Item 3 (00RW) — The Clowder Project

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Style	Class	Font	Theorem Environments
Style 1	`book`	Alegreya Sans	`tcbthm`
Style 2	`book`	Alegreya Sans	`amsthm`
Style 3	`book`	Arno*	`amsthm`
Style 4	`book`	Computer Modern	`amsthm`

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

Lax Preservation of Colimits. We have an inclusion of sets
\[ \bigcup _{i\in I}R_{-1}\webleft (U_{i}\webright )\subset R_{-1}\webleft (\bigcup _{i\in I}U_{i}\webright ), \]

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

\[ \begin{gathered} R_{-1}\webleft (U\webright )\cup R_{-1}\webleft (V\webright ) \subset R_{-1}\webleft (U\cup V\webright ),\\ \emptyset \subset R_{-1}\webleft (\emptyset \webright ), \end{gathered} \]

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

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