gh-39160: Add _make_named_class_key for several categories, miscellaneous changes
    
Fixes #39154

Let `C = Category.join([NumberFields(), QuotientFields(),
MetricSpaces()])`, then there are 4 categories being created

* `Modules(C)`
* `VectorSpaces(C)`
* `Modules(C).Filtered()`
* `VectorSpaces(C).Filtered()`

The last 2 have type `FilteredModulesCategory`, which inherit from
`RegressiveCovariantConstructionCategory, Category_over_base_ring` which
in turn inherits from `CategoryWithParameters`. Subclasses of
`CategoryWithParameters` need to override `_make_named_class_key`
correctly so that category A and B can only return same key if their
parent class, element class, subcategory class etc. are identical. But
`FilteredModulesCategory` takes `_make_named_class_key` from
`Category_over_base_ring`, which returns the base category (`C`) for the
key.

With this change, the returned key is e.g. `Modules(C).parent_class` and
`VectorSpaces(C).parent_class` respectively, which is different.

Side note: by how things are implemented in SageMath, `Modules(C)`
actually returns the same thing as `VectorSpaces(C)`. The above diagram
is only for illustration purpose.

Side note 2: `Modules(C).Filtered()` is currently a subcategory of
`VectorSpaces(C)`… so the above is not a diamond, rather it's a linear
chain.

Also make `Modules(Fields())).Filtered()` ≤ `VectorSpaces(Fields())`,
and a few other changes e.g. make `_test_category_graph` works for hom
categories.

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### ⌛ Dependencies

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URL: #39160
Reported by: user202729
Reviewer(s): Tobias Diez

Author: Release Manager

src/sage/categories/category.py

@@ -943,6 +943,22 @@ def all_super_categories(self, proper=False):
             appropriate. Simply because lazy attributes are much
             faster than any method.
 
+        .. NOTE::
+
+            This is not the same as the concept of super category in mathematics.
+            In fact, this is not even the opposite relation of :meth:`is_subcategory`::
+
+                sage: A = VectorSpaces(QQ); A
+                Category of vector spaces over Rational Field
+                sage: B = VectorSpaces(QQ.category()); B
+                Category of vector spaces over (number fields and quotient fields and metric spaces)
+                sage: A.is_subcategory(B)
+                True
+                sage: B in A.all_super_categories()
+                False
+
+        .. SEEALSO:: :meth:`_test_category_graph`
+
         EXAMPLES::
 
             sage: C = Rings(); C
@@ -1379,7 +1395,16 @@ def _test_category_graph(self, **options):
             method resolution order of the parent and element
             classes. This method checks this.
 
-        .. TODO:: currently, this won't work for hom categories.
+        Note that if
+        :meth:`~sage.structure.category_object.CategoryObject._refine_category_`
+        is called at unexpected times, the invariant might be false. Most
+        commonly, this happens with rings like ``Zmod(n)`` or ``SR``, where
+        a check like ``Zmod(n) in Fields()`` is needed (which checks the primality
+        of `n`) to refine their category to be a subcategory of fields.
+
+        .. SEEALSO::
+
+            :meth:`CategoryWithParameters._make_named_class_key`
 
         EXAMPLES::
 
@@ -1615,6 +1640,11 @@ def subcategory_class(self):
             sage: isinstance(AlgebrasWithBasis(QQ), cls)
             True
 
+        .. NOTE::
+
+            See the note about :meth:`_test_category_graph` regarding Python
+            class hierarchy.
+
         TESTS::
 
             sage: cls = Algebras(QQ).subcategory_class; cls
@@ -1667,6 +1697,11 @@ def parent_class(self):
         :class:`~sage.categories.bimodules.Bimodules`,
         :class:`~sage.categories.category_types.Category_over_base` and
         :class:`sage.categories.category.JoinCategory`.
+
+        .. NOTE::
+
+            See the note about :meth:`_test_category_graph` regarding Python
+            class hierarchy.
         """
         return self._make_named_class('parent_class', 'ParentMethods')
 
@@ -1713,6 +1748,11 @@ def element_class(self):
             0
 
         .. SEEALSO:: :meth:`parent_class`
+
+        .. NOTE::
+
+            See the note about :meth:`_test_category_graph` regarding Python
+            class hierarchy.
         """
         return self._make_named_class('element_class', 'ElementMethods')
 
@@ -1757,7 +1797,7 @@ def required_methods(self):
     # Operations on the lattice of categories
     def is_subcategory(self, c):
         """
-        Return ``True`` if ``self`` is naturally embedded as a subcategory of `c`.
+        Return ``True`` if there is a natural forgetful functor from ``self`` to `c`.
 
         EXAMPLES::
 
@@ -2046,13 +2086,18 @@ def _with_axiom(self, axiom):
         Return the subcategory of the objects of ``self`` satisfying
         the given ``axiom``.
 
