← Demystifying local state classifiers
FirstVersion.tex
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\end{tikzcd}
}
\newcommand{\Iso}{\mathrm{Iso}}
% \newcommand{\mono}{\mathrm{mono}}
% \title{Demystifying local state classifiers:\\
% % local state classifier in
% total categories with factorization systems}
\title{Local state classifiers relative to factorisation systems}
\author{Ryuya Hora, Yuto Ikeda}
\thanks{ZEN University. \url{ryuya_hora@zen.ac.jp}}
% \date{\today}
\subjclass[2020]{MSC}
\keywords{Keywords}
\begin{document}
\begin{abstract}
The notion of a local state classifier was defined to be a colimit of all monomorphisms, which was originally introduced in the topos-theoretic context. This note aims to demystify the fact that such a large colimit exists in many ``traditional categories,'' including $\Set, \mathbf{Grp}, \mathbf{Ring}, \Top,$ and $\Sh(\C,J)$, by providing a simple and conceptual existence theorem.
The key observation is that a local state classifier is closely related to two classical notions in category theory: orthogonal factorization systems and total categories.
% There are two key observations.
First, some calculations of a local state classifier are naturally generalized to a colimit of the right class of an orthogonal factorization system (from the epi-mono factorization system in a topos). Second, the original sheaf-theoretic construction of a local state classifier works in an arbitrary \textit{total} category, i.e., a category whose Yoneda embedding admits a left adjoint.
% including all locally presentable categories.
\end{abstract}
\maketitle
\tableofcontents
% \section{Quotient maps}
\section{Introduction: Local state classifier and its construction problem}
In \cite{hora2024internal}, the notion of a local state classifier was introduced. For a category $\C$, we write $\C_{\mono}$ for the subcategory of $\C$ consisting of all objects and all monomorphisms of $\C$.
\begin{definition}[{\cite[Definition 3.5]{hora2024internal}}]
A \demph{local state classifier} $\Xi$ of a category $\C$ is the colimit of all monomorphisms if it exists.
\[
\Xi\coloneqq \colim (\C_{\mono}\rightarrowtail \C)
\]
\end{definition}
Even if $\C$ is small-cocomplete, $\C$ might not admit a local state classifier, since the indexing category $\C_{\mono}$ is not small. In the paper \cite{hora2024internal}, the author proved that any Grothendieck topos admits a local state classifier by concretely constructing it.
\begin{proposition}[{\cite[Proposition 3.21]{hora2024internal}}]\label{prop:OldConstructionOfLSCinGrothendieckTopos}
For a small site $(\C,J)$, the local state classifier $\Xi$ of the sheaf topos $\Sh(\C, J)$ is given by the sheafification $\Xi=\a\Xi_0$ of the presheaf
\[
\Xi_0 \colon c \mapsto \{\text{quotient objects of }\a\yo(c)\},
\]
where $\a\colon \PSh(\C) \to \Sh(\C,J)$ denotes the sheafification functor and $\yo\colon \C \to \PSh(\C)$ denotes the Yoneda embedding.
\end{proposition}
However, this construction is not fully satisfactory in the sense that it is too technical and does not provide any conceptual understanding. The aim of this paper is to demystify the construction
% of a local state classifier
by clarifying connections with two classical categorical notions:
\begin{itemize}
\item total categories (\Cref{sec:TotalCategories}), and
\item orthogonal factorization systems (\Cref{sec:OFSandLSC}).
\end{itemize}
In the following sections, we will recall them one by one.
\section{Total categories}\label{sec:TotalCategories}
This section aims to recall the definition and basic properties of total categories. Nothing in this section is a new result.
\subsection{Definition and examples}
\begin{notation}
In this paper, $\Set$ denotes the category of all small sets and functions. For any (possibly large) category $\C$, $\PSh(\C)$ denotes the functor category from $\C^{\op}$ to $\Set$\footnote{Technically, one can take two Grothendieck universes $u_0 \in u_1$. We write $\Set$ for the $u_1$-small category of $u_0$-small sets. Then, for any $u_1$-small category $\C$, $\PSh(\C)$ denotes the functor category $[\C^\op,\Set]$, i.e., the $u_1$-small category of $u_0$-small set valued presheaves on $\C$.
}.
