Documentation

As the method names suggest, there's a category of functions:

instance Category (->) where
  id = \x -> x
  f . g = \x -> f (g x)

Isomorphisms form a category as well:

data Iso a b = Iso (a -> b) (b -> a)

instance Category Iso where
  id = Iso id id
  Iso f1 g1 . Iso f2 g2 = Iso (f1 . f2) (g2 . g1)

Natural transformations are another important example:

newtype f ~> g = NatTransform (forall x. f x -> g x)

instance Category (~>) where
  id = NatTransform id
  NatTransform f . NatTransform g = NatTransform (f . g)

Using the TypeData language extension, we can also make a category where k isn't Type, but a custom kind Door instead:

type data Door = DoorOpen | DoorClosed

data Action (before :: Door) (after :: Door) where
  DoNothing :: Action door door
  OpenDoor :: Action start DoorClosed -> Action start DoorOpen
  CloseDoor :: Action start DoorOpen -> Action start DoorClosed

instance Category Action where
  id = DoNothing

  DoNothing . action = action
  OpenDoor rest . action = OpenDoor (rest . action)
  CloseDoor rest . action = CloseDoor (rest . action)