[Haskell-cafe] Representation of lenses

[Tom Ellis]

On Tue, Jan 27, 2015 at 05:18:41PM +0000, David Turner wrote:
> I believe that the types (a -> s, s -> a -> a) and (forall f.  Functor f
> => (s -> f s) -> a -> f a) are isomorphic.  Is that right?

Yes (with the caveat that I don't actually know how to prove this).

> It's also possible that you can't get the full generality of (forall
> f. (s -> f t) -> a -> f b) lenses with getter/setter pairs

No, they are equivalent.

> So, my question is: what is the advantage of representing lenses in
> the way that they are?

The advantage that's most obvious to me is polymorphism.  You can use a lens 

    forall f.  Functor f => (s -> f s) -> a -> f a

where the callee expects a traversal

    forall f.  Applicative f => (s -> f s) -> a -> f a

This is very convenient in practice.  Perhaps there are other practical
advantages that someone else can explain.

Tom