Item 4 (00RF) — The Clowder Project

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

Oplax Preservation of Limits. We have an inclusion of sets
\[ R_{*}\webleft (\bigcap _{i\in I}U_{i}\webright )\subset \bigcap _{i\in I}R_{*}\webleft (U_{i}\webright ), \]

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

\[ \begin{gathered} R_{*}\webleft (U\cap V\webright ) \subset R_{*}\webleft (U\webright )\cap R_{*}\webleft (V\webright ),\\ R_{*}\webleft (A\webright ) \subset B, \end{gathered} \]

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

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