Mikael Johansson wrote:
A way to categorify elements of objects in a cartesian closed category (such as that that sufficiently restricted Haskell takes place in) are to view entities of type A as maps () -> A.
Dan Weston wrote:
This rather inconveniently clashes with the fact that A and () -> A are two distinct types in Haskell.
Not really. It's just that in the terminology of category theory, there is no direct way to talk about "elements" of an object. In general, objects do not need to be sets, so there is no notion of an "element". In our case, where the objects - Haskell types - happen to be sets, we get our hands on "elements" of an object by using the trick of taking elements of the set of morphisms from () to the type. With functions, we have the opposite situation. Functions are first-class in Haskell. So for each function of type A -> B there are two representations in the category: as an "element" of the object A->B, and as an element of the set of morphisms from A to B.
I take it by your argument that curried and uncurried functions, being isomorphic, are represented by the same object in your category?
No. (A,B)->C and A->B->C are two different types, so they are two different objects in the category. The two objects are isomorphic via curry and uncurry. -Yitz