\begin{filecontents*}{localrefs.bib}
@article{joyal1977remarques,
  author  = {Joyal, Andre},
  title   = {Remarques sur la theorie des jeux a deux personnes},
  journal = {Gazette des sciences mathematiques du Quebec},
  year    = {1977}
}
@article{laird2013constructing,
  author  = {Laird, James and Manzonetto, Giulio and McCusker, Guy},
  title   = {Constructing categories and models of game semantics},
  journal = {Mathematical Structures in Computer Science},
  year    = {2013}
}
@article{honsell2009conway,
  author  = {Honsell, Furio and Lenisa, Marina},
  title   = {Conway games and hypergames},
  journal = {Electronic Notes in Theoretical Computer Science},
  year    = {2009}
}
@article{honsell2011conway,
  author  = {Honsell, Furio and Lenisa, Marina},
  title   = {Conway Games, algebraically and coalgebraically},
  journal = {Preprint},
  year    = {2011}
}
@article{bavsic2024categories,
  author  = {Bavsic, Boro and others},
  title   = {Categories of impartial rulegraphs and gamegraphs},
  journal = {Preprint},
  year    = {2024}
}
@article{blute2006differential,
  author  = {Blute, Richard and Cockett, Robin and Seely, Robert},
  title   = {Differential categories},
  journal = {Mathematical Structures in Computer Science},
  year    = {2006}
}
@article{cockett2019integral,
  author  = {Cockett, Robin and Lemay, Jean-Simon Pacaud},
  title   = {Integral categories and calculus categories},
  journal = {Mathematical Structures in Computer Science},
  year    = {2019}
}
@article{loregian2021differential,
  author  = {Loregian, Fosco and Trimble, Todd},
  title   = {Differential 2-rigs},
  journal = {Preprint},
  year    = {2021}
}
@article{bouton1901nim,
  author  = {Bouton, Charles L.},
  title   = {Nim, a game with a complete mathematical theory},
  journal = {Annals of Mathematics},
  year    = {1901}
}
@book{siegel2013combinatorial,
  author    = {Siegel, Aaron N.},
  title     = {Combinatorial Game Theory},
  publisher = {American Mathematical Society},
  year      = {2013}
}
@article{joyal1981theorie,
  author  = {Joyal, Andre},
  title   = {Une theorie combinatoire des series formelles},
  journal = {Advances in Mathematics},
  year    = {1981}
}
\end{filecontents*}

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\title[A Rota--Baxter equation for winning games]{A Rota--Baxter Equation for Winning Games}
\subtitle{Games as recursive coalgebras and differential invariants}
\author[R. Hora]{Ryuya Hora}
\institute{Assistant professor at ZEN university}
\date{Differentiation in category theory and program semantics\\Kyoto University, April 6, 2026}

%AI suggestion: If this is for the actual 2026 talk, double-check the institute line on the title slide.
%AI suggestion: In a short talk, the short-survey slide can move to the appendix and you can start directly from the Nim phenomenon.
%AI suggestion: If you need one extra minute for the main theorem, merge the universal-example slide with the Nim-sum rig slide.

\begin{document}

\begin{frame}
  \titlepage
  \vspace{-0.4em}
  \begin{center}
    \small Partially based on joint work in progress with Ryo Suzuki.
  \end{center}
  \begin{center}
    \small Keywords: \icol{Rota--Baxter equation}, combinatorial games, recursive \dcol{coalgebra}, \dcol{differential} $2$-rig
  \end{center}
\end{frame}

\section{Front matter}

\begin{frame}{Where I come from}
\Large
I've been interested in toposes, \textbf{\dcol{games}, and \dcol{coalgebras}}.\\[0.7em]

\begin{figure}
\centering
\begin{tikzpicture}
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  \node[anchor=west, font=\normalsize] at ($(search.west)+(0.22,0)$) {recusion};
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  \node[anchor=west, text=mygray, font=\small] at (-3.82,-1.05) {So I kept the joke and moved on.};
\end{tikzpicture}
\end{figure}
\end{frame}

\begin{frame}{Where this talk comes from}
\small
\begin{columns}[T,totalwidth=\textwidth]
\begin{column}{0.56\textwidth}
\begin{itemize}
  \item My motivation comes from \textbf{Combinatorial Game Theory}.
  \item I am \textbf{not} a specialist in \dcol{differential} categories in the usual sense.
  \item So this talk is partly a proposal about how game recursion should be organized categorically.
\end{itemize}
\end{column}
\begin{column}{0.40\textwidth}
\begin{proposition}[What I hope to learn here]
\begin{itemize}
  \item the right map of prior work,
  \item links to \dcol{differential} / Cartesian differential categories,
  \item links to game semantics and linear logic,
  \item where the \icol{Rota--Baxter} viewpoint fits.
\end{itemize}
\end{proposition}
\end{column}
\end{columns}

