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\newcommand{\Lsc}{Local state classifier}
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% \AtBeginSection[]
% {
%   \begin{frame}
%     \frametitle{Table of Contents}
%     \tableofcontents[currentsection]
%   \end{frame}
% }
\title{Topos: the unexpected bridge between Logic and Geometry}
\author{Ryuya Hora}
% \institute{Overleaf}
\date{February 7, 2023}


\begin{document}

\frame{\titlepage}

\begin{frame}
\frametitle{Aim}
\begin{enumerate}
    \item To non-experts, to explain a general idea of topos theory.
    \item To students in the Department of Mathematics, to explain my discovery: internal parametrization of hyperconnected quotients.
\end{enumerate}
\end{frame}

\begin{frame}
\frametitle{topic of this presentation}

\begin{center}
    Mathematics\\
    \rotatebox{90}{$\subset$}\\
    Category Theory\\
    \rotatebox{90}{$\subset$}\\
    Topos Theory\\
    \rotatebox{90}{$\subset$}\\
    Internal Parametrization\\
    \rotatebox{90}{$\in$}\\
    My Discovery!\\
\end{center}

\end{frame}

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

\section{Category Theory}
\begin{frame}
\frametitle{Category Theory(1/3)}
One (typically emphasized) aspect of category theory is
\textbf{mathematics of analogies}.

It was originally invented to make a bridge between geometry and algebra and now has become a fundamental language in modern mathematics
\end{frame}

\begin{frame}
\frametitle{Category Theory(2/3)}
Even outside mathematics, a recently emerged area, \emph{applied category theory}, is used in
\begin{itemize}
    \item mathematical physics,
    \item philosophy,
    \item computer science,
    \item linguistics,
    \item cognitive science,
\end{itemize}
and others.
\end{frame}

\begin{frame}
\frametitle{Category Theory(3/3)}
\begin{block}{Discussion Topic}
    Is category theory really useful outside of mathematics?
\end{block}
\end{frame}

\section{Topos Theory}
\begin{frame}
\frametitle{Topos Theory(1/3)}
Topos is a category that satisfies several properties.
Today, we will focus on the following two aspects of topos (among 13 famous aspects).
\begin{itemize}
    \item Topos is a space. (\textbf{Geometric} aspect)
    \item Topos is a universe. (\textbf{Logical} aspect)
\end{itemize}
\end{frame}

\begin{frame}
\frametitle{Topos Theory(2/3)}
\textbf{Topos is a space}, from a geometric viewpoint.

Topos is originally defined as \dq{a generalized space} by Grothendieck in his work toward the Weil conjecture.

We can discuss geometric concepts of topos, including connectedness, subspace (subtopos), quotient, continuous function.
% Roughly speaking, the notion of topos is a generalization of the notion of topological space.
\end{frame}

\begin{frame}
\frametitle{Topos Theory(3/3)}
\textbf{Topos is a universe}, from a logical viewpoint.
In other words, we can do \dq{internal mathematics} in a fixed topos. We can do group theory, poset theory, and ring theory in a topos.

Cohen's proof of the Independence of the continuum hypothesis, which constructs a set-theoretic universe, has a topos-theoretic paraphrase.
\end{frame}

\section{Internal Parametrization}
\begin{frame}
\frametitle{Internal Parametrization(1/3)}
One of the most fundamental facts in topos theory is the \textbf{internal parametrization of subtoposes}, which states that there is a natural bijective correspondence between the following two concepts:
\begin{itemize}
    \item geometric (and external) notion, \emph{subtoposes}
    \item logical (and internal) notion, \emph{modal operators}
\end{itemize}
\end{frame}

\begin{frame}
\frametitle{Internal Parametrization(2/3)}
A \textbf{subtopos} of a topos $\E$ is defined as a morphism (of toposes) \emph{into} $\E$. Geometrically, it corresponds to the notion of subspace.

