On Sun, Feb 1, 2009 at 8:26 AM, Ben Moseley
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So, the idea is that any polymorphic Haskell function (including Data constructors) can be seen as a natural transformation - so a "function" from any object (ie type) to an arrow (ie function). So, take "listToMaybe :: [a] -> Maybe a" ... this can be seen as a natural transformation from the List functor ([] type constructor) to the Maybe functor (Maybe type constructor) which is a "function" from any type "a" (eg 'Int') to an arrow (ie Haskell function) eg "listToMaybe :: [Int] -> Maybe Int".
Aha, hadn't thought of that. In other terms, a natural transformation is how one gets from one lifted value to different lift of the same value - from a lift to a hoist, as it were. Just calling it a function doesn't do it justice. Very enlightening, thanks. I'm beginning to think Category Theory is just about the coolest thing since sliced bread. If I understand correctly, we can "represent" a data constructor in two ways (at least): qua functor: Dcon : a -> T a qua nat trans Dcon : Id a -> T a and thanks to the magic of CT we can think of these as "equivalent" (involving some kind of isomorphism). BTW, I should mention I'm not a mathematician, in case it's not blindingly obvious. My interest is literary - I'm just naive enough to think this stuff could be explained in plain English, but I'm not quite there yet. Thanks much, -gregg