Profunctors

Formally, the class Profunctor represents a profunctor from Hask -> Hask.

Intuitively it is a bifunctor where the first argument is contravariant and the second argument is covariant.

You can define a Profunctor by either defining dimap or by defining both lmap and rmap.

If you supply dimap, you should ensure that:

dimap id id ≡ id

If you supply lmap and rmap, ensure:

 lmap id ≡ id
 rmap id ≡ id

If you supply both, you should also ensure:

dimap f g ≡ lmap f . rmap g

These ensure by parametricity:

 dimap (f . g) (h . i) ≡ dimap g h . dimap f i
 lmap (f . g) ≡ lmap g . lmap f
 rmap (f . g) ≡ rmap f . rmap g

Generalizing UpStar of a strong Functor

Minimal complete definition: first' or second'

Note: Every Functor in Haskell is strong.

http://takeichi.ipl-lab.org/~asada/papers/arrStrMnd.pdf

The generalization of DownStar of a "costrong" Functor

Minimal complete definition: left' or right'

Note: We use traverse and extract as approximate costrength as needed.

Lift a Functor into a Profunctor (forwards).

Lift a Functor into a Profunctor (backwards).

Wrap an arrow for use as a Profunctor.