← Topos with enough projectives
Ver1__20241220.tex
\documentclass{amsart}
\usepackage[left=2cm, right=2cm]{geometry}
\usepackage[utf8]{inputenc}
\usepackage{amsfonts, amsthm, amssymb, mathtools,etoolbox}
\usepackage{blindtext}
\usepackage[colorlinks=true, urlcolor=blue, linkcolor=blue, citecolor=blue]{hyperref}
\usepackage{tikz,tikz-cd}
\usepackage{cleveref}
\usepackage{array}
\usepackage[style=alphabetic,sorting=nyt]{biblatex}
\renewbibmacro{in:}{}
% \addbibresource{biblio.bib}
\addbibresource{CommonBiblio20240922.bib}
\tikzset{pullback/.style={minimum size=1.2ex,path picture={
\draw[opacity=1,black,-,#1] (-0.5ex,-0.5ex) -- (0.5ex,-0.5ex) -- (0.5ex,0.5ex);%
}}}
\theoremstyle{plain}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{todo}[theorem]{Todo}
\newtheorem{conjecture}[theorem]{Conjecture}
\newtheorem{fact}[theorem]{Fact}
\theoremstyle{definition}
\newtheorem{example}[theorem]{Example}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{remark}[theorem]{Remark}
\newtheorem{notation}[theorem]{Notation}
\newtheorem{question}[theorem]{Question}
\newtheorem{idea}[theorem]{Idea}
\newcommand{\dq}[1]{``#1"}
\newcommand{\memo}[1]{\textcolor{red}{memo: #1}}
\newcommand{\invmemo}[1]{\textcolor{blue}{memo: #1}}
\newcommand{\para}[1]{\paragraph{\textbf{#1}}}
\newcommand{\N}{\mathbb{N}}
\newcommand{\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{\id}{\mathrm{id}}
\newcommand{\op}{\mathrm{op}}
\newcommand{\ob}{\mathrm{ob}}
\newcommand{\Set}{\mathbf{Set}}
\newcommand{\FinSet}{\mathbf{FinSet}}
\newcommand{\PSh}{\mathbf{PSh}}
\newcommand{\Sh}{\mathbf{Sh}}
\newcommand{\Cont}{\mathbf{Cont}}
\newcommand{\Func}[2]{[#1,#2]}
\newcommand{\abs}[1]{\left|#1\right|}
\newcommand{\demph}[1]{\textbf{#1}}
\font\maljapanese=dmjhira at 2.5ex
\newcommand{\yo}{\textrm{\!\maljapanese\char"48}}
\renewcommand{\P}{\mathcal{P}}
\newcommand{\G}{\mathcal{G}}
\newcommand{\Ck}{\C_{\kappa}}
\title{Topos with enough projectives}
\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 note towards site characterization of topoi with enough projectives.
% \end{abstract}
\maketitle
% \tableofcontents
\begin{itemize}
\item The question is asked at \href{https://mathoverflow.net/questions/316588/topos-with-enough-projectives}{[mathoverflow 316588]} by Morgan Rogers.
\item Ryo Suzuki asks a similar question in the context of internal choice principle in the topos of light condensed sets.
\end{itemize}
In this note, a regular cardinal means an infinite regular cardinal.
\section{Preliminaries on projective objects}
A subobject $\iota \colon S\rightarrowtail X$ is called \demph{retract}, if $\iota$ is a split monomorphism, and is called \demph{summand} ($=$ complemented) if $\iota$ is an injection map of a coproduct diagram $S\rightarrowtail X \leftarrowtail S'$.
\begin{lemma}[Closure properties of projective objects]\label{lem:ClosednessOfProjectives}
For a category $\C$, projective objects satisfy the following closure properties.
\begin{itemize}
\item A retract of a projective object is projective.
\item A small coproduct of projective objects is projective.
\item If the category $\C$ is extensive, a summand of a projective object is projective.
\end{itemize}
\end{lemma}
% \begin{proof}
% We only prove the last statement.
