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r""" Covariant Functorial Constructions
A *functorial construction* is a collection of functors `(F_{Cat})_{Cat}` (indexed by a collection of categories) which associate to a sequence of parents `(A, B, ...)` in a category `Cat` a parent `F_{Cat}(A, B, ...)`. Typical examples of functorial constructions are :obj:`cartesian_product` and :obj:`tensor_product`.
The category of `F_{Cat}(A, B, ...)`, which only depends on `Cat`, is called the (functorial) construction category.
A functorial construction is *(category)-covariant* if for every categories `Cat` and `SuperCat`, the category of `F_{Cat}(A, B, ...)` is a subcategory of the category of `F_{SuperCat}(A, B, ...)` whenever `Cat` is a subcategory of `SuperCat`. A functorial construction is *(category)-regressive* if the category of `F_{Cat}(A, B, ...)` is a subcategory of `Cat`.
The goal of this module is to provide generic support for covariant functorial constructions. In particular, given some parents `A`, `B`, ..., in respective categories `Cat_A`, `Cat_B`, ..., it provides tools for calculating the best known category for the parent `F(A,B,...)`. For examples, knowing that Cartesian products of semigroups (resp. monoids, groups) have a semigroup (resp. monoid, group) structure, and given a group `B` and two monoids `A` and `C` it can calculate that `A \times B \times C` is naturally endowed with a monoid structure.
See :class:`CovariantFunctorialConstruction`, :class:`CovariantConstructionCategory` and :class:`RegressiveCovariantConstructionCategory` for more details.
AUTHORS:
- Nicolas M. Thiery (2010): initial revision """ #***************************************************************************** # Copyright (C) 2010 Nicolas M. Thiery <nthiery at users.sf.net> # # Distributed under the terms of the GNU General Public License (GPL) # http://www.gnu.org/licenses/ #****************************************************************************** from sage.misc.cachefunc import cached_function, cached_method from sage.misc.lazy_attribute import lazy_class_attribute from sage.misc.lazy_import import LazyImport from sage.categories.category import Category from sage.structure.sage_object import SageObject from sage.structure.unique_representation import UniqueRepresentation from sage.structure.dynamic_class import DynamicMetaclass
class CovariantFunctorialConstruction(UniqueRepresentation, SageObject): r""" An abstract class for construction functors `F` (eg `F` = Cartesian product, tensor product, `\QQ`-algebra, ...) such that:
- Each category `Cat` (eg `Cat=` ``Groups()``) can provide a category `F_{Cat}` for parents constructed via this functor (e.g. `F_{Cat} =` ``CartesianProductsOf(Groups())``).
- For every category `Cat`, `F_{Cat}` is a subcategory of `F_{SuperCat}` for every super category `SuperCat` of `Cat` (the functorial construction is (category)-covariant).
- For parents `A`, `B`, ..., respectively in the categories `Cat_A`, `Cat_B`, ..., the category of `F(A,B,...)` is `F_{Cat}` where `Cat` is the meet of the categories `Cat_A`, `Cat_B`, ...,.
This covers two slightly different use cases:
- In the first use case, one uses directly the construction functor to create new parents::
sage: tensor() # todo: not implemented (add an example)
or even new elements, which indirectly constructs the corresponding parent::
sage: tensor(...) # todo: not implemented
- In the second use case, one implements a parent, and then put it in the category `F_{Cat}` to specify supplementary mathematical information about that parent.
The main purpose of this class is to handle automatically the trivial part of the category hierarchy. For example, ``CartesianProductsOf(Groups())`` is set automatically as a subcategory of ``CartesianProductsOf(Monoids())``.
In practice, each subclass of this class should provide the following attributes:
- ``_functor_category`` - a string which should match the name of the nested category class to be used in each category to specify information and generic operations for elements of this category.
- ``_functor_name`` - an string which specifies the name of the functor, and also (when relevant) of the method on parents and elements used for calling the construction.
TODO: What syntax do we want for `F_{Cat}`? For example, for the tensor product construction, which one of the followings do we want (see chat on IRC, on 07/12/2009):
- ``tensor(Cat)`` - ``tensor((Cat, Cat))`` - ``tensor.of((Cat, Cat))`` - ``tensor.category_from_categories((Cat, Cat, Cat))`` - ``Cat.TensorProducts()``
The syntax ``Cat.TensorProducts()`` does not supports well multivariate constructions like ``tensor.of([Algebras(), HopfAlgebras(), ...])``. Also it forces every category to be (somehow) aware of all the tensorial construction that could apply to it, even those which are only induced from super categories.
