← venn-to-topos
Main_venn_topos_rewrite.tex
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\usepackage{array,booktabs}
\usepackage{amsmath,amssymb,mathtools,amsthm}
\usepackage{mathrsfs}
\usepackage{etoolbox}
\usepackage{tikz}
\usepackage{graphicx}
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\definecolor{toposblue}{RGB}{0,72,165}
\definecolor{toposorange}{RGB}{236,120,0}
\definecolor{toposgreen}{RGB}{0,130,90}
\definecolor{toposred}{RGB}{190,45,45}
\definecolor{lightblue}{RGB}{225,238,255}
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\definecolor{line}{RGB}{214,216,218}
% Color grammar: green = logic, orange = geometry/space. Bridges stay neutral.
\colorlet{logic}{toposgreen}
\colorlet{spacec}{toposorange}
\colorlet{toposc}{spacec}
\colorlet{bridge}{muted}
\colorlet{danger}{toposred}
\colorlet{logicLight}{lightgreen}
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\newcount\lgtablelevel
\newcommand{\setlogicgeometrylevel}[1]{%
\lgtablelevel=4
\ifstrequal{#1}{intro}{\lgtablelevel=0}{}%
\ifstrequal{#1}{venn}{\lgtablelevel=1}{}%
\ifstrequal{#1}{stone}{\lgtablelevel=2}{}%
\ifstrequal{#1}{locale}{\lgtablelevel=3}{}%
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\newcommand{\lgrow}[3]{%
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\textcolor{logic}{#2} & \textcolor{spacec}{#3} \\
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\setlength{\tabcolsep}{4pt}
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\begin{tabular}{>{\raggedright\arraybackslash}p{0.52\textwidth}>{\raggedright\arraybackslash}p{0.36\textwidth}}
\toprule
\textcolor{logic}{\textbf{論理}} & \textcolor{spacec}{\textbf{幾何}} \\
\midrule
\lgrow{1}{\textcolor{logic}{命題}}{(開)部分(集合)}
\lgrow{1}{\textcolor{logic}{論理}積}{共通部分}
\lgrow{1}{\textcolor{logic}{論理}和}{和(集合)}
\lgrow{1}{\textcolor{logic}{モデル}}{\textcolor{spacec}{点}}
\lgrow{1}{\textcolor{logic}{論理}包含}{(\textcolor{spacec}{幾何}的)包含}
\lgrow{1}{完全性}{\textcolor{spacec}{点}の存在}
\lgrow{2}{有限\textcolor{logic}{ブール代数}}{有限集合}
\lgdivider
\lgrow{2}{\textcolor{logic}{ブール代数}}{\textcolor{spacec}{Stone空間}}
\lgdivider
\lgrow{3}{\textcolor{logic}{直観主義}(無限)\textcolor{logic}{命題}\textcolor{logic}{論理}}{\textcolor{spacec}{Locale}(滑らかな空間)}
\lgrow{4}{\textcolor{logic}{量化子}}{層}
\lgrow{4}{\textcolor{toposc}{トポス}(\textcolor{logic}{理論})}{\textcolor{toposc}{トポス}(空間)}
\bottomrule
\end{tabular}
\endgroup
}
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>={Latex[length=2.4mm,width=2.0mm]},
world/.style={circle,draw=spacec,fill=spaceLight,very thick,minimum size=9mm,inner sep=0pt,font=\small\bfseries},
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\title[\texorpdfstring{{\color{spacec}ベン図}から{\color{toposc}トポス}へ}{ベン図からトポスへ}]{\texorpdfstring{{\color{spacec}ベン図}から{\color{toposc}トポス}へ}{ベン図からトポスへ}}
\subtitle{}
\author{洞龍弥}
\institute[]{\texorpdfstring{数理空間{\color{toposc}トポス} 新歓 2026}{数理空間トポス 新歓 2026}}
\date[2026年5月23日]{2026年5月23日}
\begin{document}
% 1
\begin{deckplainframe}[complete]{LQS}
\begin{tikzpicture}[remember picture,overlay]
\node[anchor=center,inner sep=0pt] at (current page.center) {%
\includegraphics[width=\paperwidth,height=\paperheight]{figures/title_logic_geometry_world.png}%
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(current page.south west) rectangle ([yshift=28mm]current page.south east);
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{\LARGE\bfseries\inserttitle}\\[1.3mm]
{\normalsize\insertauthor\quad \insertinstitute\quad \insertdate}%
};
\end{tikzpicture}
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% 2
\begin{deckframe}[complete]{SXS}{目次}
\tableofcontents
\end{deckframe}
% 3
\begin{deckframe}[incomplete]{DJT}{対応表: {\color{logic}論理}と{\color{spacec}幾何}}
\centering
\logicgeometrytable{intro}
\end{deckframe}
