#10963: More functorial constructions
-------------------------+-------------------------------------------------
Reporter: | Owner: stumpc5
nthiery | Status: needs_work
Type: | Milestone:
enhancement | Resolution:
Priority: major | Merged in:
Component: | Reviewers: Simon King
categories | Work issues: Reduce startup time by 5%. Avoid
Keywords: | "recursion depth exceeded (ignored)". Trivial
Authors: | doctest fixes.
Nicolas M. Thiéry | Dependencies: #11224, #8327, #10193, #12895,
Report Upstream: N/A | #14516, #14722, #13589
Branch: |
Stopgaps: |
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Comment (by nthiery):
Replying to [comment:57 SimonKing]:
> OK, that's a considerable change. In the "good" old times, a
> category C was (by definition) a sub-category of another category D,
> if and only if D was contained in `C.all_super_categories()`. So,
> you say this shall change (or already has?).
This was already like this for join categories. E.g. with plain sage
5.11.beta3:
{{{
sage: C1 = Category.join([Magmas(), CommutativeAdditiveMonoids()])
sage: C2 = Rings()
sage: C2.is_subcategory(C1)
True
}}}
> But then, I still don't see why this should be implemented by a plain
join category.
>
> Do we agree that there is a category `Magmas().Commutative()`, such
> that all information on `Algebras(ZZ).Commutative()` is provided by
> `Algebras(ZZ)` together with `Magmas().Commutative()`?
Those two pieces are indeed sufficient to recover the category:
{{{
sage: C = Algebras(ZZ) & Magmas().Commutative(); C
Category of commutative algebras over Integer Ring
}}}
But the join calculation is non trivial since Sage discovers by
introspection that there is a specific category for commutative rings,
so we get:
{{{
sage: C.super_categories()
[Category of algebras over Integer Ring, Category of commutative
rings]
}}}
Granted, the example is not so great since the commutative rings
category is actually currently empty; so we could think about removing
it, though it's likely to eventually contain something. For a more
interesting example we can look at:
Maybe a better example is:
{{{
sage: Rings().Finite().super_categories()
[Category of rings, Category of finite monoids]
}}}
And some good tests (compare with the sources!):
{{{
sage: from sage.categories.category_with_axiom import
TestObjectsOverBaseRing
sage: C = TestObjectsOverBaseRing(QQ)
sage:
C.Facade().Commutative().FiniteDimensional().Finite().Unital().super_categories()
[Category of finite finite dimensional test objects over base ring over
Rational Field,
Category of finite commutative test objects over base ring over Rational
Field,
Category of facade commutative test objects over base ring over Rational
Field,
Category of finite dimensional commutative unital test objects over base
ring over Rational Field]
sage: C.Facade().Commutative().Finite().Unital().super_categories()
[Category of finite commutative test objects over base ring over Rational
Field,
Category of facade commutative test objects over base ring over Rational
Field,
Category of unital test objects over base ring over Rational Field]
}}}
> Sure, we could then implement `Algebras(ZZ).Commutative()` by a
`JoinCategory`.
>
> But then, I would expect that we can have a class which is similar
> to `JoinCategory` but is specially designed and thus faster. After
> all, creating the join of a list of categories should be more
> complicated then adding a list of "axiom categories" (such as
> `Magmas().Commutative()` and `Magmas().Division()` and
> `Sets().Finite()`) to a given category (such as `Rings()`).
I guess I don't see at this point what can be made really
simpler/lighter for a join category when it comes from adding axioms
to a category. I still believe we can't spare the join calculation.
Cheers,
Nicolas
--
Ticket URL: <http://trac.sagemath.org/ticket/10963#comment:61>
Sage <http://www.sagemath.org>
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