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}
\title{Topoi of automata (SLACS 2025):\\
Local state classifier for automata theory
}
\author{Ryuya Hora}
\institute{University of Tokyo}
\date{31 October, 2025}


\begin{document}

\begin{frame}

    \begin{columns}
        \begin{column}{0.8 \textwidth}
        \titlepage
        \end{column}
        \begin{column}{0.2\textwidth}
             \begin{figure}
                 \centering
                 \includegraphics[width=1\linewidth]{images/QrcodeForNotes.png}
                 \begin{center}
                    Slides (Local state classifier for automata theory)
                 \end{center}

             \end{figure}
        \end{column}
    \end{columns}
    \begin{itemize}
                \item \textit{Topoi of automata I} [\cite{hora2024topoi}]
                \item\textit{Topoi of automata II} [\cite{hora2025topoiII}]
                \item \textit{Internal Parameterization of Hyperconnected Quotients} [\cite{hora2024internal}]
            \end{itemize}
\end{frame}

\begin{frame}{Table of Contents}
    \tableofcontents
\end{frame}

\section{Introduction}

\begin{frame}{Motivation: Not Generalization, but Geometrization!}
    My motivation is neither generalisation nor abstraction.

    I want to provide a new geometric method/intuition, just like Grothendieck did to number theory!
    \begin{figure}
        \centering
        \includegraphics[width=1\linewidth]{images/ToposSubsumesMany.jpeg}
    \end{figure}

\end{frame}

\begin{frame}{Naive question: Can we imitate Grothendieck?}
\begin{columns}
    \begin{column}{0.2\textwidth}
        \begin{figure}
            \centering
            \includegraphics[width=1\linewidth]{images/portrait-placeholder.pdf}
        \end{figure}
    \end{column}
    \begin{column}{0.8\textwidth}
    \begin{itemize}
        \item Grothendieck defined \demph{topoi} (sites, and schemes) to introduce infinite/continuous geometric methods into number theory, which is traditionally thought of as a finite/discrete subject.
        \pause

        \item The theory of automata and regular languages is also (pro)finite/discrete combinatorial theory. Is it possible to introduce infinite/continuous geometric methods into this setting?
    \end{itemize}
    \pause

    \end{column}
\end{columns}
\begin{itemize}
    \item Primitive word conjecture $\leftarrow$ Topos-Galois theory
    \item Generalized star-height $\leftarrow$ Topos-cohomology
\end{itemize}
\end{frame}

\begin{frame}{Original Motivation: Lawvere's open problems in topos theory}
In 2009, W.Lawvere posed $7$ open problems in topos theory [{\cite{lawvere2025open}}]:
    \begin{columns}
        \begin{column}{0.5\textwidth}
            \begin{itemize}
                \item[1] \demph{Quotient Toposes}
                {\scriptsize [\cite{kamio2024solution}]}
                \item[2] Subqotients and Idempotents
                \item[3] Boundaries of classes of models
                \item[4] The jump operator on levels within a topos
                {\scriptsize [\cite{hora2025lawveresfourthopenproblem}]}
            \end{itemize}
        \end{column}
        \begin{column}{0.5\textwidth}
            \begin{itemize}
                \item[5] Coverings that admit averaging and microlinearity
                \item[6] How strong is the adjointness of fractional exponents?
                \item[7] The algebra of time
            \end{itemize}
        \end{column}
    \end{columns}
    \hspace{5pt}
    I defined \demph{local state classifiers} as a partial solution to the first problem.

