Under the analogy that $\{ \mathsf{t},\mathsf{f}\} $ should be the $\webleft (-1\webright )$-categorical analogue of $\mathsf{Sets}$, we may view the powerset of a set as a decategorification of the category of presheaves of a category (or of the category of copresheaves):
- The powerset of a set $X$ is equivalently (Item 2 of Proposition 2.5.1.1.4) 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.
