alternatives-convert-products-to-sums

Alternatives convert products to sums

You may know that Applicatives are product-preserving functors. That is, we have functions

pure :: Applicative f => () -> f ()
uncurry (liftA2 (,)) :: Applicative f => (f a, f b) -> f (a, b)

Next we will consider what sort of functors convert products to sums. We need a function which converts an empty product to an empty sum:

zero :: () -> f Void

(where Void is uninhabited). We also need a function which converts a pair product into a pair sum:

pair :: (f a, f b) -> f (Either a b)

Let’s simplify zero by removing the argument and fmapping absurd :: Void -> a:

empty = fmap absurd (zero ()) :: f a

It would also be nice to simplify our interface to pair. We can get a function of a somewhat nicer type

<|> = curry (fmap (either id id) . pair) :: f a -> f a -> f a

which can be proved equivalent to pair in the sense that

pair (a, b) = fmap Left a <|> fmap Right b

But these two functions

empty :: f a
(<|>) :: f a -> f a -> f a

are exactly what is required for an Alternative instance, and so Alternatives are functors which convert products to sums.