A \textbf{modal operator} of a topos $\E$ is defined as a morphism \emph{in} $\E$. Logically, of course, it corresponds to the notion of modality, \dq{mode of truth.}

The internal parametrization enables us to classify all subtoposes of a topos $\E$, by studying internal mathematics in $\E$!
\end{frame}

\begin{frame}
\frametitle{Internal Parametrization(3/3)}
In contrast to the case of subtoposes, \textbf{internal parametrization of quotient toposes} is still unknown. It is the first problem of what is called Lawvere's open problems in topos theory.


% \begin{block}{Question: Lawvere's open problem}
% Is there a suitable object of $\E$ that parameterizes connected quotients of $\E$?
% \end{block}
\end{frame}

\section{My Discovery}
\begin{frame}
\frametitle{My Discovery(1/3)}
The main theorem of my forthcoming paper is \textbf{Internal parametrization of hyperconnected quotients}. It gives a partial solution to the open problem since hyperconnected quotients are a special (and theoretically important) class of quotients.

In order to realize the internal parametrization, it is necessary to define \dq{internal correspondent.} In the paper, I define a new concept \dq{a local state classifier} and describe hyperconnected quotients in terms of internal mathematics related to a local state classifier. 
\end{frame}

\begin{frame}
\frametitle{My Discovery(2/3)}
\begin{block}{Definition}
\textbf{A local state classifier} of a category $\C$ is \[\colim(\mo{\C}\to \C).\]
\end{block}
\end{frame}


% \begin{frame}
% \frametitle{A Local State Classifier(2/3}
% A local state classifier of a category $\C$ is
% \begin{enumerate}
%     \item an object $\Xi$ of a category $\C$
%     \item equipped with a family of morphisms $\{\xi_X \colon X\to \Xi\}_{X \in \ob{\C}}$ from all objects of $\C$,
% \end{enumerate}
% that satisfies the following two conditions:
% \end{frame}

% \begin{frame}[fragile]
% \frametitle{Concrete definition (2/2)}

% \begin{itemize}
%     \item (\dq{$\xi$ commutes with mono}): \\
%     For each monomorphism $\iota: U\subob X$ in $\C$, 
% \[
% \begin{tikzcd}[column sep =tiny]
% U\ar[rr,"\iota",\mono]\ar[rd,"\xi_{U}"']&&X\ar[ld,"\xi_{X}"]\\
% &\Xi&
% \end{tikzcd}
% \]is commutative.
%     \item It is universal. (details omitted)
% \end{itemize}
% \end{frame}

\begin{frame}
\frametitle{My Discovery(3/3)}

\begin{center}
    \begin{tikzpicture}[scale = 0.75]
    \small
    %U and its inclusion into X
    \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 $x$};
    
    % X and its "element."
    \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,>->] (-1.5+0.1,3.3) -- (1-0.1,3.3);
    \draw (-0.8,3.3)circle(0) node[above]{$\iota$};
    \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 [black,thick,->] (2,3-1.6) -- (0.2,0.1-1);
    \draw (1+0.5,0.3)circle(0) node[right]{$\xi_X$};
    
    %L
    \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]{$\Xi$};
    % \filldraw [black] (-0.2,-1.7) circle(0.04);
    \draw (0,-2)circle(0) node[]{\tiny $\xi_U (x) = \xi_X (x)$};
    % \draw[black,thick](0,0) .. controls (1,1) and (2,-1) .. (3,0) ;

    \end{tikzpicture}
\end{center}
\end{frame}



%Example 1: Pointed set
\begin{frame}
\frametitle{Example 1: Pointed set (1/3)}
What is the local state classifier of the category of pointed sets $\Set_{\ast}$? 

