecolor{purple}{rgb}{0.502, 0, 0.502}
\definecolor{silver}{rgb}{0.753, 0.753, 0.753}
\definecolor{teal}{rgb}{0, 0.502, 0.502}

% Because of conflicts, \space and \mathop are converted to
% \itexspace and \operatorname during preprocessing.

% itex: \space{ht}{dp}{wd}
%
% Height and baseline depth measurements are in units of tenths of an ex while
% the width is measured in tenths of an em.
\makeatletter
\newdimen\itex@wd%
\newdimen\itex@dp%
\newdimen\itex@thd%
\def\itexspace#1#2#3{\itex@wd=#3em%
\itex@wd=0.1\itex@wd%
\itex@dp=#2ex%
\itex@dp=0.1\itex@dp%
\itex@thd=#1ex%
\itex@thd=0.1\itex@thd%
\advance\itex@thd\the\itex@dp%
\makebox[\the\itex@wd]{\rule[-\the\itex@dp]{0cm}{\the\itex@thd}}}
\makeatother

% \tensor and \multiscript
\makeatletter
\newif\if@sup
\newtoks\@sups
\def\append@sup#1{\edef\act{\noexpand\@sups={\the\@sups #1}}\act}%
\def\reset@sup{\@supfalse\@sups={}}%
\def\mk@scripts#1#2{\if #2/ \if@sup ^{\the\@sups}\fi \else%
  \ifx #1_ \if@sup ^{\the\@sups}\reset@sup \fi {}_{#2}%
  \else \append@sup#2 \@suptrue \fi%
  \expandafter\mk@scripts\fi}
\def\tensor#1#2{\reset@sup#1\mk@scripts#2_/}
\def\multiscripts#1#2#3{\reset@sup{}\mk@scripts#1_/#2%
  \reset@sup\mk@scripts#3_/}
\makeatother

% \slash
\makeatletter
\newbox\slashbox \setbox\slashbox=\hbox{$/$}
\def\itex@pslash#1{\setbox\@tempboxa=\hbox{$#1$}
  \@tempdima=0.5\wd\slashbox \advance\@tempdima 0.5\wd\@tempboxa
  \copy\slashbox \kern-\@tempdima \box\@tempboxa}
\def\slash{\protect\itex@pslash}
\makeatother

% math-mode versions of \rlap, etc
% from Alexander Perlis, "A complement to \smash, \llap, and lap"
%   http://math.arizona.edu/~aprl/publications/mathclap/
\def\clap#1{\hbox to 0pt{\hss#1\hss}}
\def\mathllap{\mathpalette\mathllapinternal}
\def\mathrlap{\mathpalette\mathrlapinternal}
\def\mathclap{\mathpalette\mathclapinternal}
\def\mathllapinternal#1#2{\llap{$\mathsurround=0pt#1{#2}$}}
\def\mathrlapinternal#1#2{\rlap{$\mathsurround=0pt#1{#2}$}}
\def\mathclapinternal#1#2{\clap{$\mathsurround=0pt#1{#2}$}}

% Renames \sqrt as \oldsqrt and redefine root to result in \sqrt[#1]{#2}
\let\oldroot\root
\def\root#1#2{\oldroot #1 \of{#2}}
\renewcommand{\sqrt}[2][]{\oldroot #1 \of{#2}}

% Manually declare the txfonts symbolsC font
\DeclareSymbolFont{symbolsC}{U}{txsyc}{m}{n}
\SetSymbolFont{symbolsC}{bold}{U}{txsyc}{bx}{n}
\DeclareFontSubstitution{U}{txsyc}{m}{n}

% Manually declare the stmaryrd font
\DeclareSymbolFont{stmry}{U}{stmry}{m}{n}
\SetSymbolFont{stmry}{bold}{U}{stmry}{b}{n}

% Manually declare the MnSymbolE font
\DeclareFontFamily{OMX}{MnSymbolE}{}
\DeclareSymbolFont{mnomx}{OMX}{MnSymbolE}{m}{n}
\SetSymbolFont{mnomx}{bold}{OMX}{MnSymbolE}{b}{n}
\DeclareFontShape{OMX}{MnSymbolE}{m}{n}{
    <-6>  MnSymbolE5
   <6-7>  MnSymbolE6
   <7-8>  MnSymbolE7
   <8-9>  MnSymbolE8
   <9-10> MnSymbolE9
  <10-12> MnSymbolE10
  <12->   MnSymbolE12}{}

