Coverage for local/lib/python2.7/site-packages/sage/categories/category.py: 73%

[<class '__main__.Ds.parent_class'>, <class '__main__.Cs.parent_class'>, <class '__main__.Bs.parent_class'>, <class '__main__.As.parent_class'>, <... 'object'>]

Note that that two categories in the same class need not have the

same ``super_categories``. For example, ``Algebras(QQ)`` has

``VectorSpaces(QQ)`` as super category, whereas ``Algebras(ZZ)``

only has ``Modules(ZZ)`` as super category. In particular, the

constructed parent class and element class will differ (inheriting,

or not, methods specific for vector spaces)::

sage: Algebras(QQ).parent_class is Algebras(ZZ).parent_class

False

sage: issubclass(Algebras(QQ).parent_class, VectorSpaces(QQ).parent_class)

True

On the other hand, identical hierarchies of classes are,

preferably, built only once (e.g. for categories over a base ring)::

sage: Algebras(GF(5)).parent_class is Algebras(GF(7)).parent_class

True

sage: F = FractionField(ZZ['t'])

sage: Coalgebras(F).parent_class is Coalgebras(FractionField(F['x'])).parent_class

True

We now construct a parent in the usual way::

sage: class myparent(Parent):

....: def __init__(self):

....: Parent.__init__(self, category=Ds())

....: def g(self):

....: return "myparent"

....: class Element:

....: pass

sage: D = myparent()

sage: D.__class__

sage: D.__class__.__bases__

(<class '__main__.myparent'>, <class '__main__.Ds.parent_class'>)

sage: D.__class__.mro()

[<class '__main__.myparent_with_category'>,

<class '__main__.myparent'>,

<type 'sage.structure.parent.Parent'>,

<type 'sage.structure.category_object.CategoryObject'>,

<type 'sage.structure.sage_object.SageObject'>,

<class '__main__.Ds.parent_class'>,

<class '__main__.Cs.parent_class'>,

<class '__main__.Bs.parent_class'>,

<class '__main__.As.parent_class'>,

<... 'object'>]

sage: D.fA()

'A'

sage: D.fB()

'B'

sage: D.fC()

'C'

sage: D.fD()

'D'

sage: D.f()

'C'

sage: D.g()

'myparent'

sage: D.element_class

sage: D.element_class.mro()

[<class '__main__.myparent_with_category.element_class'>,

<class __main__.Element at ...>,

<class '__main__.Ds.element_class'>,

<class '__main__.Cs.element_class'>,

<class '__main__.Bs.element_class'>,

<class '__main__.As.element_class'>,

<... 'object'>]

TESTS::

sage: import __main__

sage: __main__.myparent = myparent

sage: __main__.As = As

sage: __main__.Bs = Bs

sage: __main__.Cs = Cs

sage: __main__.Ds = Ds

sage: loads(dumps(Ds)) is Ds

True

sage: loads(dumps(Ds())) is Ds()

True

sage: loads(dumps(Ds().element_class)) is Ds().element_class

True

.. automethod:: _super_categories

.. automethod:: _super_categories_for_classes

.. automethod:: _all_super_categories

.. automethod:: _all_super_categories_proper

.. automethod:: _set_of_super_categories

.. automethod:: _make_named_class

.. automethod:: _repr_

.. automethod:: _repr_object_names

.. automethod:: _test_category

.. automethod:: _with_axiom

.. automethod:: _with_axiom_as_tuple

.. automethod:: _without_axioms

.. automethod:: _sort

.. automethod:: _sort_uniq

.. automethod:: __classcall__

.. automethod:: __init__

"""

@staticmethod

def __classcall__(cls, *args, **options):

"""

Input mangling for unique representation.

Let ``C = Cs(...)`` be a category. Since :trac:`12895`, the

class of ``C`` is a dynamic subclass ``Cs_with_category`` of

``Cs`` in order for ``C`` to inherit code from the

``SubcategoryMethods`` nested classes of its super categories.

The purpose of this ``__classcall__`` method is to ensure that

reconstructing ``C`` from its class with

``Cs_with_category(...)`` actually calls properly ``Cs(...)``

and gives back ``C``.

.. SEEALSO:: :meth:`subcategory_class`

EXAMPLES::

sage: A = Algebras(QQ)

sage: A.__class__

sage: A is Algebras(QQ)

True

sage: A is A.__class__(QQ)

True

"""

if isinstance(cls, DynamicMetaclass):

cls = cls.__base__

return super(Category, cls).__classcall__(cls, *args, **options)

def __init__(self, s=None):

"""

Initializes this category.

EXAMPLES::

sage: class SemiprimitiveRings(Category):

....: def super_categories(self):

....: return [Rings()]

....:

....: class ParentMethods:

....: def jacobson_radical(self):

....: return self.ideal(0)

....:

sage: C = SemiprimitiveRings()

sage: C

Category of semiprimitive rings

sage: C.__class__

.. NOTE::

Specifying the name of this category by passing a string

is deprecated. If the default name (built from the name of

the class) is not adequate, please use

:meth:`_repr_object_names` to customize it.

"""

assert s is None

self.__class__ = dynamic_class("{}_with_category".format(self.__class__.__name__),

(self.__class__, self.subcategory_class, ),

cache = False, reduction = None,

doccls=self.__class__)

@lazy_attribute

def _label(self):

"""

A short name of ``self``, obtained from its type.

EXAMPLES::

sage: Rings()._label

'Rings'

"""

t = str(self.__class__.__base__)

t = t[t.rfind('.')+1:]

return t[:t.rfind("'")]

# TODO: move this code into the method _repr_object_names once passing a string is not accepted anymore

@lazy_attribute

def __repr_object_names(self):

"""

Determine the name of the objects of this category

from its type, if it has not been explicitly given

at initialisation.

EXAMPLES::

sage: Rings()._Category__repr_object_names

'rings'

sage: PrincipalIdealDomains()._Category__repr_object_names

'principal ideal domains'

sage: Rings()

Category of rings

"""

i = -1

s = self._label

while i < len(s)-1:

for i in range(len(s)):

if s[i].isupper():

s = s[:i] + " " + s[i].lower() + s[i+1:]

break

return s.lstrip()

def _repr_object_names(self):

"""

Return the name of the objects of this category.

EXAMPLES::

sage: FiniteGroups()._repr_object_names()

'finite groups'

sage: AlgebrasWithBasis(QQ)._repr_object_names()

'algebras with basis over Rational Field'

"""

return self.__repr_object_names

def _short_name(self):

"""

Return a CamelCase name for this category.

EXAMPLES::

sage: CoxeterGroups()._short_name()

'CoxeterGroups'

sage: AlgebrasWithBasis(QQ)._short_name()

'AlgebrasWithBasis'

Conventions for short names should be discussed at the level

of Sage, and only then applied accordingly here.

"""

return self._label

@classmethod

def an_instance(cls):

"""

Return an instance of this class.

EXAMPLES::

sage: Rings.an_instance()

Category of rings

Parametrized categories should overload this default

implementation to provide appropriate arguments::

sage: Algebras.an_instance()

Category of algebras over Rational Field

sage: Bimodules.an_instance()

Category of bimodules over Rational Field on the left and Real Field with 53 bits of precision on the right

sage: AlgebraIdeals.an_instance()

Category of algebra ideals in Univariate Polynomial Ring in x over Rational Field

"""

return cls()

def __call__(self, x, *args, **opts):

"""

Construct an object in this category from the data in ``x``,

or throw ``TypeError`` or ``NotImplementedError``.

If ``x`` is readily in ``self`` it is returned unchanged.

Categories wishing to extend this minimal behavior should

implement :meth:`._call_`.

EXAMPLES::

sage: Rings()(ZZ)

Integer Ring

"""

if x in self:

return x

return self._call_(x, *args, **opts)

def _call_(self, x):

"""

Construct an object in this category from the data in ``x``,

or throw ``NotImplementedError``.

EXAMPLES::

sage: Semigroups()._call_(3)

Traceback (most recent call last):

...

NotImplementedError

"""

raise NotImplementedError

def _repr_(self):

"""

Return the print representation of this category.

EXAMPLES::

sage: Sets() # indirect doctest

Category of sets

"""

return "Category of {}".format(self._repr_object_names())

def _latex_(self):

r"""

Returns the latex representation of this category.

