Endofunction as Monoid

Question

I\'m trying this (for learning purposes):

{-# LANGUAGE FlexibleInstances #-}

instance Monoid (a -> a) where
  mempty = id
  mappend f g = f . g
         


        

Answer 1

The Haskell Category class offers methods to work with categories whose objects are precisely the Haskell types of some kind. Specifically,

class Category c where
  id :: c x x
  (.) :: c y z -> c x y -> c x z

The names of the methods should look very familiar. Notably,

instance Category (->) where
  id x = x
  f . g = \x -> f (g x)

You probably know that monoids are semigroups with identities, expressed in Haskell using

class Monoid a where
  mappend :: a -> a -> a
  mempty :: a

But another mathematical perspective is that they're categories with exactly one object. If we have a monoid, we can easily turn it into a category:

-- We don't really need this extension, but
-- invoking it will make the code below more useful.
{-# LANGUAGE PolyKinds #-}

import Control.Category
import Data.Monoid
import Prelude hiding ((.), id)

newtype Mon m a b = Mon m

instance Monoid m => Category (Mon m) where
  id = Mon mempty
  Mon x . Mon y = Mon (x `mappend` y)

Going the other way is a little bit trickier. One way to do it is to choose a kind with exactly one type, and look at categories whose sole object is that type (prepare for yucky code, which you can skip if you like; the bit below is less scary). This shows that we can freely convert between a Category whose object is the type '() in the () kind and a Monoid. The arrows of the category become the elements of the monoid.

{-# LANGUAGE DataKinds, GADTs, PolyKinds #-}

data Cat (c :: () -> () -> *) where
  Cat :: c '() '() -> Cat c
instance Category c => Monoid (Cat c) where
  mempty = Cat id
  Cat f `mappend` Cat g = Cat (f . g)

But this is yucky! Ew! And pinning things down so tightly doesn't usually accomplish anything from a practical perspective. But we can get the functionality without so much mess, by playing a little trick!

{-# LANGUAGE GADTs, PolyKinds #-} 

import Control.Category
import Data.Monoid
import Prelude hiding ((.), id)

newtype Cat' (c :: k -> k -> *) (a :: k) (b :: k) = Cat' (c a b)

instance (a ~ b, Category c) => Monoid (Cat' c a b) where
  mempty = Cat' id
  Cat' f `mappend` Cat' g = Cat' (f . g)

Instead of confining ourselves to a Category that really only has one object, we simply confine ourselves to looking at one object at a time.

The existing Monoid instance for functions makes me sad. I think it would be much more natural to use a Monoid instance for functions based on their Category instance, using the Cat' approach:

instance a ~ b => Monoid (a -> b) where
  mempty = id
  mappend = (.)

Since there's already a Monoid instance, and overlapping instances are evil, we have to make do with a newtype. We could just use

newtype Morph a b = Morph {appMorph :: a -> b}

and then write

instance a ~ b => Monoid (Morph a b) where
  mempty = Morph id
  Morph f `mappend` Morph g = Morph (f . g)

and for some purposes maybe this is the way to go, but since we're using a newtype already we usually might as well drop the non-standard equality context and use Data.Monoid.Endo, which builds that equality into the type:

newtype Endo a = Endo {appEndo :: a -> a}

instance Monoid (Endo a) where
  mempty = Endo id
  Endo f `mappend` Endo g = Endo (f . g)