\section[ベン図]{ベン図}
% 4
\begin{deckframe}[complete]{MAA}{目次}
\tableofcontents[currentsection]
\end{deckframe}
% 5
\begin{deckframe}[incomplete]{HTU}{オイラーの手紙}
\vspace{7mm}
{\fontsize{15}{18}\selectfont
\setlength{\baselineskip}{18pt}%
\makebox[\textwidth][l]{これらの丸い図形，いやむしろ}\\
\makebox[\textwidth][l]{(どのような形を与えようとも}\\
\makebox[\textwidth][l]{かまわないのですから) \textbf{\textcolor{spacec}{空間}}は，...}\\[-1mm]
\makebox[0pt][l]{\hspace{0.805\textwidth}\raisebox{-3.35cm}[0pt][0pt]{\includegraphics[height=3.7cm]{euler_letters/leonhard_euler_handmann_1753.jpg}}}%
\makebox[0.78\textwidth][l]{\textbf{\textcolor{logic}{論理}学}の中で自慢げに語られ，}\\
\makebox[0.78\textwidth][l]{多大な苦労をして証明される神秘の全てを，}\\
\makebox[0.78\textwidth][l]{私たちに明らかにしてくれるのです．}
\par}
{\footnotesize ドイツ王女への手紙，引用者訳，強調}
\end{deckframe}
% 6
\begin{deckframe}[complete]{IPU}{{\color{logic}論理}パズル(1/2) 嘘つきは誰か？}
\begin{center}
\begin{tikzpicture}[x=1cm,y=1cm]
\foreach \x/\name/\speech in {
-4.1/A/{\color{logic}「B が嘘つき」},
0.0/B/{\color{logic}「C が嘘つき」},
4.1/C/{\color{logic}\mbox{「AもBも正直」}}
}{
\draw[very thick,draw=ink,fill=paper] (\x,1.05) circle (.33);
\draw[very thick,draw=ink,fill=paper,rounded corners=8pt] (\x-.52,-0.05) rectangle (\x+.52,.72);
\node[font=\large\bfseries] at (\x,.25) {\name};
\node[font=\large,align=center,text width=4.05cm] at (\x,-1.18) {\speech};
}
\end{tikzpicture}
\end{center}
\end{deckframe}
% 7
\begin{deckframe}[incomplete]{XQZ}{{\color{logic}論理}パズル(2/2) {\color{spacec}ベン図}で``{\color{spacec}視覚的}に"理解する}
\begin{center}
\begin{tikzpicture}[x=1cm,y=1cm]
\foreach \x/\name/\speech in {
-4.1/A/{\color{logic}「B は嘘つき」},
0.0/B/{\color{logic}「C は嘘つき」},
% 4.1/C/{\color{logic}「Bが正直ならAも正直」}
4.1/C/{\color{logic}\mbox{「AもBも正直」}}
}{
\draw[very thick,draw=ink,fill=paper] (\x,1.05) circle (.33);
\draw[very thick,draw=ink,fill=paper,rounded corners=8pt] (\x-.52,-0.05) rectangle (\x+.52,.72);
\node[font=\large\bfseries] at (\x,.25) {\name};
\node[font=\large,align=center,text width=4.05cm] at (\x,-1.18) {\speech};
}
\end{tikzpicture}
\begin{tikzpicture}[scale=.88]
\begin{scope}[xshift=-4.15cm]
% \node[font=\bfseries\color{spacec}] at (0,1.22) {A $\leftrightarrow$ Bが嘘};
\draw[very thick,rounded corners=2pt,draw=line,fill=white] (-1.45,-1.18) rectangle (1.45,1.02);
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\begin{scope}
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\draw[very thick,spacec] (-.38,.12) circle (.72);
\draw[very thick,spacec] (.38,.12) circle (.72);
\draw[very thick,spacec] (0,-.48) circle (.72);
\node at (-.78,.62) {A};
\node at (.78,.62) {B};
\node at (0,-.98) {C};
\fill[black!50!spacec] (.72,.24) circle (2.6pt);
\end{scope}
\begin{scope}
% \node[font=\bfseries\color{spacec}] at (0,1.22) {B $\leftrightarrow$ Cが嘘};
\draw[very thick,rounded corners=2pt,draw=line,fill=white] (-1.45,-1.18) rectangle (1.45,1.02);
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\draw[very thick,spacec] (-.38,.12) circle (.72);
\draw[very thick,spacec] (.38,.12) circle (.72);
\draw[very thick,spacec] (0,-.48) circle (.72);
\node at (-.78,.62) {A};
\node at (.78,.62) {B};
\node at (0,-.98) {C};
\fill[black!50!spacec] (.72,.24) circle (2.6pt);
\end{scope}
\begin{scope}[xshift=4.15cm]
% C <-> (A \land B)
% 背景をまず真として塗る
\fill[spaceLight] (-1.45,-1.18) rectangle (1.45,1.02);
% C の中は一度抜く
\fill[white] (0,-.48) circle (.72);
% A \cap B は C の外では偽にしたいので抜く
\begin{scope}
\clip (-.38,.12) circle (.72); % A の中
\fill[white] (.38,.12) circle (.72); % A \cap B を白く抜く
\end{scope}
% C の中では A \cap B だけ真なので塗り戻す
\begin{scope}
\clip (0,-.48) circle (.72); % C の中
\begin{scope}
\clip (-.38,.12) circle (.72); % A の中
\fill[spaceLight] (.38,.12) circle (.72); % A \cap B \cap C
\end{scope}
\end{scope}
% 枠線
\draw[very thick,rounded corners=2pt,draw=line]
