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
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