\documentclass[dvipdfmx,14pt,notheorems,aspectratio=169]{beamer} \usepackage{array,booktabs,float} \usepackage{amsmath,amssymb,amsthm,mathtools,mathrsfs} \usepackage{graphicx,graphics} \usepackage{url} \usepackage[utf8]{inputenc} \usepackage{newunicodechar} \usepackage{pgfplots} \pgfplotsset{compat=1.18} \usepackage{tikz} \usetikzlibrary{arrows.meta,calc,positioning,fit,backgrounds} \usepackage{tikz-cd} \usepackage{eqparbox} \usepackage{bxdpx-beamer} \usepackage{pxjahyper} \usepackage[style=authoryear]{biblatex} \addbibresource{Horabiblio.bib} \graphicspath{{./images/}{images/}} \tikzset{>={Latex[width=3mm,length=3mm]}} \usetheme{Darmstadt} \usecolortheme{seahorse} % \usecolortheme{spruce} % \usecolortheme[named=olive!35!black]{structure} \setbeamertemplate{items}[default] \setbeamertemplate{navigation symbols}{} \setbeamerfont{frametitle}{size=\large,series=\bfseries} \setbeamerfont{block title}{size=\small} \setbeamerfont{block body}{size=\small} \AtBeginSection[] { \begin{frame} \frametitle{Table of Contents} \tableofcontents[currentsection] \end{frame} } % 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}} \makeatletter \newcounter{eqtabular} \newsavebox\eqtabularbox \newcolumntype{C}{>{\begin{lrbox}{\eqtabularbox}}c<{ \end{lrbox}\eqmakebox[\the\c@eqtabular @eqtabular]{\unhcopy\eqtabularbox}}} \newcolumntype{R}{>{\begin{lrbox}{\eqtabularbox}}c<{ \end{lrbox}\eqmakebox[\the\c@eqtabular @eqtabular][r]{\unhcopy\eqtabularbox}}} \newenvironment{eqtabular}[2][c] {\stepcounter{eqtabular}\tabular[#1]{#2}} {\endtabular} \makeatother \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} \begin{document} % \begin{frame} % \begin{columns} % \column{0.8\textwidth} % \titlepage % \column{0.18\textwidth} % \begin{figure} % \centering % \includegraphics[width=1\linewidth]{QrcodeForNotes.png} % \begin{center} % Slides % \end{center} % \end{figure} % \begin{tikzpicture}[scale=0.35] % \begin{scope}[shift={(0,0)}] % \foreach \x in {0,...,8} {\foreach \y in {0,...,8} {\draw[gray!50] (\x,\y) rectangle (\x+1,\y+1);}} % \fill[black] (2,3) rectangle (3,4); % \fill[black] (3,4) rectangle (4,5); % \fill[black] (4,2) rectangle (5,3); % \fill[black] (4,3) rectangle (5,4); % \fill[black] (4,4) rectangle (5,5); % \end{scope} % \end{tikzpicture} % \end{columns} % \end{frame} \begin{frame} \centering % \vspace{0.8em} \begin{beamercolorbox}[ wd=\textwidth, sep=0.75em, center, rounded=true, shadow=true ]{title} {\usebeamerfont{title}\inserttitle\par} \vspace{0.45em} {\usebeamerfont{subtitle}\usebeamercolor[fg]{subtitle}\insertsubtitle\par} \end{beamercolorbox} % \vspace{1.2em} % \begin{minipage}{0.72\linewidth} \begin{columns} \column{0.3\textwidth} \begin{figure} \centering \begin{tikzpicture}[scale=1] \fill[blue!25] (2,3) rectangle (3,4); \fill[blue!25] (3,4) rectangle (4,5); \fill[blue!25] (4,2) rectangle (5,3); \fill[blue!25] (4,3) rectangle (5,4); \fill[blue!25] (4,4) rectangle (5,5); \end{tikzpicture} \end{figure} \column{0.35\textwidth} \centering {\usebeamerfont{author}\insertauthor} \vspace{0.5em} {\usebeamerfont{institute}\insertinstitute} \vspace{0.5em} {\usebeamerfont{date}\insertdate} \column{0.3\textwidth} \centering \includegraphics[width=0.95\linewidth]{QrcodeForNotes.png} {Slides} \end{columns} \end{frame} \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} \begin{tikzpicture}[scale=0.37] \begin{scope}[shift={(0,0)}] \node[font=\bfseries] at (4.5,8.7) {0}; \foreach \x in {0,...,8} {\foreach \y in {0,...,8} {\draw[gray!50] (\x,\y) rectangle (\x+1,\y+1);}} \fill[black] (2,3) rectangle (3,4); \fill[black] (3,4) rectangle (4,5); \fill[black] (4,2) rectangle (5,3); \fill[black] (4,3) rectangle (5,4); \fill[black] (4,4) rectangle (5,5); \end{scope} \begin{scope}[shift={(10,0)}] \node[font=\bfseries] at (4.5,8.7) {1}; \foreach \x in {0,...,8} {\foreach \y in {0,...,8} {\draw[gray!50] (\x,\y) rectangle (\x+1,\y+1);}} \fill[black] (3,2) rectangle (4,3); \fill[black] (3,4) rectangle (4,5); \fill[black] (4,3) rectangle (5,4); \fill[black] (4,4) rectangle (5,5); \fill[black] (5,3) rectangle (6,4); \end{scope} \begin{scope}[shift={(20,0)}] \node[font=\bfseries] at (4.5,8.7) {2}; \foreach \x in {0,...,8} {\foreach \y in {0,...,8} {\draw[gray!50] (\x,\y) rectangle (\x+1,\y+1);}} \fill[black] (3,4) rectangle (4,5); \fill[black] (4,2) rectangle (5,3); \fill[black] (4,4) rectangle (5,5); \fill[black] (5,3) rectangle (6,4); \fill[black] (5,4) rectangle (6,5); \end{scope} \begin{scope}[shift={(30,0)}] \node[font=\bfseries] at (4.5,8.7) {4}; \foreach \x in {0,...,8} {\foreach \y in {0,...,8} {\draw[gray!50] (\x,\y) rectangle (\x+1,\y+1);}} \fill[black] (3,4) rectangle (4,5); \fill[black] (4,5) rectangle (5,6); \fill[black] (5,3) rectangle (6,4); \fill[black] (5,4) rectangle (6,5); \fill[black] (5,5) rectangle (6,6); \end{scope} \end{tikzpicture} \[ \Pow(\Z^2)\to\Pow(\Z^2) \] \end{frame} \begin{frame}{出発点: 「グライダーが右上に動く」} \centering \begin{tikzpicture}[scale=0.37] \begin{scope}[shift={(0,0)}] \node[font=\bfseries] at (4.5,8.7) {0}; \foreach \x in {0,...,8} {\foreach \y in {0,...,8} {\draw[gray!50] (\x,\y) rectangle (\x+1,\y+1);}} \fill[black] (2,3) rectangle (3,4); \fill[black] (3,4) rectangle (4,5); \fill[black] (4,2) rectangle (5,3); \fill[black] (4,3) rectangle (5,4); \fill[black] (4,4) rectangle (5,5); \end{scope} \begin{scope}[shift={(10,0)}] \node[font=\bfseries] at (4.5,8.7) {1}; \foreach \x in {0,...,8} {\foreach \y in {0,...,8} {\draw[gray!50] (\x,\y) rectangle (\x+1,\y+1);}} \fill[black] (3,2) rectangle (4,3); \fill[black] (3,4) rectangle (4,5); \fill[black] (4,3) rectangle (5,4); \fill[black] (4,4) rectangle (5,5); \fill[black] (5,3) rectangle (6,4); \end{scope} \begin{scope}[shift={(20,0)}] \node[font=\bfseries] at (4.5,8.7) {2}; \foreach \x in {0,...,8} {\foreach \y in {0,...,8} {\draw[gray!50] (\x,\y) rectangle (\x+1,\y+1);}} \fill[black] (3,4) rectangle (4,5); \fill[black] (4,2) rectangle (5,3); \fill[black] (4,4) rectangle (5,5); \fill[black] (5,3) rectangle (6,4); \fill[black] (5,4) rectangle (6,5); \end{scope} \begin{scope}[shift={(30,0)}] \node[font=\bfseries] at (4.5,8.7) {4}; \foreach \x in {0,...,8} {\foreach \y in {0,...,8} {\draw[gray!50] (\x,\y) rectangle (\x+1,\y+1);}} \fill[black] (3,4) rectangle (4,5); \fill[black] (4,5) rectangle (5,6); \fill[black] (5,3) rectangle (6,4); \fill[black] (5,4) rectangle (6,5); \fill[black] (5,5) rectangle (6,6); \draw[red,very thick,-{Latex[length=2.5mm]}] (1.5,1.5) -- (2.5,2.5); \end{scope} \end{tikzpicture} \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}