Proposition 4.4.3.1.1 (01JG): Adjointness of Powersets I

We have an adjunction

witnessed by a bijection

\[ \underbrace{\mathsf{Sets}^{\mathsf{op}}\webleft (\mathcal{P}\webleft (X\webright ),Y\webright )}_{\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\mkern 5mu\mathsf{Sets}\webleft (Y,\mathcal{P}\webleft (X\webright )\webright )} \cong \mathsf{Sets}\webleft (X,\mathcal{P}\webleft (Y\webright )\webright ), \]

natural in $X\in \operatorname {\mathrm{Obj}}\webleft (\mathsf{Sets}\webright )$ and $Y\in \operatorname {\mathrm{Obj}}\webleft (\mathsf{Sets}^{\mathsf{op}}\webright )$.

We have

$\mathsf{Sets}^{\mathsf{op}}\webleft (\mathcal{P}\webleft (A\webright ),B\webright )$	$\mathord {\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}}$	$\mathsf{Sets}\webleft (B,\mathcal{P}\webleft (A\webright )\webright )$
	$\mathord {\cong }$	$\mathsf{Sets}\webleft (B,\mathsf{Sets}\webleft (A,\{ \mathsf{t},\mathsf{f}\} \webright )\webright )$	(by Item 2 of Proposition 4.5.1.1.4)
	$\mathord {\cong }$	$\mathsf{Sets}\webleft (A\times B,\{ \mathsf{t},\mathsf{f}\} \webright )$	(by Item 2 of Proposition 4.1.3.1.3)
	$\mathord {\cong }$	$\mathsf{Sets}\webleft (A,\mathsf{Sets}\webleft (B,\{ \mathsf{t},\mathsf{f}\} \webright )\webright )$	(by Item 2 of Proposition 4.1.3.1.3)
	$\mathord {\cong }$	$\mathsf{Sets}\webleft (A,\mathcal{P}\webleft (B\webright )\webright )$,	(by Item 2 of Proposition 4.5.1.1.4)

where all bijections are natural in $A$ and $B$.¹

¹Here we are using Item 3 of Proposition 4.5.1.1.4.

The Clowder Project

4.4.3 Adjointness of Powersets I