This discussion started at #126, where we discussed a proposed linear Foldable implementation and the fact that it is not implied by linear Traversable; and ended up concluding that what we need is a variant of the existing Traversable which uses data Applicatives instead of control Applicatives.

See the thread at #126 for a longer discussion, but I'm quoting the critical comment there, from @aspiwack:

We've established that being “traversable” by Control.Applicative-s is a common occurrence, but doesn't imply Foldable.

However, being “traversable” by Data.Applicative-s, a less common occurrence, does imply Foldable (indeed Const m is a Data.Applicative (for m a monoid)).

So, here is the thought. Why don't we give a name to the equivalent of traverse, but with Data.Applicative and add it to the Foldable type class (I'd call it fold but it's already taken, let's find another name)?

This issue to discuss whether we indeed want another Traversable using data Applicatives; and if we do, bikeshed the details (mainly the name, I guess).