[Haskell-cafe] Basic question concerning the category Hask (was: concerning data constructors)

[Jonathan Cast]

On 2 Jan 2008, at 5:49 AM, Yitzchak Gale wrote:

> Hi Andrew,
>
> Andrew Bromage wrote:
>> I still say it "isn't a set" in the same way that a group "isn't a  
>> set".
>> Haskell data types have structure that is respected by Haskell
>> homomorphisms.  Sets don't.
>
> Ah, that's certainly true. But what is that additional structure?
>
> In categories that have a forgetful functor to Set, the interesting
> part of their structure comes from the fact that their
> morphisms are only a proper subset of the morphisms
> in Set.
>
> So in what way are Set morphisms restricted from being
> Hask morphisms?

The normal view taken by Haskellers is that the denotations of  
Haskell types are CPPOs.  So:

(1) Must be monotone
(2) Must be continuous

(Needn't be strict, even though that messes up the resulting category  
substantially).

jcc