\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);% }}} \usepackage{quiver} \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*{definition*}{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{\Q}{\mathbb{Q}} \newcommand{\R}{\mathbb{R}} \newcommand{\C}{\mathcal{C}} \newcommand{\D}{\mathcal{D}} \newcommand{\E}{\mathcal{E}} \newcommand{\F}{\mathcal{F}} \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{\Set}{\mathbf{Set}} \newcommand{\Group}{\mathbf{Grp}} \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]{\textit{#1}} \font\maljapanese=dmjhira at 2.5ex % \newcommand{\yo}{\textrm{\!\maljapanese\char"48}} \newcommand{\yo}{\mathrm{y}} \newcommand{\Stab}{\mathrm{Stab}} \newcommand{\mono}{\mathrm{mono}} \newcommand{\A}{\Sigma} \newcommand{\MA}{{{\Sigma}^{\ast}}} \newcommand{\proMA}{\widehat{\MA}} \newcommand{\FA}{F_{\A}} \newcommand{\pFA}{\hat{\FA}} % \newcommand{\Aset}{\PSh (\MA)} \newcommand{\Aset}{\A\text{-}\Set} \newcommand{\Lan}{\mathcal{L}} \newcommand{\Reg}{\mathcal{R}} \newcommand{\Pow}{\mathcal{P}} \newcommand{\of}{\mathrm{o.f.}} \newcommand{\pof}{{p_{\of}}} \newcommand{\fAtmt}{\Atmt_{\f}} \newcommand{\ofAtmt}{\Atmt_{\of}} % \newcommand{\fAset}{{\Aset}_{\f}} \newcommand{\fAset}{\A\text{-}\FinSet} \newcommand{\ofAset}{{\Aset}_{\of}} \newcommand{\Cont}{\mathbf{Cont}} \newcommand{\colim}{\mathbf{colim}} % \title[Normalization operator in a category]{Normalization operator in a category and the local state classifier of hyperconnected quotient topoi} \title{Normalization of a subgroup, in a category, and of a word-congruence} \author{Ryuya Hora} \thanks{Graduate School of Mathematical Sciences, University of Tokyo. \url{hora@ms.u-tokyo}} % \date{\today} \subjclass[2020]{18B25} \keywords{Topos, normalization, hyperconnected geometric morphism, local state classifier} \begin{document} \begin{abstract} This paper provides an abstract definition of a normalization operator motivated by topos theory and its application to algebraic language theory. We first define a normalization operator $\Xi \to \Xi$ in any category that admits a colimit of all monomorphisms $\Xi$, which we call a local state classifier. In the category of group actions for a group $G$, this operator coincides with the usual normalization operator receiving a subgroup $H\subset G$ and returning its normalizer subgroup $\Nor_G(H)\subset G$. % continues the author’s study of the colimit of all monomorphisms in a category, called the local state classifier $\Xi$, and its use in analyzing hyperconnected geometric morphisms from a given topos. Utilizing this generalized normalization operator, we then provide a way to construct a local state classifier of a hyperconnected quotient of a given topos. % that has a local state classifier. These results serve as preparation for a topos-theoretic study of regular languages, congruences of words, and syntactic monoids. \end{abstract} % \begin{abstract} % This paper provides an abstract definition of a normalization operator motivated by topos theory and its application to algebraic language theory. % continues the author’s study of the colimit of all monomorphisms in a category, called the local state classifier $\Xi$, and its use in analyzing hyperconnected geometric morphisms from a given topos. % We first define a normalization operator $\Xi \to \Xi$ in any category that admits a local state classifier, in particular, in any Grothendieck topos. In the category of group actions for a group $G$, this operator coincides with the usual normalization operator receiving a subgroup $H\subset G$ and returning its normalizer subgroup $\Nor_G(H)\subset G$. % We then describe how to construct a local state classifier of a given hyperconnected quotient of a given topos. % % that has a local state classifier. % These results serve as preparation for a topos-theoretic study of regular languages, congruences of words, and syntactic monoids. % \end{abstract} \maketitle \tableofcontents \section{Introduction} \subsection{Abstract definitions of a normalizer subgroup}\label{ssec:AbstractDefinitionOfNormalizer} For a group $G$ and a subgroup $H\subset G$, the \demph{normalizer} of $H$ is defined by \[ \Nor_{G}(H) \coloneqq \{g\in G \mid g^{-1}Hg=H\}. \] In the context of categorical algebra, the normalizer is abstractly described in \cite{gray2014normalizers} as follows: In a category $\C$ with a zero object, prototypically the category of groups $\C = \Group$, a monomorphism $m \colon S \rightarrowtail X$ is said to be normal, if it is a kernel of some arrow from $X$. A normalizer of a subobject $m \colon H \rightarrowtail G$ in $\C$ is defined to be the terminal object of the category of factorizations of $m$ as a normal monomorphism followed by a monomorphism (\cite[Definition 2.1]{gray2014normalizers}). This is a natural way to generalize $\Nor_{G}(H)$ as the maximal subgroup of $G$ that makes the inclusion $H\subset \Nor_{G}(H)$ normal. % \cite{gray2014normalizers} In this paper, we provide another abstract description of normalization — in a way that was (at least to the author) completely unexpected. Instead of considering the category of groups $\Group$, % normalizer of each subgroup of each group separately, we consider the category of right $G$-actions, i.e. the presheaf category $\PSh(G)$ on a given group $G$. Regarding $\Sub_{\Group}(S)$ as a right $G$-set with the right conjugate action $H\ast g \coloneqq g^{-1}Hg$, we will categorically describe the operation \begin{equation}\label{eq:NormalizationOperatorForAGroup} \Nor_G \colon \Sub_{\Group}(G) \to \Sub_{\Group}(G) \text{ in } \PSh(G). \end{equation} % which is in fact a morphism in $\PSh(G)$ since $\Nor_{G}\left(g^{-1}Hg\right)=g^{-1} \Nor_G(H)g$. % where $\Sub(G)$ denotes the set of all subgroups of $G$. % which is a morphism in $\PSh(G)$, % , since we have % \begin{equation}\label{eq:NormalizerOperatorIsNaturalTransformation} % \Nor_{G}\left(g^{-1}Hg\right)=g^{-1} \Nor_G(H)g % \end{equation} % , for a fixed group $G$. First, we take the colimit of all monomorphisms in the category $\PSh(G)$, which nontrivially exists, and let us write $\Xi$ for it. The colimit cocone is a family of morphisms from all objects in the category, for which we will write $\xi_{X}\colon X \to \Xi$ for every $X \in \ob(\PSh(G))$. Then the \dq{self-referential} component of the colimit cocone $\xi_{\Xi}\colon \Xi \to \Xi$ coincides with the normalizer operator $\Nor_G$ (\Cref{exmp:CaseOfGroup})! % The first observation is that the object $\Sub_{\Group}(G)$ is the colimit of all monomorphisms in $\PSh(G)$, (which is explained in \cite[Example 3.10.]{hora2024internal}). In other words, % it is % % there is a family of morphisms $\{\xi_{X}\colon X \to \Sub_{\Group}(G)\}_{X\in \ob (\PSh(G))}$ that makes $\Sub_{\Group}(G)$ % the colimit of the large diagram $\PSh(G)_{\mono}\rightarrowtail \PSh(G)$, where $\PSh(G)_{\mono}$ denotes the wide subcategory of $\PSh(G)$ that consists of all monomorphisms % \[ % \Sub_{\Group}(G) = \colim (\PSh(G)_{\mono}\rightarrowtail \PSh(G)). % \] % Furthermore, its colimit cocone, which is a family of morphisms from all objects in $\PSh(G)$, is given by the stabilizer operator % \[ % \Stab_X :X \to \Sub_{\Group}(G) % % \colon x \mapsto \{g\in G \mid xg=x\} % . % \] % % each component $\xi_X \colon X \to \Sub_{\Group}(G)$ sends an element $x\in X$ of a $G$-set $X$ to its stabilizer $\xi_X(x) = \Stab_G(x)\in \Sub_{\Group}(G)$. % Therefore, we conclude that the normalization operator $\Nor_G$ coincides with the \demph{self-referential} component of the colimit \[\Nor_G=\Stab_{\Sub_{\Group}(G)}\colon \Sub_{\Group}(G) \to \Sub_{\Group}(G)\] % at the object $X= \Sub_{\Group}(G)$. % % $\xi_{\Sub_{\Group}(G)}$, % % \[ % % \Nor_G =\xi_{\Sub_{\Group}(G)}\colon \Sub_{\Group}(G) \to \Sub_{\Group}(G). % % \] In general, we can define the normalizer operator as follows: % in any category that admits a colimit of all monomorphisms (\Cref{def:NormalizationOperator}). \begin{definition*}[Paraphrase of \Cref{def:NormalizationOperator}] For a category $\E$ that admits the colimit of all monomorphisms $\Xi$ with the colimit cocone $\{\xi_{X}\colon X \to \Xi\}_{X\in \ob(\E)}$, the \demph{normalizer operator} in $\E$ is the endomorphism $\xi_{\Xi}\colon \Xi \to \Xi.