    Today, I will focus on another point of view.
\end{frame}



\begin{frame}{Today's motivation}
I will try to convey the key idea, and I decided \demph{not to talk about topos theory.}
\begin{itemize}
    \item The colimit of all monomorphisms $\Xi$ exists in surprisingly many categories!
    \item $\Xi$ is the key to connect category theory and word-congruences.
    \item This is a theoretical background of the reason why I started to think about topos-theoretic automata theory.
\end{itemize}
    \begin{question}
    How can we deal with word-congruences in categorical settings?
\end{question}
\end{frame}

\section{LSCs: Definition}
\begin{frame}{Definition of LSC}
\begin{definition}[local state classifier {[\cite{hora2024internal}]}]
    A \demph{local state classifier} $\Xi$ of a category $\C$ is a colimit of all monomorphisms:
    \[\Xi = \colim (\mo{\C}\rightarrowtail \C).\]

    The associated cocone is referred to as $\{\xi_{X}\colon X \to\Xi\}_{X \in \ob{\C}}$.
\end{definition}
\[
\begin{tikzcd}[column sep =tiny, ampersand replacement = \&]
U\ar[rr,"\iota",rightarrowtail]\ar[rd,"\xi_{U}"']\&\&X\ar[ld,"\xi_{X}"]\\
\&\Xi\&
\end{tikzcd}
\]
\end{frame}





\begin{frame}{Toy Example (1/3): sets}
\begin{example}[$\Set$]
    The local state classifier of $\Set$ is the terminal object $\Xi = \{\ast\}$.
\end{example}
Consider an arbitrary cocone $\{\alpha_X : X \to L\}$ of ${\Set}_{\mathrm{inj}}\rightarrowtail \Set$. Then, for any set $X$ and its element $x\in X$, we have
\[
\begin{tikzcd}[column sep =tiny, ampersand replacement = \&]
\{\ast\}\ar[rr,"\text{pick}_x",rightarrowtail]\ar[rd,"\alpha_{\{\ast\}}"']\&\&X\ar[ld,"\alpha_{X}"]\\
\&L\&
\end{tikzcd}
\]
and therefore $\alpha_{X}(x)= \alpha_{\{\ast\}}(\ast)$.

\end{frame}


\begin{frame}{Toy Example (2/3): pointed sets}
What is the local state classifier of the category of pointed sets $\Set_{\ast}$?

\begin{center}
    \begin{tikzpicture}
        \draw[very thick, black] (-2,0) ellipse (1 and 2);
        \filldraw[black] (-2,1.5) circle (0.1);
        \filldraw[black] (-2,0.5) circle (0.1);
        \filldraw[red] (-2,-0.5) circle (0.1);
        \filldraw[black] (-2,-1.5) circle (0.1);
        \draw[very thick, black] (2,0) ellipse (1 and 1.5);
        \filldraw[red] (2,1) circle (0.1);
        \filldraw[black] (2,0) circle (0.1);
        \filldraw[black] (2,-1) circle (0.1);
        \draw[black, ->, very thick] (-1.8,1.5) -- (1.8,1);
        \draw[black, ->, very thick] (-1.8,0.5) -- (1.8,-1);
        \draw[red, ->, very thick] (-1.8,-0.5) -- (1.8,1);
        \draw[black, ->, very thick] (-1.8,-1.5) -- (1.8,0);
    \end{tikzpicture}
\end{center}
\end{frame}

\begin{frame}{Toy Example (2/3): pointed sets}
What is the local state classifier of the category of pointed sets $\Set_{\ast}$?

It is
\begin{center}

    \begin{tikzpicture}
        \draw[very thick, black] (2,0) ellipse (1 and 1);
        \draw (2,1) circle(0) node[above]{$\Xi$};
        \filldraw[red] (2,0.5) circle (0.1);
        \filldraw[black] (2,-0.5) circle (0.1);
    \end{tikzpicture}
\end{center}
equipped with ...
\end{frame}

\begin{frame}{Toy Example (2/3): pointed sets}
 cocone maps
\begin{center}
    \begin{tikzpicture}
        \draw[very thick, black] (-2,0) ellipse (1 and 2);
        \filldraw[black] (-2,1.5) circle (0.1);
        \filldraw[black] (-2,0.5) circle (0.1);
        \filldraw[red] (-2,-0.5) circle (0.1);
        \filldraw[black] (-2,-1.5) circle (0.1);

