What's the relationship between profunctors and arrows?

Question

Apparently, every Arrow is a Strong profunctor. Indeed ^>> and >>^ correspond to lmap and rmap. And fir

Answer 1

What profunctors lack compared to arrows is the ability to compose them. If we add composition, will we get an arrow?

MONOIDS

This is exactly the question tackled in section 6 of "Notions of Computation as Monoids," which unpacks a result from the (rather dense) "Categorical semantics for arrows". "Notions" is a great paper because while it dives deep into category theory it (1) doesn't assume the reader has more than a cursory knowledge of abstract algebra and (2) illustrates most of the migraine-inducing mathematics with Haskell code. We can briefly summarize section 6 of the paper here:

Say we have

class Profunctor p where
  dimap :: (contra' -> contra) -> (co -> co') -> p contra co -> p contra' co'

Your bog-standard, negative-and-positive dividin' encoding of profunctors in Haskell. Now this data type,

data (⊗) f g contra co = forall x. (f contra x) ⊗ (g x co)

as implemented in Data.Profunctor.Composition, acts like composition for profunctor. We can, for example, demonstrate a lawful instance Profunctor:

instance (Profunctor f, Profunctor g) => Profunctor (f ⊗ g) where
  dimap contra co (f ⊗ g) = (dimap contra id f) ⊗ (dimap id co g)

We will hand-wave the proof that it is lawful for reasons of time and space.

OK. Now the fun part. Say we this typeclass:

class Profunctor p => ProfunctorMonoid p where
  e :: (a -> b) -> p a b
  m :: (p ⊗ p) a b -> p a b

This is, with a lot more hand-waving, a way of encoding the notion of profunctor monoids in Haskell. Specifically this is a monoid in the monoidal category Pro, which is a monoidal structure for the functor category [C^op x C, Set] with ⊗ as the tensor and Hom as its unit. So there's a lot of ultraspecific mathematical diction to unpack here but for that you should just read the paper.

We then see that ProfunctorMonoid is isomorphic to Arrow ... almost.

instance ProfunctorMonoid p => Category p where
  id = dimap id id
  (.) pbc pab = m (pab ⊗ pbc)

instance ProfunctorMonoid p => Arrow p where
  arr = e
  first = undefined

instance Arrow p => Profunctor p where
  lmap = (^>>)
  rmap = (>>^)

instance Arrow p => ProfunctorMonoid p where
  e = arr
  m (pax ⊗ pxb) = pax >> pxb

Of course we are ignoring the typeclass laws here but, as the paper shows, they do work out fantastically.

Now I said almost because crucially we were unable to implement first. What we have really done is demonstrated an isomorphism between ProfunctorMonoid and pre-arrows .The paper calls Arrow without first a pre-arrow. It then goes on to show that

class Profunctor p => StrongProfunctor p where
  first :: p x y -> p (x, z) (y, z)

class StrongProfunctor p => StrongProfunctorMonoid p where
  e :: (a -> b) -> p a b
  m :: (p ⊗ p) a b -> p a b

is necessary and sufficient for the desired isomorphism to Arrow. The word "strong" comes from a specific notion in category theory and is described by the paper in better writing and richer detail than I could ever muster.

So to summarize:

• A monoid in the category of profunctors is a pre-arrow, and vice versa. (A previous version of the paper used the term "weak arrows" instead of pre-arrows, and that's OK too I guess.)

• A monoid in the category of strong profunctors is an arrow, and vice versa.

• Since monad is a monoid in the category of endofunctors we can think of the SAT analogy Functor : Profunctor :: Monad : Arrow. This is the real thrust of the notions-of-computation-as-monoids paper.

• Monoids and monoidal categories are gentle sea creatures that appear everywhere, and it's a shame that some students will go through computer science or software engineering education without being taught monoids.

• Category theory is fun.

• Haskell is fun.