\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}} % \newcommand{\mono}{\mathrm{mono}} % \title{Demystifying local state classifiers:\\ % % local state classifier in % total categories with factorization systems} \title{Local state classifier characterizes the Axiom of choice} \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{LSC and set theoretic models} \section{LSC and etendue}\label{sec:etendue} \begin{conjecture} A Grothendieck topos $\E$ is an étendue if and only if its local state classifier $\Xi$ admits the minimum element $\bot \colon 1_{\E} \to \Xi$. \end{conjecture} \begin{conjecture}[Subconjecture] A Grothendieck topos $\E$ satisfies the internal axiom of choice if and only if $\E$ is Boolean and its local state classifier $\Xi$ admits the minimum element $\bot \colon 1_{\E} \to \Xi$. \end{conjecture} Due to the following fact, studying LSC of \'{e}tendues is close to studying all Grothendieck topoi. \begin{fact}[{\cite[][Theorem 3.1]{rosenthal1982quotient}}] For every Grothendieck topos $\F$, there is an \'{e}tendue $\E$ and a hyperconnected geometric morphism $\E\to \F$. \end{fact} \memo{How is this Rosenthal's covering theorem rephrased by LSC?} The concept of topos as a "generalized locale" plays the role of a “space with rich self-automorphisms,” or, in other words, a “folded space.” In the order structure of LSC, a more “collapsed” state is considered larger, while a more “unfolded” state is considered smaller. For instance, in the topos of graphs, a loop edge is larger than a non-loop edge. In the topos of group actions, a trivial action on a point is larger than a free action. So, when does LSC have a minimum (global) element? The slogan would be, “the existence of the unfolding,” and this is none other than an étendue! \begin{conjecture} \label{Conj:etendue} For a Grothendieck topos $\E$, the following conditions are equivalent (?) \begin{itemize} \item its LSC $\Xi$ has a bottom $\bot \colon 1 \to \Xi$ \item $\E$ is \'{e}tendue. \end{itemize} \end{conjecture} Plan: This conjecture will be proven by rewriting \cite{kock1991presentations} in terms of LSC. \begin{fact} A Grothendieck topos $\E$ is \begin{itemize} \item \'{e}tendue if and only if it has a site $(\C,J)$, where all morphisms of $\C$ are monic. \cite{kock1991presentations} \item Boolean \'{e}tendue iff $\E$ satisfies the internal axiom of choice. \end{itemize} A presheaf topos $\PSh(\C)$ is \'{e}tendue if and only if all morphisms of $\C$ are monic. \cite{rosenthal1981etendues} \end{fact} \begin{proposition} \Cref{Conj:etendue} is true for \begin{itemize} \item localic topoi \item presheaf topoi \end{itemize} \end{proposition} \begin{proof} For a localic topoi $\E$, it's trivial since $\E$ is \'{e}tendue and its LSC is terminal. For a presheaf, the LSC $\Xi$ is the presheaf of all quotient objects of the representables. By the concrete calculation, $\Xi$ has a bottom, if and only if $\yo(f)\colon \yo(c) \to \yo(d)$ is monic for every $f\colon c\to d$, which means every morphism in $\C$ is monic. \end{proof} See \cref{rmk:ExternallyPrincipal} as well. \begin{example} $\Cont(\hat{\Z})$ is not \'{e}tendue, since it does not satisfy the internal axiom of choice \cite{freyd1980axiom, freyd1990categories}. Its lsc does not have the bottom, see \cref{exmp:LSCofLOOPS}. \end{example} \subsection{Rewriting Kock and Meordijk} This subsection aims to rewrite \cite{kock1991presentations} in terms of a local state