#10963: More functorial constructions
-------------------------------------+-------------------------------------
Reporter: nthiery | Owner: stumpc5
Type: enhancement | Status: needs_review
Priority: major | Milestone: sage-6.1
Component: categories | Resolution:
Keywords: days54 | Merged in:
Authors: Nicolas M. Thiéry | Reviewers: Simon King, Frédéric
Report Upstream: N/A | Chapoton
Branch: | Work issues:
public/ticket/10963 | Commit:
Dependencies: #11224, #8327, | eb7b486c6fecac296052f980788e15e2ad1b59e4
#10193, #12895, #14516, #14722, | Stopgaps:
#13589, #14471, #15069, #15094, |
#11688, #13394, #15150, #15506 |
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Comment (by vbraun):
Another important issue that I think still needs discussion are the
relations between different categories-with-axioms. It seems to me,
thought I can't find it spelled out in the docs, that we want to allow
arbitrary relations of the type
{{{
category1 * axiom2 = category3 * axiom4 * axiom5
}}}
I'm writing this as multiplication to stress the commutativity of axioms
and the formal analogy with radical toric (or binomial) ideals. As usual
in the presence of relations, one can either work with equivalence classes
or normal forms (unique representatives). There is some talk about
manually specifying some distinguished representative/default construction
on this ticket, but I don't understand why that would be desirable.
To figure out all relations, we clearly need a Groebner basis for
relations. There are some well-known facts about Buchberger's algorithm
that ought be of importance to us:
* It should not be implemented recursively
* The "greedy" approach does not work: S-polynomials involving high-degree
terms can and will give rise to lower-degree generators. In other words,
you cannot expect to arrive at the normal form by removing axioms at each
step.
* Being explicit about term orders is key to the implementation
There is also a consistency issue about user-supplied axioms: They must
not induce further relations for the categories and axioms that Sage ships
with.
--
Ticket URL: <http://trac.sagemath.org/ticket/10963#comment:409>
Sage <http://www.sagemath.org>
Sage: Creating a Viable Open Source Alternative to Magma, Maple, Mathematica,
and MATLAB
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