← spacetime-game-of-life
main0315.tex
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% color scheme for the opposition: time vs space
\definecolor{timeblue}{RGB}{50,110,120}
\definecolor{spaceorange}{RGB}{200,105,40}
\newcommand{\Time}[1]{\textcolor{timeblue}{#1}}
\newcommand{\Space}[1]{\textcolor{spaceorange}{#1}}
\newcommand{\TimeTerm}{\Time{時間}}
\newcommand{\SpaceTerm}{\Space{空間}}
\newcommand{\TimeEvo}{\Time{時間発展}}
\newcommand{\SpaceConcept}{\Space{空間概念}}
\newcommand{\SpaceTimeSlogan}{\text{\Space{Space}}\rtimes\text{\Time{Time}}}
\newcommand{\N}{\Time{\mathbb{N}}}
\newcommand{\Z}{\mathbb{Z}}
\newcommand{\Q}{\mathbb{Q}}
\newcommand{\R}{\mathbb{R}}
\newcommand{\E}{\mathcal{E}}
\newcommand{\Pow}{\mathcal{P}}
\newcommand{\Set}{\mathsf{Set}}
\newcommand{\PSh}{\mathsf{PSh}}
% \newcommand{\dSet}{\sigma\text{-}\mathsf{Set}}
% \newcommand{\dPSh}{\sigma\text{-}\mathsf{PSh}}
\newcommand{\dSet}{\Time{\PSh(\N)}}
% \newcommand{\dPSh}{\Time{\mathsf{DPSh}}}
\newcommand{\dPSh}{\mathsf{DPSh}}
\newcommand{\Sh}{\mathsf{Sh}}
\newcommand{\Int}{\mathrm{Int}}
\newcommand{\GoL}{\mathsf{GoL}}
\newcommand{\con}{\mathrm{con}}
\newcommand{\op}{\mathrm{op}}
\newcommand{\Two}{\mathsf{2}}
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\title{\texorpdfstring{\Space{Space}$\rtimes$\Time{Time} for Conway's Game of Life}{A space rtimes time for Conway's Game of Life}}
\subtitle{\Time{時間}に依存した\Space{空間}をrelative toposで捉える}
\author{洞龍弥}
\institute[]{東京大学数理科学研究科\,博士2年}
\date[CSCAT]{CSCAT2026}
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{Slides}
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\input{概要}
\begin{frame}{Table of Contents}
\tableofcontents[]
\end{frame}
\section{導入: Game of Lifeにおける\Space{空間}と\Time{時間}}
\begin{frame}{Conway's Game of Life}
\centering
$\Z^2$ の各点が，\Space{近傍}の明滅だけを見て \TimeEvo{}する．
\vspace{0.7em}
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\[
\Pow(\Z^2)\to\Pow(\Z^2)
\]
\end{frame}
\begin{frame}{出発点: 「グライダーが右上に動く」}
\centering
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\vspace{0.45em}
\begin{block}{問題意識}
\centering
この「右上に動く」という言明を可能にしている$\Z^2$の{幾何的構造}は何か？
\end{block}
\centering
$\to$ ただの\Space{位相空間}でも，ただの\Time{離散力学系}($=$\Time{$\N$-set})でもない！
\end{frame}
\begin{frame}{ただの\Space{位相空間}でも，ただの\Time{離散力学系}でもない！}
% \footnotesize
\begin{columns}[T,totalwidth=\textwidth]
\column{0.485\textwidth}
\begin{block}{空間だけ}
$\Z^2$に位相(or graph)を入れて，ただの\Space{空間}として捉える?
$\to$ \Time{時間発展}が見えにくい
\end{block}
\column{0.485\textwidth}
\begin{block}{時間だけ}
単に\Time{離散力学系}($=$集合と自己写像)として捉える?
