\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{\mono}{rightarrowtail} \newcommand{\epi}{twoheadrightarrow} \newcommand{\toMono}{\rightarrowtail} \newcommand{\Gal}{\mathrm{Gal}} \newcommand{\toEpi}{\twoheadrightarrow} \newcommand{\Quo}{\mathrm{Quo}} \newcommand{\A}{\mathcal{A}} \title{More on LSC} \author{Ryuya Hora} \thanks{ZEN University. \url{ryuya_hora@zen.ac.jp}} % \date{\today} \subjclass[2020]{MSC} \keywords{Keywords} \begin{document} \begin{abstract} This is a personal note on the notion of a local state classifier. This includes some new theorems and conjectures. \end{abstract} \maketitle \tableofcontents The original paper is \cite{hora2024internal}. \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{LSC of hyperquotient} \begin{definition}[Normalizer morphism] \label{def:normalizer} In a topos $\E$ with a local state classifier $\Xi$, we call the morphism $\xi_{\Xi}\colon \Xi\to \Xi$ \demph{the normalizer morphism}. % and write it as $\Nor_{\E} \coloneqq \xi_{\Xi}$. \end{definition} \begin{example}[Prototypical example] In the group action topos $\PSh(G)$ for a group $G$, the local state classifier $\Xi$ is the set of subgroups with the right conjugate actions \cite[][Example 3.10]{hora2024internal}. The normalizer morphism $\xi_{\Xi}$ sends a subgroup $H\subset G$ to its normalizer group $H\subset \Nor_{G}(H) \subset G$. \end{example} \begin{remark} The normalization morphism is NOT a semilattice homomorphism on $\Xi$. It does not preserve even the order structure. For example, in the topos $\PSh(S_3)$ of the symmetric group $S_3$-actions, the subgroup $H \coloneqq \langle(1,2)\rangle$ is larger than $\{e\} \subset G$, but $\Nor_{S_3}(H)= H \subsetneq S_3 =\Nor_{S_3}(\{e\})$. \end{remark} The next conjecture is a generalization of the inclusion relation $H \subset \Nor_G(H)$. \begin{proposition}[Normalization lemma] \label{prop:NormalizationLemma} In a topos $\E$ with a local state classifier $\Xi$, the morphism $\xi_{\Xi}\colon \Xi\to \Xi$ is equal to or larger than $\id_{\Xi}$ \[ \id_{\Xi} \leq \xi_{\Xi} \] as an element of semilatttice $\E(\Xi, \Xi)$ \end{proposition} \begin{proof} % To prove $\id_{\Xi}\leq \xi_{\Xi}$, we need to prove the commutativity of the following diagram: % \[ % \begin{tikzcd} % &\Xi\ar[ld,"{\langle\id_{\Xi}, \xi_{\Xi}\rangle}"'] % \ar[rd,"\id_{\Xi}"]&\\ % \Xi\times \Xi\ar[rr,"\land"]&&\Xi. % \end{tikzcd} % \] % Since $\Xi$ is a colimit, it suffices to prove the following commutativity for each object $X\in \ob(\E)$. % \[ % \begin{tikzcd} % &X\ar[d,"\xi_X"]\ar[ldd, bend right, ""']&\\ % &\Xi\ar[ld,"{\langle\id_{\Xi}, \xi_{\Xi}\rangle}"'] % \ar[rd,"\id_{\Xi}"]&\\ % \Xi\times \Xi\ar[rr,"\land"]&&\Xi. % \end{tikzcd} % \] To prove $\id_{\Xi}\leq \xi_{\Xi}$, we need to prove that the composite of \[ \begin{tikzcd}[column sep = 50pt] \Xi\ar[r,"{\langle\id_{\Xi}, \xi_{\Xi}\rangle}"]& \Xi\times \Xi\ar[r,"\land"]&\Xi. \end{tikzcd} \] is the identity. Since $\Xi$ is a colimit, it suffices to prove the following commutativity for each object $X\in \ob(\E)$. \[ \begin{tikzcd}[column sep = 50pt] X\ar[d,"\xi_X"']\ar[rr,bend left, "\xi_X"]&&\Xi\ar[d,equal]\\ \Xi\ar[r,"{\langle\id_{\Xi}, \xi_{\Xi}\rangle}"]& \Xi\times \Xi\ar[r,"\land"]&\Xi \end{tikzcd} \] By the definition of $\land$ operation and the fact that ${\langle\id_X, \xi_X\rangle}$ is a (split) monomorphism, we have the next commutative diagram. \[ \begin{tikzcd}[column sep = 50pt] X\ar[d,"\xi_X"']\ar[rr,bend left, "\xi_X"]\ar[r, "{\langle\id_X, \xi_X\rangle}", tail]&X\times \Xi\ar[d,"{\xi_X \times \xi_{\Xi}}"]\ar[r,"\xi_{X\times \Xi}"]&\Xi\ar[d,equal]\\ \Xi\ar[r,"{\langle\id_{\Xi}, \xi_{\Xi}\rangle}"]& \Xi\times \Xi\ar[r,"\land"]&\Xi \end{tikzcd} \] This completes the proof. \end{proof} \begin{corollary} \label{cor:FilterLivesInHQuotient} For a topos $\E$ with a local state classifier $\Xi$, an internal filter $F\rightarrowtail \Xi$ is an object of the corresponding hyperconnected quotient. \end{corollary} \begin{proof} Since the filter $F$ is upward closed, \cref{prop:NormalizationLemma} completes the proof. \end{proof} % \begin{conjecture} % If a topos $\E$ has a local state classifier $\Xi$ and $\F$ is a hyperquotient corresponding to a filter $F\rightarrowtail \Xi$, then $F$ is the local state classifier of $\F$. % \end{conjecture} \begin{question} In the case of groups, the normal subgroups are in one-to-one correspondence with the essential quotients. How about general topoi? Can we generalize it using the normalizer operator? \end{question} \begin{remark} \label{rmk:NotationOdXiF} For a hyperconnected geometric morphism $f\colon \E \to \F$ corresponding to an internal filter $F\rightarrowtail \Xi$, the morphism $\xi_Z \colon Z \to \Xi$ for $Z\in \ob(\F)$ lists along $F\rightarrowtail \Xi$ (see \cite{hora2024internal}). By abuse of notation, the lift $Z\to F$ is also denoted by $\xi_Z$. In this notation, \cite{hora2024internal} also shows that the following diagram is a pullback square \[ \begin{tikzcd} \G X\ar[r, "\xi_{\G X}"]\ar[d, "\epsilon_X", tail]& F\ar[d,tail]\\ X\ar[r , "\xi_X"']& \Xi \end{tikzcd} \] for every $X\in \ob(\E)$. \end{remark} \begin{lemma} \label{lem:JointlyEpimorphic} For a hyperconnected geometric morphism $f\colon \E \to \F$ corresponding to an internal filter $F\rightarrowtail \Xi$, % Let $f\colon \E \to \F$ be a hyperconnected geometric morphism, $\G\coloneqq f^{\ast}f_{\ast}\colon \E \to \E$ be the corresponding lex comonad, and $\epsilon\colon \G \to \id_{\E}$ be its counit. the (possibly large) family of morphisms $\{\xi_{\G X} \colon \G X \to F\}_{X\in \ob(\E)}$ is jointly epimorphic, where $\G$ denotes the corresponding lex comonad. \end{lemma} \begin{proof} Take an arbitrary subobject $S\rightarrowtail F$ such that every arrow in $\{\xi_{\G X} \colon\G X \to F\}_{X\in \ob(\E)}$ lifts along $S\rightarrowtail F$. It suffices to prove that $S=F$. (If there are two morphisms $f,g \colon F \to Z$ that are not distinguished by any morphisms in the family, then one can take their equalizer as $S$.) For every object $X \in \ob(\E)$, by the pullback square (\cref{rmk:NotationOdXiF}) \[ \begin{tikzcd} \G X\ar[r, "\xi_{\G X}"]\ar[d, "\epsilon_X", tail]& F\ar[d,tail]\\ X\ar[r , "\xi_X"']& \Xi \end{tikzcd} \] and its epi-mono factorization \[ \begin{tikzcd} \G X\ar[r, two heads]\ar[rr, bend left, "\xi_{\G X}"]\ar[d, "\epsilon_X", tail]&\Image(\xi_X)\land F\ar[r, tail]\ar[d, tail]& F\ar[d,tail]\\ X\ar[rr, bend right , "\xi_X"']\ar[r, two heads]&\Image(\xi_X)\ar[r, tail]& \Xi, \end{tikzcd} \] we have an inequality \[ \Image(\xi_X)\land F = \Image(\xi_{\G X}) \leq S \] in the Heyting algebra $\Sub(\Xi)$. This implies that $\Image(\xi_X) \leq (F \mathbin{\rightarrow} S)$. Since $\{\xi_X \colon X \to \Xi\}_{X \in \ob(\E)}$ is jointly epimorphic (because it is a colimit cocone), we have $\Xi = F \mathbin{\rightarrow} S$, i.e., $F\leq S$. This completes the proof. \end{proof} \begin{lemma} \label{lem:sgtExtNaturality} For a monomorphism $m \colon X\rightarrowtail Y$ in a topos, \[ \begin{tikzcd} X\ar[r,"\sgt_{X}", tail]\ar[d,tail, "m"]&PX\\ Y\ar[r,"\sgt_{Y}", tail]&PY\ar[u,"m^{-1}"'] \end{tikzcd} \] commutes. \end{lemma} \begin{proof} Consider the following commutative diagram: \[ \begin{tikzcd}[row sep = 10 pt] X\times X\ar[rd,"\delta_X"]\ar[dd,"m\times m"', tail]&\\ & \Omega\\ Y \times Y\ar[ru, "\delta_Y"'] \end{tikzcd} \] and take the transposes of it. \end{proof} \begin{lemma} \label{lem:sgtNaturality} For a morphism $f\colon X\to Y$ in a topos, \[ \begin{tikzcd} X\ar[r,"\sgt_{X}", tail]\ar[d, "f"]&PY\ar[d,"\exists_f"]\\ Y\ar[r,"\sgt_Y", tail]&PY \end{tikzcd} \] commutes. \end{lemma} \begin{proof} Both maps are corresponding to the subobject \[ \langle\id_X, f \rangle \colon X \rightarrowtail X\times Y \] as an element of $\E(X, PY) \cong \Sub(X\times Y)$. \end{proof} \begin{lemma} \label{lem:ExtensionLemma} Let $f\colon \E \to \F$ be a hyperconnected geometric morphism, $\G\coloneqq f^{\ast}f_{\ast}\colon \E \to \E$ be the corresponding lex comonad, and $\epsilon\colon \G \to \id_{\E}$ be its counit. For a locally determined cocone $\{\phi_Z\colon Z\to L\}_{Z \in \ob(\F)}$, \[ \begin{tikzcd} \{\psi_X\colon X\ar[r,"\sgt_X"] &PX \ar[r,"{\epsilon_X}^{-1}"]&P\G X\ar[r,"\exists_{\phi_{\G X}}"] & PL\}_{X\in \ob(\E)} \end{tikzcd} \] defines a locally determined cocone on $\E$. Furthermore, $\psi$ is an extension of $\phi$ in the sense of \[ \begin{tikzcd} \G X \ar[r, "\phi_{\G X}"]\ar[d, tail, "\epsilon_X"]& L\ar[d, "\sgt_L", tail]\\ X\ar[r, "\psi_X"]& PL \end{tikzcd} \] \end{lemma} \begin{proof} For any monomorphism $m\colon X\rightarrowtail Y$ in $\E$, we have a commutative diagram \[ \begin{tikzcd}[row sep = 10pt] X\ar[r,"\sgt_X"] \ar[dd, "m", tail]&PX \ar[r,"{\epsilon_X}^{-1}"]\ar[dd, "\exists_m"]&P\G X\ar[rd,"\exists_{\phi_{\G X}}"]\ar[dd,"\exists_{\G m}"]&\\ &&&PL\\ Y\ar[r,"\sgt_X"] &PY \ar[r,"{\epsilon_X}^{-1}"]&P\G Y\ar[ru,"\exists_{\phi_{\G Y}}"'] & \end{tikzcd} \] since the left square commutes by \cref{lem:sgtNaturality}, the right triangle commutes by the assumption of $\phi$ being locally determined, and the middle square commutes by the Beck-Chevalley condition for the pullback square \[ \begin{tikzcd} \G X \ar[r, "\G m", tail]\ar[d,"\epsilon_X", tail]&\G Y\ar[d,"\epsilon_Y", tail]\\ X\ar[r,"m", tail] & Y. \end{tikzcd} \] This proves that $\psi$ is locally determined. To prove \[ \begin{tikzcd} \G X \ar[rrr, "\phi_{\G X}"]\ar[d, tail, "\epsilon_X"]&&& L\ar[d, "\sgt_L", tail]\\ X\ar[r,"\sgt_X"'] &PX \ar[r,"{\epsilon_X}^{-1}"']&P\G X\ar[r,"\exists_{\phi_{\G X}}"'] & PL \end{tikzcd} \] is commutative, we have \[ \begin{tikzcd} \G X \ar[rrr, "\phi_{\G X}"]\ar[rrd, "\sgt_{\G X}", tail]\ar[d, tail, "\epsilon_X"]&&& L\ar[d, "\sgt_L", tail]\\ X\ar[r,"\sgt_X"'] &PX \ar[r,"{\epsilon_X}^{-1}"']&P\G X\ar[r,"\exists_{\phi_{\G X}}"'] & PL. \end{tikzcd} \] \Cref{lem:sgtExtNaturality} and \cref{lem:sgtNaturality} complete the proof. \end{proof} \begin{theorem} For a hyperconnected geometric morphism $f\colon \E \to \F$ corresponding to an internal filter $F\rightarrowtail \Xi$, the family of morphisms $\{\xi_{Z} \colon Z \to F\}_{Z\in \ob(\F)}$ is a local state classifier of the topos $\F$. \end{theorem} \begin{proof} By \cref{rmk:NotationOdXiF} and \cref{cor:FilterLivesInHQuotient}, this family of morphisms is in the category $\F$. Since the embedding $f^{\ast}\colon \F \to \E$ is fully faithful and lex, the family of morphisms is locally determined. So, we need to prove that this family has the universality as a colimit of all monomorphisms in $\F$. Since \cref{lem:JointlyEpimorphic} ensure the uniqueness part of the desired universality, we will prove the existence part. Take an arbitrary cocone $\{\phi_Z \colon Z\to L\}_{Z\in \ob(\F)}$ under all monomorphisms in $\F$. Let $\{\psi_X\colon X \to PL\}_{X\in \ob(\E)}$ be the locally determined cocone given in \cref{lem:ExtensionLemma}. By the universality of the local state classifier $\Xi$, we have a unique morphism $\gamma \colon \Xi \to PL$ such that \[ \begin{tikzcd} % [column sep = 10pt] [row sep = 50pt] &X\ar[ld,"\xi_X"']\ar[rd,"\psi_X"]&\\ \Xi\ar[rr,"\gamma"]&&PL \end{tikzcd} \] commutes for every $X\in \ob(\E)$. Due to \cref{rmk:NotationOdXiF} and the latter part of \cref{lem:ExtensionLemma}, the following diagram is also commutative. \[ \begin{tikzcd} % [column sep = 10pt] [row sep = 50pt] &\G X\ar[ld, "\xi_{\G X}"']\ar[d,"\epsilon_X", tail]\ar[rd, "\phi_{\G X}"]&\\ F\ar[d,tail] &X\ar[ld,"\xi_X"']\ar[rd,"\psi_X"]&L\ar[d,"\sgt_L", tail]\\ \Xi\ar[rr,"\gamma"]&&PL \end{tikzcd} \] Now that, it remains to prove that $F\rightarrowtail \Xi \to PL$ lifts along $\sgt_L$: \[ \begin{tikzcd} % [column sep = 10pt] [row sep = 50pt] &\G X\ar[ld, "\xi_{\G X}"']\ar[rd, "\phi_{\G X}"]&\\ F\ar[d,tail] \ar[rr, dashed, "?"]&&L\ar[d,"\sgt_L", tail]\\ \Xi\ar[rr,"\gamma"]&&PL. \end{tikzcd} \] Take the characteristic morphism of $L \rightarrowtail PL$ as \[ \begin{tikzcd} % [column sep = 10pt] [row sep = 50pt] &\G X\ar[ld, "\xi_{\G X}"']\ar[rd, "\phi_{\G X}"]&&\\ F\ar[d,tail] \ar[rr, dashed, "?"]&&L\ar[d,"\sgt_L", tail]\ar[rr, "!"]&&1\ar[d,"\true", tail]\\ \Xi\ar[rr,"\gamma"]&&PL\ar[rr,"\chi"]&&\Omega. \end{tikzcd} \] We prove that the composite $F\rightarrowtail \Xi \to PL \to \Omega$ is the true morphism $\true_{F}\colon F \to \Omega$. \Cref{lem:JointlyEpimorphic} reduces this problem to proving the composition $\G X \to F \to \Xi \to PL \to \Omega$ is $\true_{\G X}$ for every $X \in \G X$, and this follows from the commutativity of the perimeter of \[ \begin{tikzcd} % [column sep = 10pt] [row sep = 50pt] &\G X\ar[ld, "\xi_{\G X}"']\ar[rd, "\phi_{\G X}"]\ar[rrrd, bend left, "!"]&&\\ F\ar[d,tail] \ar[rr, dashed, "?"]&&L\ar[d,"\sgt_L", tail]\ar[rr, "!"]&&1\ar[d,"\true", tail]\\ \Xi\ar[rr,"\gamma"]&&PL\ar[rr,"\chi"]&&\Omega. \end{tikzcd} % \begin{tikzcd} % % [column sep = 10pt] % [row sep = 50pt] % &\G X\ar[ld, "\xi_{\G X}"']\ar[rrrd, bend left, "!"]&&\\ % F\ar[d,tail] &&&&1\ar[d,"\true", tail]\\ % \Xi\ar[rr,"\gamma"]&&PL\ar[rr,"\chi"]&&\Omega. % \end{tikzcd} \] \end{proof} \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{remark} % This is a set of atoms equipped with a (non-trivial) action. Is it possible to generalize this to an arbitrary atomic Grothendieck topos? % \end{remark} \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} \begin{example}[LSC of nominal sets] \end{example} \begin{question} How is this generalized to topological groupoids? \end{question} % \section{Monoid structure} % \begin{conjecture} % For an LSC $\Xi$ of an OFS $(E,M)$ on a monoidal closed category $\C$, if the tensoring preserves the class $M$, then $\Xi$ admits the canonical monoid structure on it. % \end{conjecture} \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 same thing to say that \[ \begin{tikzcd} \Xi_X\ar[r]\ar[d, twoheadrightarrow]\arrow[rd, phantom, "\lrcorner", very near start]&{\leq_{\Xi}}\ar[d, twoheadrightarrow, "\pi_2"]\\ X\ar[r, "\xi_X"]&\Xi, \end{tikzcd} \] \begin{example}[Group action topos] Let $G$ be a group and $PSh(G)$ be its presheaf topos of right actions. For any subgroup $H< G$, the slice topos $\PSh(G) / (H\backslash G)$ over the transitive action $H \backslash G$ is equivalent to $\PSh(H)$. Therefore, the LSC of $\PSh(G) / (H\backslash G)$ is given by the set of all subgroups of $H$. This example exemplifies the \dq{geometric} intuition behind this conjecture. \memo{Some ...} \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{Classification of smaller classes of hyperconnected geometric morphisms \memo{ongoing}} \begin{question} \label{question:subclassclassification} Which class of hyperconnected quotients corresponds to which class of filters? \end{question} \begin{table}[ht] \centering \begin{tabular}{|c|c|c|} \hline Hyperconnected& $=$ \cite{hora2024internal}& Filter\\ \hline Hyperconnected \'{e}tendue& ?