        \draw[very thick, black] (2,0) ellipse (1 and 1);
        \filldraw[red] (2,0.5) circle (0.1);
        \filldraw[black] (2,-0.5) circle (0.1);

        \draw[black, ->, very thick] (-1.8,1.5) -- (1.8,-0.5);
        \draw[black, ->, very thick] (-1.8,0.5) -- (1.8,-0.5);
        \draw[red, ->, very thick] (-1.8,-0.5) -- (1.8,0.5);
        \draw[black, ->, very thick] (-1.8,-1.5) -- (1.8,-0.5);
        \draw (0,2) circle(0) node[above]{$\xi_X$};
    \end{tikzpicture}
\end{center}
\end{frame}

\begin{frame}{Toy Example (3/3): colored sets}
What is the local state classifier of the category of $3$-colored sets $\Set/{\{\text{R,G,B}\}}$?

\begin{center}
    \begin{tikzpicture}[scale =0.85]
        \draw[very thick, black] (-2,0) ellipse (1 and 2.5);
        \filldraw[red] (-2,2) circle (0.1);
        \filldraw[green] (-2,1) circle (0.1);
        \filldraw[blue] (-2,0) circle (0.1);
        \filldraw[red] (-2,-1) circle (0.1);
        \filldraw[blue] (-2,-2) circle (0.1);
        \draw[very thick, black] (2,0) ellipse (1 and 2);
        \filldraw[green] (2,1.5) circle (0.1);
        \filldraw[blue] (2,0.5) circle (0.1);
        \filldraw[green] (2,-0.5) circle (0.1);
        \filldraw[red] (2,-1.5) circle (0.1);
        \draw[red, ->, very thick]   (-1.8,2)  -- (1.8,-1.5);
        \draw[green, ->, very thick] (-1.8,1)  -- (1.8,-0.5);
        \draw[blue, ->, very thick]  (-1.8,0)  -- (1.8,0.5);
        \draw[red, ->, very thick]   (-1.8,-1) -- (1.8,-1.5);
        \draw[blue, ->, very thick]  (-1.8,-2) -- (1.8,0.5);
    \end{tikzpicture}
\end{center}

\end{frame}
\begin{frame}{Toy Example (3/3): colored sets}
What is the local state classifier of the category of $3$-colored sets $\Set/{\{\text{R,G,B}\}}$?
It is
\begin{center}
    \begin{tikzpicture}[scale =0.9]
        \draw[very thick, black] (2,0) ellipse (1 and 1.5);
        \filldraw[red]   (2,1) circle (0.1);
        \filldraw[green] (2,0) circle (0.1);
        \filldraw[blue]  (2,-1) circle (0.1);
        \draw (2,1.5) circle(0) node[above]{$\Xi$};
    \end{tikzpicture}
\end{center}
equipped with ...

\end{frame}
\begin{frame}{Toy Example (3/3): colored sets}
morphisms that classify elements by their colors.
\begin{center}
    \begin{tikzpicture}[scale =0.85]
        \draw[very thick, black] (-2,0) ellipse (1 and 2.5);
        \filldraw[red] (-2,2) circle (0.1);
        \filldraw[green] (-2,1) circle (0.1);
        \filldraw[blue] (-2,0) circle (0.1);
        \filldraw[red] (-2,-1) circle (0.1);
        \filldraw[blue] (-2,-2) circle (0.1);
        \draw[very thick, black] (2,0) ellipse (1 and 1.5);
        \filldraw[red]   (2,1) circle (0.1);
        \filldraw[green] (2,0) circle (0.1);
        \filldraw[blue]  (2,-1) circle (0.1);
        \draw[red, ->, very thick]   (-1.8,2)  -- (1.8,1);
        \draw[green, ->, very thick] (-1.8,1)  -- (1.8,0);
        \draw[blue, ->, very thick]  (-1.8,0)  -- (1.8,-1);
        \draw[red, ->, very thick]   (-1.8,-1) -- (1.8,1);
        \draw[blue, ->, very thick]  (-1.8,-2) -- (1.8,-1);
        \draw (0,2) circle(0) node[above]{$\xi_X$};
    \end{tikzpicture}
\end{center}

