[Haskell-cafe] Natural Transformations and fmap

[Brent Yorgey]

On Mon, Jan 23, 2012 at 09:06:52PM -0800, Ryan Ingram wrote:
> On Mon, Jan 23, 2012 at 8:05 PM, Daniel Fischer <
> daniel.is.fischer at googlemail.com> wrote:
> 
> > On Tuesday 24 January 2012, 04:39:03, Ryan Ingram wrote:
> > > At the end of that paste, I prove the three Haskell monad laws from the
> > > functor laws and "monoid"-ish versions of the monad laws, but my proofs
> > > all rely on a property of natural transformations that I'm not sure how
> > > to prove; given
> > >
> > >     type m :-> n = (forall x. m x -> n x)
> > >     class Functor f where fmap :: forall a b. (a -> b) -> f a -> f b
> > >     -- Functor identity law: fmap id = id
> > >     -- Functor composition law fmap (f . g) = fmap f . fmap g
> > >
> > > Given Functors m and n, natural transformation f :: m :-> n, and g :: a
> > > -> b, how can I prove (f . fmap_m g) = (fmap_n g . f)?
> >
> > Unless I'm utterly confused, that's (part of) the definition of a natural
> > transformation (for non-category-theorists).
> >
> 
> Alright, let's pretend I know nothing about natural transformations and
> just have the type declaration
> 
> type m :-> n = (forall x. m x -> n x)
> 
> And I have
> f :: M :-> N
> g :: A -> B
> instance Functor M -- with proofs of functor laws
> instance Functor N -- with proofs of functor laws
> 
> How can I prove
>   fmap g. f :: M A -> N B
>   =
>   f . fmap g :: M A -> N B
> 
> I assume I need to make some sort of appeal to the parametricity of
> M :-> N.

This is in fact precisely the "free theorem" you get from the
parametricity of f.  Parametricity means that f must act "uniformly"
for all x -- which is an intuitive way of saying that f really is a
natural transformation.

-Brent