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RE: [Axiomdeveloper] Curiosities with Axiom mathematical structures
From: 
Bill Page 
Subject: 
RE: [Axiomdeveloper] Curiosities with Axiom mathematical structures 
Date: 
Wed, 8 Mar 2006 14:36:26 0500 
BTW. I've replied just on the axiomdeveloper since the email
lists seem to be functioning (more or less) properly again.
When replying to something that includes a Cc: to Axiom Wiki
email address@hidden, I think we need to keep
in mind that the email will be appended to a page named by the
bracket in the subject, [axiomdeveloper] in this case. Since
this name automatically occurs on all emails forwarded from
the axiomdeveloper list, this means the corresponding page on
the Axiom Wiki can get unmanageably long...
On March 7, 2006 2:53 PM William Sit wrote:
> ...
> Ralf Hemmecke wrote:
> > There is Chapter 9 of Doye's PhD thesis
> > http://portal.axiomdeveloper.org/refs/articles/doyealdorphd.pdf
> > which deals in part with the renaming.
> >
> > From what I understand, it simply makes two mathematically
> > distinct things (operator symbols and operator names) distinct
> > in the programming language. (Though I don't quite like the syntax.)
>
> ...
> By the way, I don't follow his example at bottom of p. 127:
> If \phi is a functor (he said homomorphism, but that is wrong,
> because a list and a set are from different categories),
Doye's use of the work homomorphism is correct in the context
of ordersorted algebra. Doye is not using the language of
category theory in either the mathematical sense or in Axiom's
sense.
http://en.wikipedia.org/wiki/Homomorphism
A homomorphism is just a map from one algebra to another that
preserves the abstract algebraic structure. From Doye's point of
view both lists and sets are algebras in this sense).
> he wants the equation to hold:
>
> \phi(#([1,1]) = #(\phi([1,1]))
>
This is just the definition of preservation of algebraic structure.
> The right hand side makes sense, with answer 1, but the left
> hand side does not, since #([1,1]) = 2 is a number, not a list.
In Doye's formalism both number 2 and the list [1,1] are part
of the algebraic structure of 'List'. phi maps both of these to
the corresponding structures in 'Set'.
> He claimed \phi(2) = 2.
Yes, the '2' in 'List' maps to the '2' in 'Set'.
> In addition, there is no requirement that a functor commutes
> with operations. A functor should, in addition to taking
> objects (lists) to objects (sets), also has to take a morphism
> *between* source objects (lists) to another morphism *between*
> the image objects (sets). The map '#' is not a morphism in the
> category of Lists, nor of Sets.
'#' is an operator in both the 'List' and 'Set' ordersorted
algebras.
> If one must, then '#' is a functor from the category of Lists
> (and also for Sets) to the category whose objects are
> nonnegative integers. But the composition of functors \phi
> \circ # does not make sense.
No, this is not what Doye has in mind.
> What Doye has in mind is the following diagram:
>
> #
> Lists > NNI
>  
> \phi   id
> v # v
> Sets > NNI
>
> But there is no reason to expect this to be a commutative
> diagram of functors. So I think his example illustrates nothing.
>
No. The diagram should look like this:
#: Lists > NNI

\phi 

v
#: Sets > NNI
Where we have the algebra 'List' on top and the algebra 'Set'
on the bottom. We should think of \phi as mapping everything
from 'List' to 'Set', i.e. %_list > %_set, NNI_list > NNI_set,
and #_list > #_set.
Regards,
Bill Page.
 Re: [Axiomdeveloper] Curiosities with Axiom mathematical structures, (continued)
 Re: [Axiomdeveloper] Curiosities with Axiom mathematical structures, Gabriel Dos Reis, 2006/03/13
 Re: [Axiomdeveloper] Curiosities with Axiom mathematical structures, Ralf Hemmecke, 2006/03/14
 Re: [Axiomdeveloper] Curiosities with Axiom mathematical structures, Gabriel Dos Reis, 2006/03/13
 Re: [Axiomdeveloper] Curiosities with Axiom mathematical structures, Ralf Hemmecke, 2006/03/14
 Re: [Axiomdeveloper] Curiosities with Axiom mathematical structures, Gabriel Dos Reis, 2006/03/13
 Re: [Axiomdeveloper] Curiosities with Axiom mathematical structures, Ralf Hemmecke, 2006/03/08
 Re: [Axiomdeveloper] Curiosities with Axiom mathematical structures, Ralf Hemmecke, 2006/03/08
 Re: [Axiomdeveloper] Curiosities with Axiom mathematical structures, William Sit, 2006/03/09
 Re: [Axiomdeveloper] Curiosities with Axiom mathematical structures, Ralf Hemmecke, 2006/03/09
 Re: [Axiomdeveloper] Curiosities with Axiom mathematical structures, William Sit, 2006/03/10
 RE: [Axiomdeveloper] Curiosities with Axiom mathematical structures,
Bill Page <=
 Re: [Axiomdeveloper] Curiosities with Axiom mathematical structures, William Sit, 2006/03/09
 Re: [Axiomdeveloper] Curiosities with Axiom mathematical structures, Gabriel Dos Reis, 2006/03/13
 Re: [Axiomdeveloper] Curiosities with Axiom mathematical structures, Gabriel Dos Reis, 2006/03/13
 Re: [Axiomdeveloper] Curiosities with Axiom mathematical structures, William Sit, 2006/03/14
 Re: [Axiomdeveloper] Curiosities with Axiom mathematical structures, Martin Rubey, 2006/03/02
 Re: [Axiomdeveloper] Curiosities with Axiom mathematical structures, Gabriel Dos Reis, 2006/03/02
 Re: [Axiomdeveloper] Curiosities with Axiom mathematical structures, Martin Rubey, 2006/03/04
 RE: [Axiomdeveloper] Curiosities with Axiom mathematical structures, Bill Page, 2006/03/04