Unlike a Functor, a Monad can change shape?

Question

I\'ve always enjoyed the following intuitive explanation of a monad\'s power relative to a functor: a monad can change shape; a functor cannot.

For example: length

Answer 1

The simplest type of a function satisfying the requirement I can imagine is this:

enigma :: Monad m => m () -> m ()

One can implement it in one of the following ways:

enigma1 m = m -- not changing the shape

enigma2 _ = return () -- changing the shape

This was a very simple change -- enigma2 just discards the shape and replaces it with the trivial one. Another kind of generic change is combining two shapes together:

foo :: Monad m => m () -> m () -> m ()
foo a b = a >> b

The result of foo can have shape different from both a and b.

A third obvious change of shape, requiring the full power of the monad, is a

join :: Monad m => m (m a) -> m a
join x = x >>= id

The shape of join x is usually not the same as of x itself.

Combining those primitive changes of shape, one can derive non-trivial things like sequence, foldM and alike.