It's serendipitous that I'm seeing this blog post on the front page today, because I'm currently writing an article discussing the free monad.
In addition to the free monad presented in this post, there is a variant, called the "freer" monad, based on the "bind" operation instead of the "join" operation:
data Freer f a where
Pure :: a -> Freer f a
Bind :: f a -> (a -> Freer f b) -> Freer f b
When thinking of monads as giving the semantics of some computational strategy, it's easier to define them in terms of "bind" instead of "join." This way of defining monads is sometimes called a "Kleisli triple" because it is better suggestive of "Kleisli arrows," or functions of the signature `a -> m b`. The "bind" operation defines how to compose a monadic computation with its continuation, and from this perspective, the "freer" monad resembles an abstract syntax tree.
Originally, Eugenio Moggi proposed monads as a technique for specifying the denotational semantics of programming languages. All Java programs "really" happen in the IO + Either monads, because all Java programs may perform IO and throw exceptions. To my understanding, free monads are the monad that OCaml 5 runs in, because they give the semantics for effect handlers (or resumable exceptions).
> All Java programs "really" happen in the IO + Either monads
People say things like this all the time, and I think it's a vacuous assertion. While there is probably some very narrow view where returning early with an error, throwing an exception, and binding in Either are the same thing, such a view ignores a lot of important context (e.g. all of imperative programming). This is why you have to qualify it as "IO + Either", but that doesn't say anything at all because everything is possible in IO.