\cref{Conj:etendue}& Externally principal filter \\ \hline Hyperconnected and essential& ?\cref{sec:hyperessential}& Internally principal filter \\ \hline Atomic connected& ? \cite{henry2018localic}&?\\ \hline \end{tabular} \end{table} \section{Hyperconnected essential quotients \memo{ongoing}} \label{sec:hyperessential} There are several motivations for classifying hyperconnected and essential quotient topoi: \begin{itemize} \item Just as one example of \cref{question:subclassclassification}. \item As a topos-theoretic study of the notion of syntactic monoids. \item A topos-theoretic way to deal with congruences of a category. \end{itemize} \begin{definition} A \demph{congruence} of a category $\C$ is an equivalence relation $\equiv$ of the set of morphisms $\Mor(\C)$ that induces a bijective-on-objects and full functor $\C \to \C/{\equiv} $. In other words, it is an equivalence relation $\equiv$ with \begin{itemize} \item $f\equiv g \implies \dom(f) = \dom(g) \land \cod(f)=\cod(g) $, \item $f\equiv g \implies f\circ h \equiv g\circ h$, if defined, and \item $f\equiv g \implies k\circ f \equiv k\circ g$, if defined. \end{itemize} \end{definition} \begin{lemma} Every congruence $\equiv$ induces an essential and hyperconnected geometric morphism $\PSh(\C) \to \PSh(\C/{\equiv})$. \end{lemma} \begin{conjecture} The above lemma provides a one-to-one correspondence between hyperconnected essential quotients of $\PSh(\C)$ and congruences on $\C$. \end{conjecture} See \cite{el2002simultaneously} for the classification of essential quotients of a presheaf topos. \begin{question} Consider the internal object parameterizing the filters of $\Xi$. \end{question} \begin{proposition} \label{prop:PresheafCongruence} For a small category $\C$, there is a one-to-one correspondence between \begin{itemize} \item A congruence on $\C$. \item An internal filter $F\rightarrowtail \Xi$ such that $F(c) \subset \Xi(c)$ is a principal filter for each $c\in \ob(\C)$. \end{itemize} \end{proposition} \begin{proof} The latter corresponds to a family of quotient objects $\{\yo(c) \twoheadrightarrow q_c\}$ such that \[ \begin{tikzcd} \yo(c) \ar[r, "\yo(f)"]\ar[d,twoheadrightarrow]& \yo(d) \ar[d,twoheadrightarrow]\\ q_c\ar[r, dashed, "\exists"]& q_d, \end{tikzcd} \] which corresponds to the notion of congruence. \end{proof} \begin{remark} \label{rmk:ExternallyPrincipal} This is NOT the same as a point $1 \to \Xi$. The point corresponds to a congruence $\equiv$ such that all morphisms in $\C/{\equiv}$ are monic. See \cref{sec:etendue}. \end{remark} \begin{question} How is \cref{prop:PresheafCongruence} generalized to a classification theorem of hyperconnected essential geometric morphisms for Grothedieck topoi? \begin{itemize} \item Is this related to the locale completion? \item Is this an internal notion of being principal? \memo{Maybe NO?} \item How is this related to the notion of normalizer (\cref{def:normalizer})? \end{itemize} \end{question} % \section{Preservation by functor} % \begin{question} % If % \begin{itemize} % \item a fully faithful functor $F \colon \E \to \F$ preserves and reflects (or maybe creates) monomorphisms, % \item $\F$ has a LSC $\Xi$, % \item $FX \to \Xi$ are jointly surjective (density?), and % \item $\Xi$ is in the image of $F$ % \end{itemize} % then is $\Xi$ also a LSC of $\E$? % \end{question} \section{Relation to \'{e}tendue \memo{ongoing}} \label{sec:etendue} Due to the next 