\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} \tableofcontents 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}\label{cor:ExtensiveSite} 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} \input{TheInverseDirection} \appendix \input{ExtensiveCategories} \printbibliography \end{document}