#10963: More functorial constructions
-------------------------------------+-------------------------------------
Reporter: nthiery | Owner: stumpc5
Type: enhancement | Status: needs_review
Priority: major | Milestone: sage-6.1
Component: categories | Resolution:
Keywords: days54 | Merged in:
Authors: Nicolas M. Thiéry | Reviewers: Simon King, Frédéric
Report Upstream: N/A | Chapoton
Branch: | Work issues:
public/ticket/10963 | Commit:
Dependencies: #11224, #8327, | eb7b486c6fecac296052f980788e15e2ad1b59e4
#10193, #12895, #14516, #14722, | Stopgaps:
#13589, #14471, #15069, #15094, |
#11688, #13394, #15150, #15506 |
-------------------------------------+-------------------------------------
Comment (by SimonKing):
Replying to [comment:431 nthiery]:
> Now, I am actually surprised that you get that many. Can you give a
> couple examples where this fails?
The `TestCategoryModel()` from [attachment:consistency.py] returns (among
other things) a list of all category classes for which `an_instance()`
does not return an instance of the class. This list seems to be stable.
Side-note: The other tests do not seem stable yet: They differ from run to
run. Anyway, I'll work on it.
Here are the classes and the errors I am getting:
{{{
0 sage.combinat.ncsym.bases.NCSymDualBases AttributeError
1 sage.categories.modules_with_basis.ModulesWithBasis.DualObjects
TypeError
2 sage.categories.algebra_functor.AlgebrasCategory AssertionError
3 sage.categories.additive_magmas.AdditiveMagmas.Algebras AssertionError
4 sage.categories.hopf_algebras.HopfAlgebras.DualCategory
NotImplementedError
5 sage.categories.additive_semigroups.AdditiveSemigroups.Algebras
AssertionError
6 sage.categories.category_types.AbelianCategory NotImplementedError
7 sage.combinat.ncsf_qsym.qsym.QuasiSymmetricFunctions.Bases
AttributeError
8 sage.combinat.descent_algebra.DescentAlgebraBases AttributeError
9 sage.categories.quotients.QuotientsCategory TypeError
10 sage.combinat.ncsf_qsym.ncsf.NonCommutativeSymmetricFunctions.Bases
AttributeError
11 sage.categories.additive_magmas.AdditiveCommutative.Algebras
AssertionError
12 sage.categories.realizations.RealizationsCategory TypeError
13 sage.combinat.ncsym.bases.MultiplicativeNCSymBases AttributeError
14 sage.categories.sets_cat.Sets.CartesianProducts TypeError
15 sage.categories.graded_modules.GradedModulesCategory KeyError
16 sage.categories.sets_cat.Sets.Subobjects TypeError
17 sage.categories.cartesian_product.CartesianProductsCategory TypeError
18 sage.categories.magmas.Magmas.Realizations TypeError
19 sage.categories.hopf_algebras.HopfAlgebras.Realizations TypeError
20 sage.categories.groups.Groups.Algebras AssertionError
21 sage.categories.magmas.Magmas.Algebras AssertionError
22 sage.categories.isomorphic_objects.IsomorphicObjectsCategory TypeError
23 sage.categories.sets_with_partial_maps.SetsWithPartialMaps.HomCategory
TypeError
24
sage.combinat.ncsf_qsym.generic_basis_code.GradedModulesWithInternalProduct.Realizations
TypeError
25 sage.combinat.sf.sfa.SymmetricFunctionsBases AttributeError
26 sage.categories.category.HomCategory TypeError
27 sage.categories.sets_cat.Sets.Subquotients TypeError
28 sage.categories.algebras_with_basis.AlgebrasWithBasis.CartesianProducts
TypeError
29
sage.categories.finite_enumerated_sets.FiniteEnumeratedSets.IsomorphicObjects
TypeError
30 sage.categories.sets_cat.Sets.Realizations TypeError
31 sage.categories.realizations.Category_realization_of_parent
NotImplementedError
32 sage.categories.category_with_axiom.BrokenTestObjects