\end{frame}





\begin{frame}{Intuition: why local?}
Let $\{\xi_X \colon X\to \Xi\}_{X\in \ob{\C}}$ be a local state classifier.
\vspace{-20pt}
\begin{columns}
    \begin{column}{0.4\textwidth}
        \begin{center}
    \begin{tikzpicture}[xscale = 1, yscale=0.75]
    \small
    \draw [black,thick](-2,0-1.5) -- (2,0-1.5) -- (2,-1-1.5) -- (-2,-1-1.5) -- cycle;
    \draw (0,-2.5)circle(0) node[below]{$\Xi$};
    \draw (0,-2)circle(0) node[right]{\tiny $\xi_X (x)$};

    \draw [black, thick] (2,3) ellipse (2.5 and 1.5);
    \draw (2,3+1.5)circle(0) node[above]{$X$};
    \filldraw [black] (1.5,3.3) circle(0.04);
    \draw (1.5,3.3)circle(0) node[above]{\tiny $x$};

    \draw [black,thick,->] (2,3-1.6) -- (0.2,0.1-1);
    \draw (1+0.5,0.3)circle(0) node[right]{$\xi_X$};

    \pause

    \draw [black,thick] (-2,3.3) circle (0.5);
    \draw (-2,3.3+0.5)circle(0) node[above]{$U$};
    \draw [black, dotted, thick] (1.5,3.3) circle (0.5);
    \filldraw [black] (-2,3.3) circle(0.04);
    \draw (-2,3.3)circle(0) node[above]{\tiny $x$};

    \draw [black,thick,>->] (-1.5+0.1,3.3) -- (1-0.1,3.3);
    \draw (-0.8,3.3)circle(0) node[above]{$\iota$};

    \pause
    \draw [black,thick,->] (-2,3.3-0.6) -- (-0.2,0.1-1);
    \draw (-1-0.5,1.2)circle(0) node[left]{$\xi_U$};


    \draw (0,-2)circle(0) node[]{\tiny $=$};
    \draw (0,-2)circle(0) node[left]{\tiny $\xi_U (x)$};


    \end{tikzpicture}
\end{center}
    \end{column}
    \begin{column}{0.4\textwidth}
        The value of $\xi_{X}(x)$ can be calculated in arbitraly small neighborhood of $x$.
    \end{column}
\end{columns}

\end{frame}

\section{LSCs: More Examples}




\begin{frame}{More Examples (1/4): directed graphs}
    What is the local state classifier of the category of directed graphs $\DirGraph$?
\begin{center}
    \includegraphics[scale =0.19]{images/Graphs2.png}
\end{center}
\end{frame}

\begin{frame}{More Examples (1/4): directed graphs}
    The local state classifier of the topos of directed graphs $\DirGraph$ is given by the \dq{$2$-bouquet.}
    \begin{enumerate}
    \item $\Xi$ is
    $
    \begin{tikzcd}[ampersand replacement = \&, scale = 2]
    \bullet\ar[loop left,"\bL"]\ar[loop right,"\bN ."]
    \end{tikzcd}
    $
    \item $\xi_X$ detects whether a given edge is a loop or not.
\end{enumerate}
\begin{center}
    \includegraphics[scale =0.13]{images/Graphs2.png}
\end{center}
\end{frame}

\begin{frame}{More Examples (2/4): group actions for a group $G$}
What is the local state classifier of $\PSh(G)$?\\
$=$ What is the \dq{local data} for each element of a $G$-set $X \curvearrowleft G$?
\pause


