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

From: Ralf Hemmecke
Subject: Re: [Axiom-developer] Curiosities with Axiom mathematical structures
Date: Tue, 14 Mar 2006 15:57:22 +0100
User-agent: Thunderbird 1.5 (X11/20051201)

On 03/14/2006 01:54 AM, Gabriel Dos Reis 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). | > | | I keep trying to answer these questions but I am not sure I
| would like to classify myself as an "expert" in category theory.
| :) But here goes ...
| | In category theory **Mon** (** means written in bold face font)
| consists of all monoids (as objects) and all monoid homomorphisms
| as morphisms. This does not say anything directly about what
| operation are present "inside" the objects of the category.

Well, Bill, are saying that I cannot have (1) as a category? 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.

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

Who wouldn't one consider MYMON as a category?

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.


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