Proposition 2.4.5.1.1 (01JL): Powersets as Free Cocompletions: Universal Property

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`

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

The powerset $\webleft (\mathcal{P}\webleft (X\webright ),\subset \webright )$ of $X$ 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;

satisfies the following universal property:

Given another pair $\webleft (Y,f\webright )$ consisting of
- A suplattice $\webleft (Y,\preceq \webright )$;
- A function $f\colon X\to Y$;
there exists a unique morphism of suplattices

\[ \webleft (\mathcal{P}\webleft (X\webright ),\subset \webright )\overset {\exists !}{\to }\webleft (Y,\preceq \webright ) \]

making the diagram

commute.

This is a rephrasing ofProposition 2.4.5.1.2, which we prove below.¹

¹Here we only remark that the unique morphism of suplattices in the statement is given by the left Kan extension $\text{Lan}_{\chi _{X}}\webleft (f\webright )$ of $f$ along $\chi _{X}$.