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\title{local state classifier relative to factorization system}
\author{Ryuya Hora}
\thanks{ZEN University. \url{ryuya_hora@zen.ac.jp}}
% \date{\today}
\subjclass[2020]{MSC}
\keywords{Keywords}


\begin{document}
\begin{abstract}
\end{abstract}
\maketitle

\tableofcontents

% \section{Quotient maps}

\section{Introduction}
In \cite{hora2024internal}, the notion of a local state classifier was introduced as follows:
\begin{definition}[\cite{hora2024internal}]
    A \demph{local state classifier} $\Xi$ of a category $\C$ is the colimit of all monomorphisms
    \[
    \Xi\coloneqq \colim (\C_{\mathrm{mono}}\rightarrowtail  \C)
    \]
\end{definition}

\section{The presheaf of quotient objects and its univerality}




\begin{definition}
    Let $(\EC,\MC)$ be an orthogonal factorization system on a category $\C$. 
    For each $c\in \ob(\E)$, let $\QuoE(c)$ denote the class of all isomorphism classes of morphisms in $\EC$ from the object $c$ regarded as objects in $c/\C$.
    Assuming that $\C$ is \demph{$\EC$-cowell-powered}, which means that $\QuoE(c)$ is small for any $c\in \ob(\C)$, we define the functor
    \[
    \QuoE\colon \E^{\op} \to \Set
    \]
    by the factorization (\memo{write it later}).
    % \[
    % \QuoE(f\colon X \to Y)(e\colon Y \twoheadrightarrow Q) =
    % \]
\end{definition}


What can we say about this presheaf $\QuoE\colon \C^{\op} \to \Set$? 
% This functor $\QuoE$ is rarely representable:
\begin{remark}[$\QuoE$ is rarely representable]
In contrast to the situation that a $\MC$-subobject classifier represents the presheaf of subobjects (cf. topos, or quasitopos), the presheaf of quotient objects is rarely representble:
    If $\QuoE$ is representable, then there exists an object $Q\in \E$ such that every object $X\in \ob(\E)$ admits an $\MC$-morphism into $Q$. \memo{Can we say $\MC=\C$?}
\end{remark}
\memo{If we do not assume properness, what is naturally defined might be a pseudo functor $\E^{\op} \to \mathbf{CAT}$.}
\begin{remark}
    \cite{kenney2006copower}
\end{remark}

But $\QuoE$ has a nice universal property, which is quite similar to that of a local state classifier:
\[
    \QuoE = \colim(\MC \rightarrowtail\C\xhookrightarrow{\yo}\Func{\C^{\op}}{\Set}).
    \]


Since colimits in a functor category are calculated pointwise (if it exists in each point), we first study the functor
\[
\MC \rightarrowtail\C\xhookrightarrow{\yo}\Func{\C^{\op}}{\Set} \xrightarrow{\ev_c}\Set,
\]
which coincides with the covariant hom-functor $\C(c,-)$, but with the restricted domain $\MC$
\[
\MC \xrightarrow{\C(c,-)} \Set.
\]

% \begin{lemma}\label{lem:DecompositionIntoPresentables}\memo{false, we need to consider the groupoid action of $(X/\EC)_0$}
%     The functor
%     $
%     \C(c,-)\colon \MC \to \Set
%     $
%      is decomposed into the (small) coproduct of representables as follows:
%      \[
%      \C(c,-) \cong \coprod_{[q\colon c \twoheadrightarrow d]\in \QuoE(c)} \MC(d,-).
%      \]
%      In particular, the functor $
%     \C(c,-)\colon \MC \to \Set
%     $ admits a colimit, which is given by the set $\QuoE(c)$ equipped with the colimit cocone
%     \[
%     \C(c, c') \to \Quo_E(c) \colon f \mapsto \text{(the $\EC$-part of the factorization of $f$)}.
%     \]
% \end{lemma}
% \begin{proof}
%     The isomorphism is nothing other than the unique existence of the $(\EC,\MC)$-factorization \memo{check maybe we need properness?}. The latter holds since the colimit of a representable functor is a singleton.
% \end{proof}

