Good examples of Not a Functor/Functor/Applicative/Monad?

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While explaining to someone what a type class X is I struggle to find good examples of data structures which are exactly X.

So, I request examples for:

A type constructor which is not a Functor.
A type constructor which is a Functor, but not Applicative.
A type constructor which is an Applicative, but is not a Monad.
A type constructor which is a Monad.

I think there are plenty examples of Monad everywhere, but a good example of Monad with some relation to previous examples could complete the picture.

I look for examples which would be similar to each other, differing only in aspects important for belonging to the particular type class.

If one could manage to sneak up an example of Arrow somewhere in this hierarchy (is it between Applicative and Monad?), that would be great too!

Edit Report

My style may be cramped by my phone, but here goes.

newtype Not x = Kill {kill :: x -> Void}

cannot be a Functor. If it were, we'd have

kill (fmap (const ()) (Kill id)) () :: Void

and the Moon would be made of green cheese.

Meanwhile

newtype Dead x = Oops {oops :: Void}

is a functor

instance Functor Dead where
  fmap f (Oops corpse) = Oops corpse

but cannot be applicative, or we'd have

oops (pure ()) :: Void

and Green would be made of Moon cheese (which can actually happen, but only later in the evening).

(Extra note: Void, as in Data.Void is an empty datatype. If you try to use undefined to prove it's a Monoid, I'll use unsafeCoerce to prove that it isn't.)

Joyously,

newtype Boo x = Boo {boo :: Bool}

is applicative in many ways, e.g., as Dijkstra would have it,

instance Applicative Boo where
  pure _ = Boo True
  Boo b1 <*> Boo b2 = Boo (b1 == b2)

but it cannot be a Monad. To see why not, observe that return must be constantly Boo True or Boo False, and hence that

join . return == id

cannot possibly hold.

Oh yeah, I nearly forgot

newtype Thud x = The {only :: ()}

is a Monad. Roll your own.

Plane to catch...

answered 2011-08-28T12:09:58.697

Good examples of Not a Functor/Functor/Applicative/Monad?

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