Hi, Glad to hear that Javier took up the last category files.
I don't have a strong opinion whether OrderedSets (or Monoids) should have total or partial ordering. However, since there is a possible ambiguity, my preference is to not use OrderedX as an alias for either PartiallyOrderedX or TotallyOrderedX. Specifically for PartiallyOrderedSet, Poset is standard. Regarding a FreeModules category: in general the subset (or subcollection) of free objects in a category does not form a nice or natural category. The category should determine what structures the morphisms should preserve such as binary or unary operations like +, *, -, not, xor, <. The morphisms should be implemented to respect this structure and ensure integrity of the category. Free objects play an important role in a category, but they are not preserved by taking kernels, quotients, or other standard constructions. They are the first (Sage/Python classes of) objects in the category on which morphisms (to any object in the category) can and should be implemented. The full category should contain all of the natural and interesting objects that one wants to study. Some objects X and Y may not have good computational models or we may not be able to created their elements, and similarly we may not be able to construct effective elements of Hom(X,Y) for all objects in the category. The various classes of objects should implement the free objects by special constructors (= functors from sets), but they don't fill out a nice category in themselves. For this reason, I don't think CategoryWithBasis (or more generally CategoryWith[Free]Generators) is a natural category -- unless the generators are structures intended to be preserved, as is the case with a category of pointed sets. Rather than having unnatural subcategories, one should have functions like Cat.is_free(X) or X.is_free(), then, if True, X.generators() or X.basis(), for identifying such objects and easily constructing their morphisms. Cheers, David --~--~---------~--~----~------------~-------~--~----~ 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 URL: http://www.sagemath.org -~----------~----~----~----~------~----~------~--~---
