← Topos with enough projectives
Ver1__20241215.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}
\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}
In this note, a regular cardinal means an infinite regular cardinal.
\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}
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.
% \[
% w(X) \coloneqq \sup \left\{\kappa\mid \text{there is a coproduct decomposition} \coprod_{\alpha \in \kappa} X_{\alpha} \text{ where }X_{\alpha}\not \cong 0\right\}
% \]
\end{definition}
\begin{example}
For a locally connected topos $\E$, an object $X$ is $\kappa$-narrow if and only if $|\pi_0(X)|< \kappa$.
% , where $\pi_0$ is the left adjoint of the locally constant sheaf functor $\Set \to \E$.
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 full subcategory 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{proposition}\label{prop:StructureOfCk}
For a Grothendieck topos $\E$ and a regular cardinal $\kappa$, the full subcategory $\Ck \subset \E$ is essentially small and $\kappa$-extensive. Furthermore, the canonical topology $J_{\text{can}}$ on $\Ck$ coincides with the $\kappa$-extensive topology $J_{\kappa\text{-ext}}$.
\end{proposition}
\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.
\end{proof}
% \begin{lemma}[$\kappa$-extensivity]\label{lem:extensivity}
% For a Grothendieck topos $\E$ and a regular cardinal $\kappa$, the full subcategory $\Ck \subset \E$ has the following properties.
% \begin{itemize}
% \item $\Ck$ is closed under summands,
% \item $\Ck$ is closed under $\kappa$-small coproducts,
% \item Hence $\Ck$ inherits the $\kappa$-extensivity of $\E$.
% \end{itemize}
% \end{lemma}
% The following lemma is easy, but essential.
% \begin{lemma}[]
% For a Grothendieck topos $\E$, a regular cardinal $\kappa$, and a small family of morphisms $\{f_i \colon Q_i \to P\}_{i \in I}$ in $\Ck$, the following conditions are equivalent:
% \begin{itemize}
% \item $\{f_i \colon Q_i \to P\}_{i \in I}$ is jointly epimorphic in $\E$,
% \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 thorough some $f_i\colon Q_i \to P$.
% \end{itemize}
% \end{lemma}
% \begin{proof}
% \end{proof}
\begin{lemma}
If a Grothendieck topos $\E$ has enough projectives, the category $\Ck$ is essentially small for any regular cardinal $\kappa$.
\end{lemma}
\begin{proof}
\end{proof}
Let $\E$ be a Grothendieck topos with enough projectives. We can take a generating set of projective objects $G$. Let $\kappa$ be a regular cardinal such that every $g \in G$ is $\kappa$-narrow.
\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 a(n essentially) small $\kappa$-extensive category.
\item Every object in $\C$ is projective in $\E$.
\item $\C$ satisfies the external axiom of choice. \memo{Check: the notion of epimorphisms might be changed}
\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}
What about the converse? Does $\Sh(\C, J)$ have enough projectives if $(\C,J)$ is a $\kappa$-extensive site that satisfies 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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