Haskell: Composition of morphisms in monoidal categories
I have the following definitions for a monoidal category class (Similar to the standard library, but providing inverses of the necessary natural isomorphisms):
class (Category r, Category s, Category t) => Bifunctor p r s t | p r -> s t, p s -> r t, p t -> r s where
bimap :: r a b -> s c d -> t (p a c) (p b d)
--
class (Bifunctor b k k k) => Associative k b where
associate :: k (b (b x y) z) (b x (b y z))
associateInv :: k (b x (b y z)) (b (b x y) z)
--
class (Bifunctor b k k k) => HasIdentity k b i | k b -> i
class (Associative k b, HasIdentity k b i) => Monoidal k b i | k b -> i where
idl :: k (b i a) a
idr :: k (b a i) a
idlInv :: k a (b i a)
idrInv :: k a (b a i)
--
The problem with composing morphisms in a monoidal category using (.) is that the objects may be associated differently. For instance Monoidal Hask (,) () , we might want to compose a morphism of type x -> ((a, b), c) with a morphism of type ((a, ()), (b, c)) -> y . To make the types fit, the natural isomorphism given by bimap idrInv id . associate bimap idrInv id . associate has to be applied.
Does the Haskell type system enable an automatic way of determining an appropriate isomorphism, based on the desired domain and and codomain type? I can't figure out how to do it.
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