Item 1 (0040) — The Clowder Project

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`

Functoriality. The assignments $U,V,\webleft (U,V\webright )\mapsto U\cap V$ define functors
\begin{gather*} \begin{aligned} U\cap - & \colon \webleft (\mathcal{P}\webleft (X\webright ),\subset \webright ) \to \webleft (\mathcal{P}\webleft (X\webright ),\subset \webright ),\\ -\cap V & \colon \webleft (\mathcal{P}\webleft (X\webright ),\subset \webright ) \to \webleft (\mathcal{P}\webleft (X\webright ),\subset \webright ),\\ \end{aligned}\\ -_{1}\cap -_{2} \colon \webleft (\mathcal{P}\webleft (X\webright )\times \mathcal{P}\webleft (X\webright ),\subset \times \subset \webright ) \to \webleft (\mathcal{P}\webleft (X\webright ),\subset \webright ), \end{gather*}

where $-_{1}\cap -_{2}$ is the functor where
- Action on Objects. For each $\webleft (U,V\webright )\in \mathcal{P}\webleft (X\webright )\times \mathcal{P}\webleft (X\webright )$, we have
  
  \[ \webleft [-_{1}\cap -_{2}\webright ]\webleft (U,V\webright )\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}U\cap V. \]
- Action on Morphisms. For each pair of morphisms
  
  \begin{align*} \iota _{U} & \colon U\hookrightarrow U',\\ \iota _{V} & \colon V\hookrightarrow V’ \end{align*}
  
  of $\mathcal{P}\webleft (X\webright )\times \mathcal{P}\webleft (X\webright )$, the image
  
  \[ \iota _{U}\cap \iota _{V}\colon U\cap V\hookrightarrow U'\cap V' \]
  
  of $\webleft (\iota _{U},\iota _{V}\webright )$ by $\cap $ is the inclusion
  
  \[ U\cap V\subset U'\cap V' \]
  
  i.e. where we have
  - If $U\subset U'$ and $V\subset V'$, then $U\cap V\subset U'\cap V'$.
and where $U\cap -$ and $-\cap V$ are the partial functors of $-_{1}\cap -_{2}$ at $U,V\in \mathcal{P}\webleft (X\webright )$.

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