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 … Read more
(N.B. this combines a bit from both mine and @Gabriel’s comments above.) It’s possible for every inhabitant of the Fixed point of a Functor to be infinite, i.e. let x = (Fix (Id x)) in x === (Fix (Id (Fix (Id …)))) is the only inhabitant of Fix Identity. Free differs immediately from Fix in … Read more
You are doing nothing wrong. fix id is an infinite loop. When we say that fix returns the least fixed point of a function, we mean that in the domain theory sense. So fix (\x -> 2*x-1) is not going to return 1, because although 1 is a fixed point of that function, it is … Read more