Note: for each functorial construction, there probably is one (or several) largest categories on which it applies. For example, the :func:`~sage.categories.cartesian_product.CartesianProducts` construction makes only sense for concrete categories, that is subcategories of ``Sets()``. Maybe we want to model this one way or the other. """
def category_from_parents(self, parents): """ Return the category of `F(A,B,...)` for `A,B,...` parents.
INPUT:
- self: a functor F - parents: a list (or iterable) of parents.
EXAMPLES::
sage: E = CombinatorialFreeModule(QQ, ["a", "b", "c"]) sage: tensor.category_from_parents((E, E, E)) Category of tensor products of vector spaces with basis over Rational Field """ # Should we pass a set of categories to reduce the cache size? # But then this would impose that, for any constructor, the # category of the result does not depend on the order/repetition # of the categories of the parents
@cached_method def category_from_categories(self, categories): """ Return the category of `F(A,B,...)` for `A,B,...` parents in the given categories.
INPUT:
- ``self``: a functor `F` - ``categories``: a non empty tuple of categories
EXAMPLES::
sage: Cat1 = Rings() sage: Cat2 = Groups() sage: cartesian_product.category_from_categories((Cat1, Cat1, Cat1)) Join of Category of rings and ... and Category of Cartesian products of monoids and Category of Cartesian products of commutative additive groups
sage: cartesian_product.category_from_categories((Cat1, Cat2)) Category of Cartesian products of monoids """
def category_from_category(self, category): """ Return the category of `F(A,B,...)` for `A,B,...` parents in ``category``.
INPUT:
- ``self``: a functor `F` - ``category``: a category
EXAMPLES::
sage: tensor.category_from_category(ModulesWithBasis(QQ)) Category of tensor products of vector spaces with basis over Rational Field
# TODO: add support for parametrized functors """
def _repr_(self): """ EXAMPLES::
sage: tensor # indirect doctest The tensor functorial construction """
def __call__(self, args, **kwargs): """ Functorial construction application
INPUT: - ``self``: a covariant functorial construction `F` - ``args``: a tuple (or iterable) of parents or elements
Returns `F(args)`
EXAMPLES::
sage: E = CombinatorialFreeModule(QQ, ["a", "b", "c"]); E.rename("E") sage: tensor((E, E, E)) E # E # E """
class FunctorialConstructionCategory(Category): # Should this be CategoryWithBase? """ Abstract class for categories `F_{Cat}` obtained through a functorial construction """
@lazy_class_attribute def _base_category_class(cls): """ Recover the class of the base category.
OUTPUT:
A *tuple* whose single entry is the base category class.
.. WARNING::
This is only used for functorial construction categories that are not implemented as nested classes, and won't work otherwise.
.. SEEALSO:: :meth:`__classcall__`
EXAMPLES::
sage: GradedModules._base_category_class (<class 'sage.categories.modules.Modules'>,) sage: GradedAlgebrasWithBasis._base_category_class (<class 'sage.categories.algebras_with_basis.AlgebrasWithBasis'>,)
The reason for wrapping the base category class in a tuple is that, often, the base category class implements a :meth:`__classget__` method which would get in the way upon attribute access::
sage: F = GradedAlgebrasWithBasis sage: F._foo = F._base_category_class[0] sage: F._foo Traceback (most recent call last): ... AssertionError: base category class for <...AlgebrasWithBasis'> mismatch; expected <...Algebras'>, got <...GradedAlgebrasWithBasis'>
We note that because ``Algebras.WithBasis`` is not lazily imported on startup (see :trac:`22955`), the test fails at a different point in the code. However, if this import becomes lazy again, then the following error will be generated and can replace the above::
sage: F._foo # not tested Traceback (most recent call last): ... ValueError: could not infer axiom for the nested class <...AlgebrasWithBasis'> of <...GradedAlgebrasWithBasis'>
.. TODO::
The logic is very similar to that implemented in :class:`CategoryWithAxiom._base_category_class`. Find a way to refactor this to avoid the duplication. """
@staticmethod def __classcall__(cls, category=None, *args): """ Make ``XXXCat(**)`` a shorthand for ``Cat(**).XXX()``.
EXAMPLES::
sage: GradedModules(ZZ) # indirect doctest Category of graded modules over Integer Ring sage: Modules(ZZ).Graded() Category of graded modules over Integer Ring sage: Modules.Graded(ZZ) Category of graded modules over Integer Ring sage: GradedModules(ZZ) is Modules(ZZ).Graded() True
.. SEEALSO:: :meth:`_base_category_class`
.. TODO::
The logic is very similar to that implemented in :class:`CategoryWithAxiom.__classcall__`. Find a way to refactor this to avoid the duplication. """ else:
@staticmethod def __classget__(cls, base_category, base_category_class): r""" Special binding for covariant constructions.
This implements a hack allowing e.g. ``category.Subquotients`` to recover the default ``Subquotients`` method defined in ``Category``, even if it has been overriden by a ``Subquotients`` class.