% Declare specific arrows from txfonts without loading the full package
\makeatletter
\def\re@DeclareMathSymbol#1#2#3#4{%
    \let#1=\undefined
    \DeclareMathSymbol{#1}{#2}{#3}{#4}}
\re@DeclareMathSymbol{\neArrow}{\mathrel}{symbolsC}{116}
\re@DeclareMathSymbol{\neArr}{\mathrel}{symbolsC}{116}
\re@DeclareMathSymbol{\seArrow}{\mathrel}{symbolsC}{117}
\re@DeclareMathSymbol{\seArr}{\mathrel}{symbolsC}{117}
\re@DeclareMathSymbol{\nwArrow}{\mathrel}{symbolsC}{118}
\re@DeclareMathSymbol{\nwArr}{\mathrel}{symbolsC}{118}
\re@DeclareMathSymbol{\swArrow}{\mathrel}{symbolsC}{119}
\re@DeclareMathSymbol{\swArr}{\mathrel}{symbolsC}{119}
\re@DeclareMathSymbol{\nequiv}{\mathrel}{symbolsC}{46}
\re@DeclareMathSymbol{\Perp}{\mathrel}{symbolsC}{121}
\re@DeclareMathSymbol{\Vbar}{\mathrel}{symbolsC}{121}
\re@DeclareMathSymbol{\sslash}{\mathrel}{stmry}{12}
\re@DeclareMathSymbol{\bigsqcap}{\mathop}{stmry}{"64}
\re@DeclareMathSymbol{\biginterleave}{\mathop}{stmry}{"6}
\re@DeclareMathSymbol{\invamp}{\mathrel}{symbolsC}{77}
\re@DeclareMathSymbol{\parr}{\mathrel}{symbolsC}{77}
\makeatother

% \llangle, \rrangle, \lmoustache and \rmoustache from MnSymbolE
\makeatletter
\def\Decl@Mn@Delim#1#2#3#4{%
  \if\relax\noexpand#1%
    \let#1\undefined
  \fi
  \DeclareMathDelimiter{#1}{#2}{#3}{#4}{#3}{#4}}
\def\Decl@Mn@Open#1#2#3{\Decl@Mn@Delim{#1}{\mathopen}{#2}{#3}}
\def\Decl@Mn@Close#1#2#3{\Decl@Mn@Delim{#1}{\mathclose}{#2}{#3}}
\Decl@Mn@Open{\llangle}{mnomx}{'164}
\Decl@Mn@Close{\rrangle}{mnomx}{'171}
\Decl@Mn@Open{\lmoustache}{mnomx}{'245}
\Decl@Mn@Close{\rmoustache}{mnomx}{'244}
\makeatother

% Widecheck
\makeatletter
\DeclareRobustCommand\widecheck[1]{{\mathpalette\@widecheck{#1}}}
\def\@widecheck#1#2{%
    \setbox\z@\hbox{\m@th$#1#2$}%
    \setbox\tw@\hbox{\m@th$#1%
       \widehat{%
          \vrule\@width\z@\@height\ht\z@
          \vrule\@height\z@\@width\wd\z@}$}%
    \dp\tw@-\ht\z@
    \@tempdima\ht\z@ \advance\@tempdima2\ht\tw@ \divide\@tempdima\thr@@
    \setbox\tw@\hbox{%
       \raise\@tempdima\hbox{\scalebox{1}[-1]{\lower\@tempdima\box
\tw@}}}%
    {\ooalign{\box\tw@ \cr \box\z@}}}
\makeatother

% \mathraisebox{voffset}[height][depth]{something}
\makeatletter
\NewDocumentCommand\mathraisebox{moom}{%
\IfNoValueTF{#2}{\def\@temp##1##2{\raisebox{#1}{$\m@th##1##2$}}}{%
\IfNoValueTF{#3}{\def\@temp##1##2{\raisebox{#1}[#2]{$\m@th##1##2$}}%
}{\def\@temp##1##2{\raisebox{#1}[#2][#3]{$\m@th##1##2$}}}}%
\mathpalette\@temp{#4}}
\makeatletter

