Opened 7 years ago

Last modified 3 years ago

#10667 needs_work enhancement

Morphisms and Objects of Categories

Reported by: SimonKing Owned by: nthiery
Priority: major Milestone: sage-6.4
Component: categories Keywords: objects morphisms containment sd34
Cc: niles, jpflori Merged in:
Authors: Simon King Reviewers:
Report Upstream: N/A Work issues: Cope with non-unique number fields
Branch: Commit:
Dependencies: #9138, #11115, #11780 Stopgaps:

Description

Purpose

Introduce a framework for testing whether or not something is a morphism in a category. See the discussion on sage-algebra. Here is a summary of the discussion.

Methods for categories

Categories C should have a method C.has_morphism(f) answering whether f is a morphism in C. By symmetry, we want a method C.has_object(X), answering whether X is an object in C.

Note that we want X in C to be true if and only if X is an object of C (so, it is synonymous to C.has_object(X)). This currently is not always the case:

sage: P.<x,y> = QQ[]
sage: f = P.hom(reversed(P.gens()))
sage: f in Rings().hom_category()
True

but of course f is not an object of the hom-category (it is only contained in an object of the hom-category).

Class/Set of objects and morphisms

It would be nice to have container classes for the objects and for the morphisms of a category. Then, f in C.morphisms() would be a very natural notation for C.has_morphism(f), and X in C.objects() would be another way of saying X in C.

Of course, since f in C.morphisms() and f in C.objects() are nice notations, they should be as fast as possible -- otherwise, people wouldn't use them.

Further discussion should be put in comments to this ticket.

Attachments (1)

trac10667-morphisms_and_objects.patch (223.3 KB) - added by SimonKing 6 years ago.
Implement the classes of morphisms and objects of a category; improve performance; some fixes relating with morphisms; TestSuite for all categories; add a HomCategory class to any category

Download all attachments as: .zip

Change History (72)

comment:1 follow-up: Changed 7 years ago by SimonKing

If a category has not its own implementation of a hom-category, currently the join of the hom-categories of its super-categories is chosen. Hence, we have

sage: CommutativeRings().hom_category()
Category of hom sets in Category of rings
sage: WeylGroups().hom_category()
Category of hom sets in Category of sets

I don't like that. One problem is that, for test suites, one would like to have a sample object -- but there is no way to create a sample object for a join of arbitrary categories.

Moreover, the "hom sets in the Category of rings" are simply wrong for the category of commutative rings.

Instead, I suggest to walk through the list of all super categories of self, take the first that has the attribute HomCategory (i.e., has a custom implementation of a hom category), and insert self as argument for that HomCategory:

sage: HopfAlgebrasWithBasis(QQ).hom_category()
Category of hom sets in Category of hopf algebras with basis over Rational Field
sage: WeylGroups().hom_category()
Category of hom sets in Category of weyl groups
sage: type(HopfAlgebrasWithBasis(QQ).hom_category())
<class 'sage.categories.modules_with_basis.ModulesWithBasis.HomCategory'>
sage: type(WeylGroups().hom_category())
<class 'sage.categories.sets_cat.Sets.HomCategory'>

Of course, it may happen that several super categories have different custom implementation of hom categories, and we pick just one. But I think this should be taken care of manually, as join categories have a serious drawback, IMHO.

comment:2 in reply to: ↑ 1 Changed 7 years ago by nthiery

Hi Simon!

Replying to SimonKing:

If a category has not its own implementation of a hom-category, currently the join of the hom-categories of its super-categories is chosen. Hence, we have

sage: CommutativeRings().hom_category()
Category of hom sets in Category of rings
sage: WeylGroups().hom_category()
Category of hom sets in Category of sets

I don't like that. ...

Yeah. As mentioned in the code and in the road map [1], HomCategory? is just plain broken and needs a full refactoring. I just used the occasion to create a ticket with design suggestions: #10668.

If you want to improve things in this direction, please jump right away on #10668; it might actually not be that much work, and every intermediate step would be just a work around and a waste of time.

Thanks again for your continuous motion toward improving Sage in this area!

Cheers,

Nicolas

[1] http://trac.sagemath.org/sage_trac/wiki/CategoriesRoadMap

comment:3 Changed 7 years ago by niles

  • Cc niles added

cc me! Thanks :)

comment:4 follow-up: Changed 7 years ago by SimonKing

  • Authors set to Simon King

Depends on #10496, #10659, #8611, #10467

I wont to get the patch finally off my plate. So, here it is, although it isn't finished yet.

My patch provides the following:

sage: C = Rings()
sage: P.<x,y> = QQ[]
sage: f = P.hom(reversed(P.gens()))
sage: C.has_morphism(f)
True
sage: C.morphisms()
Class of morphisms in category of rings
sage: f in C.morphisms()
True
# Currently, a category is acknowledged as "small"
# iff it is a sub-category of FiniteEnumeratedSets()
sage: FiniteFields().morphisms()
Set of morphisms in category of finite fields
sage: f in FiniteFields().morphisms()
False
sage: P in C
True
sage: C.objects()
Class of objects in category of rings
sage: P in C.objects()
True
sage: FiniteFields().objects()
Set of objects in category of finite fields

Note that I interprete Objects() (the top-category in Sage) as the "category of all classes", although this definition probably is not water-proof:

sage: C.objects().category()
Category of objects
sage: FiniteFields().objects().category()
Category of sets

Some categories have a custom containment test, e.g., the category of fields. The containment test of C.objects() automatically tests whether C has a custom containment, and uses it if it is the case:

sage: PolynomialRing(QQ,[]).category()
Category of commutative rings
sage: PolynomialRing(QQ,[]) in Fields()
True
sage: PolynomialRing(QQ,[]) in Fields().objects()
True

The documentation for the new containers for objects and morphisms are added to the Sage reference manual -- please have a look.

SageObject versus CategoryObject

SageObject and CategoryObject were almost identical. In particular, SageObject provided a method category(), that by default returned the "category of objects". In addition, the specifition says that X is an object of X.category(), i.e., X in X.category().

But that approach yields to quite unnatural constructions. For example, 1.category() used to be the "category of elements of Integer Ring", whatever that means. Worse, one used to have

# Unpatched behaviour: Bug
sage: ZZ.hom([1]) in Rings().hom_category()
True

In other words, a ring homomorphism is considered a homset of the category of rings - of course, it should just be an element of a homset:

# With the patch:
sage: ZZ.hom([1]) in Rings().hom_category()
False
sage: ZZ.hom([1]) in ZZ.Hom(ZZ)
True
sage: ZZ.hom([1]) in Rings().morphisms()
True

Fixing that bug required to re-structure SageObject and CategoryObject:

  • I removed category() and _test_category() from SageObject and moved it to CategoryObject (which directly inherits from SageObject anyway).
  • I made Element and Map inherit from SageObject, not from CategoryObject, and removed the custom category() for Element. Of course, Parent still inherits from CategoryObject.

Note that by this change, it is now impossible to define nonsense such as Hom(2,3) (2 and 3 used to be objects in a category, so, there was a hom-set!).

Hom-categories

Compare #10668: This part of my patch is not finished, yet. I suppose that eventually this ticket here will depend on #10668.

Just for now, I implemented what I described in my previous post. Otherwise, many tests from the new test suites described below would fail.

Test Suites

The test suites of categories have been extended to test against the specification of the new features. In particular, the containers for morphisms and objects provide a test suite. The test suites for C.morphisms() and C.objects() and C.hom_category() are added to the test suite of C.

The patch adds a method an_object() to categories, that is used for additional tests. The default is to return example(), but this is not provided in all cases. The purpose of an_object() is narrower than that of example(), which is supposed to provide a concise instructive (but not necessarily very efficient) implementation. In contrast, an_object() may return an object of a subcategory, if nothing else is available. The test suite of C.an_object() becomes part of the test suite of C.

