What monads can be expressed as Free over some functor?

Question

The documentation for Free says:

A number of common monads arise as free monads,

• Given data Empty a, Free Empty

Answer 1

All monads can be expressed as the Free Monad. Their additional properties (like MonadFix and Cont) could be stripped off that way because the Free Monad is not a Free MonadFix or Free Cont.

The general way is to define Functor's fmap in terms of liftM and then wrap Free around that Functor. The resulting Monad can be reduced to the previous/actual Monad by specifying how the functions return and join (Pure and Impure) have to be mapped to the actual Monad.