+        Note that this is a private method thus should not be directly
+        used, see below.
+
         INPUT:
 
         - ``axiom`` -- string, the name of an axiom
 
         EXAMPLES::
 
-            sage: Sets()._with_axiom("Finite")
+            sage: Sets()._with_axiom("Finite")  # not idiomatic
+            Category of finite sets
+            sage: Sets().Finite()  # recommended
             Category of finite sets
 
             sage: type(Magmas().Finite().Commutative())
@@ -2068,7 +2113,7 @@ def _with_axiom(self, axiom):
             sage: Sets()._with_axiom("Associative")
             Category of sets
 
-        .. WARNING:: This may be changed in the future to raising an error.
+        .. WARNING:: This may be changed in the future to raise an error.
         """
         return Category.join(self._with_axiom_as_tuple(axiom))
 
@@ -2718,6 +2763,10 @@ def _make_named_class(self, name, method_provider, cache=False, **options):
 
         It is assumed that this method is only called from a lazy
         attribute whose name coincides with the given ``name``.
+        Currently, this means :meth:`Category.subcategory_class`,
+        :meth:`Category.parent_class` or :meth:`element_class`.
+
+        Subclasses need to implement :meth:`_make_named_class_key`.
 
         OUTPUT:
 
@@ -2810,6 +2859,10 @@ def _make_named_class(self, name, method_provider, cache=False, **options):
             pass
         result = Category._make_named_class(self, name, method_provider,
                                             cache=cache, **options)
+        if key[2] != self._make_named_class_key(name):
+            # the object in the parameter may have had its category refined, which might modify the key
+            # throw result away and recompute
+            return self._make_named_class(name, method_provider, cache=cache, **options)
         self._make_named_class_cache[key] = result
         return result
 
@@ -2818,6 +2871,50 @@ def _make_named_class_key(self, name):
         r"""
         Return what the element/parent/... class depend on.
 
+        This method starts as an optimization to allow different related
+        categories to share the Python types, see :issue:`11935`.
+        However, because of the guarantees stated in :meth:`Category._test_category_graph`,
+        the following rules must be followed.
+
+        - If two categories have different lists of supercategories, they must return
+          different keys::
+
+            sage: Zmod(5) in Fields()
+            True
+            sage: Algebras(Zmod(5)).all_super_categories()
+            [..., Category of vector spaces over Ring of integers modulo 5, ...]
+            sage: Zmod(6) in Fields()
+            False
+            sage: Algebras(Zmod(6)).all_super_categories()  # of course don't have category of vector spaces
+            [..., Category of modules over Ring of integers modulo 6, ...]
+            sage: # therefore:
+            sage: Algebras(Zmod(5))._make_named_class_key("parent_class") != Algebras(Zmod(6))._make_named_class_key("parent_class")
+            True
+            sage: Algebras(Zmod(5)).parent_class != Algebras(Zmod(6)).parent_class
+            True
+
+        - If category ``A`` is a supercategory of category ``B``,
+          and category ``B`` uses the optimization, then so must ``A``.
+
+          For example, ``Modules(ZZ)`` is a supercategory of ``Algebras(ZZ)``,
+          and ``Algebras(ZZ)`` implements the optimization::
+
+            sage: from sage.categories.category import CategoryWithParameters
+            sage: isinstance(Algebras(ZZ), CategoryWithParameters)
+            True
+            sage: Algebras(ZZ).parent_class is Algebras(ZZ.category()).parent_class
+            True
+            sage: Modules(ZZ) in Algebras(ZZ).all_super_categories()
+            True
+
+          This forces ``Modules(ZZ)`` to also implement the optimization::
+
+            sage: Modules(ZZ).parent_class is Modules(ZZ.category()).parent_class
+            True
+
+        As a complication, computing the exact category might require some potentially
+        expensive test. See :meth:`Category._test_category_graph` for more details.
+
         INPUT:
 
         - ``name`` -- string; the name of the class as an attribute
@@ -2826,6 +2923,9 @@ def _make_named_class_key(self, name):
         .. SEEALSO::
 
             - :meth:`_make_named_class`
+
+            The following can be read for typical implementations of this method.
+
             - :meth:`sage.categories.category_types.Category_over_base._make_named_class_key`
             - :meth:`sage.categories.bimodules.Bimodules._make_named_class_key`
             - :meth:`JoinCategory._make_named_class_key`
@@ -3064,6 +3164,9 @@ def _with_axiom(self, axiom):
         """
         Return the category obtained by adding an axiom to ``self``.
 
+        As mentioned in :meth:`Category._with_axiom`, this method should not be used directly
+        except in internal code.
+
         .. NOTE::
 
             This is just an optimization of
@@ -3073,7 +3176,9 @@ def _with_axiom(self, axiom):
         EXAMPLES::
 
             sage: C = Category.join([Monoids(), Posets()])
-            sage: C._with_axioms(["Finite"])
+            sage: C._with_axioms(["Finite"])  # not idiomatic
+            Join of Category of finite monoids and Category of finite posets
+            sage: C.Finite()  # recommended
             Join of Category of finite monoids and Category of finite posets
 
         TESTS:

src/sage/categories/filtered_modules.py

@@ -67,6 +67,54 @@ def _repr_object_names(self):
         """
         return "filtered {}".format(self.base_category()._repr_object_names())
 
