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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}
\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 condensed abelian groups.
\end{itemize}

\section{Preliminaries on projective objects}

\begin{definition}
    For a category $\C$, an object $X$ is said to be \demph{projective}, if \memo{write}
\end{definition}



\begin{lemma}\label{lem:RetractsOfProjective}
    A retract of a projective object is projective.
\end{lemma}

\begin{lemma}\label{lem:coproductOfProjectives}
    A small coproduct of projective objects is projective.
\end{lemma}

\begin{lemma}\label{lem:SummandsOfProjectives}
    In a category $\C$ is a topos (or more generally, any extensive category), a summand of a projective object is projective.
\end{lemma}
\begin{proof}
    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}

% \begin{lemma}
%     (Assuming the axiom of choice,) 
% \end{lemma}

% \begin{definition}
%     A category $\E$ has (externamlly) \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}


\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:coproductOfProjectives} implies that 
    \[
    \coprod_{\lambda\in \Lambda} P_\lambda \twoheadrightarrow X
    \]
     is the morphism from a projective object.
\end{proof}

% \begin{lemma}
%     If a Grothendiekc topos $\E$ has enough projectives, then there is regular cardinal $\kappa$ and a $\kappa$-extensive small full subcategory $\C \hookrightarrow \E$ such that
%     \begin{itemize}
%         % \item every object (in $\C$) is projective (in $\E$).
%         % \item $\C$ is closed under taking summands
%         % \item $\C$ is closed under taking $\kappa$-small corpoducts.
%         % \item $\C$ is $\kappa$-extenive
%         \item $\C$ satisfies the external axiom of chice.
%     \end{itemize}

%     Furthermore, the canonical topology induced on $\C$ coincides with the $\kappa$-extensive topology.
    
%     % for a object $c\in \ob(\C)$, a sieve $S \subset \C/c$ over $c$ is jointly epimorphic in $\E$ (i.e., covering with respect to the canonical topology) if and only if $S$ contains a $\kappa$-coproduct coinclusions $\{S_\lambda \rightarrowtail c\}_{\lambda\in \Lambda},\; (|\Lambda|<\kappa$).
% \end{lemma}

% \begin{definition}
%     For a Grothendieck topos (or any infinitary extensive category) $\E$, the \demph{width} of an object $X$ is \demph{}
% \end{definition}

\section{Width of an object}

\begin{definition}
    The \demph{width} of an object $X\in \ob(\E)$ of a Grothendieck topos (or in any infinitary extensive category) $\E$ is the supremum of the size of non-trivial coproduct decomposition. The width of an object $X$ is denoted by $w(X)$.
    \[
    w(X) \coloneqq \sup \left\{\kappa\mid \text{there is a coproduct decomposition} \coprod_{\alpha \in \kappa} X_{\alpha} \text{ where }X_{\alpha}\not \cong 0\right\}
    \]
    
    % the minimum cardinal $\kappa$ such that for any coproduct decomposition $X\cong \coprod_{\lambda \in \Lambda}X_{\lambda}$, the number of non-empty components $\{\lambda\in \Lambda \mid X_{\lambda}\not \cong 0\}$ is less than or equal to $\kappa$.
\end{definition}

\begin{example}
    For a locally connected topos $\E$, the width of an object $X$ is the cardinality of $\pi_0 (X)$, where $\pi_0$ is the left adjoint of the locally constant sheaf functor $\Set \to \E$. For example, the width of a set $X$ in the topos of sets coincides with its cardinality.
\end{example}

% \begin{example}
    
% \end{example}

\begin{example}
    The width of the cantor set $2^{\N}$ in the topos $\Sh(2^\N)$ is $\aleph_0$.
\end{example}

\memo{right Kan extension?}

% \begin{conjecture}
% % For a regular cardinal $\kappa$, in a $\kappa$-extensive category $\E$, we have
% % \[
% % w\left(\coprod_{\lambda \in \Lambda} X_\lambda\right) = \sum_{\lambda\in \Lambda} w({X_\lambda})
% % \]
% % if $|\Lambda|< \kappa$.

% In an infinitary extensive category $\E$, we have
% \[
% w\left(\coprod_{\lambda \in \Lambda} X_\lambda\right) = \sum_{\lambda\in \Lambda} w({X_\lambda}).
% \]
% \end{conjecture}
% \begin{proof}
% The coproduct $\coprod_{\lambda \in \Lambda} X_\lambda$ will be denoted by $X$ in this proof.

%     $w\left(X\right) \leq \sum_{\lambda\in \Lambda} w({X_\lambda})$:
%     Take an arbitrary decomposition $X = \coprod_{i \in I}Y_i$, where $Y_i \not \cong 0$. We prove $|I|\leq \sum_{\lambda\in \Lambda} w({X_\lambda})$.
%     For each $\lambda \in \Lambda$, consider the set 
%     $I_\lambda \coloneqq \{i\in I \mid X_{\lambda} \times_{X} Y_i \not \cong 0\}$. By the infiniary extensivity of $\E$ implies that
%     \[
%     X \cong \coprod_{\lambda \in \Lambda} \coprod_{i \in I_{\lambda}} X_{\lambda} \times_X Y_i.
%     \]
%     Furthermore, since $Y_i \not \cong 0$ and $\E$ is extensive, the canonical function
%     \[
%     \coprod_{\lambda\in \Lambda } I_{\lambda} \twoheadrightarrow I
%     \]
%     is a surjection. This proves that

