RE: [Axiom-developer] Re: [Axiom-math] Are Fraction and Complex domains.

[Page, Bill]

On Thursday, May 11, 2006 7:14 PM Gabriel Dos Reis wrote:
> 
> Ralf Hemmecke writes:
> 
> [...]
> 
> | But of course, I could live with that identification if it 
> | is clearly documented that ()->Cat can be identified with Cat.
> | Where are our category experts? I believe there is a distinction
> | here, n'est pas?
> 
> From Category Theory point of view, a constant x of type T 
> is the same as the (unique) morphism x : 1 -> T, where 1 is
> the one-point set.

The relevant object here is the initial object 0 in a complete
Cartesian closed category (CCCC), not the terminal object 1.

http://en.wikipedia.org/wiki/Initial_object

It is quite reasonable I think to let () denote the initial
object in the CCCC associated with the categorical semantics
of Aldor.

The definition of initial is that: there exists precisely one
morphism () → X for each object X. So in a sense any "categorical"
distinction between the morphism and co-domain object is not
likely to be of much interest.

> 
> Now, I also understand that beyond the name, Aldor's categories
> are not mathematical categories; so...

I think that is an over statement. There is a considerable
similarity between categories in Aldor and mathematical
categories. But that is irrelevant. Any "sufficiently complex"
programming language must have the structure of a complete
Cartesian closed category.

> 
> My practice of functional programming suggests that the
> identification is useful in many cases, than keeping the
> artifice.  But YMMV.
> 

I think this identification can be justified from a mathematical
point of view.

Regards,
Bill Page.