What is applicative?

It is Monoidal functor:

class Functor f => Monoidal f where
  unit :: f ()
  (**) :: f a -> f b -> f (a,b)
  (<*>) :: f (a -> b) -> f a -> f b
  
-- The two operators are equivalent:
f <*> a = apply <$> f ** a
  where apply (f, a) = f a
  
a ** b = (,) <$> b <*> a
LawProof
Naturalityfmap (f *** g) (u ** v) = fmap f u ** fmap g v
Left idunit ** v ≝ v
Right idu ** unit ≝ u
Associativityu ** (v ** w) ≝

Notes:

(***) :: (a -> b) -> (c -> d) -> (a, c) -> (b, d)
f *** g = \(x,y) -> (f x, g y)

≝ refers to isomorphic - f ((), a) is isomorphic to f a.

What is naturality?

It says that if I combine the contexts of 2 applicative functors, then apply functions to the result values

It should be the same as me applying functions to each applicative, then combining them.

Examples of Invalid naturality

Highly contrived example..

Assume monoidal instance for Int, with 1 and (+) as empty and (<>).

fmap _ (Right _) = Left empty
fmap f (Right 2 ** Left 1) = fmap f Left 1
                           = Left 1
fmap f (Right 2) ** fmap g (Left 1) = Left 1 ** Left 1
                                    = Left 2

References:

logo