Similarly, I introduce a method a_morphism. By default, it takes the output of an_object(), tries to create an automorphism by reverting the list of generators, and if that fails then it returns the identity automorphism. The latter sounds trivial, but actually I found several cases where the identity automorphism was provided with the wrong category. This led to the following bug fixes:

  • In sage.categories.homset.Hom, there was an assymmetry between the categories of the domain and the codomain. I suggest to choose the meet of both categories. However, note that there was a comment like this:
       To avoid creating superfluous categories, homsets are in the
       homset category of the lowest category which currently says
       something specific about its homsets.
    
    which meant that the endomorphisms of the rational field used to belong to the "Category of hom sets in Category of rings". I didn't observe any problems changing it into the "Category of hom sets in Category of quotient fields".
  • Without the patch:
    sage: F = GF(5); MS = MatrixSpace(F,2,2)
    sage: G = MatrixGroup([MS([1,1,0,1])])
    sage: H = MatrixGroup([MS([1,0,1,1])])
    sage: phi = G.hom(H.gens())
    sage: phi.category_for()
    Category of groups
    sage: H.category()
    Category of finite groups
    sage: G.category()
    Category of finite groups
    
    Hence, the morphism belongs to a category that is too broad. With the patch:
    sage: F = GF(5); MS = MatrixSpace(F,2,2)
    sage: G = MatrixGroup([MS([1,1,0,1])])
    sage: H = MatrixGroup([MS([1,0,1,1])])
    sage: phi = G.hom(H.gens())
    sage: phi.category_for()
    Category of finite groups
    
    In several cases I have not been able to find any proper use case in Sage for a given category. In these cases, I have not been able to provide an_object(), so that in these cases I have to skip some tests from the test suites:
  • JoinCategory (I guess it is impossible to construct a generic object of the join of two arbitrary categories)
  • AbstractCategory (Do I understand correctly that the abstract category for the category of ZZ-modules would be the catogory of modules? I.e., one abstracts the base ring away?)
  • Schemes:
    # Bug, not fixed in the patch
    sage: Spec(QQ).category()
    Category of sets
    
  • UniqueFactorizationDomains
    # Bug, not fixed in the patch
    sage: ZZ in UniqueFactorizationDomains()
    False
    
  • AlgebraModules:
    sage: QQ['x'] in Algebras(QQ)
    True
    # Bug?
    sage: QQ['x']^3 in AlgebraModules(QQ['x'])
    False
    
  • GSets (Is there any G-set in Sage that knows that it is a G-set?)
  • DualObjectsCategory
  • Sets().Subquotients()
  • FiniteDimensional...: Most categories whose name starts with FiniteDimensional are not in use.
  • PartiallyOrderedMonoids
  • MonoidAlgebras
  • TensorProductsCategory
  • I added a minimal implementation of pointed sets:
    sage: from sage.categories.examples.pointed_sets import PointedSet
    sage: S = PointedSet([1,2,3],2)
    sage: S
    {1, 2, 3} -> 2
    sage: S is loads(dumps(S))
    True
    sage: S == PointedSet([1,2,3],3)
    False
    
  • I fixed the category of partially ordered sets.

Without patch:

sage: from sage.combinat.posets.posets import Poset
sage: elms = [1,2,3,4,5,6,7]
sage: rels = [[1,2],[3,4],[4,5],[2,5]]
sage: Poset((elms, rels), cover_relations = True).category()
Category of sets

With the patch, we obtain Category of partially ordered sets.

  • The category of matrix algebras has not been used. I added the obvious example:
    sage: MatrixSpace(QQ,2).category()
    Category of matrix algebras over Rational Field
    
    which used to be the category of algebras (not: matrix algebras).

Groupoids

Groupoids are considered as a category with a single object. However, this object did not exist. The patch provides it, modeled as an empty set:

sage: G = SymmetricGroup(5)
sage: C = Groupoid(G)
sage: O = C.an_object(); O
Unique object of Groupoid with underlying set SymmetricGroup(5)
sage: len(O)
0
sage: O.an_element()
Traceback (most recent call last):
...
EmptySetError: 

The elements of G correspond to morphisms of its groupoid. I suggest to actually consider them as morphisms (which is stronger than saying they correspond to morphisms):

sage: G.an_element() in C.morphisms()
True

Note that this point of view is needed in order to have a functorial approach towards actions, namely: If we want to view a group action of G on a set S as a functor from the groupoid of G to the category containing S as an object, then

  1. we need that functors can map morphisms (see #8807), and
  1. we need that group elements are considered as morphisms.

Actually, that was the starting point for my work on this ticket.

On the other hand, I do think that considering G as a homset of Groupoid(G) is not a very clean solution. But I believe this could be addressed on a different ticket, as this one is already too long.

Need for Speed

Of course, testing containment in a category C or in C.objects() or in C.morphisms() should be as fast as possible. I did the following:

  • I added a shortpath to C.__contains__ and C.objects().__contains__ for the common case that the category of the argument is C.
  • C.objects() and C.morphisms() are cached methods. By #8611, the overhead is now pretty small anyway.
  • Containment of an object O in a category C relies on testing whether O.category() is a sub-category of C. This is cached, by #8611. In addition, I remove the overhead entirely, by directly accessing the cache.
  • The containers for objects and morphisms are implemented in Cython. The default containment test is copied from the category, in order to reduce the overhead of calling a Python function. Therefore, F in O (where O = C.objects()) is sometimes even a little quicker than F in C:
    sage: F = GF(5)
    sage: C = Rings()
    sage: O = C.objects()
    sage: F in O
    True
    sage: timeit('F in O',number=100000)
    100000 loops, best of 3: 5.2 µs per loop
    sage: timeit('F in C',number=100000)
    100000 loops, best of 3: 5.38 µs per loop
    

Here are some timings. Their purpose is to show that X in C did not slow down (in fact, there is a speed-up in one special case), and that X in C.objects() has almost no overhead compared to X in C.

Setting:

sage: G = SymmetricGroup(5)
sage: P.<x,y> = QQ[]
sage: F = PolynomialRing(QQ,[])
sage: C1 = Rings()
sage: C2 = G.category()
sage: C3 = Fields()

Sanity tests:

# test that X in C.objects() is the same as X in C
# For C1:
sage: P in C1
True
sage: P in C1.objects()
True
sage: G in C1
False
sage: G in C1.objects()
False
sage: F in C1
True
sage: F in C1.objects()
True

# For C2, which is a join (that's a special case):
sage: P in C2
False
sage: P in C2.objects()
False
sage: G in C2
True
sage: G in C2.objects()
True
sage: F in C2
False
sage: F in C2.objects()
False

# For C3 (having a custom containment test):
sage: P in C3
False
sage: P in C3.objects()
False
sage: G in C3
False
sage: G in C3.objects()
False
sage: F in C3
True
sage: F in C3.objects()
True

Timings without the new patch (but with #10496, #10659, #8611 and #10467):

# containment in C1
sage: timeit('P in C1',number=100000)
100000 loops, best of 3: 11.1 µs per loop
sage: timeit('G in C1',number=100000)
100000 loops, best of 3: 4.32 µs per loop
sage: timeit('F in C1',number=100000)
100000 loops, best of 3: 11.9 µs per loop
# containment in C2
sage: timeit('P in C2',number=100000)
100000 loops, best of 3: 11.1 µs per loop
sage: timeit('G in C2',number=100000)
100000 loops, best of 3: 4.2 µs per loop
sage: timeit('F in C2',number=100000)
100000 loops, best of 3: 11.5 µs per loop
# containment in C3 (custom test for fields!)
sage: timeit('P in C3',number=100000)
100000 loops, best of 3: 16.1 µs per loop
sage: timeit('G in C3',number=100000)
100000 loops, best of 3: 20.5 µs per loop
sage: timeit('F in C3',number=100000)
100000 loops, best of 3: 17.9 µs per loop

Timings with the patch, including the new syntax X in C.objects():

# containment in C1
sage: timeit('P in C1',number=100000)
100000 loops, best of 3: 9.29 µs per loop
sage: timeit('P in C1.objects()',number=100000)
100000 loops, best of 3: 9.85 µs per loop
sage: timeit('G in C1',number=100000)
100000 loops, best of 3: 1.91 µs per loop
sage: timeit('G in C1.objects()',number=100000)
100000 loops, best of 3: 2.45 µs per loop
sage: timeit('F in C1',number=100000)
100000 loops, best of 3: 9.51 µs per loop
sage: timeit('F in C1.objects()',number=100000)
100000 loops, best of 3: 10.2 µs per loop
# containment in C2
sage: timeit('P in C2',number=100000)
100000 loops, best of 3: 9.2 µs per loop
sage: timeit('P in C2.objects()',number=100000)
100000 loops, best of 3: 9.85 µs per loop
# using the shortpath, as G.category() is C2
sage: timeit('G in C2',number=100000)
100000 loops, best of 3: 836 ns per loop
sage: timeit('G in C2.objects()',number=100000)
100000 loops, best of 3: 1.52 µs per loop
sage: timeit('F in C2',number=100000)
100000 loops, best of 3: 9.52 µs per loop
sage: timeit('F in C2.objects()',number=100000)
100000 loops, best of 3: 10.3 µs per loop
# containment in C3 (custom test for fields!)
sage: timeit('P in C3',number=100000)
100000 loops, best of 3: 14 µs per loop
sage: timeit('P in C3.objects()',number=100000)
100000 loops, best of 3: 15.9 µs per loop
sage: timeit('G in C3',number=100000)
100000 loops, best of 3: 15.7 µs per loop
sage: timeit('G in C3.objects()',number=100000)
100000 loops, best of 3: 18.3 µs per loop
sage: timeit('F in C3',number=100000)
100000 loops, best of 3: 15.6 µs per loop
sage: timeit('F in C3.objects()',number=100000)
100000 loops, best of 3: 17.3 µs per loop

