← Notes on advances of LSC
factorisation systems__20251109.tex
\documentclass{amsart}
\usepackage[left=2cm, right=2cm]{geometry}
\usepackage[utf8]{inputenc}
\usepackage{amsfonts, amsthm, amssymb, mathtools,etoolbox}
\usepackage{blindtext}
\usepackage[colorlinks=true, urlcolor=blue, linkcolor=blue, citecolor=blue]{hyperref}
\usepackage{tikz,tikz-cd}
\usepackage{cleveref}
\usepackage{array}
\usepackage[style=alphabetic,sorting=nyt]{biblatex}
\renewbibmacro{in:}{}
% \addbibresource{biblio.bib}
\addbibresource{CommonBiblio20240922.bib}
\tikzset{pullback/.style={minimum size=1.2ex,path picture={
\draw[opacity=1,black,-,#1] (-0.5ex,-0.5ex) -- (0.5ex,-0.5ex) -- (0.5ex,0.5ex);%
}}}
\usetikzlibrary{calc}
\theoremstyle{plain}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{todo}[theorem]{Todo}
\newtheorem{conjecture}[theorem]{Conjecture}
\newtheorem{fact}[theorem]{Fact}
\theoremstyle{definition}
\newtheorem{example}[theorem]{Example}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{remark}[theorem]{Remark}
\newtheorem{notation}[theorem]{Notation}
\newtheorem{question}[theorem]{Question}
\newtheorem{idea}[theorem]{Idea}
\newcommand{\dq}[1]{``#1"}
\newcommand{\memo}[1]{\textcolor{red}{memo: #1}}
% \newcommand{\invmemo}[1]{\textcolor{blue}{memo: #1}}
\newcommand{\invmemo}[1]{}
\newcommand{\para}[1]{\paragraph{\textbf{#1}}}
\newcommand{\N}{\mathbb{N}}
\newcommand{\Nor}{\mathrm{N}}
\newcommand{\Z}{\mathbb{Z}}
\newcommand{\R}{\mathbb{R}}
\newcommand{\HQ}{\mathcal{HQ}}
\newcommand{\C}{\mathcal{C}}
\newcommand{\D}{\mathcal{D}}
\newcommand{\E}{\mathcal{E}}
\newcommand{\F}{\mathcal{F}}
\renewcommand{\S}{\mathcal{S}}
\newcommand{\G}{\mathbb{G}}
\newcommand{\id}{\mathrm{id}}
\newcommand{\op}{\mathrm{op}}
\newcommand{\ob}{\mathrm{ob}}
\newcommand{\true}{\mathrm{true}}
\newcommand{\Image}{\mathrm{Im}}
\newcommand{\Sub}{\mathrm{Sub}}
\newcommand{\Mor}{\mathrm{Mor}}
\newcommand{\cod}{\mathrm{cod}}
\newcommand{\dom}{\mathrm{dom}}
\newcommand{\Set}{\mathbf{Set}}
\newcommand{\Cont}{\mathbf{Cont}}
\newcommand{\FinSet}{\mathbf{FinSet}}
\newcommand{\Sh}{\mathbf{Sh}}
\newcommand{\PSh}{\mathbf{PSh}}
\newcommand{\Top}{\mathbf{Top}}
\newcommand{\sgt}{\{\cdot\}}
% \newcommand{\Func}[2]{[#1,#2]}
\newcommand{\abs}[1]{\left|#1\right|}
\newcommand{\demph}[1]{\textbf{#1}}
% \font\maljapanese=dmjhira at 2.5ex
% \newcommand{\yo}{\textrm{\!\maljapanese\char"48}}
% \newcommand{\yo}{y}
\newcommand{\yo}{Y}
\newcommand{\mono}{\mathrm{mono}}
\newcommand{\epi}{twoheadrightarrow}
\newcommand{\toMono}{\rightarrowtail}
\newcommand{\Gal}{\mathrm{Gal}}
\newcommand{\toEpi}{\twoheadrightarrow}
\newcommand{\Quo}{\mathrm{Quo}}
\newcommand{\A}{\mathcal{A}}
\newcommand{\EC}{\mathbf{E}}
\newcommand{\MC}{\mathbf{M}}
\newcommand{\QuoE}{\Quo_{\EC}}
\newcommand{\SubM}{\Sub_{\MC}}
\DeclareMathOperator*{\colim}{colim}
\newcommand{\ev}{\mathrm{ev}}
\newcommand{\1}{\mathbf{1}}
\renewcommand{\a}{\mathbf{a}}
\newcommand{\ADJ}[4]
{
\begin{tikzcd}[ampersand replacement = \&, column sep = small]
{#1}
\ar[rr, shift right=1.3ex, "{#2}"']
\&\perp\&
{#3}
\ar[ll, shift right=1.3ex,"{#4}"']
\end{tikzcd}
}
\newcommand{\Iso}{\mathrm{Iso}}
% \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}
\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:
% }
% \subsection{
Local state classifier and its construction problem}
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
\[
\Xi\coloneqq \colim (\C_{\mathrm{mono}}\rightarrowtail \C)
\]
if it exists.
\end{definition}
Even if $\C$ is (small) cocomplete, $\C$ might not admit a local state classifier, since the indexing category $\C_{\mathrm{mono}}$ is not small. In the same paper \cite{hora2024internal}, the author proved that any Grothendieck topoi admits a local state classifier by concretely constructing it.
\begin{proposition}[\cite{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 following 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, because 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, and
\item orthogonal factorization systems.
\end{itemize}
From the next section, we will see them one by one.
\section{Total categories}
This section aims to recall the definition and basic properties of total categories. Nothing in this section could be 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 $u_0, u_1$. $\Set$ is the $u_1$-small category of $u_0$-small sets. Then, for any $u_1$-small category $\C$, $\PSh(\C)$ is 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 $\C \xhookrightarrow{\yo}\PSh(\C)$ 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 means a 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}
\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}
\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}