DAY
Kleisli Endomorphisms


Apr 8, 2017

After listening to the latest Magic Read-along episode "You should watch this" (which you should go listen to now) I got caught up thinking about Brian's idea of an Endomorphism version of Kleisli composition for use with Redux, it's actually a very similar model to what I'm using in my event framework for event listeners so I figured I'd try to formalize the pattern and recognize some of the concepts involved. They talk about the idea of a Redux-reducer, which is usually of type s -> Action -> s, it takes a state and an action and returns a new state. He then re-arranged the arguments to Action -> s -> s. He then recognized this as Action -> Endo s (an Endo-morphism is just any function from one type to itself: a -> a). He would take his list of reducers and partially apply them with the Action, yielding a list of type Endo s where s is the state object the reducer operates over. At this point we can use the Monoid instance Endo has defined, so we foldmap with Endo to combine the list of reducers into a sort of pipeline where each function feeds into the next; the Endo instance of Monoid is just function composition over functions which return the same type as their input.

This cleans up the interface of the reducers a fair amount, but what about an alternate kind of Endo which uses Kleisli composition instead of normal function composition? Kleisli composition often referenced as (>=>); takes two functions which return monads and composes them together using the underlying bind/flatmap of the Monad. The type of Kleisli composition is: (>=>) :: Monad m => (a -> m b) -> (b -> m c) -> a -> m c. If we could define a nice Endo-style monoid over this type then we could compose reducers like we did above, but also allow the functions to perform monadic effects (which is a bad idea in Redux, but there are other times this would be useful, imagine running a user through a pipeline of transformations which interact with a database or do some error handling). We can easily define this instance like so:

import Control.Monad
newtype KEndo m a = KEndo
  { getKEndo :: (a -> m a) }
  
instance Monad m => Monoid (KEndo m a) where
  mempty = KEndo return
  (KEndo a) `mappend` (KEndo b) = KEndo (a >=> b)  

This is great, now if we have a list of functions of some type [User -> Writer Error User] or something we can use foldmap to combine them into a single function! It works like this:

actions :: [User -> Writer Error User]
actions = [...]

pipeline :: User -> Writer Error User
pipeline = getKEndo . foldMap KEndo $ actions

The whole Kleisli Endo thing is a cool idea; but this thing has actually been done before! It's actually the same as the StateT state monad transformer from mtl; let's see how we can make the comparison. A generic Endo is of type s -> s, this is isomorphic to s -> ((), s), aka State s (). The trick is that the Kleisli Endo (s -> m s or by isomorphism s -> m ((), s)) can actually be generalized over the () to s -> m (a, s) which incidentally matches runStateT :: StateT s m a -> s -> m (a, s) from mtl!

So basically KEndo is isomorphic to StateT, but we'd still like a monoid instance for it, Gabriel shows a monoid over the IO monad in "Applied category theory and abstract algebra", the Monoid he shows actually generalizes to any monad as this instance:

instance (Monad m, Monoid a) => Monoid (m a) where
  mempty = return mempty
  ma `mappend` mb = do
    a <- ma
    b <- mb
    return (a `mappend` b)

So that means we can use this instance for StateT (which is a monad). Since () is a trivial monoid (where every mappend just returns ()) the simple case is State s () which was our KEndo of s -> ((), s) but now we have the Monoid instance, which behaves the same as the KEndo instance, so we don't need KEndo anymore. If we want to allow arbitrary effects we use the Transformer version: StateT s m () where m is a monad containing any additional effects we want. In addition to being able to add additional effects we also gain the ability to aggregate information as a monoid! If you decided you wanted your reducers to also aggregate some form of information, then they'd be: Monoid a => Action -> s -> (a, s), which is Action -> State s a, and if a is a monoid, then the monoid instance of State acts like Endo, but also aggregates the 'a's along the way!

Lastly we recognize that in the case of the Redux Reducers, if we have a whole list of reducers: Action -> State s () then we can rephrase it as the ReaderT Monad: ReaderT Action (State s) (), which maintains all of the nice monoids we've set up so far, and becomes even more composable!