EXAMPLES::

sage: latex(Sets()) # indirect doctest

\mathbf{Sets}

sage: latex(CommutativeAdditiveSemigroups())

\mathbf{CommutativeAdditiveSemigroups}

"""

return "\\mathbf{%s}"%self._short_name()

# The convention for which hash function to use should be decided at the level of UniqueRepresentation

# The implementation below is bad (hash independent of the base ring)

# def __hash__(self):

# """

# Returns a hash for this category.

# Currently this is just the hash of the string representing the category.

# EXAMPLES::

# sage: hash(Algebras(QQ)) #indirect doctest

# 699942203

# sage: hash(Algebras(ZZ))

# 699942203

# """

# return hash(self.__category) # Any reason not to use id?

def _subcategory_hook_(self, category):

"""

Quick subcategory check.

INPUT:

- ``category`` -- a category

OUTPUT:

- ``True``, if ``category`` is a subcategory of ``self``.

- ``False``, if ``category`` is not a subcategory of ``self``.

- ``Unknown``, if a quick check was not enough to determine

whether ``category`` is a subcategory of ``self`` or not.

The aim of this method is to offer a framework to add cheap

tests for subcategories. When doing

``category.is_subcategory(self)`` (note the reverse order of

``self`` and ``category``), this method is usually called

first. Only if it returns ``Unknown``, :meth:`is_subcategory`

will build the list of super categories of ``category``.

This method need not to handle the case where ``category`` is

``self``; this is the first test that is done in

:meth:`is_subcategory`.

This default implementation tests whether the parent class of

``category`` is a subclass of the parent class of ``self``.

This is most of the time a complete subcategory test.

.. WARNING::

This test is incomplete for categories in

:class:`CategoryWithParameters`, as introduced by

:trac:`11935`. This method is therefore overwritten by

:meth:`~sage.categories.category.CategoryWithParameters._subcategory_hook_`.

EXAMPLES::

sage: Rings()._subcategory_hook_(Rings())

True

"""

return issubclass(category.parent_class, self.parent_class)

def __contains__(self, x):

"""

Membership testing

Returns whether ``x`` is an object in this category, that is

if the category of ``x`` is a subcategory of ``self``.

EXAMPLES::

sage: ZZ in Sets()

True

"""

try:

c = x.category()

except AttributeError:

return False

return c.is_subcategory(self)

@staticmethod

def __classcontains__(cls, x):

"""

Membership testing, without arguments

INPUT:

- ``cls`` -- a category class

- ``x`` -- any object

Returns whether ``x`` is an object of a category which is an instance

of ``cls``.

EXAMPLES:

This method makes it easy to test if an object is, say, a

vector space, without having to specify the base ring::

sage: F = FreeModule(QQ,3)

sage: F in VectorSpaces

True

sage: F = FreeModule(ZZ,3)

sage: F in VectorSpaces

False

sage: F in Algebras

False

TESTS:

Non category objects shall be handled properly::

sage: [1,2] in Algebras

False

"""

try:

c = x.categories()

except AttributeError:

return False

return any(isinstance(cat, cls) for cat in c)

def is_abelian(self):

"""

Returns whether this category is abelian.

An abelian category is a category satisfying:

- It has a zero object;

- It has all pullbacks and pushouts;

- All monomorphisms and epimorphisms are normal.

Equivalently, one can define an increasing sequence of conditions:

- A category is pre-additive if it is enriched over abelian groups

(all homsets are abelian groups and composition is bilinear);

- A pre-additive category is additive if every finite set of objects

has a biproduct (we can form direct sums and direct products);

- An additive category is pre-abelian if every morphism has both a

kernel and a cokernel;

- A pre-abelian category is abelian if every monomorphism is the

kernel of some morphism and every epimorphism is the cokernel of

some morphism.

EXAMPLES::

sage: Modules(ZZ).is_abelian()

True

sage: FreeModules(ZZ).is_abelian()

False

sage: FreeModules(QQ).is_abelian()

True

sage: CommutativeAdditiveGroups().is_abelian()

True

sage: Semigroups().is_abelian()

Traceback (most recent call last):

NotImplementedError: is_abelian

"""

raise NotImplementedError("is_abelian")

##########################################################################

# Methods related to the category hierarchy

##########################################################################

def category_graph(self):

r"""

Returns the graph of all super categories of this category

EXAMPLES::

sage: C = Algebras(QQ)

sage: G = C.category_graph()

sage: G.is_directed_acyclic()

True

sage: G.girth()

"""

return category_graph([self])

@abstract_method

def super_categories(self):

"""

Return the *immediate* super categories of ``self``.

OUTPUT:

- a duplicate-free list of categories.

Every category should implement this method.

EXAMPLES::

sage: Groups().super_categories()

[Category of monoids, Category of inverse unital magmas]

sage: Objects().super_categories()

[]

.. NOTE::

Since :trac:`10963`, the order of the categories in the

result is irrelevant. For details, see

:ref:`category-primer-category-order`.

.. NOTE::

Whenever speed matters, developers are advised to use the

lazy attribute :meth:`_super_categories` instead of

calling this method.

"""

@lazy_attribute

def _all_super_categories(self):

r"""

All the super categories of this category, including this category.

Since :trac:`11943`, the order of super categories is

determined by Python's method resolution order C3 algorithm.

.. SEEALSO:: :meth:`all_super_categories`

.. note:: this attribute is likely to eventually become a tuple.

.. note:: this sets :meth:`_super_categories_for_classes` as a side effect

EXAMPLES::

sage: C = Rings(); C

Category of rings

sage: C._all_super_categories

[Category of rings, Category of rngs, Category of semirings, ...

Category of monoids, ...

Category of commutative additive groups, ...

Category of sets, Category of sets with partial maps,

Category of objects]

"""

(result, bases) = C3_sorted_merge([cat._all_super_categories

for cat in self._super_categories] +

[self._super_categories],

category_sort_key)

if not sorted(result, key = category_sort_key, reverse=True) == result:

warn("Inconsistent sorting results for all super categories of {}".format(

self.__class__))

self._super_categories_for_classes = bases

return [self] + result

@lazy_attribute

def _all_super_categories_proper(self):

r"""

All the proper super categories of this category.

Since :trac:`11943`, the order of super categories is

determined by Python's method resolution order C3 algorithm.

.. SEEALSO:: :meth:`all_super_categories`

.. note:: this attribute is likely to eventually become a tuple.

EXAMPLES::

sage: C = Rings(); C

Category of rings

sage: C._all_super_categories_proper

[Category of rngs, Category of semirings, ...

Category of monoids, ...

Category of commutative additive groups, ...

Category of sets, Category of sets with partial maps,

Category of objects]

"""

return self._all_super_categories[1:]

@lazy_attribute

def _set_of_super_categories(self):

"""

The frozen set of all proper super categories of this category.

.. note:: this is used for speeding up category containment tests.

.. SEEALSO:: :meth:`all_super_categories`

EXAMPLES::

sage: Groups()._set_of_super_categories

frozenset({Category of inverse unital magmas,

Category of unital magmas,

Category of magmas,

Category of monoids,

Category of objects,

Category of semigroups,

Category of sets with partial maps,

Category of sets})

sage: sorted(Groups()._set_of_super_categories, key=str)

[Category of inverse unital magmas, Category of magmas, Category of monoids,

Category of objects, Category of semigroups, Category of sets,

Category of sets with partial maps, Category of unital magmas]

TESTS::

sage: C = HopfAlgebrasWithBasis(GF(7))

sage: C._set_of_super_categories == frozenset(C._all_super_categories_proper)

True

"""

return frozenset(self._all_super_categories_proper)

def all_super_categories(self, proper=False):

"""

Returns the list of all super categories of this category.

INPUT:

- ``proper`` -- a boolean (default: ``False``); whether to exclude this category.

Since :trac:`11943`, the order of super categories is

determined by Python's method resolution order C3 algorithm.

.. note::

Whenever speed matters, the developers are advised to use

instead the lazy attributes :meth:`_all_super_categories`,

:meth:`_all_super_categories_proper`, or

:meth:`_set_of_super_categories`, as

appropriate. Simply because lazy attributes are much

faster than any method.

