Types of functors

Types of functors

• special functors

  • endofunctor, a functor from a category to itself

  • identity functor

  • full functor

  • faithful functor

  • forgetful functor

  • constant functor

  • power-set functor

  • dual-set functor

  • monad: endofunctor with some additional structure

• by variance

  • contravariant (functor)

  • covariant functor is the full name for any functor

  • bifunctor

  • profunctor

An endofunctor maps a category 𝒞 to itself, F : 𝒞 → 𝒞.

An identity functor is type of endofunctor: it maps a category 𝒞 to itself, 𝒞 ⟼ 𝒞, by mapping each object in 𝒞 to itself and each arrow in 𝒞 to itself. Iᴄ : 𝒞 → 𝒞 such that ∀a ∈ Obj(𝒞). a ⟼ a ⋀ ∀f ∈ Arr(𝒞). f ⟼ f

A faithful functor reflects epis and monos.

Constant functor ∆d : 𝒞 → 𝒟 for fixed d ∈ 𝒟, ∆d : a ⟼ d, f ⟼ idᴅ

Power-set functor 𝒫 : 𝗦𝗲𝘁 → 𝗦𝗲𝘁 sends subsets to their image under maps. Let A,B ∈ 𝗦𝗲𝘁, f : A → B and S ⊂ A, then: 𝒫A = 𝒫(A), 𝒫f : 𝒫(A) → 𝒫(B), S ⟼ f(S)

Many categories that represent algebras i.e. sets endowed with additional structure (e.g. groups, vector spaces, rings, topological spaces, etc.) there is a forgetful functor going back to 𝗦𝗲𝘁, where objects are sent to their carrier sets. There is also a forgetful functor F : 𝗖𝗮𝘁 → Graph, sending each category to the graph defined by its objects and arrows.

Dual-set functor

∗ : Vect → Vect
  : W ⟼ W∗
  : (f : V → W) ⟼ (f∗ : W∗ → V∗)

This is an example of a contravariant functor, a functor from Vect to Vectᵒᵖ, the category with reversed arrows and composition rules.