Category Theory for Programmers Chapter 8: Functoriality

Last chapter was about functors, but this is about how we can make them useful. In particular, combining functors via bifunctors. A reminder of what a bifunctor is:

class Bifunctor f where
  bimap :: (a -> c) -> (b -> d) -> f a b -> f c d
  bimap g h = first g . second h
  first :: (a -> c) -> f a b -> f c b
  first g = bimap g id
  second :: (b -> d) -> f a b -> f a d
  second = bimap id
  1. Show the data type data Pair a b = Pair a b is a bifunctor. Specifically, we show that if a or b is a functor, then Pair is a functor if

    instance Bifunctor (Pair a b) where
      bimap f g (Pair a b) = Pair (f a) (g b)
  2. Show isomorphism between Maybe and

    type Maybe' a = Either (Const () a) (Identity a)
    maybeToEither :: Maybe a -> Maybe' a
    maybeToEither Nothing = Left (Const ())
    maybeToEither (Just x) = Right (Identity x)
    eitherToMaybe :: Maybe' a -> Maybe a
    eitherToMaybe (Left _) = Nothing
    eitherToMaybe (Right (Identity x)) = Just x
  3. Show that PreList is an instance of Bifunctor,

    data PreList a b = Nil | Cons a b

    Let’s make this look like some things we know are functors and bifunctors already.

    data PreList a b = Maybe (a, b)
                     = Either (Const () c) (Identity a, Identity b)

    I don’t really know where to go with this other than we see it’s functors and bifunctors all the way down. But it’s probably easiest to just implement the interface outright.

  4. Show K2, Fst, and Snd define bifunctors in a and b.

    interface Bifunctor (K2 c a b) where
      bimap _ _ (K2 c) = K2 c
    interface Bifunctor (Fst a b) where
      bimap f _ (Fst a) = Fst (f a)
    interface Bifunctor (Snd a b) where
      bimap _ g (Snd a b) = Snd a (g b)

    As for the rest of this, a couple good resources about profunctors are this link and this video.