Kleisli Arrows

An arrow suggests and implies a direction. It is a symbol for the universal notion of a single step (of a process) or a individual completed transformation.

Here a and b are of the same level of abstraction (usually a Set)

a -> b

Here it crosses an abstraction boundary or a resulting value is of a different kind of abstraction.

a -> m b

This notion is, in turn, has been generalized as

pure :: a -> f a

which captures a general notion of placing into.

or even more generally

return :: a -> m a

And then we have a couple of convectional hacks to apply these abstract entities to each other or compose them.

type Applicative :: (* -> *) -> Constraint
class Functor f => Applicative f where
  ...

type Monad :: (* -> *) -> Constraint
class Applicative m => Monad m where
  ...

Notice the arrows (kinds).