[Haskell-cafe] type and data constructors in CT

[Ben Moseley]

You can view a polymorphic unary type constructor of type ":: a -> T"  
as a polymorphic function.

In general, polymorphic functions correspond roughly to natural  
transformations (in this case from the identity functor to T).

--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
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