I already tried this on Reddit, but there's no sign of a response - maybe it's the wrong place, maybe I'm too impatient. Anyway...

In a learning problem I've been messing around with, I realised I needed a typeclass for functions with operations for applying, composing etc. Reasons...

1. It can be convenient to treat a representation of a function as if it were the function itself, so that applying the function implicitly uses an interpreter, and composing functions derives a new description.

2. Once you have a typeclass for functions, you can have derived typeclasses for special kinds of functions - in my case, I want invertible functions.

For example, functions that apply integer offsets could be represented by an ADT containing an integer. Applying those functions just means adding the integer. Composition is implemented by adding the wrapped integers. The inverse function has the integer negated. The identity function wraps zero. The constant function cannot be provided because there's no suitable representation for it.

Of course it doesn't need to spell things as if it the values were genuine Haskell functions, but once I had the idea, I thought a library like that must already exist and maybe even using the standard spellings. But I can't find such a typeclass in the Haskell library.

I found the Data.Function module, but there's no typeclass - just some common functions that are also available from the Prelude.

So - why isn't there a typeclass for functions? Is it "just because there isn't" or "because it's not so useful as you think"? Or maybe there's a fundamental problem with the idea?

The biggest possible problem I've thought of so far is that function application on actual functions would probably have to be special-cased by the compiler to avoid a looping problem - in order to apply this function I need to apply the function application function, and to do that I need to call the function application function, and to do that...