$ % \[ % \xi_{\Xi}\colon \Xi \to \Xi. % \] \end{definition*} Such an object $\Xi$ exists in all Grothendieck topoi, and in particular, all presheaf categories. Therefore, our generalized normalization operator appears in contexts that are completely different from group theory. As examples, we will see the normalization operator for directed graphs (\Cref{exmp:ToposOfDirectedGraphTwo}) and free monoid ($=$ words) actions (\Cref{sec:MotivationgExample}). We also generalize the obvious inequality $H \subset \Nor_G(H)$ for a general context (\Cref{prop:NormalizationLemma}), which plays a central role in the proof of the main theorem. \subsection{Local state classifier in a hyperconnected quotient topos}\label{ssec:MotivationFromLSC} % \memo{We want to know h.c.g.m.} The theoretical motivation of this paper comes from topos theory, especially the study of \demph{hyperconnected geometric morphisms}. Since \cite{johnstone1981factorization} introduced the notion of hyperconnected geometric morphisms, topos theory has heavily utilized it. For example, in the study of topological monoid actions \cite{rogers2023toposes}, hyperconnected geometric morphisms play a central role. As explained in \Cref{ssec:MotivationFromLanguages} and \Cref{sec:MotivationgExample}, these structures are also important in the topos-theoretic approach to automata theory. Referring to the terminology `quotient topos' in \cite{lawvere2025open}, we call (an equivalence class of) a hyperconnected geometric morphism from a topos $\E$ a \demph{hyperconnected quotient} of $\E$. A way to enumerate all hyperconnected quotients is firstly given by \cite{rosenthal1982quotient} using generators of a given Grothendieck topos. % % That paper introduces the notion of a \demph{local state classifier} $\Xi$ defined as the colimit of all monomorphisms \cite[][Definition 3.4]{hora2024internal}. % \memo{It suffices to know LSC} In order to obtain a simpler classification of hyperconnected quotients, the colimit of all monomorphisms $\Xi$ is studied under the name of \demph{local state classifier} in the author's paper \cite{hora2024internal} . The main theorem of the paper \cite{hora2024internal} states that if a topos has a local state classifier $\Xi$, then hyperconnected quotients of $\E$ are in one-to-one correspondence with internal filters of $\Xi$. This result provides a convenient way to classify all hyperconnected quotients of a broader class of topoi, including all Grothendieck topoi. Therefore, we need a way to calculate the local state classifier of a given topos. % \memo{Calculating LSC is not trivial} However, explicitly describing the local state classifier $\Xi$ is not always easy. Although the local state classifier of a presheaf topos is explicitly given by $\Xi(c) = \{\text{quotient objects of $\yo(a)$}\}$ \cite[][Example 3.22.]{hora2024internal}, it is not easy to describe a local state classifier of a non-presheaf topos. % Even though the description of $\Xi$ is given in terms of sheafification, that is not very convenient for a non-presheaf topos. % However, it is not easy to describe a local state classifier of a non-presheaf topos. % \memo{we provides a way!} The main theorem of the present paper (\Cref{thm:MainTheorem}) provides a new method for describing the local state classifier of a hyperconnected quotient of a known topos. As corollaries, we obtain the description of the local state classifier of the topos of continuous actions $\Cont(G)$ for a given topological group $G$ (\Cref{cor:LSCofTopologicalGroups}) and that of the topos of orbit-finite $\A$-sets $\ofAset$ (\Cref{cor:LocalStateClassifierOfOrbitfiniteAset}). The generalized normalization operator is utilized in its proof. % To prove the main theorem, we introduce the notion of \demph{normalization operator} $\xi_{\Xi}\colon \Xi \to \Xi$ in any category with a local state classifier $\Xi$. This abstractly defined operator coincides with the usual normalization of subgroups in the topos of group actions. \subsection{Word combinatorics in algebraic language theory}\label{ssec:MotivationFromLanguages} The last and most concrete motivation is a topos-theoretic approach to algebraic language theory, and this paper is intended to be a theoretical preparation for the upcoming paper `Topoi of automata II'. In automata theory, it is crucial to consider right congruences on the words $\MA$ for a given alphabet $\A$. The set of all right congruences provides the local state classifier of the topos $\Aset \coloneqq \PSh(\Sigma^*)$, and plays a central role in the ongoing theory of topoi of automata (especially for the theory of congruences and syntactic monoids). Here, since $\Aset$ is a presheaf topos, it is easy to describe its local state classifier. However, in order to capture finiteness related to algebraic language theory, we need to consider the topoi of topological (or, in many cases, profinite) monoid actions (see \cite{hora2024topoi}) and its local state classifier. Here, the main theorem of the present paper is useful, since every topos of topological monoid actions is a hyperconnected quotient of a monoid action topos, as studied in \cite{rogers2023toposes}. % \para{Structure of the paper} % For the reader's convenience, we will recall those notions in \Cref{sec:Preliminaries}, which serves as a short summary of the paper \cite{hora2024internal}. \subsection*{Acknowledgement} The author would like to thank his supervisor Ryu Hasegawa for his continuous support and suggestions. He is also grateful to Matias Menni for his discussion on the notion of local state classifier, and to the members of the category theory reading group at RIMS. He was supported by JSPS KAKENHI Grant Number JP24KJ0837 and FoPM, WINGS Program, the University of Tokyo. \section{Preliminaries on Hyperconnected quotients and local state classifier}\label{sec:Preliminaries} This section is a $2$-page summary of the paper \cite{hora2024internal}, which defines and studies the notion of a local state classifier. \subsection{Hyperconnected quotients} This subsection aims to recall the preliminaries on hyperconnected geometric morphisms. See \cite{johnstone1981factorization} or \cite[][A.4.6]{johnstone2002sketchesv1} for more details. % We will also explain the notion of local state classifier from \cite{hora2024internal}. \begin{definition}[Hyperconnected geometric morphisms] A geometric morphism $f\colon \E \to \F$ is said to be \demph{hyperconnected} if it is connected (i.e. $f^{\ast}\colon \F \to \E$ is fully faithful) and its counit $\epsilon_X\colon f^{\ast}f_{\ast}\to \id_{\E}$ is monic. \end{definition} In this paper, a \demph{hyperconnected quotiet} of a topos $\E$ means (an equivalence class of) a hyperconnected geometric morphism from $\E$. Since $f^{\ast}$ is fully faithful for a hyperconnected quotient $f\colon \E \to \F$, we can regard $\F$ as a (replete) full subcategory of $\E$. With this identification, we will write `$X\in \ob(\E)$ belongs to $\F$' for `$X\in \ob(\E)$ belongs to the essential image of $f^{\ast}$' in this paper. This does not cause a problem, since we will not distinguish two mutually equivalent hyperconnected quotients. \subsection{Local state classifier} In this subsection, we will briefly explain the notion of local state classifier. For more proofs, informal explanations, and examples, see the original article \cite{hora2024internal}. \subsubsection{Definition} \begin{definition}[{\cite[][Definition 3.4]{hora2024internal}}] The \demph{local state classifier} of a category $\E$ is the colimit of all monomorphisms of $\E$, if it exists. In other words, it is an object $\Xi$ equipped with a family of morphisms $\{\xi_X \colon X \to \Xi\}_{X\in \ob(\E)}$, such that they form a colimit cocone under the faithful embedding functor $\E_{\mono}\rightarrowtail \E$. \end{definition} The definition of a local state classifier is quite transcendental, and even a (small) cocomplete category might not admit a local state classifier. However, we can prove the following proposition: \begin{proposition}[{\cite[][Section 3.16.]