EXAMPLES::
sage: Sets.Subquotients <class 'sage.categories.sets_cat.Sets.Subquotients'> sage: Sets().Subquotients Cached version of <function Subquotients at ...>
This method also initializes the attribute ``_base_category_class`` if not already set::
sage: Sets.Subquotients._base_category_class (<class 'sage.categories.sets_cat.Sets'>,)
It also forces the resolution of lazy imports (see :trac:`15648`)::
sage: type(Algebras.__dict__["Graded"]) <type 'sage.misc.lazy_import.LazyImport'> sage: Algebras.Graded <class 'sage.categories.graded_algebras.GradedAlgebras'> sage: type(Algebras.__dict__["Graded"]) <type 'sage.misc.classcall_metaclass.ClasscallMetaclass'>
.. TODO::
The logic is very similar to that implemented in :class:`CategoryWithAxiom.__classget__`. Find a way to refactor this to avoid the duplication. """ else: "base category class for {} mismatch; expected {}, got {}".format( cls, cls._base_category_class[0], base_category_class)
# Workaround #15648: if Sets.Subquotients is a LazyImport object, # this forces the substitution of the object back into Sets # to avoid resolving the lazy import over and over cls._functor_category)
@classmethod @cached_function def category_of(cls, category, *args): """ Return the image category of the functor `F_{Cat}`.
This is the main entry point for constructing the category `F_{Cat}` of parents `F(A,B,...)` constructed from parents `A,B,...` in `Cat`.
INPUT:
- ``cls`` -- the category class for the functorial construction `F` - ``category`` -- a category `Cat` - ``*args`` -- further arguments for the functor
EXAMPLES::
sage: sage.categories.tensor.TensorProductsCategory.category_of(ModulesWithBasis(QQ)) Category of tensor products of vector spaces with basis over Rational Field
sage: sage.categories.algebra_functor.AlgebrasCategory.category_of(FiniteMonoids(), QQ) Join of Category of finite dimensional algebras with basis over Rational Field and Category of monoid algebras over Rational Field and Category of finite set algebras over Rational Field """ # TODO: find a better test # the purpose is to test whether ``category`` implements that functor else:
def __init__(self, category, *args): """ TESTS::
sage: from sage.categories.covariant_functorial_construction import CovariantConstructionCategory sage: class FooBars(CovariantConstructionCategory): ....: _functor_category = "FooBars" ....: _base_category_class = (Category,) sage: Category.FooBars = lambda self: FooBars.category_of(self) sage: C = FooBars(ModulesWithBasis(ZZ)) sage: C Category of foo bars of modules with basis over Integer Ring sage: C.base_category() Category of modules with basis over Integer Ring sage: latex(C) \mathbf{FooBars}(\mathbf{ModulesWithBasis}_{\Bold{Z}}) sage: import __main__; __main__.FooBars = FooBars # Fake FooBars being defined in a python module sage: TestSuite(C).run() """
def base_category(self): """ Return the base category of the category ``self``.
For any category ``B`` = `F_{Cat}` obtained through a functorial construction `F`, the call ``B.base_category()`` returns the category `Cat`.
EXAMPLES::
sage: Semigroups().Quotients().base_category() Category of semigroups """
def extra_super_categories(self): """ Return the extra super categories of a construction category.
Default implementation which returns ``[]``.
EXAMPLES::
sage: Sets().Subquotients().extra_super_categories() [] sage: Semigroups().Quotients().extra_super_categories() [] """
def super_categories(self): """ Return the super categories of a construction category.
EXAMPLES::
sage: Sets().Subquotients().super_categories() [Category of sets] sage: Semigroups().Quotients().super_categories() [Category of subquotients of semigroups, Category of quotients of sets] """ self.extra_super_categories(), as_list = True)
def _repr_object_names(self): """ EXAMPLES::
sage: Semigroups().Subquotients() # indirect doctest Category of subquotients of semigroups """
def _latex_(self): """ EXAMPLES::
sage: latex(Semigroups().Subquotients()) # indirect doctest \mathbf{Subquotients}(\mathbf{Semigroups}) sage: latex(ModulesWithBasis(QQ).TensorProducts()) \mathbf{TensorProducts}(\mathbf{WithBasis}_{\Bold{Q}}) sage: latex(Semigroups().Algebras(QQ)) \mathbf{Algebras}(\mathbf{Semigroups}) """
class CovariantConstructionCategory(FunctorialConstructionCategory): """ Abstract class for categories `F_{Cat}` obtained through a covariant functorial construction """
@classmethod def default_super_categories(cls, category, *args): """ Return the default super categories of `F_{Cat}(A,B,...)` for `A,B,...` parents in `Cat`.