% udots (taken from yhmath)
\makeatletter
\def\udots{\mathinner{\mkern2mu\raise\p@\hbox{.}
\mkern2mu\raise4\p@\hbox{.}\mkern1mu
\raise7\p@\vbox{\kern7\p@\hbox{.}}\mkern1mu}}
\makeatother

%% Fix array
\newcommand{\itexarray}[1]{\begin{matrix}#1\end{matrix}}
%% \itexnum is a noop
\newcommand{\itexnum}[1]{#1}

%% Renaming existing commands
\newcommand{\underoverset}[3]{\underset{#1}{\overset{#2}{#3}}}
\newcommand{\widevec}{\overrightarrow}
\newcommand{\darr}{\downarrow}
\newcommand{\nearr}{\nearrow}
\newcommand{\nwarr}{\nwarrow}
\newcommand{\searr}{\searrow}
\newcommand{\swarr}{\swarrow}
\newcommand{\curvearrowbotright}{\curvearrowright}
\newcommand{\uparr}{\uparrow}
\newcommand{\downuparrow}{\updownarrow}
\newcommand{\duparr}{\updownarrow}
\newcommand{\updarr}{\updownarrow}
\newcommand{\gt}{>}
\newcommand{\lt}{<}
\newcommand{\map}{\mapsto}
\newcommand{\embedsin}{\hookrightarrow}
\newcommand{\Alpha}{A}
\newcommand{\Beta}{B}
\newcommand{\Zeta}{Z}
\newcommand{\Eta}{H}
\newcommand{\Iota}{I}
\newcommand{\Kappa}{K}
\newcommand{\Mu}{M}
\newcommand{\Nu}{N}
\newcommand{\Rho}{P}
\newcommand{\Tau}{T}
\newcommand{\Upsi}{\Upsilon}
\newcommand{\omicron}{o}
\newcommand{\lang}{\langle}
\newcommand{\rang}{\rangle}
\newcommand{\Union}{\bigcup}
\newcommand{\Intersection}{\bigcap}
\newcommand{\Oplus}{\bigoplus}
\newcommand{\Otimes}{\bigotimes}
\newcommand{\Wedge}{\bigwedge}
\newcommand{\Vee}{\bigvee}
\newcommand{\coproduct}{\coprod}
\newcommand{\product}{\prod}
\newcommand{\closure}{\overline}
\newcommand{\integral}{\int}
\newcommand{\doubleintegral}{\iint}
\newcommand{\tripleintegral}{\iiint}
\newcommand{\quadrupleintegral}{\iiiint}
\newcommand{\conint}{\oint}
\newcommand{\contourintegral}{\oint}
\newcommand{\infinity}{\infty}
\newcommand{\bottom}{\bot}
\newcommand{\minusb}{\boxminus}
\newcommand{\plusb}{\boxplus}
\newcommand{\timesb}{\boxtimes}
\newcommand{\intersection}{\cap}
\newcommand{\union}{\cup}
\newcommand{\Del}{\nabla}
\newcommand{\odash}{\circleddash}
\newcommand{\negspace}{\!}
\newcommand{\widebar}{\overline}
\newcommand{\textsize}{\normalsize}
\renewcommand{\scriptsize}{\scriptstyle}
\newcommand{\scriptscriptsize}{\scriptscriptstyle}
\newcommand{\mathfr}{\mathfrak}
\newcommand{\statusline}[2]{#2}
\newcommand{\tooltip}[2]{#2}
\newcommand{\toggle}[2]{#2}

% Theorem Environments
\theoremstyle{plain}
\newtheorem{theorem}{Theorem}
\newtheorem{lemma}{Lemma}
\newtheorem{prop}{Proposition}
\newtheorem{cor}{Corollary}
\newtheorem*{utheorem}{Theorem}
\newtheorem*{ulemma}{Lemma}
\newtheorem*{uprop}{Proposition}
\newtheorem*{ucor}{Corollary}
\theoremstyle{definition}
\newtheorem{defn}{Definition}
\newtheorem{example}{Example}
\newtheorem*{udefn}{Definition}
\newtheorem*{uexample}{Example}
\theoremstyle{remark}
\newtheorem{remark}{Remark}
\newtheorem{note}{Note}
\newtheorem*{uremark}{Remark}
\newtheorem*{unote}{Note}