$\to$ \Space{空間}情報が見えにくい
\end{block}
\end{columns}
\vspace{1em}
\centering
\textbf{どちらも不十分！}
\vspace{0.15em}
\begin{alertblock}{スローガン}
$\Z^2$ の点たちは，\TimeEvo{}における相互作用を通じて，\Space{空間}的なつながりを得ている．
\end{alertblock}
このスローガンを数学的に定式化したい．
\end{frame}
\input{概要}
\section{Pretopology と Dynamical presheaf}
\begin{frame}{Pretopology(1/3) \Space{近傍} $N_{(x,y)}$}
% \small
% \Space{空間}的側面と\Time{時間}的側面を兼ね備えた(less categoricalな)数学的記述を1つ与えよう．
\Space{空間}的側面と\Time{時間}的側面を兼ね備えた素朴な(less categoricalな)数学的記述を1つ与えよう．
\vspace{0.35em}
\begin{columns}[T,totalwidth=\textwidth]
\column{0.68\textwidth}
% \[
% N_{(x,y)}=
% \{(x+i,y+j)\in\Z^2 \mid i,j\in\{-1,0,1\}\}
% \]
\begin{block}{Def (近傍 $N_{(x,y)}$)}
各点 $(x,y)\in\Z^2$ に対し，
隣接する$9$マスからなる集合
$N_{(x,y)}\subset\Z^2$ を
\Space{近傍}と呼ぶ．
\end{block}
\vspace{0.4em}
Game of Lifeでは，\Time{$1$ステップ後}の各点$(x,y)$の状態は，\Space{近傍}$N_{(x,y)}$の状態だけから決まる．
\column{0.28\textwidth}
\centering
\begin{tikzpicture}[scale=0.82]
\foreach \a in {0,...,4} {
\foreach \b in {0,...,4} {
\draw[gray!50] (\a,\b) rectangle ++(1,1);
}
}
\foreach \a/\b in {1/1,1/2,1/3,2/1,2/2,2/3,3/1,3/2,3/3} {
\fill[spaceorange!20] (\a,\b) rectangle ++(1,1);
}
\fill[spaceorange!55] (2,2) rectangle ++(1,1);
\draw[spaceorange!85!black,very thick] (1,1) rectangle (4,4);
\node at (2.5,2.5) {$\scriptstyle (x,y)$};
\node[anchor=west] at (3.05,4.45) {$N_{(x,y)}$};
\end{tikzpicture}
\end{columns}
\end{frame}
\begin{frame}{Pretopology(2/3) 内部 $\Int(S)$}
\small
\begin{columns}[T,totalwidth=\textwidth]
\column{0.6\textwidth}
\begin{block}{Def (内部)}
$S\subset\Z^2$について，その\textbf{内部}を次で定める．
\[
\Int(S)=\{(x,y)\in\Z^2 \mid N_{(x,y)}\subset S\}
\]
\end{block}
Game of Lifeは，
% $S\subset\Z^2$上で現在の状態が分かっているとき，$\Int(S)$の
% \Time{1 step}先の状態がわかる．つまり
% % \vspace{0.25em}
% % \begin{exampleblock}{仮の記述}
各$S\subset\Z^2$に対して
\[
\delta_S\colon \Pow(\Space{S})\Time{\to}\Pow(\Space{\Int(S)})
\]
という写像を定めている．
% \end{exampleblock}
\column{0.37\textwidth}
\centering
\begin{tikzpicture}[scale=0.38]
% grid
\foreach \x in {0,...,14} {
\foreach \y in {0,...,14} {
\draw[gray!35] (\x,\y) rectangle ++(1,1);
}
}
% S
\foreach \x/\y in {
1/13,7/13,10/13,7/12,8/12,9/12,5/11,6/11,7/11,9/11,10/11,13/11,
5/10,10/10,11/10,3/9,4/9,5/9,10/9,3/8,9/8,10/8,11/8,12/8,
3/7,4/7,7/7,8/7,9/7,4/6,5/6,7/6,5/5,7/5,8/5,3/4,4/4,5/4,6/4,7/4,
4/3,5/3,6/3,4/2,4/1,10/3,11/3
}{
\fill[spaceorange!40] (\x,\y) rectangle ++(1,1);
}
% Int(S)
\foreach \x/\y in {
8/11,6/10,7/10,8/10,9/10,6/9,8/9,9/9,
4/8,5/8,6/8,7/8,8/8,5/7,6/7,6/6,6/5
}{
\fill[spaceorange!70] (\x,\y) rectangle ++(1,1);
}
% Int^2(S)
\foreach \x/\y in {7/9}{
\fill[spaceorange!100] (\x,\y) rectangle ++(1,1);
}
% labels (外側に配置)
% \node[anchor=west, spaceorange!45] at (11.2,3) {$S$};
% \node[anchor=west, spaceorange!75] at (9.2,5) {$\Int(S)$};