For the category of right $G$-actions $\PSh(G)$ for a group $G$,
\begin{enumerate}
    \item $\Xi$ is the set of all subgroups $\SubGrp{(G)}$ (equipped with the conjugate action). % $H \mapsto gHg^{-1}$.
    \item $\xi_X$ sends each element $x\in X$ to its stabilizer subgroup.
\end{enumerate}
\end{frame}

\begin{frame}{More Examples (3/4): presheaves}
Let $\C$ be a small category. The local state classifier of the presheaf topos $\PSh{(\C)}$ is the presheaf of quotient objects of representable presheaves:
\[\Xi \colon \C^{\op}\to \Set\colon c \mapsto \{y(c) \twoheadrightarrow C\}\]

\pause
\begin{remark}[Dual similarity with the Subobject classifier]
    It is a kind of dual to the subobject classifier, which is the presheaf of subobjects of representable presheaves:
\[\Omega \colon \C^{\op}\to \Set\colon c \mapsto \{y(c) \leftarrowtail C\}\]
\end{remark}

\end{frame}

\begin{frame}{More Examples (4/4): Other categories}
The following categories have LSCs.
\begin{itemize}
    \item The category of simplicial sets $\mathbf{sSet}$
    \item The topos of Joyal's species $\Species$ (cf. [\cite{fiore2024stabilized}])
    \item The category of coalgebraic automata $\Coalg_{\T}=\Atmt$.
    \item The Schanuel topos $\Cont(\Aut{\N})\cong \Sh(\mo{\FinSet}^{\op}, J_{\mathrm{at.}})$

    \item $\mathbf{Ab}$,  $\mathbf{SemiLattice}$, and any other categories of algebras.
    \item Any Grothendieck topoi including $\Sh(X)$ for a topological space $X$.
    \item Any locally presentable categories, including $\mathbf{Poset}, \mathbf{Cat}, \dots$
\end{itemize}
\end{frame}

\section{Atmt: LSC of \texorpdfstring{$\Aset$}{Aset}}

\begin{frame}{$\A$-sets}
    We fix a finite set $\Sigma$.
    A \demph{$\A$-set} is a pair $(Q, \delta)$ of a set $Q$ and a function $\delta\colon Q \times \A \to Q$.
    \begin{definition}
        $\Aset$ is
        defined by $\Aset \coloneqq \PSh(\MA)$.
    \end{definition}
    \begin{figure}
        \centering
        \includegraphics[width=1\linewidth]{images/Aset.jpeg}
    \end{figure}
\end{frame}




\begin{frame}{The $\A$-set of right congruences}
A \demph{right congruence} is an equivalence relation $\sim$ on $\MA$ such that
\[
\forall u,v,w\in \MA,\; u\sim v \implies uw\sim vw.
\]
\begin{definition}[The $\A$-set of right congruences $\XNi$]
    \begin{description}
    \item[Element] Right congruence
    ($=$ quotient object of the representable $\MA$.)
    \item[$\A$-action] $u \mathrel{{({\sim}*w)}} v \iff wu\sim wv.$
\end{description}
\end{definition}

\end{frame}

\begin{frame}{Right congruences form LSC}



    For any $\A$-set $(Q, \delta)$, we define a morphism
\[\xi_{(Q, \delta)} \colon (Q, \delta) \to \Xi\]
by sending $q\in Q$ to the right congruence $\xi_{(Q, \delta)}(q)\in \Xi$:
    \[
    u \mathrel{\xi_{(Q, \delta)}(q)}v \iff q*u=q*v.
    \]

    \begin{proposition}[LSC of $\Aset$, Universality of right congruences]
    The $\A$-set of right congruences $\Xi$, together with the cocone $\{\xi_{(Q, \delta)}\colon (Q, \delta) \to \XNi\}_{(Q, \delta)\in \ob (\Aset)}$
    is the local state classifier of $\Aset$.
\end{proposition}
\end{frame}

\begin{frame}{Toy example of LSC calculation}
  \begin{example}[The $\A$-set of languages $\Lan$]
    \begin{description}
    \item[Element] Language
    \item[$\A$-action]
    $L * w \coloneqq \{v\in \MA\mid wv\in L\}$ (Brzozowski derivative)
    \item[Universality] $\Aset((Q, \delta), \Lan)\cong \Pow(Q)$
\end{description}
\end{example}