%-------------------------------------------------------------------

\begin{document}

%-------------------------------------------------------------------


\section*{skeletal geometric morphism}

\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{remark}{Remark}\dotfill \pageref*{remark} \linebreak
\noindent\hyperlink{related_entries}{Related entries}\dotfill \pageref*{related_entries} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

\textbf{Skeletal geometric morphisms} are those [[geometric morphisms]] that preserve [[double negation|double negation sheaves]] and therefore play a role in the descriptions of classes of toposes like e.g. [[Boolean topos|Boolean]] or [[De Morgan toposes]] in whose definitions the negation participates.

The notion of a skeletal geometric morphism can be viewed as a weakening of the notion of [[open geometric morphism]].

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

A [[geometric morphism]] $f :  \mathcal{F} \to \mathcal{E}$ is called \emph{skeletal} if the following equivalent conditions hold:

\begin{itemize}%
\item $f$ restricts to a geometric morphism $Sh_{\neg\neg}(\mathcal{F}) \to Sh_{\neg\neg}(\mathcal{E})$ .


\item The [[inverse image]] $f^\ast$ preserves $\neg\neg$-dense monomorphisms.


\item For any subobject $A'\rightarrowtail A$ in $\mathcal{E}$: $\neg\neg f^\ast(A')=\neg f^\ast(\neg A')$ in $Sub_\mathcal{F}(f^\ast(A))$.



\end{itemize}
\hypertarget{example}{}\subsection*{{Example}}\label{example}

\begin{itemize}%
\item Inverse images of [[open geometric morphism|open geometric morphisms]] are [[Heyting functors]], hence commute with $\neg$ and, therefore, \emph{open geometric morphisms are skeletal}. In particular, geometric morphisms with [[Boolean topos|Boolean]] codomain are open (\hyperlink{J02}{Johnstone 2002, p.612}), hence skeletal.


\item [[dense subtopos|Dense subtoposes]] $i:Sh_j(\mathcal{E})\hookrightarrow \mathcal{E}$ are precisely those subtoposes with $Sh_{\neg\neg}(Sh_j(\mathcal{E}))=Sh_{\neg\neg}(\mathcal{E})$ (cf. \href{dense+subtopos#negdense}{this proposition}) and, therefore, are skeletal.



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

The following two propositions concern skeletal [[geometric embedding|inclusions]] (cf. Johnstone (\hyperlink{J02}{2002, p.1007})):

\begin{prop}
\label{}\hypertarget{}{}
An inclusion $i:Sh_j(\mathcal{E})\hookrightarrow \mathcal{E}$ is skeletal iff $ext(j)$ , the $j$-closure of $0\rightarrowtail 1$ , is a $\neg\neg$-closed [[subterminal object]] of $\mathcal{E}$.

\end{prop}
\textbf{Proof}: First, notice that in general for a topology $j$ a subterminal $U$ is $j$-closed iff it is a $j$-sheaf since $1$ is always a $j$-sheaf. Hence $0$ is $\neg\neg$-closed precisely when it is a $\neg\neg$-sheaf.

Now assume $i$ skeletal. Since $ext(j)$ is a $\neg\neg$-sheaf in $Sh_j(\mathcal{E})$ and $i$ preserves them, it is also a $\neg\neg$-sheaf in $\mathcal{E}$.

Conversely, assume $ext(j)$ is a $\neg\neg$-sheaf in $\mathcal{E}$. Since it is also a $j$-sheaf, it is contained in $Sh_j(\mathcal{E})\cap Sh_{\neg\neg}(\mathcal{E})$ but this coincides with $Sh_{\neg\neg}(Sh_j(\mathcal{E}))$ because as a subtopos of the Boolean $Sh_{\neg\neg}(\mathcal{E})$ the intersection $Sh_j(\mathcal{E})\cap Sh_{\neg\neg}(\mathcal{E})$ is Boolean and $Sh_j(\mathcal{E})\cap Sh_{\neg\neg}(\mathcal{E})$, since it contains $ext(j)$, is dense in $Sh_j(\mathcal{E})$ and there can be only one such dense Boolean subtopos. $\qed$

\begin{prop}
\label{}\hypertarget{}{}
The class $\Sigma$ of skeletal [[geometric embedding|inclusions]] is the smallest class $\Gamma$ of geometric morphisms such that:

\begin{itemize}%
\item $\Gamma$ contains [[open subtopos|open inclusions]] and,


\item $\Gamma$ is closed under precomposition with [[dense subtopos|dense inclusions]]: from $g$ dense, $f\in\Gamma$ and $f,g$ composable, follows $fg\in\Gamma$. $\qed$



