> For the complete documentation index, see [llms.txt](https://mandober.gitbook.io/math-debrief/llms.txt). Markdown versions of documentation pages are available by appending `.md` to page URLs; this page is available as [Markdown](https://mandober.gitbook.io/math-debrief/450-category-theory/10-functors/powerset.md).

# Powerset functor

A powerset functor, `𝒫 : 𝗦𝗲𝘁 → 𝗦𝗲𝘁`, sends subsets to their image under maps.

∀A,B ∈ Obj(𝗦𝗲𝘁), f : A → B and S ⊂ A:\
𝒫 A = 𝒫(A),\
𝒫 f : 𝒫(A) → 𝒫(B), S ⟼ f(S)

𝗦𝗲𝘁 is a category of sets. The objects are all small sets, and the arrows are functions between sets.