← Topos with enough projectives
Ver3__ver20250202.tex
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\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}
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% Function to draw horizontal lines
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\draw (-#1,2*#1) -- (#1,2*#1);
}
% First set of horizontal hierarchy lines using a loop
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\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}