{hora2024internal}}]\label{prop:ExistenceForGrothendieck} Every Grothendieck topos $\E$ has a local state classifier. \end{proposition} \subsubsection{Inducing full subcategories} % But How is a local state classifier related to the classification of hyperconnected quotients? Since $\Xi$ is just an object of $\E$ and a hyperconnected quotient is a (very nice) subcategory of $\E$, they might seem unrelated. The answer is, in short, that we can construct a full subcategory of $\E$ from any subobject of $\Xi$. % Here is an answer: Let $\E$ be a category with a local state classifier $\Xi$. For any subobject $\iota_F\colon F \rightarrowtail\Xi$ % of the local state classifier of a category $\E$ , we can define a full subcategory $\E_F \hookrightarrow \E$ by \begin{equation}\label{eq:FullSubCondition} X\in \ob(\E_F) \iff \begin{tikzcd} & F\ar[d, rightarrowtail, "\iota_F"]\\ X\ar[r,"\xi_X"']\ar[ru, dashed, "\exists"]&\Xi. \end{tikzcd} \end{equation} % \[ % X\in \ob(\E_F) % \iff % \begin{tikzcd} % & F\ar[d, rightarrowtail]\\ % X\ar[r,"\xi_X"']\ar[ru, dashed, "\exists"]&\Xi. % \end{tikzcd} % \] In other words, we define the full subcategory $\E_F$ of $\E$, specifying objects by \[ \ob(\E_F) \coloneqq \{X\in \ob(\E)\mid \text{ the morphism $\xi_X$ factors through $F\rightarrowtail \Xi$}\}. \] In this note, % we write $\iota_F \colon F \rightarrowtail\Xi$ for the embedding morphism (for a fixed subobject $F$ of $\Xi$). F for each object $X\in \ob(\E_F)$, we write $\xi_X^F\colon X \to F$ for the unique lift of $\xi_X$ along $\iota_F$ \[ \begin{tikzcd} & F\ar[d, rightarrowtail, "\iota_F"]\\ X\ar[r,"\xi_X"']\ar[ru, "\xi_X^F"]&\Xi. \end{tikzcd} \] % Let us summarize the definitions around the main theorem of \cite{hora2024internal} without any proofs. \subsubsection{The order structure} Although the definition of a local state classifier makes sense for any categories, it behaves better in cartesian closed categories. First and foremost, in a cartesian closed category, the local state classifier acquires a canonical semilattice structure as follows: \begin{proposition}[{\cite[][Proposition 3.27.]{hora2024internal}}]\label{prop:SemilatticeStructure} If a cartesian closed category (in particular, an elementary topos) $\E$ admits a local state classifier $\{\xi_X\colon X\to \Xi\}_{X\in \ob (\E)}$, there exists a unique internal $\land$-semilattice structure on $\Xi$ such that the diagram \[ \begin{tikzcd}[column sep =5pt] &X_1\times \dots \times X_n \ar[ld, "(\xi_{X_1}) \times \dots \times (\xi_{X_n})"']\ar[rd, "\xi_{(X_1 \times \dots \times X_n)}"]&\\ \Xi^n\ar[rr,"\land"']&&\Xi \end{tikzcd} \] commutes for any finite sequence of objects $X_1, \dots, X_n \in \ob(\E)$. \end{proposition} This internal semilattice structure induces a semilattice structure on each homset $\E(X,\Xi)$ for each object $X\in \ob(\E)$. Therefore, each homset $\E(X, \Xi)$ admits a natural partial order defined by $f\leq g \iff f\land g =f$. A subobject $\iota_F \colon F \rightarrowtail \Xi$ is said to be an \demph{internal filter}, if each subset $\E(X,F) \rightarrowtail \E(X,\Xi)$ is a filter in the usual sense (i.e., upward closed and closed under finite meets $\top, \land$). (In \cite{hora2024internal}, the author adopts a diagrammatic definition of an internal filter so that it makes sense even for locally large categories.) \subsubsection{The classification theorem} The paper \cite{hora2024internal} proves that, if the category $\E$ is an elementary topos % with a local state classifier $\Xi$ and the subobject $F\rightarrowtail\Xi$ is an internal filter, % an \demph{internal filter} (with respect to the internal $\land$-semilattice structure of $\Xi$ (\Cref{prop:SemilatticeStructure})), the resulting full subcategory $\E_F$ is also an elementary topos, and the embedding $\E_F \hookrightarrow \E$ admits a right adjoint defining % is an inverse image functor of a hyperconnected geometric morphism $f_F\colon \E \to \E_F$. The main theorem of \cite{hora2024internal} (\Cref{thm:OldMainTheorem}) states that this construction $F \mapsto \E_F$ provides a bijective correspondence between the internal filters of $\Xi$ and the hyperconnected quotients of $\E$. % \Cref{thm:OldMainTheorem} states that, % if the category $\E$ is an elementary topos with a local state classifier $\Xi$, the above construction $F \mapsto \E_F$ provides a bijective correspondence between the internal filters of the internal $\land$-semilattice $\Xi$ and the hyperconnected quotients of $\E$. Notice that the embedding $\E_F \hookrightarrow \E$ serves as the inverse image functor of the corresponding hyperconnected geometric morphism $f_F\colon \E \to \E_F$. % The following theorem is the main theorem of \cite{hora2024internal}. \begin{theorem}[{\cite[][Theorem 4.1]{hora2024internal}\footnote{In \cite{hora2024internal}, there is another correspondant, internal semilattice homomorphisms $\Xi \to \Omega$.}}]\label{thm:OldMainTheorem} If an elementary topos $\E$ has a local state classifier $\Xi$, % then $\Xi$ has an internal semilattice structure, % and there exists a bijective correspondence between the following data: \begin{itemize} \item Hyperconnected quotients of the topos $\E$. % \item Internal semilattice homomorphisms $\Xi \to \Omega$. \item Internal filters of the local state classifier $\Xi$. \end{itemize} \end{theorem} % \begin{remark}[External description of internal filter] % \end{remark} % \begin{remark}[Description of the corresponding comonad and its counit map]\label{rmk:NotationOdXiF} The paper \cite{hora2024internal} also provides the description of the corresponding lex comonad $\G \coloneqq f^{*}f_* \colon \E \to \E$ with its counit $\epsilon \colon \G \to \id_{\E}$ \[ \begin{tikzcd} {\;}\ar[rr,phantom, ""'{name=F}]& \E_F \ar[rd,"f^*"]&{\;} \\ \E \ar[ru,"f_*"]\ar[rr, "\G", ""'{name=U}]\ar[rr, bend right =50, "\id_\E"', ""{name=W}]& & \E, \ar[to=U, from=F, phantom, "\rotatebox{90}{$\coloneqq$}"] \ar[to=W, from=U, Rightarrow, "\epsilon"] \end{tikzcd} \] which states that the following diagram is a pullback square \begin{equation}\label{eq:PullbackDescriptionOfTheCounitAndComonad} \begin{tikzcd} \G X\ar[r, "\xi^F_{\G X}"]\ar[d, "\epsilon_X", tail]\ar[rd, phantom, "\lrcorner", very near start]& F\ar[d,tail, "\iota_F"]\\ X\ar[r , "\xi_X"']& \Xi \end{tikzcd} \end{equation} for every $X\in \ob(\E)$. % \end{remark} \section{The statement of the main theorem} In order to motivate the following sections, let us state the main theorem first. \begin{theorem}\label{thm:MainTheorem} Let $\E$ be an elementary topos with a local state classifier $\Xi$, and $F \rightarrowtail \Xi$ be an internal filter. Then, the family of morphisms $\{\xi_{Z}^F \colon Z \to F\}_{Z\in \ob(\E_F)}$ is a local state classifier of the induced hyperconnected quotient topos $\E_{F}$. \end{theorem} % There are implicit non-triviality here. Notice that the above theorem implicitly states that the filter $F$ belongs to the % induced hyperconnected quotient full subcategory $\E_F$. This \dq{self-referential} phenomenon, $F \in \ob(\E_F)$, is not trivial. In fact, without the assumption that $F$ is a filter, there are a lot of counter-examples. \begin{example}[The topos of graphs: {[Not being a loop] is a loop.} (1/2)] \label{exmp:ToposOfGraphsOne} Let us consider the topos of directed graphs $\E \coloneqq \PSh(\rightrightarrows)$. As explained in \cite[][Toy Example 5.3.]