If I find out that something is a Monoid or Monad, I get all to use all kinds of fun functions, like foldMap, sequence or even mapM. They make me happy.
What do I get if I find out that something is a Category? Do I get anything fun besides overloading id and (.) ?
Do I get anything fun besides overloading id and (.) ?
Well, you also (should) get the laws associated with them:
"identity/left" forall p .
id . p = p
"identity/right" forall p .
p . id = p
"association" forall p q r .
(p . q) . r = p . (q . r)
As a side note:
The reason that you don't get that many fun functions like mapM or foldMap is, that saying something is a category is actually saying very little about it. The most useful functions I guess are the ones defined in Control.Category, which sometimes make code easier to read: >>> and <<<