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\title{A space$\rtimes$time for Conway's Game of Life}
\author{Ryuya Hora}
\institute{Graduate School of Mathematical Sciences, University of Tokyo}
\date{}

\newcommand{\N}{\mathbb{N}}
\newcommand{\Z}{\mathbb{Z}}
\newcommand{\Pow}{\mathcal{P}}
\newcommand{\Set}{\mathbf{Set}}
\newcommand{\Cat}{\mathbf{Cat}}
\newcommand{\dSet}{\sigma\text{-}\mathbf{Set}}
\newcommand{\PSh}{\mathbf{PSh}}
\newcommand{\dPSh}{\sigma\text{-}\mathbf{PSh}}
\newcommand{\Sh}{\mathbf{Sh}}
\newcommand{\Int}{\mathrm{Int}}
\newcommand{\GoL}{\mathbf{GoL}}
\newcommand{\con}{\mathrm{con}}
\newcommand{\2}{\mathbf{2}}
\newcommand{\demph}[1]{\textbf{#1}}

\begin{document}

\begin{frame}
  \titlepage
\end{frame}

\begin{frame}{Starting question}
\begin{block}{Motivation}
Conway's Game of Life is a discrete model of biological behavior on the lattice $\Z^2$ with time evolution.
\end{block}

\vspace{0.4em}
\begin{alertblock}{Main question}
What is the \demph{intrinsic geometry} inside Conway's Game of Life?
\end{alertblock}

\vspace{0.4em}
Typical geometric language already appears in practice:
\begin{itemize}
  \item ``two gliders approach each other,''
  \item ``a pattern moves,''
  \item ``a local configuration generates a global behavior.''
\end{itemize}

So there is geometry here --- but not obviously ordinary topology.
\end{frame}

\begin{frame}{Why ordinary topology is not enough}
Two basic features distinguish the Game of Life from an ordinary topological space:
\begin{columns}[T,totalwidth=\textwidth]
\column{0.48\textwidth}
\begin{block}{Discrete}
The underlying space is the countable lattice $\Z^2$.
\end{block}

\column{0.48\textwidth}
\begin{block}{Dynamic}
The state evolves by a function
\[
\Pow(\Z^2) \to \Pow(\Z^2).
\]
\end{block}
\end{columns}

\vspace{0.5em}
Naive answers miss one side of the story:
\begin{itemize}
  \item $\prod_{(x,y)\in\Z^2}\Set$ remembers the cells but forgets time,
  \item $\dSet$ remembers time but forgets the spatial organization.
\end{itemize}
\end{frame}

\begin{frame}{Why topos theory?}
\begin{itemize}
  \item Topos theory is flexible enough to treat both continuous and discrete geometry.
  \item It also naturally accommodates dynamics via presheaf topoi.
  \item So the question becomes:
\end{itemize}

\begin{alertblock}{Reformulated question}
In what topos does Conway's Game of Life naturally live?
\end{alertblock}

\vspace{0.6em}
\centering
\begin{tikzpicture}[scale=0.9]
    \draw[thick] (0,0) circle (3.1cm);
    \node at (1.7,2.1) {topoi};
    \draw[thick] (-1.7,-0.2) circle (1.35cm);
    \node[align=center] at (-1.7,1.45) {topological\\spaces};
    \node at (-1.7,0.1) {$\Sh(X)$};
    \draw[-{Latex[length=2mm]}] (-1.7,-0.15) -- (-1.7,-1.15);
    \node[right] at (-1.7,-0.65) {$\Gamma$};
    \node at (-1.7,-1.45) {$\Set$};
    \node at (1.6,0.2) {\Large ?};
    \draw[-{Latex[length=2mm]}] (1.6,-0.05) -- (1.6,-1.15);
    \node[right] at (1.6,-0.65) {$\gamma$};
    \node at (1.6,-1.45) {$\dSet$};
\end{tikzpicture}
\end{frame}

\begin{frame}{The base topos of discrete dynamical systems}
\begin{definition}
A \demph{discrete dynamical system} is a pair $(X,f)$ of a set $X$ and a map $f\colon X\to X$.
\end{definition}

The category of these systems is denoted by $\dSet$.