NotImplementedError
33 sage.categories.category.CategoryWithParameters NotImplementedError
34 sage.categories.magmas.Commutative.Algebras AssertionError
35 sage.categories.vector_spaces.VectorSpaces.DualObjects TypeError
36 sage.categories.hopf_algebras.HopfAlgebras.Morphism NotImplementedError
37 sage.categories.category_types.Category_in_ambient TypeError
38 sage.categories.objects.Objects.HomCategory TypeError
39
sage.combinat.ncsf_qsym.ncsf.NonCommutativeSymmetricFunctions.MultiplicativeBasesOnGroupLikeElements
AttributeError
40 sage.categories.schemes.Schemes.HomCategory TypeError
41
sage.combinat.ncsf_qsym.ncsf.NonCommutativeSymmetricFunctions.MultiplicativeBases
AttributeError
42 sage.categories.semigroups.Semigroups.Subquotients TypeError
43 sage.categories.sets_cat.Sets.Algebras AssertionError
44 sage.categories.coalgebras.Coalgebras.TensorProducts TypeError
45 sage.categories.hecke_modules.HeckeModules.HomCategory TypeError
46 sage.categories.category_types.Category_module NotImplementedError
47 sage.categories.algebras.Algebras.CartesianProducts TypeError
48 sage.categories.algebras_with_basis.AlgebrasWithBasis.TensorProducts
TypeError
49 sage.combinat.ncsym.bases.NCSymOrNCSymDualBases AttributeError
50 sage.categories.subquotients.SubquotientsCategory TypeError
51 sage.categories.algebras.Algebras.TensorProducts TypeError
52 sage.categories.coalgebras.Coalgebras.DualObjects TypeError
53 sage.categories.magmas.Unital.Algebras AssertionError
54 sage.categories.additive_magmas.AdditiveUnital.Algebras AssertionError
55 sage.categories.category_types.Category_ideal NotImplementedError
56 sage.categories.modules_with_basis.ModulesWithBasis.HomCategory
TypeError
57 sage.categories.tensor.TensorProductsCategory TypeError
58 sage.categories.algebras.Algebras.DualObjects TypeError
59 sage.categories.semigroups.Semigroups.Quotients TypeError
60
sage.categories.hopf_algebras_with_basis.HopfAlgebrasWithBasis.TensorProducts
TypeError
61 sage.categories.finite_sets.FiniteSets.Subquotients TypeError
62 sage.categories.category_types.Category_over_base_ring
NotImplementedError
63 sage.categories.category_with_axiom.SmallTestObjects
NotImplementedError
64 sage.categories.hopf_algebras.HopfAlgebras.TensorProducts TypeError
65
sage.combinat.ncsf_qsym.ncsf.NonCommutativeSymmetricFunctions.MultiplicativeBasesOnPrimitiveElements
AttributeError
66
sage.categories.covariant_functorial_construction.CovariantConstructionCategory
TypeError
67 sage.categories.magmas.Magmas.CartesianProducts TypeError
68 sage.categories.semigroups.Semigroups.CartesianProducts TypeError
69 sage.categories.dual.DualObjectsCategory TypeError
70 sage.categories.modules.Modules.EndCategory TypeError
71 sage.categories.finite_sets.FiniteSets.Algebras AssertionError
72 sage.algebras.iwahori_hecke_algebra.IwahoriHeckeAlgebra._BasesCategory
AttributeError
73 sage.categories.coalgebras.Coalgebras.WithRealizations TypeError
74 sage.categories.subobjects.SubobjectsCategory TypeError
75 sage.categories.sets_cat.Sets.WithRealizations TypeError
76 sage.categories.additive_magmas.AdditiveUnital.WithRealizations
TypeError
77
sage.categories.finite_enumerated_sets.FiniteEnumeratedSets.CartesianProducts
TypeError
78 sage.categories.modules_with_basis.ModulesWithBasis.CartesianProducts
TypeError
79 sage.categories.semigroups.Semigroups.Algebras AssertionError
80 <class 'sage.categories.modules.Modules'> yields instance of <class
'sage.categories.vector_spaces.VectorSpaces_with_category'>
