Is there a monad that doesn't have a corresponding monad transformer (except IO)?

Question

So far, every monad (that can be represented as a data type) that I have encountered had a corresponding monad transformer, or could have one. Is there such a monad that can

Answer 1

Here's a hand-wavy I'm-not-quite-sure answer.

Monads can be thought of as the interface of imperative languages. return is how you inject a pure value into the language, and >>= is how you splice pieces of the language together. The Monad laws ensure that "refactoring" pieces of the language works the way you would expect. Any additional actions provided by a monad can be thought of as its "operations."

Monad Transformers are one way to approach the "extensible effects" problem. If we have a Monad Transformer t which transforms a Monad m, then we could say that the language m is being extended with additional operations available via t. The Identity monad is the language with no effects/operations, so applying t to Identity will just get you a language with only the operations provided by t.

So if we think of Monads in terms of the "inject, splice, and other operations" model, then we can just reformulate them using the Free Monad Transformer. Even the IO monad could be turned into a transformer this way. The only catch is that you probably want some way to peel that layer off the transformer stack at some point, and the only sensible way to do it is if you have IO at the bottom of the stack so that you can just perform the operations there.