Composition and Kleisli composition are Monoids

Created: 2023-02-15 09:26

Function composition is a Monoid:

Associativity

f . (g . h) = (f . g) . h

Identity

f . id = id . f = f

Kleisli composition is composition lifted to a Monadic context:

(.)   :: (b ->   c) -> (a ->   b) -> (a ->   c)
(<=<) :: (b -> m c) -> (a -> m b) -> (a -> m c)

And thus it’s also a Monoid, where pure is the identity.

Associativity

f <=< (g <=< h) = (f <=< g) <=< h

Identity

f <=< pure = pure <=< f = f