\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}} \newcommand{\J}[1]{J_{#1\text{-ext}}} \newcommand{\Jk}{\J{\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} In this paper, we provide a site characterization of topoi with enough projectives. More concretely, we prove that a Grothendieck topos has enough projective objects if and only if it is a sheaf topos over a $\kappa$-extensive site for some regular cardinal $\kappa$. These observations lead us to generalize the notion of Gaeta topoi to $\kappa$-Gaeta topoi. We will see that $2$-Gaeta topoi are precisely presheaf topoi. \end{abstract} \maketitle % \tableofcontents \memo{cite: Collapsed Toposes and Cartesian Closed Varieties by Peter Johnstone} \memo{These topoi might be called $\kappa$-Gaeta topos. See \href{https://ncatlab.org/nlab/show/Gaeta+topos}{[Gaeta topos]}. For the finite regular cardinal $2$, $2$-Gaeta topos is presheaf topoi. See also \href{https://inmabb.criba.edu.ar/revuma/pdf/v67n2/v67n2a01.pdf}{[DECIDABLE OBJECTS AND MOLECULAR TOPOSES]} } \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. \memo{We can allow $\kappa=2$.} \subsection*{Acknowledgement} \memo{Prof. Takeshi Saito, Ryo Suzuki, Morgan Rogers, Matias Menni} \section{Introduction} \begin{figure}[ht] \begin{center} \begin{tikzpicture} % V-shape structure as a cycle % \draw[very thick] (-8,16) -- (0,0) -- (8,16) -- cycle; % \draw[thick] (-10,20) -- (0,0) -- (10,20); % \draw[very thick] (-8,16) -- (0,0) -- (8,16); \draw[very thick] (-6,12) -- (0,0) -- (6,12); % Function to draw horizontal lines \newcommand{\drawHierarchyLine}[1]{ \draw (-#1,2*#1) -- (#1,2*#1); } % First set of horizontal hierarchy lines using a loop \foreach \x in {1,2,3,4,5,6,7} { \pgfmathsetmacro{\val}{4*(1-(1/2)^\x)} \drawHierarchyLine{\val}; } % Second set of horizontal hierarchy lines using a loop \foreach \x in {1,2,3} { \pgfmathsetmacro{\val}{2+4*(1-(1/2)^\x)} \drawHierarchyLine{\val}; } \node at (2 +0.7,4) {$\kappa=2$}; \node at (0,2.7) {Presheaf topoi}; \node at (3 +0.7,6) {$\kappa=\aleph_0$}; \node at (0, 5) {Gaeta topoi}; \node at (3.5 +0.7,7) {$\kappa=\aleph_1$}; \node at (3.75+0.7,7.5) {$\kappa=\aleph_2$}; % \node at (4 +1,8) {$\kappa=\aleph_{\omega +1}$}; \node at (0,9) {$\vdots$}; \node at (5 +1,10) {$\kappa=\aleph_{\omega +1}$}; \node at (5.5 +1,11) {$\kappa=\aleph_{\omega +2}$}; \node at (0,12) {$\vdots$}; \end{tikzpicture} \end{center} \caption{Hierarchy of topoi with enough projectives} \label{fig:v_hierarchy} \end{figure} \section{\texorpdfstring{$\kappa$}{kappa}-extensive site and \texorpdfstring{$\kappa$}{kappa}-Gaeta topoi} % Recall that for a small extensive category $\C$, the sheaf topos over $\C$ equipped with its extensive topology is called \demph{Gaeta topos}. \memo{cite \cite{marmolejo2019level}.} % In this section, we generalize Gaeta topos to $\kappa$-Gaeta topos with a parameter $\kappa$, which is a fixed regular cardinal. % \begin{description} % \item[$\kappa=2$] $2$-Gaeta topoi are precisely presheaf topoi. \memo{check} % \item[$\kappa=\aleph_0$] $\aleph_0$-Gaeta topoi are precisely Gaeta topoi. % \end{description} For a (possibly finite) cardinal $\kappa$, we mean `less than $\kappa$' by the word `$\kappa$-small.' \begin{definition} A (possibly finite) cardinal $\kappa$ is \demph{regular}, if \begin{itemize} \item $1\in \kappa$, and \item for any $\kappa$-small family of $\kappa$-small