% \node[anchor=west, spaceorange!100] at (7.2,7) {$\Int^2(S)$};
\node[anchor=west, black] at (5,1) {$\Int^2(S)\subsetneq \Int(S)\subsetneq S$};
% \node[anchor=west, spaceorange!45] at (15.4-2,1) {$S$};
% \node[anchor=west, spaceorange!75] at (11.2-2,1) {$\Int(S)\subsetneq$};
% \node[anchor=west, spaceorange!100] at (7-2,1) {$\Int^2(S)\subsetneq$};
\end{tikzpicture}
\end{columns}
\end{frame}
\begin{frame}{Pretopology(3/3) \texorpdfstring{$\coloneqq$}{coloneqq} \Time{動的な}\Space{Topology}}
\footnotesize
\begin{columns}[T,totalwidth=\textwidth]
\column{0.5\textwidth}
\begin{block}{Def (pretopology)}
集合$X$上の\textbf{pretopology}とは，写像
\[
\Int\colon\Pow(X)\to\Pow(X)
\]
であって
\begin{itemize}
\item $\Int(S\cap T)=\Int(S)\cap\Int(T)$
\item $\Int(X)=X$
\item $\Int(S)\subset S$
\end{itemize}
を満たすもの．
\end{block}
\column{0.47\textwidth}
\begin{exampleblock}{Example (Game of Life)}
先ほどの$\Int$は，$\Z^2$上のpretopologyである．
\end{exampleblock}
\begin{exampleblock}{Example (Ordinary topology)}
Ordinary \Space{topology}は冪等な
\[
\Int^2=\Int
\]
pretopologyと同義．
\end{exampleblock}
冪等性$\Int^2=\Int$を仮定しない!
$\to$ \textbf{\Time{動的な}\Space{topology}}
\end{columns}
\end{frame}
\begin{frame}{Dynamical presheaf over a pretopology}
\small
\begin{columns}
% [T,totalwidth=\textwidth]
\column{0.55\textwidth}
\begin{block}{Def (Dynamical presheaf $\dPSh$)}
Pretopology $(X,\Int)$ 上の \textbf{dynamical presheaf} $(F,\delta)$とは，
presheaf
\[
F\colon \Pow(X)^{\op}\to\Set
\]
と，整合的な写像の族
\[
\delta=\{\delta_S\colon F(S)\to F(\Int(S))\}_{S\subset X}
\]
の組のことをいう．\\
$\dPSh(X,\Int)\coloneqq\text{Dynamical presheafの圏}$
\end{block}
% \vspace{0.45em}
\column{0.5\textwidth}
\centering
\begin{tikzcd}[ampersand replacement=\&]
F(S) \arrow[r,"\delta_S"] \arrow[d,"\mathrm{res}"',""{name=A}]
\& F(\Int(S)) \arrow[d,"\mathrm{res}",""'{name=B}]\ar[from=A,to=B,phantom,"\text{整合的}"] \\
F(T) \arrow[r,"\delta_T"']
\& F(\Int(T))
\end{tikzcd}
\vspace{0.35em}
\begin{exampleblock}{Example (Game of Life)}
Game of Lifeはpretopology $(\Z^2,\Int)$上のdynamical presheaf
\[\GoL \in \dPSh(\Z^2, \Int)\]
を与える．
\end{exampleblock}
\end{columns}
\end{frame}
\input{概要}
\section{A \Space{Space}\texorpdfstring{$\rtimes$}{rtimes}\Time{Time} as a relative topos}
\begin{frame}{Relative topos (1/2): Definition}
\small
\begin{columns}[T,totalwidth=\textwidth]
\column{0.75\textwidth}
\begin{block}{Def (Relative topos)}
Topos $\mathcal S$上の\textbf{relative topos}とは，topos $\mathcal{\E}$とgeometric morphism
\[
\gamma\colon \mathcal E\to\mathcal S
\]
の組$(\E, \gamma))$のこと．
\end{block}
\vspace{0.45em}
\begin{exampleblock}{Example (Ordinary geometry)}
Grothendieck toposは $\Set$ 上の relative topos と見なせる
\[
\gamma\colon \Space{\Sh(X)}\to\Set
\]
\end{exampleblock}
\column{0.2\textwidth}
\centering
\vspace{1.5em}
