### alternatives-convert-products-to-sums

`Alternative`

s convert products to sums

You may know that `Applicative`

s 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)`

Letâ€™s simplify `zero`

by removing the argument and `fmap`

ping `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 `Alternative`

s are functors which convert products to sums.