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 \[ \begin{tikzcd} A' \ar[r, \mono, "f'"]\ar[d, \epi, "q"]& \gamma^* I \times B\ar[d, "\mathrm{proj}"]\\ A\ar[r, "f"] & B, \end{tikzcd} \] 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 \[ \begin{tikzcd} &U_{\lambda}\ar[ld, \mono]\ar[rd, \mono, "{f|_{U_{\lambda}}}"]&\\ A\ar[rd, "\xi_A"']\ar[rr,"f"]&&B\ar[ld, "\xi_B"]\\ &\Xi& \end{tikzcd} \] 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 $\E$ 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} Asuuming 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 \[ \begin{tikzcd} X\ar[r, twoheadrightarrow, "!"]\ar[d, "\iota", rightarrowtail]\arrow[rd, "\xi_X"]&1\ar[d, "\bot", rightarrowtail]\\ B\ar[r, twoheadrightarrow, "\xi_B"]&\Xi \end{tikzcd} \] \section{Relationship with locality of a topos (Inspired by Matias Menni)} \begin{proposition}[Menni 2024] For a small category $\C$ with a terminal object, the LSC of the presheaf topos $\PSh(\C)$ has a unique point. \end{proposition} This can be generalized to any local topoi. \begin{question} Is there any way to characterize local topoi in terms of LSC? \end{question} There are a lot of non-local topoi, in which LSC has only one point. For example, LSC of any localic topoi is terminal, but localic topoi is not local in general. \section{Module structure of subtopoi and hyperconnected quotients} \para{History} \begin{itemize} \item[1940] In Birkhoff's book on lattice theory, the notion of \demph{m-lattice} (multiplicative monoid) is introduced. \cite{birkhoff1940lattice} \item[1995] As far as the author knows, Dikranjan and Tholen's book firstly introduces the notion of \demph{productive closure operator}, as the closure operator preserving product structure \cite{dikranjan1995categorical}. \item[2006] \cite{hosseini2006relation} defines the notion of \demph{weak Lawvere–Tierney topology} on a topos. \item[2009] In the second problem of \cite{Open240411Lawvere} asks about the interaction between subtopoi and quotient topoi. \item[2021] Khanjanzadeh and Madanshekaf prove that \demph{weak productive topologies form an $l$-monoid}. \cite{khanjanzadeh2021weak} \item[2021] Observation of subtopoi in \cite{menni2021hyperconnected} should be related. \item[2024] \cite{hora2024internal} gives an internal parameterization of hyperconnected quotients, which are acted on by the monoid of productive weak topologies. \end{itemize} \para{The semiring of productive weak Lawvere-Tierney topologies} \begin{definition}[\cite{dikranjan1995categorical, khanjanzadeh2021weak}] A \demph{productive weak Lawvere-Tierney topologies} (in short, we call it a \demph{productive operator}) on a topos $\E$ is an internal semilattice homomorphism from $\Omega$ to $\Omega$ itself. The set (or possibly a class) of all productive topologies is denoted by $\R_{\E}$. \end{definition} \begin{theorem}[{\cite[][Theorem 4.1 (i)]{khanjanzadeh2021weak}}] The set $\R_{\E}$ with the ``addition" $\land$ and ``multiplication" $\circ$ forms an additively idempotent (possibly non-commutative) semiring. \end{theorem} \begin{remark} Every commutative monoid's endo-homomorphisms form a semiring in the same way. \end{remark} \begin{example} For the topos of sets $\Set$, the semiring $\R_{\Set}$ is the semiring $\{0,1\}$ with $1+1=1$. \end{example} \begin{example} For the topos of discrete dynamical systems $\E= \PSh(\N)$, the semiring $\R_{\E}$ is the min-plus algebra on $\{0<1< \cdots < \infty < \infty'\}$. \end{example} Obviously, this semiring has complete information on the poset of subtopoi. \begin{proposition} The poset of all (multiplicatively) idempotent elements\footnote{$e\leq f \iff ef=e$} is contravariantly