The powerset of a set is a decategorification of the category of presheaves of a category: while[1]
- The powerset of a set $X$ is equivalently (Item 1 and Item 2 of Proposition 2.4.3.1.6) the set
\[ \mathsf{Sets}\webleft (X,\{ \mathsf{t},\mathsf{f}\} \webright ) \]
of functions from $X$ to the set $\{ \mathsf{t},\mathsf{f}\} $ of classical truth values.
- The category of presheaves on a category $\mathcal{C}$ is the category
\[ \mathsf{Fun}\webleft (\mathcal{C}^{\mathsf{op}},\mathsf{Sets}\webright ) \]
of functors from $\mathcal{C}^{\mathsf{op}}$ to the category $\mathsf{Sets}$ of sets.
