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Morphisms and Objects of Categories #10667

Open simon-king-jena opened 13 years ago

simon-king-jena commented 13 years ago

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.

Depends on #9138 Depends on #11115 Depends on #11780

CC: @nilesjohnson @jpflori

Component: categories

Keywords: objects morphisms containment sd34

Work Issues: Cope with non-unique number fields

Author: Simon King

Issue created by migration from https://trac.sagemath.org/ticket/10667

simon-king-jena commented 13 years ago
comment:1

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.

nthiery commented 13 years ago
comment:2

Hi Simon!

Replying to @simon-king-jena:

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

9343d2e0-59ba-4406-bd4f-c78e4cf1230e commented 13 years ago
comment:3

cc me! Thanks :)

simon-king-jena commented 13 years ago
comment:4

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:

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:

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

  2. 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:

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
simon-king-jena commented 13 years ago

Author: Simon King

simon-king-jena commented 13 years ago
comment:5

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?

nthiery commented 13 years ago
comment:6

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).

simon-king-jena commented 13 years ago
comment:7

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!

simon-king-jena commented 13 years ago
comment:8

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

simon-king-jena commented 13 years ago
comment:9

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.

simon-king-jena commented 13 years ago
comment:10

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.

simon-king-jena commented 13 years ago
comment:11

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.

simon-king-jena commented 13 years ago
comment:12

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 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.

simon-king-jena commented 13 years ago

Dependencies: #9138, #11115

simon-king-jena commented 13 years ago
comment:13

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
simon-king-jena commented 13 years ago
comment:14

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

simon-king-jena commented 13 years ago

Work Issues: doctests

simon-king-jena commented 13 years ago
comment:15

Replying to @simon-king-jena:

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.

simon-king-jena commented 13 years ago
comment:16

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:
nthiery commented 13 years ago
comment:17

Replying to @simon-king-jena:

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!

simon-king-jena commented 13 years ago
comment:18

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()].

  2. 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!

simon-king-jena commented 13 years ago
comment:19

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?

simon-king-jena commented 13 years ago
comment:20

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.

nthiery commented 13 years ago
comment:21

Replying to @simon-king-jena:

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

simon-king-jena commented 13 years ago
comment:22

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.

simon-king-jena commented 13 years ago
comment:23

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()?

nthiery commented 13 years ago
comment:24

Replying to @simon-king-jena:

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.

nthiery commented 13 years ago
comment:25

Replying to @simon-king-jena:

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).

simon-king-jena commented 13 years ago
comment:26

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().

simon-king-jena commented 13 years ago
comment:27

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.

simon-king-jena commented 13 years ago
comment:28

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.

simon-king-jena commented 13 years ago
comment:29

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

?

nthiery commented 13 years ago
comment:30

Hi Simon!

Replying to @simon-king-jena:

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,

nthiery commented 13 years ago
comment:31

Replying to @simon-king-jena:

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.

nthiery commented 13 years ago
comment:32

Replying to @simon-king-jena:

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 :-)

simon-king-jena commented 13 years ago
comment:33

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
simon-king-jena commented 13 years ago
comment:34

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]
simon-king-jena commented 13 years ago
comment:35

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.

simon-king-jena commented 13 years ago
comment:36

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.

simon-king-jena commented 13 years ago
comment:37

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.

simon-king-jena commented 13 years ago
comment:38

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...

nthiery commented 13 years ago
comment:39

Replying to @simon-king-jena:

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...

:-)

nthiery commented 13 years ago
comment:40

Replying to @simon-king-jena:

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?

simon-king-jena commented 13 years ago
comment:41

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.

simon-king-jena commented 13 years ago
comment:42

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.

simon-king-jena commented 13 years ago
comment:43

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.

simon-king-jena commented 13 years ago

Changed work issues from doctests to Cartesian products

simon-king-jena commented 13 years ago
comment:44

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.

simon-king-jena commented 13 years ago

Changed dependencies from #9138, #11115 to #9138, #11115, #11780

simon-king-jena commented 13 years ago
comment:45

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