Or, directly testing containment in the class of objects:

sage: O1 = C1.objects()
sage: O2 = C2.objects()
sage: O3 = C3.objects()
sage: timeit('P in O1',number=100000)
100000 loops, best of 3: 8.88 µs per loop
sage: timeit('G in O1',number=100000)
100000 loops, best of 3: 1.56 µs per loop
sage: timeit('F in O1',number=100000)
100000 loops, best of 3: 9.31 µs per loop
sage: timeit('P in O2',number=100000)
100000 loops, best of 3: 8.94 µs per loop
sage: timeit('G in O2',number=100000)
100000 loops, best of 3: 704 ns per loop
sage: timeit('F in O2',number=100000)
100000 loops, best of 3: 9.29 µs per loop
sage: timeit('P in O3',number=100000)
100000 loops, best of 3: 14.9 µs per loop
sage: timeit('G in O3',number=100000)
100000 loops, best of 3: 17.1 µs per loop
sage: timeit('F in O3',number=100000)
100000 loops, best of 3: 16.5 µs per loop

comment:5 Changed 7 years ago by SimonKing

  • Status changed from new to needs_info

So, what's the status of the ticket?

I need more info!

First thing: I am still not happy with the groupoids. But can this perhaps be solved in a different ticket?

Second and more urgent thing? Why does my example of pointed sets not inherit from PointedSets().parent_class? What did I do wrong? I asked on sage-support, but didn't receive a reply.

The problem is:

sage: from sage.categories.examples.pointed_sets import PointedSet
sage: S = PointedSet([1,2,3],2)
sage: S.category()
Category of pointed sets
sage: S.__class__
<class 'sage.categories.examples.pointed_sets.PointedSet_with_category'>

So, the category is initialised. But:

sage: isinstance(S,PointedSets().parent_class)
False

What goes wrong in my implementation?

comment:6 in reply to: ↑ 4 Changed 7 years ago by nthiery

Hi Simon!

I have only browsed quickly through this yet. I'll try to have a look soon at the broken parent you mention in the other comment. Just some small comments before heading for my bed.

  • In sage.categories.homset.Hom, there was an assymmetry between the categories of the domain and the codomain. I suggest to choose the meet of both categories. However, note that there was a comment like this:
       To avoid creating superfluous categories, homsets are in the
       homset category of the lowest category which currently says
       something specific about its homsets.
    
    which meant that the endomorphisms of the rational field used to belong to the "Category of hom sets in Category of rings". I didn't observe any problems changing it into the "Category of hom sets in Category of quotient fields".

I wrote this comment. This won't break indeed, but there might eventually be a penalty in creating that many categories. I need to think back about it, but this comment might become irrelevant since we are going to break the inheritance in-between hom categories.

  • AbstractCategory (Do I understand correctly that the abstract category for the category of ZZ-modules would be the catogory of modules? I.e., one abstracts the base ring away?)

As I said, don't bother understanding: that's going to be removed. If it gets in your way, just kill the beast, and remove right now everything about AbstractCategory? (typically by taking over the appropriate bits of the patch I mentioned).

comment:7 Changed 7 years ago by SimonKing

  • Status changed from needs_info to needs_review

Thanks to Nicolas for his comments on sage-algebra explaining why my example of pointed sets didn't work well. It is now fixed. Hence, ready for review!

comment:8 Changed 7 years ago by SimonKing

I updated the patch, so that it now applies to sage-4.6.2.alpha4. There are no dependencies.

comment:9 Changed 7 years ago by SimonKing

Since the patchbot keeps complaining that the patch did not apply to good old sage-4.6.1 and since I just verified once again that the patch cleanly applies to sage-4.6.2.alpha4, I replaced the patch by an identical copy and hope that it pushes the patchbot to try it another time with the new sage version.

comment:10 Changed 6 years ago by SimonKing

  • Status changed from needs_review to needs_work

On #9054, William expressed his anger about category containment tests being too slow. That reminded me of the ticket here. Since the patches do not apply, it needs work. But I guess it is worth while to resume work on that ticket.

comment:11 Changed 6 years ago by SimonKing

I have updated the patch, so that it should apply against sage-4.7.2.alpha2. I did not run tests, yet. Here are some timings:

sage: from sage.rings.commutative_ring import is_CommutativeRing
sage: %timeit is_CommutativeRing(QQ)
625 loops, best of 3: 1.09 µs per loop
sage: C = CommutativeRings().objects()
sage: %timeit QQ in C
625 loops, best of 3: 3.82 µs per loop

is_CommutativeRing simply tests whether QQ is an instance of sage.rings.ring.CommutativeRing, which is of course very fast (but not very reliable from a mathematical point of view.

Anyway, I try to squeeze QQ in C a bit more.

comment:12 Changed 6 years ago by SimonKing

  • Dependencies set to #9138, #11115
  • Status changed from needs_work to needs_review

I created a new version of my patch. The aim is to make the performance of containment tests even better. I did the following, compared with the old patch:

  • I make the patch dependent on #9138 and #11115 (both help to come to speed).
  • I've put power series rings into the category and coercion framework.
  • I introduced base() for join categories: If at least one of the underlying categories has a base and if there is no conflict with different bases, then the join shall have that base as well. That is needed because some algebras have a join category by #9138.
  • GroupAlgebras? should not only be Hopf algebras with basis but also group algebras. Hence, I made it a join of the two.
  • I implemented is_supercategory (there has only been is_subcategory), and use it to make containment tests even faster. It depends on #11115, because I use the Cython class of cached methods.

I guess the best news is that the containment test via category framework can now compete with a pure class check, if that class check is done in Python. I take, for example, commutative rings:

sage: from sage.rings.commutative_ring import is_CommutativeRing
sage: is_CommutativeRing??
Type:           function
Base Class:     <type 'function'>
String Form:    <function is_CommutativeRing at 0x118c488>
Namespace:      Interactive
File:           /mnt/local/king/SAGE/sandbox/sage-4.7.2.alpha2/local/lib/python2.6/site-packages/sage/rings/commutative_ring.py
Definition:     is_CommutativeRing(R)
Source:
def is_CommutativeRing(R):
    return isinstance(R, CommutativeRing)
sage: is_CommutativeRing(QQ)
True
sage: s = SymmetricGroup(4)
sage: is_CommutativeRing(s)
False
sage: %timeit is_CommutativeRing(QQ)
625 loops, best of 3: 1.09 µs per loop
sage: %timeit is_CommutativeRing(s)
625 loops, best of 3: 3.51 µs per loop

Since is_CommutativeRing just tests the class, it is supposed to be very fast. But let us compare with the generic containment test in the category of commutative rings and in the class of objects of that category:

sage: C = CommutativeRings()
sage: O = C.objects()
sage: QQ in C
True
sage: QQ in O
True
sage: s in C
False
sage: s in O
False
sage: %timeit QQ in C
625 loops, best of 3: 4.62 µs per loop
# Timing in sage-4.6.2: 12.9 µs per loop
sage: %timeit QQ in O
625 loops, best of 3: 1.5 µs per loop
sage: %timeit s in C
625 loops, best of 3: 4.69 µs per loop
# Timing in sage-4.6.2: 10.2 µs per loop
sage: %timeit s in O
625 loops, best of 3: 1.46 µs per loop

Hence, is_CommutativeRing(s) is slower than s in O, where O = CommutativeRings().objects().

The reason for that speedup is Cython. While is_CommutativeRing is a Python function, the objects of a category are implemented in Cython. Moreover, category containment is tested by the cached method is_supercategory, which also is in Cython by #9138.

