On Wed, Feb 22, 2012 at 1:40 AM, wren ng thornton
It's a category-theoretic product, but not for the category of domains. Let Set be the category of sets and set-theoretic functions. And let pDCPO be the category of (pointed) domains and their homomorphisms.
The (domain-theoretic) domain product was discussed in my previous message to MigMit. It's a category-theoretic product in Set (among other categories) because the necessary morphisms exist and they satisfy the necessary equations. Moreover, in Set the domain product coincides with the cartesian product (we just forget about the orderings on the input domains and the resulting product). Hence, since the cartesian product is a category-theoretic product for Set, we know that the domain product must be a category-theoretic product in Set.
Well, like Miguel, I don't see the problem with choosing the ordering: (a,b) <= (a', b') iff a <= a' and b <= b' This makes (a0, b0) the bottom element. The only difference with Haskell's tuple types is that it lacks an extra element below (a0, b0); it's unlifted. It also seems to me that this _is_ the correct product for domains, unless I'm still sketchy on what you mean by domain. I don't think it matters that we're only considering strict homomorphisms.
Conversely, the smash product is a category-theoretic product in pDCPO, but not in Set. Since every domain homomorphism must map bottoms to bottoms, it follows that f(d0) = a0 and g(d0) = b0. From this we have the necessary continuity to ensure that C, pi_A, and pi_B all exist in pDCPO. However, since there exist set-theoretic functions f and g which do not have that special property, the smash product is going to lose information about the non-bottom component of the product and so it cannot satisfy the necessary category-theoretic equations (in Set).
The smash product is not a product for domains (in general), either.
You are not allowed to throw away the components for such a product,
because given a homomorphism f that maps everything to a0, pi_B .