\vspace{0.3em}
\begin{center}
\emph{I am coming mainly from the game side, but trying to speak to this audience.}
\end{center}
\end{frame}

\begin{frame}{The dichotomy in this talk}
\begin{figure}
\centering
\begin{tikzpicture}[x=1cm,y=1cm,>=Latex,thick]
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  \node[text=mygray, font=\bfseries] at (0,2.05) {vs};
  \node[text=icol, font=\bfseries] at ( 2,2.05) {Integration};

  \node[text=dcol, font=\bfseries, align=center] (L1) at (-4.9,  1.20) {Play forward};
  \node[text=dcol, font=\bfseries, align=center] (L2) at (-4.9,  0.00) {Coalgebra};
  \node[text=dcol, font=\bfseries, align=center] (L3) at (-4.9, -1.20) {Leibniz rule};

  \node[text=icol, font=\bfseries, align=center] (R1) at ( 4.9,  1.20) {Analyze backward};
  \node[text=icol, font=\bfseries, align=center] (R2) at ( 4.9,  0.00) {Algebra};
  \node[text=icol, font=\bfseries, align=center] (R3) at ( 4.9, -1.20) {Rota--Baxter\\equation};

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  \draw[->, line width=1.5pt, draw=icol] ( 2,-1.95) -- ( 2,1.55);
\end{tikzpicture}
\end{figure}
\end{frame}

%AI suggestion: If you prefer a more aggressive opening, move this survey slide after the first section instead of before it.
\begin{frame}{Short survey: games $\times$ category theory is not new}
\footnotesize
\begin{proposition}[Important positioning]
I know that \textbf{games + category theory} is already a rich story.
This talk isolates the narrower interface between
\[
\text{impartial combinatorial games}
\qquad\text{and}\qquad
\text{\dcol{differential} / \icol{Rota--Baxter} ideas}.
\]
\end{proposition}

\begin{columns}[T,totalwidth=\textwidth]
\begin{column}{0.32\textwidth}
\begin{example}[Game semantics]
\begin{itemize}
  \item strategies as morphisms,
  \item linear logic / programming semantics.
\end{itemize}
\end{example}
\end{column}
\begin{column}{0.32\textwidth}
\begin{example}[\dcol{Coalgebraic} games]
\begin{itemize}
  \item Conway games / hypergames,
  \item recursive viewpoints on impartial games.
\end{itemize}
\end{example}
\end{column}
\begin{column}{0.32\textwidth}
\begin{example}[\icol{Differential} side]
\begin{itemize}
  \item differential categories,
  \item integral / calculus categories.
\end{itemize}
\end{example}
\end{column}
\end{columns}

\vspace{0.15em}
{\scriptsize Representative references: game semantics \parencite{joyal1977remarques,laird2013constructing}; coalgebraic games \parencite{honsell2009conway,honsell2011conway,bavsic2024categories}; differential side \parencite{blute2006differential,cockett2019integral,loregian2021differential}.}
\end{frame}

\section[Play forward $\leftrightarrow$ Analyze backward]{Play vs Analysis: Generalized Bouton's theorem is \texorpdfstring{\icol{Integration}}{Integration}!}
\subsection{Preliminaries: Winning Nim!}

\begin{frame}{Rule of $n$-heap Nim}
\small
\begin{itemize}
  \item $n$ heaps of stones are given.
  \item Two players take turns choosing one heap and removing at least one stone from that heap.
  \item The player who is unable to take a stone loses.
\end{itemize}

\begin{figure}
\centering
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\end{figure}
\end{frame}