\end{itemize}
\end{prop}
The following exhibits the link between skeletal morphisms and Booleanness:

\begin{prop}
\label{Boolean_skeletal}\hypertarget{Boolean_skeletal}{}
A topos $\mathcal{E}$ is Boolean iff all geometric morphisms $\mathcal{F}\to\mathcal{E}$ to $\mathcal{E}$ are skeletal.

\end{prop}
\textbf{Proof}: When $\mathcal{E}$ is Boolean it coincides with $Sh_{\neg\neg}(\mathcal{E})$ hence $\neg\neg$-sheaves of $\mathcal{F}$ trivially have to land there.

Conversely, assume all $\mathcal{F}\to\mathcal{E}$ are skeletal. By [[Barr's theorem]], $\mathcal{E}$ receives a surjective $f:\mathcal{B}\to\mathcal{E}$ from a Boolean topos. $f$ being skeletal and surjective implies that $im(f)=\mathcal{E}$ is Boolean. $\qed$

A pullback characterisation of [[open geometric morphisms]] from Johnstone (\hyperlink{J06}{2006, cor. 4.9}):

\begin{prop}
\label{}\hypertarget{}{}
A [[geometric morphism]] $f:\mathcal{F}\to\mathcal{E}$ is open iff the pullback of any [[bounded geometric morphism]] with codomain $\mathcal{E}$ is skeletal. $\qed$

\end{prop}
\hypertarget{remark}{}\subsection*{{Remark}}\label{remark}

Skeletal morphisms between [[frame|frames]] are studied in \hyperlink{Banaschewski_Pultr94}{Banaschewski-Pultr (1994},\hyperlink{Banaschewski_Pultr96}{1996}), called \emph{weakly open} there.

The equivalent concept for topological spaces appears in \hyperlink{Mioduszewski_Rudolf69}{Mioduszewski-Rudolf (1969)}.

It is possible to define an analogous concept of \emph{m-skeletal geometric morphism} using the [[De Morganization|De Morgan topology]] on a topos $\mathcal{E}$ instead of $\neg\neg$.

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

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


\item [[dominant geometric morphism]]


\item [[proper geometric morphism]]


\item [[double negation]]


\item [[De Morganization]]


\item [[De Morgan topos]]



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

\begin{itemize}%
\item B. Banaschewski, A. Pultr, \emph{Variants of openness} , Appl. Cat. Struc. \textbf{2} (1994) pp.1-21.


\item B. Banaschewski, A. Pultr, \emph{Booleanization} , Cah. Top. G\'e{}om. Diff. Cat. \textbf{XXXVII} no.1 (1996) pp.41-60. (\href{http://numdam.mathdoc.fr/numdam-bin/fitem?id=CTGDC_1996__37_1_41_0}{numdam})


\item [[Peter Johnstone]], \emph{Factorization theorems for geometric morphisms II} , pp.216-233 in LNM \textbf{915} Springer Heidelberg 1982.


\item [[Peter Johnstone]], \emph{[[Sketches of an Elephant]] vol.II} , Oxford UP 2002. (section D4.6, pp.1006-1010)


\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{The Gleason Cover of a Realizability Topos} , TAC \textbf{28} no.32 (2013) pp.1139-1152. (\href{http://www.tac.mta.ca/tac/volumes/28/32/28-32abs.html}{abstract})


\item J. Mioduszewski, L. Rudolf, \emph{H-closed and extremally disconnected Hausdorff spaces} , Dissertationes Math. \textbf{66} 1969. (\href{http://pldml.icm.edu.pl/pldml/element/bwmeta1.element.desklight-99c47193-1b38-4485-9b3f-62fc71f8f83b}{toc})



\end{itemize}
[[!redirects skeletal geometric morphisms]] [[!redirects skeletal morphism]]



\end{document}