+    def _make_named_class_key(self, name):
+        r"""
+        Return what the element/parent/... classes depend on.
+
+        .. SEEALSO::
+
+            - :meth:`.CategoryWithParameters._make_named_class_key`
+
+        EXAMPLES::
+
+            sage: Modules(ZZ).Filtered()._make_named_class_key('element_class')
+            <class 'sage.categories.modules.Modules.element_class'>
+
+        Note that we cannot simply return the base as in
+        :meth:`.Category_over_base._make_named_class_key` because of the following
+        (see :issue:`39154`)::
+
+            sage: VectorSpacesQQ = VectorSpaces(QQ); VectorSpacesQQ
+            Category of vector spaces over Rational Field
+            sage: # ModulesQQ = Modules(QQ)  # doesn't work because...
+            sage: Modules(QQ) is VectorSpacesQQ
+            True
+            sage: ModulesQQ = VectorSpacesQQ.super_categories()[0]; ModulesQQ
+            Category of modules over Rational Field
+            sage: VectorSpacesQQ.Filtered()
+            Category of filtered vector spaces over Rational Field
+            sage: ModulesQQ.Filtered()
+            Category of filtered modules over Rational Field
+            sage: VectorSpacesQQ.Filtered()._make_named_class_key('parent_class')
+            <class 'sage.categories.vector_spaces.VectorSpaces.parent_class'>
+            sage: ModulesQQ.Filtered()._make_named_class_key('parent_class')
+            <class 'sage.categories.modules.Modules.parent_class'>
+            sage: assert (VectorSpacesQQ.Filtered()._make_named_class_key('parent_class') !=
+            ....:         ModulesQQ.Filtered()._make_named_class_key('parent_class'))
+            sage: VectorSpacesQQ.Filtered().parent_class
+            <class 'sage.categories.vector_spaces.VectorSpaces.Filtered.parent_class'>
+            sage: ModulesQQ.Filtered().parent_class
+            <class 'sage.categories.filtered_modules.FilteredModules.parent_class'>
+
+        Nevertheless, as explained in :meth:`.Category_over_base._make_named_class_key`,
+        ``Modules(QQ).Filtered()`` and ``Modules(QQ.category()).Filtered()`` must have
+        the same parent class::
+
+            sage: Modules(QQ).Filtered().parent_class == Modules(QQ.category()).Filtered().parent_class
+            True
+        """
+        return getattr(self._base_category, name)
+
 
 class FilteredModules(FilteredModulesCategory):
     r"""
@@ -122,8 +170,9 @@ def extra_super_categories(self):
         """
         from sage.categories.modules import Modules
         from sage.categories.fields import Fields
+        from sage.categories.category import Category
         base_ring = self.base_ring()
-        if base_ring in Fields():
+        if base_ring in Fields() or (isinstance(base_ring, Category) and base_ring.is_subcategory(Fields())):
             return [Modules(base_ring)]
         else:
             return []

src/sage/categories/homset.py

@@ -396,7 +396,7 @@ def Hom(X, Y, category=None, check=True):
     """
     # This should use cache_function instead
     # However some special handling is currently needed for
-    # domains/docomains that break the unique parent condition. Also,
+    # domains/codomains that break the unique parent condition. Also,
     # at some point, it somehow broke the coercion (see e.g. sage -t
     # sage.rings.real_mpfr). To be investigated.
     global _cache

src/sage/categories/homsets.py

@@ -10,13 +10,13 @@
 # *****************************************************************************
 
 from sage.misc.cachefunc import cached_method
-from sage.categories.category import Category, JoinCategory
+from sage.categories.category import Category, JoinCategory, CategoryWithParameters
 from sage.categories.category_singleton import Category_singleton
 from sage.categories.category_with_axiom import CategoryWithAxiom
 from sage.categories.covariant_functorial_construction import FunctorialConstructionCategory
 
 
-class HomsetsCategory(FunctorialConstructionCategory):
+class HomsetsCategory(FunctorialConstructionCategory, CategoryWithParameters):
 
     _functor_category = "Homsets"
 
@@ -155,6 +155,17 @@ def base(self):
                 return C.base()
         raise AttributeError("This hom category has no base")
 
+    def _make_named_class_key(self, name):
+        r"""
+        Return what the element/parent/... classes depend on.
+
+        .. SEEALSO::
+
+            - :meth:`CategoryWithParameters`
+            - :meth:`CategoryWithParameters._make_named_class_key`
+        """
+        return getattr(self.base_category(), name)
+
 
 class HomsetsOf(HomsetsCategory):
     """

src/sage/groups/perm_gps/permgroup_named.py

@@ -3514,7 +3514,7 @@ class SmallPermutationGroup(PermutationGroup_generic):
         [ 2  0 -1  2  0 -1]
         sage: def numgps(n): return ZZ(libgap.NumberSmallGroups(n))
         sage: all(SmallPermutationGroup(n,k).id() == [n,k]
-        ....:     for n in [1..64] for k in [1..numgps(n)])
+        ....:     for n in [1..64] for k in [1..numgps(n)])  # long time (180s)
         True
         sage: H = SmallPermutationGroup(6,1)
         sage: H.is_abelian()