\end{notation}
\begin{definition}\memo{See e.g. \cite{street1978yoneda, tholen1980note, street1984family}}
A locally small category $\C$ is said to be \demph{total} if its Yoneda embedding $\yo_{{\C}}\colon \C \hookrightarrow\PSh(\C)$ admits a left adjoint $L\colon \PSh(\C) \to \C$.
\end{definition}
\[
\begin{tikzcd}[ampersand replacement = \&, column sep = small]
{\C}
\ar[rr, shift right=1.3ex, "{\yo_{\C}}"', hookrightarrow,]
\&\perp\&
{\PSh(\C)}
\ar[ll, shift right=1.3ex,"{\exists L}"', dashed]
\end{tikzcd}
\]
In this note, `total category' always means `locally small total category.'
\begin{quote}\cite{street1978yoneda}
Total categories are precisely the categories, algebraic and topological, at which traditional category theory was aimed.
\end{quote}
\begin{example}[Easiest example]
The easiest category for which one can check the totality is the terminal category $\1$. The Yoneda embedding
\[
\yo_{\1}\colon \1 \hookrightarrow \PSh(\1)\cong\Set
\]
admits a left adjoint, which is given by the unique functor $L\colon \Set \to \1$.
% The category of sets is \cite{rosebrugh1994adjoint}
\end{example}
\begin{example}[Complete lattice]
Every small complete lattice $P$ is total. In fact, one can show that the functor
\[
L\colon\PSh(P) \to P \colon F \mapsto \sup\{p\in P\mid F(p) \neq \emptyset\}
\]
is a left adjoint to the Yoneda embedding.
\end{example}
\begin{example}[Presheaf categories]
The first non-small example of a total category is a presheaf category. Let $J$ be an arbitrary small category. Then, one can prove that the pre-composition functor ${-}\circ\yo_{J}^{\op} \colon \PSh(\PSh(J)) \to \PSh(J)$ is a left adjoint to the Yoneda embedding functor $\yo_{\PSh(J)} \colon \PSh(J) \hookrightarrow \PSh(\PSh(J))$
\begin{equation}\label{eq:PresheafIsTotal}
\begin{tikzcd}[ampersand replacement = \&, column sep = small]
{\PSh(J)}
\ar[rr, shift right=1.3ex, "{\yo_{\PSh(J)}}"', hookrightarrow,]
\&\perp\&
{\PSh(\PSh(J))}.
\ar[ll, shift right=1.3ex,"{-}\circ\yo_{J}^{\op}"']
\end{tikzcd}
\end{equation}
(In the case $J=\1$, this is a part of the adjoint $5$-tuple in \cite{rosebrugh1994adjoint}.)
\end{example}
\begin{example}[Reflective subcategory of a total category] If a category $\C$ is a reflective subcategory of a total category $\D$
\[
\begin{tikzcd}[ampersand replacement = \&, column sep = small]
{\C}
\ar[rr, shift right=1.3ex, "{\iota}"', hookrightarrow]
\&\perp\&
{\D},
\ar[ll, shift right=1.3ex,"{\a}"']
\end{tikzcd}
\]
then we have a chain of adjunction
\[
\begin{tikzcd}[ampersand replacement = \&, column sep = small]
{\C}
\ar[rr, shift right=1.3ex, "{\iota}"', hookrightarrow]
% \ar[rrrrrr, shift right=1.3ex, "{\yo_\C}"', hookrightarrow, bend right]
\&\perp\&
{\D}
\ar[ll, shift right=1.3ex,"{\a}"']
\ar[rr, shift right=1.3ex, "{\yo_{\D}}"', hookrightarrow]
\&\perp\&
{\PSh(\D)}
\ar[ll, shift right=1.3ex,"{L}"']
\ar[rr, shift right=1.3ex, "{{-}\circ \iota^{\op}}"']
\&\perp\&
{\PSh(\C)}.