\begin{frame}[t]{Bouton's winning strategy (1/2) Nim sum}
\footnotesize
\begin{columns}[T,totalwidth=\textwidth]
\begin{column}{0.58\textwidth}
\begin{definition}[Nim-sum]
The \emph{Nim-sum} $\nimsum$ is bit-wise xor, i.e. the abelian group structure on $\N$ induced by the binary expansion
\[
\N \cong \bigoplus_{k=0}^{\infty} \Z/2\Z.
\]
\end{definition}
\begin{example}
$5\nimsum 7 = \left(101\right)_2 \nimsum \left(111\right)_2 = \left(010\right)_2 = 2$.
\end{example}
\end{column}
\begin{column}{0.38\textwidth}
\begin{figure}
\centering
\begin{tikzpicture}[thick,>=Latex,scale=0.85]
  \node at (-1.0,0.65) {$3=$};
  \node at (-0.2,0.65) {$0$};
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  \node at ( 1.0,0.65) {$1$};
  \node at (-1.25,0.0) {$\nimsum$};
  \node at (-1.0,-0.65) {$5=$};
  \node at (-0.2,-0.65) {$1$};
  \node at ( 0.4,-0.65) {$0$};
  \node at ( 1.0,-0.65) {$1$};
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  \node at (-0.2,-1.7) {$1$};
  \node at ( 0.4,-1.7) {$1$};
  \node at ( 1.0,-1.7) {$0$};
\end{tikzpicture}
\end{figure}
\end{column}
\end{columns}
\end{frame}

\begin{frame}{Bouton's winning strategy (2/2)}
\small
\begin{theorem}[Bouton, 1901]
A state $\left(a_1,\dots,a_n\right)$ of $n$-heap Nim is winning for the previous player iff
\[
a_1\nimsum\cdots\nimsum a_n = 0.
\]
\end{theorem}

\begin{proposition}[Phenomenon]
The only non-trivial part of the proof is the identity
\[
\mex\left(S\right)\nimsum \mex\left(T\right)
= \mex\left(\left(\mex\left(S\right)\nimsum T\right)\cup \left(S\nimsum \mex\left(T\right)\right)\right).
\]
This already looks like an \icol{integration law}, namely a \icol{Rota--Baxter equation}.
\end{proposition}
\end{frame}

\subsection{Impartial Games}

\begin{frame}{Definition of games}
\small
\begin{columns}[T,totalwidth=\textwidth]
\begin{column}{0.78\textwidth}
\begin{definition}[(Impartial) game]
A game $\mathbb{X}=\left(X,\rel\right)$ is a pair of a (possibly infinite) set $X$ and a binary relation $\rel \subset X\times X$ satisfying:
\begin{enumerate}
  \item \emph{finite options}: $\#\left\{x'\in X\mid x\rel x'\right\}$ is finite for every $x\in X$;
  \item \emph{finite time}: there is no infinite path $x_0\rel x_1\rel x_2\rel \cdots$.
\end{enumerate}
\end{definition}

\begin{example}[$\Nim{n}$: $n$-heap Nim]
The game $\Nim{n}=\left(\N^n,\rel\right)$ is defined by
\[
\left(a_i\right)_{1\leq i\leq n} \rel \left(b_i\right)_{1\leq i\leq n}
\iff \exists i\, \left(a_i>b_i \land a_j=b_j\ \left(j\neq i\right)\right).
\]
\end{example}
\end{column}
\begin{column}{0.18\textwidth}
\begin{figure}
\centering
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\end{figure}
\end{column}
\end{columns}
\end{frame}

\begin{frame}{$\times$: Box product of games}
\small
\begin{definition}[Box product {\tiny a.k.a. Conway's addition}]
The box product of two games $\mathbb{X}=\left(X,\rel\right)$ and $\mathbb{Y}=\left(Y,\rel\right)$ is the game $\mathbb{X}\ConAdd\mathbb{Y}=\left(X\times Y,\rel_{\ConAdd}\right)$ with
\[
\left(x,y\right)\rel_{\ConAdd}\left(x',y'\right)
\iff \left(x\rel x' \land y=y'\right) \lor \left(x=x' \land y\rel y'\right).
\]
\end{definition}

\begin{figure}
\centering
\begin{tikzpicture}[>=Latex, scale=0.42]
  \slantedgrid{3}{0}{-8}{0}{X}
  \node at (-7,-4) {$\mathbb{X}$};
  \node at (-4.3,-1) {$\ConAdd$};
  \slantedgrid{0}{4}{0}{0.5}{Y}
  \node at (-1.5,-4) {$\mathbb{Y}$};
  \node at (2.3,-1) {$=$};
  \slantedgrid{3}{4}{7}{2}{Z}
  \node at (6.5,-4) {$\mathbb{X}\ConAdd\mathbb{Y}$};
\end{tikzpicture}
\caption{An example of box product.}
\label{fig:boxproduct}
\end{figure}
\end{frame}