\vspace{0.5em}
\begin{block}{Key fact}
\[
\dSet \simeq \PSh(\N),
\]
so $\dSet$ is itself a topos.
\end{block}

\vspace{0.5em}
Hence a relative topos over $\dSet$ should be interpreted as a kind of \demph{dynamical geometry}.
\end{frame}

\begin{frame}{Relative topos viewpoint}
\begin{definition}
A \demph{relative topos} over a base topos $\mathcal S$ is a topos $\mathcal E$ equipped with a geometric morphism
\[
\gamma\colon \mathcal E \to \mathcal S.
\]
\end{definition}

\vspace{0.5em}
For an ordinary Grothendieck topos, the base is $\Set$ and $\Gamma(X)$ is the set of global sections.

\vspace{0.5em}
For a relative topos over $\dSet$:
\begin{itemize}
  \item an object $X\in \mathcal E$ has a \demph{discrete dynamical system of global sections} $\gamma_*(X)$;
  \item this is exactly the kind of output we want for the Game of Life.
\end{itemize}

\begin{alertblock}{Target}
Find $\mathcal E\to\dSet$ and an object $\GoL\in\mathcal E$ such that $\gamma_*(\GoL)$ is the usual Game of Life evolution.
\end{alertblock}
\end{frame}

\begin{frame}{Conway's Game of Life as a map}
Write $\2=\{0,1\}$, where $1=$ alive and $0=$ dead.
For a set $X$, identify $\Pow(X)$ with the set of functions $X\to\2$.

\vspace{0.5em}
For $(x,y)\in\Z^2$, let
\[
N_{(x,y)}=\{(x+i,y+j)\mid i,j\in\{-1,0,1\}\}.
\]

\begin{block}{Definition}
Conway's Game of Life is the map
\[
\con\colon \Pow(\Z^2)\to\Pow(\Z^2)
\]
where $\con(f)(x,y)=1$ iff either
\begin{itemize}
  \item $f(x,y)=1$ and $\#(N_{(x,y)}\cap f^{-1}(1))\in\{3,4\}$, or
  \item $f(x,y)=0$ and $\#(N_{(x,y)}\cap f^{-1}(1))=3$.
\end{itemize}
\end{block}
\end{frame}

\begin{frame}{A geometric phenomenon: the glider}
\centering
\begin{tikzpicture}[scale=0.42]
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        \fill[black] (2,3) rectangle (3,4);
        \fill[black] (3,4) rectangle (4,5);
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        \fill[black] (4,3) rectangle (5,4);
        \fill[black] (4,4) rectangle (5,5);
    \end{scope}
    % Frame 2
    \begin{scope}[shift={(10.5,0)}]
        \node at (4.5, 8.8) {\textbf{2}};
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        \fill[black] (5,3) rectangle (6,4);
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    % Frame 3
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    % Frame 4
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\end{tikzpicture}

\vspace{0.4em}
A glider suggests that the relevant geometry must talk about \demph{local rules}, \demph{motion}, and \demph{time evolution} simultaneously.
\end{frame}

\begin{frame}{Local rule vs. global evolution}
The update rule is defined \demph{locally}:
\begin{itemize}
  \item to compute the next value at $(x,y)$ we only inspect the $3\times 3$ neighborhood $N_{(x,y)}$.
\end{itemize}

But the entire configuration changes \demph{globally}:
\begin{itemize}
  \item one step is still a map on all of $\Pow(\Z^2)$.
\end{itemize}

\vspace{0.5em}
This suggests replacing ordinary topology by a weaker notion of ``interior'' that already contains time evolution.
\end{frame}

\begin{frame}{A non-idempotent interior operator}
Given $S\subset \Z^2$, define
\[
\Int(S)=\{(x,y)\in \Z^2\mid N_{(x,y)}\subset S\}.
\]

Interpretation:
\begin{itemize}
  \item if a state is known on $S$, then one Game-of-Life step is defined on $\Int(S)$;
  \item thus $\Int(S)$ is the part of $S$ where the local rule can be evaluated.
\end{itemize}

\vspace{0.3em}
\centering
\begin{tikzpicture}[scale=0.38]
  \foreach \x in {0,...,8} {\foreach \y in {0,...,8} {\draw[gray!50] (\x,\y) rectangle ++(1,1);}}
  \fill[blue!20] (1,1) rectangle (8,8);
  \fill[blue!55] (2,2) rectangle (7,7);
  \fill[blue!85] (3,3) rectangle (6,6);
  \node at (9.7,6.6) {$S$};
  \node at (10.4,4.7) {$\Int(S)$};
  \node at (10.8,3.0) {$\Int^2(S)$};
\end{tikzpicture}