Caveat: I did not run the full tests, yet, and I would like to try and remove some custom containment tests in the category framework, that tend to be slower than the generic test and might not be needed with #9138.

comment:13 Changed 6 years ago by SimonKing

I forgot to mention that I also improved is_Ring.

With sage-4.7.2.alpha1 plus #9138 and #11115:

sage: from sage.rings.ring import is_Ring
sage: MS = MatrixSpace(QQ,2)
sage: %timeit is_Ring(QQ)
625 loops, best of 3: 5.1 µs per loop
sage: %timeit is_Ring(MS)
625 loops, best of 3: 17.3 µs per loop
sage: C = Rings()
sage: %timeit QQ in C
625 loops, best of 3: 4.18 µs per loop
sage: %timeit MS in C
625 loops, best of 3: 4.31 µs per loop

With sage-4.7.2.alpha2 plus #9138 and #11115 and the patch from here:

sage: from sage.rings.ring import is_Ring
sage: MS = MatrixSpace(QQ,2)
sage: %timeit is_Ring(QQ)
625 loops, best of 3: 259 ns per loop
sage: %timeit is_Ring(MS)
625 loops, best of 3: 17.5 µs per loop
sage: C = Rings().objects()
sage: %timeit QQ in C
625 loops, best of 3: 1.49 µs per loop
sage: %timeit MS in C
625 loops, best of 3: 1.57 µs per loop

comment:14 follow-up: Changed 6 years ago by SimonKing

  • Milestone set to sage-5.0

I leave it as "needs review", but I think I have to adopt the Cython improvements on morphism containment tests as well.

comment:15 in reply to: ↑ 14 Changed 6 years ago by SimonKing

  • Status changed from needs_review to needs_work
  • Work issues set to doctests

Replying to SimonKing:

I leave it as "needs review", but I think I have to adopt the Cython improvements on morphism containment tests as well.

Nope, it wouldn't easily work for the morphisms.

It turns out that I have to fix many doctests.

comment:16 follow-up: Changed 6 years ago by SimonKing

It is a very bad error, and I don't know at which point I introduced it. It is about incompatible method resolution orders:

sage: class Foo(Homset, Objects().HomCategory(Objects()).parent_class): pass
....: 
---------------------------------------------------------------------------
TypeError                                 Traceback (most recent call last)

/mnt/local/king/SAGE/sandbox/sage-4.7.2.alpha2/devel/sage-main/<ipython console> in <module>()

TypeError: Error when calling the metaclass bases
    Cannot create a consistent method resolution
order (MRO) for bases Homset, Objects.HomCategory.parent_class

It seems that the problem is in the order in which the two classes are presented:

sage: class Foo(Objects().HomCategory(Objects()).parent_class, Homset): pass
....: 
sage:

comment:17 in reply to: ↑ 16 ; follow-up: Changed 6 years ago by nthiery

Replying to SimonKing:

It is a very bad error, and I don't know at which point I introduced it. It is about incompatible method resolution orders:

sage: class Foo(Homset, Objects().HomCategory(Objects()).parent_class): pass
order (MRO) for bases Homset, Objects.HomCategory.parent_class

Yeah, this kind of error can be quite tricky indeed. This is the very technical bit where I am bit uneasy about the future scaling of categories using dynamic classes. The only way to avoid such errors sanely is to specify general rules about the order of the base classes of a class. There are very minimal comments about that at the end of the primer. Reducing the risk of this kind of issue is also one of the goal of #10963 (the more is done automatically, the higher are the chances of consistency).

It seems that the problem is in the order in which the two classes are presented:

sage: class Foo(Objects().HomCategory(Objects()).parent_class, Homset): pass
....: 
sage:

Here, I would say that the rule is that category code (in particular what comes from a_category.parent_class) should always come after concrete classes.

Good luck!

By the way: congrats on all your category optimization work. I love it!

comment:18 in reply to: ↑ 17 Changed 6 years ago by SimonKing

Replying to nthiery:

The only way to avoid such errors sanely is to specify general rules about the order of the base classes of a class.

I thought of that. But it could slow things down.

Here, I would say that the rule is that category code (in particular what comes from a_category.parent_class) should always come after concrete classes.

It is not very much concrete. The real error reduces to what I have shown. But in fact, the class Homset comes from sage.categories.objects.HomCategory.ParentMethods. To be precise:

sage: from sage.structure.dynamic_class import dynamic_class
sage: C = Objects().hom_category()
sage: dynamic_class('bla', (C.parent_class,), C.ParentMethods)

And going further down, the above is caused by some lines in the parent_class lazy attribute:

        return dynamic_class("%s.parent_class"%self.__class__.__name__,
                             tuple(cat.parent_class for cat in self.super_categories()),
                             self.ParentMethods,
                             reduction = (getattr, (self, "parent_class")))

In my case, cat.parent_class is a sub-class of self.ParentMethods (self being C above). Here is the relation with my patch:

sage: C = ChainComplexes(ZZ)
sage: HC = C.hom_category()
sage: HC.super_categories()
[Category of hom sets in Category of objects]
# it was [Category of sets] without my patch!

And finally, that comes from

sage: HC.base_category
Category of chain complexes over Integer Ring
# it used to be Category of objects without my patch

Since [category.hom_category() for category in self.base_category.super_categories()] is part of the super categories, the change of the base category was the ultimate cause of the error.

However, I wouldn't like to change the base_category. After all, the base_category is the category to which the homsets belong. Here, the homsets belong to the category of chain complexes over Integer Ring. Thus, "category of objects" is plainly wrong.

So, for now, I see two ways to fix it:

  1. Change HomCategory.super_categories. It should only return self.extra_super_categories() + Sets() (after all, homsets are sets), but not [category.hom_category() for category in self.base_category.super_categories()].
  1. Change parent_class, so that not all of tuple(cat.parent_class for cat in self.super_categories()) is included, but only the bits that are no sub-classes of self.ParentMethods.

Is it really mathematically correct that the hom-category of a category C is a subcategory of the hom-category of any super-category of C?

For example, if C is the category of unital K-algebras (K some field) then C is a subcategory of the category of K-vectorspaces. The homsets of K-vectorspaces are K-vectorspaces. But the homsets of unital K-algebras do not form K-vectorspaces, isn't it?

At least, computationally, option 1 is faster than option 2. And when you confirm that option 2 (which is the status quo!) is actually mathematically wrong then it is clear what I will do.

By the way: congrats on all your category optimization work. I love it!

Thank you!

comment:19 follow-up: Changed 6 years ago by SimonKing

Indeed the hom category of algebras is attributed as a subcategory of the category of vectorspaces:

sage: Algebras(QQ).hom_category().is_subcategory(VectorSpaces(QQ))
True

Isn't that plainly wrong?

comment:20 Changed 6 years ago by SimonKing

It seems to me that the implementation can not so easily be cleaned.

In some cases, we do want that the hom category of a category inherits stuff from the hom category of a super category - simply in order to avoid code duplication. For example, VectorSpaces(...).hom_category() does (and should) inherit from Modules(...).hom_category().

In other cases, we do not want that inheritance. For example, we do not want that Algebras(...).hom_category() inherits from VectorSpaces(...).hom_category().

Indeed, we currently have

sage: Algebras(QQ).hom_category().extra_super_categories()
[Category of sets]

I tried to understand why we have the above answer. We have

sage: Algebras(QQ).hom_category().extra_super_categories.__module__
'sage.categories.rings'

So, the method is inherited from the hom category of the category of rings.

Why is it (correctly) not inherited from the hom category of the category of Q-modules?

sage: Modules(QQ).hom_category().extra_super_categories()
[Category of vector spaces over Rational Field]

It seems to me that the correct inheritence is just due to the fact that Algebras(...).super_categories() returns first Rings() and then Modules(...)

sage: Algebras(QQ).super_categories()
[Category of rings, Category of vector spaces over Rational Field]

If that list would be returned in the opposite order, then Algebras(QQ).hom_category() would pick up the extra_super_categories method from Modules(QQ).hom_category(), which would not be correct.

I think that inheritance being dependent on the order of a list of super categories is very much error prone and difficult to debug.

comment:21 in reply to: ↑ 19 ; follow-up: Changed 6 years ago by nthiery

Replying to SimonKing:

Indeed the hom category of algebras is attributed as a subcategory of the category of vectorspaces:

sage: Algebras(QQ).hom_category().is_subcategory(VectorSpaces(QQ))
True

Isn't that plainly wrong?

