[Haskell-cafe] Categories (cont.)

[wren ng thornton]

On 12/20/12 7:07 PM, Christopher Howard wrote:
> On 12/20/2012 03:59 AM, wren ng thornton wrote:
>> In order to fake this theory in Haskell we can do:
>>
>>      newtype MonoidCategory a i j = MC a
>>
>>      instance Monoid a => Category (MonoidCategory a) where
>>          id          = MC mempty
>>          MC f . MC g = MC (f `mappend` g)
>>
>> This is a fake because technically (MonoidCategory A X Y) is a different
>> type than (MonoidCategory A P Q), but since the indices are phantom
>> types, we (the programmers) know they're isomorphic. From the category
>> theory side of things, we have K*K many copies of the monoid where K is
>> the cardinality of the kind "*". We can capture this isomorphism if we
>> like:
>>
>>      castMC :: MonoidCategory a i j -> MonoidCategory a k l
>>      castMC (MC a) = MC a
>>
>> but Haskell won't automatically insert this coercion for us; we gotta do
>> it manually. In more recent versions of GHC we can use data kinds in
>> order to declare a kind like:
>>
>>      MonoidCategory :: * -> () -> () -> *
>>
>> which would then ensure that we can only talk about (MonoidCategory a ()
>> ()). Unfortunately, this would mean we can't use the Control.Category
>> type class, since this kind is more restrictive than (* -> * -> * -> *).
>> But perhaps in the future that can be fixed by using kind polymorphism...
>
> Finally... I actually made some measurable progress, using these
> "phantom types" you mentioned:

Yep, everything should work fine with the phantom types. The only 
problem is, as I mentioned, that you're not really getting the category 
associated with the underlying monoid, you're getting a whole bunch of 
copies of that monoid (one copy for each instantiation of the phantom 
types).

-- 
Live well,
~wren