Dear and Javier, dear John,
On Wed, Aug 26, 2009 at 12:44:52AM -0700, javier wrote:
> On Aug 24, 1:33 pm, "Nicolas M. Thiery" <nicolas.thi...@u-psud.fr>
> wrote:
> > - 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?
>
> This is an "easy to spot" problem. The deep underlying problem is the
> universal property.
>
> > Otherwise said, the category of NonUnitalAlgebras (which is not yet
> > implemented in Sage) indeed has adirectsum?
>
> No, as John points the problem is with the universal property. The
> "filler map" required by such property always exists (and it is
> unique) at the level of the underlying vector spaces, but there is no
> way to guarantee that it is a morphism of algebras (in general, it
> isn't).
>
> Back to terminology, what it is implemented is a categorical product,
> which in this category (algebras) coincides with the vector spaces
> direct product. If we want to implement a categorical coproduct, then
> we are aiming for the free product, not any "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?
>
> It is also called direct product, same as in groups (where we have
> direct and semi-direct products).
Thanks for the feedback, http://en.wikipedia.org/wiki/Direct_product
finished to clear my confusion. I was being co-silly :-)
So in the long run, we definitely need both the direct sum and the
direct product functorial constructions, with appropriate
"inheritance" to share whatever is valid for both direct products and
direct sums. For the moment, I'll just leave a note about this in the
code (for the moment, this feature is essentially used nowhere).
A little question: if V and W are two vector spaces which turn out to
be also in stronger categories, some with direct sums, and some
without, what should the following do:
sage: direct_sum(V, W)
One 100% safe option is to raise an error.
Another would be to return the direct sum (as vector spaces) of V and
W, which is also a direct product (that would be another story if the
sum was infinite), and to endow it with whatever operations from the
other categories applying either to direct sums or to direct products.
This second option, if at all safe, sounds more friendly for non
category experts.
> PS: I am cross-posting this to sage-combinat-devel. Maybe we should
> move the discussion there?
Crosspost is good, since we need feedback from everyone.
Cheers,
Nicolas
--
Nicolas M. Thiéry "Isil" <nthi...@users.sf.net>
http://Nicolas.Thiery.name/
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