\begin{frame}{$\iop$: \icol{Mex} and Grundy number}
\footnotesize
\begin{columns}[T,totalwidth=\textwidth]
\begin{column}{0.62\textwidth}
\begin{definition}[mex]
For a finite subset $S\subset \N$, its mex is
\[
\mex\left(S\right)=\min\left(\N\setminus S\right).
\]
\end{definition}

\begin{definition}[Grundy number]
For a game $\mathbb{X}=\left(X,\rel\right)$ and a state $x\in X$, its Grundy number is recursively defined by
\[
\G{\mathbb{X}}{x}=\mex\left(\left\{\G{\mathbb{X}}{x'}\mid x\rel x'\right\}\right).
\]
\end{definition}

\begin{proposition}[Grundy number is enough to win]
A state $x$ is winning iff $\G{\mathbb{X}}{x}=0$.
\end{proposition}
\end{column}
\begin{column}{0.34\textwidth}
\begin{figure}
\centering
\begin{tikzpicture}[scale=0.78]
  \node[state,label=above:$x$] (x) at (0,1.8) {};
  \node[state,label=left:$x_1$] (x1) at (-1.45,0.85) {};
  \node[state,label=above:$x_2$] (x2) at (0,0.85) {};
  \node[state,label=right:$x_3$] (x3) at (1.45,0.85) {};
  \node[pstate,label=left:$0$] (z1) at (-1.8,-0.7) {};
  \node[nstate,label=below:$1$] (z2) at (-0.7,-0.7) {};
  \node[pstate,label=below:$0$] (z3) at (0.7,-0.7) {};
  \node[nstate,label=right:$2$] (z4) at (1.9,-0.7) {};
  \draw[gameedge] (x) -- (x1);
  \draw[gameedge] (x) -- (x2);
  \draw[gameedge] (x) -- (x3);
  \draw[gameedge] (x1) -- (z1);
  \draw[gameedge] (x1) -- (z2);
  \draw[gameedge] (x2) -- (z2);
  \draw[gameedge] (x2) -- (z3);
  \draw[gameedge] (x3) -- (z3);
  \draw[gameedge] (x3) -- (z4);
\end{tikzpicture}
\caption{Recursive calculation from terminal positions.}
\end{figure}
\end{column}
\end{columns}
\end{frame}

\begin{frame}{Generalized Bouton's theorem}
\small
\begin{theorem}[Generalized Bouton's theorem]
For two games $\mathbb{X}=\left(X,\rel\right)$ and $\mathbb{Y}=\left(Y,\rel\right)$, we have
\[
\G{\mathbb{X}\ConAdd\mathbb{Y}}{\left(x,y\right)}
=
\G{\mathbb{X}}{x}\nimsum \G{\mathbb{Y}}{y}.
\]
\end{theorem}

\begin{proof}[The only non-trivial part of the proof]
\[
\mex\left(S\right)\nimsum \mex\left(T\right)
= \mex\left(\left(\mex\left(S\right)\nimsum T\right)\cup \left(S\nimsum \mex\left(T\right)\right)\right).
\]
\end{proof}
This is already strikingly similar to the \icol{Rota--Baxter equation}
\[
\left(\Int f\right)\left(\Int g\right)
= \Int\left(\left(\Int f\right)g + f\left(\Int g\right)\right).
\]
\end{frame}

\section[Coalgebras $\leftrightarrow$ Algebras]{Coalgebra vs Algebra: A locally finitely presentable category of games = recursive \texorpdfstring{\dcol{$\Pf$-coalgebras}}{Pfin-coalgebras}}
\subsection{Preliminaries: Coalgebra and recursion}

\begin{frame}{Algebra/Coalgebra of an endofunctor}
\small
\begin{definition}[$T$-algebras and $T$-coalgebras]
For a category $\mathcal{C}$ and an endofunctor $T\colon \mathcal{C}\to \mathcal{C}$:
\begin{itemize}
  \item A \icol{$T$-algebra} is a pair $\left(A,\alpha\right)$ with a morphism $\alpha\colon TA \mathrel{\icol{\to}} A$.
  \item A \dcol{$T$-coalgebra} is a pair $\left(X,\theta\right)$ with a morphism $\theta\colon X \rel TX$.
\end{itemize}
\end{definition}

\begin{example}
In this talk, the key case is $\mathcal{C}=\Set$ and $T=\Pf$, where
\[
\Pf\left(X\right)=\left\{S\subset X\mid \#S<\infty\right\}.
\]
\end{example}
\end{frame}

