存在量化类型类

Question

What is the type-class equivalent to the following existentially quantified dictionary, inspired by the Pipe type:

{-# LANGUAGE ExistentialQuantification, PolymorphicComponents #-}

data PipeD p = forall cat . PipeD {
    isoI        :: forall a b   m r . Iso (->) (p a b m r) (cat m r a b),
    categoryI   :: forall       m r . (Monad m) => CategoryI (cat m r)  ,
    monadI      :: forall a b   m   . (Monad m) => MonadI (p a b m)     ,
    monadTransI :: forall a b       . MonadTransI (p a b)               }

The rough idea I'm going for is trying to say that given the (PipeLike p) constraint, we can then infer (MonadTrans (pab), Monad (pabm) and (using pseudo-code) (Category "\\ab -> pabmr") .

The CategoryI and MonadI are just the dictionary equivalents of those type-classes that I use to express the idea that Category , Monad , and MonadTrans are (sort of) super-classes of this PipeLike type.

The Iso type is just the following dictionary storing an isomorphism:

data Iso (~>) a b = Iso {
    fw :: a ~> b ,
    bw :: b ~> a }

Answer 1

If this is indeed a type class, then the dictionary value is determined solely by the type p . In particular, the type cat is determined solely by p . This can be expressed using an associated data type . In a class definition, an associated data type is written like a data definition without a right-hand side.

Once you express cat as a type, the other members can easily be changed to type classes, as I've shown for Monad and MonadTrans . Note that I prefer to use explicit kind signatures for complicated kinds.

{-# LANGUAGE TypeFamilies, FlexibleInstances, UndecidableInstances #-}

class Pipe (p :: * -> * -> (* -> *) -> * -> *) where
  data Cat p :: (* -> *) -> * -> * -> * -> *
  isoI      :: forall a b m r. Iso (->) (p a b m r) (Category p m r a b)
  categoryI :: forall a b m.   Monad m => CategoryI (Category p m r)

instance (Pipe p, Monad m) => Monad (p a b m)

instance Pipe p => MonadTrans (p a b)