Re: [Haskell-cafe] Monad laws in presence of bottoms

On 2/21/12 11:54 AM, Dan Doel wrote:

On Tue, Feb 21, 2012 at 10:44 AM, wren ng thornton wrote:

...

That's a similar sort of issue, just about whether undefined == (undefined,undefined) or not. If the equality holds then tuples would be domain products[1], but domain products do not form domains! ... [1] Also a category-theoretic product.

This doesn't make much sense to me, either. If it's a category theoretic product in a category of domains, then the product must be a domain, no?

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 category-theoretic product of A and B is just some triple (C, pi_A : C -> A, pi_B : C -> B) such that forall D, forall f : D -> A, and forall g : D -> B, there exists a unique h : D -> C such that f = pi_A . h and g = pi_B . h ---in other words, it's the limit of a functor from the category with two objects and no morphisms which maps one object to A and the other object to B. 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. However, the domain product is not a category-theoretic product in pDCPO. First off, the objects in pDCPO are domains, but since the domain product of A and B has no bottom it can't be a domain, so it can't be an object in pDCPO. Moreover, the morphisms in pDCPO will preserve the domain structure--- but there's no way to do that for a map from some domain C to the domain product of A and B, because there's no way to map the bottom of C to the bottom of the domain product (because there isn't one!). 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). For more on the category-theoretic study of domain theory, see http://www.cs.bham.ac.uk/~axj/pub/papers/handy1.pdf. You can search the pdf for "pointed dcpo" to jump to the relevant sections. Of course there are many different category-theoretic formalizations of domain theory, pDCPO is just one of them. To get a survey of the terrain, you may want to take a look at http://seclab.web.cs.illinois.edu/wp-content/uploads/2011/04/Gunter85.pdf. For a more direct introduction to domain theory and its application to lazy languages, see Geoffrey Burn's _Lazy Functional Languages: Abstract Interpretation and Compilation_. -- Live well, ~wren