EXAMPLES::

sage: C = Rings(); C

Category of rings

sage: C.all_super_categories()

[Category of rings, Category of rngs, Category of semirings, ...

Category of monoids, ...

Category of commutative additive groups, ...

Category of sets, Category of sets with partial maps,

Category of objects]

sage: C.all_super_categories(proper = True)

[Category of rngs, Category of semirings, ...

Category of monoids, ...

Category of commutative additive groups, ...

Category of sets, Category of sets with partial maps,

Category of objects]

sage: Sets().all_super_categories()

[Category of sets, Category of sets with partial maps, Category of objects]

sage: Sets().all_super_categories(proper=True)

[Category of sets with partial maps, Category of objects]

sage: Sets().all_super_categories() is Sets()._all_super_categories

True

sage: Sets().all_super_categories(proper=True) is Sets()._all_super_categories_proper

True

"""

if proper:

return self._all_super_categories_proper

return self._all_super_categories

@lazy_attribute

def _super_categories(self):

"""

The immediate super categories of this category.

This lazy attribute caches the result of the mandatory method

:meth:`super_categories` for speed. It also does some mangling

(flattening join categories, sorting, ...).

Whenever speed matters, developers are advised to use this

lazy attribute rather than calling :meth:`super_categories`.

.. NOTE::

This attribute is likely to eventually become a tuple.

When this happens, we might as well use :meth:`Category._sort`,

if not :meth:`Category._sort_uniq`.

EXAMPLES::

sage: Rings()._super_categories

[Category of rngs, Category of semirings]

"""

return sorted(_flatten_categories(self.super_categories(),JoinCategory), key = category_sort_key, reverse=True)

@lazy_attribute

def _super_categories_for_classes(self):

"""

The super categories of this category used for building classes.

This is a close variant of :meth:`_super_categories` used for

constructing the list of the bases for :meth:`parent_class`,

:meth:`element_class`, and friends. The purpose is ensure that

Python will find a proper Method Resolution Order for those

classes. For background, see :mod:`sage.misc.c3_controlled`.

.. SEEALSO:: :meth:`_cmp_key`.

.. NOTE::

This attribute is calculated as a by-product of computing

:meth:`_all_super_categories`.

EXAMPLES::

sage: Rings()._super_categories_for_classes

[Category of rngs, Category of semirings]

"""

self._all_super_categories

return self._super_categories_for_classes

##########################################################################

# Methods handling of full subcategories

##########################################################################

def additional_structure(self):

"""

Return whether ``self`` defines additional structure.

OUTPUT:

- ``self`` if ``self`` defines additional structure and

``None`` otherwise. This default implementation returns

``self``.

A category `C` *defines additional structure* if `C`-morphisms

shall preserve more structure (e.g. operations) than that

specified by the super categories of `C`. For example, the

category of magmas defines additional structure, namely the

operation `*` that shall be preserved by magma morphisms. On

the other hand the category of rings does not define additional

structure: a function between two rings that is both a unital

magma morphism and a unital additive magma morphism is

automatically a ring morphism.

Formally speaking `C` *defines additional structure*, if `C`

is *not* a full subcategory of the join of its super

categories: the morphisms need to preserve more structure, and

thus the homsets are smaller.

By default, a category is considered as defining additional

structure, unless it is a :ref:`category with axiom

<category-primer-axioms>`.

EXAMPLES:

Here are some typical structure categories, with the

additional structure they define::

sage: Sets().additional_structure()

Category of sets

sage: Magmas().additional_structure() # `*`

Category of magmas

sage: AdditiveMagmas().additional_structure() # `+`

Category of additive magmas

sage: LeftModules(ZZ).additional_structure() # left multiplication by scalar

Category of left modules over Integer Ring

sage: Coalgebras(QQ).additional_structure() # coproduct

Category of coalgebras over Rational Field

sage: Crystals().additional_structure() # crystal operators

Category of crystals

On the other hand, the category of semigroups is not a

structure category, since its operation `+` is already defined

by the category of magmas::

sage: Semigroups().additional_structure()

Most :ref:`categories with axiom <category-primer-axioms>`

don't define additional structure::

sage: Sets().Finite().additional_structure()

sage: Rings().Commutative().additional_structure()

sage: Modules(QQ).FiniteDimensional().additional_structure()

sage: from sage.categories.magmatic_algebras import MagmaticAlgebras

sage: MagmaticAlgebras(QQ).Unital().additional_structure()

As of Sage 6.4, the only exceptions are the category of unital

magmas or the category of unital additive magmas (both define

a unit which shall be preserved by morphisms)::

sage: Magmas().Unital().additional_structure()

Category of unital magmas

sage: AdditiveMagmas().AdditiveUnital().additional_structure()

Category of additive unital additive magmas

Similarly, :ref:`functorial construction categories

<category-primer-functorial-constructions>` don't define

additional structure, unless the construction is actually

defined by their base category. For example, the category of

graded modules defines a grading which shall be preserved by

morphisms::

sage: Modules(ZZ).Graded().additional_structure()

Category of graded modules over Integer Ring

On the other hand, the category of graded algebras does not

define additional structure; indeed an algebra morphism which

is also a module morphism is a graded algebra morphism::

sage: Algebras(ZZ).Graded().additional_structure()

Similarly, morphisms are requested to preserve the structure

given by the following constructions::

sage: Sets().Quotients().additional_structure()

Category of quotients of sets

sage: Sets().CartesianProducts().additional_structure()

Category of Cartesian products of sets

sage: Modules(QQ).TensorProducts().additional_structure()

This might change, as we are lacking enough data points to

guarantee that this was the correct design decision.

.. NOTE::

In some cases a category defines additional structure,

where the structure can be useful to manipulate morphisms

but where, in most use cases, we don't want the morphisms

to necessarily preserve it. For example, in the context of

finite dimensional vector spaces, having a distinguished

basis allows for representing morphisms by matrices; yet

considering only morphisms that preserve that

distinguished basis would be boring.

In such cases, we might want to eventually have two

categories, one where the additional structure is

preserved, and one where it's not necessarily preserved

(we would need to find an idiom for this).

At this point, a choice is to be made each time, according

to the main use cases. Some of those choices are yet to be

settled. For example, should by default:

- an euclidean domain morphism preserve euclidean

division? ::

sage: EuclideanDomains().additional_structure()

Category of euclidean domains

- an enumerated set morphism preserve the distinguished

enumeration? ::

sage: EnumeratedSets().additional_structure()

- a module with basis morphism preserve the distinguished

basis? ::

sage: Modules(QQ).WithBasis().additional_structure()

.. SEEALSO::

This method together with the methods overloading it

provide the basic data to determine, for a given category,

the super categories that define some structure (see

:meth:`structure`), and to test whether a category is a

full subcategory of some other category (see

:meth:`is_full_subcategory`). For example, the category of

Coxeter groups is not full subcategory of the category of

groups since morphisms need to preserve the distinguished

generators::

sage: CoxeterGroups().is_full_subcategory(Groups())

False

The support for modeling full subcategories has been

introduced in :trac:`16340`.

"""

return self

@cached_method

def structure(self):

r"""

Return the structure ``self`` is endowed with.

This method returns the structure that morphisms in this

category shall be preserving. For example, it tells that a

ring is a set endowed with a structure of both a unital magma

and an additive unital magma which satisfies some further

axioms. In other words, a ring morphism is a function that

preserves the unital magma and additive unital magma

structure.

In practice, this returns the collection of all the super

categories of ``self`` that define some additional structure,

as a frozen set.

EXAMPLES::

sage: Objects().structure()

frozenset()

sage: def structure(C):

....: return Category._sort(C.structure())

sage: structure(Sets())

(Category of sets, Category of sets with partial maps)

sage: structure(Magmas())

(Category of magmas, Category of sets, Category of sets with partial maps)

In the following example, we only list the smallest structure

categories to get a more readable output::

sage: def structure(C):

....: return Category._sort_uniq(C.structure())

sage: structure(Magmas())

(Category of magmas,)

sage: structure(Rings())

(Category of unital magmas, Category of additive unital additive magmas)

sage: structure(Fields())

(Category of euclidean domains,)

sage: structure(Algebras(QQ))

(Category of unital magmas,

Category of right modules over Rational Field,

Category of left modules over Rational Field)

sage: structure(HopfAlgebras(QQ).Graded().WithBasis().Connected())

(Category of hopf algebras over Rational Field,

Category of graded modules over Rational Field)

This method is used in :meth:`is_full_subcategory` for

deciding whether a category is a full subcategory of some

other category, and for documentation purposes. It is computed

recursively from the result of :meth:`additional_structure`

on the super categories of ``self``.

