Item 3 (00SG) — The Clowder Project

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*To be replaced with Linus Romer's Elemaints when it is released.

Preservation of Colimits. We have an equality of sets
\[ R^{-1}\webleft (\bigcup _{i\in I}U_{i}\webright )=\bigcup _{i\in I}R^{-1}\webleft (U_{i}\webright ), \]

natural in $\webleft\{ U_{i}\webright\} _{i\in I}\in \mathcal{P}\webleft (B\webright )^{\times I}$. In particular, we have equalities

\[ \begin{gathered} R^{-1}\webleft (U\webright )\cup R^{-1}\webleft (V\webright ) = R^{-1}\webleft (U\cup V\webright ),\\ R^{-1}\webleft (\emptyset \webright ) = \emptyset , \end{gathered} \]

natural in $U,V\in \mathcal{P}\webleft (B\webright )$.

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