\begin{tikzpicture}[>=Latex,scale=1.5]
\node[draw,rounded corners,fill=blue!10,minimum width=2.8cm,minimum height=0.88cm] (E) at (0,0) {$\mathcal E$};
\node[draw,rounded corners,fill=blue!20,minimum width=2.8cm,minimum height=0.88cm] (S) at (0,-2.0) {$\mathcal S$};
\draw[->,very thick] (E) -- node[right] {$\gamma$} (S);
\node[align=center] at (0,0.7) {\small $\mathcal{S}$-relative topos};
\node[align=center] at (0,-2.7) {\small base topos};
\end{tikzpicture}
\end{columns}
\end{frame}
\begin{frame}{Relative topos (2/2): relative presheaf topos}
\small
\begin{columns}[T,totalwidth=\textwidth]
\column{0.75\textwidth}
\begin{block}{Fact (Internal presheafはrelative toposをなす)}
Topos $\mathcal{S}$とそのinternal category $\mathfrak{D}$について，$\mathcal{S}$-internal presheafの圏$\PSh_{\mathcal{S}}(\mathfrak{D})$は$\mathcal{S}$-relative toposである．
\end{block}
$\mathcal{S}$自身がpresheaf toposであるとき，次が成立する．
\begin{exampleblock}{Example (Presheaf-internal presheaf = Presheaf)}
Small category $C$と$\PSh(C)$-internal category $\mathfrak{D}\colon C^\op \to \mathsf{Cat}$について，
% internal presheafの圏
$\PSh_{\PSh(C)}(\mathfrak{D})$は
\[
\PSh_{\PSh(C)}(\mathfrak{D}) \simeq \PSh(\mathfrak{D} \rtimes C)
\]
とかける．(cf. [\cite{johnstone2002sketchesv1}])
\end{exampleblock}
\column{0.22\textwidth}
\vspace{1em}
\centering
\begin{tikzpicture}[>=Latex,scale=1.5]
\node[draw,rounded corners,fill=blue!10,minimum width=2.8cm,minimum height=0.88cm] (E) at (0,0) {$\PSh_{\mathcal{S}}(\mathfrak{D})$};
\node[draw,rounded corners,fill=blue!20,minimum width=2.8cm,minimum height=0.88cm] (S) at (0,-2.0) {$\mathcal{S}$};
\draw[->,very thick] (E) -- node[right] {$\gamma$} (S);
\node[align=center] at (0,0.7) {\small $\mathcal{S}$-relative\\ presheaf topos};
\node[align=center] at (0,-2.5) {\small base topos};
\node[align=left] at (1.3,-2) {$\ni \mathfrak{D}$};
\end{tikzpicture}
\end{columns}
\end{frame}
\begin{frame}{\texorpdfstring{$\GoL$}{GoL}は\Time{離散力学系}-relative presheaf(1/3): $\mathcal{S}\coloneqq \dSet$}
\begin{columns}[T,totalwidth=\textwidth]
\column{0.56\textwidth}
\vspace{1em}
\begin{block}{Def (離散力学系)}
\textbf{\Time{離散力学系}}とは，集合 $A$ と自己写像 $s\colon A\to A$ の組 $(A,s)$．
% \Time{離散力学系}の圏を$\dSet$ と書く．
% \[
% \dSet\simeq\PSh(\N)
% \]$\dSet$ 自体が 1 つのtoposである．
\end{block}
\vspace{1em}
\Time{離散力学系}の圏$\dSet$はtoposである
\column{0.38\textwidth}
\centering
\begin{tikzpicture}[>=Latex,scale=1.2]
% \node[draw,rounded corners,fill=gray!10,minimum width=2.8cm,minimum height=0.88cm] (E) at (0,0) {??};
\node[draw,rounded corners, fill=blue!10,minimum width=2.8cm,minimum height=0.88cm] (E) at (0,0) {??};
\node[draw,rounded corners,fill=timeblue!20,minimum width=2.8cm,minimum height=0.88cm] (S) at (0,-2.0) {$\dSet$};
\draw[->,very thick] (E) -- node[right] {$\gamma$} (S);