"""

result = { D for C in self.super_categories() for D in C.structure() }

if self.additional_structure() is not None:

result.add(self)

return frozenset(result)

def is_full_subcategory(self, other):

"""

Return whether ``self`` is a full subcategory of ``other``.

A subcategory `B` of a category `A` is a *full subcategory* if

any `A`-morphism between two objects of `B` is also a

`B`-morphism (the reciprocal always holds: any `B`-morphism

between two objects of `B` is an `A`-morphism).

This is computed by testing whether ``self`` is a subcategory

of ``other`` and whether they have the same structure, as

determined by :meth:`structure` from the

result of :meth:`additional_structure` on the super

categories.

.. WARNING::

A positive answer is guaranteed to be mathematically

correct. A negative answer may mean that Sage has not been

taught enough information (or can not yet within the

current model) to derive this information. See

:meth:`full_super_categories` for a discussion.

.. SEEALSO::

- :meth:`is_subcategory`

- :meth:`full_super_categories`

EXAMPLES::

sage: Magmas().Associative().is_full_subcategory(Magmas())

True

sage: Magmas().Unital().is_full_subcategory(Magmas())

False

sage: Rings().is_full_subcategory(Magmas().Unital() & AdditiveMagmas().AdditiveUnital())

True

Here are two typical examples of false negatives::

sage: Groups().is_full_subcategory(Semigroups())

False

sage: Groups().is_full_subcategory(Semigroups()) # todo: not implemented

True

sage: Fields().is_full_subcategory(Rings())

False

sage: Fields().is_full_subcategory(Rings()) # todo: not implemented

True

.. TODO::

The latter is a consequence of :class:`EuclideanDomains`

currently being a structure category. Is this what we

want? ::

sage: EuclideanDomains().is_full_subcategory(Rings())

False

"""

return self.is_subcategory(other) and \

len(self.structure()) == \

len(other.structure())

@cached_method

def full_super_categories(self):

"""

Return the *immediate* full super categories of ``self``.

.. SEEALSO::

- :meth:`super_categories`

- :meth:`is_full_subcategory`

.. WARNING::

The current implementation selects the full subcategories

among the immediate super categories of ``self``. This

assumes that, if `C\subset B\subset A` is a chain of

categories and `C` is a full subcategory of `A`, then `C`

is a full subcategory of `B` and `B` is a full subcategory

of `A`.

This assumption is guaranteed to hold with the current

model and implementation of full subcategories in

Sage. However, mathematically speaking, this is too

restrictive. This indeed prevents the complete modelling

of situations where any `A` morphism between elements of

`C` automatically preserves the `B` structure. See below

for an example.

EXAMPLES:

A semigroup morphism between two finite semigroups is a finite

semigroup morphism::

sage: Semigroups().Finite().full_super_categories()

[Category of semigroups]

On the other hand, a semigroup morphism between two monoids is

not necessarily a monoid morphism (which must map the unit to

the unit)::

sage: Monoids().super_categories()

[Category of semigroups, Category of unital magmas]

sage: Monoids().full_super_categories()

[Category of unital magmas]

Any semigroup morphism between two groups is automatically a

monoid morphism (in a group the unit is the unique idempotent,

so it has to be mapped to the unit). Yet, due to the

limitation of the model advertised above, Sage currently can't

be taught that the category of groups is a full subcategory of

the category of semigroups::

sage: Groups().full_super_categories() # todo: not implemented

[Category of monoids, Category of semigroups, Category of inverse unital magmas]

sage: Groups().full_super_categories()

[Category of monoids, Category of inverse unital magmas]

"""

return [C for C in self.super_categories()

if self.is_full_subcategory(C)]

##########################################################################

# Test methods

##########################################################################

def _test_category_graph(self, **options):

"""

Check that the category graph matches with Python's method resolution order

.. note::

By :trac:`11943`, the list of categories returned by

:meth:`all_super_categories` is supposed to match with the

method resolution order of the parent and element

classes. This method checks this.

.. todo:: currently, this won't work for hom categories.

EXAMPLES::

sage: C = HopfAlgebrasWithBasis(QQ)

sage: C.parent_class.mro() == [X.parent_class for X in C._all_super_categories] + [object]

True

sage: C.element_class.mro() == [X.element_class for X in C._all_super_categories] + [object]

True

sage: TestSuite(C).run() # indirect doctest

"""

tester = self._tester(**options)

tester.assertTrue(self.parent_class.mro() == [C.parent_class for C in self._all_super_categories] + [object])

tester.assertTrue(self.element_class.mro() == [C.element_class for C in self._all_super_categories] + [object])

def _test_category(self, **options):

r"""

Run generic tests on this category

.. SEEALSO:: :class:`TestSuite`.

EXAMPLES::

sage: Sets()._test_category()

Let us now write a couple broken categories::

sage: class MyObjects(Category):

....: pass

sage: MyObjects()._test_category()

Traceback (most recent call last):

...

NotImplementedError: <abstract method super_categories at ...>

sage: class MyObjects(Category):

....: def super_categories(self):

....: return tuple()

sage: MyObjects()._test_category()

Traceback (most recent call last):

...

AssertionError: Category of my objects.super_categories() should return a list

sage: class MyObjects(Category):

....: def super_categories(self):

....: return []

sage: MyObjects()._test_category()

Traceback (most recent call last):

...

AssertionError: Category of my objects is not a subcategory of Objects()

"""

from sage.categories.objects import Objects

from sage.categories.sets_cat import Sets

tester = self._tester(**options)

tester.assertTrue(isinstance(self.super_categories(), list),

"%s.super_categories() should return a list"%self)

tester.assertTrue(self.is_subcategory(Objects()),

"%s is not a subcategory of Objects()"%self)

tester.assertTrue(isinstance(self.parent_class, type))

tester.assertTrue(all(not isinstance(cat, JoinCategory) for cat in self._super_categories))

if not isinstance(self, JoinCategory):

tester.assertTrue(all(self._cmp_key > cat._cmp_key for cat in self._super_categories))

tester.assertTrue(self.is_subcategory( Category.join(self.super_categories()) )) # Not an obviously passing test with axioms

for category in self._all_super_categories_proper:

if self.is_full_subcategory(category):

tester.assertTrue(any(cat.is_subcategory(category)

for cat in self.full_super_categories()),

"Every full super category should be a super category"

"of some immediate full super category")

if self.is_subcategory(Sets()):

tester.assertTrue(isinstance(self.parent_class, type))

tester.assertTrue(isinstance(self.element_class, type))

_cmp_key = _cmp_key

##########################################################################

# Construction of the associated abstract classes for parents, elements, ...

##########################################################################

def _make_named_class(self, name, method_provider, cache=False, picklable=True):

"""

Construction of the parent/element/... class of ``self``.

INPUT:

- ``name`` -- a string; the name of the class as an attribute of

``self``. E.g. "parent_class"

- ``method_provider`` -- a string; the name of an attribute of

``self`` that provides methods for the new class (in

addition to those coming from the super categories).

E.g. "ParentMethods"

- ``cache`` -- a boolean or ``ignore_reduction`` (default: ``False``)

(passed down to dynamic_class; for internal use only)

- ``picklable`` -- a boolean (default: ``True``)

ASSUMPTION:

It is assumed that this method is only called from a lazy

attribute whose name coincides with the given ``name``.

OUTPUT:

A dynamic class with bases given by the corresponding named

classes of ``self``'s super_categories, and methods taken from

the class ``getattr(self,method_provider)``.

.. NOTE::

- In this default implementation, the reduction data of

the named class makes it depend on ``self``. Since the

result is going to be stored in a lazy attribute of

``self`` anyway, we may as well disable the caching in

``dynamic_class`` (hence the default value

``cache=False``).

- :class:`CategoryWithParameters` overrides this method so

that the same parent/element/... classes can be shared

between closely related categories.

