Item 10 (008U) — The Clowder Project

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Style	Class	Font	Theorem Environments
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
\[ \bigcup _{i\in I}f_{!}\webleft (U_{i}\webright )\subset f_{!}\webleft(\bigcup _{i\in I}U_{i}\webright), \]

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

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

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

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