Is there a van Laarhoven representation of `Optional`

Question

Many types of optics have a van Laarhoven representation.

For example, a Lens of type Lens s t a b can be represented as:

 Functor f => (a -> f b) -> s -> f t

Similarly a Traversal, can be represented in a similar way, swapping the Functor constraint for Applicative:

 Applicative f => (a -> f b) -> s -> f t

Several optics frameworks, such as Monocle and Arrow define a type called Optional.

In Monocle's Optics heirarchy Optional fits between Lens and Traversal

As I understand it: If a Traversal is like a Lens that may have zero to many targets, then an Optional is like a Lens that may have zero to one targets.

In Monocle, Optional is defined as a pair of functions:

getOrModify :: s -> Either t a 
set :: (b, s) -> t

Comments in the Monocle source code suggest that it's also possible to represent an Optional "as a weaker PLens and weaker PPrism"

Is it possible to represent an Optional as a van Laarhoven function?

Larimer

Answer 1

There would be a way to represent it if the Functor/Applicative/Monad hierarchy were more fine-grained. In particular:

class Functor f => Pointed f where
    pure :: a -> f a

type Optional s t a b = forall f. Pointed f => (a -> f b) -> s -> f t

Note that the type would probably be named Affine in the lens library if that was neatly in the class hierarchy.

Cubbyhole

This is conveniently completely fine for my use case, which is a "just for fun" Scala Optics Library where I've written the Functor Hierarchy from scratch, and can therefore do whatever I like with it. Am I right in thinking that this suggests that If I also had a another type-class between Functor and Applicative with liftA2, but no pure, then that would define an optic with one to many targets? Does this have a name? – Larimer

It does suggest that. I have no idea what the optic would be called. – Cubbyhole

@Larimer That intermediate class is called Apply in the "semigroupoids" package. hackage.haskell.org/package/semigroupoids-5.3.3/docs/… – Wadleigh

... and the corresponding optic in the lens package is called Traversal1 – Pennington

@Cubbyhole How can we verify that this type Optional s t a b is indeed the correct representation for that optic, and that all the appropriate laws hold? This seems like a magical trick. Why was it Pointed and not, say, Copointed? For another given optic, how can we guess the correct profunctor representation? – Bilek

@Bilek If you write a bunch of van Laarhoven lenses and traversals by hand (very highly recommended - they're not magic), you eventually get a handle on how the operations correspond to the number of targets present in a data structure. fmap handles a structure with a single target. pure handles a structure with no targets. (<*>) combines substructures that have each handled targets. From this you can observe that something with only fmap and pure will let you handle 0 or 1 targets, but not more than 1. And that's Pointed. – Cubbyhole

As for Profunctor optics... Dunno. I've never really seen the appeal, so I haven't written enough of them by hand to see what the blocks are. My impression from the little I have done is that they don't seem as amenable to piecewise breakdowns. (That impression may be wrong.) – Cubbyhole