**Introduction**

As Dan Doel points out here, the gadget `Free` that turns a functor into a monad is itself a kind of monad, though not the usual kind of monad we find in Haskell.
I'll call it a higher order monad and you can find a type class corresponding to this in various places including an old version of Ed Kmett's category-extras. I'll borrow some code from there.
I hunted around and couldn't find an implementation of `Free` as an instance of this class so I thought I'd plug the gap.

> {-# LANGUAGE RankNTypes, FlexibleContexts, InstanceSigs, ScopedTypeVariables #-}To make things unambiguous I'll implement free monads in the usual way here:

> import Control.Monad > import Data.Monoid

> data Free f a = Pure a | Free (f (Free f a))The usual Haskell typeclass

> instance Functor f => Functor (Free f) where > fmap f (Pure a) = Pure (f a) > fmap f (Free a) = Free (fmap (fmap f) a)

> instance Functor f => Monad (Free f) where > return = Pure > Pure a >>= f = f a > Free a >>= f = Free (fmap (>>= f) a)

`Monad`corresponds to monads in the category of types and functions,

`Hask`. We're going to want monads in the category of endomorphisms of

`Hask`which I'll call

`Endo`.

The objects in `Endo` correspond to Haskell's `Functor`.
The arrows in `Endo` are the natural transformations between these functors:

> type Natural f g = (Functor f, Functor g) => forall a. f a -> g aSo now we are led to consider functors in

`Endo`.

> class HFunctor f whereA functor in

`Endo`must map functors in

`Hask`to functors in

`Hask`. So if

`f`is a functor in

`Endo`and

`g`is a functor in

`Hask`, then

`f g`must be another functor in

`Hask`. So there must be an

`fmap`associated with this new functor. There's an associated

`fmap`for every

`g`and we collect them all into one big happy natural family:

> ffmap :: Functor g => (a -> b) -> f g a -> f g bBut note also that by virtue of being a functor itself,

`f`must have its own

`fmap`type function associated with it. The arrows in

`Endo`are natural transformations in

`Hask`so the

`fmap`for

`HFunctor`must take arrows in

`Endo`to arrows in

`Endo`like so:

> hfmap :: (Functor g, Functor h) => Natural g h -> Natural (f g) (f h)Many constructions in the category

`Hask`carry over to

`Endo`. In

`Hask`we can form a product of type types

`a`and

`b`as

`(a, b)`. In

`Endo`we form the product of two functors

`f`and

`g`as

> data Product f g a = Product (f (g a))Note that this product isn't commutative. We don't necessarily have an isomorphism from

`Product f g`to

`Product g f`. (This breaks many attempts to transfer constructions from

`Hask`to

`Endo`.) We also won't explicitly use

`Product`because we can simply use the usual Haskell composition of functors inline.

We can implement some functions that act on product types in both senses of the word "product":

> left :: (a -> c) -> (a, b) -> (c, b) > left f (a, b) = (f a, b)(Compare with what I wrote here.)

> right :: (b -> c) -> (a, b) -> (a, c) > right f (a, b) = (a, f b)

> hleft :: (Functor a, Functor b, Functor c) => Natural a c -> a (b x) -> c (b x) > hleft f = f

> hright :: (Functor a, Functor b, Functor c) => Natural b c -> a (b x) -> a (c x) > hright f = fmap f

We have something in `Endo` a bit like the type with one element in `Hask`, namely the identity functor.
The product of a type `a` with the one element type in `Hask` gives you something isomorphic to `a`.
In `Endo` the product is composition for which the identity functor is the identity.
(Two different meanings of the word "identity" there.)

We also have sums.
For example, if we define a functor like so

> data F a = A a | B a awe can think of

`F`as a sum of two functors: one with a single constructor

`A`and another with constructor

`B`.