(-1.45,-1.18) rectangle (1.45,1.02);
% 円の輪郭
\draw[very thick,spacec] (-.38,.12) circle (.72);
\draw[very thick,spacec] (.38,.12) circle (.72);
\draw[very thick,spacec] (0,-.48) circle (.72);
% ラベル
\node at (-.78,.62) {A};
\node at (.78,.62) {B};
\node at (0,-.98) {C};
\fill[black!50!spacec] (.72,.24) circle (2.6pt);
\end{scope}
\end{tikzpicture}
\end{center}
% \hfill\vocab{答え: AとCが本当，Bが嘘}
\begin{center}
\vocab{答え: AとCが嘘，Bが正直}
\end{center}
\end{deckframe}
% 8
\begin{deckframe}[incomplete]{YPS}{{\color{logic}モデル}$=${\color{spacec}点}}
\begin{center}
{\large\bfseries
無矛盾なこと $\iff$ \textcolor{spacec}{点}が存在する
}
\end{center}
\vspace{-3mm}
\begin{center}
\begin{tikzpicture}[x=1cm,y=1cm]
\node[draw=line,fill=paper,rounded corners=2pt,inner xsep=5pt,inner ysep=2pt,font=\large\bfseries]
at (-2.8,.34) {A: \mbox{「B が嘘つき」}};
\node[draw=line,fill=paper,rounded corners=2pt,inner xsep=5pt,inner ysep=2pt,font=\large\bfseries]
at (2.8,.34) {B: \mbox{「C が嘘つき」}};
\node[draw=line,fill=paper,rounded corners=2pt,inner xsep=5pt,inner ysep=2pt,font=\large\bfseries]
at (0,-.55) {C: \mbox{「AもBも正直」}};
\end{tikzpicture}
\end{center}
\vspace{-5mm}
\begin{center}
\begin{tikzpicture}[x=1cm,y=1cm,scale=.88]
% universe
\fill[spaceLight] (-2.3,-1.25) rectangle (2.3,1.45);
% A, B を一度白で抜く
\fill[white] (-.55,.12) circle (.95);
\fill[white] (.55,.12) circle (.95);
% A \cap B を塗り戻す
\begin{scope}
\clip (-.55,.12) circle (.95);
\fill[spaceLight] (.55,.12) circle (.95);
\end{scope}
% 枠線
\draw[very thick,rounded corners=3pt,draw=line]
(-2.3,-1.25) rectangle (2.3,1.45);
% 円の輪郭
\draw[very thick,spacec] (-.55,.12) circle (.95);
\draw[very thick,spacec] (.55,.12) circle (.95);
% ラベル
\node[font=\bfseries] at (-1.15,.9) {$A$};
\node[font=\bfseries] at (1.15,.9) {$B$};
% model points
\fill[black!65!spacec] (0,.12) circle (3pt);
\node[font=\scriptsize,above] at (0,.22) {$M_1$};
\fill[black!65!spacec] (1.55,-.85) circle (3pt);
\node[font=\scriptsize,above] at (1.55,-.75) {$M_2$};
\end{tikzpicture}
\end{center}
\vspace{-3mm}
\begin{center}
\begin{tikzpicture}[x=1cm,y=1cm]
\begin{scope}
\clip (-1.35,0) circle (.62);
\node at (-1.35,-.18)
{\includegraphics[height=2.05cm]{figures/people/saul_kripke_public_domain.jpg}};
\end{scope}
\draw[very thick,draw=ink] (-1.35,0) circle (.62);
\draw[very thick,draw=line,fill=paper]
(-.73,.08) -- (-.38,.28) -- (-.38,-.12) -- cycle;
\draw[very thick,draw=line,fill=paper,rounded corners=6pt]
(-.38,-.36) rectangle (2.05,.48);
\node[font=\Large\bfseries] at (.83,.06) {可能世界};
\end{tikzpicture}
\end{center}
\end{deckframe}
% 9
\begin{deckframe}[incomplete]{ZSU}{対応表: {\color{logic}論理}と{\color{spacec}幾何}}
\centering
\logicgeometrytable{venn}
\end{deckframe}
% 10
\begin{deckframe}[incomplete]{ICE}{{\color{logic}モデル}が存在しない{\color{logic}論理}パズル}
\begin{center}
\begin{tikzpicture}[x=1cm,y=1cm,scale=.98]
\foreach \x/\name/\speech in {
-3.7/A/{\color{logic}「B は嘘つき」},
0.0/B/{\color{logic}「C は嘘つき」},
3.7/C/{\color{logic}「A が正直なら B も正直」}
}{
\draw[very thick,draw=ink,fill=paper] (\x,0.72) circle (.33);
\draw[very thick,draw=ink,fill=paper,rounded corners=8pt]
(\x-.52,-0.4) rectangle (\x+.52,.38);
\node[font=\large\bfseries] at (\x,-0.1) {\name};
\node[font=\small,align=center,text width=2.35cm] at (\x,-1.15) {\speech};
}
\end{tikzpicture}
\vspace{-2mm}
\begin{tikzpicture}[scale=.86]
% A <-> not B
\begin{scope}[xshift=-4.15cm]
\draw[very thick,rounded corners=2pt,draw=line,fill=white]
(-1.45,-1.18) rectangle (1.45,1.02);
\fill[spaceLight] (-.38,.12) circle (.72);
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\fill[white] (.38,.12) circle (.72);
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\draw[very thick,spacec] (-.38,.12) circle (.72);
\draw[very thick,spacec] (.38,.12) circle (.72);