\begin{proposition}[Universality of $\Quo$]\memo{We can deduce the restricted versions from this} Assuming the local smallness of $\C$,
    the presheaf $\QuoE\colon \C^{\op} \to \Set$ provides the colimit 
    \[
    \QuoE = \colim(\MC \rightarrowtail\C\xhookrightarrow{\yo}\Func{\C^{\op}}{\Set})
    \]
\end{proposition}
\begin{proof}
    This follows from \cref{lem:DecompositionIntoPresentables}. A direct proof is also given in the appendix.
\end{proof}

\begin{remark}If all morphisms in $\EC$ are epic,
    the functor
    $
    \C(c,-)\colon \MC \to \Set
    $
     is decomposed into the (small) coproduct of representables as follows:
     \[
     \C(c,-) \cong \coprod_{[q\colon c \twoheadrightarrow d]\in \QuoE(c)} \MC(d,-).
     \]
     In particular, the functor $
    \C(c,-)\colon \MC \to \Set
    $ admits a colimit, which is given by the set $\QuoE(c)$ equipped with the colimit cocone
    \[
    \C(c, c') \to \Quo_E(c) \colon f \mapsto \text{(the $\EC$-part of the factorization of $f$)}.
    \]
    \begin{proof}
    The isomorphism is nothing other than the unique existence of the $(\EC,\MC)$-factorization \memo{check maybe we need properness?}. The latter holds since the colimit of a representable functor is a singleton.
\end{proof}
\end{remark}



\section{Local state classifier relative to factorization system}
\begin{definition}
    For a category $\C$ and a wide subcategory $\MC \rightarrowtail\C$, the \demph{$\MC$-local state classifier} is the colimit of the embedding functor $\MC \rightarrowtail \C$ if it exists.
\end{definition}

\begin{example}
If $\MC$ is the class of all monomorphisms, then the $\MC$-local state classifier is the local state classifier in \cite{hora2024internal}.
\end{example}

\begin{example}
    The $\C$-local state classifier is the terminal object of $\C$. See \cite{riehl2017category} and \cite{menni2025nonsingular}. \memo{This example is not boring. The monoid structure induced on the terminal object might be useful in game theory or in other coalgebraic contexts.}
\end{example}

\begin{definition}\memo{cite paper}
    A locally small category $\C$ is said to be \demph{total} if its yoneda embedding $\C \xhookrightarrow{\yo}\Func{\C^{\op}}{\Set}$ admits a left adjoint.
\end{definition}

In this note, total category means a locally small total category.

\begin{theorem}
    % If a locally small category $\C$ with a factorization system $(\EC,\MC)$ satisfies the following conditions:
    % \begin{itemize}
    %     \item $\C$ is total,
    %     \item $\C$ is $\EC$-cowell-powered, 
    % \end{itemize}
    For any total category $\C$ with $\EC$-cowell powered factorization system $(\EC, \MC)$, $\C$ admits a $\MC$-local state classifier $\Xi_\MC$. Furthermore, the $\MC$-local state classifier $\Xi_\MC$ is given by 
    \[
    \Xi_\MC = L(\QuoE),
    \]
    where $L\colon \Func{\C^{\op}}{\Set}\to \C$ denotes the left adjoint to the yoneda embedding.
\end{theorem}

Any locally presentable categories have LSC, due to the strong-epi mono factorization system.


