On Sun, May 24, 2009 at 4:47 PM, David Roe <[email protected]> wrote: > 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.
OK, so you just don't want elements to be objects in a category by *default*. However, if one does a little more work then elements can be objects in a category. That sounds reasonable to me. William > 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 >> >> > > > > > -- William Stein Associate Professor of Mathematics University of Washington http://wstein.org --~--~---------~--~----~------------~-------~--~----~ 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 -~----------~----~----~----~------~----~------~--~---
