Monoidal functors and applicatives

monoidal-functors.hs

{-# LANGUAGE UndecidableInstances #-}
{-# LANGUAGE FlexibleInstances #-}
-- Tuesday, June 02, 2020
-- Monoidial functors and Applicatives
import Prelude hiding (Applicative(..), (**))

-- Lax monoidial functors
-- Laws:
-- Identity:      unit ** x     ~= x ** unit ~= unit
-- Associativity: u ** (v ** w) ~= (u ** v) ** w
class Functor f => Monoidal f where
  unit :: f ()
  (**) :: f a -> f b -> f (a,b)

-- Applicative functors
-- Identity: p   ure id <*> v      = v
-- Homomorphism: pure f <*> pure x = pure (f x)
-- Interchange:  u <*> pure y      = pure ($ y) <*> u
-- Composition:  u <*> (v <*> w)   = pure (.) <*> u <*> v <*> w
class Functor f => Applicative f where
  pure :: a -> f a
  (<*>) :: f (a -> b) -> f a -> f b

infixl 4 <*>

-- Proof that we can implement pure and (<*>) in terms of unit and
-- (**), and vice versa
instance (Functor f, Monoidal f) => Applicative f where
  pure x = fmap (const x) unit
  f <*> x = fmap (uncurry ($)) (f ** x)

instance (Functor f, Applicative f) => Monoidal f where
  unit = pure ()
  fa ** fb = (,) <$> fa <*> fb