{hora2024internal}, its local state classifier $\Xi$ looks like \[ \Xi = \left ( \begin{tikzcd}[scale=3] \bullet\ar[loop left,"\text{[Being a loop]}"]\ar[loop right,"\text{[Not being a loop]}"] \end{tikzcd} \right ). \] For a directed graph $X = (s,t\colon E \rightrightarrows V)$ in $\E$, the graph morphism $\xi_X$ sends every vertex to the unique vertex of $\Xi$, and sends each morphism $e\in E$ to either [Being a loop] or [Not being a loop] detecting whether the edge $e$ is a loop or not. Let us consider a subgraph \[ F = \left ( \begin{tikzcd}[scale=3] \bullet\ar[loop right,"\text{[Not being a loop]}"] \end{tikzcd} \right ), \] which is not an internal filter. Then the induced full subcategory $\E_F$ consists of graphs whose edges are not loops (i.e., $s(e) \neq t(e)$ for any $e\in E$). Obviously, $F$ itself does not belongs to the subcategory, since the edge [Not being a loop] is a loop! Thus we obtain an example of the situation $F\notin \ob(\E_F)$. \end{example} % and is the main topic of the next section. % % non-triv % In some situations, this theorem helps one to describe a local state classifier of a non-presheaf topos. % \memo{Can I cite kit?} % \memo{Can I write Myhill-Nerode Theorem?} % The main content of the next section, the normalization operator, is a kind of measurement % This will be proven in \Cref{cor:FilterLivesInHQuotient} in a little bit more general form. The main concept in the next section, \demph{the normalization operator,} precisely describes such a \dq{self-referential aspect} of the local state classifier (see also \Cref{exmp:ToposOfDirectedGraphTwo}). By using this and the order structure of $\Xi$, \Cref{cor:FilterLivesInHQuotient} shows that for an upward closed $F$, the condition $F \in \ob(\E_F)$ holds. \section{Normalization operator in a category with a local state classifier} The aim of this section is to define and study what we call the normalization operator of a category. \subsection{Definition and examples} \begin{definition}[Normalization operator]\label{def:NormalizationOperator} For a category $\E$ that admits a local state classifier $\Xi$, \demph{the normalization operator} \[\xi_{\Xi}\colon \Xi \to \Xi\] is the component of the colimit cocone $\{\xi_X \colon X \to\Xi\}_{X\in \ob(\E)}$ for the object $\Xi$. % and write it as $\Nor_{\E} \coloneqq \xi_{\Xi}$. \end{definition} \begin{example}\label{exmp:CaseOfGroup} This terminology is inspired by the case of group action topos (see \cite[][Example 3.10]{hora2024internal} for details). In the topos of right $G$-actions $\PSh(G)$ for a group $G$, the local state classifier $\Xi$ is the set of subgroups equipped with the right conjugate actions. \[ H\cdot g \coloneqq g^{-1}Hg \text{ in }\Xi \] Each component of the colimit cocone $\{\xi_X \colon X \to\Xi\}_{X\in \ob(\E)}$ sends an element $x\in X$ of a $G$-set $X$ to its stabilizer subgroup. \[ \xi_X(x) = \{g\in G\mid x\cdot g = x\} \] Therefore, the normalization operator $\xi_{\Xi} \colon \Xi \to \Xi$ sends a subgroup $H\in \Xi$ to its normalizer group $\Nor_{G}(H) \in \Xi$ \[ \xi_{\Xi}\colon H \mapsto \{g\in G \mid g^{-1}Hg= H\} = \Nor_G (H). \] \end{example} \begin{example}[The topos of graphs: {[Not being a loop] is a loop.} (2/2)]\label{exmp:ToposOfDirectedGraphTwo} The normalization operator $\xi_\Xi \colon \Xi \to \Xi$ in the topos of directed graphs $\E = \PSh(\rightrightarrows)$ sends both of two loops [Being a loop] and [Not being a loop] in $\Xi$ \[ \Xi = \left ( \begin{tikzcd}[scale=3] \bullet\ar[loop left,"\text{[Being a loop]}"]\ar[loop right,"\text{[Not being a loop]}"] \end{tikzcd} \right ) \] to the edge [Being a loop]. In particular, we have $\xi_{\Xi}(\text{[Not being a loop]}) = \text{[Being a loop]}$, which captures the self-referential statement \dq{[Not being a loop] is a loop.} \end{example} \begin{example}[Localic topos: trivial case]\label{exmp:localic} A Grothendieck topos $\E$ is localic if and only if its local state classifier $\Xi$ is a terminal object. In such a localic case, including the sheaf topos $\E=\Sh(X)$ over a topological space $X$, the normalization operator $\xi_{\Xi}\colon \Xi \to \Xi$ trivially coincides with the identity morphism $\id_{\Xi}$. \end{example} The author does not know when the identity map $\xi_{\Xi}$ coincides with the identity. In the group action topos $\PSh(G)$, the normalization operator cannot be the identity unless $G$ is trivial since $\xi_{\Xi}(\{e\})=G$. \Cref{exmp:localic} implies that such a class of topoi includes all localic topoi. The next example shows that it is broader than localic topoi. \begin{example}[The topos of idempotent functions: $\xi_{\Xi}=\id_{\Xi}$]\label{exmp:ToposOfIdempotents} Let us consider the topos of idempotent functions % $\E\coloneqq \PSh(\mathbb{F}_2,1,\times) $, $\E\coloneqq \PSh(\langle x\mid x^2=x\rangle)$, which is equivalent to the topos of actions of the monoid $(\mathbb{F}_2,1,\times)$. An object of this topos is a pair $(X, \sigma\colon X \to X)$ of a set $X$ and an idempotent endofunction $\sigma^2=\sigma$. The local state classifier $\Xi$ of this topos is the $2$-element set \[ \Xi = \{\text{[fixed]}, \text{[not fixed]}\} \] equipped with the idempotent morphism \[ \begin{tikzcd}[row sep = 10pt] \text{[not fixed]} \ar[r, mapsto, "\sigma"]&\text{[fixed]}\\ \text{[fixed]}\ar[r, mapsto, "\sigma"]&\text{[fixed]}. \end{tikzcd} \] Each component $\xi_{(X, \sigma)}\colon X \to \Xi$ sends a fixed point $\sigma(x)=x$ to $\text{[fixed]}$ and a non-fixed point $\sigma(x)\neq x$ to $\text{[not fixed]}$. Therefore, $\xi_{\Xi}$ coincides with the identity function % $\xi_{\Xi}=\id_{\Xi}$, \[ \xi_{\Xi}=\id_{\Xi} \] since $\text{[fixed]}$ is fixed and $\text{[not fixed]}$ is not fixed. \end{example} % \begin{example}[Monoid actions and Word actions] % \end{example} While the examples we have seen tend to be idempotent, the normalization operator is usually not idempotent at all. \begin{example}[The topos of species: Non-idempotent normalization operator]\label{exmp:DihedralGroup} For the $4$th Dihedral group $D_4 \coloneqq \langle \sigma, \tau\mid \sigma^4=1, \tau^2=1, \tau\sigma = \sigma^3\tau\rangle$, the normalization operator $\xi_{\Xi}$ in $\PSh(D_4)$, which coincides with the group-theoretic one, is visualized in the following diagram. % https://q.uiver.app/#q=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 \[\begin{tikzcd} && {D_4} \\ & {\langle\tau, \sigma^2\rangle} & {\langle \sigma\rangle} & {\langle\sigma\tau, \sigma^2\rangle} \\ {\langle\tau\rangle} & {\langle\sigma^2\tau\rangle} & {\langle\sigma^2\rangle} & {\langle\sigma\tau\rangle} & {\langle\sigma^3 \tau\rangle} \\ && {\langle\rangle} \arrow[color={rgb,255:red,92;green,92;blue,214}, squiggly, from=1-3, to=1-3, loop, in=55, out=125, distance=10mm] \arrow[no head, from=2-2, to=1-3] \arrow[color={rgb,255:red,92;green,92;blue,214}, curve={height=-6pt}, squiggly, from=2-2, to=1-3] \arrow[no head, from=2-3, to=1-3] \arrow[color={rgb,255:red,92;green,92;blue,214}, curve={height=6pt}, squiggly, from=2-3, to=1-3] \arrow[no head, from=2-4, to=1-3] \arrow[color={rgb,255:red,92;green,92;blue,214}, curve={height=6pt}, squiggly, from=2-4, to=1-3] \arrow[no head, from=3-1, to=2-2] \arrow[color={rgb,255:red,92;green,92;blue,214}, curve={height=-6pt}, squiggly, from=3-1, to=2-2] \arrow[no head, from=3-2, to=2-2] \arrow[color={rgb,255:red,92;green,92;blue,214}, curve={height=-6pt}, squiggly, from=3-2, to=2-2] \arrow[color={rgb,255:red,92;green,92;blue,214}, curve={height=-6pt}, squiggly, from=3-3, to=1-3] \arrow[no head, from=3-3, to=2-2] \arrow[no head, from=3-3, to=2-3] \arrow[no head, from=3-3, to=2-4] \arrow[no head, from=3-4, to=2-4] \arrow[color={rgb,255:red,92;green,92;blue,214}, curve={height=6pt}, squiggly, from=3-4, to=2-4] \arrow[no head, from=3-5, to=2-4] \arrow[color={rgb,255:red,92;green,92;blue,214}, curve={height=6pt}, squiggly, from=3-5, to=2-4] \arrow[color={rgb,255:red,92;green,92;blue,214}, curve={height=18pt}, squiggly, from=4-3, to=1-3] \arrow[no head, from=4-3, to=3-1] \arrow[no head, from=4-3, to=3-2] \arrow[no head, from=4-3, to=3-3] \arrow[no head, from=4-3, to=3-4] \arrow[no