\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}