- The bases of the named class may also contain the named

classes of some indirect super categories, according to

:meth:`_super_categories_for_classes`. This is to

guarantee that Python will build consistent method

resolution orders. For background, see

:mod:`sage.misc.c3_controlled`.

.. SEEALSO:: :meth:`CategoryWithParameters._make_named_class`

EXAMPLES::

sage: PC = Rings()._make_named_class("parent_class", "ParentMethods"); PC

sage: type(PC)

sage: PC.__bases__

(<class 'sage.categories.rngs.Rngs.parent_class'>,

<class 'sage.categories.semirings.Semirings.parent_class'>)

Note that, by default, the result is not cached::

sage: PC is Rings()._make_named_class("parent_class", "ParentMethods")

False

Indeed this method is only meant to construct lazy attributes

like ``parent_class`` which already handle this caching::

sage: Rings().parent_class

Reduction for pickling also assumes the existence of this lazy

attribute::

sage: PC._reduction

(<built-in function getattr>, (Category of rings, 'parent_class'))

sage: loads(dumps(PC)) is Rings().parent_class

True

TESTS::

sage: class A: pass

sage: class BrokenCategory(Category):

....: def super_categories(self): return []

....: ParentMethods = 1

....: class ElementMethods(A):

....: pass

....: class MorphismMethods(object):

....: pass

sage: C = BrokenCategory()

sage: C._make_named_class("parent_class", "ParentMethods")

Traceback (most recent call last):

...

AssertionError: BrokenCategory.ParentMethods should be a class

sage: C._make_named_class("element_class", "ElementMethods")

doctest:...: UserWarning: BrokenCategory.ElementMethods should not have a super class

sage: C._make_named_class("morphism_class", "MorphismMethods")

"""

cls = self.__class__

if isinstance(cls, DynamicMetaclass):

cls = cls.__base__

class_name = "%s.%s"%(cls.__name__, name)

method_provider_cls = getattr(self, method_provider, None)

if method_provider_cls is None:

# If the category provides no XXXMethods class,

# point to the documentation of the category itself

doccls = cls

else:

# Otherwise, check XXXMethods

assert inspect.isclass(method_provider_cls),\

"%s.%s should be a class"%(cls.__name__, method_provider)

mro = inspect.getmro(method_provider_cls)

if len(mro) > 2 or (len(mro) == 2 and mro[1] is not object):

warn("%s.%s should not have a super class"%(cls.__name__, method_provider))

# and point the documentation to it

doccls = method_provider_cls

if picklable:

reduction = (getattr, (self, name))

else:

reduction = None

return dynamic_class(class_name,

tuple(getattr(cat,name) for cat in self._super_categories_for_classes),

method_provider_cls, prepend_cls_bases = False, doccls = doccls,

reduction = reduction, cache = cache)

@lazy_attribute

def subcategory_class(self):

"""

A common superclass for all subcategories of this category (including this one).

This class derives from ``D.subcategory_class`` for each super

category `D` of ``self``, and includes all the methods from

the nested class ``self.SubcategoryMethods``, if it exists.

.. SEEALSO::

- :trac:`12895`

- :meth:`parent_class`

- :meth:`element_class`

- :meth:`_make_named_class`

EXAMPLES::

sage: cls = Rings().subcategory_class; cls

sage: type(cls)

``Rings()`` is an instance of this class, as well as all its subcategories::

sage: isinstance(Rings(), cls)

True

sage: isinstance(AlgebrasWithBasis(QQ), cls)

True

TESTS::

sage: cls = Algebras(QQ).subcategory_class; cls

sage: type(cls)

"""

return self._make_named_class('subcategory_class', 'SubcategoryMethods',

cache=False, picklable=False)

@lazy_attribute

def parent_class(self):

r"""

A common super class for all parents in this category (and its

subcategories).

This class contains the methods defined in the nested class

``self.ParentMethods`` (if it exists), and has as bases the

parent classes of the super categories of ``self``.

.. SEEALSO::

- :meth:`element_class`, :meth:`morphism_class`

- :class:`Category` for details

EXAMPLES::

sage: C = Algebras(QQ).parent_class; C

sage: type(C)

By :trac:`11935`, some categories share their parent

classes. For example, the parent class of an algebra only

depends on the category of the base ring. A typical example is

the category of algebras over a finite field versus algebras

over a non-field::

sage: Algebras(GF(7)).parent_class is Algebras(GF(5)).parent_class

True

sage: Algebras(QQ).parent_class is Algebras(ZZ).parent_class

False

sage: Algebras(ZZ['t']).parent_class is Algebras(ZZ['t','x']).parent_class

True

See :class:`CategoryWithParameters` for an abstract base class for

categories that depend on parameters, even though the parent

and element classes only depend on the parent or element

classes of its super categories. It is used in

:class:`~sage.categories.bimodules.Bimodules`,

:class:`~sage.categories.category_types.Category_over_base` and

:class:`sage.categories.category.JoinCategory`.

"""

return self._make_named_class('parent_class', 'ParentMethods')

@lazy_attribute

def element_class(self):

r"""

A common super class for all elements of parents in this category

(and its subcategories).

This class contains the methods defined in the nested class

``self.ElementMethods`` (if it exists), and has as bases the

element classes of the super categories of ``self``.

.. SEEALSO::

- :meth:`parent_class`, :meth:`morphism_class`

- :class:`Category` for details

EXAMPLES::

sage: C = Algebras(QQ).element_class; C

sage: type(C)

By :trac:`11935`, some categories share their element

classes. For example, the element class of an algebra only

depends on the category of the base. A typical example is the

category of algebras over a field versus algebras over a

non-field::

sage: Algebras(GF(5)).element_class is Algebras(GF(3)).element_class

True

sage: Algebras(QQ).element_class is Algebras(ZZ).element_class

False

sage: Algebras(ZZ['t']).element_class is Algebras(ZZ['t','x']).element_class

True

These classes are constructed with ``__slots__ = []``, so they

behave like extension types::

sage: E = FiniteEnumeratedSets().element_class

sage: from sage.structure.misc import is_extension_type

sage: is_extension_type(E)

True

.. SEEALSO:: :meth:`parent_class`

"""

return self._make_named_class('element_class', 'ElementMethods')

@lazy_attribute

def morphism_class(self):

r"""

A common super class for all morphisms between parents in this

category (and its subcategories).

This class contains the methods defined in the nested class

``self.MorphismMethods`` (if it exists), and has as bases the

morphism classes of the super categories of ``self``.

.. SEEALSO::

- :meth:`parent_class`, :meth:`element_class`

- :class:`Category` for details

EXAMPLES::

sage: C = Algebras(QQ).morphism_class; C

sage: type(C)

"""

return self._make_named_class('morphism_class', 'MorphismMethods')

def required_methods(self):

"""

Returns the methods that are required and optional for parents

in this category and their elements.

EXAMPLES::

sage: Algebras(QQ).required_methods()

{'element': {'optional': ['_add_', '_mul_'], 'required': ['__nonzero__']},

'parent': {'optional': ['algebra_generators'], 'required': ['__contains__']}}

"""

return { "parent" : abstract_methods_of_class(self.parent_class),

"element" : abstract_methods_of_class(self.element_class) }

# Operations on the lattice of categories

def is_subcategory(self, c):

"""

Returns True if self is naturally embedded as a subcategory of c.

EXAMPLES::

sage: AbGrps = CommutativeAdditiveGroups()

sage: Rings().is_subcategory(AbGrps)

True

sage: AbGrps.is_subcategory(Rings())

False

The ``is_subcategory`` function takes into account the

base.

sage: M3 = VectorSpaces(FiniteField(3))

sage: M9 = VectorSpaces(FiniteField(9, 'a'))

sage: M3.is_subcategory(M9)

False

Join categories are properly handled::

sage: CatJ = Category.join((CommutativeAdditiveGroups(), Semigroups()))

sage: Rings().is_subcategory(CatJ)

True

sage: V3 = VectorSpaces(FiniteField(3))

sage: POSet = PartiallyOrderedSets()

sage: PoV3 = Category.join((V3, POSet))

sage: A3 = AlgebrasWithBasis(FiniteField(3))

sage: PoA3 = Category.join((A3, POSet))

sage: PoA3.is_subcategory(PoV3)

True

sage: PoV3.is_subcategory(PoV3)

True

sage: PoV3.is_subcategory(PoA3)

False

"""

if c is self:

return True

subcat_hook = c._subcategory_hook_(self)

if subcat_hook is Unknown:

return c in self._set_of_super_categories

return subcat_hook

def or_subcategory(self, category = None, join = False):

"""

Return ``category`` or ``self`` if ``category`` is ``None``.