\ar[ll, shift right=1.3ex,"{{-}\circ \a^{\op}}"', hookrightarrow]
\end{tikzcd}
\]
Furthermore, its right adjoint part sends an object $c\in \ob(\C)$ to a presheaf $\D(\iota {-}, \iota c) $, which is naturally isomorphic to $\C(-,c)=\yo_{\C}(c)$ since $\iota$ is fully faithful. This proves that the Yoneda embedding $\yo_{\C}$ admits a left adjoint
\begin{equation}\label{eq:TotalityOfReflectiveSubcategory}
\begin{tikzcd}[ampersand replacement = \&, column sep = small]
{\C}
\ar[rr, shift right=1.3ex, "{\iota}"', hookrightarrow]
\ar[rrrrrr, shift right=1.3ex, "{\yo_\C}"', hookrightarrow, bend right =15pt]
\&\perp\&
{\D}
\ar[ll, shift right=1.3ex,"{\a}"']
\ar[rr, shift right=1.3ex, "{\yo_{\D}}"', hookrightarrow]
\&\perp\&
{\PSh(\D)}
\ar[ll, shift right=1.3ex,"{L}"']
\ar[rr, shift right=1.3ex, "{{-}\circ \iota^{\op}}"']
\&\perp\&
{\PSh(\C)},
\ar[ll, shift right=1.3ex,"{{-}\circ \a^{\op}}"', hookrightarrow]
\end{tikzcd}
\end{equation}
which means that $\C$ is total.
\end{example}
\begin{example}
\memo{cite} It is known that any reflective subcategory of a total category is total, and that any presheaf category is total. Therefore, all locally presentable categories, including all Grothendieck topoi and all algebraic categories, are total. Other examples include $\Top$.
\end{example}
\subsection{Small and large colimits in a total category}
Since a (possibly large) colimit in a reflective subcategory is given by the reflector functor, the calculation of a colimit of $J \xrightarrow{F} \C$ is reduced to that of $J \xrightarrow{F} \C \xrightarrow{\yo_\C} \PSh(\C)$ for a total category $\C$.
\begin{lemma}\label{lem:colimitAfterYoneda}
For any locally small category $\C$ and any (possibly large) diagram $F \colon J \to \C$, the colimit of
% $\yo_{\C}\circ F \colon J \to \C \to \PSh(\C)$
$J \xrightarrow{F} \C \xrightarrow{\yo_\C} \PSh(\C)$
is given by the presheaf
\[
\pi_{0}({-}\downarrow F)\colon c \mapsto \pi_0(c\downarrow F)
\]
if $\pi_0(c\downarrow F)$ is small for each $c\in \C$.
\end{lemma}
\begin{proof}
A colimit in a functor category is given by objectwise colimits as soon as the objectwise colimits exist. Therefore, our problem is reduced to the calculation of the colimit of the functor
% Since the functor $J \xrightarrow{F} \C \xrightarrow{\yo_\C} \PSh(\C)$ is equal to
\[
\begin{tikzcd}
J\ar[r, "F"] &\C \ar[r, "{\yo_{\C}}"] \ar[rr, bend right, "{\C(c,-)}"']& \PSh(\C)\ar[r, "\ev_c"] &\Set.
\end{tikzcd}
\]
In general, the colimit of a functor $G\colon J \to \Set$ is given by $\pi_0\left(\int G\right)$ if it is small. This completes the proof since $\int \C(c,-)\circ F = c\downarrow F$.
\end{proof}
\begin{example}[Final functors and the colimit of Yoneda]
If $F\colon J \to \C$ is final (i.e., $\pi_0(c\downarrow F)$ is singleton), then the colimit of $J \xrightarrow{F} \C \xrightarrow{\yo_\C} \PSh(\C)$ is given by the terminal presheaf. In particular, if $F= \id_{\C}\colon J=\C \to \C$, the colimit of the Yoneda embedding is the terminal presheaf.
\end{example}
\begin{proposition}[colimits in total categories]\label{prop:colimitsInTotalCategories}
For any total category $\C$ with $L \dashv \yo_{\C}$ and any (possibly large) diagram $F\colon J \to \C$, if the comma category $c \downarrow F$ has a small number of connected components for each $c\in \C$, then $F$ admits a colimit, which is given by
\[
\colim_{J} F = L\left(\pi_0(c\downarrow F)\right).
\]
\end{proposition}
\section{Orthogonal factorization system and its local state classifier}\label{sec:OFSandLSC}
\subsection{Definition and examples}
This subsection 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}[Trivial example]
For any category $\C$, the pair $(\EC=\Mor, \MC=\Iso)$ is an orthogonal factorization system. Dually, the pair $(\EC=\Iso, \MC=\Mor)$ is also an orthogonal factorization system.