\draw[very thick,spacec] (0,-.48) circle (.72);
\node at (-.78,.62) {A};
\node at (.78,.62) {B};
\node at (0,-.98) {C};
\end{scope}
% B <-> not C
\begin{scope}
\draw[very thick,rounded corners=2pt,draw=line,fill=white]
(-1.45,-1.18) rectangle (1.45,1.02);
\fill[spaceLight] (.38,.12) circle (.72);
\fill[spaceLight] (0,-.48) circle (.72);
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\fill[white] (0,-.48) circle (.72);
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\draw[very thick,spacec] (-.38,.12) circle (.72);
\draw[very thick,spacec] (.38,.12) circle (.72);
\draw[very thick,spacec] (0,-.48) circle (.72);
\node at (-.78,.62) {A};
\node at (.78,.62) {B};
\node at (0,-.98) {C};
\end{scope}
% C <-> (A => B)
\begin{scope}[xshift=4.15cm]
\draw[very thick,rounded corners=2pt,draw=line,fill=white]
(-1.45,-1.18) rectangle (1.45,1.02);
% C の外では反転:
% not(A => B) = A \cap B^c
% したがって C^c \cap A \cap B^c を塗る
\begin{scope}
\clip (-.38,.12) circle (.72); % A の中
\fill[spaceLight] (-1.45,-1.18) rectangle (1.45,1.02);
\fill[white] (.38,.12) circle (.72); % B の中を抜く
\fill[white] (0,-.48) circle (.72); % C の中を抜く
\end{scope}
% C の中では A => B:
% C \cap (A^c \cup B) を塗る
\begin{scope}
\clip (0,-.48) circle (.72); % C の中
\fill[spaceLight] (-1.45,-1.18) rectangle (1.45,1.02);
% 反例 A \cap B^c だけ白で抜く
\begin{scope}
\clip (-.38,.12) circle (.72); % A の中
\fill[white] (-1.45,-1.18) rectangle (1.45,1.02);
\fill[spaceLight] (.38,.12) circle (.72); % A \cap B を塗り戻す
\end{scope}
\end{scope}
\draw[very thick,rounded corners=2pt,draw=line]
(-1.45,-1.18) rectangle (1.45,1.02);
\draw[very thick,spacec] (-.38,.12) circle (.72);
\draw[very thick,spacec] (.38,.12) circle (.72);
\draw[very thick,spacec] (0,-.48) circle (.72);
\node at (-.78,.62) {A};
\node at (.78,.62) {B};
\node at (0,-.98) {C};
\end{scope}
\end{tikzpicture}
\end{center}
\begin{center}
\vocab{答え: \textcolor{logic}{モデル}は存在しない}
\end{center}
\end{deckframe}
\section{Stone双対}
% 11
\begin{deckframe}[incomplete]{VFO}{目次}
\tableofcontents[currentsection]
\end{deckframe}
% 12
\begin{deckframe}[incomplete]{ZJS}{{\color{logic}論理}パズル: この人から1億円もらおう！}
絶対\textcolor{logic}{約束}守るマンがやってきて，言った．
\begin{center}
\begin{tikzpicture}[x=1cm,y=1cm]
% 人
\draw[very thick,draw=ink,fill=paper]
(-5.15,0.65) circle (.42);
\draw[very thick,draw=ink,fill=paper,rounded corners=9pt]
(-5.8,-1.2) rectangle (-4.5,0.15);
\node[font=\large\bfseries] at (-5.15,-.55) {絶};
% 吹き出し本体
\node[
draw=ink,
very thick,
fill=paper,
rounded corners=10pt,
align=center,
text width=9.2cm,
inner sep=9pt,
font=\Large\bfseries
] (speech) at (1.0,0.25)
{\textcolor{logic}{正しい}ことを言ったら1000円あげるよ，\\
けど\textcolor{logic}{間違った}ことを言ったら1000円あげない};
% 吹き出しのしっぽ
% \draw[very thick,draw=ink,fill=paper]
% (-3.55,0.05) -- (-2.95,0.22) -- (-3.45,-0.28) -- cycle;
\end{tikzpicture}
\end{center}
どう\textcolor{logic}{発言}すれば，1億円をもらえるか？ (メタ可)
\vfill
{\centering\tiny\color{muted}出典（改変）: R. M. Smullyan, \textit{Forever Undecided}, Knopf, 1987; cf. \textit{Logical Labyrinths}, A K Peters, 2008.\par}
\end{deckframe}
% 13
\begin{deckframe}[incomplete]{ZWI}{{\color{logic}論理}パズル: 普通の解法}
「あなたは，私に1000円をくれず，かつ1億円もくれない．」
\vspace{-1mm}
\begin{center}
\begin{tikzpicture}[
x=1cm,y=1cm,
nom/.style={
rounded corners=3pt,
draw=ink,
fill=paper,
very thick,
align=center,
font=\small\bfseries,
inner sep=6pt,
minimum width=3.3cm
},
badleaf/.style={
rounded corners=3pt,
draw=danger,
fill=dangerLight,
very thick,
align=center,
font=\small\bfseries,
inner sep=6pt,
minimum width=2.7cm
},
arr/.style={->,very thick,draw=bridge}
]
\node[nom] (root) at (0,1.35) {\textcolor{logic}{発言}の\textcolor{logic}{真偽}};
\node[badleaf] (truecase) at (-3.9,-0.4)
{\textcolor{logic}{約束}に矛盾};
\node[nom] (falsecase) at (2.0,-0.4)
{1000円か1億円もらえる};
\node[badleaf] (false1000) at (0.0,-2.65)
