← Notes on advances of LSC
LSCandAC.tex
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% \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.}
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