← Topos with enough projectives
Ver2 (Contains some errors)__ver20240130.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}
% This is a note towards site characterization of topoi with enough projectives.
% \end{abstract}
\maketitle
% \tableofcontents
\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.
}
\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 $\kappa=2$.}
\section{Preliminaries on projective objects}
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.
\end{definition}
\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}
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 \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 $J$ 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{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 a Grothendieck topos $\E$, 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 $\Ck$ 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.
The last statement follows from the fact that every object $X$ in a Grothendieckt topos is $\kappa$-narrow for sufficiently large cardinal $\kappa$.
\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}
In order to investigate a site characterization for topoi with enough projectives, studying the properties of the site $(\Ck, J_{\kappa\text{-ext}})$ is
% \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}
\section{Disjointly generated site witnesses that its sheaf topos has enough projectives}
% \subsection{Disjointly generated sieves and sites}
\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}[Extensive site without the initial object]\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}
\memo{This may 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}
\section{Conclusion}
\begin{theorem}
For a \memo{non-degenerate? Or allow $\kappa=0$?} 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}
% \section{\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}.
% In this section, we generalize Gaeta topos to $\kappa$-Gaeta topos with a parameter $\kappa$, which is a fixed regular cardinal.
% \begin{description}
% \item[$\kappa=0$] The only $0$-Gaeta topos is the initial topos $1 \simeq \Sh(\emptyset)$.
% \item[$\kappa=1$] The only $1$-Gaeta topos is the terminal topos $\Set \simeq \Sh(1)$.
% \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 possible finite cardinal $\kappa$, we mean `less than $\kappa$' by the word `$\kappa$-small.'
% \begin{definition}
% A (possibly finite) cardinal $\kappa$ is \demph{regular}, if 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{definition}
% First regular cardinals are $\kappa = 0,1,2, \aleph_0, \aleph_1, \dots, \aleph_{\omega+1}, \dots $.
% \begin{definition}
% For a (possibly finite) 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[$\kappa =0$] Every category is $0$-extenisve, since there are no $0$-small set.
% \item[$\kappa =1,2$] A category $\C$ is $1$-extensive, if and only if $\C$ 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{definition}
% For a (possibly finite) regular cardinal $\kappa$ and a small $\kappa$-extensive category $\C$, the associated \demph{$\kappa$-extensive topology} is the Grothendieck topology generated by $\kappa$-small families $\{X_i \to X\}_{i\in I}$ such that the canonical map
% \[
% \coprod_{i\in I} X_i\to X
% \]
% is an isomorphism.
% \end{definition}
% \begin{description}
% \item[$\kappa =0$] For a small category ($=$ $0$-extensive category) $\C$, its $0$-extensive topology
% \item[$\kappa =1,2$] A category $\C$ is $1$-extensive, if and only if $\C$ 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}
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