What's the cocone component
$
\xi_{\Lan}\colon \Lan \to \Xi$?
\pause
\begin{proposition}[A categorical derivation of Nerode congruence]
    For any language $L$, $\xi_{\Lan}(L)$ coincides with the \demph{Nerode congruence} of $L$.
\end{proposition}
$u \mathrel{\xi_{\Lan}(L)} v \iff L*u=L*v \iff\left( \forall w\in \MA \; uw\in L \iff  vw\in L\right )$
\end{frame}



\section{Atmt: Covariety of \texorpdfstring{$\Aset$}{Aset}}



\begin{frame}{Idea: $\A$-set classes $\to$ language classes}
    From a full subcategory $\F\subset \Aset$ ,
    we can consider the \demph{induced language class} $\LC_\F\subset \Lan$.
    \[
    \LC_{\F}\coloneqq\{L \in \Lan\mid \text{some $\A$-set in $\F$ recognizes $L$} \} .
    \]

\begin{example}[regular languages]
$\AFinSet$ is the category of finite $\A$-sets.
        $\LC_{\AFinSet}$ is the set of regular languages.
\end{example}
\begin{figure}
    \centering
    \includegraphics[width=0.7\linewidth]{images/0523Covariety.jpeg}
\end{figure}


\end{frame}



\begin{frame}{Q: What kind of full subcategories should we consider?}
    It suffices to consider \demph{nice full subcategories.} but which?

    In category theory, we tend to consider reflective subcategories:
    \begin{itemize}
        \item Subtopoi? ($=$ lex reflective subcat)
        \item Birkhoff's HSP varieties? ($=$ subquotient-closed reflective sub)
        \item Orthogonality classes?
    \end{itemize}
    \pause
    We need to go the "opposite" direction\footnote{This is the reason why it's related to the open problem of \demph{quotient topoi}.}, coreflective ones!

\end{frame}

\begin{frame}{A: Covariety of $\Aset$}
    \begin{definition}[Covariety]
        A \demph{covariety}\footnote{This terminology is due to Peter Johnstone.This is defined for any Grothendieck topos.} of $\Aset$ is a full subcategory of $\Aset$ that is closed under
        \begin{itemize}
            \item subobjects,
            \item quotient objects, and
            \item coproducts.
        \end{itemize}
    \end{definition}
    Every covariety is \demph{coreflective.}
    \begin{lemma}[Covarieties are enough.]
        For any full subcategory $\F \subset \Aset$, there exists the minimum covariety $\overline{\F} \subset \Aset$ that contains $\F$. Furthermore, we have $\LC_{\F}=\LC_{\overline{\F}}$.
    \end{lemma}
\end{frame}

\begin{frame}{Example: What is $\overline{\AFinSet}$? Orbit-finiteness}
    \begin{definition}[$\ofAset$]
        A $\A$-set $(Q, \delta\colon Q\times \A \to Q)$ is said to be \demph{orbit-finite} if, for any state $q\in Q$, its orbit $q \MA \coloneqq\{qw \mid w\in \MA\}$ is a finite set.
    \end{definition}
Let $\ofAset$ denote the full subcategory of $\Aset$ that consists of all orbit-finite $\A$-sets.
    \begin{figure}
        \centering
        \includegraphics[width=0.55\linewidth]{images/Orbit-finiteness.jpeg}
    \end{figure}
    \begin{example}
    $\overline{\AFinSet}= \ofAset\hookrightarrow \Aset$ is a covariety such that $\LC_{\ofAset}= \Reg$.
    \end{example}
\end{frame}

