#10963: Axioms and more functorial constructions
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Reporter: nthiery | Owner: stumpc5
Type: enhancement | Status: needs_info
Priority: major | Milestone: sage-6.2
Component: categories | Resolution:
Keywords: days54 | Merged in:
Authors: Nicolas M. Thiéry | Reviewers: Simon King, Frédéric
Report Upstream: N/A | Chapoton
Branch: | Work issues: merge with #15801
public/ticket/10963-doc- | once things stabilize
distributive | Commit:
Dependencies: #11224, #8327, | cd4f5f92cd415902ea2118292954a5685c6f8cfd
#10193, #12895, #14516, #14722, | Stopgaps:
#13589, #14471, #15069, #15094, |
#11688, #13394, #15150, #15506, |
#15757, #15759, #15919 |
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Comment (by nthiery):
Hi Darij!
Sorry for being slow; I have been traveling lately. And thanks for all
the proofreading and improvements!
I just pushed some proofreading.
Replying to [comment:603 darij]:
> OK, now I see what you mean by "covariant" :) The problem is that the
quotient is a certain *universal* homomorphic image, and a quotient in B
(despite still being a homomorphic image in A) will not always have this
universality property in A.
And I see what you mean by "quotient" :-) It's more general than what
I had in mind. In Sage, the category of quotients in A is really about
objects Y of A that are endowed with a distinguished surjective
A-morphism X->Y from some object X of A (and a section of this
morphism). See the description in:
sage: S = Sets()
sage: S.Subquotients?
Maybe the name "Quotients" is misleading, and should be replaced (in
some later ticket) by something like "HomomorphicImages".
In this restricted setting do you still see any issue?
> I fear this will come up in practice a lot, e.g., when you consider Lie
algebras which have a grading and a filtration, and the filtration is
*not* the one you would get from the grading. (These appear everywhere.)
So I'm not very happy to have "graded Xes" as a subcategory of "Xes"...
> Anyway I've added some doc now. Is it on the right track?
I tried to improve the description of what subcategories are.
One thing is that "GradedXXX" is not about the objects of XXX that can
be endowed with a grading (like e.g. the trivial grading), but about
objects of XXX endowed with a distinguished grading. Thus in the above
case you could have an unrelated category (say FilteredLieAlgebras),
and a construction taking a graded lie algebra and endowing it with
the filtration induced by its grading, but also other constructions
endowing it with other filtrations.
Cheers,
Nicolas
--
Ticket URL: <http://trac.sagemath.org/ticket/10963#comment:607>
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