head, from=4-3, to=3-5] \end{tikzcd}\] % \href{https://q.uiver.app/#q=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}{[quiver]} % https://q.uiver.app/#q=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 % \[\begin{tikzcd} % && {D_4} \\ % & {\langle\tau, \sigma^2\rangle} & {\langle \sigma\rangle} & {\langle\sigma\tau, \sigma^2\rangle} \\ % {\langle\tau\rangle} & {\langle\sigma^2\tau\rangle} & {\langle\sigma^2\rangle} & {\langle\sigma\tau\rangle} & {\langle\sigma^3 \tau\rangle} \\ % && {\langle\rangle} % \arrow[from=2-2, to=1-3] % \arrow[from=2-3, to=1-3] % \arrow[from=2-4, to=1-3] % \arrow[from=3-1, to=2-2] % \arrow[from=3-2, to=2-2] % \arrow[from=3-3, to=2-2] % \arrow[from=3-3, to=2-3] % \arrow[from=3-3, to=2-4] % \arrow[from=3-4, to=2-4] % \arrow[from=3-5, to=2-4] % \arrow[from=4-3, to=3-1] % \arrow[from=4-3, to=3-2] % \arrow[from=4-3, to=3-3] % \arrow[from=4-3, to=3-4] % \arrow[from=4-3, to=3-5] % \end{tikzcd}\] Therefore, $\xi_{\Xi}$ is not idempotent nor order-preserving. The same argument works for $\PSh(S_4)$ and hence for the topos of species $\PSh(\FinSet_{\mathrm{bij}})$. This implies that the normalization operator in the topos of species $\PSh(\FinSet_{\mathrm{bij}})$ is not idempotent. \end{example} \subsection{The normalization lemma} At first glance, the normalization operator has nothing to do with the order structure ($=$ the semilattice structure) of $\Xi$. % is NOT a semilattice homomorphism on $\Xi$. For example, it does not preserve even the order structure (see \Cref{exmp:DihedralGroup}). % 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 $\Nor_{S_3}(H)= H \subsetneq S_3 =\Nor_{S_3}(\{e\})$. % \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} However, there is an obvious inclusion relation \[H \subset \Nor_G(H)\] for the normalizer of a subgroup $H \subset G$, which can be generalized as follows: % The next proposition is a generalization of the inclusion relation \begin{proposition}[Normalization lemma] \label{prop:NormalizationLemma} In a cartesian closed category $\E$ with a local state classifier $\Xi$, the morphism $\xi_{\Xi}\colon \Xi\to \Xi$ is equal to or larger than $\id_{\Xi}$ % \[ % \id_{\Xi} \leq \xi_{\Xi} % \] \[ \begin{tikzcd}[column sep=50pt] \Xi\ar[r, bend left, ""'{name=A}, "\id_\Xi"]\ar[r, bend right, "\xi_\Xi"', ""{name=B}] \ar[from=A, to=B, phantom, "\rotatebox{90}{$\geq$}"] &\Xi \end{tikzcd} \] with respect to the $\land$-semilattice structure on $\E(\Xi, \Xi)$. \end{proposition} \begin{proof} 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 any cartesian closed category $\E$ with a local state classifier $\Xi$, and any internal filter (or more generally, any upward closed subobject) $F\rightarrowtail \Xi$, we have \[ F \in \ob(\E_F), \] i.e., $F$ belongs to the induced full subcategory $\E_F \hookrightarrow \E$. % is an object of the corresponding hyperconnected quotient $\E \twoheadrightarrow \E_F$. \end{corollary} \begin{proof} % Let $\iota_{F} \colon F \rightarrowtail\Xi$ denote the inclusion map. Due to the equivalence (\ref{eq:FullSubCondition}), it suffices to prove that $\xi_F \colon F \to \Xi$ lifts along $\iota_{F}\colon F \rightarrowtail \Xi$. \[ F\in \ob(\E_F) \iff \begin{tikzcd} & F\ar[d, rightarrowtail, "\iota_{F}"]\\ F\ar[r,"\xi_F"']\ar[ru, dashed, "\exists"]&\Xi \end{tikzcd} \] Since $\iota_{F}$ trivially lifts along itself and the internal filter $F$ is upward closed, it is enough to prove the inequality $\iota_{F} \leq \xi_F$. This follows from the following diagram and the inequality of \Cref{prop:NormalizationLemma}. % $\xi_F$ also lifts along $\iota_{F}$. % The inequality $\id_\Xi \leq \xi_{\Xi}$ (\Cref{prop:NormalizationLemma}) implies $\iota_{F} \leq \xi_F$ due to the following diagram. \[ % \begin{tikzcd}[column sep=50pt] % F\ar[r, bend left, ""'{name=A}, "\iota_{F}"]\ar[r, bend right, "\xi_F"', ""{name=B}] \ar[from=A, to=B, phantom, "\rotatebox{90}{$\geq$}"] &\Xi % \end{tikzcd} % = \begin{tikzcd}[column sep=50pt] F \ar[r,"\iota_{F}", rightarrowtail]\ar[rr, bend right=50, "\xi_F"',""{name=C}]&\Xi\ar["\rotatebox{90}{$=$}", to={C}, phantom]\ar[r, bend left, ""'{name=A}, "\id_\Xi"]\ar[r, bend right, "\xi_\Xi"', ""{name=B}] \ar[from=A, to=B, phantom, "\rotatebox{90}{$\geq$}"] &\Xi \end{tikzcd} \] \end{proof} \section{Local state classifier in a hyperconnected quotient} The goal of this section is to prove \Cref{thm:MainTheorem}. In this section, we fix the following data: \begin{itemize} \item $\E$ is an elementary topos with a local state classifier $\Xi$. \item $F$ is an internal filter of $\Xi$. \item $\E_F$ is the corresponding hyperconnected quoteint of $\E$. \item $\G$ is the corresponding (lex idempotent) comonad on $\E$, with monic counit $\{\epsilon_X \colon \G X \rightarrowtail X\}_{X\in \ob(\E)}$ \end{itemize} Due to \Cref{thm:OldMainTheorem} and \Cref{prop:ExistenceForGrothendieck}, this situation covers all hyperconnected geometric morphisms between Grothendieck topoi. % Due to \Cref{cor:FilterLivesInHQuotient}, for any internal filter $F \rightarrowtail \Xi$, all morphisms $\{\xi_X^F\colon X \to F \}_{X\in \ob(\E_F)}$, which are morphisms in $\E$ a priori, belong to the full subcategory $\E_F$. Due to \Cref{cor:FilterLivesInHQuotient}, we know that all components of the family $\{\xi_{Z}^F \colon Z \to F\}_{Z\in \ob(\E_F)}$ belong to the full subcategory $\E_F$. As next lemma shows, it is not hard to prove that it is a cocone. % \memo{Warning: The constructions, like powerset construction, are done in the larger topos $\E$.} \begin{lemma}[Being a cocone]\label{lem:BeingCocone} The family $\{\xi_{Z}^F \colon Z \to F\}_{Z\in \ob(\E_F)}$ is a cocone under the functor ${(\E_F)}_{\mono} \to \E_F$. \end{lemma} \begin{proof} Let $m\colon Z\rightarrowtail Z'$ be an arbitrary monomorphism in the category $\E_F$. Since the embedding $\E_F \hookrightarrow \E$ preserves finite limits, $m$ remains to be monic in the ambient topos $\E$. Therefore, we have the commutativity of the outer perimeter of the following diagram: \[ \begin{tikzcd} Z\ar[rr,"m", rightarrowtail]\ar[rd, "\xi^F_{Z}"']\ar[rdd, "\xi_Z"', bend right]&&Z'\ar[ld, "\xi^F_{Z'}"] \ar[ldd, "\xi_Z'", bend left]\\ &F\ar[d,rightarrowtail, "\iota_F"]&\\ &\Xi.& \end{tikzcd} \] Since $\iota_F$ is monic, this implies the commutativity of the inner triangle $ \xi_{Z'}^F \circ m = \xi^F_{Z}$, which completes the proof. \end{proof} In the rest of this section, we will prove the universality of the family $\{\xi_{Z}^F \colon Z \to F\}_{Z\in \ob(\E_F)}$ as a colimit of $(\E_F)_{\mono} \to \E_F$. What we can use is the fact that the cocone $\{\xi_X \colon X \to \Xi\}_{X\in \ob(\E)}$ is a (large) colimit cocone $\E_{\mono}\to \E$. So we will convert the situations in $\E_F$ to the larger category $\E$ and reduce the required universality of $F \in \ob(\E_F)$ to that of $\Xi \in \ob(\E)$. First, we will prove the uniqueness part of the universality of $F$. Recall that a (possibly large) family of morphisms $\{f_\lambda \colon X_\lambda \to Y\}_{\lambda \in \Lambda}$ is said to be \demph{jointly epimorphic} if, for any parallel morphisms $g,h\colon Y \rightrightarrows Z$, the implication $ (\forall \lambda \in \Lambda\; g\circ f_\lambda = h\circ f_\lambda) \implies (g=h) $ holds. % holds. % \[ % (\forall \lambda \in \Lambda\; g\circ f_\lambda = h\circ f_\lambda) \iff (g=h) % \] % holds. % implies $g=h$. In the following proof, we will not assume that the topos $\E$ is a Grothendieck topos. So we cannot use the complete lattice structure of the subobject lattice of $\Xi$. Instead of it, we use the Heyting algebra structure of it, which makes sense in an arbitrary elementary topos. \begin{lemma}[Universality (1/2): Uniqueness] \label{lem:JointlyEpimorphic} The cocone $\{\xi_{Z}^F \colon Z \to F\}_{Z\in \ob(\E_F)}$ is jointly epimorphic (in $\E$, and hence in $\E_F$). \end{lemma} \begin{proof} Take an arbitrary subobject $S\rightarrowtail F$ such that every arrow in $\{\xi^F_{Z} \colon Z \to F\}_{Z\in \ob(\E_F)}$ lifts along $S\rightarrowtail F$. It suffices to prove $S=F$ (due to \Cref{lem:JointlyEpimorphicFamilyAndSubobject}). % since for any two morphisms $g, h \colon F \to Z$ that are not distinguished by any morphisms in the family, every morphism $\xi^F_Z$ factors through their equalizer $S \rightarrowtail F \rightrightarrows Z$. % % which implies $g=h$. % Notice that each morphism $\xi^F_Z \colon Z \to F$ factors through $S$, i.e., By the lifting assumption on $S$, the inequality \[ \Image(\xi^F_Z) \leq S \text{ in } \Sub_{\E}(\Xi) \] holds for every $Z\in \ob(\E_F)$. We also have the equality \[ \Image(\xi_X)\land F = \Image(\xi^F_{\G X}) \text{ in } \Sub_{\E}(\Xi) \] for every object $X \in \ob(\E)$, since the epi-mono factorization of $\E$, which is pullback stable, decomposes the pullback square (\ref{eq:PullbackDescriptionOfTheCounitAndComonad}) \[ \begin{tikzcd} \G X\ar[r, "\xi_{\G X}^F"]\ar[d, "\epsilon_X", tail] \ar[rd, phantom, "\lrcorner", very near start]& F\ar[d,tail, "\iota_F"]\\ X\ar[r , "\xi_X"']& \Xi \end{tikzcd} \] into the following pullback diagram \[ \begin{tikzcd} \G X\ar[r, two heads]\ar[rr, bend left, "\xi_{\G X}^F"]\ar[d, "\epsilon_X", tail]\ar[rd, phantom, "\lrcorner", very near start]& % \Image(\xi_X)\land F \Image(\xi^F_{\G X}) \ar[r, tail]\ar[d, tail]\ar[rd, phantom, "\lrcorner", very near start]& F\ar[d,tail, "\iota_F"]\\ X\ar[rr, bend right , "\xi_X"']\ar[r, two heads]&\Image(\xi_X)\ar[r, tail]& \Xi. \end{tikzcd} \] Combining the above two (in)equalities in $\Sub_{\E}(\Xi)$, we obtain an inequality \[ \Image(\xi_X)\land F \leq S \text{ in } \Sub_{\E}(\Xi) \] for each object $X\in \ob(\E)$. Since the poset $\Sub_{\E}(\Xi)$ is a Heyting algebra, this is equivalent to the inequality \[ \Image(\xi_X) \leq (F \mathbin{\rightarrow} S) \text{ in } \Sub_{\E}(\Xi). \] This means that the colimit cocone $\{\xi_X\colon X \to \Xi\}_{X\in \ob(\E)}$ factors throgh the subobject $(F\to S) \in \Sub_{\E}(\Xi)$. Since the colimit cocone $\{\xi_X \colon X \to \Xi\}_{X \in \ob(\E)}$ is jointly epimorphic, we have \[ \Xi = \top= (F \mathbin{\rightarrow} S) \text{ in } \Sub_{\E}(\Xi). \] (due to \Cref{lem:JointlyEpimorphicFamilyAndSubobject}), i.e., $F\leq S$. Since $S\leq F$ holds by definition, this completes the proof. \end{proof} Lastly, we need to prove the existence part of the universality of $F$, using the universality of $\Xi$. This is the tricky part, since in order to use the universality of $\Xi$, we need to construct a cocone under $\E_{\mono} \to \E$ from a given cocone $\{\phi_Z\colon Z\to L\}_{Z \in \ob(\E_F)}$ under $(\E_F)_{\mono} \to \E_F$. In other words, we need to extend the index set of cocone from $\ob(\E_F)$ to $\ob(\E)$. The first idea for the extension problem is to use the counit $\epsilon_X \colon \G X \rightarrowtail X$. Since $\G X$ is an object of $\E_F$, we can canonically associate an object $\G X$ of $\E_F$ with each object $X$ of $\E$. But here is another problem. Although we want to construct a family of morphisms \textbf{from} all objects $X$ in $\E$, what the counit provides is morphisms \textbf{to} objects of $\E$. \[ \begin{tikzcd} \G X \ar[r,"\phi_{\G X}"]\ar[d, rightarrowtail, "\epsilon_X"]& L\\ X\ar[ru, "?"', dashed] & \end{tikzcd} \] To inverse the direction of the arrow, we need the next idea, to use the powerset object. Recall that any morphism $f\colon X \to Y$ in a topos induces three morphisms between the powerset objects: \[ \begin{tikzcd}[column sep = 70pt] PX \ar[r,"\exists_f ", bend left] \ar[r, "\forall_f"', bend right ] & PY. \ar[l, "f^{-1}"'] \end{tikzcd} \] Combining these ideas, we obtain a cocone under $\E_{\mono} \to \E$ as follows: \[ \begin{tikzcd} \G X \ar[r,"\phi_{\G X}"]\ar[d, rightarrowtail, "\epsilon_X"]& L\\ X\ar[ru, "?"', dashed] & \end{tikzcd} \xrightarrow{\text{taking powerobjects}} \begin{tikzcd} P\G X \ar[r,"\exists_{\phi_{\G X}}"]& PL\\ PX \ar[u, "{\epsilon_X}^{-1}", twoheadrightarrow] &\\ X\ar[u, "\sgt_X", rightarrowtail] \ar[ruu, "!"', dashed] \end{tikzcd} \] Here, we use the morphism $\exists_{\phi_{\G X}}$, not $\forall_{\phi_{\G X}}$, because of its nicer properties like \Cref{lem:sgtNaturality}. In what follows, the notation $PX$ means the powerset object in the topos $\E$, not in the topos $\E_F$. \begin{lemma} \label{lem:ExtensionLemma} Let $\{\phi_Z\colon Z\to L\}_{Z \in \ob(\E_F)}$ be a cocone under the diagram $(\E_F)_{\mono}\to \E_F$. Then the family of morphisms \[ \begin{tikzcd} \{\psi_X\colon X\ar[r,"\sgt_X"] &PX \ar[r,"{\epsilon_X}^{-1}"]&P\G X\ar[r,"\exists_{\phi_{\G X}}"] & PL\}_{X\in \ob(\E)} \end{tikzcd} \] defines a cocone under the diagram $\E_{\mono} \to \E$. Furthermore, $\psi$ is an extension of $\phi$ in the sense that the diagram % \[ % \begin{tikzcd} % \G X \ar[r, "\phi_{\G X}"]\ar[d, tail, "\epsilon_X"]& L\ar[d, "\sgt_L", tail]\\ % X\ar[r, "\psi_X"]& PL % \end{tikzcd} % \] \begin{equation}\label{eq:PsiExtendsPhi} \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{equation} commutes. \end{lemma} \begin{proof} For any monomorphism $m\colon X\rightarrowtail Y$ in $\E$, we have a commutative diagram \[ \begin{tikzcd}[row sep = 10pt] X\ar[r,"\sgt_X"] \ar[dd, "m", tail]&PX \ar[r,"{\epsilon_X}^{-1}"]\ar[dd, "\exists_m"]&P\G X\ar[rd,"\exists_{\phi_{\G X}}"]\ar[dd,"\exists_{\G m}"]&\\ &&&PL\\ Y\ar[r,"\sgt_X"] &PY \ar[r,"{\epsilon_Y}^{-1}"]&P\G Y\ar[ru,"\exists_{\phi_{\G Y}}"'] & \end{tikzcd} \] since the left square commutes by \Cref{lem:sgtNaturality}, the right triangle commutes by the assumption of $\phi$ being a cocone under $(\E_F)_{\mono} \to \E_{F}$, and the middle square commutes by the Beck-Chevalley condition for the pullback square \[ \begin{tikzcd} \G X \ar[r, "\G m", tail]\ar[d,"\epsilon_X", tail]\ar[rd, phantom, "\lrcorner", very near start]&\G Y\ar[d,"\epsilon_Y", tail]\\ X\ar[r,"m", tail] & Y. \end{tikzcd} \] This proves that $\psi$ is a cocone under $\E_{\mono}\to \E$. To prove the latter part, which states the diagram \[ \begin{tikzcd} \G X \ar[rrr, "\phi_{\G X}"]\ar[d, tail, "\epsilon_X"]&&& L\ar[d, "\sgt_L", tail]\\ X\ar[r,"\sgt_X"'] &PX \ar[r,"{\epsilon_X}^{-1}"']&P\G X\ar[r,"\exists_{\phi_{\G X}}"'] & PL \end{tikzcd} \] 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:sgtNaturality} and \Cref{lem:sgtExtNaturality} complete the proof. \end{proof} \begin{proof}[Proof of \Cref{thm:MainTheorem}] In (\Cref{cor:FilterLivesInHQuotient} and) \Cref{lem:BeingCocone}, we have already observed that the family of morphisms $\{\xi_{Z}^F \colon Z \to F\}_{Z\in \ob(\E_F)}$ is a cocone under the functor $(\E_{F})_{\mono} \to \E_F$. % Furthermore, we have seen that the cocone is jointly surjective in \Cref{lem:JointlyEpimorphic}. It suffices to prove that this family has the universality as a colimit of all monomorphisms in $\E_F$. Since \Cref{lem:JointlyEpimorphic} ensures the uniqueness part of the desired universality, we will prove the existence part. Take an arbitrary cocone $\{\phi_Z \colon Z\to L\}_{Z\in \ob(\E_F)}$ under the functor $(\E_{F})_{\mono} \to \E_F$. Let $\{\psi_X\colon X \to PL\}_{X\in \ob(\E)}$ be the cocone under the functor $\E_{\mono} \to \E$ 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 the diagram (\ref{eq:PullbackDescriptionOfTheCounitAndComonad}) and (\ref{eq:PsiExtendsPhi}), % the latter part of \Cref{lem:ExtensionLemma}, the following diagram is also commutative. \begin{equation}\label{eq:block} \begin{tikzcd} % [column sep = 10pt] [row sep = 50pt] &\G X\ar[ld, "\xi_{\G X}^F"']\ar[d,"\epsilon_X", tail]\ar[rd, "\phi_{\G X}"]&\\ F\ar[d,tail, "\iota_F"'] &X\ar[ld,"\xi_X"']\ar[rd,"\psi_X"]&L\ar[d,"\sgt_L", tail]\\ \Xi\ar[rr,"\gamma"]&&PL \end{tikzcd} \end{equation} Theorefore, it is sufficient to prove that $\gamma \circ \iota_F\colon F\rightarrowtail \Xi \to PL$ lifts along $\sgt_L$ \[ \begin{tikzcd} % [column sep = 10pt] [row sep = 50pt] &\G X\ar[ld, "\xi_{\G X}^F"']\ar[rd, "\phi_{\G X}"]&\\ F\ar[d,tail, "\iota_F"'] \ar[rr, dashed, "?"]