classifier. \begin{definition}[\cite{kock1991presentations}] For a geometric morphism $\gamma \colon \E \to \S$ between two elementary topoi, a morphism $f \colon A \to B$ is said to be \demph{locally monic} relative to $\gamma$, if there exist the diagram \[ \begin{tikzcd} A' \ar[r, rightarrowtail, "f'"]\ar[d, \epi, "q"]& \gamma^* I \times B\ar[d, "\mathrm{proj}"]\\ A\ar[r, "f"] & B, \end{tikzcd} \] where $f'$ is monic and $q$ is epic. \end{definition} For a Grothendieck topos $\E$, a morphism $f$ in $\E$ is said to be locally monic if it is locally monic relative to the global section geometric morphism $\gamma \colon \E \to \Set$. For a morphism $f\colon A \to B$, $f$ is locally monic if and only if there is a covering $\{U_{\lambda}\toMono A\}_{\lambda\in \Lambda}$ such that each restricted morphism \[ % f\restriction_{U_\lambda} \colon U_{\lambda} \toMono A \to B f|_{U_\lambda} \colon U_{\lambda} \toMono A \to B \] is monic. \begin{conjecture} A morphism $f\colon A \to B$ in a Grothendieck topos $\E$ is locally monic (relative to the global section geometric morphism) if and only if \[ \begin{tikzcd}[column sep = 10pt] A \ar[rr,"f"]\ar[rd,"\xi_A"'] &&B\ar[ld, "\xi_B"]\\ &\Xi& \end{tikzcd} \] commutes. \end{conjecture} Colloquially, this conjecture states that ‘locally monic’ is equivalent to saying that it preserves local states without collapsing them. See \cref{lem:FoldingLemma}. This conjecture holds for localic topoi and presheaf topoi. \begin{lemma} If a morphism $f\colon A \to B$ in a Grothendieck topos $\E$ is locally monic, then \[ \begin{tikzcd}[column sep = 10pt] A \ar[rr,"f"]\ar[rd,"\xi_A"'] &&B\ar[ld, "\xi_B"]\\ &\Xi& \end{tikzcd} \] commutes. \end{lemma} \begin{proof} There is a covering $\{U_{\lambda}\toMono A\}_{\lambda\in \Lambda}$ such that each restricted morphism \[ % f\restriction_{U_\lambda} \colon U_{\lambda} \toMono A \to B f|_{U_\lambda} \colon U_{\lambda} \toMono A \to B \] is monic. Then, the outside square of \[ \begin{tikzcd} &U_{\lambda}\ar[ld, rightarrowtail]\ar[rd, rightarrowtail, "{f|_{U_{\lambda}}}"]&\\ A\ar[rd, "\xi_A"']\ar[rr,"f"]&&B\ar[ld, "\xi_B"]\\ &\Xi& \end{tikzcd} \] commutes. Since $\{U_{\lambda}\toMono A\}_{\lambda\in \Lambda}$ is jointly epimorphic, this completes the proof. \end{proof} (For the converse question: relationship with the existence of reduced subobjects coverings) \subsection{Torsion-free objects} \begin{lemma}\label{lem:TorsionFreeAsBottom} For an object $X$ of a topos $\E$ with a local state classifier $\Xi$, the following conditions are equivalent: \begin{itemize} \item $\pi\colon \Xi_X \to X$ is the terminal object of $\E/X$ \item For any object $Y$ and any map $f\colon Y \to X$, the composite map $\xi_X \circ f\colon Y \to \Xi$ % \in \E(Y, \Xi)$ is an bottom element in the $\land$-semilattice $\E(Y, \Xi)$ \end{itemize} \end{lemma} \begin{conjecture} The above condition should be also equivalent to \begin{itemize} \item $\E/X$ is localic \item $X$ is torsion-free in the sense of \cite{kock1991presentations}. \end{itemize} \end{conjecture} \subsection{Inhabitedness} \begin{lemma} If an object $X$ satisfies the condition in \Cref{lem:TorsionFreeAsBottom}, then the map $\xi_X \colon X \to \Xi$ factors through the support of $X$: \[ \begin{tikzcd} X\ar[d, twoheadrightarrow]\ar[rd, "\xi_X"]&\\ T\ar[d, rightarrowtail]\ar[r, rightarrowtail, dashed, "\exists"]& \Xi\\ 1_\E& \end{tikzcd} \] \end{lemma} \begin{proof} Due to \cref{lem:TorsionFreeAsBottom}, the