Yes it is plain wrong. We had discussed this early this Spring, and we even both have patches fixing this (using a different syntax) :-) See #10668.

This property only holds for full subcategories, and last time we discussed that we were looking for a syntax specifying when a category is a full subcategory of another one.

Cheers,

Nicolas

comment:22 in reply to: ↑ 21 ; follow-up: Changed 6 years ago by SimonKing

Replying to nthiery:

Yes it is plain wrong. We had discussed this early this Spring, and we even both have patches fixing this (using a different syntax) :-) See #10668.

Ouch! I completely forgot about that other ticket!

This property only holds for full subcategories, and last time we discussed that we were looking for a syntax specifying when a category is a full subcategory of another one.

OK. Then I wonder what I should do here.

The purpose of this ticket is to provide containers for the morphisms and objects of a category, and to provide an acceptable speed. It is not the purpose to refactor hom categories - because that is to be done in #10668.

Hence, for now, I suggest that I will restrict myself on getting the tests pass. I guess it will be possible in a couple of days, and may require to change HomCategory.super_categories in the way I suggested above. But apart from that, I will not aim at refactoring everything.

comment:23 follow-up: Changed 6 years ago by SimonKing

Another mathematical question:

I thought that any hom category is a sub-category of the category of sets.

Currently, HomCategory(Objects()).super_categories() returns Objects(), which is a bug anyway, because it does not return a list! But should it return [Sets()]? The same answer for HomCategory(SetsWithPartialMaps()).super_categories()?

comment:24 in reply to: ↑ 22 Changed 6 years ago by nthiery

Replying to SimonKing:

OK. Then I wonder what I should do here.

The purpose of this ticket is to provide containers for the morphisms and objects of a category, and to provide an acceptable speed. It is not the purpose to refactor hom categories - because that is to be done in #10668.

Hence, for now, I suggest that I will restrict myself on getting the tests pass. I guess it will be possible in a couple of days, and may require to change HomCategory.super_categories in the way I suggested above. But apart from that, I will not aim at refactoring everything.

This sounds good. I just hope that there are not too many things that currently depends on that (buggy most of the time, but from time to time correct) inheritance.

comment:25 in reply to: ↑ 23 Changed 6 years ago by nthiery

Replying to SimonKing:

Another mathematical question:

I thought that any hom category is a sub-category of the category of sets.

Currently, HomCategory(Objects()).super_categories() returns Objects(), which is a bug anyway, because it does not return a list! But should it return [Sets()]? The same answer for HomCategory(SetsWithPartialMaps()).super_categories()?

Yes it should definitely be fixed to be a list. Now, I would tend to be safe and stick to [Objects()], unless some abstract category expert is absolutely convinced that [Sets()] is always correct (I doubt it).

comment:26 Changed 6 years ago by SimonKing

Personally, I believe that Objects() is not a category in the proper meaning of the word. I think for any category C and any objects A,B of C, then Hom(A,B) must by definition be a set.

But you are right, in case of doubt one should use Objects(), not Sets().

comment:27 follow-up: Changed 6 years ago by SimonKing

No, after all, I think that Sets() is correct.

We already have

sage: O = Objects()
sage: H = O.HomCategory(O)
sage: H.super_categories()
[Category of sets]

and a comment in the doc string of H.super_categegories:

            """
            This declares that any homset `Hom(A, B)` for `A` and `B`
            in the category of objects is a set.
            This more or less assumes that the category is locally small.
            See http://en.wikipedia.org/wiki/Category_(mathematics)

            EXAMPLES::

                sage: Objects().hom_category().super_categories()
                [Category of sets]
            """

So, that should be fine.

comment:28 follow-up: Changed 6 years ago by SimonKing

Currently, I'm having trouble with getting an appropriate class for the homsets.

You know that rings have a specially designed class for their homsets:

sage: Rings().HomCategory
<class 'sage.categories.rings.Rings.HomCategory'>
sage: Rings().HomCategory(Rings()).parent_class.__module__
'sage.categories.rings'

By #9944 and #9138, polynomial rings are (commutative) algebras and not just rings. The category of algebras does not define their own HomCategory class.

However, two of its super categories have special HomCategory, namely

sage: Modules(ZZ).HomCategory
<class 'sage.categories.modules.Modules.HomCategory'>
sage: Rings().HomCategory
<class 'sage.categories.rings.Rings.HomCategory'>

Wouldn't it be a good idea to create a lazy attribute HomCategory for sage.categories.category.Category, that returns a dynamic class formed by all the hom category classes of the super categories?

Hence, what I suggest means that Algebras(ZZ).HomCategory would be a sub-class of both Rings().HomCategory and Modules(ZZ).HomCategory. At least in this example, it would work, regardless of the order:

sage: class Foo(Rings().HomCategory, Modules(ZZ).HomCategory): pass
....: 
sage: class Foo(Modules(ZZ).HomCategory, Rings().HomCategory): pass
....: 

As a dynamic class, we would probably have

sage: from sage.structure.dynamic_class import dynamic_class
sage: from sage.categories.category import HomCategory
sage: dynamic_class('FooHomCategory', (Rings().HomCategory, Modules(ZZ).HomCategory, HomCategory))
<class 'sage.categories.rings.FooHomCategory'>

But note that putting HomCategory in front of the tuple or providing it as second argument after the tuple will not work.

I think that this would be a very clean solution. The method resolution order of the dynamic class would, if I understand correctly, first pick up the stuff defined for rings, then the stuff defined for modules, and finally the generic stuff of HomCategory.

comment:29 follow-up: Changed 6 years ago by SimonKing

Perhaps a related question: In sage/categories/rings.py, we have

    class HomCategory(HomCategory):
        class ParentMethods:
            def __new__(cls, X, Y, category):
                from sage.rings.homset import RingHomset
                return RingHomset(X, Y, category = category)
            def __getnewargs__(self):
                return (self.domain(), self.codomain(), self.category())

Wouldn't it be possible to simply have

    class HomCategory(HomCategory):
        class ParentMethods(RingHomset):
            pass

?

comment:30 in reply to: ↑ 28 ; follow-ups: Changed 6 years ago by nthiery

Hi Simon!

Replying to SimonKing:

Wouldn't it be a good idea to create a lazy attribute HomCategory for sage.categories.category.Category, that returns a dynamic class formed by all the hom category classes of the super categories?

If I remember correctly, that's more or less what you had implemented for #10668 :-) Of course (and you had taken care of this), unless one is having a full subcategory, one should have this inheritance only for elements of the hom category (i.e. morphisms), not for the homsets or the category.

So ... Do you see a temporary quick fix to have #10667 work for the moment, before we go on to #10668?

Cheers,

comment:31 in reply to: ↑ 29 Changed 6 years ago by nthiery

Replying to SimonKing:

Perhaps a related question: In sage/categories/rings.py, we have

    class HomCategory(HomCategory):
        class ParentMethods:
            def __new__(cls, X, Y, category):
                from sage.rings.homset import RingHomset
                return RingHomset(X, Y, category = category)
            def __getnewargs__(self):
                return (self.domain(), self.codomain(), self.category())

Wouldn't it be possible to simply have

    class HomCategory(HomCategory):
        class ParentMethods(RingHomset):
            pass

?

Somehow both are wrong; I had just put that here to make the damn thing work for the moment: the category really should not be dealing with the concrete classes used to implement the Homsets. That's Hom's job at best, but we need to design a proper protocol for this.

comment:32 in reply to: ↑ 27 Changed 6 years ago by nthiery

Replying to SimonKing:

No, after all, I think that Sets() is correct. and a comment in the doc string of H.super_categegories:

            """
            This declares that any homset `Hom(A, B)` for `A` and `B`
            in the category of objects is a set.
            This more or less assumes that the category is locally small.
            See http://en.wikipedia.org/wiki/Category_(mathematics)

            EXAMPLES::

                sage: Objects().hom_category().super_categories()
                [Category of sets]
            """

ROTFL. I wrote that comment. So I guess I should agree with it :-)

comment:33 Changed 6 years ago by SimonKing

Yes, it seems to work nicely with lazy attribute plus dynamic class!!