{\textcolor{logic}{約束}に矛盾};
\node[nom] (false100m) at (4.0,-2.65)
{$1$億円 GET};
\draw[arr] (root) -- node[left,font=\small] {真} (truecase);
\draw[arr] (root) -- node[right,font=\small] {偽} (falsecase);
\draw[arr] (falsecase) -- node[left,font=\small] {$1000$円} (false1000);
\draw[arr] (falsecase) -- node[right,font=\small] {$1$億円} (false100m);
\end{tikzpicture}
\end{center}
\end{deckframe}
% 14
\begin{deckframe}[incomplete]{UOL}{{\color{logic} 代数}的解法(準備): {\color{logic} 論理}の{\color{logic}代数}}
\begin{columns}
\column{0.5\textwidth}
\begin{definition}[\textcolor{logic}{ブール代数}(雑)]
\textcolor{logic}{ブール代数}とは，足し算と掛け算ができて\footnote{環}，
\[x^2=x\]を満たすもの．
\end{definition}
\column{0.5\textwidth}
\begin{example}
\textcolor{logic}{命題}の集合\footnote{ある\textcolor{logic}{構文}的システムを固定して考える}は，\textcolor{logic}{ブール代数}．
\begin{center}
\begin{tabular}{c|c}\hline
0 & $\bot$ \\\hline
$p+q$ & $p \not\leftrightarrow q$ \\\hline
1 & $\top$ \\\hline
$\times$ & $p \land q$ \\ \hline
\end{tabular}
\end{center}
\end{example}
\end{columns}
\end{deckframe}
% 15
\begin{deckframe}[incomplete]{BFK}{1億円パズル A {\color{logic}代数}}
もらえる額$n$が満たしてほしい\textcolor{logic}{条件}：
\[
\left(
\propvar{\textcolor{logic}{発言}}
\leftrightarrow
\propvar{$n=10^3$}
\right)
=
\propvar{$n=10^8$}
\]
\begin{align*}
1+
\propvar{$n=10^3$}
+
\propvar{\textcolor{logic}{発言}}
&=
\propvar{$n=10^8$}
\\
\propvar{\textcolor{logic}{発言}}
&=
\propvar{$n=10^8$}
+
\propvar{$n=10^3$}
+1\\
&=
\propvar{$n=10^8$}
\leftrightarrow
\propvar{$n=10^3$}\\
&= \propvar{$n\neq10^3, 10^8$}
\end{align*}
\end{deckframe}
% 16
\begin{deckframe}[incomplete]{TMZ}{思想: {\color{spacec}カントール空間}は，無限の{\color{logic}選択}の配置空間}
\begin{center}
\begin{tikzpicture}[xscale=1.02,yscale=1.12,
bit/.style={circle,draw=ink!65,fill=paper,inner sep=0pt,minimum size=3.6mm},
chosen/.style={circle,draw=spacec,fill=spaceLight,very thick,inner sep=0pt,minimum size=5.0mm},
edgelabel/.style={font=\scriptsize\color{ink},inner sep=1.2pt,fill=paper,fill opacity=.94,text opacity=1}
]
\node[bit] (r) at (0,2.45) {};
\node[chosen] (a0) at (-3.20,1.30) {};
\node[bit] (a1) at (3.20,1.30) {};
\node[chosen] (b00) at (-4.80,0.15) {};
\node[bit] (b01) at (-1.60,0.15) {};
\node[bit] (b10) at (1.60,0.15) {};
\node[bit] (b11) at (4.80,0.15) {};
\node[bit] (c000) at (-5.60,-1.00) {};
\node[chosen] (c001) at (-4.00,-1.00) {};
\node[bit] (c010) at (-2.40,-1.00) {};
\node[bit] (c011) at (-0.80,-1.00) {};
\node[bit] (c100) at (0.80,-1.00) {};
\node[bit] (c101) at (2.40,-1.00) {};
\node[bit] (c110) at (4.00,-1.00) {};
\node[bit] (c111) at (5.60,-1.00) {};
\foreach \u/\v in {r/a0,r/a1,a0/b00,a0/b01,a1/b10,a1/b11,b00/c000,b00/c001,b01/c010,b01/c011,b10/c100,b10/c101,b11/c110,b11/c111}
\draw[thick,line] (\u) -- (\v);
\draw[very thick,spacec] (r) -- (a0) -- (b00) -- (c001);
\draw[thick,spacec,densely dashed] (c001) -- (-4.00,-1.60);
\draw[thick,line,densely dashed] (-4.00,-1.60) -- (-4.50,-2.05);
\draw[thick,line,densely dashed] (-4.00,-1.60) -- (-3.50,-2.05);
\node[edgelabel] at (-1.95,2.15) {$0\notin S$};
\node[edgelabel] at (1.95,2.15) {$0\in S$};
\node[edgelabel] at (-4.55,1.02) {$1\notin S$};
\node[edgelabel] at (-2.15,1.02) {$1\in S$};
\node[edgelabel] at (-5.55,-0.35) {$2\notin S$};
\node[edgelabel] at (-3.90,-0.35) {$2\in S$};
\node[tinylabel,spacec] at (-4.00,-2.34) {$3,4,5,\ldots$ はまだ自由};
\node[font=\small\bfseries] at (1.25,-2.34)
{\textcolor{logic}{発言}の\textcolor{spacec}{配置空間}};
\end{tikzpicture}
\end{center}
\end{deckframe}
% 17
\begin{deckframe}[incomplete]{HSL}{定理: Stone双対}
\begin{center}
{\large\bfseries \textcolor{logic}{ブール代数} $\longleftrightarrow$ \textcolor{spacec}{Stone空間}}
\end{center}
\vspace{-2mm}
\begin{theorem}[Stone双対]
\textcolor{logic}{ブール代数}と\textcolor{spacec}{Stone空間}は\textcolor{logic}{理論}として同じ\footnote{反変圏同値}．
