Item 12 (008W) — 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.

Symmetric Lax Monoidality With Respect to Unions. The direct image with compact support function of Item 1 has a symmetric lax monoidal structure
\[ \webleft (f_{!},f^{\otimes }_{!},f^{\otimes }_{!|\mathbb {1}}\webright ) \colon \webleft (\mathcal{P}\webleft (X\webright ),\cup ,\text{Ø}\webright ) \to \webleft (\mathcal{P}\webleft (Y\webright ),\cup ,\text{Ø}\webright ), \]

being equipped with inclusions

\[ \begin{gathered} f^{\otimes }_{!|U,V} \colon f_{!}\webleft (U\webright )\cup f_{!}\webleft (V\webright ) \hookrightarrow f_{!}\webleft (U\cup V\webright ),\\ f^{\otimes }_{!|\mathbb {1}} \colon \text{Ø}\hookrightarrow f_{!}\webleft (\text{Ø}\webright ), \end{gathered} \]

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

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