I have (to be a doctest):

            sage: A = Algebras(ZZ)
            sage: H = A.hom_category()   #indirect doctest
            sage: H
            Category of hom sets in Category of algebras over Integer Ring
            sage: isinstance(H, Rings().HomCategory)
            True
            sage: isinstance(H, Modules(ZZ).HomCategory)
            True

comment:34 Changed 6 years ago by SimonKing

PS:

I forgot to add that the super categories of the hom category are fine as well. We have:

sage: A = Algebras(ZZ)
sage: H = A.hom_category()   #indirect doctest
sage: H.super_categories()
[Category of hom sets in Category of objects]
sage: H.an_object()
Set of Homomorphisms from Univariate Polynomial Ring in x over Integer Ring to Univariate Polynomial Ring in x over Integer Ring
sage: H.an_object().category()
Category of hom sets in Category of algebras over Integer Ring
sage: H.an_object().category().super_categories()
[Category of hom sets in Category of objects]

comment:35 in reply to: ↑ 30 Changed 6 years ago by SimonKing

Replying to nthiery:

So ... Do you see a temporary quick fix to have #10667 work for the moment, before we go on to #10668?

I don't know yet. I am still walking my way accross the mine field of doctest errors. For example, the idea to provide a lazy attribute dynamic class for HomCategory is simply a means to enable

sage: R = QQ['z0','z1','z2','z3']
sage: R.hom(R.gens())
Ring endomorphism of Multivariate Polynomial Ring in z0, z1, z2, z3 over Rational Field
  Defn: z0 |--> z0
        z1 |--> z1
        z2 |--> z2
        z3 |--> z3

That easy example would have failed in the previous version of my patch.

comment:36 follow-up: Changed 6 years ago by SimonKing

The tests in devel/sage/doc pass. That encourages me to post my current patch version, so that you can already have a look at it (when you have the time; I know, probably you don't...).

However, I keep it as "needs work", since I did not run the devel/sage/sage/, and since I need to check whether I really tested and documented all new functionality.

comment:37 in reply to: ↑ 30 Changed 6 years ago by SimonKing

Replying to nthiery:

So ... Do you see a temporary quick fix to have #10667 work for the moment, before we go on to #10668?

Meanwhile I think that a quick fix based on a modification of the current patch will be doable. I get 242 doctest errors related with Steenrod algebras. They seem to be caused by a wrong method resolution order, and I suppose that it can be fixed by changing the order on the list of super categories for some category. The remaining test failures seem to be harmless.

comment:38 follow-up: Changed 6 years ago by SimonKing

Well, mostly harmless. Some involve to implement the category framework for uni- and multivariate power series rings. Multivariate power series rings was a recent addition - so, why has it not been done in the first place?

It seems that the 242 Steenrod errors are mostly gone. At least, TestSuite(SteenrodAlgebra(2)).run() works.

Time to call it a day...

comment:39 in reply to: ↑ 38 Changed 6 years ago by nthiery

Replying to SimonKing:

Well, mostly harmless. Some involve to implement the category framework for uni- and multivariate power series rings. Multivariate power series rings was a recent addition - so, why has it not been done in the first place?

Still way too much code using prehistoric stuff; so devs and reviewers don't take the right examples to start from.

It seems that the 242 Steenrod errors are mostly gone. At least, TestSuite(SteenrodAlgebra(2)).run() works.

Yippee!

Time to call it a day...

:-)

comment:40 in reply to: ↑ 36 ; follow-up: Changed 6 years ago by nthiery

Replying to SimonKing:

The tests in devel/sage/doc pass. That encourages me to post my current patch version, so that you can already have a look at it (when you have the time; I know, probably you don't...).

I really should take the time. At this point, I am so much behind with your patches that I am thinking we should have a face to face review sprint. Alas, I don't have yet the schedule for my classes this fall to see whether I could come to the Sage days at KL. Are you planning to come to France anytime soon?

comment:41 in reply to: ↑ 40 Changed 6 years ago by SimonKing

Replying to nthiery:

Alas, I don't have yet the schedule for my classes this fall to see whether I could come to the Sage days at KL. Are you planning to come to France anytime soon?

I will be in Kaiserslautern, but apart from that I have no plans at all. But there should be some travel money available from my project...

For the record: Steenrod algebra tests pass fully. I am still having trouble to find the right order of base classes for dynamic classes. Probably it would not be possible at all. Thus, with my current patch (not posted), I catch the resulting type error, and return a generic class (such as Objects().HomCategory) for implementing the hom category of a category.

comment:42 follow-up: Changed 6 years ago by SimonKing

I am making some progress.

Testsuites are really a good thing! By adding tests for morphisms, I found a couple of bugs. And that's why my patch can not just be a short work-around (unless I make the Testsuites skip some tests). It will contain fixes in different parts of sage.

Just one example:

sage: E = CombinatorialFreeModule(ZZ, [1,2,3])
sage: F = CombinatorialFreeModule(ZZ, [2,3,4])
sage: H = Hom(E, F)
sage: TestSuite(H).run()
Failure in _test_additive_associativity:
Traceback (most recent call last):
  File "/mnt/local/king/SAGE/broken/local/lib/python2.6/site-packages/sage/misc/sage_unittest.py", line 275, in run
    test_method(tester = tester)
  File "/mnt/local/king/SAGE/broken/local/lib/python2.6/site-packages/sage/categories/commutative_additive_semigroups.py", line 80, in _test_additive_associativity
    tester.assert_((x + y) + z == x + (y + z))
TypeError: unsupported operand type(s) for +: 'ModuleMorphismByLinearity' and 'ModuleMorphismByLinearity'
------------------------------------------------------------
Failure in _test_an_element:
Traceback (most recent call last):
  File "/mnt/local/king/SAGE/broken/local/lib/python2.6/site-packages/sage/misc/sage_unittest.py", line 275, in run
    test_method(tester = tester)
  File "/mnt/local/king/SAGE/broken/local/lib/python2.6/site-packages/sage/categories/sets_cat.py", line 388, in _test_an_element
    tester.assertEqual(self(an_element), an_element, "element construction is not idempotent")
...

The reason is that dumps(H.zero()) fails:

sage: dumps(H.zero())
---------------------------------------------------------------------------
PicklingError                             Traceback (most recent call last)

/home/king/SAGE/work/categories/objects/<ipython console> in <module>()

/mnt/local/king/SAGE/broken/local/lib/python2.6/site-packages/sage/structure/sage_object.so in sage.structure.sage_object.dumps (sage/structure/sage_object.c:8274)()

/mnt/local/king/SAGE/broken/local/lib/python2.6/site-packages/sage/structure/sage_object.so in sage.structure.sage_object.SageObject.dumps (sage/structure/sage_object.c:2183)()

PicklingError: Can't pickle <type 'function'>: attribute lookup __builtin__.function failed

There are more of those bugs.

The good news: I think I found a stable way to get the method resolution order of hom category parent classes right.

I even tried to get rid of the odd explicit __new__ method in sage.categories.rings.Rings.HomCategory.ParentMethods. But I am not sure if I will succeed.

comment:43 follow-up: Changed 6 years ago by SimonKing

  • Work issues changed from doctests to Cartesian products

Just a status report: I got rid of the __new__ method. Instead, I produce a __classcall__ method, similar to what is done in UniqueRepresentation (and in fact I make sage.categories.rings.Rings.HomCategory.ParentClass inherit from UniqueRepresentation).

I have already mentioned that I added some methods to the Cartesian product categories, so that some test suites actually passed. Next, I fixed another problem with Cartesian products: It has not been possible to create Cartesian products of algebras. It neither worked in sage-4.6.2 nor in sage-4.7.2.alpha1+#9138, but failed with different errors.

sage: C = cartesian_product([ZZ['x'], ZZ['y']])
<BOOM>

After my patches and the addition of the base_ring method to Cartesian product categories, the problem arose with __init_extra__ method in sage.categories.algebras.Algebras.ParentMethods: The Cartesian product of algebras over a ring R is an algebra over R (apparently acting diagonally). The __init_extra__ tries to create a coercion from the R to the cartesian product. However, that ended in an infinite recursion. I solved it by adding a from_base_ring method, that is understood by __init_extra__.

The remaining problem concerns summation of elements of Cartesian products. Multiplication is defined, via sage.categories.magmas.Magmas.CartesianProduct.ParentMethods.product. But summation is missing. I guess it should be implemented in sage.categories.AdditiveMagmas.CartesianProduct.ParentMethods.summation.

comment:44 Changed 6 years ago by SimonKing

Helas.