\end{theorem}
\end{deckframe}
% 18
\begin{deckframe}[incomplete]{SVC}{視点: {\color{logic}モデル} = {\color{spacec}点}}
\begin{columns}[T,totalwidth=\textwidth]
\begin{column}{.43\textwidth}
\centering
\begin{tikzpicture}[scale=.95]
\node[logicbox,minimum width=3.1cm] (b) at (0,1.40) {\textcolor{logic}{ブール代数} $B$};
\node[logicbox,minimum width=3.1cm] (two) at (0,-1.35) {$2=\{\bot,\top\}$};
\draw[arrow] (b) -- node[right,font=\small\bfseries\color{logic}] {評価} (two);
\end{tikzpicture}
\end{column}
\begin{column}{.53\textwidth}
\centering
\begin{tikzpicture}[scale=.50,
bit/.style={circle,draw=ink!65,fill=white,inner sep=0pt,minimum size=3.0mm},
chosen/.style={circle,draw=spacec,fill=spaceLight,very thick,inner sep=0pt,minimum size=4.2mm}
]
\node[spacebox,minimum width=2.4cm] (cantor) at (4.40,2.35) {$2^{\mathbb N}$};
\node[bit] (r) at (0,2.10) {};
\node[chosen] (a0) at (-2.8,1.05) {};
\node[bit] (a1) at (2.8,1.05) {};
\node[bit] (b00) at (-4.2,0.00) {};
\node[chosen] (b01) at (-1.4,0.00) {};
\node[bit] (b10) at (1.4,0.00) {};
\node[bit] (b11) at (4.2,0.00) {};
\node[bit] (c000) at (-4.9,-1.05) {};
\node[bit] (c001) at (-3.5,-1.05) {};
\node[chosen] (c010) at (-2.1,-1.05) {};
\node[bit] (c011) at (-0.7,-1.05) {};
\node[bit] (c100) at (0.7,-1.05) {};
\node[bit] (c101) at (2.1,-1.05) {};
\node[bit] (c110) at (3.5,-1.05) {};
\node[bit] (c111) at (4.9,-1.05) {};
\foreach \u/\v in {r/a0,r/a1,a0/b00,a0/b01,a1/b10,a1/b11,b00/c000,b00/c001,b01/c010,b01/c011,b10/c100,b10/c101,b11/c110,b11/c111}
\draw[thick,draw=line] (\u) -- (\v);
\draw[very thick,spacec] (r) -- (a0) -- (b01) -- (c010);
\draw[very thick,logic,rounded corners=2pt] (-5.2,-1.70) rectangle (-0.35,1.35);
\node[spacebox,minimum width=2.4cm] (one) at (4.40,-2.30) {$1$};
\draw[arrow] (one) -- node[right,font=\small\bfseries\color{spacec}] {点} (cantor);
\end{tikzpicture}
\end{column}
\end{columns}
\end{deckframe}
% 19
\begin{deckframe}[incomplete]{GYK}{対応表: {\color{logic}論理}と{\color{spacec}幾何}}
\centering
\logicgeometrytable{stone}
\end{deckframe}
\section{Locale}
% 20
\begin{deckframe}[incomplete]{YCI}{目次}
\tableofcontents[currentsection]
\end{deckframe}
% 21
\begin{deckframe}[incomplete]{QNF}{{\color{logic}論理}の不満，{\color{spacec}幾何}の不満}
\begin{center}
{\large\bfseries Topos へ向かう，ふたつの不満}
\end{center}
\vspace{-2mm}
\begin{columns}[T,totalwidth=.98\textwidth]
\begin{column}{.48\textwidth}
\centering
\setlength{\fboxsep}{6pt}%
\fcolorbox{logic}{logicLight}{%
\begin{minipage}[c][4.55cm][c]{.90\linewidth}
\centering
\begin{tikzpicture}[x=1cm,y=1cm]
\path[use as bounding box] (-2.35,-2.00) rectangle (2.35,2.00);
\node[font=\Large\bfseries\color{logic},align=center,anchor=base] at (0,1.42)
{\textcolor{logic}{論理}の不満};
\node[font=\large\bfseries,align=center,anchor=base] at (0,.55)
{\textcolor{logic}{命題}\textcolor{logic}{論理}: \textcolor{logic}{量化子}なし};
\node[font=\Large\bfseries\color{logic},align=center,anchor=base] at (0,-.32)
{$\exists\qquad\forall$};
\node[font=\large\bfseries,align=center,anchor=base] at (0,-1.35)
{現代数学: \textcolor{logic}{量化子}だらけ};
\end{tikzpicture}
\end{minipage}}
\end{column}
\begin{column}{.48\textwidth}
\centering
\setlength{\fboxsep}{6pt}%
\fcolorbox{spacec}{spaceLight}{%
\begin{minipage}[c][4.55cm][c]{.90\linewidth}
\centering
\begin{tikzpicture}[x=1cm,y=1cm]
\path[use as bounding box] (-2.35,-2.00) rectangle (2.35,2.00);
\node[font=\Large\bfseries\color{spacec},align=center,anchor=base] at (0,1.42)
{\textcolor{spacec}{幾何}の不満};
\node[font=\large\bfseries,align=center,anchor=base] at (0,.55)
{空間がバラバラ！};
\begin{scope}[shift={(-.90,-.32)},scale=.62]
\draw[very thick,dashed,ink!45] (0,.16) circle (.46);
\draw[very thick,ink!45] (-.34,-.18) -- (.34,.50);
\end{scope}
\begin{scope}[shift={(.95,-.32)},scale=.62]