&&L\ar[d,"\sgt_L", tail]\\ \Xi\ar[rr,"\gamma"]&&PL, \end{tikzcd} \] since $\sgt_L$ is monic (and every object in $\E_{F}$ is isomorphic to an object of the form of $\G X$). Take the characteristic morphism of $\sgt_{L}\colon L \rightarrowtail PL$ as \[ \begin{tikzcd} % [column sep = 10pt] [row sep = 50pt] &\G X\ar[ld, "\xi_{\G X}^F"']\ar[rd, "\phi_{\G X}"]&&\\ F\ar[d,tail, "\iota_F"'] \ar[rr, dashed, "?"]&&L\ar[d,"\sgt_L", tail]\ar[rr, "!"]\ar[rrd, phantom, "\lrcorner", very near start]&&1\ar[d,"\true", tail]\\ \Xi\ar[rr,"\gamma"]&&PL\ar[rr,"\chi_L"]&&\Omega. \end{tikzcd} \] By the universality of the pullback, it suffices to % We prove that the composite $F\rightarrowtail \Xi \to PL \to \Omega$ coincides with the true morphism $\true_{F}\colon F \to \Omega$. The joint surjectivity of $\{\xi_{\G X}^F\colon\G X \to F \}_{X\in \ob(\E_F)}$, which is an immediate corollary of \Cref{lem:JointlyEpimorphic}, reduces it to proving that the composition $\G X \to F \to \Xi \to PL \to \Omega$ coincides with $\true_{\G X}$ for every $X \in \ob(\E)$. This follows from the commutativity of the perimeter of \[ % \begin{tikzcd} % % [column sep = 10pt] % [row sep = 50pt] % &\G X\ar[ld, "\xi_{\G X}^F"']\ar[rd, "\phi_{\G X}"]\ar[rrrd, bend left, "!"]\ar[d,"\epsilon_X"]&&\\ % F\ar[d,tail, "\iota_F"'] &X\ar[ld, "\xi_X"']\ar[rd,"\psi_X"]&L\ar[d,"\sgt_L", tail]\ar[rr, "!"]\ar[rrd, phantom, "\lrcorner", very near start]&&1\ar[d,"\true", tail]\\ % \Xi\ar[rr,"\gamma"]&&PL\ar[rr,"\chi_L"]&&\Omega. % \end{tikzcd} \begin{tikzcd} % [column sep = 10pt] [row sep = 50pt] &\G X\ar[ld, "\xi_{\G X}^F"']\ar[rd, "\phi_{\G X}"]\ar[rrrd, bend left, "!"]&&\\ F\ar[d,tail, "\iota_F"'] % \ar[rrd, phantom, "(\ref{eq:block})"] &(\ref{eq:block})&L\ar[d,"\sgt_L", tail]\ar[rr, "!"]\ar[rrd, phantom, "\lrcorner", very near start]&&1\ar[d,"\true", tail]\\ \Xi\ar[rr,"\gamma"]&&PL\ar[rr,"\chi_L"]&&\Omega. \end{tikzcd} \] This completes the proof. \end{proof} Let us give an example of our main theorem. \begin{corollary}[Local state classifier of a continuous group action topos]\label{cor:LSCofTopologicalGroups} For a topological group $G$, the local state classifier $\Xi$ of the Grothendieck topos of continuous $G$-actions $\Cont(G)$ is given by % \[ % \Xi=\{H\subset G\mid H: \text{open subgroup of }G\}, % \] \[ \Xi=\{\text{open subgroups of }G\}, \] equipped with the right conjugate action \[ H*g \coloneqq g^{-1}Hg. \] \end{corollary} \begin{proof} The Grothendieck topos of continuous $G$-actions, which is denoted by $\Cont(G)$, is a hyperconnected quotient of the presheaf topos $\PSh(G^{\delta})$ on the underlying (discrete) group $G^{\delta}$ \[ h\colon \PSh(G^{\delta}) \twoheadrightarrow \Cont(G). \] The local state classifier of the presheaf topos $\PSh(G^{\delta})$ is the set of all subgroups, and each component $\xi_X$ of the colimit cocone sends an element to its stabilizer. A $G^{\delta}$-set belongs to $\Cont(G)$ if and only if the stabilizer subgroups for any elements are open. Therefore, the internal filter $F$ that corresponds to the hyperconnected quotient $\Cont(G)$ is the set of all open subgroups of $G$. (One can easily verify that this is in fact an internal filter.) Therefore, we complete the proof by applying \Cref{thm:MainTheorem}. \end{proof} \section{Motivating example: the topos of word actions}\label{sec:MotivationgExample} This section aims to briefly explain a motivating example: the topos of word actions $\Aset$. The following contents are intended to be included in `Topoi of Automata II,' but the author believes that presenting a nontrivial concrete example will help the reader’s understanding. So the author will provide a brief overview of the part related to the Normalizer. More detailed explanations and applications will be given in the upcoming paper, `Topoi of Automata II.' \subsection{Right congruences form the local state classifier of the topos \texorpdfstring{$\Aset$}{Aset}} For a set $\A$, which we call \demph{alphabet}, we consider its free monoid $\MA$ and its presheaf topos $\Aset \coloneqq \PSh(\MA)$. An object of $\Aset$ can be regarded as a pair of a set $Q$ and a function $\delta \colon Q\times \A \to Q$. % , which respects the monoid structure of $\MA$. This is the simplest topos of the four topoi in the author's paper \cite{hora2024topoi}. In that paper, the $\A$-set of languages $\Lan \in \ob(\Aset)$ is defined as the set of all languages $\Lan \coloneqq \Pow(\MA)$ equipped with the \demph{left quotient action} \[ L\ast u \coloneqq \{v\in \MA\mid uv\in L\}, \] which is also denoted by $u^{-1}L$. What is the local state classifier of the topos $\Aset$? By the general formula for the local state classifier of a presheaf topos, we can conclude that $\Xi$ is the $\A$-set of all \demph{right congruences} \[ \Xi = \left\{\text{equivalence relation }{\sim}\text{ on }\MA\mid \forall u,v,w\in \MA,\; u\sim v \implies uw \sim vw\right\}, \] equipped with the right $\A$-action \[ u\mathrel{({\sim} * w)}v \iff wu \sim wv. \] % Here, a right congruence is an equivalence relation $\sim$ on the set $\MA$, such that % \[ % u\sim v \implies uw \sim vw % \] % holds for any $u,v, w\in \MA$. What makes $\Xi$ interesting is its colimit cocone $\{\xi_{(Q, \delta)}\colon (Q, \delta)\to \Xi\}$. Each colimit cocone $\xi_{(Q, \delta)}\colon (Q, \delta)\to \Xi$ sends a state $q$ to the right congruence $\xi_{(Q, \delta)}(q)$ defined by \[ u \mathrel{(\xi_{(Q, \delta)}(q) )} v \iff qu=qv. \] In particular, the morphism $\xi_{\Lan}\colon \Lan \to \Xi$ sends a language $L\in \Lan$ to % the induced congruence $\xi_{\Lan}(L)$ coincides with what's called the \demph{Nerode congruence} \[ u \mathrel{(\xi_{\Lan}(L))}v \iff (L \ast u = L \ast v) \iff \left(\forall w\in \MA, (uw\in L \iff vw\in L)\right), \] which is used for automata minimalization In \cite{hora2024topoi}, the hyperconnected geometric morphism $h \colon \Aset \twoheadrightarrow \ofAset$ to the topos of \demph{orbitwise finite $\A$-sets} plays a central role. Here, an orbitwise finite $\A$-set is a $\A$-set $(Q, \delta)$ such that for each element $q\in Q$, its orbit $\{qw\mid w\in \MA\}$ is finite. By construction, the corresponding internal filter $F_{\of} \rightarrowtail \Xi$ is given by \begin{equation}\label{eq:DefinitionOfFof} F_{\of}= \{{\sim \in \Xi \mid |\MA/{\sim}}|<\infty\}. \end{equation} % \[ % F_{\of}= \{{\sim \in \Xi \mid |\MA/{\sim}}|<\infty\}. % \] One reason why this particular topos $\ofAset$, which corresponds to the filter $F_{\of}$, captures the notion of regular languages (see \cite{hora2024topoi}) is the fact that the Myhill-Nerode theorem implies that a language $L$ is regular if and only if $\xi_{\Lan}(L) \in F_{\of}$. The main theorem of the present paper (\Cref{thm:MainTheorem}) proves that the local state classifier of the topos $\ofAset$ is given by $F_{\of}$. \begin{corollary}\label{cor:LocalStateClassifierOfOrbitfiniteAset} The local state classifier of $\ofAset$ is given by $F_{\of}$ (\cref{eq:DefinitionOfFof}), equipped with the morphisms $\{\xi_{(Q, \delta)}\colon (Q, \delta)\to F_{\of}\}_{(Q, \delta)\in \ob(\ofAset)}$, where \[ u\mathrel{(\xi_{(Q,\delta)}(q))}v \iff qu=qv. \] \end{corollary} % Notice that the Myhill-Nerode theorem implies that a language $L$ is regular if and only if $\xi_{\Lan}(L) \in F_{\of}$. \subsection{Two-sided congruence and Syntactic monoids} In order to define the \demph{syntactic monoid} of a language, it does not suffice to consider only right congruences. We need \demph{two-sided congruences} % or equivalently, monoid congruences on $\MA$. Recall that an equivalence relation ${\sim} \subset \MA\times \MA$ is said to be a two-sided congruence, if it satisfies \[ u\sim v \implies (\forall w,w',\; wuw' \sim wvw'). \] In algebraic language theory, the \demph{syntactic monoid} $M_L$ of a language $L$ is defined to be a quotient monoid $M_L\coloneqq \MA/{\cong_L}$ of $\MA$, where $\cong_L$ is the two-sided congruence defined by \[ u \cong_L v \iff \left (\forall w,w', \; wuw' \in L \iff wvw'\in L\right). \] We can rewrite this in terms of the Nerode congruence $\xi_{\Lan}(L)$ as follows: % and the internal lattice structure\footnote{Rigorously speaking, we use the compltete lattice structure of $\Xi$.