#10963: More functorial constructions
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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 SimonKing):
Replying to [comment:409 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.
It is clear that this can't work in full mathematical generality (since in
principle there is an infinity of potential axioms to consider). But in a
CAS, it could actually work. Let's discuss it:
- At any point in time, we have a finite list of axioms (it may grow in
future, though).
- At any point in time, we have a finite set of "basic categories": By
this, I mean categories that provide the above mentioned "features":
`Sets` (provides `__contains__`), `Magmas` (provides (`__mul__`),
`AdditiveMagmas` (provides `__add__`) and so on.
- The union of the "basic categories" and the axioms generates a
commutative monoid: Multiplying categories means forming the join,
multiplying with axioms means applying them.
- If I understand correctly, Nicolas has introduced an ordering on the set
of categories anyway. In any case, it is clear that we *can* introduce an
ordering on the commutative monoid.
- The relations are, as you remark, binomial. Thus, the word problem in
our commutative monoid modulo relations can be solved by means of Gröbner
bases.
So far, the approach looks good. However, here is a problem: How do we
model the fact that the axiom `Distributive` does not apply to `Magmas`
and does not apply to `AdditiveMagmas`, but does apply to
`Magmas*AdditiveMagmas`?
Perhaps we actually do not need to model this fact in or commutative
monoid. Any categorial construction corresponds to an element of the
monoid. Two constructions result in the same category if and only if the
normal forms (modulo relations) of the corresponding monoid elements
coincide. Hence,
each standard monomial is a potential label of a category. However, it
should
be fine to assume that only a subset of the standard monomials actually
occurs
as label: `Magmas*Distributive` does not occur as label of a category, but
`Magmas*AdditiveMagmas*Distributive` does occur.
--
Ticket URL: <http://trac.sagemath.org/ticket/10963#comment:410>
Sage <http://www.sagemath.org>
Sage: Creating a Viable Open Source Alternative to Magma, Maple, Mathematica,
and MATLAB
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