\foreach \x in {-0.70,-0.45,-0.20,0.20,0.45,0.70}
\fill[spacec] (\x,.28) circle (2.5pt);
\foreach \x in {-0.62,-0.50,-0.38,-0.18,-0.06,0.06,0.18,0.38,0.50,0.62}
\fill[spacec] (\x,-.15) circle (1.7pt);
\draw[very thick,spacec] (-.78,.54) -- (-.14,.54);
\draw[very thick,spacec] (.14,.54) -- (.78,.54);
\end{scope}
\end{tikzpicture}
\end{minipage}}
\end{column}
\end{columns}
\end{deckframe}
% 22
\begin{deckframe}[incomplete]{XKZ}{{\color{spacec}幾何}の不満: 空間がバラバラすぎる！}
\begin{center}
\begin{tikzpicture}[scale=.98]
\node[logicbox,minimum width=3.8cm,minimum height=1.35cm,
font=\Large\bfseries\color{logic}] (logic) at (-3.9,0.35)
{\textcolor{logic}{直観主義}\textcolor{logic}{論理}};
\node[spacebox,minimum width=4.6cm,minimum height=1.35cm,
font=\Large\bfseries\color{spacec}] (smooth) at (3.8,0.35)
{滑らかな空間も\\扱いたい};
\draw[arrow] (smooth.west) -- node[above,font=\small\bfseries\color{bridge}]{対応する\textcolor{logic}{論理}} (logic.east);
\end{tikzpicture}
\end{center}
\end{deckframe}
% 23
\begin{deckframe}[incomplete]{JOY}{定義: {\color{spacec}Locale}}
\begin{center}
{\large\bfseries 空間を，\textcolor{spacec}{開集合}=\textcolor{logic}{命題}で再定義する}
\end{center}
\vspace{-5mm}
\begin{columns}[T,onlytextwidth]
\begin{column}{.58\textwidth}
\begin{definition}[\textcolor{spacec}{Locale}]
\textcolor{spacec}{Locale}\footnote{厳密には\textcolor{logic}{代数}側は Frame，\textcolor{spacec}{Locale} はその双対圏．}
とは，任意 join と有限 meet を持ち，分配法則
\[
a\wedge\bigvee_i b_i=\bigvee_i(a\wedge b_i)
\]
を満たす順序構造．
\end{definition}
\end{column}
\begin{column}{.38\textwidth}
\centering
\begin{tikzpicture}[x=1.03cm,y=1cm,scale=1.86,line cap=round,line join=round]
\path[use as bounding box] (-1.31,-1.83) rectangle (1.31,1.31);
\draw[very thick,draw=spacec,fill=spaceLight,rounded corners=3pt]
(-1.30,-1.78) rectangle (1.30,-1.43);
\draw[very thick,draw=spacec]
(-.58,-1.43) -- (-.28,-.86)
(.58,-1.43) -- (.28,-.86);
\draw[very thick,draw=spacec,fill=paper,rounded corners=2pt]
(-1.08,-.86) rectangle (1.02,-.58);
\draw[thick,draw=spacec] (-.78,-.72) -- (.72,-.72);
\draw[line width=5.0pt,draw=spacec]
(-.54,-1.40) .. controls (-1.18,-.40) and (-.95,.62) .. (-.15,1.02);
\draw[line width=2.0pt,draw=spaceLight]
(-.49,-1.30) .. controls (-.98,-.42) and (-.80,.46) .. (-.12,.90);
\draw[very thick,draw=spacec,fill=spaceLight,rounded corners=2pt]
(-.66,1.05) -- (-.10,1.28) -- (.05,1.04) -- (-.51,.81) -- cycle;
\draw[very thick,draw=spacec,fill=spaceLight,rounded corners=3pt]
(-.34,.23) -- (.26,.48) -- (-.06,1.24) -- (-.66,.99) -- cycle;
\draw[very thick,draw=spacec,fill=paper,rounded corners=2pt]
(.17,.23) -- (.58,.40) -- (.42,.04) -- (.02,-.13) -- cycle;
\draw[very thick,draw=spacec] (-.50,-.58) -- (-.50,.05);
\draw[very thick,draw=spacec,fill=spaceLight] (-.50,.09) circle (.18);
\draw[thick,draw=spacec!65,fill=spaceLight!75] (-.04,-.45) ellipse (.55 and .16);
\draw[thick,draw=spacec!65] (-.04,-.45) ellipse (.18 and .05);
\end{tikzpicture}
\end{column}
\end{columns}
\end{deckframe}
% 24
\begin{deckframe}[incomplete]{VHV}{{\color{logic}命題}: 滑らかな空間も全て\\[-0.4ex]{propositional geometric {\color{logic}theory} の {\color{spacec}classifying} {\color{spacec}space}!}}
\begin{center}
{\large\bfseries \textcolor{logic}{直観主義}的(geom.)\textcolor{logic}{命題}\textcolor{logic}{論理} $\leftrightarrow$ \textcolor{spacec}{Locale}}
\end{center}
\vspace{4mm}
\begin{theorem}[\textcolor{spacec}{Locale}の標語]
全ての \textcolor{spacec}{Locale} は，(join-infinitary)\textcolor{logic}{命題}\textcolor{logic}{論理}と対応する．
\end{theorem}
\end{deckframe}
% 25
\begin{deckframe}[incomplete]{PGB}{思想: {\color{logic}排中律}は連続性に必要}
\begin{center}
{\large\bfseries \textcolor{logic}{排中律}が，空間をバラバラにする}
\end{center}
\vspace{-1mm}
\begin{center}
\begin{tikzpicture}[scale=1.0]
\draw[->,very thick,spacec] (-4.8,0) -- (4.8,0) node[right] {$\mathbb{R}$};