\end{example}
\begin{example}[Regular categories, topoi, abelian categories, and categories of algebras]
For any regular category $\C$, the pair $(\EC=\text{regular epi}, \MC= \text{mono})$ is an orthogonal factorization system. In particular,
\begin{itemize}
\item for any elementary topos $\C$, we have epi-mono factorization system,
\item for any abelian category $\C$, we have epi-mono factorization system, and
\item for any category of models of an algebraic theory $\C$, we have $(\text{Surj. hom.}, \text{inj. hom.})$-factorization system.
\end{itemize}
\end{example}
\begin{example}[quasitopoi]
\end{example}
\begin{example}[Locally presentable categories.]
For any locally presentable category $\C$, the pair $(\EC=\text{strong epi}, \MC= \text{mono})$ is an orthogonal factorization system. (\cite{adamek1994locally})
\end{example}
\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}
For the later reference, we recall some basic properties of orthogonal factorization systems.
\begin{lemma}\label{lem:BasicPropertiesOfOFS}
For any category $\C$ equipped with an orthogonal factorization system $(\EC, \MC)$, $\EC$ is pushout stable and $\MC$ is pullback stable.
\end{lemma}
\subsection{Local state classifier relative to factorization system}
\begin{notation}
For an orthogonal factorization system $(\EC,\MC)$ on a category $\C$, the wide subcategory of $\C$ consisting of all morphisms in $\EC$ (resp. $\MC$) is denoted by $\C_\EC$ (resp. $\C_\MC$).
\end{notation}
\begin{definition}
For a category $\C$ equipped with an orthogonal factorization system $(\EC,\MC)$, the \demph{$\MC$-local state classifier} is the colimit of the embedding functor $\iota_{\MC}\colon \C_\MC \rightarrowtail \C$, 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}
For any category $\C$, the pair $(\EC=\Iso, \MC=\Mor)$ is an orthogonal factorization system.
The $\Mor$-local state classifier is the colimit of the identity functor, which is known to be the same as the terminal object of $\C$ (See \cite{riehl2017category} and \cite{menni2025nonsingular}). Therefore, a category $\C$ admits a $\Mor$-local state classifier if and only if $\C$ admits a terminal object.
\invmemo{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}
\section{Existence theorem}\label{sec:ExistenceTheorem}
\subsection{The presheaf of quotient objects and its univesality}
Given that \cref{prop:colimitsInTotalCategories} holds, in order to know $\Xi = \colim(\iota_{\MC}\colon \C_{\MC}\rightarrowtail \C)$, what we need to know is $\pi_0\left(c\downarrow \iota_\MC\right)$. Here, we can utilize the structure of orthogonal factorization system.
\begin{definition}
Let $(\EC,\MC)$ be an orthogonal factorization system on a category $\C$.
For each $c\in \ob(\C)$, let $\QuoE(c)$ denote the class of all isomorphism classes of $c/\C_{\EC}$.
% morphisms in $\EC$ from the object $c$ regarded as objects in $c/\C$.
\end{definition}
\begin{notation}
Assuming that $\C$ is \demph{$\EC$-cowell-powered}, which means that $\QuoE(c)$ is small for any $c\in \ob(\C)$, we define a functor
\[
\QuoE\colon \E^{\op} \to \Set
\]
using the $(\EC,\MC)$-factorization (\memo{write it later}).
% \[
% \QuoE(f\colon X \to Y)(e\colon Y \twoheadrightarrow Q) =
% \]
\end{notation}
\begin{lemma}\label{lem:QuoAsPizero}
For any category $\C$ equipped with an orthogonal factorization system $(\EC,\MC)$ and an object $c\in \ob(\C)$, we have a natural bijection between two classes $\pi_0\left(c\downarrow \iota_{\MC}\right)$ and $\QuoE(c)$.
\end{lemma}
\begin{proof}
For any morphism $f\colon c \to d$ in $\C$, we define $q_f\in \QuoE(c)$ as (the isomorphism class of) the $\EC$-part of its $(\EC,\MC)$-factorization. This function $q_{-}\colon \ob\left(c\downarrow \iota_{\MC}\right)\to \QuoE(c)$ is well-defined since the factorization is unique up to a (unique) isomorphism.
