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Applicative functors are more powerful than Functors but not as powerful as monads. They provide a way to partially apply a function inside an Applicative context to parameters also inside the same context.

Confused? Well hopefully after this article you won’t be.

As we saw earlier, a Functor is just a Haskell typeclass with the single required fmap function.

class Functor f where
  fmap :: (a -> b) -> f a -> f b

We can apply a function from a -> b to a value wrapped inside of a functor context.

Now what if we want to apply a function to multiple arguments that are all wrapped inside of a context? Such as summing two numbers wrapped in a Maybe.

(Just 1) + (Just 2) -- ?

If we apply fmap to Just 1 with the (+) function we get back another function wrapped in the Maybe context.

ghci> :t (+) <$> Just 1
(+) <$> Just 1 :: Num a => Maybe (a -> a)

But now we are stuck. We need to apply a function inside of the Maybe context to a number inside the Maybe context.

This is where Applicatives come in!

They allow us to take a function which operates on normal unwrapped values, and apply it to parameters which are all wrapped in a context.

The typeclass definition requires all Applicatives to be Functors (meaning they all implement the fmap function).

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

An Applicative must implement the pure and (<*>) functions.

pure just takes a normal unwrapped value and wraps it into an Applicative context.

(<*>) takes a function wrapped in an Applicative context and a parameter also wrapped. It then unwraps both, applies a to the function, and wraps the result back up.

Lets see how it is implemented for the Maybe type.

instance Applicative Maybe where
  pure x = Just x
  (Just f) <*> (Just x) = Just (f x)
  _ <*> _ = Nothing

If either Maybe types are Nothing then Nothing is returned. Otherwise, each value is unwrapped (via pattern matching), the function is applied to the contents of second Maybe, and the result is wrapped back up as a Maybe.

Since all functions in Haskell are curried, we can partially apply a function of \(n\) arguments to \(n\) parameters wrapped in the Applicative context.

Using our above example, summing two numbers wrapped in a Maybe would look like this

ghci> (+) <$> Just 1 <*> Just 2
Just 3

We first use <$> to partially apply + to Just 1. Functor stuff. But then we use <*> to take the wrapped function and apply it to the wrapped parameter.

Alternatively we can use the pure function to wrap (+) inside of an Applicative.

ghci> pure (+) <*> Just 1 <*> Just 2
Just 3

What happens if one of the parameters is Nothing?

ghci> (+) <$> Just 1 <*> Nothing
Nothing

This is awesome! We don’t have to worry about checking whether or not our parameters are valid. The (<*>) function does that for us under the hood.