Item 6 (00RH) — The Clowder Project

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Style 1	`book`	Alegreya Sans	`tcbthm`
Style 2	`book`	Alegreya Sans	`amsthm`
Style 3	`book`	Arno*	`amsthm`
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*To be replaced with Linus Romer's Elemaints when it is released.

Symmetric Oplax Monoidality With Respect to Intersections. The direct image function of Item 1 has a symmetric oplax monoidal structure
\[ \webleft (R_{*},R^{\otimes }_{*},R^{\otimes }_{*|\mathbb {1}}\webright ) \colon \webleft (\mathcal{P}\webleft (A\webright ),\cap ,A\webright ) \to \webleft (\mathcal{P}\webleft (B\webright ),\cap ,B\webright ), \]

being equipped with inclusions

\[ \begin{gathered} R^{\otimes }_{*|U,V} \colon R_{*}\webleft (U\cap V\webright ) \subset R_{*}\webleft (U\webright )\cap R_{*}\webleft (V\webright ),\\ R^{\otimes }_{*|\mathbb {1}} \colon R_{*}\webleft (A\webright ) \subset B, \end{gathered} \]

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

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