What is the difference between Fix, Mu and Nu in Ed Kmett's recursion scheme package

In Ed Kmett's recursion-scheme package, there are three declarations:

newtype Fix f = Fix (f (Fix f))

newtype Mu f = Mu (forall a. (f a -> a) -> a)

data Nu f where 
  Nu :: (a -> f a) -> a -> Nu f

What is the difference between those three data types?

Mu represents a recursive type as its fold, and Nu represents it as its unfold. In Haskell, these are isomorphic, and are different ways to represent the same type. If you pretend that Haskell doesn't have arbitrary recursion, the distinction between these types becomes more interesting: Mu f is the least (initial) fixed point of f, and Nu f is its greatest (terminal) fixed point.

A fixed point of f is a type T an isomorphism between T and f T, i.e. a pair of inverse functions in :: f T -> T, out :: T -> f T. The type Fix just uses Haskell's built-in type recursion to declare the isomorphism directly. But you can implement in/out for both Mu and Nu.

For a concrete example, pretend for a moment that you can't write recursive values. The inhabitants of Mu Maybe , i.e. values :: forall r. (Maybe r -> r) -> r, are the naturals, {0, 1, 2, ...}; the inhabitants of Nu Maybe, i.e. values :: exists x. (x, x -> Maybe x), are the conaturals {0, 1, 2, ..., ∞}. Think about the possible values of these types to see why Nu Maybe has an extra inhabitant.

If you want to get some intuition for these types, it can be a fun exercise to implement the following without recursion (roughly in increasing order of difficulty):

• zeroMu :: Mu Maybe, succMu :: Mu Maybe -> Mu Maybe
• zeroNu :: Nu Maybe, succNu :: Nu Maybe -> Nu Maybe, inftyNu :: Nu Maybe
• muTofix :: Mu f -> Fix f, fixToNu :: Fix f -> Nu f
• inMu :: f (Mu f) -> Mu f, outMu :: Mu f -> f (Mu f)
• inNu :: f (Nu f) -> Nu f, outNu :: Nu f -> f (Nu f)

You can also try to implement these, but they require recursion:

• nuToFix :: Nu f -> Fix f, fixToMu :: Fix f -> Mu f

Mu f is the least fixed point, and Nu f is the greatest, so writing a function :: Mu f -> Nu f is very easy, but writing a function :: Nu f -> Mu f is hard; it's like swimming against the current.

(At one point I was meaning to write a more detailed explanation of these types, but it might be a little too long for this format.)