Robert, Thanks for the elaboration. I hope that this follow-up is not too far off topic and I don't want to distract you (or any of the Sage developers) from your main tasks. I wrote this mostly as notes and questions to myself - but if you or anyone else have some time or the inclination to correct or comment as time permits, that would be great.
On Tue, Jun 3, 2008 at 10:04 PM, Robert Bradshaw wrote: > > On Jun 3, 2008, at 4:50 PM, Bill Page wrote: > >> How does it [categories in Sage] relate to the >> concept of "parent" - which seems equally ill-defined to me? > > A Parent is an Object in the category of Sets, though in the context > of coercion one usually assumes one can some kind of arithmetic > (binary operations) on their elements. > ... > On Jun 3, 2008, at 7:11 PM, David Harvey wrote: > ... >> 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)? Ok (and thanks also for the clarification, David). There are of course two different uses of "object" here: 1) object of some category, 2) Python object. All Python objects have a 'type', i.e. belong to some Python class. So in Sage 3.0.2 I observe for example: sage: type(1) <type 'sage.rings.integer.Integer'> sage: parent(1) Integer Ring sage: type(parent(1)) <type 'sage.rings.integer_ring.IntegerRing_class'> sage: category(parent(1)) Category of rings sage: type(category(parent(1))) <class 'sage.categories.category_types.Rings'> and sage: type(1.0) <type 'sage.rings.real_mpfr.RealNumber'> sage: parent(1.0) Real Field with 53 bits of precision sage: type(parent(1.0)) <type 'sage.rings.real_mpfr.RealField'> sage: category(parent(1.0)) Category of fields sage: type(category(parent(1.0))) <class 'sage.categories.category_types.Fields'> These seem consistent to me, albeit rather complex. However I am not sure I understand the following: sage: parent(IntegerRing()) <type 'sage.rings.integer_ring.IntegerRing_class'> sage: parent(RealField()) <type 'sage.rings.real_mpfr.RealField'> Could you explain why the parent of an object of some category is a type? > >> What is the relationship to inheritance in Python? Is the intention >> to give all mathematical objects defined in Sage some categorical >> structure? What about categories themselves as mathematical >> structures - e.g. a category is a kind of directed graph with additional >> structure? > > A big push of this model is to divorce the relationship between > inheritance (which primarily has to do with implementation) and > mathematical categorization. At first blush it still seems strange to me to distinguish between the parent of an object and the type of an object. The instances of a type are often considered elements. Is the invention of 'parent' in Sage due to some inherent limitation of Python classes? Do you really want to say more abstractly that a certain type *represents* a certain parent? E.g. Does 'sage.rings.real_mpfr.RealNumber' represent 'RealField()'? Can there be other representations of 'RealField()' in Sage? How can I find out for example if 'sage.rings.integer.Integer' represents 'IntegerRing()'? >From my experience with Axiom I am sympathetic to the separation of implementation and specification but in Axiom "domains" are both parents and types. A domain can inherit it's implementation from another domain (via add). A domain can also serve as the representation for some other domain (via Rep). For example IntegerMod(5) is the representation of PrimeField(5). IntegerMod(5) in turn is represented by SingleInteger, etc. In Axiom, at the deepest level all types are ultimately provided by Lisp. These types are not really Axiom domains and so I suppose the situation is analogous to Python types and Sage parents. Axiom categories on the other hand form a lattice. Domains belong to categories by assertion or by virtue of the subdomain relationship. In Axiom there is an operator named 'has' which provides the same information as 'category' in Sage so that: x has y is just 'y == category(x)'. For example 'Float has Field' is equivalent to: sage: Fields() == category(RealField()) True Although in Axiom a domain may satisfy more than one category. Will this be possible in Sage in the future? What information does Sage actually have about a given category? For example, what does Sage know about the relationship between Rings() and Fields()? E.g. functors? None of this has much to do with "category theory" as such. Categories in Axiom are only vaguely related to categories in Mathematics. So I suppose from your description that categories in Sage will play a similar role but will be more strongly related to the mathematical concept? > This will allow categories to play a larger role (though work in that > area can be done after the merge as it is not disruptive). For > example, Z/nZ[x] is implemented the same whether or not p is > a prime, but the category it lives in (and hence the methods one > can do on it) vary. Are you suggesting that Z/pZ should be considered a field just because p is prime? sage: category(Zmod(7)) Category of rings sage: Zmod(7).is_field() True Perhaps an alternative would be to provide a coercion to some object whose parent is in Fields()? > As another example, matrix spaces are algebras iff they are square, > but one doesn't want to have a special class for square matrices over > <insert optimized type here>. I am not sure what you mean by "matrix spaces" as such but the category of matrices does make sense to me. See for example: http://unapologetic.wordpress.com/2008/06/02/the-category-of-matrices-i Maybe it is interesting because in this category the morphisms are matrices and the objects are just the row and column dimensions - non-negative integers. How would this fit into the category model in Sage? There is a monoidal sub-category of square matrices. >Generic algorithms can be put on the categories such > that if g is in category C, and C has a method x, then g.x() will > work. Do you mean 'category(parent(g))==C'? How will such a generic method x of C operate on g? Do you have some example coding that implements this idea? Regards, Bill Page. --~--~---------~--~----~------------~-------~--~----~ To post to this group, send email to sage-devel@googlegroups.com 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 -~----------~----~----~----~------~----~------~--~---