\section{Classification of $M$-pullback stable family of $M$-subobjects.}
\memo{Consider $M$-subobject classifier}
\appendix
\section{Orthogonal factorization system}
This section aims to recall the basic properties of orthogonal factorization systems.
\begin{definition}\label{def:orthogonalfs}
    An \demph{orthogonal factorization system} on a category $\C$ is a pair $(\EC, \MC)$ of subclasses $\EC, \MC \subset \Mor (\C)$ that satisfies the following conditions.
    \begin{itemize}
        \item Both $\EC, \MC$ contains all isomorphisms.
        \item Both $\EC, \MC$ are closed under compositions.
        \item Any morphism in $\C$ is uniquely decomposed into a morphism in $\EC$ followed by one in $\MC$ up to unique isomorphisms.
    \end{itemize}
\end{definition}

\begin{example}[Graphical example]\label{exmp:GraphicalExampleOfOFS}
    Let $\R^2$ denote the poset (since a category) whose objects are elements of $\R^2$ and morphisms are inequalities in the product poset $(\R, \leq )^2$. Then the following classes
    \begin{itemize}
        \item $\EC=\{(x,y) \leq (x',y')\mid x=x'\}$\text{: vertical ones}
        \item $\MC=\{(x,y) \leq (x',y')\mid y=y'\}$ \text{: horizontal ones}
    \end{itemize}
    define an orthogonal factorization system on the category $\R^2$.
    Trivially, both classes contain all isomorphisms, which are all identities, and closed under compositions. The unique decomposition of $f: (x,y)\leq (x',y')$ is given by $(x,y) \underset{\EC}{\leq} (x,y')\underset{\MC}{\leq} (x',y')$ (\Cref{fig:DecompostionOFS}).
        \begin{figure}[htbp]\label{fig:DecompostionOFS}
  \centering
  \begin{tikzpicture}[scale=0.7]
    % Grid
    \draw[very thin, gray] (-3.5,-3.5) grid (3.5,3.5);

    % Axes
    \draw[->, thick] (-4,0) -- (4,0) node[anchor=west]{$x$};
    \draw[->, thick] (0,-4) -- (0,4) node[anchor=south]{$y$};

    \draw[->, thick, black] (-3,-2) -- (3,3) node[anchor=west]{};
    \draw[->, thick, blue]  (-3,-2) -- (-3,3) node[anchor=east]{$\EC$};
    \draw[->, thick, red]  (-3,3) -- (3,3) node[anchor=west]{$\MC$};
  \end{tikzpicture}
\end{figure}
    
    \begin{figure}[htbp]
  \centering
  \begin{tikzpicture}[scale=0.7]
    % Grid
    \draw[very thin, gray] (-3.5,-3.5) grid (3.5,3.5);

    % Axes
    \draw[->, thick] (-4,0) -- (4,0) node[anchor=west]{$x$};
    \draw[->, thick] (0,-4) -- (0,4) node[anchor=south]{$y$};

    % Horizontal arrow (on the line y=2)
    \draw[->, thick, blue] (-3,-2) -- (1,-2) node[anchor=west]{};
    
    % Vertical arrow (on the line x=-2)
    \draw[->, thick, red] (3,-1) -- (3,3) node[anchor=south]{};
  \end{tikzpicture}
\end{figure}

\end{example}

\section{Preliminaries on total categories}

\section{A direct proof of the universality of $\QuoE$}

In order to clarify the statement, let us write down the colimit cocone $\{\zeta_c \colon \yo(c) \to \QuoE\}$.

Each component $\zeta_{c,d} \colon \C(d,c) \to \QuoE (d)$ sends a morphism $f\colon d \to c$ to (the equivalence class of) the $\EC$-part of the factorization,
\[
\begin{tikzcd}
    d \ar[rr, "f"] \ar[rd, twoheadrightarrow, "\zeta_{c,d}(f)"']&&c\\
    &\bullet \ar[ru, "m_f"', rightarrowtail]&
\end{tikzcd}
\]
where $m_f$ denotes the $\MC$-part.
% Rigorously speaking, $\xi_{c,d}(f)$ is the equivalence class of the $\MC$-part.

