[Haskell-cafe] determine a type by universal property

[Joachim Breitner]

Hi,

Am Donnerstag, den 17.01.2019, 14:33 +0100 schrieb Olaf Klinke:
> I have two type signatures where I conjecture the only functors satisfying these
> are the Identity functor and functors of the form (,)s. Can anyone give hints as
> how to tackle a proof?
> 
> Signature 1: What functors t admit a function
> forall f. Functor f => t (f a) -> f (t a)

What about the functor 

   data Void1 a

It seems I can write

   g :: forall f. Functor f => t (f a) -> f (t a)
   g x = case x of {}

but Void1 is not the identity. (I guess it is `(,) Void`, if you want…)

So if you allow the latter, let’s try a proof. Assume we have t, and 

   g :: forall f. Functor f => t (f a) -> f (t a)

We want to prove that there is an isomorphism from t to ((,) s) for
some type s. Define

   s = t ()

(because what else could it be.) Now we need functions

   f1 :: forall a. t a -> (t (), a)
   f2 :: forall a. (t (), a) -> t a

that are isomorphisms. Here is one implementation:
  
   f1 :: forall a. t a -> (t (), a)
   f1 = swap  . g . fmap (λx → (x,()))

   {- note:
     x1 :: t a
     x2 :: t ((,) a ())
     x2 = fmap (λx → (x,())) x1
     x3 :: (,) a (t ())
     x3 = g x2
     x4 :: (t (), a)
     x4 = swap x3
   -}

   f2 :: forall a. (t (), a) -> t a
   f2 (s, x) = x <$ s

Now, are these isomorphisms? At this point I am not sure how to
proceed… probably invoking the free theorem of g? But even that will
not work nicely. Consider

   type T = (,) Integer
   g :: forall f. Functor => T (f a) -> f (T a)
   g (c, fx) = (,) (c + 1) <$> fx

here we count the number of invocations of g. Surely with this g,
f2 . f1 cannot be the identity.


Cheers,
Joachim
-- 
Joachim Breitner
  mail at joachim-breitner.de
  http://www.joachim-breitner.de/

-------------- next part --------------
A non-text attachment was scrubbed...
Name: signature.asc
Type: application/pgp-signature
Size: 833 bytes
Desc: This is a digitally signed message part
URL: <http://mail.haskell.org/pipermail/haskell-cafe/attachments/20190117/0c842bfd/attachment.sig>