I fixed the Cartesian products, but now I have a couple of hundred errors in elliptic curves. The problem is that non-unique parents occur in sage.libs.singular.ring.singular_ring_new when constructing a multivariate polynomial ring over a number field.

I have no idea why that problem is invisible without my patch.

It seems that this ticket is a can of worms. I will keep the current bug fixes in my patch (sorry that I didn't submit it yet), but from now on new bug fixes will give rise to new tickets.

comment:45 Changed 6 years ago by SimonKing

  • Dependencies changed from #9138, #11115 to #9138, #11115, #11780

The ticket for the non-unique polynomial rings is #11780.

Changed 6 years ago by SimonKing

Implement the classes of morphisms and objects of a category; improve performance; some fixes relating with morphisms; TestSuite for all categories; add a HomCategory class to any category

comment:46 Changed 6 years ago by SimonKing

I have a patch at #11780 that solves the problem. I am thus also submitting my current patch on morphisms and objects. It contains many new features that I will describe later. It is still "needs work", since the patch contains some uncommented code that should eventually be deleted, and since there will certainly be a couple of "trivial" doctest errors, namely for tests that concern the super categories of a category (I had to re-order the super categories in order to get the method resolution orders right).

But if you want to play with it: Go ahead (and don't forget to apply #11780 first...).

comment:47 Changed 6 years ago by SimonKing

Nicolas, I have a question on the category of schemes. In sage.categories.schemes.Schemes.HomCategory, there is a comment saying

            FIXME: what category structure is there on Homsets of schemes?
            The result above is wrong, and should be fixed during the next
            homsets overhaul.

Is there any answer? What is the category structure?

comment:48 Changed 6 years ago by SimonKing

Helas. The number of errors has decreased with the new patch. However, there remain numerous errors of the same kind ("there is some non-unique parent and thus the coercion system complains"). #11780 fixed many of these errors, but not all.

comment:49 in reply to: ↑ 42 Changed 6 years ago by nthiery

Replying to SimonKing:

I am making some progress.

Testsuites are really a good thing!

:-)

The good news: I think I found a stable way to get the method resolution order of hom category parent classes right.

Nice!

comment:50 in reply to: ↑ 43 ; follow-up: Changed 6 years ago by nthiery

Replying to SimonKing:

Just a status report: I got rid of the __new__ method. Instead, I produce a __classcall__ method,

Sounds good. I guess I had not yet implemented classcall when I wrote the new workarounds.

similar to what is done in UniqueRepresentation (and in fact I make sage.categories.rings.Rings.HomCategory.ParentClass inherit from UniqueRepresentation).

I am not absolutely sure about this: as for parents, it is recommended for Homsets to have unique representation, but I am not sure this is currently *required* and *enforced*. So this may open a larger can of worm than this ticket can handle. This might be the issue you encountered with polynomials.

After my patches and the addition of the base_ring method to Cartesian product categories, the problem arose with __init_extra__ method in sage.categories.algebras.Algebras.ParentMethods: The Cartesian product of algebras over a ring R is an algebra over R (apparently acting diagonally). The __init_extra__ tries to create a coercion from the R to the cartesian product. However, that ended in an infinite recursion. I solved it by adding a from_base_ring method, that is understood by __init_extra__.

Cool!

The remaining problem concerns summation of elements of Cartesian products. Multiplication is defined, via sage.categories.magmas.Magmas.CartesianProduct.ParentMethods.product. But summation is missing. I guess it should be implemented in sage.categories.AdditiveMagmas.CartesianProduct.ParentMethods.summation.

Thanks for implementing this missing piece!

Cheers,

Nicolas

comment:51 in reply to: ↑ 50 ; follow-up: Changed 6 years ago by SimonKing

Replying to nthiery:

Replying to SimonKing: I am not absolutely sure about this: as for parents, it is recommended for Homsets to have unique representation, but I am not sure this is currently *required* and *enforced*.

First of all, Homsets are cached. But using unique representation, it is less easy to break the cache.

And I think that we should use any opportunity to reduce the number of violations of the unique parent assumption. After all, it is a matter of efficiency.

This might be the issue you encountered with polynomials.

Actually it was. The problem was that (for the sake of explicit documentation) some tests create a polynomial ring directly, not using the PolynomialRing constructor. My solution: I introduced a parent method for rings, that removes the ring from the homset cache, and use it after any test that creates a non-unique parent. Of course, it is for internal use only.

comment:52 in reply to: ↑ 51 ; follow-up: Changed 6 years ago by nthiery

Replying to SimonKing:

First of all, Homsets are cached. But using unique representation, it is less easy to break the cache.

I am glad that UniqueRepresentation? works well :-)

And I think that we should use any opportunity to reduce the number of violations of the unique parent assumption. After all, it is a matter of efficiency.

Agreed, the more unique parents, the better. But you don't have to fix all of Sage misfeatures in just this patch :-)

Besides, I am still not yet sure that we want to strictly enforce 100% unique parents. There might be occasional exceptions -- I don't know, things like temporarily created parents or what not -- where we might want to not have uniqueness.

Actually it was. The problem was that (for the sake of explicit documentation) some tests create a polynomial ring directly, not using the PolynomialRing constructor. My solution: I introduced a parent method for rings, that removes the ring from the homset cache, and use it after any test that creates a non-unique parent. Of course, it is for internal use only.

Ok. I could see other use cases. Should this be a method of UniqueRepresentation? -- of course still for internal use ?

Cheers,

Nicolas

comment:53 in reply to: ↑ 52 ; follow-up: Changed 6 years ago by SimonKing

Replying to nthiery:

I am glad that UniqueRepresentation? works well :-)

I am not 100% certain that they work well. At least for getting the tests of elliptic curves pass, we probably need #11670 (uniqueness of number fields). And note that (currently) I only introduce UniqueRepresentation to homsets of rings. I did not try to have it for all parents.

Agreed, the more unique parents, the better. But you don't have to fix all of Sage misfeatures in just this patch :-)

But all that were uncovered by new tests introduced with this patch.

Besides, I am still not yet sure that we want to strictly enforce 100% unique parents. There might be occasional exceptions -- I don't know, things like temporarily created parents or what not -- where we might want to not have uniqueness.

Why would one not want uniqueness for temporarily created parents? When the same parent is frequently created, then it is more efficient to just use a cache. Or are you concerned that one creates too many different parents that will stay in cache forever?

Ok. I could see other use cases. Should this be a method of UniqueRepresentation? -- of course still for internal use ?

No, what I just wrote can't be a method of UniqueRepresentation. Here is the purpose of what I wrote: Let X be a ring; X._remove_from_homset_cache() removes Hom(X,Y) and Hom(Y,X) from cache, for any ring Y.

Hence, it is not X.__class__.__classcall__.cache that is cleared, but X.category().hom_category().parent_class.__classcall__.cache. And an item is removed from the cache not if X is the value of that item, but if X appears in the key of the item.

But I think it would be a good idea to add a X._reduce_from_cache() method to UniqueRepresentation: It would remove any item of X.__class__.__classcall__.cache whose value is (equal to) X, and then it would try X._reduce_from_homset_cache() as well (which would of course only be available for rings).

comment:54 Changed 6 years ago by SimonKing

It turns out that #11670 will not solve the problem. But it seems to me that I come closer to a solution: In contrast to many other cases, hom sets of number fields are not unique parents. With my patch, they are even less unique. Here is a show case:

sage-4.6.2

sage: N = NumberField(x^12 - 4*x^11 + 6*x^10 - 5*x^9 + 5*x^8 - 9*x^7 + 21*x^6 - 9*x^5 + 5*x^4 - 5*x^3 + 6*x^2 - 4*x + 1, 'a') 
sage: Hom(N,N) is Hom(N,N)
False
sage: Hom(N,N) == Hom(N,N)
True

With my patch, we still have

sage: N = NumberField(x^12 - 4*x^11 + 6*x^10 - 5*x^9 + 5*x^8 - 9*x^7 + 21*x^6 - 9*x^5 + 5*x^4 - 5*x^3 + 6*x^2 - 4*x + 1, 'a') 
sage: Hom(N,N) is Hom(N,N)
False

but then

sage: Hom(N,N) == Hom(N,N)
False

I don't know yet why this is the case, because the __cmp__ function of number field homsets did not change.

comment:55 follow-up: Changed 6 years ago by SimonKing

Aha! I understand!

Hom(N,N) reduces to N._Hom_(N), which directly constructs a number field hom set.

