The Clowder Project

where $-_{1}\cup -_{2}$ 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
  \[ \webleft [-_{1}\cup -_{2}\webright ]\webleft (U,V\webright )\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}U\cup V. \]

• 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

  \[ \iota _{U}\cup \iota _{V}\colon U\cup V\hookrightarrow U'\cup V' \]

  of $\webleft (\iota _{U},\iota _{V}\webright )$ by $\cup $ is the inclusion

  \[ U\cup V\subset U'\cup V' \]

  i.e. where we have

  • If $U\subset U'$ and $V\subset V'$, then $U\cup V\subset U'\cup V'$.