\node[align=center] at (0,0.7) {\small $\dSet$-relative topos};
\node[align=center] at (0,-2.6) {\small base topos};
\end{tikzpicture}
\end{columns}
% \vspace{1em}
{
$\to$ \textbf{$\dSet$-relative topos theory}では
base topos $\mathcal{S}=\dSet$に\Time{時間発展}が組み込まれている！}
\footnotesize(cf. [\cite{tomasic2020topos}])
\end{frame}
\begin{frame}{\texorpdfstring{$\GoL$}{GoL}は\Time{離散力学系}-relative presheaf(2/3): メイン主張}
\small
\begin{columns}[T,totalwidth=\textwidth]
\column{0.65\textwidth}
Pretopology $\Int\colon \Space{\Pow(X)}\Time{\to}\Space{\Pow(X)}$は順序を保つ:
\begin{block}{Lemma (Pretopologyはinternal poset)}
$(\Space{\Pow(X)},\Int)$ は $\dSet$-\textbf{internal \Space{poset}}．
\end{block}
\begin{alertblock}{Thm (Dynamical presheaf as relative presheaf)}
\[
\dPSh(X,\Int)\simeq \PSh_{\dSet}(\Space{\Pow(X)},\Int)
\]
\end{alertblock}
\column{0.3\textwidth}
\centering
\begin{tikzpicture}[>=Latex,scale=1.2]
% \node[draw,rounded corners,fill=gray!10,minimum width=2.8cm,minimum height=0.88cm] (E) at (0,0) {??};
\node[draw,rounded corners, fill=blue!10,minimum width=2.8cm,minimum height=0.88cm] (E) at (0,0) {$\PSh_{\dSet}(\Space{\Pow(X)},\Int)$};
\node[draw,rounded corners,fill=timeblue!20,minimum width=2.8cm,minimum height=0.88cm] (S) at (0,-2.0) {$\dSet$};
\draw[->,very thick] (E) -- node[right] {$\gamma$} (S);
\node[align=center] at (0,0.7) {\small $\dSet$-relative topos};
\node[align=center] at (0,-2.6) {\small base topos};
\node[align=left] at (1.8,-2) {$\ni \Space{\Pow(X)}$};
\end{tikzpicture}
\end{columns}
\begin{exampleblock}{Cor (Game of lifeはspace$\rtimes$time上のpresheaf)}
\[
\GoL \in\dPSh(\Z^2,\Int)\simeq \PSh_{\dSet}(\Space{\Pow(\Z^2)}, \Int)\simeq \PSh(\Space{\Pow(\Z^2)}\rtimes_{\Int}\N)
\]
\end{exampleblock}
\end{frame}
\begin{frame}{\texorpdfstring{$\GoL$}{GoL}は\Time{離散力学系}-relative presheaf(3/3): 気持ち}
\begin{exampleblock}{Cor (Game of lifeはspace$\rtimes$time上のpresheaf) 再掲}
\[
\GoL \in\PSh_{\dSet}(\Space{\Pow(\Z^2)}, \Int)\simeq \PSh(\Space{\Pow(\Z^2)}\rtimes_{\Int}\N)
\]
\end{exampleblock}
\vspace{0.25em}
\begin{columns}[T,totalwidth=\textwidth]
\column{0.46\textwidth}
\centering
\textbf{ordinary \Space{geometry}}
\[
% X\leadsto
\Space{\Sh(X)}\to\Set
\]
\column{0.50\textwidth}
\centering
\textbf{\Time{Time}-dependent \Space{geometry}}
\[
% (X,\Int)\leadsto
\PSh_{\dSet}(\Space{\Pow(X)},\Int)\to\dSet
\]
\end{columns}
\vspace{0.25em}
\begin{center}
\large
\textbf{
\Time{時間発展}の内部で\Space{空間}を考えると，}\\
{単純な並列\Space{Space}$\times$\Time{Time}ではなく}\\
\textbf{
依存関係\Space{Space}$\rtimes$\Time{Time}を捉えられる！}
\end{center}
\end{frame}
% \begin{frame}{観察(1es$\Time{Time} の上の presheaf}
\begin{frame}{観察(1/2): \texorpdfstring{$\Space{\Pow(\Z^2)}\rtimes_{\Int}\N$}{rtimes}はどんな圏？}
\begin{exampleblock}{Cor (Game of lifeはspace$\rtimes$time上のpresheaf) 再掲}\[
