Primarily because it's not clear what the morphisms are. At least, I don't think that elements should default to a category of all elements of their parent with trivial morphisms: I don't see that as being useful.
Now, it's certainly true that one can have categories where the objects are usually represented by elements in Sage. But I'd say that the best way to implement the category with objects the positive integers and a unique morphism from m to n if m divides n would be to create a new class inheriting from CategoryObject, given an integer at runtime. David On Sun, May 24, 2009 at 4:37 PM, William Stein <[email protected]> wrote: > > On Sun, May 24, 2009 at 4:30 PM, David Roe <[email protected]> wrote: > > > >> > >> >> Bill Page wrote: > >> >>> How does it related to the concept of "parent" - which seems equally > >> >>> ill-defined to me? > >> > >> > On Jun 3, 2008, at 10:04 PM, Robert Bradshaw wrote: > >> >> A Parent is an Object in the category of Sets, > >> > >> David Harvey wrote: > >> > huh? Don't you mean to say something more like "a parent is an object > >> > of a concrete category", i.e. a category C with a faithful functor > >> > f : C -> Set, such that the "elements" (as understood by Sage) of the > >> > parent P are exactly the elements of f(P)? > >> > >> I rather like David's definition because it is careful to make a > >> connection to category theory. Unfortunately, as far as I can tell > >> Sage does not directly implement this notion. I mean: there is no such > >> identifiable concrete category in Sage as such, even if it is true > >> that a parent "behaves" in this manner. In any case, perhaps it would > >> be helpful if 'parent' was replaced with > > > > I still like the name parent, but I agree with David's definition (and > have > > recently written basically the same thing in the documentation for the > > category patch): a parent is an object in a concrete category. Currently > it > > inherits from CategoryObject, which implements an arbitrary object (not > > necessarily in a concrete category). > > > >> > >> sage: concrete_category(1) > >> Integer Ring > >> > >> But there is a further complication since there already is some kind > >> of "category" associated with elements: > >> > >> sage: category(1) > >> Category of elements of Integer Ring > >> > >> I do not understand what this really is yet. It does not seem to me > >> that the elements of Integer Ring actually form a category as such. > > > > My understanding was that this dates from when Sage tried to give every > > object a category. I think it should change: elements don't have a > > category, their parents do. > > Why? Why do you think elements can't be objects in a category? > > -- William > > > > --~--~---------~--~----~------------~-------~--~----~ To post to this group, send email to [email protected] To unsubscribe from this group, send email to [email protected] For more options, visit this group at http://groups.google.com/group/sage-devel URLs: http://www.sagemath.org -~----------~----~----~----~------~----~------~--~---