81 sage.categories.modules.Modules.HomCategory TypeError
82 sage.categories.category_with_axiom.TestObjectsOverBaseRing.Commutative
TypeError
83 sage.categories.monoids.Monoids.WithRealizations TypeError
84
sage.categories.covariant_functorial_construction.RegressiveCovariantConstructionCategory
TypeError
85 sage.categories.rings.Rings.HomCategory TypeError
86 sage.categories.category_singleton.Category_singleton AssertionError
87 sage.categories.category.Category NotImplementedError
88 sage.categories.monoids.Monoids.Subquotients TypeError
89 sage.combinat.ncsym.bases.NCSymBases AttributeError
90 sage.categories.coalgebras.Coalgebras.Realizations TypeError
91 sage.categories.with_realizations.WithRealizationsCategory TypeError
92 sage.categories.sets_cat.Sets.IsomorphicObjects TypeError
93 sage.categories.sets_cat.Sets.Quotients TypeError
94 sage.categories.monoids.Monoids.CartesianProducts TypeError
95 sage.combinat.ncsf_qsym.generic_basis_code.BasesOfQSymOrNCSF
AttributeError
96 sage.categories.modules_with_basis.ModulesWithBasis.TensorProducts
TypeError
97 sage.categories.magmas.Magmas.Subquotients TypeError
98 sage.categories.category.JoinCategory TypeError
99
sage.categories.graded_hopf_algebras_with_basis.GradedHopfAlgebrasWithBasis.WithRealizations
TypeError
100
sage.algebras.iwahori_hecke_algebra.IwahoriHeckeAlgebra_nonstandard._BasesCategory
AttributeError
101 sage.categories.sets_cat.Sets.HomCategory TypeError
102 sage.categories.monoids.Monoids.Algebras AssertionError
103 sage.categories.category_types.Category_over_base NotImplementedError
104
sage.categories.commutative_additive_groups.CommutativeAdditiveGroups.Algebras
AssertionError
105 sage.categories.category_with_axiom.BrokenTestObjects.Commutative
NotImplementedError
106 <class 'sage.categories.modules_with_basis.ModulesWithBasis'> yields
instance of <class
'sage.categories.vector_spaces.VectorSpaces.WithBasis_with_category'>
107 sage.categories.category_with_axiom.TestObjectsOverBaseRing.Unital
TypeError
108
sage.categories.category_with_axiom.TestObjectsOverBaseRing.FiniteDimensional
TypeError
109 sage.categories.category_with_axiom.BrokenTestObjects.Finite
NotImplementedError
110 sage.categories.category_with_axiom.Commutative.Facade TypeError
111 <class
'sage.categories.finite_dimensional_modules_with_basis.FiniteDimensionalModulesWithBasis'>
yields instance of <class
'sage.categories.category.JoinCategory_with_category'>
112 sage.categories.category_with_axiom.Commutative.FiniteDimensional
TypeError
113 <class 'sage.categories.modules.Modules.FiniteDimensional'> yields
instance of <class 'sage.categories.category.JoinCategory_with_category'>
114 sage.categories.category_with_axiom.SmallTestObjects.Finite
NotImplementedError
115 sage.categories.category_with_axiom.Commutative.Finite TypeError
116 sage.categories.category_with_axiom.FiniteDimensional.Finite TypeError
117 sage.categories.category_with_axiom.Commutative.Finite
NotImplementedError
118 sage.categories.category_with_axiom.FiniteDimensional.Unital TypeError
119 sage.categories.category_with_axiom.Finite.Commutative
NotImplementedError
120 sage.categories.category_with_axiom.Unital.Commutative TypeError
}}}
In examples 80, 106, 111 and 113, `C.an_instance()` does return something,
but it does not return an instance of C.
And please don't forget:
{{{
sage: type(sage.categories.category_with_axiom.SmallTestObjects.Connected)
<type 'int'>
sage:
type(sage.categories.category_with_axiom.SmallTestObjects.Commutative)
<type 'classobj'>
}}}
which are bugs, too.