We can now think about reproducing an `Endo` flavoured version of lists.
The usual definition is isomorphic to:

> data List a = Nil | Cons a (List a)And it has a

`Monoid`instance:

> instance Monoid (List a) where > mempty = Nil > mappend Nil as = as > mappend (Cons a as) bs = Cons a (mappend as bs)We can try to translate that into

`Endo`. The

`Nil`part can be thought of as being an element of a type with one element so it should become the identity functor. The

`Cons a (List a)`part is a product of

`a`and

`List a`so that should get replaced by a composition. So we expect to see something vaguely like:

List' a = Nil' | Cons' (a (List' a))That's not quite right because

`List' a`is a functor, not a type, and so acts on types. So a better definition would be:

List' a b = Nil' b | Cons' (a (List' a b))That's just the definition of

`Free`. So free monads are lists in

`Endo`. As everyone knows :-) monads are just monoids in the category of endofunctors. Free monads are also just free monoids in the category of endofunctors.

So now we can expect many constructions associated with monoids and lists to carry over to monads and free monads.

An obvious one is the generalization of the singleton map `a -> List a`:

> singleton :: a -> List a > singleton a = Cons a NilAnother is the generalization of

> hsingleton :: Natural f (Free f) > hsingleton f = Free (fmap Pure f)

`foldMap`. This can be found under a variety of names in the various free monad libraries out there but this implementation is designed to highlight the similarity between monoids and monads:

> foldMap :: Monoid m => (a -> m) -> List a -> m > foldMap _ Nil = mempty > foldMap f (Cons a as) = uncurry mappend $ left f $ right (foldMap f) (a, as)The similarity here isn't simply formal. If you think of a list as a sequence of instructions then

> fold :: Monoid m => List m -> m > fold = foldMap id

> hFoldMap :: (Functor f, Functor m, Monad m) => Natural f m -> Natural (Free f) m > hFoldMap _ (Pure x) = return x > hFoldMap f (Free x) = join $ hleft f $ hright (hFoldMap f) x

> hFold :: Monad f => Natural (Free f) f > hFold = hFoldMap id

`foldMap`interprets the sequence of instructions like a computer program. Similarly

`hFoldMap`can be used to interpret programs for which the free monad provides an abstract syntax tree.

You'll find some of these functions here by different names.

Now we can consider `Free`. It's easy to show this is a `HFunctor` by copying a suitable definition for `List`:

> instance Functor List where > fmap f = foldMap (singleton . f)We can define

> instance HFunctor Free where > ffmap = fmap > hfmap f = hFoldMap (hsingleton . f)

`HMonad`as follows:

> class HMonad m where > hreturn :: Functor f => f a -> m f a > hbind :: (Functor f, Functor g) => m f a -> Natural f (m g) -> m g aBefore making

`Free`an instance, let's look at how we'd make

`List`an instance of

`Monad`

> instance Monad List where > return = singleton > m >>= f = fold (fmap f m)And now the instance I promised at the beginning.

> instance HMonad Free where > hreturn = hsingleton > hbind m f = hFold (hfmap f m)I've skipped the proofs that the monad laws hold and that

`hreturn`and

`hbind`are actually natural transformations in

`Endo`. Maybe I'll leave those as exercises for the reader.

**Update**

After writing this I tried googling for "instance HMonad Free" and I found this by haasn. There's some other good stuff in there too.

There's a similar trick for Prompt, the free monad for a type constructor.

ReplyDeleteThere's a similar trick for Prompt, the free monad for a unary type constructor.

ReplyDeleteWhats that :< symbol in the liked source mean?

ReplyDeleteSlight typo, where it says "the category of endomorphisms", it should say "the category of endofunctors", where the objects are endofunctors and the morphisms are natural transformations.

ReplyDeleteWhat you are looking for is the free package on hackage.

ReplyDeleteControl.Monad.Free is there alongside free Applicatives and Alternatives and other encodings of the free monad and free monad transformer.

The current closest thing to a home for HMonad and HFunctor is probably index-core by Gabriel Gonzales.

ReplyDeletehttp://hackage.haskell.org/package/index-core-1.0.1/docs/Control-IMonad-Core.html

I have work partially done on a replacement for indexed, but it is currently hung up waiting for GHC to fix Any so that it doesn't inhabit every kind, so we can have proper product kinds.

https://github.com/ekmett/indexed