INPUT:

- ``category`` -- a sub category of ``self``, tuple/list thereof,

or ``None``

- ``join`` -- a boolean (default: ``False``)

OUTPUT:

- a category

EXAMPLES::

sage: Monoids().or_subcategory(Groups())

Category of groups

sage: Monoids().or_subcategory(None)

Category of monoids

If category is a list/tuple, then a join category is returned::

sage: Monoids().or_subcategory((CommutativeAdditiveMonoids(), Groups()))

Join of Category of groups and Category of commutative additive monoids

If ``join`` is ``False``, an error if raised if category is not a

subcategory of ``self``::

sage: Monoids().or_subcategory(EnumeratedSets())

Traceback (most recent call last):

...

ValueError: Subcategory of `Category of monoids` required; got `Category of enumerated sets`

Otherwise, the two categories are joined together::

sage: Monoids().or_subcategory(EnumeratedSets(), join=True)

Category of enumerated monoids

"""

if category is None:

return self

if isinstance(category, (tuple, list)):

category = Category.join(category)

assert isinstance(category, Category)

if join:

return Category.join([self, category])

else:

if not category.is_subcategory(self):

raise ValueError("Subcategory of `{}` required; got `{}`".format(self, category))

return category

def _is_subclass(self, c):

"""

Same as is_subcategory, but c may also be the class of a

category instead of a category.

EXAMPLES::

sage: Fields()._is_subclass(Rings)

True

sage: Algebras(QQ)._is_subclass(Modules)

True

sage: Algebras(QQ)._is_subclass(ModulesWithBasis)

False

"""

assert( isinstance(c, Category) or (issubclass(c.__class__, type) and issubclass(c, Category)) )

if isinstance(c, Category):

return self.is_subcategory(c)

else:

return any(isinstance(cat, c) for cat in self._all_super_categories)

@cached_method

def _meet_(self, other):

"""

Returns the largest common subcategory of self and other:

EXAMPLES::

sage: Monoids()._meet_(Monoids())

Category of monoids

sage: Rings()._meet_(Rings())

Category of rings

sage: Rings()._meet_(Monoids())

Category of monoids

sage: Monoids()._meet_(Rings())

Category of monoids

sage: VectorSpaces(QQ)._meet_(Modules(ZZ))

Category of commutative additive groups

sage: Algebras(ZZ)._meet_(Algebras(QQ))

Category of rings

sage: Groups()._meet_(Rings())

Category of monoids

sage: Algebras(QQ)._meet_(Category.join([Fields(), ModulesWithBasis(QQ)]))

Join of Category of rings and Category of vector spaces over Rational Field

Note: abstractly, the category poset is a distributive

lattice, so this is well defined; however, the subset of those

categories actually implemented is not: we need to also

include their join-categories.

For example, the category of rings is *not* the join of the

category of abelian groups and that of semi groups, just a

subcategory of their join, since rings further require

distributivity.

For the meet computation, there may be several lowest common

sub categories of self and other, in which case, we need to

take the join of them all.

FIXME:

- If A is a subcategory of B, A has *more* structure than B,

but then *less* objects in there. We should choose an

appropriate convention for A<B. Using subcategory calls

for A<B, but the current meet and join call for A>B.

"""

if self is other: # useful? fast pathway

return self

elif self.is_subcategory(other):

return other

elif other.is_subcategory(self):

# Useful fast pathway; try:

# %time L = EllipticCurve('960d1').prove_BSD()

return self

else:

return Category.join(self._meet_(sup) for sup in other._super_categories)

@staticmethod

def meet(categories):

"""

Returns the meet of a list of categories

INPUT:

- ``categories`` - a non empty list (or iterable) of categories

.. SEEALSO:: :meth:`__or__` for a shortcut

EXAMPLES::

sage: Category.meet([Algebras(ZZ), Algebras(QQ), Groups()])

Category of monoids

That meet of an empty list should be a category which is a

subcategory of all categories, which does not make practical sense::

sage: Category.meet([])

Traceback (most recent call last):

...

ValueError: The meet of an empty list of categories is not implemented

"""

categories = tuple(categories)

if not categories:

raise ValueError("The meet of an empty list of categories is not implemented")

result = categories[0]

for category in categories[1:]:

result = result._meet_(category)

return result

@cached_method

def axioms(self):

"""

Return the axioms known to be satisfied by all the objects of ``self``.

Technically, this is the set of all the axioms ``A`` such that, if

``Cs`` is the category defining ``A``, then ``self`` is a subcategory

of ``Cs().A()``. Any additional axiom ``A`` would yield a strict

subcategory of ``self``, at the very least ``self & Cs().A()`` where

``Cs`` is the category defining ``A``.

EXAMPLES::

sage: Monoids().axioms()

frozenset({'Associative', 'Unital'})

sage: (EnumeratedSets().Infinite() & Sets().Facade()).axioms()

frozenset({'Enumerated', 'Facade', 'Infinite'})

"""

return frozenset(axiom

for category in self._super_categories

for axiom in category.axioms())

@cached_method

def _with_axiom_as_tuple(self, axiom):

"""

Return a tuple of categories whose join is ``self._with_axiom()``.

INPUT:

- ``axiom`` -- a string, the name of an axiom

This is a lazy version of :meth:`_with_axiom` which is used to

avoid recursion loops during join calculations.

.. NOTE:: The order in the result is irrelevant.

EXAMPLES::

sage: Sets()._with_axiom_as_tuple('Finite')

(Category of finite sets,)

sage: Magmas()._with_axiom_as_tuple('Finite')

(Category of magmas, Category of finite sets)

sage: Rings().Division()._with_axiom_as_tuple('Finite')

(Category of division rings,

Category of finite monoids,

Category of commutative magmas,

Category of finite additive groups)

sage: HopfAlgebras(QQ)._with_axiom_as_tuple('FiniteDimensional')

(Category of hopf algebras over Rational Field,

Category of finite dimensional modules over Rational Field)

"""

if axiom in self.axioms():

return (self, )

axiom_attribute = getattr(self.__class__, axiom, None)

if axiom_attribute is None:

# If the axiom is not defined for this category, ignore it

# This uses the following invariant: the categories for

# which a given axiom is defined form a lower set

return (self,)

if axiom in self.__class__.__base__.__dict__:

# self implements this axiom

from .category_with_axiom import CategoryWithAxiom

if inspect.isclass(axiom_attribute) and issubclass(axiom_attribute, CategoryWithAxiom):

return (axiom_attribute(self),)

warn(("Expecting {}.{} to be a subclass of CategoryWithAxiom to"

" implement a category with axiom; got {}; ignoring").format(

self.__class__.__base__.__name__, axiom, axiom_attribute))

# self does not implement this axiom

result = (self, ) + \

tuple(cat

for category in self._super_categories

for cat in category._with_axiom_as_tuple(axiom))

hook = getattr(self, axiom+"_extra_super_categories", None)

if hook is not None:

assert inspect.ismethod(hook)

result += tuple(hook())

return _sort_uniq(result)

@cached_method

def _with_axiom(self, axiom):

"""

Return the subcategory of the objects of ``self`` satisfying

the given ``axiom``.

INPUT:

- ``axiom`` -- a string, the name of an axiom

EXAMPLES::

sage: Sets()._with_axiom("Finite")

Category of finite sets

sage: type(Magmas().Finite().Commutative())

sage: Magmas().Finite().Commutative().super_categories()

[Category of commutative magmas, Category of finite sets]

sage: Algebras(QQ).WithBasis().Commutative() is Algebras(QQ).Commutative().WithBasis()

True

When ``axiom`` is not defined for ``self``, ``self`` is returned::

sage: Sets()._with_axiom("Associative")

Category of sets

.. WARNING:: This may be changed in the future to raising an error.