\begin{lemma}
    Each $\zeta_{c} \colon \yo(c) \to \QuoE$ is a natural transformation of the two presheaves $\yo(c)$ and $\QuoE$.
\end{lemma}
\begin{proof}
    What we need to prove is that, for any composable maps
    \[
    d_1 \xrightarrow{g} d_0 \xrightarrow{f} c
    \]
    we have $\zeta_{c, d_1}(fg) = \zeta_{c, d_0}(f)*g$.
    % \[
    % \begin{tikzcd}
    %     d_1 \ar[r, "g"] \ar[d, twoheadrightarrow, "\zeta_{c, d_1}(fg)"']&d_0\ar[d, "\zeta_{c,d_1}(f)"', twoheadrightarrow]\\
    %     \bullet \ar[r, "\exists", rightarrowtail, dashed] & \bullet.
    % \end{tikzcd}
    % \]
    % This follows since
    % \[
    % \begin{tikzcd}
    %     d_1 \ar[r, "g"] \ar[d, twoheadrightarrow, "\zeta_{c, d_1}(fg)"']&d_0\ar[d, "\zeta_{c,d_0}(f)"', twoheadrightarrow]\ar[dd, "f", bend left]\\
    %     \bullet \ar[r, "\exists", rightarrowtail, dashed] 
    %     % \ar[rd, "m_{fg}"', rightarrowtail]
    %     & \bullet\ar[d, rightarrowtail, "m_f"']\\
    %     &c
    % \end{tikzcd}
    % \]
    % we have
    This follows since
    \[
    \begin{tikzcd}
        d_1 \ar[r, "g"] \ar[d, twoheadrightarrow, "\zeta_{c, d_1}(fg)"']&d_0\ar[d, "\zeta_{c,d_1}(f)"', twoheadrightarrow]\\
        \bullet \ar[r, "\exists", rightarrowtail, dashed] & \bullet.
    \end{tikzcd}
    \]
    This follows since the following diagram proves that $\zeta_{c, d_0}(f)*g$ provides the $\EC$-part of the morphism $fg$.
    \[
    \begin{tikzcd}
        d_1 \ar[r, "g"] \ar[d, twoheadrightarrow, "\zeta_{c, d_0}(f)*g"']&d_0\ar[d, "\zeta_{c,d_0}(f)"', twoheadrightarrow]\ar[dd, "f", bend left]\\
        \bullet \ar[r,  rightarrowtail] 
        % \ar[rd, "m_{fg}"', rightarrowtail, bend right]
        & \bullet\ar[d, rightarrowtail, "m_f"']\\
        &c
    \end{tikzcd}
    \]
\end{proof}

Then, we will see that $\{\zeta_c \colon \yo(c) \to \QuoE\}$ is a cocone. Notice that a family of natural transformations $\{\alpha_c\colon \yo(c) \to P\}_{c\in \ob (\C)}$ is a cocone under the functor $\MC \rightarrowtail\C\xhookrightarrow{\yo}\Func{\C^{\op}}{\Set}$ if and only if, for any composable maps
\[
\begin{tikzcd}
    d \ar[r, rightarrow, "{f}"]& c_0 \ar[r, rightarrowtail ,"{m\in \MC}"]& c_1,
\end{tikzcd}
\]
where $m$ is an $\MC$-morphism, the equation
\begin{equation}\label{eq:coconecondition}
    \alpha_{c_0,d}(f) = \alpha_{c_1,d}(mf).
\end{equation}
holds.
\[
\begin{tikzcd}
    \C(d,c_0)\ar[rd, "\alpha_{c_0,d}"']\ar[rr, "m_{*}"]&&\C(d,c_1)\ar[ld, "\alpha_{c_1, d}"]\\
    &P(d)&
\end{tikzcd}
\]