% Assuming that a projective object $X$ is decomposed into $P\cong X+Y$, we prove that $X$ is projective. For an emimorphism $e\colon A \twoheadrightarrow B$ and a morphism $f\colon X\to B$, we consider the lifting problem of
% \[
% \begin{tikzcd}
% &A+Y\ar[d,"e+\id_{Y}", twoheadrightarrow]\\
% P\cong X+Y\ar[r,"f+ \id_{Y}"']\ar[ru,"l",dashed]& B+Y.
% \end{tikzcd}
% \]
% The extensivity of the category $\C$ implies that $\iota$ is a coproduct of $s \colon X \to A$ and $\id_Y$, which provides the lift
% \[
% \begin{tikzcd}
% &A\ar[d,"e", twoheadrightarrow]\\
% X\ar[r,"f"']\ar[ru,"s",dashed]& B.
% \end{tikzcd}
% \]
% \end{proof}
\section{topos with enough projectives has projective generating sets}
\begin{definition}
A category $\E$ has (externally) \demph{enough projectives,} if, for any object $X\in \ob(\E)$, there exists a projective object $P$ and an epimorphism $P \twoheadrightarrow X$.
\end{definition}
\begin{question}
When does a Grothendieck topos have enough projectives (in terms of geometry, the internal logic, or site)?
\end{question}
\begin{proposition}[Morgan Rogers {\href{https://mathoverflow.net/questions/316588/topos-with-enough-projectives}{[mathoverflow 316588]}}]
For a Grothendieck topos $\E$, the following conditions are equivalent:
\begin{enumerate}
\item $\E$ has enough projectives.
\item $\E$ has a (small) generating set $\P$ consisting of projective objects.
\end{enumerate}
\end{proposition}
\begin{proof}
We first prove $(1) \implies (2)$. Fix a generating set $\G$ of $\E$. If $\E$ has enough projectives for each object $g\in \G$, we can take a projective object $P_g$ equipped with an epimorphism $P_g \twoheadrightarrow g$. This shows that $\P \coloneqq \{P_g \mid g \in \G\}$ is a generating set consisting of projective objects.
Next, we prove $(2) \implies (1)$. For any object $X \in \E$, the assumption implies that there exists a jointly epimorphic small family of morphisms from projective objects $\{a_\lambda \colon P_\lambda \to X\}_{\lambda\in \Lambda}$, where $P_\lambda \in \P$. \Cref{lem:ClosednessOfProjectives} implies that
\[
\coprod_{\lambda\in \Lambda} P_\lambda \twoheadrightarrow X
\]
is the morphism from a projective object.
\end{proof}
This proposition suggests a possibility of site characterization.
\section{\texorpdfstring{$\kappa$}{kappa}-narrow objects}
To construct a nice site from a given topos with enough projectives, we consider the notion of being $\kappa$-narrow.
\begin{definition}\label{def:narrow}
An object $X\in \ob(\E)$ of an infinitary extensive category $\E$ is said to be \demph{$\kappa$-narrow} for an infinite cardinal number $\kappa$ if for any $\kappa$-coproduct decomposition
\[
X \cong \coprod_{\alpha \in \kappa} X_{\alpha}
\]
there exists $\alpha \in \kappa$ such that $X_{\alpha}$ is initial.
\end{definition}
\begin{example}
For a locally connected topos $\E$, an object $X$ is $\kappa$-narrow if and only if $|\pi_0(X)|< \kappa$.
In particular, a set $X$ is $\kappa$-narrow (in the topos of sets $\Set$) if and only if $|X|<\kappa$.
\end{example}
\begin{example}
The cantor set $2^{\N}$ in the topos $\Sh(2^\N)$ is $\aleph_0$-narrow.
\end{example}
% \memo{right Kan extension?}
\begin{proposition}[Closure properties of $\kappa$-narrow objects]\label{prop:closednessOfNarrowObjects}
For an infinitary extensive category $\E$ and an infinite cardinal number $\kappa$,
\begin{itemize}
\item A $\kappa$-small coproduct of $\kappa$-narrow objects is $\kappa$-arrow if $\kappa$ is regular.
