[Haskell-cafe] Functor and Haskell
Dan Doel
dan.doel at gmail.com
Thu Apr 23 15:34:58 EDT 2009
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 at Int :: [Int] -> Maybe Int
listToMaybe at Char :: [Char] -> Maybe Char
listToMaybe at 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.
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