Right. Note, your original formulation---saying that a Functor instance maps from Hask to some subcategory of Hask (e.g. all types of the form Maybe X) was not wrong. It's just not a very useful way to think about it; as Rein described it's more useful to think of Functor instances as functors Hask -> Hask. -Brent On Tue, Apr 29, 2014 at 01:05:13PM -0600, Dimitri DeFigueiredo wrote:
Wow, this makes it very clear!
So, the object mapping is provided by the type constructor and the arrow mapping by fmap.
Thanks :-)
Em 29/04/14 12:35, Rein Henrichs escreveu:
Yes, this is reasonable, but it might be useful to be a bit more explicit about the categories involved.
An instance of Functor is a functor which maps objects and arrows in the category of Hask (of Haskell types and functions) to... objects and arrows in the category of Hask. It is an endofunctor. The object mapping must map a type to a type and therefore must have the kind * -> *. This means that it is a type constructor, e.g., Maybe. The arrow mapping must map arrows to arrows, which means it must map functions to functions. fmap provides this mapping and the Functor laws in Haskell are of course equivalent to the laws from category theory.
And while the inclusion of bottom is problematic for Hask, it is still possible to reason about it in a "morally correct" way (http://www.cse.chalmers.se/~nad/publications/danielsson-et-al-popl2006.pdf http://www.cse.chalmers.se/%7Enad/publications/danielsson-et-al-popl2006.pdf).
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