sets $\{X_i\}_{i\in I}$, \[ |I|< \kappa \text{ and }\forall i \in I\; |X_i|< \kappa \] then their sum $\coprod_{i\in I} X_i$ is also $\kappa$-small. \end{itemize} \end{definition} The sequence of regular cardinals starts with $\kappa = 2, \aleph_0, \aleph_1, \dots, \aleph_{\omega+1}, \dots $. \memo{The condition $1\in \kappa$ is natural, in the sense that [Small sheaves paper]} \begin{definition} For a regular cardinal $\kappa$, a small category $\C$ is said to be $\kappa$-extensive, if $\C$ admits \begin{itemize} \item $\kappa$-small coproducts, and \item pullbacks of $\kappa$-small coproduct inclusion maps (along arbitrary maps), \end{itemize} and furthermore, $\kappa$-small coproducts are \begin{itemize} \item disjoint, and \item pullback stable. \end{itemize} \end{definition} \begin{description} \item[$\kappa =2$] A category $\C$ is $2$-extensive, if and only if $\C$ has a strict initial object. % \memo{cite schulman's small sheaf paper} % \item[$\kappa =2$] A category $\C$ is $2$-extensive, if and only if $1$ \item[$\kappa= \aleph_0$] The $\aleph_0$-extensivity is the usual extensivity as studied in \cite{carboni1993introduction}. \end{description} \begin{notation}\label{not:ExtensiveTopology} For a regular cardinal $\kappa$ and a small $\kappa$-extensive category $\C$, we write $\Jk$ for the collection of sieves $S$ that contain a $\kappa$-small coproduct diagram. In other words, a sieve $S$ belongs to $\Jk$ if and only if there is a $\kappa$-small family $\{X_i \to X\}_{i\in I} \subset S$ such that the canonical map \[ \coprod_{i\in I} X_i\to X \] is an isomorphism. \end{notation} (The following lemma is known. For example, \memo{[UltracategoriesAppendix]} proves this for the cases where $\kappa= \aleph_0$, and $\C$ has pullbacks. We prove the following in a general form.) \begin{lemma}[Description of $\kappa$-extensive topology]\label{lem:ExtensiveTopology} For a regular cardinal $\kappa$ and a small $\kappa$-extensive category $\C$, the collection $\Jk$ is a Grothendieck topology on $\C$. % a sieve $S$ on an object $X\in \ob{(\C)}$ is a covering sieve if and only if $S$ contains a $\kappa$-small coproduct diagram, i.e., a $\kappa$-small family $\{X_i \to X\}_{i\in I} \subset S$ such that the canonical map % \[ % \coprod_{i\in I} X_i\to X % \] % is an isomorphism. \end{lemma} \begin{proof} We need to check the three axioms. \memo{We need to cite something} \begin{description} \item[Trivial cover] For any object $X\in \ob(\C)$, the trivial sieve $\langle\id_X\rangle$ belongs to $\Jk$, since $\id_X\colon X\to X$ is a $\kappa$-small coproduct diagram. (We used the assumption $1\in \kappa$.) \item[Stability] This follows from the $\kappa$-extensivity of $\C$, especially from the condition that $\kappa$-small coproducts are pullback stable. \item[Transitivity] This follows from (the second condition of) the regularity of $\kappa$. \end{description} \end{proof} \begin{definition}\label{def:extensivetopologyAndSite} For a regular cardinal $\kappa$ and a small $\kappa$-extensive category $\C$, we call the Grothendieck topology $\Jk$ (\Cref{not:ExtensiveTopology}, \Cref{lem:ExtensiveTopology}) \demph{the $\kappa$-extensive topology} associated to $\C$. We call the site $(\C, \Jk)$ \demph{the $\kappa$-extensive site} associated to $\C$. \end{definition} \begin{definition}[$\kappa$-Gaeta topos]\label{def:GaetaTopos} For a regular