[Haskell-cafe] Relating functors in Category Theory to Functor

[Frank Atanassow]

On Jun 29, 2004, at 6:46 PM, Iavor S. Diatchki wrote:

>>> In Haskell, natural transformations are polymorphic functions, tau 
>>> :: f a -> g a. For example, maybeToList :: Maybe a -> [a].
>>>
> actually i think this is a good approximation.  not all polymorphic 
> functions are natural transformations, but "simple" ones usually are.

I think you have it backwards, unless you are thinking of a more 
general notion of polymorphism than parametric polymorphism.

Ad hoc polymorphic functions may or may not be natural; you have to 
verify the naturality condition in each case.

But every parametrically polymorphic function is a natural 
transformation, though the converse fails: not every natural 
transformation is parametrically polymorphic. In particular, some 
natural transformations are generic functions (polytypic), and their 
components (instantiations at a type) are not all instances of a single 
algorithm.

I'm not sure if every natural transformation on endofunctors on a 
category modelling a Haskell-like language is denoted by either a 
parametric or generic term.

Regards,
Frank