[Haskell-cafe] Functor and Haskell

[Daryoush Mehrtash]

Thanks this was helpful.

In many of Conal Elliot's writings I see that he shows that his semantic
function is a natural transformation.  Is that just basically showing the
polymorphic nature of his semantic functions, or are there other benifits
you get by showing a particular function is a natural transformation?

Daryoush

On Thu, Apr 23, 2009 at 12:34 PM, Dan Doel <dan.doel at gmail.com> wrote:

> 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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