cardinal $\kappa$, we call a Grothendieck topos $\E$ a \demph{$\kappa$-Gaeta topos}, if there exists a small $\kappa$-extensive category $\C$ such that $\E \simeq \Sh(\C,\Jk)$. \end{definition} \begin{proposition}\label{prop:DescriptionOfExtensiveSheaf} For a regular cardinal $\kappa$ and a small $\kappa$-extensive category $\C$, a presheaf $T \colon \C^{\op} \to \Set$ is a $\Jk$-sheaf if and ony if it sends $\kappa$-small coproducts to $\kappa$-small products. \[ T\left(\coprod_{i\in I} X_i\right) \cong \prod_{i\in I}T\left( X_i\right) \] \end{proposition} \begin{proof} We first assume that $T$ is a $\Jk$-sheaf, and prove that $T$ preserves $\kappa$-small coproducts. Since the empty sieve covers the strict initial object $\emptyset\in \ob(\C)$, we have $T(\emptyset)=1$. For a $\kappa$-small coproduct diagram $\{X_i \to X\}_{i\in I} $, let $S$ denote the sieve on $X$ generated by the coproduct inclusions. By the $\kappa$-extensivity of $\C$ (especially by the disjointness of $\kappa$-small coproducts), the sieve $S\rightarrowtail \yo(X)$ (regarded as an object of $\PSh(\C)$) is the pushout of % \[ % \begin{tikzcd} % &\yo(X_i)&&\\ % \yo(\emptyset)\ar[ru, rightarrowtail] \ar[r, rightarrowtail]\ar[rd, rightarrowtail]\ar[rdd, rightarrowtail] % &\yo(X_{i'})&S\ar[r, rightarrowtail]&\yo(X)\\ % &\yo(X_{i''})&&\\ % &\vdots&&\\ % \end{tikzcd} % \] % \[ % \begin{tikzcd} % &\yo(X_i)\ar[rd]&\\ % \yo(\emptyset)\ar[ru, rightarrowtail] \ar[r, rightarrowtail]\ar[rd, rightarrowtail]\ar[rdd, rightarrowtail] % &\yo(X_{i'})\ar[r]&S\\ % &\yo(X_{i''})\ar[ru]&\\ % &\vdots \ar[ruu]&\\ % \end{tikzcd} % \] \[ \begin{tikzcd} &\yo(X_i)\\ \yo(\emptyset)\ar[ru, rightarrowtail] \ar[r, rightarrowtail]\ar[rd, rightarrowtail]\ar[rdd, rightarrowtail] &\yo(X_{i'})\\ &\yo(X_{i''})\\ &\vdots \\ \end{tikzcd} \] in the presheaf topos $\PSh(\C)$. Thus the glueing condition of $S\rightarrowtail \yo(X)$ states that \[ \begin{tikzcd} &T(X_i)\ar[rd]&\\ T(X)\ar[ru]\ar[r]\ar[rd]\ar[rdd] &T(X_{i'})\ar[r] & T(\emptyset)\\ &T(X_{i''})\ar[ru]&\\ &\vdots \ar[ruu]&\\ \end{tikzcd} \] is a pullback diagram (in $\Set$). Thus, $T(\emptyset)\cong 1$ implies $ T(X) \cong \prod_{i\in I}T\left( X_i\right) $. Conversely, we assume that $T$ sends $\kappa$-small coproducts to $\kappa$-small products. We prove that $T$ is a $\Jk$-sheaf. Let $S$ be a $\Jk$-covering sieve on an object $X\in \ob(\C)$, and $\{t_f \in T(Y)\}_{f\colon Y \to X\in S}$ is a $S$-matching family. By definition of $\Jk$, it contains a $\kappa$-small coproduct diagram $\{\iota_i \colon X_i \to X\}_{i\in I}$. By the assumption, we obtain $t\in T(X)$ that corresponds to $(t_{\iota_{i}})$ via the canonical bijection $T(X) \cong \prod_{i\in I}T\left( X_i\right)$. In other words, $t\in T(X)$ is the unique elemnt such that $t_{\iota_i} = t\iota_i$ for every $i \in I$. Since $t$ is the unique candidate for the amalgamation of the $\{t_f\}_{f\in S}$, it suffices to prove $t_f = tf \in T(Y)$ for any $f\colon Y \to X \in S$. We define $Y_i$ by the following pullback square \[ \begin{tikzcd} Y_i\ar[d, rightarrowtail, "\tau_i"] \ar[rd, phantom, very near start, "\lrcorner"]\ar[r, "f_i"]& X_i\ar[d, rightarrowtail, "\iota_{i}"]\\ Y \ar[r, "f"] & X. \end{tikzcd} \] (Notice that the existence of those shape of pullbacks is ensured by the $\kappa$-extensivity.) Since we have $T(Y) \cong \prod_{i\in I}T\left( Y_i\right)$, it suffices to