We prove that this correspondence defines a well-defined function $q_{-}\colon\pi_0\left(c\downarrow \iota_{\MC}\right) \to \QuoE(c)$ and that it is bijective. To prove that the function is well-defined, what we need to prove is $q_{f} = q_{m\circ f}$ for any $m\in \MC$. This follows since $\MC$ is closed under composition and $(\EC,\MC)$-factorization is unique up to isomorphisms. The surjectivity follows since for any $e\in \EC$, $q_e$ is the isomorphism class of $e$. \invmemo{technically, here we use the fact that $\MC$ contains all identities.} Finally, we prove the injectivity. Suppose that $f_0\colon c\to d_0$ and $f_1\colon c\to d_1$ are sent to the same element $q_{f_0}=q_{f_1} \in \QuoE(c)$. Then we obtain the following commutative diagram
\[
\begin{tikzcd}[column sep =100pt]
&\bullet\ar[dd, "\cong"', "\sigma"]\ar[r,rightarrowtail]&d_0\\
c\ar[ru, twoheadrightarrow]\ar[rd, twoheadrightarrow]\ar[rru, bend left, "f_0"]\ar[rrd, bend right, "f_1"']&&\\
&\bullet \ar[r,rightarrowtail]&d_1,
\end{tikzcd}
\]
where $\sigma$ is the unique isomorphism. Since $\MC$ contains all isomorphisms, this proves that $f_0$ and $f_1$ are in the same connected component in $c \downarrow \iota_{\MC}$.
\end{proof}
\begin{remark}[Easier proof of \Cref{lem:QuoAsPizero} for proper cases]
In practical examples, \cref{lem:QuoAsPizero} is more easily and conceptually proven. An orthogonal factorization system $(\EC, \MC)$ is called \demph{proper} if both $\EC\subset \textrm{Epi}$ and $\MC \subset \textrm{Mono}$ hold.
If the considered orthogonal factorization system is proper\footnote{Actually, we only uses $\EC\subset \textrm{Epi}$ here.}, which often is the case in practical situations, then
% the comma category $c\downarrow \iota_{\MC}$
the functor
$
\C(c,\iota_{\MC}{-})\colon \C_{\MC} \rightarrowtail \C\to \Set
$
is decomposed into the coproduct of representables as follows:
\[
\C(c,\iota_{\MC}{-}) \cong \coprod_{[q\colon c \twoheadrightarrow d]\in \QuoE(c)} \C_{\MC}(d,-).
\]
In particular, the comma category $c\downarrow\iota_{\MC}$ is just the coproduct of slice categories:
\[c\downarrow\iota_{\MC} = \int \C(c,\iota_{\MC}{-}) = \coprod_{[q\colon c \twoheadrightarrow d]\in \QuoE(c)} \int \C_{\MC}(d,-) = \coprod_{[q\colon c \twoheadrightarrow d]\in \QuoE(c)} d/\C_{\MC}\]
Its set of connected components obviously coincides with $\QuoE(c)$, since every slice category is connected.
% 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}
\begin{proposition}[Universality of $\Quo$]\label{prop:UniversalityOfQuo}
% \memo{We can deduce the restricted versions from this}
For any locally small category $\C$ equipped with an $\EC$-cowell-powered orthogonal factorization system $(\EC, \MC)$,
% Assuming the local smallness of $\C$,
the presheaf $\QuoE\colon \C^{\op} \to \Set$ provides the colimit of the functor $\C_{\MC} \rightarrowtail\C\xhookrightarrow{\yo}\PSh(\C)$.
\[
\QuoE = \colim(\C_{\MC} \rightarrowtail\C\xhookrightarrow{\yo}\PSh(\C))
\]
\end{proposition}
\begin{proof}
This follows from \Cref{prop:colimitsInTotalCategories} and \Cref{lem:QuoAsPizero}. Technically, one needs to check that the right $\C$-actions in the two presheaves $\QuoE$ and $\pi_0\left(c\downarrow \iota_{\MC}\right)$ are equal. This is done by observing that the action of a morphism $f\colon c_0 \to c_1$ is given by the $\pi_0$ of the pre-composition functor
$
\left(c_1 \downarrow \iota_{\MC} \right)\xrightarrow{-\circ f}\left(c_0 \downarrow \iota_{\MC} \right).