\vspace{0.3em}
Crucially, in general $\Int^2\neq\Int$.
\end{frame}

\begin{frame}{Pretopological spaces}
\begin{definition}
A \demph{pretopology} on a set $X$ is a map
\[
\Int\colon \Pow(X)\to\Pow(X)
\]
such that
\begin{itemize}
  \item $\Int$ preserves finite meets:
  \[
  \Int(S\cap T)=\Int(S)\cap\Int(T), \qquad \Int(X)=X,
  \]
  \item and $\Int(S)\subset S$ for every $S\subset X$.
\end{itemize}
\end{definition}

A topological space is the special case where $\Int$ is idempotent.

\vspace{0.4em}
\begin{alertblock}{Slogan}
Pretopological spaces are a \demph{dynamic version of topological spaces}.
\end{alertblock}
\end{frame}

\begin{frame}{The pretopology behind the Game of Life}
The lattice $\Z^2$ comes with the graph relation given by the eight neighboring directions.

\vspace{0.4em}
\centering
\begin{tikzpicture}[scale=0.9]
\foreach \x in {-1,0,1,2} {
    \foreach \y in {-1,0,1,2} {
        \fill (\x, \y) circle (2.2pt);
        \foreach \dx/\dy in {1/0,0/1,-1/0,0/-1,1/1,1/-1,-1/1,-1/-1} {
            \draw[gray!70] (\x,\y) -- ++(\dx,\dy);
        }
    }
}
\node[anchor=east] at (-1.7,0.5) {$\cdots$};
\node[anchor=west] at (2.7,0.5) {$\cdots$};
\node[anchor=north] at (0.5,-1.7) {$\vdots$};
\node[anchor=south] at (0.5,2.7) {$\vdots$};
\end{tikzpicture}

\vspace{0.5em}
For this graph, the operator $\Int(S)$ precisely selects those vertices whose whole neighborhood stays inside $S$.
\end{frame}

\begin{frame}{Dynamical presheaves}
\begin{definition}
A \demph{dynamical presheaf} on a pretopological space $(X,\Int)$ is a presheaf
\[
F\colon \Pow(X)^{\mathrm{op}}\to\Set
\]
with maps
\[
\sigma_S\colon F(S)\to F(\Int(S))
\]
compatible with restrictions.
\end{definition}

\vspace{0.5em}
That is, for $T\subset S$, the square
\[
\begin{array}{ccc}
F(S) & \xrightarrow{\ \sigma_S\ } & F(\Int(S)) \\
\downarrow & & \downarrow \\
F(T) & \xrightarrow{\ \sigma_T\ } & F(\Int(T))
\end{array}
\]
commutes.

\vspace{0.5em}
So a dynamical presheaf is an ordinary presheaf together with one-step evolution.
\end{frame}

\begin{frame}{The Game of Life as a dynamical presheaf}
Define $\GoL$ on $(\Z^2,\Int)$ by
\[
\GoL(S)=\Pow(S).
\]

Restriction maps are ordinary restrictions of functions $S\to\2$.

\vspace{0.5em}
The evolution map is the restricted Conway map
\[
\con\colon \Pow(S)\to\Pow(\Int(S)).
\]

\begin{block}{Meaning}
From a configuration known on $S$, the local rule determines the next configuration on the smaller region $\Int(S)$.
\end{block}

Hence Conway's Game of Life is naturally an object
\[
\GoL\in \dPSh(\Z^2,\Int).
\]
\end{frame}

\begin{frame}{A category encoding space and time}
The pretopological space $(X,\Int)$ determines a category
\[
\Pow(X)\rtimes_{\Int} \N.
\]

\begin{block}{Objects}
Subsets $U\subset X$.
\end{block}

\begin{block}{Morphisms}
A morphism $U\to V$ is a number $n\in\N$ such that
\[
U\subset \Int^n(V).
\]
\end{block}

\begin{block}{Composition}
\[
(U\xrightarrow{n}V,\; V\xrightarrow{m}W)\mapsto U\xrightarrow{n+m}W.
\]
\end{block}

This is the semidirect-product category of \demph{space} (subsets) and \demph{time} ($\N$).
\end{frame}

\begin{frame}{Main categorical identification}
\begin{theorem}
For any pretopological space $(X,\Int)$,
\[
\dPSh(X,\Int)\simeq \PSh\bigl(\Pow(X)\rtimes_{\Int}\N\bigr).
\]
\end{theorem}

\vspace{0.5em}
So the category of dynamical presheaves is just a presheaf topos.

