\section{\texorpdfstring{$\kappa$}{kappa}-extensive topology}


In this appendix, $\kappa$ is a fixed infinite regular cardinal. The word `$\kappa$-small,' always means `less than $\kappa$.'
% \begin{definition}
%     A categroy $\C$ is said to be \demph{$\kappa$-extensive} if, $\C$ admits all $\kappa$-small coproducts, for any $\kappa$-small family of objects $\{X_\lambda\}_{\lambda\in \Lambda}$, the canonical comparison functor
%     \[
%     \C/\left({\coprod_{\lambda\in \Lambda} X_\lambda}\right) \to \prod_{\lambda\in \Lambda}\C/X_{\lambda}
%     \]
%     is an equivalence of categories.
% \end{definition}

\begin{definition}
    A categroy $\C$ is said to be \demph{$\kappa$-extensive} if, 
    \begin{itemize}
        \item $\C$ admits all $\kappa$-small coproducts,
        \item $\C$ admits pullbacks of $\kappa$-small coproduct injection along arbitrary morphisms, and
        \item $\kappa$-small coproducts are disjoint and pullback stable.
    \end{itemize}
\end{definition}

In the case where $\kappa=\aleph_0$, $\aleph_0$-extensive category is simply called \demph{extensive.} Extensive categories are extensively studied in \cite{carboni1993introduction}.