\item A summand of a $\kappa$-narrow object is $\kappa$-narrow.
\end{itemize}
\end{proposition}
\begin{proof}
Let $X$ be a $\kappa$-small coproduct of $\kappa$-narrow objects $\{X_\lambda\}_{\lambda \in \Lambda}\; (|\Lambda|<\kappa)$.
\[
X = \coprod_{\lambda \in \Lambda} X_\lambda
\]
Take an arbitrary $\kappa$-coproduct decomposition $X = \coprod_{\alpha \in \kappa}Y_{\alpha}$. The infiniary extensivity of $\E$ implies that
\[
X = \coprod_{\lambda \in \Lambda} X_\lambda \cong \coprod_{\lambda \in \Lambda} \coprod_{\alpha\in \kappa} X_{\lambda} \times_X Y_{\alpha}.
\]
For each $\lambda\in \Lambda$, we define $I_{\lambda} \coloneqq \{\alpha\in \kappa \mid X_{\lambda} \times_X Y_{\alpha} \text{ is not initial.}\}$
Since each $X_\lambda$ is $\kappa$-narrow, we have $|I_{\lambda}|<\kappa$. The regularity of $\kappa$ implies that $\bigcup_{\lambda \in \Lambda} I_{\lambda} \subsetneq \kappa$. For an element $\alpha \in \kappa \setminus \left(\bigcup_{\lambda \in \Lambda} I_{\lambda}\right)$, we have
\[
Y_{\alpha} \cong \coprod_{\lambda \in \Lambda} X_{\lambda} \times_X Y_{\alpha} \cong \coprod_{\lambda \in \Lambda} \emptyset \cong \emptyset.
\]
This completes the proof of the former statement.
The latter statement is easier to prove.
\end{proof}
The following lemma is easy, but essential.
\begin{lemma}\label{lem:ProjectiveDecompositionLemma}
For an infinitary extensive caetgory $\E$, $\kappa$-narrow projective object $P$ for a regular cardinal $\kappa$, and a small (but not-necessarily $\kappa$-small) family of morphisms $\{f_i \colon X_i \to P\}_{i \in I}$, the following conditions are equivalent:
\begin{itemize}
\item $\{f_i \colon X_i \to P\}_{i \in I}$ is jointly epimorphic.
\item There exists a $\kappa$-small corpoduct decomposition $\coprod_{\lambda \in \Lambda}P_\lambda \; (|\Lambda|<\kappa)$ such that every inclusion $P_{\lambda} \rightarrowtail \P$ factors through some $f_i\colon X_i \to P$.
\end{itemize}
\end{lemma}
\begin{proof}
It is easy to prove that the latter condition implies the former. We prove the opposite.
Assume that $\{f_i \colon X_i \to P\}_{i \in I}$ is jointly epimorphic. Then we obtain the canonical epimorphism $\sum_{i \in I}f_i \colon \coprod_{i \in I} X_i \twoheadrightarrow P$. The projectivity of $P$ ensures the existence of a section $ \coprod_{i \in I} X_i \leftarrowtail P\colon s$. Since $\E$ is infinitary extensive, the morphism $s$ induces the $I$-coproduct decomposition $P \cong \coprod_{i\in I}P_i$ by the pullback diagram
\[
\begin{tikzcd}
X_i \ar[d, rightarrowtail]&P_i\ar[l, rightarrowtail]\ar[d, rightarrowtail]\ar[dl, phantom, very near start, "\llcorner"]\\
\coprod_{i \in I} X_i \ar[r, bend right, twoheadrightarrow, "\sum_{i \in I}f_i"']& P.\ar[l, rightarrowtail, "s"']
\end{tikzcd}
\]
This implies that each inclusion $P_i \rightarrowtail P$ factors through $f_i \colon X_i \to P$. Since $P$ is $\kappa$-narrow, the subset $\Lambda \coloneqq \{i \in I \mid P_i\not \cong \emptyset\}$ is $\kappa$-small, and we have a $\kappa$-small coproduct decomposition $P \cong \coprod_{\lambda \in \Lambda}P_{\lambda}$. This completes the proof.