\begin{frame}{covariety classification (1/2)}
But how can we calculate covarieties of $\Aset$? (Even the smallness of the number of them is non-trivial a priori.)
\pause
\begin{columns}
    \begin{column}{\textwidth}
        \begin{theorem}[Classification of covarieties]
        For any Grothendieck topos $\E$, (in particular $\E=\Aset$,) there is a bijective correspondence between
        \begin{itemize}
            \item covarieties of $\E$, and
            \item upward closed\footnote{$\Xi$ admits a unique order structure such that $\xi_X (x)\land \xi_{Y}(y) = \xi_{X\times Y} (x,y)$.} subobjects $U \rightarrowtail\Xi$.
        \end{itemize}
    \end{theorem}
    This theorem reduces the study of covarieties of $\Aset$ to the word combinatorics of congruences $\Xi$.
    \end{column}
\end{columns}
\end{frame}

\begin{frame}{covariety classification (2/2)}
    For a given upward closed subobject $U \rightarrowtail \Xi$, the corresponding covariety $\F_{U}$ is given by
        \[
           \ob(\F_U)\coloneqq \left \{ X\in \ob(\E)\;\middle |
           \begin{tikzcd}[ampersand replacement = \&]
                \& U\ar[d, rightarrowtail]\\
                X\ar[r,"\xi_X"']\ar[ru, dashed, "\exists"]\&\XNi
            \end{tikzcd}
            \right \}.
        \]

        \begin{example}[$\ofAset \xleftrightarrow{\textrm{corresponding}} U_{\of}$ ]
            The covariety $\ofAset$ corresponds to the subobject
            $U_{\of}\coloneqq \{{\sim}\in \Xi \mid \MA/{\sim}\text{ is finite}\}$.
        \end{example}
\end{frame}



\begin{frame}[shrink]{What I cannot talk: Other examples of calculation with LSC}
\small
$\Xi$ translates abstract definitions into concrete word combinatorics:
\begin{itemize}
    \pause
    \item \demph{Myhill-Nerode theorem} $\LC_{\F}=\xi_{\Lan}^{-1} (U)$ for each covariety $\F$.
    \begin{itemize}
        \item $\LC_{\ofAset}= \xi_{\Lan}^{-1}( U_{\of}) $ (cf. $\xi_{\Lan}\colon \Lan \to \Xi$ is the nerode congruence.)
    \end{itemize}


    \pause
    \item \demph{Syntactic congruence} ${\cong_L} = \bigwedge_{w\in \MA} \left (\xi_{\Lan}(L)*w\right )$ in $\Xi$.
    \begin{itemize}
        \item
        We need $\xi_{\Xi}\colon \Xi \to \Xi$ to capture the behavior of $\xi_{\Lan}(L)$.
    \end{itemize}

    \pause
    \item \demph{Syntactic monoids} $\Gal(C)$.
    \begin{itemize}
        \item Prodiscrete for arbitrary $C$: \alert{work in progress; not established}.
        \item $\Gal(\Reg)=\proMA$: the profinite monoid of profinite words
        \item
        $\Gal(\{L\}) = \MA/{\cong_L} \impliedby \xi_{\Xi}(\xi_{\Lan}(L))\in U_{\of} \impliedby\xi_{\Lan}(L) \in U_{\of}$
    \end{itemize}
    \item \demph{Dyck example} \alert{(work in progress; not established)}:\newline
    $\Gal(\{\textrm{well-parenthesized}\})=\langle a,b\mid ab=1\rangle\sqcup \{*\}$ with a non-trivial topology.
    \pause
    \item \demph{Syntactic topos} $\SQ_C$ for each $C\subset \Lan$.
    \begin{itemize}
        \item Classification of hyperconnected quotient topoi ($=$ covarieties closed under finite products){\scriptsize [\cite{hora2024internal}]}
        \item \alert{Work in progress; not established:} $\SQ_C \simeq \Cont(\Gal(C))$\newline (semi-Galois theory; cf. [\cite{uramoto2025semi}]).
    \end{itemize}
\end{itemize}

\end{frame}

\begin{frame}[shrink]{References}
    \IfFileExists{\jobname.bbl}{\printbibliography}{\strut}
\end{frame}





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