"""

return Category.join(self._with_axiom_as_tuple(axiom))

def _with_axioms(self, axioms):

"""

Return the subcategory of the objects of ``self`` satisfying

the given ``axioms``.

INPUT:

- ``axioms`` -- a list of strings, the names of the axioms

EXAMPLES::

sage: Sets()._with_axioms(["Finite"])

Category of finite sets

sage: Sets()._with_axioms(["Infinite"])

Category of infinite sets

sage: FiniteSets()._with_axioms(["Finite"])

Category of finite sets

Axioms that are not defined for the ``self`` are ignored::

sage: Sets()._with_axioms(["FooBar"])

Category of sets

sage: Magmas()._with_axioms(["FooBar", "Unital"])

Category of unital magmas

Note that adding several axioms at once can do more than

adding them one by one. This is because the availability of an

axiom may depend on another axiom. For example, for

semigroups, the ``Inverse`` axiom is meaningless unless there

is a unit::

sage: Semigroups().Inverse()

Traceback (most recent call last):

...

AttributeError: 'Semigroups_with_category' object has no attribute 'Inverse'

sage: Semigroups()._with_axioms(["Inverse"])

Category of semigroups

So one needs to first add the ``Unital`` axiom, and then the

``Inverse`` axiom::

sage: Semigroups().Unital().Inverse()

Category of groups

or to specify all of them at once, in any order::

sage: Semigroups()._with_axioms(["Inverse", "Unital"])

Category of groups

sage: Semigroups()._with_axioms(["Unital", "Inverse"])

Category of groups

sage: Magmas()._with_axioms(['Commutative', 'Associative', 'Unital','Inverse'])

Category of commutative groups

sage: Magmas()._with_axioms(['Inverse', 'Commutative', 'Associative', 'Unital'])

Category of commutative groups

"""

# We repeat adding axioms until they have all been

# integrated or nothing happens

axioms = frozenset(axioms)

previous = None

result = self

while result is not previous:

previous = result

for axiom in axioms:

result = result._with_axiom(axiom)

axioms = axioms.difference(result.axioms())

return result

@cached_method

def _without_axiom(self, axiom):

r"""

Return the category with axiom ``axiom`` removed.

OUTPUT:

A category ``C`` which does not have axiom ``axiom``

and such that either ``C`` is ``self``, or adding back all the

axioms of ``self`` gives back ``self``.

.. WARNING:: This is not guaranteed to be robust.

EXAMPLES::

sage: Sets()._without_axiom("Facade")

Category of sets

sage: Sets().Facade()._without_axiom("Facade")

Category of sets

sage: Algebras(QQ)._without_axiom("Unital")

Category of associative algebras over Rational Field

sage: Groups()._without_axiom("Unital") # todo: not implemented

Category of semigroups

"""

if axiom not in self.axioms():

return self

else:

raise ValueError("Cannot remove axiom {} from {}".format(axiom, self))

def _without_axioms(self, named=False):

r"""

Return the category without the axioms that have been added

to create it.

INPUT:

- ``named`` -- a boolean (default: ``False``)

.. TODO:: Improve this explanation.

If ``named`` is ``True``, then this stops at the first

category that has an explicit name of its own. See

:meth:`.category_with_axiom.CategoryWithAxiom._without_axioms`

EXAMPLES::

sage: Sets()._without_axioms()

Category of sets

sage: Semigroups()._without_axioms()

Category of magmas

sage: Algebras(QQ).Commutative().WithBasis()._without_axioms()

Category of magmatic algebras over Rational Field

sage: Algebras(QQ).Commutative().WithBasis()._without_axioms(named=True)

Category of algebras over Rational Field

"""

return self

_flatten_categories = _flatten_categories

@staticmethod

def _sort(categories):

"""

Return the categories after sorting them decreasingly according

to their comparison key.

.. SEEALSO:: :meth:`_cmp_key`

INPUT:

- ``categories`` -- a list (or iterable) of non-join categories

OUTPUT:

A sorted tuple of categories, possibly with repeats.

.. NOTE::

The auxiliary function `_flatten_categories` used in the test

below expects a second argument, which is a type such that

instances of that type will be replaced by its super

categories. Usually, this type is :class:`JoinCategory`.

EXAMPLES::

sage: Category._sort([Sets(), Objects(), Coalgebras(QQ), Monoids(), Sets().Finite()])

(Category of monoids,

Category of coalgebras over Rational Field,

Category of finite sets,

Category of sets,

Category of objects)

sage: Category._sort([Sets().Finite(), Semigroups().Finite(), Sets().Facade(),Magmas().Commutative()])

(Category of finite semigroups,

Category of commutative magmas,

Category of finite sets,

Category of facade sets)

sage: Category._sort(Category._flatten_categories([Sets().Finite(), Algebras(QQ).WithBasis(), Semigroups().Finite(), Sets().Facade(),Algebras(QQ).Commutative(), Algebras(QQ).Graded().WithBasis()], sage.categories.category.JoinCategory))

(Category of algebras with basis over Rational Field,

Category of algebras with basis over Rational Field,

Category of graded algebras over Rational Field,

Category of commutative algebras over Rational Field,

Category of finite semigroups,

Category of finite sets,

Category of facade sets)

"""

return tuple(sorted(categories, key=category_sort_key, reverse=True))

_sort_uniq = _sort_uniq # a cythonised helper

def __and__(self, other):

"""

Return the intersection of two categories.

This is just a shortcut for :meth:`join`.

EXAMPLES::

sage: Sets().Finite() & Rings().Commutative()

Category of finite commutative rings

sage: Monoids() & CommutativeAdditiveMonoids()

Join of Category of monoids and Category of commutative additive monoids

"""

return Category.join([self, other])

def __or__(self, other):

"""

Return the smallest category containing the two categories.

This is just a shortcut for :meth:`meet`.

EXAMPLES::

sage: Algebras(QQ) | Groups()

Category of monoids

"""

return Category.meet([self, other])

_join_cache = _join_cache

@staticmethod

def join(categories, as_list=False, ignore_axioms=(), axioms=()):

"""

Return the join of the input categories in the lattice of categories.

At the level of objects and morphisms, this operation

corresponds to intersection: the objects and morphisms of a

join category are those that belong to all its super

categories.

INPUT:

- ``categories`` -- a list (or iterable) of categories

- ``as_list`` -- a boolean (default: ``False``);

whether the result should be returned as a list

- ``axioms`` -- a tuple of strings; the names of some

supplementary axioms

.. SEEALSO:: :meth:`__and__` for a shortcut

EXAMPLES::

sage: J = Category.join((Groups(), CommutativeAdditiveMonoids())); J

Join of Category of groups and Category of commutative additive monoids

sage: J.super_categories()

[Category of groups, Category of commutative additive monoids]

sage: J.all_super_categories(proper=True)

[Category of groups, ..., Category of magmas,

Category of commutative additive monoids, ..., Category of additive magmas,

Category of sets, ...]

As a short hand, one can use::

sage: Groups() & CommutativeAdditiveMonoids()

Join of Category of groups and Category of commutative additive monoids

This is a commutative and associative operation::

sage: Groups() & Posets()

Join of Category of groups and Category of posets

sage: Posets() & Groups()

Join of Category of groups and Category of posets

sage: Groups() & (CommutativeAdditiveMonoids() & Posets())

Join of Category of groups

and Category of commutative additive monoids

and Category of posets

sage: (Groups() & CommutativeAdditiveMonoids()) & Posets()

Join of Category of groups

and Category of commutative additive monoids

and Category of posets

The join of a single category is the category itself::

sage: Category.join([Monoids()])

Category of monoids

Similarly, the join of several mutually comparable categories is

the smallest one::

sage: Category.join((Sets(), Rings(), Monoids()))

Category of rings

In particular, the unit is the top category :class:`Objects`::

sage: Groups() & Objects()

Category of groups

If the optional parameter ``as_list`` is ``True``, this

returns the super categories of the join as a list, without

constructing the join category itself::

sage: Category.join((Groups(), CommutativeAdditiveMonoids()), as_list=True)

[Category of groups, Category of commutative additive monoids]

sage: Category.join((Sets(), Rings(), Monoids()), as_list=True)