(Hint: monic $\Leftrightarrow$ injective)
\begin{center}
    \begin{tikzpicture}
        \draw[very thick, black] (-2,0) ellipse (1 and 2);
        \filldraw[black] (-2,1.5) circle (0.1);
        % \filldraw[black] (-2,1) circle (0.1);
        \filldraw[black] (-2,0.5) circle (0.1);
        % \filldraw[black] (-2,0) circle (0.1);
        \filldraw[red] (-2,-0.5) circle (0.1);
        % \filldraw[black] (-2,-1) 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}
\frametitle{Example 1: Pointed set (2/3)}
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}
\frametitle{Example 1: Pointed set (3/3)}
morphisms that classify the base points.
\begin{center}
    \begin{tikzpicture}
        %left
        \draw[very thick, black] (-2,0) ellipse (1 and 2);
        % \draw (-2,2) circle(0) node[above]{$X$};
        \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);
        
        %right
        \draw[very thick, black] (2,0) ellipse (1 and 1);
        % \draw (2,2) circle(0) node[above]{$\Xi$};
        \filldraw[red] (2,0.5) circle (0.1);
        \filldraw[black] (2,-0.5) circle (0.1);
        
        %map
        \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}
\frametitle{Example 2: $3$-colored set (1/3)}
What is the local state classifier of the category of $3$-colored sets $\Set/{\{\text{R,G,B}\}}$? 

(Hint: monic $\Leftrightarrow$ injective)
\begin{center}
    \begin{tikzpicture}[scale =0.85]
        %left
        \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);
        %right
        \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);
        %map
        \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}
\frametitle{Example 2: $3$-colored set (2/3)}
It is
\begin{center}
    \begin{tikzpicture}[scale =0.9]
        %right
        \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}
\frametitle{Example 2: $3$-colored set  (3/3)}
morphisms that classify elements by their colors.
\begin{center}
    \begin{tikzpicture}[scale =0.85]
        %left
        \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);
        %right
        \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);
        %map
        \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}
\frametitle{Example 3: Directed graph}
For the category of directed graphs $\DirGraph$,
\begin{enumerate}
    \item $\Xi$ is
    $
    \begin{tikzcd}
    \bullet\ar[loop left,"\bL"]\ar[loop right,"\bN ."]
    \end{tikzcd}
    $
    \item $\xi_X$ classifies an edge of a directed graph $X$ according to whether it is a loop or not.
\end{enumerate}

\end{frame}
\begin{frame}
\frametitle{Example 4: Group action}
For the category of $G$-sets $\Set^{G}$ 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 an element $x\in X$ to its stabilizer subgroup.
\end{enumerate}
\end{frame}
  

\begin{frame}
\frametitle{Advanced Examples}
\begin{enumerate}
    \item That of the sheaf category $\Sh{X}$ over a topological space $X$ is a terminal object.
    \item Those of $\mathrm{Group}$, $\mathrm{Vect}_{K}$, $\mathrm{Poset}$, $\mathrm{Top}$ and $\mathrm{Manifold}$ are their terminal objects.
    \item That of the category of combinatorial species $\Species$ is \dq{species of symmetries}.
    \item That of an arbitrary Grothendieck topos $\E$ is constructed as ...
\end{enumerate}
\end{frame}


% \section{Applications}

% \begin{frame}
% \frametitle{Application to Topos theory}
% \begin{enumerate}
%     \item I found a concrete description of a local state classifier for all Grothendieck topos.
%     \item I proved that a local state classifier of a topos \emphora{parametrize} hyperconnected quotients. (main result)
%     \item Our result includes a solution to Lawvere's open problem for a limited class of toposes, namely Boolean toposes.
% \end{enumerate}
% \end{frame}

\section{Conclusion}
\begin{frame}
\frametitle{Conclusion}
A local state classifier is ...
\begin{enumerate}
    \item defined in terms of elementary category theory.
    \item intended to be a \dq{collection of all local states.} 
    \item used to solve a topos-theoretic problem.
    % \item By concrete calculation, we can see that it captures the idea of \dq{local} (in our context).
    \item an abstract generalization of several familiar concepts.
\end{enumerate}
\end{frame}
\end{document}

\begin{frame}
\frametitle{}

\end{frame}