\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{\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{\PSh}{\mathbf{PSh}} \newcommand{\Sh}{\mathbf{Sh}} \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}} \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{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} The notion of a local state classifier is defined to be a colimit of all monomorphisms. This note aims to prove a general existence theorem for such a large colimit. First, we generalize the notion of a local state classifier to general orthogonal factorization system (not only the epi-mono factorization system in a topos). Then, we prove that the generalized local state classifier relative to a suitable factorization system on an arbitrary total category exists 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}] 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. \subsection{Definition and examples} \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 $\C \xhookrightarrow{\yo}\Func{\C^{\op}}{\Set}$ admits a left adjoint $L\colon \Func{\C^{\op}}{\Set} \to \C$. \end{definition} In this note, total category means a locally small total category. \begin{example}[Easiest example] The easiest category that one can check the totality is the terminal category $\1$. The yoneda embedding \[ \yo_{\1}\colon \1 \hookrightarrow \Set \] admits a left adjoint, which is given by the unique functor $\Set \to \1$. % The category of sets is \cite{rosebrugh1994adjoint} \end{example} \begin{example}[Complete lattice] Let $P$ be a small complete lattice. Then, we can prove that $P$ is total by showing 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 example of a total and large category is a presheaf category. Let $J$ be an arbitrary small category. Then, one can prove that the pre-composition functor ${-}\circ\yo_{J} \colon \PSh(\PSh(J)) \to \PSh(J)$ is a left adjoint to the Yoneda embedding functor $\yo_{\PSh(J)} \colon \PSh(J) \hookrightarrow \PSh(\PSh(J))$. (In the case $J=\1$, this is a part of the adjoint $5$-tuple in \cite{rosebrugh1994adjoint}.) \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] Any total category is small cocomplete. Furthermore, For any total category $\C$ with $L \dashv \yo_{\C}$ and any (possibly large) diagram $F\colon J \to \C$, if \[ \C(c, F{-})\colon J \xrightarrow{F} \C \xrightarrow{\C(c,-)}\Set \] admits a colimit for any $c\in \C$, then there exists a colimit of $F\colon J \to \C$, which is given by \[ \colim_{j\in J} Fj = L\left( \colim_{j\in J} \C(c, F{j})\right). \] In particular, if the comma category $c \downarrow F$ has only small number of connected components for each $c\in \C$, then $F$ admits a colimit. \[ \colim_{J} F = L\left(\pi_0(c\downarrow F)\right). \] \end{proposition} \section{Orthogonal factorization system} \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} \subsection{Local state classifier relative to factorization system} \begin{notation} For an orthogonal factorization systems $(\EC,\MC)$ 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 systems $(\EC,\MC)$, the \demph{$\MC$-local state classifier} is the colimit of the embedding functor $\C_\MC \rightarrowtail \C$, 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. \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} \subsection{The presheaf of quotient objects and its univesality} \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 $c/\C_{\EC}$. % morphisms in $\EC$ from the object $c$ regarded as objects in $c/\C$. \end{definition} 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 is well-defined since the factorization is unique up to a (unique) isomorphism. \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 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{notation} 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} \memo{write} \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} \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 \texorpdfstring{$M$-}{}pullback stable family of \texorpdfstring{$M$-}{}subobjects.} \memo{Consider $M$-subobject classifier} \section{Examples} \memo{nLab: A total category is cartesian closed iff L preserves binary products (cf. Wood 1982, Thm. 9).} \begin{example}[Grothendieck quasi-topoi] \end{example} \begin{example}[local state classifier in a locally presentable categories] \end{example} \appendix \section{A direct proof of the universality of \texorpdfstring{$\QuoE$}{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}