On Thursday 23 April 2009 2:44:48 pm Daryoush Mehrtash wrote:
Thanks for this example I get the point now. (at least i think i do :) )
One more question.... This all being on the same category then the functor transformation can also be view as a simple morphism too. In this example the listToMaybe can be viewed as morphism between list and Maybe types that are both in the Hask categroy too. right? If so then what would viewing the morphism as natural transformation by you?
listToMaybe in general wouldn't be a morphism in the category, because morphisms would be from concrete types to other concrete types. [1] So, if you'll excuse some notation I just made up (with a little help from GHC core notation :)): listToMaybe@Int :: [Int] -> Maybe Int listToMaybe@Char :: [Char] -> Maybe Char listToMaybe@String :: [String] -> Maybe String are all morphisms in the alleged Hask category. Each polymorphic function (similar to the above one, at least) defines a family of morphisms like that. *But*, that's what a natural transformation is: a family of morphisms, one for each object in the category, that commute with functor application in a certain way. Thus, one can look at the fully polymorphic listToMaybe as a natural transformation: listToMaybe :: [] -> Maybe -- Dan [1] Maybe you could make up a category where polymorphic types are objects as well, but that doesn't seem to be the way people typically go about applying category theory to Haskell.