RE: [Axiom-developer] Curiosities with Axiom mathematical structures

[Page, Bill]

Ralf, 

On Tuesday, March 14, 2006 9:57 AM you wrote:
> > "Bill Page" <address@hidden> writes:
> > 
> > | On March 13, 2006 6:34 AM Ralf Hemmecke asked:
> > | > ... 
> > | > But here the question to our category theory experts:
> > | > Since Monoid is something like (*,1) would it make sense
> > | > to speak of a category (in the mathematical sense) of
> > | > 
> > | >    monoids that have * and 1 as their operations   (1)
> > | > 
> > | > ? Morphisms would respect 1 not just the identity element 
> > | > with respect to *. And for each morphism f we would have
> > | > f(a*b) = f(a)*f(b). Of course as operations the two * above
> > | > are different but in that category they have to have the same
> > | > name. (No idea whether this makes sense, but it seems that
> > | > this is the way as "Category" it is implemented in Axiom/Aldor.)
> > | > 
> > | > Then, of course, (N, +, 0) is not an object in the category 
> > | > given by (1).
> > | > 
> > | 
> ... 
> Well, Bill, are saying that I cannot have (1) as a category?

Let me try to answer again: No. :)

I think your definition (1) is not a category in the usual sense
of category theory.

> That a monoid itself is a category is true, but distracts from
> my question.
> 
> A category is a collection of objects and morphisms. Let's 
> define MYMON. Let an object be as follows:
>    A set M together with to operations *: (M, M) -> M
>    and 1: M such that the monoid axioms are satisfied.
>    (I am lazy and don't specify them here explicitly.)
> The morphisms in MYMON are the usual monoid-homomorphisms.
> 

Ok.

> And yes, if something wants to be an object in MYMON, it
> has to have * as its binary operation and not +.

This does not make much sense to me. That is the same thing
as trying to say that if some set "wants to be" an object
in MYMON it must be named M and not N. This is irrelevant
when defining a category. Both M and * are "formal parameters"
that do not have any more meaning than we wish to assign to
them.

> 
> Who wouldn't one consider MYMON as a category?
>

I think MYMON is a category. I just think your conclusion
about the role of the parameters is wrong.
 
> And yes, I consider (N, +, 0) not an object in MYMON. What
> I try to say is that Aldor has currently this view for the
> "Monoid" category.
> 

Do you mean you think the distinction is a "syntactic" one
and not a semantic one? If so, then I think I agree with
you. Aldor categories are not categories in the sense of
category theory. Nonetheless it is the nature of symbolic
calculation on a computer that we are always in the position
of trying to express semantic ideas in terms of syntax. In
the same way the series of digits:

   3 1 4 1 9 5

is not an "integer" in the mathematical sense. Nonetheless
we do use this symbolic representation of an integer to
perform calculations by hand and a similar (usually binary)
notation to do calculations inside the computer.

But I do not see how recognizing this fact gets us any
further along the path to understanding how to represent
the mathematical notion of a monoid in Axiom.

Regards,
Bill Page.