[Haskell-cafe] Ratification for my understanding of functors

[wren ng thornton]

On 1/8/11 5:23 AM, C K Kashyap wrote:
> Hi,
>
> Thanks to Luke Palmer's explanation. I got a better idea of functors.
> I just want to ratify my understanding of what a functor is -
> according to category theory, functors are arrows between categories
> ... these arrows map objets and arrows from one category into another.

Yes, provided the mappings on objects and arrows behaves in the right 
way, namely preserving identities and compositions:

     F(id_A) = id_(FA)

     F(f.g) = F(f) . F(g)

That is, functors are exactly category homomorphisms (i.e., mappings 
which preserve the categorical structure).

> So, would it be right to say that the list type constructor + the map
> function form the functor? Saying "list is a functor then is a
> convenient approximation right?

Yes, the combination of the type constructor (mapping from types/objects 
to types/objects) and the fmap instance (mapping from functions/arrows 
to functions/arrows) together constitute the functor. Saying "list is a 
functor" is a common sloppiness just like saying "list is a monad".

-- 
Live well,
~wren