\GoL \in\PSh_{\dSet}(\Space{\Pow(\Z^2)}, \Int)\simeq \PSh(\Space{\Pow(\Z^2)}\rtimes_{\Int}\N)
\]
\end{exampleblock}
\vspace{0.1em}
\begin{columns}[T,totalwidth=\textwidth]
\column{0.54\textwidth}
\begin{description}
\item[Object] $\Z^2$のsubset $\Space{U}\subset X$
\item[Morphisms] 射$\Space{U}\to \Space{V}$は自然数$n\in\N$であって$\Space{U}\subset\Int^n(\Space{V})$を満たすもの
\end{description}
\column{0.44\textwidth}
\begin{figure}
\centering
\begin{tikzpicture}[scale=1.2,>=Latex]
\draw[rounded corners=3pt,fill=spaceorange!10,draw=spaceorange!80!black]
(0,0) rectangle (4.9,3.35);
\draw[rounded corners=3pt,fill=spaceorange!17,draw=spaceorange!80!black]
(0.45,0.35) rectangle (4.45,3.0);
\draw[rounded corners=3pt,fill=spaceorange!26,draw=spaceorange!80!black]
(0.95,0.75) rectangle (3.95,2.6);
\draw[rounded corners=3pt,fill=white,draw=spaceorange!80!black]
(1.55,1.22) rectangle (3.35,2.12);
\footnotesize
\node[anchor=east] at (4.0+0.1,3.04+0.1) {$V$};
\node[anchor=east] at (3.8+0.1,2.66+0.1) {$\Int(V)$};
\node[anchor=east] at (3.6+0.1,2.24+0.1) {$\Int^2(V)$};
\node at (2.43,1.67) {$U$};
\draw[timeblue,very thick,-{Latex[length=2.5mm]}]
(4.78+0.1,3.18)
.. controls (5.55+0.1,3.08) and (5.20+0.1,2.75) ..
node[right,pos=0.52,fill=white,inner sep=0.7pt,font=\scriptsize] {$1$}
(4.30,2.78);
\draw[timeblue,very thick,-{Latex[length=2.5mm]}]
(4.36+0.1,2.80-0.2)
.. controls (5.10+0.1,2.64-0.2) and (4.84+0.1,2.16-0.2) ..
node[right,pos=0.48,fill=white,inner sep=0.7pt,font=\scriptsize] {$1$}
(3.80,2.30-0.2);
\end{tikzpicture}
\end{figure}
\end{columns}
\end{frame}
\begin{frame}{観察(2/2): 大域切断$\gamma: \PSh(\Space{\Pow(\Z^2)}\rtimes_{\Int}\N)\to \dSet$}
\begin{columns}[T,totalwidth=\textwidth]
\column{0.45\textwidth}
$\GoL\in \PSh(\Space{\Pow(\Z^2)}\rtimes_{\Int}\N)$\\
の大域切断は\\$\Pow(\Z^2)\Time{\to} \Pow(\Z^2) \in \dSet$\\である．
\column{0.56\textwidth}
\centering
\begin{tikzpicture}[>=Latex,scale=1.2]
% \node[draw,rounded corners,fill=gray!10,minimum width=2.8cm,minimum height=0.88cm] (E) at (0,0) {??};
\node[draw,rounded corners, fill=blue!10,minimum width=2.8cm,minimum height=0.88cm] (E) at (0,0) {$\PSh_{\dSet}(\Space{\Pow(\Z^2)},\Int)$};
\node[draw,rounded corners,fill=timeblue!20,minimum width=2.8cm,minimum height=0.88cm] (S) at (0,-2.0) {$\dSet$};
\draw[->,very thick] (E) -- node[right] {$\gamma$} (S);
\node[align=center] at (0,0.7) {\small $\dSet$-relative topos};
\node[align=center] at (0,-2.6) {\small base topos};
\node[align=left] at (2.7,0) {$\ni \GoL$};
\node[align=left] at (2.75,-2) {$\ni {\Pow(\Z^2)}$};
\draw[thick, {Bar[]-Latex}] (2.8,-0.3) -- node[right] {} (2.8,-1.7);
\end{tikzpicture}
\end{columns}
% {\Space{Space} を忘れて\\\Time{time evolution} だけ残す};
% relative topos から base topos への大域切断
% \[