prove that $(t_f) \tau_i = (tf) \tau_i$ for every $i\in I$. This holds since $(t_f) \tau_i = t_{f \tau_i} = t_{\iota_i f_i} = t_{\iota_i} f_i = t{\iota_i} f_i =(tf) \tau_i$. \end{proof} \begin{corollary}\label{cor:subcanonical} For a regular cardinal $\kappa$ and a small $\kappa$-extensive category $\C$, the $\kappa$-extensive topology $\Jk$ is subcanonical. \end{corollary} \begin{proof} This immediately follows from \Cref{prop:DescriptionOfExtensiveSheaf} \end{proof} The case where $\kappa=2$ is worth mentioning: \begin{corollary}\label{cor:kappaIsTwoPresheaf} A Grothendieck topos $\E$ is a $2$-Gaeta topos if and only if it is a presheaf topos. \end{corollary} \begin{proof} For any small category $\C$, we obtain the category $\C^{\triangleleft}$ by formally adding a(n new) initial object to $\C$. Since the $2$-extensivity is equivalent to the existence of a strict initial object, the resulting category $\C^{\triangleleft}$ is $2$-extensive. Conversely, one can easily observe that every $2$-extensive category $\D$ is of the form of $\D \simeq \C^{\triangleleft}$ with a small category $\C$. % there exists a small category $\D$ such that $\C$ is equivalent to the category $\D^{\triangleleft}$, $\D$ equipped with a formally added initial object $\C \simeq \D^{\triangleleft}$. Then, the equivalence of categories \[ \Sh(\C^{\triangleleft}, \J{2}) \simeq \PSh(\C) \] provided by \Cref{prop:DescriptionOfExtensiveSheaf} completes the proof. % The $2$-extensive topology on $\C\simeq \D^{\triangleleft}$ is identical to the minimum topology on $\C$ except that the strict initial object is covered by the empty sieve. A presheaf $T\colon $ \end{proof} % \begin{example}[$2$-Gaeta topoi $=$ presheaf topoi] % Take an arbitrary $2$-extensive category $\C$. The $2$-extensivity is equivalent to saying that $\C$ has a strict initial object $\emptyset$. Then, there exists a small category $\D$ and $\C$ is equivalent to the category $\D^{\triangleleft}$, $\D$ equipped with a formally added initial object $\C \simeq \D^{\triangleleft}$. The $2$-extensive topology on $\C\simeq \D^{\triangleleft}$ is identical to the minimum topology on $\C$ except that the strict initial object is covered by the empty sieve. A presheaf $T\colon $ % \end{example} \begin{example}[Condensed] \end{example} \begin{example}[Bornological] \cite{lawvere2006some}\cite{lawvere1988toposesGenerated} \end{example} \begin{example}[Coextensive algebras] \cite{lawvere2008core}\cite{marmolejo2019level} \end{example} \section{From \texorpdfstring{$\kappa$}{kappa}-extensive site to projective objects} \subsection{Sites generated by free sieves} \begin{definition}\label{def:FreeSieve} For a small category $\C$, we say a sieve $S$ on an object $X$ is \demph{free} if the sieve $S$, regarded as a presheaf on $\C$, is a small coproduct of representable presheaves. \end{definition} In other words, a sieve $S$ is free if and only if there is a famiy of elements $\{f_i \colon X_i \to X \}_{i\in I} \subset S$ satisfying the following equivalent conditions: \begin{enumerate} \item Every element $f\colon Y \to X \in S$ is decomposed into \[ \begin{tikzcd} Y\ar[rr,"g"]\ar[rd, dashed, "\exists! h"']&&X\\ &X_i \ar[ru, dashed, "\exists! i \in I\; f_i"']& \end{tikzcd} \] with a unique pair of $i\in I$ and $h\colon Y \to X_i$. % factors through exactly one $f\in I$. % \item $S$ is the disjoint union of $\langle i \rangle \; (i \in I)$, and, for every $i\in I$, the morphism $f_i\colon X_i \to X$ is monic. \item In the presheaf category $\PSh(\C)$, the canonical morphism \[ \coprod_{i\in I} \yo(X_i) \to S \] is an isomorphism. \item For any presheaf $F$, the morphism \[\PSh(\C)(S, F) \to \prod_{i\in I}F(Y_i)\]is bijective. \end{enumerate} \begin{remark} If $\C$ is Cauchy complete, a sieve (or more generally, any presheaf) $S$ is free if and only if $S$ is projective. \end{remark} \begin{definition}\label{def:GeneratedByFreeSieves} For a small category $\C$, we say a Grothendieck topology $J$ is \demph{generated by free sieves} if, for any $J$-covering sieve $S$, there exists a free $J$-covering subsieve $S'\subset S$. \end{definition} \begin{example} The trivial topology $(\C, J_{\text{triv}})$ is generated by free sieves. In fact, the maximum sieve on each object $X$ is free. \end{example} \begin{proposition}\label{prop:ExtensiveTopologyInducesFreegeneratedSite} For a regular cardinal $\kappa$ and a $\kappa$-extensive category $\C$, we consider its full subcategory $\C'$ that has all objects in $\C$ except the initial object. Then the restriction of $\Jk$ on $\C'$, which is denoted by $\Jk'$, is a Grothendieck topology on $\C'$ generated by free sieves. Furthrermore, we have $\Sh(C, J_{\kappa\text{-ext}}) \simeq \Sh(\C', \Jk')$. \end{proposition} \begin{proof} Since $(\C, \Jk)$ is subcanonical (\Cref{cor:subcanonical}) and the initial object is covered by the empty sieve, the comparison lemma \memo{check and cite} implies that $(\C',\Jk')$ is a site and $\Sh(\C, \Jk ) \simeq \Sh(\C',\Jk')$. % Since the embedding $ \C \hookrightarrow \Sh(\C, \Jk)$ preserves the initial object, the full subcategory $\C' \hookrightarrow \C \hookrightarrow \Sh(\C, \Jk)$ is generating. By the site construction of Giraud's theorem, we obtain $\Sh(\C,\Jk) \simeq \Sh(\C',J')$, where $J'$ denotes the canonical topology induced by the embedding. % A sieve $S$ on $X\in \C'$ is $J'$-covering if and only if $\{\yo(f) \colon \yo(Y) \to \yo(X)\}_{f\in Y \to X}$ is jointly epimorphic in $\Sh(\C,\Jk)$. \memo{Due to \Cref{cor:subcanonical}, this is equivalent to saying that $S$ contains $\kappa$-small coproduct.} \memo{Write with site morphism...} We prove that the site $(\C',\Jk')$ is generated by free sieves. If a sieve $S$ on an object $X\in \ob(\C')$ is a $\Jk'$-covering sieve, then $S$ contains a $\kappa$-small coproduct diagram $\{\iota_i \colon X_i \to X\}_{i\in I} \subset S$. Let $S'\subset S$ be the subsieve generated by the family $\{\iota_i \colon X_i \rightarrowtail X\}_{i\in I}$. For any morphism $f\colon Y \to X \in S'$, $f$ factors through a unique $\iota_i$, since $Y$ is not initial in $\C$ and $\kappa$-small coproducts are disjoint in $\C$. This proves that $S'$ is a free subsieve of $S$. \end{proof} \subsection{Site to projective objects} \begin{definition} We say that a site $(\C,J)$ \demph{witness} that a Grothendieck topos $\E$ has enough projectives if we have $\E\simeq \Sh(\C,J)$ and the inclusion functor \[ \Sh(\C,J) \hookrightarrow \PSh(\C) \] preserves epimorphisms. \end{definition} Notice that a site $(\C,J)$ witnesses its sheaf topos $\Sh(\C,J)$ has enough projectives if and only if, for each $x\in \ob(\C)$, its represented sheaf $\mathbf{a}\yo (x)$ is projective. A Grothendieck topos $\E$ has enough projectives objects, if and only