On Sat, Jan 31, 2009 at 1:02 PM, Ben Moseley
You can view a polymorphic unary type constructor of type ":: a -> T" as a polymorphic function.
Shouldn't that be * :: a -> T a ?
In general, polymorphic functions correspond roughly to natural transformations (in this case from the identity functor to T).
Are you saying a type constructor is a nat trans and not a functor (component)? Seems much more like a plain ol' functor mapping of object to object to me - the objects being types. Can you clarify what you mean about the correspondence with natural transformations? I admit I haven't thought through "polymorphic function", mainly because there doesn't seem to be any such beast in category theory, and to be honest I've always thought the metaphor is misleading. After all, a function by definition cannot be polymorphic. It seems like fancy name for a syntactic convenience to me - a way to express /intensionally/ a set of equations without writing them all out explicitly. Thanks, gregg
--Ben
On 31 Jan 2009, at 17:00, Gregg Reynolds wrote:
Hi,
I think I've finally figured out what a monad is, but there's one thing I haven't seen addressed in category theory stuff I've found online. That is the relation between type constructors and data constructors.
As I understand it, a type constructor Tcon a is basically the object component of a functor T that maps one Haskell type to another. Haskell types are construed as the objects of category "HaskellType". I think that's a pretty straightforward interpretation of the CT definition of functor.
But a data constructor Dcon a is an /element/ mapping taking elements (values) of one type to elements of another type. So it too can be construed as a functor, if each type itself is construed as a category.
So this gives us two functors, but they operate on different things, and I don't see how to get from one to the other in CT terms. Or rather, they're obviously related, but I don't see how to express that relation formally.
If somebody could point me in the right direction I'd be grateful. Might even write a tutorial. Can't have too many of those.
Thanks,
Gregg _______________________________________________ Haskell-Cafe mailing list Haskell-Cafe@haskell.org http://www.haskell.org/mailman/listinfo/haskell-cafe