The Functor Family: Profunctor

Jan 22, 2020

This post assumes prior knowledge of - Contravariant - Bifunctor

## Why

We've seen how types of kind * -> * can have instances for Functor or Contravariant, depending on the position of the type argument. We have also seen that types of kind * -> * -> * can have Bifunctor instances. These types are morally Functor in both type arguments. We're left with one very common type which we can't map both arguments of: a -> b. It does have a Functor instance for b, but the a is morally Contravariant (so it can't have a Bifunctor instance). This is where Profunctors come in.

Here's a list of a few common types with the instances they allow:

Type Functor Bifunctor Contravariant Profunctor
Maybe a
[a]
Either a b
(a,b)
Const a b
Predicate a
a -> b

Although there are some exceptions, you will usually see Contravariant or Profunctor instances over function types. Predicate itself is a newtype over a -> Bool, and so are most types with these instances.

Let's take a closer look at a -> b. We can easily map over the b, but what about the a? For example, given showInt :: Int -> String, what do we need to convert this function to showBool :: Bool -> String:

showInt :: Int -> String
showInt = show

showBool :: Bool -> String
showBool b = _help

We would have access to: - showInt :: Int -> String - b :: Bool and we want to use showInt, so we would need a way to pass b to it, which means we'd need a function f :: Bool -> Int and then _help would become showInt (f b).

But if we take a step back, in order to go from Int -> String to Bool -> String, we need Bool -> Int, which is exactly the Contravariant way of mapping types.

Exercise 1: Implement a mapInput function like:

mapInput :: (input -> out) -> (newInput -> input) -> (newInput -> out)

Extra credit: try a pointfree implementation as mapInput = _.

Exercise 2: Try to guess how the Profunctor class looks like. Look at Functor, Contravariant, and Bifunctor for inspiration.

class Profunctor p where

Exercise 3: Implement an instance for -> for your Profunctor class.

instance Profunctor (->) where

## How

Unlike Functor, Contravariant, and Bifunctor, the Profunctor class is not in base/Prelude. You will need to bring in a package like profunctors to access it.

class Profunctor p where
{-# MINIMAL dimap | lmap, rmap #-}
dimap :: (c -> a) -> (b -> d) -> p a b -> p c d
lmap :: (c -> a) -> p a b -> p c b
rmap :: (b -> c) -> p a b -> p a c

dimap takes two functions and is able to map both arguments in a type of kind * -> * -> *. lmap is like mapInput. second is always the same thing as fmap.

Exercise 4: implement dimap in terms of lmap and rmap.

Exercise 5: implement lmap and rmap in terms of dimap.

Exercise 6: implement the Profunctor instance for ->:

instance Profunctor (->) where
-- your pick: dimap or lmap and rmap

Exercise 7: (hard) implement the Profunctor instance for:

data Sum f g a b
= L (f a b)
| R (g a b)

instance (Profunctor f, Profunctor g) => Profunctor (Sum f g) where

Exercise 8: (hard) implement the Profunctor instance for:

newtype Product f g a b = Product (f a b, g a b)

instance (Profunctor f, Profunctor g) => Profunctor (Product f g) where

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