Alternative instance for Compose

[Keith]

With [], Maybe, and some other Alternative Functors, we have the nice property that empty <*> x = empty. (Think near-semiring.)

But with your lifted version, empty <*> Compose empty = Compose empty. The authentically empty 'empty' is still the true zero element.

-Keith
—
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On July 16, 2020 2:00:22 AM UTC, Asad Saeeduddin <masaeedu at gmail.com> wrote:
>Hello folks,
>
>base 4.14 has the following instance for `Compose f g`:
>
>instance (Alternative f, Applicative g) => Alternative (Compose f g) where
>     empty = Compose empty
>     (<|>) = coerce ((<|>) :: f (g a) -> f (g a) -> f (g a))
>       :: forall a . Compose f g a -> Compose f g a -> Compose f g a
>
>This instance doesn't really do anything with the `Applicative g` 
>constraint it is demanding. It's also kind of unclear what utility it 
>delivers, given that the resulting Alternative instance is 
>indistinguishable from the outer functor's Alternative instance. In 
>other words: `getCompose $ Compose x <|> Compose y == x <|> y`.
>
>It seems to me a more useful instance would be:
>
>instance (Applicative f, Alternative g) => Alternative (Compose f g) where
>     empty = Compose $ pure empty
>     (<|>) = _ $ liftA2 (<|>)
>
>This is also nicer in a mathematical sense: `Applicative` functors 
>correspond to lax monoidal functors from `Hask, (- , -), ()` to `Hask, 
>(-, -), ()`. We can interpret `Alternative`s as lax monoidal functors 
>from `Hask, Either - -, Void` to `Hask, (-, -), ()`. Compatible lax 
>monoidal functors compose, but if you think about the relevant "types" 
>of the functors a bit, you'll realize that while we can compose an 
>`Applicative` after an `Alternative` to get another `Alternative`, the 
>reverse does not work. Hence the instance we have today, which has no 
>choice but to just ignore the `Applicative` constraint it is demanding.
>
>Does it make sense to replace the instance we have today with the `pure 
>empty`, `liftA2 (<|>)` one?
>
>Thanks,
>Asad
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