\begin{frame}{Coalgebra-Algebra morphism and Recursive coalgebra}
\footnotesize
\begin{definition}[Coalgebra-algebra morphism]
A coalgebra-algebra morphism from a \dcol{$T$-coalgebra} $\left(X,\theta\right)$ to an \icol{$T$-algebra} $\left(A,\alpha\right)$ is a morphism $f\colon X\to A$ such that
\[
\begin{tikzcd}[ampersand replacement=\&, row sep=large, column sep=large]
X \arrow[r,"f"] \arrow[d,"{\theta}"'] \& A \\
TX \arrow[r,"{Tf}"'] \& TA \arrow[u,"{\alpha}"']
\end{tikzcd}
\]
commutes.
\end{definition}

\begin{definition}[Recursive coalgebra]
A \dcol{$T$-coalgebra} $\left(X,\theta\right)$ is \emph{recursive} if, for every \icol{$T$-algebra} $\left(A,\alpha\right)$, there exists a unique coalgebra-algebra morphism $\left(X,\theta\right)\to \left(A,\alpha\right)$.
\end{definition}
\end{frame}

\subsection{Games as recursive coalgebras}

\begin{frame}{Games = Recursive $\Pf$-coalgebras}
\small
\begin{theorem}[Games as recursive coalgebras]
The category of games is equivalent to the category of recursive \dcol{$\Pf$-coalgebras}:
\[
\Gs \simeq \RecCoalg{\Pf}.
\]
\end{theorem}

\begin{figure}
\centering
\[
\begin{tikzcd}[ampersand replacement=\&, row sep=large, column sep=huge]
\text{game } \left(X,\rel\right) \arrow[r, rightsquigarrow] \arrow[d, mapsto] 
  \& \text{recursive \dcol{$\Pf$-coalgebra}} \arrow[d, mapsto] \\
\theta_X\colon X \rel \Pf\left(X\right),
\quad x \mapsto \left\{x'\mid x\rel x'\right\}
  \arrow[r, rightsquigarrow]
  \& \text{backward recursion against the flow of play}
\end{tikzcd}
\]
\end{figure}

\begin{proposition}[Two examples]
\begin{itemize}
  \item One-heap Nim corresponds to $\nu\colon \N\to \Pf\left(\N\right)$, $\nu\left(n\right)=\left\{0,\dots,n-1\right\}$.
  \item The terminal game is the game of hereditarily finite sets, with $A\rel B$ iff $B\in A$.
\end{itemize}
\end{proposition}
\end{frame}

%AI suggestion: If the audience is less familiar with locally presentable categories, move this slide after the main theorem or send it to the appendix.
\begin{frame}{Digression: Categorical structure of games}
\small
\begin{proposition}[The category $\Gs$ behaves very well]
\begin{itemize}
  \item The category $\Gs$ is \textbf{locally finitely presentable}.
  \item In particular, $\Gs$ is complete and cocomplete.
  \item The box product $\ConAdd$ is a symmetric monoidal closed structure on $\Gs$.
\end{itemize}
\end{proposition}

\begin{proposition}[Game values are hylomorphisms]
Typical game values arise from \icol{$\Pf$-algebras}:
\[
\text{outcome } \oc,
\qquad
\text{Grundy number } \G{\mathbb{X}}{x},
\qquad
\text{birthday } \BirthDay.
\]
So a \dcol{coalgebraic} presentation of games naturally packages their recursive \icol{algebraic} values.
\end{proposition}
\end{frame}

\section[Differentiation $\leftrightarrow$ Integration]{Differentiation vs Integration: Calculus 2-rig of pointed game families}
\subsection{Differential $2$-rig of games}

\begin{frame}{Differentiation on families of pointed games}
\footnotesize
\begin{columns}[T,totalwidth=\textwidth]
\begin{column}{0.50\textwidth}
\begin{itemize}
  \item A pointed game is a pair $\left(X,x\right)$ with a chosen starting position.
  \item Let $\Gsp$ be the category of pointed games.
  \item Let $\Fam\left(\Gsp\right)$ be the free finite-coproduct completion.
\end{itemize}

Think of an object of $\Fam\left(\Gsp\right)$ as a finite family of local game situations.

