← Topos with enough projectives

QuestionToRims.tex

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\title{A question to RIMS CS}
\author{Ryuya Hora}
\thanks{Graduate School of Mathematical Sciences, University of Tokyo. \url{hora@ms.u-tokyo}}
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\begin{document}
\begin{abstract}
    This is a question about a relationship between my reserch on topoi with enough projectives and (colimit-class) presentability.
\end{abstract}
\maketitle

\section{Context}
\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} 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{theorem}[Main Theorem]\label{mainTheorem}
    For a Grothendieck topos $\E$, the following conditions are equivalent:
    \begin{enumerate}
        \item The topos $\E$ has enough projective objects.
        \item \demph{There is a generating set of $\E$ consisting of $\kappa$-narrow projective objects for a regular cardinal $\kappa$.}
        \item The topos $\E$ is equivalent to a sheaf topos over a $\kappa$-extensive site for a regular cardinal $\kappa$.
    \end{enumerate}
\end{theorem}

For more details, see my ongoing paper \href{redacted-overleaf-private-url\#c82788}{[overleaf]}.

\section{Question}

\begin{question}
    Are the conditions in \Cref{mainTheorem} equivalent to $\Phi$-presentability for some diagram class $\Phi$?
\end{question}

\begin{question}
    In particular, is `being $\kappa$-narrow and projective' equivalent to $\Phi$-presentability?
\end{question}

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