$
\memo{I will write more details on this point.}
\end{proof}
\subsection{Exsitence and construction of a local state classifier}
\begin{theorem}[$\Xi$ is the best approximation to the collection of quotients]\label{thm:LocalStateClassifierInTotalCategory}
% 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 $\C$ equipped with an $\EC$-cowell-powered factorization system $(\EC, \MC)$, $\C$ admits an $\MC$-local state classifier $\Xi_\MC$. Furthermore,
% the $\MC$-local state classifier $\Xi_\MC$
it is given by
\[
\Xi_\MC = L(\QuoE),
\]
where $L\colon \PSh(\C)\to \C$ denotes the left adjoint to the Yoneda embedding.
\end{theorem}
\begin{corollary}
Let $\C$ be a reflective subcategory of a presheaf category $\PSh(J)$ for a small category $J$
\[
\begin{tikzcd}[ampersand replacement = \&, column sep = small]
{\C}
\ar[rr, shift right=1.3ex, "{\iota}"', hookrightarrow]
\&\perp\&
{\PSh(J)}.
\ar[ll, shift right=1.3ex,"{\a}"']
\end{tikzcd}
\]
Then, for any $\EC$-cowell-powered orthogonal factorization system $(\EC, \MC)$ on $\C$, the $\MC$-local state classifier $\Xi_{\MC}$ exists. Furthermore, it is given by applying $\a$ to the presheaf $J^{\op} \to \Set: j \mapsto \QuoE(\a\yo_{J}(j))$:
% \[
% \Xi_{\MC} = \a\left(\QuoE\circ \a\circ\yo_{J}: J^{\op} \to \Set\right).
% \]
\[
\Xi_{\MC} = \a\left(J^{\op}\xrightarrow{\yo_{J}^{\op}} \PSh(J)^{\op}\xrightarrow{\a^{\op}} \C^{\op} \xrightarrow{\QuoE} \Set\right).
\]
\end{corollary}
\begin{proof}
Combining two adjunctions \ref{eq:PresheafIsTotal} and \ref{eq:TotalityOfReflectiveSubcategory}, we have a combined adjunction
\[
\begin{tikzcd}[ampersand replacement = \&, column sep = small]
{\C}
\ar[rr, shift right=1.3ex, "{\iota}"', hookrightarrow]
\ar[rrrrrr, shift right=1.3ex, "{\yo_\C}"', hookrightarrow, bend right =15pt]
\&\perp\&
{\PSh(J)}
\ar[ll, shift right=1.3ex,"{\a}"']
\ar[rr, shift right=1.3ex, "{\yo_{\PSh(J)}}"', hookrightarrow]
\&\perp\&
{\PSh(\PSh(J))}
\ar[ll, shift right=1.3ex,"{{-}\circ \yo_{J}^{\op}}"']
\ar[rr, shift right=1.3ex, "{{-}\circ \iota^{\op}}"']
\&\perp\&
{\PSh(\C)}.
\ar[ll, shift right=1.3ex,"{{-}\circ \a^{\op}}"', hookrightarrow]
\end{tikzcd}
\]
\end{proof}
Notice that this corollary subsumes \Cref{prop:OldConstructionOfLSCinGrothendieckTopos}, while this new version is much more easily proven.
\begin{corollary}
Every locally presentable category has a local state classifier.
\end{corollary}
\begin{proof}
Any locally presentable category admits the $(\textrm{StrongEpi}, \textrm{Mono})$-factorization system. It is also total and cowell-powered.
\end{proof}
\invmemo{If we do not assume properness, what is naturally defined might be just a pseudo-functor $\E^{\op} \to \mathbf{CAT}$.}
% This functor $\QuoE$ is rarely representable:
\section{Adding legs to a snake: Classification theorems}
\subsection{Classification of \texorpdfstring{$M$-}{}pullback stable family of \texorpdfstring{$M$-}{}subobjects}
\memo{Consider $M$-subobject classifier, or just $\QuoE \to \SubM$}
\begin{notation}
For a category $\C$ equipped with an $\MC$-powered orthogonal factorization system $(\EC,\MC)$, assuming that $\C$ has pullbacks along $\MC$-morphisms, we define
\[
\SubM \colon \C^{\op} \to \Set
\]
as the presheaf that sends an object $c\in \ob(\C)$ to the set of all isomorphism classes of $\C_{\MC}/c$.
