← Notes on advances of LSC
factorisation systems__20250604.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);%
}}}
\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}{\mathcal{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{\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{\mono}{rightarrowtail}
\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}
\title{local state classifier relative to factorization system}
\author{Ryuya Hora}
\thanks{ZEN University. \url{ryuya_hora@zen.ac.jp}}
% \date{\today}
\subjclass[2020]{MSC}
\keywords{Keywords}
\begin{document}
\begin{abstract}
\end{abstract}
\maketitle
\tableofcontents
% \section{Quotient maps}
\section{The presheaf of quotient objects and its univerality}
\begin{definition}
Let $(\EC,\MC)$ be an orthogonal factorization system on a category $\C$.
For each $c\in \ob(\E)$, let $\QuoE(c)$ denote the class of all isomorphism classes of morphisms in $\EC$ from the object $c$ regarded as objects in $c/\C$.
Assuming that $\C$ is \demph{$\EC$-cowell-powered}, which means that $\QuoE(c)$ is small for any $c\in \ob(\C)$, we define the functor
\[
\QuoE\colon \E^{\op} \to \Set
\]
by the factorization (\memo{write it later}).
% \[
% \QuoE(f\colon X \to Y)(e\colon Y \twoheadrightarrow Q) =
% \]
\end{definition}
What can we say about this presheaf $\QuoE\colon \C^{\op} \to \Set$? 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:
% This functor $\QuoE$ is rarely representable:
\begin{remark}[$\QuoE$ is rarely representable]
If $\QuoE$ is representable, then there exists an object $Q\in \E$ such that every object $X\in \ob(\E)$ admits a $\MC$-morphism into $Q$. \memo{Can we say $\MC=\C$?}
\end{remark}
\begin{remark}
\cite{kenney2006copower}
\end{remark}
\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}
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}
\section{Local state classifier relative to factorization system}
\begin{definition}
For a category $\C$ and a wide subcategory $\MC \rightarrowtail\C$, the \demph{$\MC$-local state classifier} is the colimit of the embedding functor $\MC \rightarrowtail \C$ if it exists.
\end{definition}
\begin{example}
If $\MC$ is the class of all monomorphisms, then the $\MC$-local state classifier is the local state classifier in \cite{hora2024internal}.
\end{example}
\begin{example}
The $\C$-local state classifier is the terminal object of $\C$. See \cite{riehl2017category} and \cite{menni2025nonsingular}. \memo{This example is not boring. The monoid structure induced on the terminal object might be useful in game theory or in other coalgebraic contexts.}
\end{example}
\begin{definition}\memo{cite paper}
A locally small category $\C$ is said to be \demph{total} if its yoneda embedding $\C \xhookrightarrow{\yo}\Func{\C^{\op}}{\Set}$ admits a left adjoint.
\end{definition}
In this note, total category means a locally small total category.
\begin{theorem}
% If a locally small category $\C$ with a factorization system $(\EC,\MC)$ satisfies the following conditions:
% \begin{itemize}
% \item $\C$ is total,
% \item $\C$ is $\EC$-cowell-powered,
% \end{itemize}
For any total category $\C$ with $\EC$-cowell powered factorization system $(\EC, \MC)$, $\C$ admits a $\MC$-local state classifier $\Xi_\MC$. Furthermore, the $\MC$-local state classifier $\Xi_\MC$ is given by
\[
\Xi_\MC = L(\QuoE),
\]
where $L\colon \Func{\C^{\op}}{\Set}\to \C$ denotes the left adjoint to the yoneda embedding.
\end{theorem}
Any locally presentable categories have LSC, due to the strong-epi mono factorization system.
\appendix
\section{Preliminaries on total categories}
\printbibliography
\end{document}