isomorphic to the poset of subtopoi of $\E$. \end{proposition} \begin{proof} This is due to the fact that a Lawvere-Tierney topology is exactly an idempotent productive operator. \end{proof} \para{Hyperconnected quotients form a left module} \begin{definition} The set (or class) of hyperconnected geometric morphisms (rigorously, their equivalence classes) is denoted by $\HQ$. \end{definition} \begin{theorem} If a topos $\E$ has a local state classifier \footnote{this assumption should be deleted by observing the counit calculations}, the poset $\HQ$ forms a meet semilattice, and is a left module of the semiring $\R_{\E}$. \end{theorem} \begin{proof} This is just an abstract non-sense of semilattice enriched categories. See \memo{Appendix:SemilatticeEnrichment}. \end{proof} \begin{question} Can this be extended to arbitrary quotient topoi? (I guess, NO.) \end{question} \begin{conjecture} \label{conjecture:Main} For a subtopos $f$ and hyperconnected quotient $h$ of a topos $\E$, $f$ contains $h$ if and only if \[f\cdot h = h.\] % where $f$ donotes the corresponding idempotent in $\R_{\E}$ and $h\in \HQ$. \end{conjecture} If so, this provides one approximate solution to the second problem in \cite{Open240411Lawvere} for the restricted class of hyperconnected quotients. \begin{remark}[Non-commutative geometry here?] Our discussion looks like a non-commutative and semilattice version of the Pierce spectrum. From this point of view, the module $\HQ$ is a kind of quasi-coherent sheaf. Furthermore, \cref{conjecture:Main} states that a subtopos $f$ contains a hyperconnected quotient $h$, if the ``support" of $h$ is in the ``clopen set" $f$. \end{remark} \section{Exponential structure} \begin{proposition} % For a topos $\E$ with a local state classifier $\Xi$ and If a pointed endofunctor \[ \begin{tikzcd}[column sep = 70pt] \E \ar[r, "\id_{\E}", ""'{name = A}, bend left]\ar[r, ""{name=B},"F"', bend right]& \ar[from = A, to =B, Rightarrow, "\alpha"]\E \end{tikzcd} \] preserves monomorphisms, then there uniquely exists a morphism $\hat{\alpha}\colon \Xi \to \Xi$ such that \[ \begin{tikzcd} X\ar[d, "\xi_X"]\ar[r, "\alpha_X"]&FX\ar[d,"\xi_{FX}"]\\ \Xi\ar[r,"\hat{\alpha}"]&\Xi \end{tikzcd} \] commutes. \end{proposition} \begin{proof} It is enough to prove that the family of morphisms \[ \begin{tikzcd} X \ar[r, "\alpha_X"] &FX\ar[r, "\xi_{FX}"]& \Xi \end{tikzcd} \] commutes with all monomorphisms. \end{proof} What morphism is induced by the exponential functor ${-}^A$? \section{Site description of LSC} \begin{example} For a topological group $G$, the topos $\Cont(G)$ admits the canonical atomic site $(C,J)$ where $C$ is the category of transitive $G$-sets (or the category of open subgroups of $G$). Since $\Xi$ is the set of all open subgroups of $G$ with the right conjugate action we have \[ \Xi(H\backslash G)=\{K\subset G\text{: open subgroup}\mid H\subset \Nor_G(K)\}. \] In particular, we have $\Xi(G\backslash G)=\{K\subset G\text{: open normal subgroup}\}$. \end{example} This example shows that $\E(X, \Xi)$ cannot be written as a subset of $\Quo(X)$. Informally, $X\to\Xi$ should correspond to \dq{local quotients}. For example, in the topos $\PSh(G)$, morphisms $1\to \Xi$ correspond to quotients of $G$, which is an étale cover $G\twoheadrightarrow 1$, that are compatible with the automorphisms over $1$. \section{Relationship with Copower object} \cite{kenney2006copower} \begin{definition} For a cowell-powered topos $\E$, we define a contravariant functor \[ \Quo_\E \colon \E^{\op} \to \Set, \] which sends an object $X$ to the set of all quotient objects of $X$. The action of morphisms is given by the epi-mono