\vspace{0.6em}
\begin{alertblock}{Consequence}
\[
\dPSh(\Z^2,\Int)
\]
is a topos containing the object $\GoL$.
\end{alertblock}
\end{frame}

\begin{frame}{Internal viewpoint over $\dSet$}
Because $\Int\colon \Pow(X)\to\Pow(X)$ is order-preserving, $(\Pow(X),\Int)$ forms an internal poset in $\dSet$.

\vspace{0.6em}
Then one has another description:
\[
\dPSh(X,\Int)\simeq \PSh_{\dSet}(\Pow(X),\Int).
\]

\begin{block}{Interpretation}
A dynamical presheaf is exactly an \demph{internal presheaf} on the internal poset $\Pow(X)$ inside the topos of discrete dynamical systems.
\end{block}

Therefore $\dPSh(X,\Int)$ is naturally a \demph{relative topos over $\dSet$}.
\end{frame}

\begin{frame}{The geometric morphism to $\dSet$}
Let
\[
\pi\colon \Pow(X)\rtimes_{\Int}\N \to \N
\]
be the projection.

This induces a geometric morphism
\[
\gamma\colon \dPSh(X,\Int)\to \dSet.
\]

\vspace{0.4em}
In fact one gets an adjoint quintuple; in particular the central part is
\[
\gamma_! = \mathrm{ev}_{\emptyset}
\;\dashv\;
\gamma^*
\;\dashv\;
\gamma_* = \mathrm{ev}_{X}.
\]
The further outer adjoints come from the embeddings of $\emptyset$ and $X$.

\vspace{0.5em}
For the Game of Life object,
\[
\gamma_*(\GoL)=\GoL(X)=\Pow(X)
\]
with evolution map exactly the usual Conway map.
\end{frame}

\begin{frame}{Conclusion}
\begin{alertblock}{Answer proposed in the note}
The Game of Life lives naturally in the relative topos
\[
\gamma\colon \dPSh(\Z^2,\Int)\to \dSet,
\]
where its global sections recover
\[
\con\colon \Pow(\Z^2)\to\Pow(\Z^2).
\]
\end{alertblock}

\vspace{0.5em}
What this captures:
\begin{itemize}
  \item locality of the rule,
  \item global time evolution,
  \item a single object combining space and dynamics.
\end{itemize}
\end{frame}

\begin{frame}{What is still missing?}
This construction is only a first step.

\begin{enumerate}
  \item \textbf{Subtopos / sheaf condition:} find a more intrinsic Grothendieck topology so that finite regions are dense and $\GoL$ becomes a genuine sheaf.
  \item \textbf{Quotients by symmetry:} account for the action of $\Z^2\rtimes D_4$.
  \item \textbf{Modal logic:} clarify the logic encoded by pretopology.
  \item \textbf{Continuous-time analogies:} replace $\N$ by $\mathbb R_{\ge 0}$.
  \item \textbf{Topoi of systems:} understand points and observations of such system topoi.
\end{enumerate}
\end{frame}

\begin{frame}{Takeaway}
\begin{center}
\Large
Topological spaces are static.\\[0.4em]
Pretopological spaces carry a one-step evolution.\\[0.6em]
\normalsize
This makes them a natural bridge from Conway's local rule\\
 to a relative topos over discrete time.
\end{center}
\end{frame}

\begin{frame}{References}
\footnotesize
\begin{thebibliography}{9}
\bibitem{johnstone}
P. T. Johnstone,
\emph{Sketches of an Elephant}, Vol. 2.

\bibitem{maclane}
S. Mac Lane and I. Moerdijk,
\emph{Sheaves in Geometry and Logic}.

\bibitem{tomasic}
I. Toma\v si\'c,
\emph{Topos Theory for Discrete-Time Dynamical Systems}.

\bibitem{awodey}
S. Awodey,
\emph{Topological semantics for modal logic} (background direction).

\bibitem{grothendieck}
A. Grothendieck,
\emph{R\'ecoltes et Semailles}.
\end{thebibliography}
\end{frame}

\begin{frame}
\centering
\Huge Thank you
\end{frame}

\end{document}