Concrete Type Example of a Functor that Fails to be an Applicative? [duplicate]

Question

From functors that are not applicatives:

A type constructor which is a Functor but not an Applicative. A simple example is a pair:

instance Functor ((,) r) where
    fmap f (x,y) = (x, f y)

But there is no way how to define its Applicative instance without imposing additional restrictions on r. In particular, there is no way how to define pure :: a -> (r, a) for an arbitrary r.

Here, pure fails to be definable for all types at once; however, for any concrete type T, one can make ((,) T) an applicative.

Question: Is there an example of a concrete functor (i.e., no type variables involved) that is a functor but not an applicative?

Answer 1

I don't have 50 reputation to comment here, so I'll try to do it as an answer:

however, for any concrete type T, one can make ((,) T) an applicative.

...

There's a theorem in mathematics that any collection with at least 2 elements can be made into a monoid. So for any concrete type T, it could in principle be made a member of Monoid, and then could in principle be made Applicative. What's wrong with this reasoning?

What about the tuple from the uninhabited type? (,) Void

It is a Functor,right?

Could you derive Applicative for it? How would pure be implemented?

Answer 2

There is nice example in reactive-banana library.

It features Event a types, which represents a single simultaneous event in time (think, an impulse), and Behavior a type, which represents a value that is available in any moment (for instance, emitting a value from the last event).

Behavior is an Applicative, because you can combine two of them - they have a value in any point of time.

Event, though, is a Functor only, because you can't combine them. Given two Events you can't be sure they will happen simultaneosly.