Item 5 (00RY) — The Clowder Project

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Style 1	`book`	Alegreya Sans	`tcbthm`
Style 2	`book`	Alegreya Sans	`amsthm`
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*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
\[ \webleft (R_{-1},R^{\otimes }_{-1},R^{\otimes }_{-1|\mathbb {1}}\webright ) \colon \webleft (\mathcal{P}\webleft (A\webright ),\cup ,\emptyset \webright ) \to \webleft (\mathcal{P}\webleft (B\webright ),\cup ,\emptyset \webright ), \]

being equipped with inclusions

\[ \begin{gathered} R^{\otimes }_{-1|U,V} \colon R_{-1}\webleft (U\webright )\cup R_{-1}\webleft (V\webright ) \subset R_{-1}\webleft (U\cup V\webright ),\\ R^{\otimes }_{-1|\mathbb {1}} \colon \emptyset \subset R_{-1}\webleft (\emptyset \webright ), \end{gathered} \]

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

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