% by the pullback actions.
% if $\C$ has pullbacks along $\MC$-morphisms.
\end{notation}
\begin{remark}[$\SubM$ is NOT dual to $\QuoE$.]
$\SubM$ is not $\QuoE$ for the opposite category $\C^{\op}$ with the opposite factorization system $(\MC^{\op}, \EC^{\op})$. In fact, $\Quo_{\MC^{\op}}\colon \C = \C^{\op\op}\to \Set$ is a covariant (not contravariant!) functor, which has the same object function $c \mapsto \Quo_{\MC^{\op}}(c) = \SubM(c)$. In the case of $\C=\Set$, $\Quo_{\MC^{\op}}$ is the covariant powerset functor, and $\SubM$ is the contravariant powerset functor.
\end{remark}
\begin{proposition}\label{prop:PresheaflevelClassificationTheorem}
For any category $\C$ equipped with an $\EC$-cowell-powered and $\MC$-well-powered orthogonal factorization system $(\EC,\MC)$ and with $\MC$-pullbacks,
there is a one-to-one correspondence between
\begin{itemize}
\item natural transformations $\QuoE \to \SubM$ in $\PSh(\C)$ and
\item families of $\MC$-subobjects $\{S_c\}_{c\in \ob(\C)}$ that is stable under $\MC$-pullbacks.
\end{itemize}
\end{proposition}
\begin{proof}
This is an immediate corollary of \Cref{prop:UniversalityOfQuo} and the Yoneda lemma.
\end{proof}
\begin{corollary}
In the situation of \Cref{prop:PresheaflevelClassificationTheorem}, if $\C$ is total and $\SubM$ is representable $\SubM\cong \C(-, \Omega_{\MC})$, then we can add
\begin{itemize}
\item morphisms $\Xi_{\MC} \to \Omega_{\MC}$ and
\item $\MC$-subobjects of $\Xi_\MC$.
\end{itemize}
to the correspondence in \Cref{prop:PresheaflevelClassificationTheorem}.
\end{corollary}
\begin{proof}
We have $\PSh(\C)(\QuoE, \SubM) = \PSh(\C)(\QuoE, \yo_{\C} (\Omega_{\MC})) \cong\C(L(\Quo_{\EC}), \Omega_{\MC}) = \C(\Xi_\MC, \Omega_{\MC}) \cong \SubM(\Xi_{\MC})$ due to $\Xi_{\MC}= L(\QuoE)$ (\Cref{thm:LocalStateClassifierInTotalCategory}).
\end{proof}
This corollary subsumes the Grothendieck topos case of the broader correspondence theorem in \cite[Proposition 4.5.][]{hora2024internal}.
\begin{remark}[$\QuoE$ is rarely representable]
In contrast to the situation that an $\MC$-subobject classifier represents the presheaf of subobjects (cf. topos, or quasitopos), the presheaf of quotient objects is rarely representable:
If $\QuoE$ is representable, then there exists an object $Q\in \ob(\C)$ such that every object $X\in \ob(\C)$ admits an $\MC$-morphism into $Q$. \memo{Can we say $\MC=\C$?}
\end{remark}
\begin{remark}
\cite{kenney2006copower}
\end{remark}
\begin{example}[Grothendieck quasitopos]
\end{example}
\subsection{Classification with additional structures}
% \section{Examples}
\begin{theorem}[{\cite[Theorem 9][]{wood1982some}}]
A total category $\C$ with $L\dashv \yo_\C$ is cartesian closed if and only if $L$ preserves binary products.
\end{theorem}
Therefore, for a total cartesian closed category $\C$, the $\MC$-local state classifier $\Xi_\MC = L(\QuoE)$ inherits the semilattice structure of $\QuoE$.
\memo{When does it preserve order structure? Related to this question, when is $L$ lex? = $\C$ is a Grothendieck topos! See \cite{street1981notions}}
\begin{example}[Grothendieck quasitopoi]
\end{example}
\begin{example}[local state classifier in a locally presentable category]
\end{example}
\printbibliography
\end{document}