Power Set

Definition

Given a set $A$, its power set is the set of all subsets of $A$:

$$\mathcal{P}(A)=\{B.B\subseteq A\}$$

Remarks

In topos theory, the analogue of powerset is referred to as the subobjects of $B$, written $\mathrm{Sub}(B)$ in $\mathrm{Set}$.

The powerset category is given by the poset of inclusion, $(\mathcal{P}(B),\subseteq )$, which could be thought of as a the slice category of the category of sets under inclusion $(\mathrm{Set},\subseteq )$.