Item 3 (00T2) — The Clowder Project

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Style 1	`book`	Alegreya Sans	`tcbthm`
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*To be replaced with Linus Romer's Elemaints when it is released.

Lax Preservation of Colimits. We have an inclusion of sets
\[ \bigcup _{i\in I}R_{!}\webleft (U_{i}\webright )\subset R_{!}\webleft (\bigcup _{i\in I}U_{i}\webright ), \]

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

\[ \begin{gathered} R_{!}\webleft (U\webright )\cup R_{!}\webleft (V\webright ) \subset R_{!}\webleft (U\cup V\webright ),\\ \text{Ø}\subset R_{!}\webleft (\text{Ø}\webright ), \end{gathered} \]

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

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