Functor package:rebase

A type f is a Functor if it provides a function fmap which, given any types a and b lets you apply any function from (a -> b) to turn an f a into an f b, preserving the structure of f. Furthermore f needs to adhere to the following:

• Identity fmap id == id
• Composition fmap (f . g) == fmap f . fmap g

Note, that the second law follows from the free theorem of the type fmap and the first law, so you need only check that the former condition holds. See https://www.schoolofhaskell.com/user/edwardk/snippets/fmap or https://github.com/quchen/articles/blob/master/second_functor_law.md for an explanation.

A Profunctor with the same input and output types can be seen as an Invariant functor.

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

Laws:

unit . counit ≡ id
counit . unit ≡ id

Laws:

proextract . promap f ≡ f . proextract
proextract . produplicate ≡ id
promap proextract . produplicate ≡ id
produplicate . produplicate ≡ promap produplicate . produplicate

ProfunctorFunctor has a polymorphic kind since 5.6.

Laws:

promap f . proreturn ≡ proreturn . f
projoin . proreturn ≡ id
projoin . promap proreturn ≡ id
projoin . projoin ≡ projoin . promap projoin

Make a Functor over the second argument of a Bifunctor.

Wrap a Functor to be used as a member of Invariant.

Wrap a Profunctor to be used as a member of Invariant2.

Every Bifunctor is also an Invariant2 functor.

Every Profunctor is also an Invariant2 functor.

Every Functor is also an Invariant functor.

A Profunctor with the same input and output types can be seen as an Invariant functor.