#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: | Stopgaps:
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Comment (by nthiery):
Replying to [comment:1 SimonKing]:
> So, the strategy would be this: The current "Category over base ring"
should accept as input no a ring, but a subcategory of the category of
rings.
Or more precisely, it would accept both kinds of input.
> A polynomial ring should be initialised in the category of "commutative
algebras over Category of quotient fields" (say), and should of course
have its own reference to the base ring (say, the rational field).
Yup. And later in the category of "polynomial rings over a quotient
field."
> Since the parent and element classes only depend on the category of the
base ring (by the current implementation), a category refinement would
probably not be needed.
Yes.
> But *if* needed, it would amount to take the join with the category
"`FixedBaseField(QQ)`, perhaps with super-categories
"`FixedBaseRing(QQ)`".
Yes, though I would write it as taking the join with Algebras(QQ).
> Additional technical problem: We not only have base rings, but (for
bimodules) left and right base rings. Should this be reflected in the
"`FixedBaseBla`" categories as well?
We will want to accept both:
{{{
Bimodules(Fields(), Fields())
}}}
and
{{{
Bimodules(QQ, QQ)
}}}
In a first step, I would not bother about mixed calls likes
{{{Bimodules(QQ, Fields)}}}/
> And another complication: We want to have categories expressing the fact
that a field (not just a ring) is acting on the objects, and then want to
compute the join. Is there an easy programmatic way to let an error be
raised when someone tries to compute the join "`Algebras(Fields()) &
FixedBaseRing(ZZ)`"?
I don't see a trivial way to do it at this point. It certainly would
be a nice feature to bark upon such constructions, but not having it
is not really worst than the current situation where we have:
{{{
sage: Algebras(QQ) & Algebras(GF(3))
Join of Category of algebras over Finite Field of size 3 and Category of
algebras over Rational Field
}}}
So this issue should not prevent us to proceed even if we don't find a
good way out.
--
Ticket URL: <http://trac.sagemath.org/ticket/15801#comment:2>
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