Applicatives

An Applicative is a Functor that can inject values and apply wrapped functions:

class Applicative f <= Functor f
  pure : a -> f a
  ap : f (a -> b) -> f a -> f b

Applicatives are great when you want to combine independent computations that live “in a container”.

pure

let x: Option i32 = pure 1 in x

The type depends on context. For example, this forces Option:

let x: Option i32 = pure 1 in x

A common pattern: building up computations

Because functions are curried, you can apply step-by-step. For Option:

ap (ap (pure (\x y -> x + y)) (Some 1)) (Some 2)

ap with Option

ap (Some ((+) 1)) (Some 41)

If either side is “missing”, the result is missing:

( ap None (Some 1)
, ap (Some ((+) 1)) None
)

Applicative style

You can build up multi-argument computations by applying step-by-step:

ap (ap (pure (\x y -> x + y)) (Some 1)) (Some 2)