\draw[very thick,spacec] (0.15,0) -- (4.2,0);
\draw[very thick,spacec] (-4.2,0) -- (-0.15,0);
\fill[paper,draw=danger,very thick] (0,0) circle (4pt);
\node[font=\large\color{logic}] at (2.2,0.55) {$U$};
\node[font=\large\color{logic}] at (-2.3,0.55) {$\neg U$};
\node[font=\large\bfseries] at (0,-1.45) {$U\lor\neg U=\mathbb{R}\setminus\{0\}$};
\end{tikzpicture}
\end{center}
\end{deckframe}
% 26
\begin{deckframe}[incomplete]{NVR}{対応表: {\color{logic}論理}と{\color{spacec}幾何}}
\centering
\logicgeometrytable{locale}
\end{deckframe}
\section{Topos}
% 27
\begin{deckframe}[incomplete]{OWD}{目次}
\tableofcontents[currentsection]
\end{deckframe}
% 28
\begin{deckframe}[incomplete]{LHS}{{\color{logic}量化子}と sheaf}
\begin{center}
{\Large\bfseries \textcolor{logic}{量化子} $\longleftrightarrow$ \textcolor{spacec}{sheaf}}
\end{center}
\vspace{-2mm}
\begin{columns}[T,totalwidth=.98\textwidth]
\begin{column}{.46\textwidth}
\centering
\begin{tikzpicture}[x=1cm,y=1cm]
\path[use as bounding box] (-2.35,-2.00) rectangle (2.35,2.00);
\node[font=\Large\bfseries\color{logic}] at (0,1.46) {論理};
\node[logicbox,minimum width=3.65cm,minimum height=1.12cm,
font=\Large\bfseries\color{logic}] at (0,.48)
{\textcolor{logic}{量化子}};
\node[font=\Large\bfseries\color{logic}] at (0,-.58)
{$\exists x\,\varphi(x)\qquad \forall x\,\varphi(x)$};
\draw[very thick,logic] (-1.78,-1.28) -- (1.78,-1.28);
\foreach \x/\lab in {-1.15/a,0/x,1.15/b}{
\fill[logic] (\x,-1.28) circle (2.4pt);
\node[font=\large\color{logic}] at (\x,-1.67) {$\lab$};
}
\end{tikzpicture}
\end{column}
\begin{column}{.50\textwidth}
\centering
\begin{tikzpicture}[x=1cm,y=1cm,line cap=round,line join=round]
\path[use as bounding box] (-2.55,-2.00) rectangle (2.55,2.00);
\node[font=\Large\bfseries\color{spacec}] at (0,1.46) {幾何};
\draw[very thick,spacec,smooth,domain=-540:540,samples=210,variable=\t]
plot ({.78*cos(\t)}, {.22 + .0016*\t + .12*sin(\t)});
\node[font=\large\color{spacec}] at (1.05,1.10) {$\R$};
\draw[very thick,spacec] (0,-.16) circle (.78);
\node[font=\Large\bfseries\color{spacec},anchor=west] at (1.02,.04) {$S^1$};
\fill[spacec] (.70,.18) circle (2.6pt);
\draw[arrow] (.78,.20) .. controls (.94,.13) and (.92,.03) .. (.70,.18);
\draw[arrow] (-.78,.18) .. controls (-.94,.10) and (-.88,.00) .. (-.70,.18);
\node[spacebox,minimum width=3.35cm,font=\large\bfseries\color{spacec}]
at (0,-1.62) {$p:\R\to S^1$\\etale bundle};
\end{tikzpicture}
\end{column}
\end{columns}
\end{deckframe}
% 29
\begin{deckframe}[incomplete]{AYV}{定理: Geometric {\color{logic}theory} = Topos}
\begin{center}
{\Large\bfseries \textcolor{logic}{理論} $\longleftrightarrow$ \textcolor{spacec}{Topos}}
\end{center}
\vspace{-1mm}
\begin{theorem}[\textcolor{spacec}{Classifying} topos]
Geometric \textcolor{logic}{theory} $T$ には \textcolor{spacec}{classifying} topos $\mathcal E_T$ が対応し，
$\mathcal E_T$ は $T$ の\textcolor{logic}{モデル}を分類する．
\end{theorem}
\end{deckframe}
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\begin{deckframe}[incomplete]{IQP}{対応表: {\color{logic}論理}と{\color{spacec}幾何}}
\centering
\logicgeometrytable{topos}
\end{deckframe}
% 31
\begin{deckframe}[incomplete]{HZH}{思想: {\color{spacec}ベン図}の進化論的起源}
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\includegraphics[height=.35\textheight]{figures/png/yamaguchi_mikan_map_icon.svg.png}
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{\fontsize{42}{42}\selectfont\(\setminus\)}
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{\fontsize{42}{42}\selectfont\(=\)}
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\includegraphics[height=.35\textheight]{figures/png/yamaguchi_mikan_without_bear_map_icon.svg.png}
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\end{center}
\end{deckframe}
\end{document}