factorization system \[ \Quo_\E(f)(\alpha \colon X\twoheadrightarrow Q) =\Image(\alpha f). \] \end{definition} \begin{example}[\cite{hora2024internal}] The local state classifier of $\PSh(\C)$ is given by $\Xi= \Quo_{\PSh(\C)}\circ \yo$. \end{example} \begin{proposition}[Definition/Proposition\cite{kenney2006copower}] For any topos $\E$ and an object $A$, the functor \[ \E^{\op}\to \Set: X \mapsto \Quo_{\E/X}(X^* A) \] is representable. \end{proposition} This representing object is called \demph{copower object} and is denoted by $QA$. \begin{conjecture}[?] For any Grothendieck topos $\E$, the local state classifier is given by the sheafification of $\Quo_{\E}\colon \E^{\op}\to \Set$ with respect to the canonical site $(\E, J_{\mathrm{joint.surj.}})$. \end{conjecture} \section{Covariety classification} \begin{definition}[\memo{This terminology is due to Peter Johnstone. Cf. Menni's shell}] A \demph{covariety} of a Grothendieck topos\footnote{this should be defined in a more general framework.} $\E$ is a full subcategory closed under \begin{itemize} \item subquotients (in particular, isomorphisms), and \item coproducts \end{itemize} \end{definition} \begin{lemma} For any upward closed subset $U \rightarrowtail \Xi$ in a Grothendieck topos $\E$, the full subcategory $\E_U \hookrightarrow \E$ is a covariety. \end{lemma} \begin{lemma} For any covariety of a Grothendieck topos $\A \hookrightarrow \E$, every object $X\in \ob(\E)$ admits the maximum subobject $A_X\rightarrowtail X$ that belongs to $\A$. The family of morphisms $\{\epsilon _X\colon A_X\rightarrowtail X\}_{X\in \ob(\E)}$ provides the monic counit that witnesses that the embedding $\A\hookrightarrow \E$ is coreflective. Furthermore, the corresponding characteristic maps $\{\chi_{A_X}\colon X \to \Omega\}_{X\in \ob(\E)}$ are locally determined. \end{lemma} The proof is the same as the original case. This induces a map $\gamma_{\A}\colon \Xi \to \Omega$. \begin{lemma} For any internal poset $P$ and its subobject $U \rightarrowtail P$, the following conditions are equivalent: \begin{itemize} \item $U$ is upward closed. \item $\chi_U\colon P \to \Omega$ is order preserving. \end{itemize} \end{lemma} \begin{lemma} For any covariety of a Grothendieck topos $\A\hookrightarrow \E$, the induced morphism $\gamma_{\A}\colon \Xi \to \Omega$ is order preserving. \end{lemma} \begin{proof} We need to prove that $\Xi \times \Xi \to \Omega\times \Omega$ sends ${\leq}_{\Xi}$ into ${\leq}_{\Omega}$. Since the morphisms $\{\xi_X \times \xi_Y \colon X\times Y \to \Xi \times \Xi\}_{(X,Y)\in \ob(\E)^2}$ are jointly surjective, it suffices to prove that for any \[ \begin{tikzcd} X &Z\ar[l, "f"']\ar[r, "g"] &Y, \end{tikzcd} \] we have \[ \xi_X f \leq \xi_Y g \implies f^*A_X \leq g^* A_Y. \] The left hand side is equivalent to saying $\xi_X f = (\xi_X f \land \xi_Y g) = \xi_{X\times Y}\langle f,g \rangle$, which implies $f^* A_X \leq \langle f,g \rangle^* A_{X\times Y}$. As we also have $A_{X\times Y} \leq \pi_2^* A_Y$, we conclude $f^* A_X \leq \langle f,g \rangle^* A_{X\times Y} \leq \langle f,g \rangle^* \pi_2^* A_Y = g^* A_Y$. \end{proof} \begin{theorem} For any Grothendieck topos $\E$, there is a natural bijective correspondence between \begin{itemize} \item covareities of $\E$, \item order-preserving maps $\Xi \to \Omega$, and \item upward closed subobjects of $\Xi$. \end{itemize} \end{theorem} \section{LSC of presentable categories} \appendix \section{Related works} \begin{itemize} \item Toyota's observations \item Menni's paper \cite{menni2021hyperconnected} \item Toby Kenny's copower objects \cite{kenney2006copower} \end{itemize} \printbibliography \end{document}