← Notes on advances of LSC
General progress__main.tex
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\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}