} of $\Xi$. \begin{align*} u \cong_L v &\iff \left (\forall w,w', \; wuw' \in L \iff wvw'\in L\right)\\ &\iff \left (\forall w,w', \; uw' \in L*w \iff vw'\in L*w\right)\\ &\iff \forall w, \; u \mathrel{(\xi_{\Lan}(L*w))} v\\ &\iff \forall w, \; u \mathrel{(\xi_{\Lan}(L)*w)} v. \end{align*} % the coarsest two-sided congruence that is finer than the Nerode congruence $\xi_{\Lan}(L)$. % Here is a general question, how can we construct the coarsest two-sided congruence that is finer than a given right congruence ${\sim}\in \Xi$? % By this description, the coarsest two-sided congruence is given by % \[ % \bigwedge_{w\in \MA} ({\sim}*w) % \] % \[ % u(\xi_{\Xi}({\sim}))v \iff ({\sim}*u = {\sim}*v) \iff (\forall w,w',\; (uw\sim uw' \iff vw\sim vw')) % \] % This structure is crucial in the automata theory. Therefore, the two-sided congruence $\cong_L$ is given by the infimum\footnote{This is an infinite infimum, which not necessarily exists in the general setting of elementary topoi. In this particular case of $\Aset$, the local state classifier $\Xi$ has an structure of complete lattice.} \[ {\cong_L} = \inf \{\xi_{\Lan}(L)*w\mid w\in \MA\} \text{ in } \Xi. % \bigwedge_{w\in \MA} \xi_{\Lan}(L)*w. \] What we will be interested in, in the upcoming paper `Topoi of automata II,' is the following condition % \[\xi_{\Lan}(L) \in F \iff {\cong_L}\in F\] \begin{equation}\label{eq:NerodeAndMyhillCongruence} \text{For any internal filter $F\subset \Xi$, we have }\xi_{\Lan}(L) \in F \iff {\cong_L}\in F. \end{equation} % for any internal filter $F\subset \Xi$. From a topos theoretic viewpoint, this equivalence \ref{eq:NerodeAndMyhillCongruence} is crucial to compare the minimal automaton and the syntactic monoid, since it states that the Nerode congruence $\xi_{\Lan}(L)$, which realizes the minimal automaton, and the syntactic congruence $\cong_L$, which realizes the syntactic monoid, behave in the same way in terms of hyperconnected quotients. Here, we need the nomalizer operator $\xi_{\Xi}$. Since any internal filter $F$ is closed under the $\MA$-action and finite infimum, the condition \begin{equation}\label{eq:RegularCongruenceCondition} \{\xi_{\Lan}(L)*w\mid w\in \MA\}\subset \Xi \text{ is a finite set.} \end{equation} % \[ % \{\xi_{\Lan}(L)*w\mid w\in \MA\}\subset \Xi % \] % is a finite set. is sufficient for ensuring the condition \ref{eq:NerodeAndMyhillCongruence}. This condition is equivalent to $\xi_{\Xi}(\xi_{\Lan}(L))\in F_{\of}$, in which the normalization operator $\xi_{\Xi}$ appears. Summarizing what we have observed, we obtain the following proposition. \begin{proposition}\label{prop:NerodeAndMyhillCongruences} For any language $L\in \Lan$ with the property $\xi_{\Xi}(\xi_{\Lan}(L))\in F_{\of}$ and any internal filter $F\subset \Xi$, the following conditions are equivalent: % \begin{itemize} % \item $\xi_{\Lan}(L) \in F$: The Nerode congruence belongs to $F$. % \item ${\cong_{L}}\in F$: The syntactic congruence belongs to $F$. % \end{itemize} \begin{itemize} \item The Nerode congruence $\xi_{\Lan}(L)$ belongs to $F$. \item The syntactic congruence ${\cong_{L}}$ belongs to $F$. \end{itemize} % we have % \[ % \xi_{\Lan}(L) \in F \iff {\cong_{L}}\in F. % \] \end{proposition} The typical examples of a language that satisfies the condition $\xi_{\Xi}(\xi_{\Lan}(L)) \in F_{\of}$ are regular languages. In its proof, we can use the normalization lemma (\Cref{prop:NormalizationLemma}). \begin{corollary}\label{cor:RegularlanguageCongruences} For any regular language $L\in \Lan$ and any internal filter $F\subset \Xi$, the following conditions are equivalent: \begin{itemize} \item The Nerode congruence $\xi_{\Lan}(L)$ belongs to $F$. \item The syntactic congruence ${\cong_{L}}$ belongs to $F$. \end{itemize} \end{corollary} \begin{proof} In order to apply \Cref{prop:NerodeAndMyhillCongruences}, it suffices to prove $\xi_{\Xi}(\xi_{\Lan}(L))\in F_{\of}$ for every regular language $L$. Since a language $L$ is regular if and only if $\xi_{\Lan}(L)\in F_{\of}$, and every internal filter $F$ is upward closed, it suffices to prove $\xi_{\Lan}(L) \leq \xi_{\Xi}(\xi_{\Lan}(L))$. This follows from the normalization lemma $\id_{\Xi}\leq \xi_{\Xi}$ (\Cref{prop:NormalizationLemma}). % \[ % % \id_{\Xi}\leq \xi_{\Xi} \text{ (\Cref{prop:NormalizationLemma})}. % \] % implies % \[ % \xi_{\Lan}(L) \leq \xi_{\Xi}(\xi_{\Lan}(L)). % \] % Since this completes the proof $\xi_{\Xi}(\xi_{\Lan}(L))\in F$. \end{proof} % we will be interested in when to \appendix \section{Preliminaries on elementary topoi} This appendix summarizes the properties of elementary topoi that are used in the main part. \begin{lemma}[Jointly epimorphic families in an elementary topos]\label{lem:JointlyEpimorphicFamilyAndSubobject} For a (possibly large) family of morphisms $\{f_\lambda \colon X_\lambda \to Y\}_{\lambda \in \Lambda}$ in a category $\E$, we consider the following two conditions: \begin{enumerate} \item $\{f_\lambda \colon X_\lambda \to Y\}_{\lambda \in \Lambda}$ is jointly epimorphic. \item If all morphisms in the family factor through a monomorphism $m\colon S\rightarrowtail Y$, then $m$ is an isomorphism. \end{enumerate} If $\E$ is balanced, i.e. every monic and epic morphism is an isomorphism, then $(1)$ implies $(2)$. If $\E$ has equalizers, then $(2)$ implies $(1)$. In particular, if $\E$ is an elementary topos, the two conditions $(1),(2)$ are equivalent. \end{lemma} \begin{proof} First, assuming that the family is jointly epimorphic and $\E$ is balanced, we prove $(2)$. Take an arbitrary monomorphism $m\colon S\rightarrowtail Y$ such that every morphism $f_\lambda$ in the family factors through $m$ as $f_{\lambda} = m\circ f^S_{\lambda}$. For any morphisms $g,h \colon Y \rightrightarrows Z$ such that $g\circ m = h\circ m$, we have $g\circ f_{\lambda} = g\circ m \circ f_{\lambda}^{S} = h\circ m \circ f_{\lambda}^{S} = h\circ f_{\lambda}$, and the $g=h$. This proves that $m$ is also epic, and hence the balancedness assumption implies that $m$ is an isomorphism. Next, assuming $(2)$ and that $\E$ has equalizers, we prove $(1)$. Take an arbitrary morphisms $g,h \colon Y \rightrightarrows Z$ such that $g\circ f_{\lambda} =h\circ f_{\lambda}$ for any $\lambda$. We prove $g=h$. Let $m\colon S\rightarrowtail Y$ be the equalizet of the two morphisms $g,h$. Then every morphism in the family factors through $m$, and the assumption $(2)$ implies that $m$ is an isomorphism. This proves that $g=h$. \end{proof} \begin{lemma} \label{lem:sgtNaturality} For any morphism $f\colon X\to Y$ in a topos, the diagram \[ \begin{tikzcd} X\ar[r,"\sgt_{X}", tail]\ar[d, "f"]&PY\ar[d,"\exists_f"]\\ Y\ar[r,"\sgt_Y", tail]&PY \end{tikzcd} \] commutes. \end{lemma} \begin{proof} Via the bojection $\E(X, PY) \cong \Sub(X\times Y)$, both of two maps corresponds to the subobject \[ \langle\id_X, f \rangle \colon X \rightarrowtail X\times Y. \] In fact, the right above part corresponds to the image of the composite \[ \begin{tikzcd} X \ar[r,rightarrowtail, "\Delta_X"] &X\times X \ar[r,"\id_X \times f"]& X\times Y, \end{tikzcd} \] and the left below part corresponds to the pullback \[ \begin{tikzcd} X\ar[r, "f"] \ar[d, rightarrowtail, "{\langle \id_X, f\rangle }"' ] \ar[rd, phantom, "\lrcorner", very near start] & Y\ar[d, "\Delta_Y", rightarrowtail]\\ X\times Y \ar[r,"f\times \id_Y"] & Y\times Y. \end{tikzcd} \] \end{proof} \begin{lemma} \label{lem:sgtExtNaturality} For any 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: \[ \begin{tikzcd}[row sep = 10 pt] X\times X\ar[rd,"\delta_X"]\ar[dd,"m\times m"', tail]&\\ & \Omega\\ Y \times Y.\ar[ru, "\delta_Y"'] \end{tikzcd} \] Taking the transposes of it with the naturality of the three-variable adjunction, we obtain the commutativity as stated. \end{proof} \printbibliography \end{document}