#15801: Categories over a base ring category
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Reporter: nthiery | Owner:
Type: enhancement | Status: new
Priority: major | Milestone: sage-6.2
Component: categories | Resolution:
Keywords: | Merged in:
Authors: | Reviewers:
Report Upstream: N/A | Work issues:
Branch: | Commit:
Dependencies: 10963 | Stopgaps:
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Changes (by SimonKing):
* dependencies: => 10963
Comment:
We want to build on top of #10963, right?
Replying to [comment:4 nthiery]:
> > One possible approach to solve that problem: The strict super-
categories of a category with fixed base ring should ''all'' be categories
with unspecified base rings. Hence, `Algebras(R)` should not have the
super-categories `Rings()` and `Modules(R)`, but `Algebras(R.category())`.
>
> This sounds reasonable. A variant would be, for a category over base
> ring C, to set {{{C(R).parent_class}}} explicitly to
> {{{C(R.category()).parent_class}}}, bypassing the
> {{{_make_named_class_key}}} business since it won't really be needed
> anymore anyway (and similarly for the element class and so on).
I think `_make_named_class_key` would still be make sense: We need to
distinguish the element and parent classes of `Algebras(Fields())` from
`Algebras(Rings())`, but probably not from `Algebras(QuotientFields())`.
And in particular (when following your suggestion below) we would still
want that `Modules(GF(5))` and `Modules(GF(7))` have the same parent and
element classes.
> I'd say that we certainly want {{{Algebras(R)}}} to be a subcategory
> of {{{Modules(R)}}}, as tested by {{{is_subcategory}}}. But we could
> imagine doing like for joins, and *not* put {{{Modules(R)}}} in
> {{{Algebras(R).<all_>super_categories()}}}.
Yes, but then one needs a category `FixedBase(R)` to be joined with! And I
think that's not so nice, as pointed out in comment:1. Or so I thought.
But:
> In fact, maybe {{{Algebras(R)}}} should really be a join category:
> that of {{{Algebras(Fields()) & Modules(R)}}}.
Hmmm. Yes, in that case one wouldn't need an ugly `FixedBase(R)`
category...
> <ramble> Here Modules(R) would play a role similar to the category
> with axiom {{{C().A()}}} associated to the category {{{C}}} defining
> the axiom {{{A}}}. We would not have the option to define the analogue
> of a ``category with axioms`` for a subcategory, but at this point I
> don't foresee a use for it either.</ramble>
Well, that's what I commented on at #10963: I was playing with the idea of
an axiom `WithFixedBase`, accepting an argument, but Volker has cut the
argument short `;-)`.
--
Ticket URL: <http://trac.sagemath.org/ticket/15801#comment:5>
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