diagram \[ \begin{tikzcd} X\times_{\Xi} X\ar[r,shift left, "\pi_1"]\ar[r,shift right, "\pi_2"']&X\ar[r, "\xi_X"]& \Xi \end{tikzcd} \] commutes. The lemma follows since the support of $X$, denoted by $T$, is the coequalizer of this diagram, since a topos is regular. \end{proof} \begin{lemma} If there exists an inhabited object that satisfies the condition in \Cref{lem:TorsionFreeAsBottom}, then $\Xi$ has a global bottom element $\bot\colon 1_\E \to \Xi$. \end{lemma} Assuming the next conjecture \begin{conjecture} In any Grothendieck topos $\E$ (or its relativization), there is an object $B$ such that $\xi_B \colon B \to \Xi$ is epic. \end{conjecture} which is closely related to \Cref{sec:Bounds}, we can construct an inhabited and torsion-free (in the sense of \Cref{lem:TorsionFreeAsBottom}) object $X$ by a pullback: \[ \begin{tikzcd} X\ar[r, twoheadrightarrow, "!"]\ar[d, "\iota", rightarrowtail]\arrow[rd, phantom, "\lrcorner", very near start]&1\ar[d, "\bot", rightarrowtail]\\ B\ar[r, twoheadrightarrow, "\xi_B"]&\Xi, \end{tikzcd} \] since we have \[ \begin{tikzcd} X\ar[r, twoheadrightarrow, "!"]\ar[d, "\iota", rightarrowtail]\arrow[rd, "\xi_X"]&1\ar[d, "\bot", rightarrowtail]\\ B\ar[r, twoheadrightarrow, "\xi_B"]&\Xi \end{tikzcd} \] \appendix \section{LSC of continuous action toposes} \begin{corollary} For a topological group $G$, the LSC of $\Cont(G)$ is the set of open subgroups of $G$, equipped with the right conjugate actions. \end{corollary} \begin{example}[LSC of loops] \label{exmp:LSCofLOOPS} The LSC of $\Cont(\hat{\Z})$ is the semilattice of positive integers with the (reversed) divisibility order and the trivial action. \end{example} \section{LSC of slice topos \memo{ongoing}}\label{sec:LSCofSliceTopos} Motivated by \cref{sec:etendue}, we will describe the LSC of the slice topos. Our starting point is the next lemma, which is proven in \cite{hora2024internal}. \begin{lemma}[{\cite{hora2024internal}}]\label{lem:FoldingLemma} For any morphism $f\colon Y \to X$, \[ \xi_Y \leq \xi_X \circ f. \] \end{lemma} This allows us to define \[ \begin{tikzcd} Y\ar[rdd, bend right, "f"'] \ar[rrd, bend left, "{\langle \xi_Y, \xi_X \circ f\rangle}"]\ar[rd, dashed, "\zeta_f"]&&\\ &P\ar[r]\ar[d, twoheadrightarrow]\arrow[rd, phantom, "\lrcorner", very near start]&{\leq_{\Xi}}\ar[d, twoheadrightarrow, "\pi_2"]\\ &X\ar[r, "\xi_X"]&\Xi \end{tikzcd} \] \begin{definition} For a topos $\E$ with a local state classifier $\Xi$ and an object $X\in \ob(\E)$, we define $\Xi_X$ by \[ \Xi_X\coloneqq \{(s,x)\in \Xi\times X\mid s\leq \xi_X(x)\} \] interpreted in the internal language in $\E$. \end{definition} This is exactly the same as the pullback \[ \begin{tikzcd} \Xi_X\ar[r]\ar[d, twoheadrightarrow]\arrow[rd, phantom, "\lrcorner", very near start]&{\leq_{\Xi}}\ar[d, twoheadrightarrow, "\pi_2"]\\ X\ar[r, "\xi_X"]&\Xi, \end{tikzcd} \] \begin{example} \end{example} \begin{conjecture} For a topos $\E$ with LSC $\Xi$, the LSC of the slice topos $\E/X$ is given by the pullback \[ \begin{tikzcd} P\ar[r]\ar[d, twoheadrightarrow]\arrow[rd, phantom, "\lrcorner", very near start]&{\leq_{\Xi}}\ar[d, twoheadrightarrow, "\pi_2"]\\ X\ar[r, "\xi_X"]&\Xi, \end{tikzcd} \] with the cocone maps $\{\zeta_f \colon Y \to P\}_{f\colon Y \to X}$ \end{conjecture} \section{Relationship with bounds \memo{ongoing}}\label{sec:Bounds} There should be some connection with the notion of bound and LSC. The reasons why I think so include \begin{itemize} \item Every Grothendieck ($\Set$-bounded) topos has a LSC, every finite presheaf ($\FinSet$-bounded) topos (over finite category) has a LSC, but $\FinSet^{\Z}$, which is not bounded over $\FinSet$ doesn't. \item Informally speaking, an object $B$ is a bound, if and only if \dq{every state of every object is a quotient state of a state of $B$.