\vspace{0.3em}
\begin{definition}[\dcol{Differential} operator]
For a pointed game $\left(X,x\right)$, define
\[
\dd\left(X,x\right)\coloneqq \left\{\left(X,x'\right)\right\}_{x\rel x'}.
\]
\end{definition}
\end{column}
\begin{column}{0.46\textwidth}
\begin{figure}
\centering
\begin{tikzpicture}[scale=0.84]
  \node[bluepanel, minimum width=2.5cm, minimum height=0.75cm] (rootbox) at (0,2.05) {$\left(X,x\right)$};
  \node[state] (x) at (0,1.15) {};
  \node[state] (x1) at (-1.2,0.15) {};
  \node[state] (x2) at (0,-0.05) {};
  \node[state] (x3) at (1.2,0.15) {};
  \draw[gameedge] (x) -- (x1);
  \draw[gameedge] (x) -- (x2);
  \draw[gameedge] (x) -- (x3);
  \node[greenpanel, fit=(x1) (x2) (x3), inner sep=7pt] (fam) {};
  \node[greenpanel, minimum width=3.1cm, minimum height=0.75cm] (d) at (0,-1.15) {$\dd\left(X,x\right)$};
  \draw[formulaarrow] (rootbox) -- (d);
  \draw[formulaarrow] (fam.south) -- (d.north);
\end{tikzpicture}
\end{figure}
\end{column}
\end{columns}
\end{frame}

\begin{frame}{Leibniz rule for Box product}
\small
For the box product, every move changes \emph{either} the left component or the right component.

\vspace{0.2em}
\begin{theorem}
In $\Fam\left(\Gsp\right)$ we have a categorified \dcol{Leibniz rule}:
\[
\dd\left(X\ConAdd Y\right)
\cong
\left(\dd X\right)\ConAdd Y
\;\sqcup\;
X\ConAdd \left(\dd Y\right).
\]
\end{theorem}

\begin{figure}
\centering
\begin{tikzpicture}[scale=0.82]
  \node[bluepanel, minimum width=2.6cm, minimum height=0.8cm] (xy) at (0,0.05) {$X\ConAdd Y$};
  \node[bluepanel, minimum width=2.7cm, minimum height=0.8cm] (dx) at (-3.05,-1.7) {$\left(\dd X\right)\ConAdd Y$};
  \node[bluepanel, minimum width=2.7cm, minimum height=0.8cm] (dy) at (3.05,-1.7) {$X\ConAdd\left(\dd Y\right)$};
  \draw[formulaarrow] (xy) -- (dx) node[midway,left] {move in $X$};
  \draw[formulaarrow] (xy) -- (dy) node[midway,right] {move in $Y$};
\end{tikzpicture}
\end{figure}

{\scriptsize This is the differential structure emphasized here; compare with differential 2-rigs \parencite{joyal1981theorie,loregian2021differential}.}
\end{frame}

\subsection{Invariants in Rota-Baxter rig}

\begin{frame}[t]{Decategorification target: calculus / Rota--Baxter rigs}
\scriptsize
\begin{definition}[\dcol{Differential} rig]
A rig $\left(A,0,1,+,\times,\dd\right)$ with
\[
\dd\left(a+b\right)=\dd a+\dd b,
\qquad
\dd\left(ab\right)=\left(\dd a\right)b + a\left(\dd b\right).
\]
\end{definition}

\begin{definition}[\icol{Integral} rig]
A rig with an operator $\Int$ satisfying
\[
1 = \Int 0,
\qquad
\left(\Int f\right)\left(\Int g\right)
= \Int\left(\left(\Int f\right)g + f\left(\Int g\right)\right).
\]
\end{definition}

\begin{definition}[Calculus rig]
A rig carrying both structures, with the fundamental theorem
\[
\dd\,\Int f = f.
\]
\end{definition}

\begin{center}
{\footnotesize\emph{Game families can be sent to such algebraic \dcol{differential} / \icol{integral} structures.}}
\end{center}
\end{frame}

\begin{frame}[t]{Rota--Baxter valued invariants of games}
\small
\begin{columns}[T,totalwidth=\textwidth]
\begin{column}{0.56\textwidth}
Let $A$ be an \icol{integral} rig. For a pointed game $\left(X,x\right)$ define recursively
\[
F_{\left(X,x\right)}\coloneqq \Int\!\left(\sum_{x\rel x'} F_{\left(X,x'\right)}\right).
\]
For a finite family, define $F$ by finite sums.