INPUT:
- ``cls`` -- the category class for the functor `F` - ``category`` -- a category `Cat` - ``*args`` -- further arguments for the functor
OUTPUT: a (join) category
The default implementation is to return the join of the categories of `F(A,B,...)` for `A,B,...` in turn in each of the super categories of ``category``.
This is implemented as a class method, in order to be able to reconstruct the functorial category associated to each of the super categories of ``category``.
EXAMPLES:
Bialgebras are both algebras and coalgebras::
sage: Bialgebras(QQ).super_categories() [Category of algebras over Rational Field, Category of coalgebras over Rational Field]
Hence tensor products of bialgebras are tensor products of algebras and tensor products of coalgebras::
sage: Bialgebras(QQ).TensorProducts().super_categories() [Category of tensor products of algebras over Rational Field, Category of tensor products of coalgebras over Rational Field]
Here is how :meth:`default_super_categories` was called internally::
sage: sage.categories.tensor.TensorProductsCategory.default_super_categories(Bialgebras(QQ)) Join of Category of tensor products of algebras over Rational Field and Category of tensor products of coalgebras over Rational Field
We now show a similar example, with the ``Algebra`` functor which takes a parameter `\QQ`::
sage: FiniteMonoids().super_categories() [Category of monoids, Category of finite semigroups] sage: sorted(FiniteMonoids().Algebras(QQ).super_categories(), key=str) [Category of finite dimensional algebras with basis over Rational Field, Category of finite set algebras over Rational Field, Category of monoid algebras over Rational Field]
Note that neither the category of *finite* semigroup algebras nor that of monoid algebras appear in the result; this is because there is currently nothing specific implemented about them.
Here is how :meth:`default_super_categories` was called internally::
sage: sage.categories.algebra_functor.AlgebrasCategory.default_super_categories(FiniteMonoids(), QQ) Join of Category of finite dimensional algebras with basis over Rational Field and Category of monoid algebras over Rational Field and Category of finite set algebras over Rational Field """ for cat in category._super_categories if hasattr(cat, cls._functor_category)])
def is_construction_defined_by_base(self): r""" Return whether the construction is defined by the base of ``self``.
EXAMPLES:
The graded functorial construction is defined by the modules category. Hence this method returns ``True`` for graded modules and ``False`` for other graded xxx categories::
sage: Modules(ZZ).Graded().is_construction_defined_by_base() True sage: Algebras(QQ).Graded().is_construction_defined_by_base() False sage: Modules(ZZ).WithBasis().Graded().is_construction_defined_by_base() False
This is implemented as follows: given the base category `A` and the construction `F` of ``self``, that is ``self=A.F()``, check whether no super category of `A` has `F` defined.
.. NOTE::
Recall that, when `A` does not implement the construction ``F``, a join category is returned. Therefore, in such cases, this method is not available::
sage: Coalgebras(QQ).Graded().is_construction_defined_by_base() Traceback (most recent call last): ... AttributeError: 'JoinCategory_with_category' object has no attribute 'is_construction_defined_by_base' """
def additional_structure(self): r""" Return the additional structure defined by ``self``.
By default, a functorial construction category ``A.F()`` defines additional structure if and only if `A` is the category defining `F`. The rationale is that, for a subcategory `B` of `A`, the fact that `B.F()` morphisms shall preserve the `F`-specific structure is already imposed by `A.F()`.
.. SEEALSO::
- :meth:`Category.additional_structure`. - :meth:`is_construction_defined_by_base`.
EXAMPLES:
sage: Modules(ZZ).Graded().additional_structure() Category of graded modules over Integer Ring sage: Algebras(ZZ).Graded().additional_structure()
TESTS::
sage: Modules(ZZ).Graded().additional_structure.__module__ 'sage.categories.covariant_functorial_construction' """ else:
class RegressiveCovariantConstructionCategory(CovariantConstructionCategory): """ Abstract class for categories `F_{Cat}` obtained through a regressive covariant functorial construction """
@classmethod def default_super_categories(cls, category, *args): """ Return the default super categories of `F_{Cat}(A,B,...)` for `A,B,...` parents in `Cat`.
INPUT:
- ``cls`` -- the category class for the functor `F` - ``category`` -- a category `Cat` - ``*args`` -- further arguments for the functor
OUTPUT:
A join category.
This implements the property that an induced subcategory is a subcategory.
EXAMPLES:
A subquotient of a monoid is a monoid, and a subquotient of semigroup::
sage: Monoids().Subquotients().super_categories() [Category of monoids, Category of subquotients of semigroups]
TESTS::
sage: C = Monoids().Subquotients() sage: C.__class__.default_super_categories(C.base_category(), *C._args) Category of unital subquotients of semigroups """ |