if it admits a site $(\C,J)$ that witnesses that $\E$ has enough projectives. \begin{lemma}[Epimorphisms in a sheaf topos]\label{lem:EpimorphismsOfSheaves} For a site $(\C, J)$, a morphism of sheaves $f\colon A \to B$ is epic in the topos $\Sh(\C,J)$ if and only if, for any $X\in \ob(\C)$ and $b\in B(X)$, there exists a $J$-covering sieve $S$ and a (not necessarily $S$-matching) family $\{a_h\in A(Y)\}_{h\colon Y\to X \in S}$, such that $bh=f_y(a_h) \in B(Y)$ for any $h \colon Y \to X \in S$. \end{lemma} \begin{proof} The morphism $f$ is an epimorphism if and only if its image coincides with $B$. Since the image of $f$ in the sheaf topos is the sheafification of the objectwise image, we obtain the above description. \end{proof} \begin{proposition}\label{prop:FreeSieveGenerationImpliesEnoughProjectives} If a site $(\C,J)$ is generated by free sieves, then $(\C,J)$ witnesses that $\Sh(\C,J)$ has enough projectives. \end{proposition} \begin{proof} Suppose that the site $(\C,J)$ is generated by free sieves. Let $f\colon A\to B$ be an epimorphism in the topos $\Sh(\C,J)$. Take an arbitrary object $X\in \ob(\C)$ and $b\in B(X)$. We will construct $a\in A(X)$ such that $f_X(a) =b$. Since $f$ is epic, \Cref{lem:EpimorphismsOfSheaves} provides a $J$-covering sieve $S$ on the object $X$, and $\{a_h\in A(Y)\}_{h\colon Y\to X \in S}$, such that $bh=f_Y(a_h) \in B(Y)$ for any $h \colon Y \to X$. Using the assumption that $(\C,J)$ is generated by free sieves, we can take a free $J$-covering subsieve $S'\subset S$, and a family of elements $\{h_{i}\colon X_i \to X\}_{i\in I} \subset S'$ that witnesses the isomorphism \[ \coprod_{i\in I} \yo(X_i) \cong S'. \] % So far, we have obtained a family $\{a_h\in A(Y)\}_{h\colon Y\to X \in I}$, such that $bh=f_y(a_h) \in B(y)$ for any $h \colon y \to x \in I$. Since $A$ is $J$-sheaf and $S'$ is free, we obtain a bijection \[ A(X) \cong \PSh(\C)(\yo(X) ,A) \cong \PSh(\C)(S',A) \cong \prod_{i\in I}A(X_i) \] and the unique element $a\in A(X)$ such that $a h_i = a_{h_i}$ for any $i\in I$. We prove that $f_X(a) = b$. Since $S'$ is a $J$-covering, it suffices to prove that $f_X(a) h_i = b h_i$ for each $i\in I$, which is verified by \[ f_{X}(a) h_i = f_{X_i}(a h_i) = f_{X_i}(a_{h_i})=b h_i. \] This completes the proof. \end{proof} \begin{corollary} If a Grothendieck topos $\E$ is a $\kappa$-Gaeta topos for a regular cardinal $\kappa$, then $\E$ has enough projectives. \end{corollary} \section{From projective objects to \texorpdfstring{$\kappa$}{kappa}-extensive sites} \subsection{Preliminaries on projective objects} We adopt the following terminology. \begin{itemize} \item A subobject $\iota \colon S\rightarrowtail X$ is called a \demph{retract} if $\iota$ is a split monomorphism. \item A subobject $\iota \colon S\rightarrowtail X$ is called a \demph{summand} % ($=$ complemented) if there exists another subobject $S' \rightarrowtail X$ such that % $\iota$ is an injection map of a coproduct diagram \[S\rightarrowtail X \leftarrowtail S'\] is a coproduct diagram. \end{itemize} \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} \subsection{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{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} \subsection{\texorpdfstring{$\kappa$}{kappa}-narrow objects} To extract a $\kappa$-extensive site from a given topos with enough projectives, we consider the