functors.txt

=================================================================================================
|                                      Mapping functions                                        |
=================================================================================================

                                   ------------------------
                                   | One mapping function |
                                   ------------------------

class Functor f where
  map :: ∀ a b. (a -> b) -> f a -> f b                             (Covariant)

class Contravariant f where
  cmap :: ∀ a b. (b -> a) -> f a -> f b                            (Contravariant)

                                   -------------------------
                                   | Two mapping functions |
                                   -------------------------

class Bifunctor f where
  bimap :: ∀ a b c d. (a -> c) -> (b -> d) -> f a b -> f c d       (Covariant, Covariant)

class Invariant f where
  imap :: ∀ a b. (a -> b) -> (b -> a) -> f a -> f b                (Covariant, Contravariant)

class Profunctor p where
  dimap :: ∀ a b c d. (b -> a) -> (c -> d) -> p a c -> p b d       (Contravariant, Covariant)

=================================================================================================
|                                      Type parameters                                          |
=================================================================================================

                                   ----------------------
                                   | One type parameter |
                                   ----------------------

class Functor f where
  map :: ∀ a b. (a -> b) -> f a -> f b                             (Covariant)

class Contravariant f where
  cmap :: ∀ a b. (b -> a) -> f a -> f b                            (Contravariant)

class Invariant f where
  imap :: ∀ a b. (a -> b) -> (b -> a) -> f a -> f b                (Covariant, Contravariant)

                                   -----------------------
                                   | Two type parameters |
                                   -----------------------

class Bifunctor f where
  bimap :: ∀ a b c d. (a -> c) -> (b -> d) -> f a b -> f c d       (Covariant, Covariant)

class Profunctor p where
  dimap :: ∀ a b c d. (b -> a) -> (c -> d) -> p a c -> p b d       (Contravariant, Covariant)

=================================================================================================
|                           Function Functors ((->) r, (->))                                    |
=================================================================================================

class Functor f where
  map :: ∀ a b. (a -> b) -> (r -> a) -> (r -> b)

instance functorFn :: Functor ((->) r a) where
  map f g = f << g

-----------------------------------------------------------------------

class Profunctor p where
  dimap :: ∀ a b c d. (b -> a) -> (c -> d) -> (a -> c) -> (b -> d)

instance profunctorFn :: Profunctor (->) where
  dimap f g h = g << h << f

=======================================================================
Everything else does not work!
=======================================================================

class Contravariant f where
  cmap :: ∀ a b. (b -> a) -> (a -> r) -> (b -> r)

instance contravariantFn :: Contravariant ((->) r) where
  OOPS (a -> r = (->) a r) does not fit; not possible

-- But could have, e.g.: a -> Int
instance contravariantFnInt :: Contravariant (a -> Int) where
  cmap f g = g << f

-----------------------------------------------------------------------

class Invariant f where
  imap :: ∀ a b. (a -> b) -> (b -> a) -> (r -> a) -> (r -> b)

instance invariantFn :: Invariant ((->) r) where
  -- imap f g h = f << h

Note: But g is never used. It's basically a Functor. Functions are
not Invariant Functors!
-----------------------------------------------------------------------

class Bifunctor f where
  bimap :: ∀ a b c d. (a -> c) -> (b -> d) -> (a -> b) -> (c -> d)

instance bifunctorFN :: Bifunctor ((->) r) where
  -- bimap f g h = COMPOSITION NOT POSSIBLE!!!

-----------------------------------------------------------------------