Item 2 (006T) — The Clowder Project

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Style 1	`book`	Alegreya Sans	`tcbthm`
Style 2	`book`	Alegreya Sans	`amsthm`
Style 3	`book`	Arno*	`amsthm`
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*To be replaced with Linus Romer's Elemaints when it is released.

Cartesian Closedness. The category $\mathcal{P}\webleft (X\webright )$ is Cartesian closed with internal Hom
\[ \mathbf{Hom}_{\mathcal{P}\webleft (X\webright )}\webleft (-_{1},-_{2}\webright )\colon \mathcal{P}\webleft (X\webright )\mkern -0.0mu^{\mathsf{op}}\times \mathcal{P}\webleft (X\webright ) \to \mathcal{P}\webleft (X\webright ) \]

given by ^[1]

\[ \mathbf{Hom}_{\mathcal{P}\webleft (X\webright )}\webleft (U,V\webright )\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\webleft (X\setminus U\webright )\cup V \]

for each $U,V\in \text{Obj}\webleft (\mathcal{P}\webleft (X\webright )\webright )$.

Footnotes

[1] For intuition regarding the expression defining $\mathbf{Hom}_{\mathcal{P}\webleft (X\webright )}\webleft (U,V\webright )$, see Remark 2.3.9.1.3.

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