% \gamma_*:\dPSh(\Z^2,\Int)\to \dSet
% \]
% は，\Space{空間}方向の情報を忘れて，
% \Time{時間}発展だけを取り出す操作になっている．
\end{frame}
\input{概要}
\section{展望とお願い}
% \begin{frame}{\Time{今後}の\Space{展望}}
% この話はまだまだ荒い！
% \begin{itemize}
% \item \textbf{Sheaf condition}: 有限領域をdenseにするGrothendieck topologyを入れねば
% \item \textbf{Symmetry}: $\Z^2\rtimes D_4$などの対称性でquotientしたい
% \item \textbf{Logic}: ($S4$でない時制論理で?)「グライダーが右上に動く」を数学的に定式化したい
% \item \textbf{Smooth version}: $\Set$ から別のtoposへ．例えば波動方程式を捉えるなら
% $\mathsf{SmoothSet}^{\R_{\geq 0}}$-relative topos を考える
% \end{itemize}
% \end{frame}
\begin{frame}{\Time{今後}の\Space{展望} この話はまだまだ荒い！}
\begin{columns}
\column{0.66\textwidth}
\begin{itemize}
\setlength{\itemsep}{1.2em}
\item \textbf{Sheaf condition}: $\GoL$を含む最小のsite $(\Space{\Pow_{\mathrm{fin}}(\Z^2)}\rtimes_{\Int} \N, J)$
\item \textbf{Logic}: (\Time{冪等でない}様相演算子で)「グライダーが\Space{右上に}\Time{動く}」を定式化
\item \textbf{Symmetry}: 対称性\Space{$\Z^2\rtimes D_4$}で割る
\item \textbf{Smooth version}: 波動方程式を\Time{$\mathsf{SmoothSet}^{\R_{\geq 0}}$}-relative toposで捉える
\end{itemize}
\column{0.35\textwidth}
\begin{figure}
\centering
\begin{tikzpicture}[>=Latex,scale=0.7]
\fill (0,0) circle (1.5pt);
% \node[below right] at (0,0) {$(\Space{x_0},\Time{0})$};
% domain of influence
\fill[gray!10] (0,0) -- (2.7,2.7) -- (-2.7,2.7) -- cycle;
% boundary lines
\draw[very thick,gray,->] (0,0) -- (2.8,2.8);
\draw[very thick,gray,->] (0,0) -- (-2.8,2.8);
% \draw[very thick,spaceorange] (-2.0,2.0) -- (2.0,2.0);
% \fill[spaceorange] (-2.0,2.0) circle (1.2pt);
% \fill[spaceorange] (2.0,2.0) circle (1.2pt);
\node[above] at (0,-1.2)
{$\Space{|x-x_0|}\le c\Time{t}$};
\draw[->,thick, spaceorange] (-3.2,0) -- (3.4,0) node[right] {$x$};
\draw[->,thick, timeblue] (0,-0.2) -- (0,3.4) node[above] {$t$};
\end{tikzpicture}
\end{figure}
\end{columns}
\end{frame}
\begin{frame}{遠いモチベーション: 進化論の数理モデル}
\small
\begin{columns}[T,totalwidth=\textwidth]
\column{0.44\textwidth}
進化論の数理モデルでは多くの場合，
\begin{itemize}
\item Population size
\item Gene frequency
\end{itemize}
などの \textbf{数値} を扱う．
\column{0.52\textwidth}
\centering
\begin{tikzpicture}
\begin{axis}[
width=6.5cm,
height=4.45cm,
xmin=0, xmax=8,
ymin=0, ymax=4.2,
axis lines=left,
xlabel={$t$},
ylabel={population},
xtick=\empty,
ytick=\empty,
enlargelimits=false,
clip=false
]
\addplot[very thick,blue!70!red,domain=0:8,samples=220] {2.2 + 1.0*sin(deg(1.2*x))};
\addplot[very thick,blue!30!red,domain=0:8,samples=220] {2.0 + 0.9*sin(deg(1.2*x - 1.2))};
\node[anchor=west] at (axis cs:8,2.9) {\scriptsize predator};
\node[anchor=west] at (axis cs:8,1.8) {\scriptsize prey};
\end{axis}
\end{tikzpicture}
\end{columns}
しかし，私が知りたいのは，
\begin{center}
「同じものが動いている」
\end{center}
と言えるための
個体性が，\Space{空間}と \Time{時間発展}しか持たない系から創発することへの数学的理解である．
\end{frame}
\end{document}