On Sun, Aug 23, 2009 at 02:48:15AM -0700, javier wrote:
> > possibly simply by browsing:
> >
> >
> > http://combinat.sagemath.org/hgwebdir.cgi/code/file/tip/sage/categories)
> >
> > and make sure they makes sense.
>
> About this, I clicked on a file at random (algebras.py) to have an
> idea on a general shape of the stuff. I was very startled to see a
> method called direct sum. The category of algebras (over a fixed based
> field) does not have a "direct sum" (or coproduct), what is
> constructed there is actually the direct product. Sure thing, as in
> the "big category" (vector spaces over the base field) there is a
> direct sum, which coincides with the direct product, but it doesn't
> descend to the category of algebras: the map A ---> A\oplus B in
> vector spaces is defined by a |---> (a,0), which this is not a
> morphism of algebras.
>
> I know that most physicists and some mathematicians abuse the term
> "direct sum" to refer to the direct product, but since this whole
> thing is called "Categories" I expect a "Category theory" level of
> precision.
>
> Is this the kind of stuff you refer to with "checking mathematical
> sanity", or am I going needlessly abstract here?
Yes, thanks much for your feedback!
Having a "Category theory" level of precision is definitely an aim of
the category framework (but practicality is also one). Many aspects
are not up to speed yet to this respect (in particular around
homsets). The category framework could (and will) definitely use
several more iterations, if at all possible from developers with
various areas of expertise.
Since we are pretty late in this second iteration on Sage categories,
and since this sounds like a relatively minor issue, I would vote for
debating it and writing down explicitly in the documentation the
current abuses. However, unless a trivial fix emerges, I would
postpone the issue for a later iteration. This to avoid delaying
further this iteration and also to limit the risks of over-engineering
(we speak of usine à gaz in French). The right design will have better
chances to emerge once practical expertise will have been gained by
the Sage community in using the category framework.
Now to the debate:
- The problem with the map a -> (a,0) is only that 1_A is mapped to
(1_A,0) which is not 1_{A\oplusB} = (1_A,1_B), right?
Otherwise said, the category of NonUnitalAlgebras (which is not yet
implemented in Sage) indeed has a direct sum?
- I am in the train right now, without appropriate references. For,
say, monoids (additive, or multiplicative), how should the
operation of taking cartesian products be called?
Cheers,
Nicolas
--
Nicolas M. Thiéry "Isil" <nthi...@users.sf.net>
http://Nicolas.Thiery.name/
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