notion of being $\kappa$-narrow. \begin{definition}\label{def:narrow} For a (possibly finite) cardinal $\kappa$, an object $X\in \ob(\E)$ of a Grothendieck topos $\E$ is said to be \demph{$\kappa$-narrow} $\kappa$ 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{remark} Being $\kappa$-narrow is analogous to the presentability of an object. An object $X$ is $\kappa$-narrow if and only if $\E(X, -)\colon \E\to \Set$ preserves the filtered colimit diagram $\kappa = \cup_{S\subset \kappa, |S|<\kappa} S$. \end{remark} \begin{example} An object $X$ is $2$-narrow if and only if $X$ is initial or connected. \end{example} \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} In the sheaf topos over the Cantor space $\Sh(2^\N)$, the terminal object is $\aleph_0$-narrow. \end{example} % \memo{right Kan extension?} \begin{proposition}[Closure properties of $\kappa$-narrow objects]\label{prop:closednessOfNarrowObjects} For a Grothendieck topos $\E$ and a regular cardinal $\kappa$, \begin{itemize} \item A $\kappa$-small coproduct of $\kappa$-narrow objects is $\kappa$-narrow. % 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 a regular cardinal $\kappa$, a Grothendieck topos $\E$, a $\kappa$-narrow projective object $P$, and a small (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} \subsection{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 $\D$ and a regular cardinal $\kappa$, the category $\Ck\subset \PSh(\D)$ is the full subcategory of $\kappa$-small coproducts of representable presheaves. For a small category $\C$, the category $\C_2 \subset \PSh(\C)$ for $\kappa=2$ is equivalent to $\overline{\C}^{\triangleleft}$, where $\overline{\C}$ denotes tha Cauchy completion of $\C$. This is the opposite construction of \Cref{cor:kappaIsTwoPresheaf}. \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 $\E$, 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 $\Ck$ inherits the $\kappa$-extensivity of $\E$. \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 $\kappa$. \memo{Well-foundedness} \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} \section{Conclusion} \begin{theorem}[Main Theorem] For a Grothendieck topos $\E$, the following conditions are equivalent: \begin{enumerate} \item $\E$ has enough projective objects. \item There is a regular cardinal $\kappa$ such that $\E$ is a $\kappa$-Gaeta topos (\Cref{def:GaetaTopos}). \item $\E$ is equivalent to a sheaf topos $E\simeq \Sh(\C,J)$, where $(\C,J)$ is a small site generated by free sieves. \end{enumerate} \end{theorem} \begin{proof} \Cref{cor:ExtensiveSite} proves the implication $(1)\implies (2)$. \Cref{prop:ExtensiveTopologyInducesFreegeneratedSite} proves the implication $(2) \implies (3)$. \Cref{prop:FreeSieveGenerationImpliesEnoughProjectives} proves the implication $(3) \implies (1)$. \end{proof} \begin{example}[Presheaves] A presheaf topos $\PSh(\C)$ has enough projective, since the trivial topology on $\C$ is generated by free sieves. \end{example} \begin{example}[Condensed sets] For a strong limit cardinal $\lambda$, the topos of $\lambda$-condensed sets has enough projectives, since the topos is equivalent to the sheaf topos over the $\aleph_0$-extensive site of $\lambda$-extremely disconnected spaces. % is extensive. \end{example} \begin{example} \memo{This may subsume \cite{dupont1989projectivity}} \end{example} \printbibliography \end{document}