The following article is just a few notes on the nature of the Free monad.

``````> {-# LANGUAGE DeriveFunctor #-}
> {-# LANGUAGE GeneralizedNewtypeDeriving #-}
> {-# LANGUAGE UndecidableInstances #-}
>
> module FreeMaybe where
>

There can be just two values of type `Maybe a`: `Nothing` and `Just a`. Now let’s look at the free monad of `Maybe a`, `Free Maybe a`:

``````> data Free f a = Pure a | Free (f (Free f a))
>
> instance Functor f => Functor (Free f) where
>     fmap f (Pure a)   = Pure (f a)
>     fmap f (Free ffa) = Free \$ fmap (fmap f) ffa
>
> instance Functor f => Monad (Free f) where
>     return = Pure
>     Pure a >>= f = f a
>     Free ffa >>= f = Free \$ fmap (>>= f) ffa
>
> instance (Show a, Show (f (Free f a))) => Show (Free f a) where
>     showsPrec d (Pure a) = showParen (d > 10) \$
>         showString "Pure " . showsPrec 11 a
>     showsPrec d (Free m) = showParen (d > 10) \$
>         showString "Free " . showsPrec 11 m``````

There are four “shapes” that values of `Free Maybe a` can take:

``````Pure a
Free Nothing
Free (Just (Free (Just (... (Free Nothing)))))
Free (Just (Free (Just (... (Free (Pure a))))))``````

In terms of whether a `Free Maybe a` represents an `a` or not, `Free Maybe a` is equivalent to `Maybe a`. However, `Maybe a` is right adjoint to `Free Maybe a`, meaning that it forgets the structure of `Free Maybe a` – namely, which of the four shapes above the value was, and how many occurences of `Free (Just` there were.

Why would you ever use `Free Maybe a`? Precisely if you cared about the number of Justs. Now, say we had a functor that carried other information:

``````> data Info a = Info { infoExtra :: String, infoData :: a }
>     deriving (Show, Functor)``````

Then `Free Info a` is isomorphic to if `infoExtra` had been `[String]`:

``````> main :: IO ()
> main = do
>     print \$ Free (Info "Hello" (Free (Info "World" (Pure "!"))))``````

Which results in:

``````>>> main
Free (Info {infoExtra = "Hello",
infoData = Free (Info {infoExtra = "World", infoData = Pure "!"})})
it :: ()``````

But now it’s also a `Monad`, even though we never defined a `Monad` instance for `Info`:

``````> main :: IO ()
> main = do
>     print \$ do
>         x <- Free (Info "Hello" (Pure "!"))
>         y <- Free (Info "World" (Pure "!"))
>         return \$ x ++ y``````

This outputs:

``````>>> foo
Free (Info {infoExtra = "Hello",
infoData = Free (Info {infoExtra = "World", infoData = Pure "!!"})})
it :: ()``````

This works because the Free monad simply accumulates the states of the various functor values, without “combining” them as a real monadic join would have done. `Free Info a` has left it up to us to do that joining later.