\vspace{0.25em}
\begin{theorem}
This assignment preserves the rig operations:
\[
F_{\mathcal{X}\sqcup\mathcal{Y}} = F_{\mathcal{X}} + F_{\mathcal{Y}},
\qquad
F_{\mathcal{X}\ConAdd\mathcal{Y}} = F_{\mathcal{X}}\times F_{\mathcal{Y}}.
\]
\end{theorem}
\end{column}
\begin{column}{0.40\textwidth}
\begin{figure}
\centering
\begin{tikzpicture}[scale=0.86]
  \node[bluepanel, minimum width=2.7cm, minimum height=0.75cm] (fx) at (0,2.1) {$F_{\left(X,x\right)}$};
  \node[greenpanel, minimum width=1.25cm, minimum height=0.68cm] (int) at (0,1.05) {$\Int$};
  \node[bluepanel, minimum width=3.2cm, minimum height=0.78cm] (sum) at (0,-1.95) {$\sum_{x\rel x'} F_{\left(X,x'\right)}$};
  \node[state] (x) at (0,-0.02) {};
  \node[state] (x1) at (-1.15,-0.95) {};
  \node[state] (x2) at (0,-1.12) {};
  \node[state] (x3) at (1.15,-0.95) {};
  \draw[gameedge] (x) -- (x1);
  \draw[gameedge] (x) -- (x2);
  \draw[gameedge] (x) -- (x3);
  \draw[formulaarrow] (sum) -- (int);
  \draw[formulaarrow] (int) -- (fx);
\end{tikzpicture}
\end{figure}
\end{column}
\end{columns}

{\scriptsize Why multiplication works: apply the \icol{Rota--Baxter} identity to the \dcol{Leibniz} rule for options of a box product.}
\end{frame}

\subsection{Examples}

\begin{frame}[t]{The universal example: game families themselves}
\footnotesize
The family of all pointed positions in the terminal game carries a calculus-rig-like structure:

\begin{center}
\renewcommand{\arraystretch}{1.08}
\begin{tabular}{@{}ll@{}}
\toprule
\textbf{\icol{analysis}} & \textbf{\dcol{game families}} \\
\midrule
$0$ & empty family \\
$1$ & the terminal pointed game \\
$+$ & disjoint union / union \\
$\times$ & gamewise box product \\
$\dd$ & take all immediate options \\
$\Int$ & adjoin a new root / braces $A\mapsto \left\{A\right\}$ \\
\bottomrule
\end{tabular}
\end{center}

\begin{figure}
\centering
\begin{tikzpicture}[scale=0.68]
  \node[bluepanel, minimum width=1.2cm, minimum height=0.62cm] (A) at (-2.8,0) {$A$};
  \node[greenpanel, minimum width=1.4cm, minimum height=0.62cm] (IA) at (0,0) {$\left\{A\right\}$};
  \node[bluepanel, minimum width=1.7cm, minimum height=0.62cm] (dA) at (2.8,0) {$\dd A$};
  \draw[formulaarrow] (A) -- node[above] {$\Int$} (IA);
  \draw[formulaarrow] (A) -- node[above] {$\dd$} (dA);
  \node[state] (a1) at (2.45,-0.82) {};
  \node[state] (a2) at (3.12,-0.82) {};
  \draw[gameedge] (dA.south) -- (a1);
  \draw[gameedge] (dA.south) -- (a2);
\end{tikzpicture}
\end{figure}
\end{frame}

\begin{frame}[t]{Nim-sum gives a concrete Rota--Baxter rig}
\footnotesize
\begin{columns}[T,totalwidth=\textwidth]
\begin{column}{0.58\textwidth}
Start from the commutative monoid $\left(\N,\nimsum,0\right)$.
Its free idempotent rig is $\Pf\left(\N\right)$ with
\[
S+T \coloneqq S\cup T,
\qquad
S\times T \coloneqq \left\{s\nimsum t\mid s\in S,\ t\in T\right\}.
\]
Define the \icol{integral} operator by
\[
\Int\left(S\right)\coloneqq \left\{\mex\left(S\right)\right\}.
\]

\begin{theorem}
This makes $\Pf\left(\N\right)$ into an \icol{integral} (\icol{Rota--Baxter}) rig.
\end{theorem}
\end{column}
\begin{column}{0.38\textwidth}
\begin{figure}
\centering
\begin{tikzpicture}[thick,>=Latex,scale=0.85]
  \node at (-1.0,0.65) {$3=$};
  \node at (-0.2,0.65) {$0$};
  \node at ( 0.4,0.65) {$1$};
  \node at ( 1.0,0.65) {$1$};
  \node at (-1.25,0.0) {$\nimsum$};
  \node at (-1.0,-0.65) {$5=$};
  \node at (-0.2,-0.65) {$1$};
  \node at ( 0.4,-0.65) {$0$};
  \node at ( 1.0,-0.65) {$1$};
  \draw[very thick] (-1.2,-1.05) -- (1.3,-1.05);
  \node at (-1.0,-1.7) {$6=$};
  \node at (-0.2,-1.7) {$1$};
  \node at ( 0.4,-1.7) {$1$};
  \node at ( 1.0,-1.7) {$0$};
\end{tikzpicture}
\end{figure}
\end{column}
\end{columns}
\end{frame}

