Definition:Power Set Functor

Definition

Covariant

The (covariant) power set functor $\PP: \mathbf{Set} \to \mathbf{Set}$ is the covariant functor which sends:

• An object $x$ to its power set $\powerset x$.
• A morphism $f: x \to y$ to the direct image mapping $\powerset f: \powerset x \to \powerset y$.

Contravariant

The contravariant power set functor $\overline \PP: \mathbf{Set} \to \mathbf{Set}$ is the contravariant functor which sends:

• An object $x$ to its power set $\powerset x$.
• A morphism $f : x \to y$ to the inverse image mapping $\map {\overline \PP} f : \powerset y \to \powerset x$.

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