The Clowder Project

The pair $\webleft (\mathcal{P}\webleft (X\webright ),\chi _{\webleft (-\webright )}\webright )$ consisting of

• The powerset of $X$ together with reverse inclusion $\mathcal{P}\webleft (X\webright )^{\mathsf{op}}=\webleft (\mathcal{P}\webleft (X\webright ),\supset \webright )$ of Definition 2.4.1.1.1;
• The characteristic embedding $\chi _{\webleft (-\webright )}\colon X\hookrightarrow \mathcal{P}\webleft (X\webright )$ of $X$ into $\mathcal{P}\webleft (X\webright )$ of Definition 2.5.4.1.1;

• Given another pair $\webleft (Y,f\webright )$ consisting of
  • An inflattice $\webleft (Y,\preceq \webright )$;
  • A function $f\colon X\to Y$;
  there exists a unique morphism of inflattices
  \[ \webleft (\mathcal{P}\webleft (X\webright ),\supset \webright )\overset {\exists !}{\to }\webleft (Y,\preceq \webright ) \]

  making the diagram

  

  commute.