} \end{itemize} \begin{conjecture} If a topos $\E$ has a local state classifier, then every bounded $\E$-topos has a local state classifier. \end{conjecture} \begin{conjecture} An object $X$ of a Grothendieck topos $\E$ is a bound, if and only if $\xi_{X}\colon X\to \Xi$ is downward unbounded, in the sense that every upward closed subobject of $\Xi$ containing $\Image(\xi_{X})$ is $\Xi$ itself. \end{conjecture} \begin{example} Even if the hyperconnected quotient generated by $B$ is $\E$ itself, $\E$ might not be a bound. For example, the object $B \coloneqq \Z/2\Z + \Z/3\Z$ in $\PSh(\Z/6\Z)$ is not a bound, but every non-trivial hyperconnected quotient does not contain $B$. \end{example} So what we need to consider is the \demph{broader correspondence} in \cite{hora2024internal}. \memo{And is related to Menni's paper \cite{menni2021hyperconnected}.} \begin{conjecture}[\memo{Proven}, This is also proven by P.T. Johnstone. Its presheaf case is proven in \cite{menni2025nonsingular}] As a restriction of \cite[][broader correspondence]{hora2024internal}, we obtain a one-to-one correspondence between \begin{itemize} \item Upward closed subobject of $\Xi$, and \item Coreflective full subcategory closed under subquotients. \item Order-preserving map $\Xi \to \Omega$ \end{itemize} \end{conjecture} \memo{It may subsume the monic skelta by Menni} \begin{example}[Galois theory]\memo{Check and generalize it} % Let $K$ be a nice field (like a field with characteristic $0$), and Let $\Gal(\overline{\mathbb{F}_{p}}/\mathbb{F}_{p})$ be the absolute Galois group of a finite field $\mathbb{F}_{p}$. The algebraic closure equipped with the action $\Gal(\overline{\mathbb{F}_{p}}/\mathbb{F}_{p})$ is an internal ring of $\Cont(\Gal(\overline{\mathbb{F}_{p}}/\mathbb{F}_{p}))$, with surjective $\xi_{\overline{\mathbb{F}_{p}}}$. For each open subgroup $S \subset \Gal(\overline{\mathbb{F}_{p}}/\mathbb{F}_{p})$ and its corresponding hyperconnected geometric morphism $\Cont(\Gal(\overline{\mathbb{F}_{p}}/\mathbb{F}_{p})) \to \PSh(\Gal(\overline{\mathbb{F}_{p}}/\mathbb{F}_{p})/S)$, the counit $K_S \rightarrowtail \overline{\mathbb{F}_{p}}$ is the embedding of the Galois-correspondant. \end{example} % \section{Presentability} % Not only every Grothendieck topos, but every category of models of an equational theory has an LSC. This might be generalized as follows: % \begin{conjecture} % Every locally presentable category has a local state classifier. % \end{conjecture} \section{Internally complete semilattice} \begin{definition} For an internal poset $P$ in a topos $\E$ and an object $I$, \demph{the $I$-indexed meet} $\land_I$ is the internal right adjoint of the diagonal morphism \[ \Delta_I \colon \Xi\to \Xi^{I}. \] \memo{check} Externally speaking, the $\land_I$ exists if and only if there exists a right adjoint to \[ {-}\circ \pi_1:\E(X, \Xi) \to \E(X\times I, \Xi) \] for any $X\in \ob(\E)$ that is natural in $X$. \end{definition} \memo{Possibly, the right condition might be the internal completeness, not the special case of it, namely the existence of the bottom object.} \printbibliography \end{document}