\section{The inverse direction}



\begin{lemma}\label{lem:EquivalenceForTheDefinitionOfDisjointGeneratedness}
    For a small category $\C$, a sieve $S$ on an object $x$, and a subset $I\subset S$, the following conditions are equivalent:
    \begin{enumerate}
        \item Every element $g\colon y \to x \in S$ lifts along exactly one $i\in I$.
        \item $S$ is a disjoint union of $\langle i \rangle \; (i \in I)$.
        \item For any presheaf $F$, the morphism 
        \[\PSh(\C)(S, F) \to \prod_{i\colon y_i \to x\in I}F(y_i)\]is bijective.
    \end{enumerate}
\end{lemma}

\begin{definition}\label{def:DisjointlyGeneratedSieve}
     For a small category $\C$, we say a sieve $S$ on an object $x$ is \demph{disjointly generated} by a subset $I\subset S$, if it satisfies the equivalent conditions in \Cref{lem:EquivalenceForTheDefinitionOfDisjointGeneratedness}.
\end{definition}

\begin{definition}\label{def:DisjointlyGeneratedTopology}
     For a small category $\C$, we say a Grothendieck topology $J$ is \demph{disjointly generated} if, for any $J$-covering sieve $S$, there exists a $J$-covering subsieve $S'\subset S$ that is disjointly generated as a sieve.
\end{definition}

\begin{example}
    A trivial topology $(\C, J_{\text{triv}})$ is disjointly generated. In fact, the maximal sieve on an object $x$ is disjointly generated by the singleton $I=\{\id_x\}$.
\end{example}

\begin{example}\label{exmp:ExtensiveTopologyInducesDisjointlyGenerated} 
% \memo{Check}
    For a $\kappa$-extensive category $\C$, we consider its full subcategory $\C'$ that has all objects in $\C$ except the initial object. Then the topology $J'$ on $\C'$, obtained by restricting $(\C, J_{\kappa\text{-ext}})$, is disjointly generated. 
    This follows from the disjointness of the coproducts.
    Furthermore, we have $\Sh(C, J_{\kappa\text{-ext}}) \simeq \Sh(\C', J)$.
\end{example}

\begin{example}
    Subsume \cite{dupont1989projectivity}
\end{example}

\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, 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 $\{a_h\in A(y)\}_{h\colon y\to x \in S}$, 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 epic, 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:DisjointlyGeneratedImpliesEnoughProjectives}
    If $(\C,J)$ is a disjointly generated site, then $(\C,J)$ witnesses that $\Sh(\C,J)$ has enough projectives.
\end{proposition}
\begin{proof}
    Let $p\colon A\to B$ 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 $p_x(a) =b$.

    Since $p$ 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 disjointly generated, we can take a disjointly generated $J$-covering subsieve $S'\subset S$, and its disjoint generator $I \subset S' \subset S$. 
    
    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 $S'$ is disjointly generated, we have a bijection
    \[
    A(x) \cong \PSh(\C)(\yo(x) ,A) \cong \PSh(\C)(S',A) \cong \prod_{h\colon y\to x\in I}A(y).
    \]
    and the unique element $a\in A(x)$ such that $a h = a_h$ for any $h\colon y \to x \in I$. 

    We prove that $f_x(a) = b$. Since $S'$ is a $J$-covering, it suffices to prove that $f_x(a) h = bh$ for each $h\in I$, which is verified by
    \[
    f_x(a) h = f_y(ah) = f_y(a_h)=bh.
    \]
    This completes the proof.
\end{proof}

\begin{theorem}
    For a Grothendieck topos $\E$, the following conditions are equivalent:
    \begin{enumerate}
        \item $\E$ has enough projective objects.
        \item $\E$ is equivalent to a sheaf topos over a $\kappa$-extensive topology $(\C,  J_{\kappa\text{-ext}})$.
        \item $\E$ is equivalent to a sheaf topos over a disjointly generated site.
    \end{enumerate}
\end{theorem}
\begin{proof}
    \Cref{cor:ExtensiveSite} proves the implication $(1)\implies (2)$. 
    \Cref{exmp:ExtensiveTopologyInducesDisjointlyGenerated} proves the implication $(2) \implies (3)$.
    \Cref{prop:DisjointlyGeneratedImpliesEnoughProjectives} proves the implication $(3) \implies (2)$.
\end{proof}

\begin{example}[Presheaves]
    A presheaf topos $\PSh(\C)$ has enough ptojective, since the trivial topology on $\C$ is disjointly generated.
\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}