data-functor-logistic: Updatable analogue of Distributive functors

Logistic is to setters as Distributive is to getters

Distributive functors are containers where getters can be enumerated as their own types.

This is the definition of the Distributive class:

class Functor g => Distributive g where
  distribute :: Functor f => f (g a) -> g (f a)

One easy-to-understand instance is Complex.

data Complex a = !a :+ !a

realPart :: Complex a -> a
realPart (x :+ _) =  x

imagPart :: Complex a -> a
imagPart (_ :+ y) =  y

instance Distributive Complex where
  distribute wc = fmap realPart wc :+ fmap imagPart wc

Given any functor-wrapped value, distribute fmaps the getters of Complex to it. distribute id instantiates it as the function ((->) r) functor. In this case, distribute id is equal to realPart :+ imagPart.

However, we cannot modify the elements this way because distribute passes getters but not setters.

Here we introduce a new Logistic class to provide settability to containers:

class Functor t => Logistic t where
  deliver :: Contravariant f => f (t a -> t a) -> t (f (a -> a))

While the type of deliver is slightly more intimidating, it's actually very close to the distribute; the Functor constraint is Contravariant instead and the contents are endomorphisms.

Here's the instance for Complex. deliver f contramaps a setter function to f for each field:

instance Logistic Complex where
  deliver f
    = contramap (\g (a :+ b) -> g a :+ b) f
    :+ contramap (\g (a :+ b) -> a :+ g b) f

Instantiating the Op contravariant functor, it is trivial to obtain a collection of setters.

newtype Op a b = Op { getOp :: b -> a }

setters :: Logistic t => t ((a -> a) -> t a -> t a)
setters = getOp <$> deliver (Op id)

ghci> let setR :+ setI = setters
ghci> setR (+1) (0 :+ 1)
1 :+ 1
ghci> setI (+1) (0 :+ 1)
0 :+ 2

deliver has a generic default implementation which works for any single-constructor products.

This class can be useful to complement Distributive. Generalisation to higher-kinded data would also be interesting.