Lax monoidal functors with a different monoidal structure

The “neat alternative presentation” for Applicative is based on the following two equivalencies

pure a = fmap (const a) unit
unit = pure ()

ff <*> fa = fmap (\(f,a) -> f a) $ ff ** fa
fa ** fb = pure (,) <*> fa <*> fb

The trick to get this “neat alternative presentation” for Applicative is the same as the trick for zipWith – replace explicit types and constructors in the interface with things that the type or constructor can be passed into to recover what the original interface was.

unit :: f ()

Is replaced with pure which we can substitute the type () and the constructor () :: () into to recover unit.

pure :: a  -> f a
pure    () :: f ()

And similarly (though not as straightforward) for substituting the type (a,b) and the constructor (,) :: a -> b -> (a,b) into liftA2 to recover **.

liftA2 :: (a -> b -> c) -> f a -> f b -> f c
liftA2    (,)           :: f a -> f b -> f (a,b)

Applicative then gets the nice <*> operator by lifting function application ($) :: (a -> b) -> a -> b into the functor.

(<*>) :: f (a -> b) -> f a -> f b
(<*>) = liftA2 ($)

To find a “neat alternative presentation” for PtS we need to find

• something we can substitute the type Void into to recover unit
• something we can substitute the type Either a b and the constructors Left :: a -> Either a b and Right :: b -> Either a b into to recover **

(If you notice that we already have something the constructors Left and Right can be passed to you can probably figure out what we can replace ** with without following the steps I used; I didn’t notice this until after I solved it)

unit

This immediately gets us an alternative to unit for sums:

empty :: f a
empty = fmap absurd unit

unit :: f Void
unit = empty

operator

We’d like to find an alternative to (**). There is an alternative to sums like Either that allows them to be written as functions of products. It shows up as the visitor pattern in object oriented programming languages where sums don’t exist.

data Either a b = Left a | Right b

{-# LANGUAGE RankNTypes #-}
type Sum a b = forall c. (a -> c) -> (b -> c) -> c

It’s what you would get if you changed the order of either‘s arguments and partially applied them.

either :: (a -> c) -> (b -> c) -> Either a b -> c

toSum :: Either a b -> Sum a b
toSum e = \forA forB -> either forA forB e

toEither :: Sum a b -> Either a b
toEither s = s Left Right

We can see that Either a b ≅ Sum a b. This allows us to rewrite the type for (**)

(**) :: f a -> f b -> f (Either a b)
(**) :: f a -> f b -> f (Sum a b)
(**) :: f a -> f b -> f ((a -> c) -> (b -> c) -> c)

Now it’s clear what ** does. It delays fmaping something onto both of its arguments, and combines the results of those two mappings. If we introduce a new operator, <||> :: f c -> f c -> f c which simply assumes that the fmaping was done already, then we can see that

fmap (\f -> f forA forB) (fa ** fb) = fmap forA fa <||> fmap forB fb

Or back in terms of Either:

fa ** fb = fmap Left fa <||> fmap Right fb
fa1 <||> fa2 = fmap (either id id) $ fa1 ** fa2

So we can express everything PtS can express with the following class, and everything that could implement PtS can implement the following class:

class Functor f => AlmostAlternative f where
    empty  :: f a
    (<||>) :: f a -> f a -> f a

This is almost certainly the same as the Alternative class, except we didn’t require that the Functor be Applicative.

Conclusion

It’s just a Functor that is a Monoid for all types. It’d be equivalent to the following:

class (Functor f, forall a. Monoid (f a)) => MonoidalFunctor f

The forall a. Monoid (f a) constraint is pseudo-code; I don’t know a way to express constraints like this in Haskell.