\end{proof}
\section{The \texorpdfstring{$\kappa$}{kappa}-extensive site of \texorpdfstring{$\kappa$-narrow}{kappa-narrow} projective objects}
\begin{definition}
For a Grothendieck topos $\E$ and a regular cardinal $\kappa$, the full subcategory of $\kappa$-narrow projective objects is denoted by $\Ck$.
\end{definition}
\begin{example}
For a Cauchy-complete small category $J$ and a regular cardinal $\kappa$, the category $\Ck\subset \PSh(J)$ is the full subcategory of $\kappa$-small coproducts of representable presheaves.
\end{example}
\begin{theorem}\label{thm:StructureOfCk}
For a Grothendieck topos $\E$ and a regular cardinal $\kappa$,
\begin{itemize}
\item the full subcategory $\Ck \subset \E$ is essentially small and $\kappa$-extensive.
\item the canonical topology $J_{\text{can}}$ on $\Ck$ coincides with the $\kappa$-extensive topology $J_{\kappa\text{-ext}}$.
\end{itemize}
Furthermore, there exists a regular cardinal $\kappa$ such that $\Ck$ is a generating full subcategory of a Grothendieck topos $\E$, if and only if the topos $\E$ has enough projectives.
\end{theorem}
\begin{proof}
First, we prove the essential smallness. Fix a generating set $G$ of the topos $\E$. \Cref{lem:ProjectiveDecompositionLemma} implies that any object in $\Ck$ is a $\kappa$-small coproduct of subobjects of objects in $G$. This implies that there are at most small number of isomorphism classes in $\Ck$.
% For any object $P$ in $\Ck$, there is a jointly epimorphic family onto $P$.
\Cref{lem:ClosednessOfProjectives} and \Cref{prop:closednessOfNarrowObjects} implies that $\Ck \hookrightarrow \E$ is closed under taking summands and $\kappa$-small coproducts. This implies that $\Ck$ is $\kappa$-extensive.
\Cref{lem:ProjectiveDecompositionLemma} implies that a sieve $S \subset \Ck({-},P)$ belongs to $J_{\text{can}}$ if and only if it contains a $\kappa$-small coproduct diagram.
The last statement follows from the fact that every object $X$ in a Grothendieckt topos is $\kappa$-narrow for sufficiently large cardinal $\kappa$.
\end{proof}
\begin{corollary}
If a Grothendieck topos $\E$ has enough projectives, we have
\[
\E \simeq \Sh(\Ck, J_{\kappa\text{-ext}})
\]
for sufficiently large regular cardinal $\kappa$,
where $\Ck$ is the essentially small $\kappa$-extensive full subcategory consisting of $\kappa$-narrow projective objects, and $J_{\kappa\text{-ext}}$ is its $\kappa$-extensive topology.
\end{corollary}
In order to investigate a site characterization for topoi with enough projectives, studying the properties of the site $(\Ck, J_{\kappa\text{-ext}})$ is
% \section{Related topics}
% \begin{example}[Condensed math]
% A compact Hausdorff space $X$ is called \demph{extremally disconnected}, if it is projective in the category of compact Hausdorff spaces.
% \end{example}
% \begin{remark}[Axiom of choice]
% A Grothendieck topos $\E$ satisfies the external axiom of choice, ``Every epimorphism has a section,"
% % \[
% % \text{Every epimorphism has a section}
% % \]
% if and only if every object in a topos $\E$ is projective.
% \end{remark}
% \begin{remark}
% Enough projectiveness for the sheaves of abelian groups.
% \end{remark}
% \begin{itemize}
% \item \href{https://arxiv.org/abs/2412.03203}{A Foundation for Synthetic Stone Duality}
% \end{itemize}
\printbibliography
\end{document}