> Have you tried the category_sample
> function?
Never heard of it before, hence: No.
> I am not sure. If there is some inconsistency, you can always work in
> a sublattice (with bottom) that contains all the involved categories,
> rather than the full lattice. And I would not be surprised if it could
> be argued that one could always choose such a sublattice with a
> category that is actually implemented in Sage.
Sublattice it is. But how big do you need to choose the sublattice in
order to
detect the inconsistency? Since we already have Gröbner bases in this
thread:
Generally you'll need to consider elements of rather high degree in order
to
detect all relations in small degree. Sure, in a boolean polynomial ring
the
order is bounded by the number of generators, as they are idempotent.
> Right. But Sage's startup would still require being able to manipulate
> such a Gröbner basis in one form or the other. And one needs to make
> sure the Gröbner basis is consistent with all the code (that is Sage's
> compilation might require a Gröbner basis computation). And that it
> can be extended dynamically if users introduce new categories in their
> own library.
Exactly. If I had to choose between a Gröbner basis computation in a
boolean
polynomial ring (which is a relatively moderate task due to idempotency)
and
the requirement to manually do local choices that are globally consistent,
I'd
do the former.
> I am certainly not saying it's not doable. But it introduces some
> complexity which has to be well motivated.
Are you talking about ''mathematical'' complexity? Then my answer is that
the
complexity of the underlying local--global problem is there, whether we
want
it or not. And I'd rather have the complexity dealt with by a mathematical
theory (commutative algebra) than by the inspiration of all future
developers
of categories.
> Sorry, I can't resist; let me use the very argument that soo many
> people have raised when saying that all that category stuff was just
> overdesign. «Before introducing non trivial design to solve a scaling
> issue, one needs to be sure there is one in practice». So far, I
> haven't had a single time where I got bothered by that.
With exception of the `Blahs.Unital.Blue` example, you mean...
> For Fields vs Division rings, the asymmetry is *very* natural. You put
> the class in Fields.Finite, because it's about stuff valid for finite
> fields. And you put in DivisionRings the theorem which tells you that
> in the context of division rings, the "Finite" axiom implies the
> "commutative" axioms.
Then consider the axiom `a*b==a*c => b==c` (I don't know an English word
for
it, thus call it "kürzbar"). There are infinite "kürzbare" rings that
aren't
division ring (e.g., the ring of integers). However, for finite rings,
being
"kürzbar" and "division" are equivalent.
So, you have a perfectly symmetric formulation:
`Rings.Finite.Kürzbar==Rings.Finite.Division`. So, how to choose a
default? In
this particular example you could argue that both are equal to
`Fields.Finite`. But generally?
Next, imagine you have several of such symmetric statements. You ''can''
do a
consistent symmetry break---but that's probably equivalent to chosing a
monomial order in a polynomial ring.
> And
> second DivisionRings().Finite() would not coincide with
> Fields().Finite(); and this is the first thing you would test.
Why not? If I do
{{{
DivisionRings.Finite = FiniteFields # lazily imported from
sage.categories.finite_fields
}}}
and
{{{
Fields.Finite = FiniteFields # lazily imported from
sage.categories.finite_fields
}}}
then of course both are the same!
The only problem is that the current ''implementation'' would complain.
And
that's what I think is a deficiency of the current implementation. Doing
the
above is natural and easy syntax, and it should be supported.
> I believe, and will work on proving formally, that the current
> implementation is perfectly well-defined and gives normal forms.
I am talking about people who want to extend the current implementation:
Add
new axioms, new basic categories, and in particular new mathematical
theorems
about categorial identities.
> No I did not do wrong. It does demonstrates *on
> purpose* a *missing feature*: namely that you currently can't use the
> Blue_extra_super_categories mechanism in the category where the Blue
> axiom is defined.
You talk about an implementation detail (`Blue_extra_super_categories`). I
am
talking about the fact that in this example (made up) mathematical axioms
do not commute.
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Ticket URL: <http://trac.sagemath.org/ticket/10963#comment:433>
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