On Aug 26, 9:30 am, Jason Grout <jason-s...@creativetrax.com> wrote: > John H Palmieri wrote: > > On Aug 26, 6:21 am, "Nicolas M. Thiery" <nicolas.thi...@u-psud.fr> > > wrote: > >> Dear and Javier, dear John,
> >> 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) > > > Could there be an optional argument: > > > sage: direct_sum(V, W, cat) > > > If cat is not present, find the "strongest" category which contains > > both V and W, and compute the direct sum there, raising an error if it > > doesn't exist. (Or just raise an error? Or raise an error, saying > > what this strongest category is, and suggesting computing the direct > > sum there? Maybe I like this last approach the best.) If cat is > > present, check that cat contains both V and W, and then compute the > > direct sum there. > > Isn't this sort of what the coercion system is all about? If you can't > do a direct sum of V and W, try coercing them into a category that can > do a direct sum? If you want to specifically compute the direct sum in > a particular category, then do an explicit cast yourself, like > direct_sum(cat(V), cat(W)), or maybe cat.direct_sum(V,W) or something. I completely agree with your last point: direct_sum(cat(V), cat(W)) or cat.direct_sum... is the way to go for explicitly specifying the category. This is better than passing an optional argument to direct_sum. I'm leaning more strongly toward the point of view that raising an error is the right approach when the direct sum doesn't exist. If V and W are objects in the same category, and that category doesn't have direct sums, then direct_sum(V,W) should raise an error. This should even happen, I think, if there is a forgetful functor to a category in which the direct sum is defined. As noted earlier, if V and W are k- algebras, then they don't have a direct sum in the category of k- algebras. If they happen to be commutative, then they do have a direct sum in the category of commutative k-algebras: V tensor_k W. They also have direct sums in the category of k-vector spaces (V x W) and in the category of sets (V disjoint union W). All of these are different, so which should you choose? It seems safest to raise an error, perhaps with the suggestion that the direct sum might exist in some other category, like the category of k-vector spaces, and so you might execute direct_sum(VectorSpaces(k)(V), VectorSpaces(k)(W)) (or whatever the command is) instead. I suppose there could be another function, "direct_sum_force" or something, which finds a category containing V and W in which the direct sum exists. But I think that if category(V) == category(W) is True, then sage: category(V) == category(direct_sum(V, W)) should return True, also, since, as Nicolas says, 'Having a "Category theory" level of precision is definitely an aim of the category framework'. John --~--~---------~--~----~------------~-------~--~----~ To post to this group, send an email to sage-devel@googlegroups.com To unsubscribe from this group, send an email to sage-devel-unsubscr...@googlegroups.com For more options, visit this group at http://groups.google.com/group/sage-devel URLs: http://www.sagemath.org -~----------~----~----~----~------~----~------~--~---