% For the later reference, we will provide a general 

\begin{lemma}
    The family of natural transformations $\{\zeta_c \colon \yo(c) \to \QuoE\}$ is a cocone under the functor $\MC \rightarrowtail\C\xhookrightarrow{\yo}\Func{\C^{\op}}{\Set}$.
\end{lemma}
\begin{proof}
For any composable maps
\[
\begin{tikzcd}
    d \ar[r, rightarrow, "{f}"]& c_0 \ar[r, rightarrowtail ,"{m\in \MC}"]& c_1,
\end{tikzcd}
\]
where $m$ is an $\MC$-morphism, the equation \cref{eq:coconecondition} 
\begin{equation*}
    \zeta_{c_0,d}(f) = \zeta_{c_1,d}(mf).
\end{equation*}
holds, since post-composition with the $\MC$-morphism $m$
\[
\begin{tikzcd}
    d \ar[rr, rightarrow, "{f}"]\ar[rd, "\zeta_{c_0, d}(f)"', twoheadrightarrow]&& c_0 \ar[r, rightarrowtail ,"{m\in \MC}"]& c_1\\
    &\bullet\ar[ru, "m_f"', rightarrowtail]&&
\end{tikzcd}
\]
does not change the $\EC$-part of its factorization.
\end{proof}

\begin{lemma}
    The family of natural transformations $\{\zeta_c \colon \yo(c) \to \QuoE\}$ is a colimit cocone of the functor $\MC \rightarrowtail\C\xhookrightarrow{\yo}\Func{\C^{\op}}{\Set}$.
\end{lemma}
\begin{proof}
    We have proven that it is a cocone. So it remains to prove the universality. Let us take an arbitrary cocone $\{\alpha_c\colon \yo(c) \to P\}_{c\in \ob (\C)}$ under the functor. We will show the unique exsitence of the cocone map $\gamma \colon \QuoE \to P$.

    First we will see the uniqueness. If there exists such $\gamma$, then we have
    \[
    \gamma_{c}(\pi \colon c \twoheadrightarrow q) = \gamma_c (\zeta_{q,c}(\pi \colon c \twoheadrightarrow q)) = \alpha_{q,c}(\pi \colon c \twoheadrightarrow q).
    \]
    since we have the commutative diagram
    \[
    \begin{tikzcd}
        &\yo(q)\ar[ld, "\zeta_q"']\ar[rd, "\alpha_q"]&\\
        \QuoE\ar[rr, "\gamma"']&&P
    \end{tikzcd}
    \]
    and hence
    \[
    \begin{tikzcd}
        &\C(c,q)\ar[ld, "\zeta_{q,c}"']\ar[rd, "\alpha_{q,c}"]&\\
        \QuoE(c)\ar[rr, "\gamma_c"']&&P(c).
    \end{tikzcd}
    \]
    This proves the uniqueness.

    It suffices to prove that the function\footnote{Technically, we need to show the well-definedness, since this definition a priori depends on the choice of $\pi$. This well-definedness follows from the fact that the right class $\MC$ of a factorization system contains all isomorphisms.} $\gamma_{c}(\pi \colon c \twoheadrightarrow q)= \alpha_{q,c}(\pi \colon c \twoheadrightarrow q).
    $ actually defines a cocone map.
    The naturality with respect to $c\in \C$ follows since, for any 
    for 
    \[
    \begin{tikzcd}
        c_0\ar[r, "\forall f"]\ar[d, "\pi*f"', twoheadrightarrow]&c\ar[d, "\pi", twoheadrightarrow]\\
        q_0\ar[r, rightarrowtail]&q,
    \end{tikzcd}
    \]
    we have
    \[
    \gamma_{c}(\pi)*f= \alpha_{q,c}(\pi)*f = \alpha_{q, c_0}(\pi f) = \alpha_{q, c_0}(m_{\pi f} \circ (\pi*f)) = \alpha_{q_0,c_0}(\pi * f) = \gamma_{c_0}(\pi*f),
    \]
    where the fourth equation follows due to \cref{eq:coconecondition}.

    \memo{The commutativity as a cocone map follows similarly by \cref{eq:coconecondition}}
\end{proof}

\printbibliography
\end{document}