The Clowder Project

where

• Action on Objects. For each $U\in \mathcal{P}\webleft (X\webright )$, we have
  \[ \webleft [\webleft (-\webright )^{\textsf{c}}\webright ]\webleft (U\webright ) \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}U^{\textsf{c}}. \]

• Action on Morphisms. For each morphism $\iota _{U}\colon U\hookrightarrow V$ of $\mathcal{P}\webleft (X\webright )$, the image
  \[ \iota ^{\textsf{c}}_{U}\colon V^{\textsf{c}}\hookrightarrow U^{\textsf{c}} \]

  of $\iota _{U}$ by $\webleft (-\webright )^{\textsf{c}}$ is the inclusion

  \[ V^{\textsf{c}}\subset U^{\textsf{c}} \]

  i.e. where we have

  • If $U\subset V$, then $V^{\textsf{c}}\subset U^{\textsf{c}}$.