Control.Category

A class for categories.

In mathematics, a category is defined as a collection of objects and a collection of morphisms between objects, together with an identity morphism id for every object and an operation (.) that composes compatible morphisms.

This class is defined in an analogous way. The collection of morphisms is represented by a type parameter cat, which has kind k -> k -> Type for some kind variable k that represents the collection of objects; most of the time the choice of k will be Type.

Examples

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)

Right-to-left composition. This is a synonym for (.), but it can be useful to make the order of composition more apparent.

Left-to-right composition. This is useful if you want to write a morphism as a pipeline going from left to right.