%     \[
%     |I| \leq \sum_{\lambda\in \Lambda} |I_{\lambda}| \leq \sum_{\lambda\in \Lambda} w(X_{\lambda}).
%     \]
% \end{proof}

\begin{conjecture}
In an infinitary extensive category $\E$, we have
\[
w\left(\coprod_{\lambda \in \Lambda} X_\lambda\right) \leq \sum_{\lambda\in \Lambda} w({X_\lambda}).
\]
\end{conjecture}
\begin{proof}
The coproduct $\coprod_{\lambda \in \Lambda} X_\lambda$ will be denoted by $X$ in this proof.
    Take an arbitrary decomposition $X = \coprod_{i \in I}Y_i$, where $Y_i \not \cong 0$. We prove $|I|\leq \sum_{\lambda\in \Lambda} w({X_\lambda})$.
    For each $\lambda \in \Lambda$, consider the set 
    $I_\lambda \coloneqq \{i\in I \mid X_{\lambda} \times_{X} Y_i \not \cong 0\}$. By the infiniary extensivity of $\E$ implies that
    \[
    X \cong \coprod_{\lambda \in \Lambda} \coprod_{i \in I_{\lambda}} X_{\lambda} \times_X Y_i.
    \]
    Furthermore, since $Y_i \not \cong 0$ and $\E$ is extensive, the canonical function
    \[
    \coprod_{\lambda\in \Lambda } I_{\lambda} \twoheadrightarrow I
    \]
    is a surjection. This proves that

    \[
    |I| \leq \sum_{\lambda\in \Lambda} |I_{\lambda}| \leq \sum_{\lambda\in \Lambda} w(X_{\lambda}).
    \]
\end{proof}



% \begin{conjecture}
%     If a Grothendieck topos $\E$ has enough projectives, then there is regular cardinal $\kappa$ and a $\kappa$-extensive small dense full subcategory $\C \hookrightarrow \E$ that satisfies the external axiom of choice. Furthermore, the canonical Grothendieck topology on $\C$ coincides with the $\kappa$-extensive topology.
% \end{conjecture}
% \begin{proof}
%     Let $G$ be a generating set consisting of projective objects.
%     Let $G'$ be the set of all summands of all objects in $G'$, which itself is an essentially small generating set. Furthermore, \cref{lem:SummandsOfProjectives} implies that $G'$ also consists of projective objects.
%     % By taking all summands of all objects in $G'$, we can assume that $G$ is closed under taking summands. 
    
%     Let $\kappa$ be a regular cardinal larger than $|G'|$, and let $\C \hookrightarrow \E$ be the full subcategory of $\E$ consisting of all $\kappa$-small coproducts of objects in $G'$. 
    
%     Then, $\C$ has the following properties.
%     \begin{itemize}
%         \item $\C$ is closed under taking $\kappa$ coproducts, since $\kappa$ is regular.
%         \item $\C$ is closed under taking summands, since a summand of coproduct is a coproduct of summands in a cocomplete topos (infinitary extensivity)\memo{check}.
%         \item $\C$ is (essentially) small $\kappa$-extensive category.
%         \item every object in $\C$ is projective in $\E$.
%         \item $\C$ satisfies the external axiom of choice.
%     \end{itemize}

%     Let $\{f_{\lambda} \colon P_\lambda \to P\}_{\lambda \in \Lambda}$ be a jointly epimorphic family of morphisms. Then, we have the canonical epimorphism
%     \[
%     \coprod_{\lambda\in \Lambda} P_\lambda \twoheadrightarrow P
%     \]
%     in $\E$.
%     Since $P$ is projective in $\E$, there is a section morphism 
%     \[
%     \coprod_{\lambda\in \Lambda} P_\lambda \leftarrowtail P
%     \]
%     Since the topos $\E$ is infinitary extensive, 
% \end{proof}

\begin{conjecture}
    If a Grothendieck topos $\E$ has enough projectives, then there is regular cardinal $\kappa$ and a $\kappa$-extensive small dense full subcategory $\C \hookrightarrow \E$ that satisfies the external axiom of choice. Furthermore, the canonical Grothendieck topology on $\C$ coincides with the $\kappa$-extensive topology.
\end{conjecture}
\begin{proof}
    Let $G$ be a generating set consisting of projective objects, and $\kappa$ be a regular cardinal larger than  $w(g)$ for any $g\in G$.
    
    We define $\C \hookrightarrow \E$ as the full subcategory of $\E$ consisting of all $\kappa$-small coproducts of summands of objects in $G'$. 
    
    Then, $\C$ has the following properties.
    \begin{itemize}
        \item $\C$ is closed under taking $\kappa$ coproducts, since $\kappa$ is regular.
        \item $\C$ is closed under taking summands, since a summand of coproduct is a coproduct of summands in a cocomplete topos (infinitary extensivity)\memo{check}.
        \item $\C$ is (essentially) small $\kappa$-extensive category.
        \item every object in $\C$ is projective in $\E$.
        \item $\C$ satisfies the external axiom of choice.
    \end{itemize}

    Let $\{f_{\lambda} \colon P_\lambda \to P\}_{\lambda \in \Lambda}$ be a jointly epimorphic family of morphisms. Then, we have the canonical epimorphism
    \[
    \coprod_{\lambda\in \Lambda} P_\lambda \twoheadrightarrow P
    \]
    in $\E$.
    Since $P$ is projective in $\E$, there is a section morphism 
    \[
    \coprod_{\lambda\in \Lambda} P_\lambda \leftarrowtail P
    \]
    Since the topos $\E$ is infinitary extensive, 
\end{proof}

How about the converse? Does $\Sh(\C, J)$ have enough projectives, if $(\C,J)$ is a $\kappa$-extensive site satisfying the external axiom of choice?

\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}


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