By consequence, Hom(N,N) does not become a unique parent, even though it inherits from UniqueRepresentation via inheritance from the category. But that inheritance takes place after creation of the hom set, so that it is too late for Rings().HomCategory.ParentMethods.__classcall__.

In other words: Via category inheritance, Hom(N,N) inherits __eq__ from UniqueRepresentation, which is used for comparison and precedes the use of the custom __cmp__ method of number field homsets. However, __eq__ expects unique parents.

We thus have:

age: N = NumberField(x^12 - 4*x^11 + 6*x^10 - 5*x^9 + 5*x^8 - 9*x^7 + 21*x^6 - 9*x^5 + 5*x^4 - 5*x^3 + 6*x^2 - 4*x + 1, 'a')
sage: H = Hom(N,N)
sage: H == Hom(N,N)
False
sage: H > Hom(N,N)
False
sage: H < Hom(N,N)
False

I guess, until number fields are truly unique parents, I should add an __eq__ to number field hom sets, in order to not have it inherited from UniqueRepresentation.

comment:56 Changed 6 years ago by SimonKing

  • Work issues changed from Cartesian products to Cope with non-unique number fields

I think I found a valid work-around: Sometimes, a number field is created with passing the option cache=False to the number field constructor. If that option is used, I suggest to call the new _remove_from_homset_cache. It seems to work!

With that change (not yet posted), we have

sage: E = EllipticCurve('389a'); P = E.heegner_point(-7, 5); P
Heegner point of discriminant -7 and conductor 5 on elliptic curve of conductor 389
sage:  z = P.point_exact(100, optimize=True)

With the old patch, one would have the following error:

AssertionError: BUG in coercion model
    Apparently there are two versions of
        Number Field in a with defining polynomial x^12 + 4*x^11 + 56*x^10 + 170*x^9 + 1130*x^8 + 2564*x^7 + 10791*x^6 + 18054*x^5 + 51340*x^4 + 57530*x^3 + 102986*x^2 + 53724*x + 35001
    in the cache.

Running doctests, and then I hope the most serious problems have finally disappeared...

comment:57 in reply to: ↑ 53 ; follow-up: Changed 6 years ago by nthiery

Replying to SimonKing:

Why would one not want uniqueness for temporarily created parents? When the same parent is frequently created, then it is more efficient to just use a cache. Or are you concerned that one creates too many different parents that will stay in cache forever?

Possibly so. Or about creating many different parents that each need a lot of input data, or maybe some non hashable data; and maybe you need each such parent only once. So the overhead in time or code complexity of guaranteeing unique representation would not be worth it.

Honestly, I don't have a specific use case, just a bad feeling about it.

But I think it would be a good idea to add a X._reduce_from_cache() method to UniqueRepresentation: It would remove any item of X.__class__.__classcall__.cache whose value is (equal to) X,

+1. I am not sure about the name though. What about something like _delete_from_cache instead?

and then it would try X._reduce_from_homset_cache() as well (which would of course only be available for rings).

UniqueRepresentation? is meant to also be used by non Parents. So I'd rather have nothing Parent-related in it. On the other hand, Parent could overload UniqueRepresentation?'s method to also call that for homsets.

Cheers,

Nicolas

comment:58 in reply to: ↑ 55 ; follow-up: Changed 6 years ago by nthiery

Hi Simon,

Replying to SimonKing:

Aha! I understand!

:-)

I guess, until number fields are truly unique parents, I should add an __eq__ to number field hom sets, in order to not have it inherited from UniqueRepresentation.

I am not very keen on having a class inherit (indirectly) from UniqueRepresentation?, and then hacking it's way around to actually not have to implement the unique representation protocole. I'd rather only inherit explicitly from UniqueRepresentation? when I mean it.

Cheers,

comment:59 in reply to: ↑ 57 ; follow-up: Changed 6 years ago by SimonKing

Replying to nthiery:

But I think it would be a good idea to add a X._reduce_from_cache() method to UniqueRepresentation: It would remove any item of X.__class__.__classcall__.cache whose value is (equal to) X,

+1. I am not sure about the name though. What about something like _delete_from_cache instead?

Sorry, I meant to write _remove_from_cache, not _reduce_from_cache.

and then it would try X._reduce_from_homset_cache() as well (which would of course only be available for rings).

UniqueRepresentation? is meant to also be used by non Parents.

And _reduce_from_homset_cache is only for those parents that happen to belong to the category of rings. That's why I write "try ... (which would ... only be available for rings)". It would not be available for non-rings, and in particular not for non-parents. So, no problem, the attribute error would be caught anyway.

Parent could overload UniqueRepresentation?'s method to also call that for homsets.

No, it could not, because most parents are no UniqueRepresentations.

comment:60 in reply to: ↑ 58 ; follow-up: Changed 6 years ago by SimonKing

Replying to nthiery:

I am not very keen on having a class inherit (indirectly) from UniqueRepresentation?, and then hacking it's way around to actually not have to implement the unique representation protocole. I'd rather only inherit explicitly from UniqueRepresentation? when I mean it.

Yes, what I did doesn't fully convince me.

With my patch, the Rings().HomCategory.ParentMethods inherits from UniqueRepresentation, and since NumberFields() is a sub-category of Rings(), the default NumberFields().hom_category().parent_class inherits from UniqueRepresentation?` as well.

But it would be possible to have a custom NumberFields().HomCategory.ParentMethods, similar to the custom Rings().HomCategory.ParentMethods.

While Rings().hom_category().parent_class with my patch inherits from UniqueRepresentation and (via classcall) from either sage.rings.homset.RingHomset_generic or sage.rings.homset.RingHomset_quo_ring, one could have NumberFields().hom_category().parent_class inherit from sage.rings.number_field.morphism.NumberFieldHomset, but not from UniqueRepresentation.

comment:61 follow-up: Changed 6 years ago by SimonKing

Next: I accidentally found that quotient rings are no unique parents either.

I guess they should be, right?

comment:62 in reply to: ↑ 61 Changed 6 years ago by SimonKing

Replying to SimonKing:

Next: I accidentally found that quotient rings are no unique parents either.

I guess they should be, right?

Dealt with on a different ticket, I mean...

comment:63 in reply to: ↑ 60 Changed 6 years ago by nthiery

Replying to SimonKing:

But it would be possible to have a custom NumberFields().HomCategory.ParentMethods, similar to the custom Rings().HomCategory.ParentMethods.

While Rings().hom_category().parent_class with my patch inherits from UniqueRepresentation and (via classcall) from either sage.rings.homset.RingHomset_generic or sage.rings.homset.RingHomset_quo_ring, one could have NumberFields().hom_category().parent_class inherit from sage.rings.number_field.morphism.NumberFieldHomset, but not from UniqueRepresentation.

Sound good.

comment:64 in reply to: ↑ 59 Changed 6 years ago by nthiery

Replying to SimonKing:

And _reduce_from_homset_cache is only for those parents that

happen to belong to the category of rings. That's why I write "try ... (which would ... only be available for rings)". It would not be available for non-rings, and in particular not for non-parents. So, no problem, the attribute error would be caught anyway.

Yes, it would work. Yet, in a perfect world, UniqueRepresentation? ought to be generalized outside of Sage, as a general purpose Python tool (even though I am not sure anyone will take the time for that). So having some logic in there specifically targetted toward homsets smells. But maybe it's just a question of finding a more general name for this hook.

No, it could not, because most parents are no UniqueRepresentations.

Good point :-)

comment:65 Changed 6 years ago by SimonKing

  • Keywords sd34 added

I am not sure if the attached patch is up to date. Unfortunately, applying it to sage-4.7.2.alpha3 gives 5 hunks that fail to apply. I hope that I'll be able to get finally a working version during the upcoming sage days 34.

comment:66 Changed 4 years ago by jdemeyer

  • Milestone changed from sage-5.11 to sage-5.12

comment:67 Changed 4 years ago by vbraun_spam

  • Milestone changed from sage-6.1 to sage-6.2

comment:68 Changed 3 years ago by vbraun_spam

  • Milestone changed from sage-6.2 to sage-6.3

comment:69 Changed 3 years ago by vbraun_spam

  • Milestone changed from sage-6.3 to sage-6.4

comment:70 Changed 3 years ago by SimonKing

Let us see if anything from this patch could be useful in the current state of the art. See #10668 and #16340.

comment:71 Changed 3 years ago by jpflori

  • Cc jpflori added
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