[kmill]

I checked that while I was writing my comment, and I don't believe there is such an isomorphism. For the identity monad, we have the following functions:

   return :: a -> a
   return = id

   join :: a -> a
   join = id

If this were isomorphic to the function composition monoid, there would be some way to interpret join as function composition, but I don't see it. Please show me if you know the isomorphism! (My thinking is a bit fuzzy on this, but it appears to me that the identity monad is actually isomorphic to a trivial monoid --- if you want, you can model the trivial monoid as the one-element set {0} under addition.)

Bind is

   (>>=) :: a -> (a -> b) -> b
   (>>=) = flip id

that is, function application. Still no function composer in sight, though.

[BoiledCabbage]

At this point I admit I'm way beyond my depth.

This article seems to imply a correlation between the fish operator (<=<) and composition. Specifically his section on Kleisli monad.

http://www.haskellforall.com/2012/08/the-category-design-pat...

But I don't know the space well enough to know if that's answering your question.

[kmill]

It's true that the Kleisli category gives a composition law from a monad, but let's not lose sight: the point isn't whether there is a conceivable way to compose things, but whether a monad is just a monoid of functions under composition.

Monads are like a monoid at the type level plus a bunch of coherence rules. For every type a, there is a "composition" m (m a) -> m a and a "unit" a -> m a. (Compare with a monoid, where there is a composition m x m -> m and a unit {1} -> m.) Also, the composition must be natural in the sense that whenever there is a map f :: a -> b then you have a bunch of "commuting squares": the composition m (m a) -> m (m b) -> m b must equal m (m a) -> m a -> m b and the composition a -> m a -> m b must equal a -> b -> m b. (Some of these maps are fmap f or fmap (fmap f).)

The naturality thing is important and shows up quite a lot, and category theory was invented to understand naturality. It's sort of a higher higher order functional programming.

[BoiledCabbage]

Thanks this was helpful. And you're right. I was blurring the concept of composition and actual functional composition.

Clearly much more to learn.