On Thu, Aug 21, 2003 at 11:32:47AM +0100, C T McBride wrote:
My point, however, is not to use <$> with that type, but the more general
class Fun f where eta :: x -> f x (<$>) :: f (s -> t) -> f s -> f t
Is there a better name for Fun? Is it ancient and venerable? Am I an ignoramus twice over?
It seems that something this useful should have a name, but I've been unable to find one. This interface was first used for parsers by Niklas Rojemo. Doaitse Swierstra has developed a lot of interesting parsers with this interface, roughly infixl 4 <$>, <*> class Functor f => Sequence f where eta :: a -> f a (<*>) :: f (a -> b) -> f a -> f b (<$>) :: (a -> b) -> f a -> f b (<$>) = fmap One would expect this to satisfy identity and associativity laws: eta f <*> v = f <$> v u <*> eta y = ($ y) <$> u u <*> (v <*> w) = ((.) <$> u <*> v) <*> w as well as naturality of eta and (<*>). Several other choices of primitives are available. One is: lift0 :: a -> f a lift1 :: (a -> b) -> f a -> f b lift2 :: (a -> b -> c) -> f a -> f b -> f c lift0 = eta lift1 = fmap lift2 f fa fb = f <$> fa <*> fb (<*>) = lift2 ($) Another interface is eta plus mult :: f a -> f b -> f (a,b) with axioms eta x `mult` v = fmap (\y -> (x,y)) v u `mult` eta y = fmap (\x -> (x,y)) u u `mult` (v `mult` w) = fmap assoc ((u `mult` v) `mult` w) where assoc ((x,y),z) = (x,(y,z)) These are all equivalent (or would be, if Haskell had true products). The last version generalizes to any symmetric monoidal category: it requires that f be a lax monoidal functor, with eta_1 and mult as the transformations, that eta is a transformation from the identity monoidal functor to f, plus the symmetry condition eta x `mult` v = fmap swap (v `mult` eta x) As you note, they're preserved under composition. They also compose with arrows (in the sense of Hughes) to make new arrows, and they're preserved by products (as are arrows).