\begin{frame}{The key identity is exactly a Rota--Baxter equation}
\small
In the rig $\Pf\left(\N\right)$,
\[
\left(\Int S\right)\left(\Int T\right)
=
\Int\left(\left(\Int S\right)T + S\left(\Int T\right)\right)
\]
becomes
\[
\left\{\mex\left(S\right)\right\}\times \left\{\mex\left(T\right)\right\}
=
\Int\left(\left(\left\{\mex\left(S\right)\right\}\times T\right)\cup \left(S\times \left\{\mex\left(T\right)\right\}\right)\right).
\]
Unpacking the product gives exactly
\[
\mex\left(S\right)\nimsum \mex\left(T\right)
=
\mex\left(\left(\mex\left(S\right)\nimsum T\right)\cup \left(S\nimsum \mex\left(T\right)\right)\right).
\]

\begin{proposition}[Interpretation]
This \icol{integral} identity is the shadow of the \dcol{Leibniz rule} for the option operator of box product.
\end{proposition}
\end{frame}

\begin{frame}[t]{Recovering Bouton's theorem}
\scriptsize
\begin{columns}[T,totalwidth=\textwidth]
\begin{column}{0.56\textwidth}
Apply the construction to pointed Nim positions.
\begin{itemize}
  \itemsep0.2em
  \item One-heap Nim satisfies
  \[
  F_{\left(\mathrm{Nim}_1,n\right)} = \left\{n\right\}.
  \]
  \item Hence for the box product of $n$ heaps,
  \[
  F_{\left(a_1,\dots,a_n\right)} = \left\{a_1\nimsum\cdots\nimsum a_n\right\}.
  \]
  \item Therefore $\left(a_1,\dots,a_n\right)$ is a $P$-position iff
  \[
  a_1\nimsum\cdots\nimsum a_n = 0.
  \]
\end{itemize}
\end{column}
\begin{column}{0.40\textwidth}
\begin{figure}
\centering
\begin{tikzpicture}[scale=0.75]
  \node[bluepanel, minimum width=2.65cm, minimum height=0.68cm] (n1) at (0,1.5) {$\left(\mathrm{Nim}_1,n\right)$};
  \node[greenpanel, minimum width=1.30cm, minimum height=0.64cm] (fn1) at (0,0.35) {$\left\{n\right\}$};
  \node[bluepanel, minimum width=2.85cm, minimum height=0.68cm] (nn) at (0,-1.05) {$\left(a_1,\dots,a_n\right)$};
  \node[greenpanel, minimum width=3.15cm, minimum height=0.68cm] (fnn) at (0,-2.25) {$\left\{a_1\nimsum\cdots\nimsum a_n\right\}$};
  \draw[formulaarrow] (n1) -- node[right] {$F$} (fn1);
  \draw[formulaarrow] (nn) -- node[right] {$F$} (fnn);
\end{tikzpicture}
\end{figure}
\end{column}
\end{columns}

\begin{proposition}[What changed conceptually?]
Instead of proving a mysterious xor identity by hand, we view it as the \icol{Rota--Baxter} image of a \dcol{differential} rule on games.
\end{proposition}
\end{frame}

\begin{frame}{Take-home messages}
\small
\begin{enumerate}
  \item \textbf{Games are recursive \dcol{coalgebras}.} The basic game values come from hylomorphisms for $\Pf$.
  \item \textbf{Box product has a \dcol{differential} flavor.} On pointed games / families, the option operator satisfies a \dcol{Leibniz} rule.
  \item \textbf{Nim-sum is a \icol{Rota--Baxter} phenomenon.} The classical mex + xor identity is the \icol{integral} image of that Leibniz rule.
\end{enumerate}

\vspace{0.65em}
\begin{proposition}[Outlook]
Partisan / probabilistic variants, a genuine chain rule, and a systematic interface with differential categories remain open.
\end{proposition}
\end{frame}

\begin{frame}[allowframebreaks]{References}
\printbibliography[heading=none]
\end{frame}

\end{document}