[Haskell-cafe] Monad laws in presence of bottoms

[wren ng thornton]

On 2/21/12 11:54 AM, Dan Doel wrote:
> On Tue, Feb 21, 2012 at 10:44 AM, wren ng thornton<wren at freegeek.org>  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