[Category of rings]

sage: Category.join((Modules(ZZ), FiniteFields()), as_list=True)

[Category of finite enumerated fields, Category of modules over Integer Ring]

sage: Category.join([], as_list=True)

[]

sage: Category.join([Groups()], as_list=True)

[Category of groups]

sage: Category.join([Groups() & Posets()], as_list=True)

[Category of groups, Category of posets]

Support for axiom categories (TODO: put here meaningfull examples)::

sage: Sets().Facade() & Sets().Infinite()

Category of facade infinite sets

sage: Magmas().Infinite() & Sets().Facade()

Category of facade infinite magmas

sage: FiniteSets() & Monoids()

Category of finite monoids

sage: Rings().Commutative() & Sets().Finite()

Category of finite commutative rings

Note that several of the above examples are actually join

categories; they are just nicely displayed::

sage: AlgebrasWithBasis(QQ) & FiniteSets().Algebras(QQ)

Join of Category of finite dimensional algebras with basis over Rational Field

and Category of finite set algebras over Rational Field

sage: UniqueFactorizationDomains() & Algebras(QQ)

Join of Category of unique factorization domains

and Category of commutative algebras over Rational Field

TESTS::

sage: Magmas().Unital().Commutative().Finite() is Magmas().Finite().Commutative().Unital()

True

sage: from sage.categories.category_with_axiom import TestObjects

sage: T = TestObjects()

sage: TCF = T.Commutative().Facade(); TCF

Category of facade commutative test objects

sage: TCF is T.Facade().Commutative()

True

sage: TCF is (T.Facade() & T.Commutative())

True

sage: TCF.axioms()

frozenset({'Commutative', 'Facade'})

sage: type(TCF)

sage: TCF = T.Commutative().FiniteDimensional()

sage: TCF is T.FiniteDimensional().Commutative()

True

sage: TCF is T.Commutative() & T.FiniteDimensional()

True

sage: TCF is T.FiniteDimensional() & T.Commutative()

True

sage: type(TCF)

sage: TCU = T.Commutative().Unital()

sage: TCU is T.Unital().Commutative()

True

sage: TCU is T.Commutative() & T.Unital()

True

sage: TCU is T.Unital() & T.Commutative()

True

sage: TUCF = T.Unital().Commutative().FiniteDimensional(); TUCF

Category of finite dimensional commutative unital test objects

sage: type(TUCF)

sage: TFFC = T.Facade().FiniteDimensional().Commutative(); TFFC

Category of facade finite dimensional commutative test objects

sage: type(TFFC)

sage: TFFC.super_categories()

[Category of facade commutative test objects,

Category of finite dimensional commutative test objects]

"""

# Get the list of categories and deal with some trivial cases

categories = list(categories)

if not categories:

if as_list:

return []

else:

# Since Objects() is the top category, it is the neutral element of join

from .objects import Objects

return Objects()

elif len(categories) == 1:

category = categories[0]

if as_list:

if isinstance(category, JoinCategory):

return category.super_categories()

else:

return categories

else:

return category

# Get the cache key, and look into the cache

# Ensure associativity and commutativity by flattening

# TODO:

# - Do we want to store the cache after or before the mangling of the categories?

# - Caching with ignore_axioms?

# JoinCategory's sorting, and removing duplicates

cache_key = _sort_uniq(_flatten_categories(categories, JoinCategory))

if not ignore_axioms:

try:

out = _join_cache[cache_key]

if as_list:

if isinstance(out, JoinCategory):

return out._super_categories

return [out]

return out

except KeyError:

pass

# Handle axioms

result = join_as_tuple(cache_key, axioms, ignore_axioms)

if as_list:

return list(result)

if len(result) == 1:

result = result[0]

else:

result = JoinCategory(result)

if not ignore_axioms:

_join_cache[cache_key] = result

return result

def category(self):

"""

Return the category of this category. So far, all categories

are in the category of objects.

EXAMPLES::

sage: Sets().category()

Category of objects

sage: VectorSpaces(QQ).category()

Category of objects

"""

from .objects import Objects

return Objects()

def example(self, *args, **keywords):

"""

Returns an object in this category. Most of the time, this is a parent.

This serves three purposes:

- Give a typical example to better explain what the category is all about.

(and by the way prove that the category is non empty :-) )

- Provide a minimal template for implementing other objects in this category

- Provide an object on which to test generic code implemented by the category

For all those applications, the implementation of the object

shall be kept to a strict minimum. The object is therefore not

meant to be used for other applications; most of the time a

full featured version is available elsewhere in Sage, and

should be used insted.

Technical note: by default ``FooBar(...).example()`` is

constructed by looking up

``sage.categories.examples.foo_bar.Example`` and calling it as

``Example()``. Extra positional or named parameters are also

passed down. For a category over base ring, the base ring is

further passed down as an optional argument.

Categories are welcome to override this default implementation.

EXAMPLES::

sage: Semigroups().example()

An example of a semigroup: the left zero semigroup

sage: Monoids().Subquotients().example()

NotImplemented

"""

if '.' in self.__class__.__name__:

# this magic should not apply to nested categories like Monoids.Subquotients

return NotImplemented

module_name = self.__module__.replace("sage.categories", "sage.categories.examples")

import sys

try:

__import__(module_name)

module = sys.modules[module_name]

except ImportError:

return NotImplemented

try:

cls = module.Example

except AttributeError:

return NotImplemented

# Add the base ring as optional argument if this is a category over base ring

if "base_ring" not in keywords:

try:

keywords["base_ring"] = self.base_ring()

except AttributeError:

pass

return cls(*args, **keywords)

def is_Category(x):

"""

Returns True if x is a category.

EXAMPLES::

sage: sage.categories.category.is_Category(CommutativeAdditiveSemigroups())

True

sage: sage.categories.category.is_Category(ZZ)

False

"""

return isinstance(x, Category)

@cached_function

def category_sample():

r"""

Return a sample of categories.

It is constructed by looking for all concrete category classes declared in

``sage.categories.all``, calling :meth:`Category.an_instance` on those and

taking all their super categories.

EXAMPLES::

sage: from sage.categories.category import category_sample

sage: sorted(category_sample(), key=str)

[Category of G-sets for Symmetric group of order 8! as a permutation group,

Category of Hecke modules over Rational Field,

Category of Lie algebras over Rational Field,

Category of additive magmas, ...,

Category of fields, ...,

Category of graded hopf algebras with basis over Rational Field, ...,

Category of modular abelian varieties over Rational Field, ...,

Category of simplicial complexes, ...,

Category of vector spaces over Rational Field, ...,

Category of weyl groups, ...

"""

import sage.categories.all

abstract_classes_for_categories = [Category]

return tuple(cls.an_instance()

for cls in sage.categories.all.__dict__.values()

if isinstance(cls, type) and issubclass(cls, Category) and cls not in abstract_classes_for_categories)

def category_graph(categories = None):

"""

Return the graph of the categories in Sage.

INPUT:

- ``categories`` -- a list (or iterable) of categories

If ``categories`` is specified, then the graph contains the

mentioned categories together with all their super

categories. Otherwise the graph contains (an instance of) each

category in :mod:`sage.categories.all` (e.g. ``Algebras(QQ)`` for

algebras).

For readability, the names of the category are shortened.

.. TODO:: Further remove the base ring (see also :trac:`15801`).

EXAMPLES::

sage: G = sage.categories.category.category_graph(categories = [Groups()])

sage: G.vertices()

['groups', 'inverse unital magmas', 'magmas', 'monoids', 'objects',

'semigroups', 'sets', 'sets with partial maps', 'unital magmas']

sage: G.plot()

Graphics object consisting of 20 graphics primitives

sage: sage.categories.category.category_graph().plot()

Graphics object consisting of ... graphics primitives

"""

from sage import graphs

if categories is None:

categories = category_sample()

# Include all the super categories

# Get rid of join categories

categories = set(cat

for category in categories

for cat in category.all_super_categories(proper=isinstance(category, JoinCategory)))

g = graphs.digraph.DiGraph()

for cat in categories:

g.add_vertex(cat._repr_object_names())

for source in categories:

# Don't use super_categories() since it might contain join categories

for target in source._super_categories:

g.add_edge([source._repr_object_names(), target._repr_object_names()])

return g

##############################################################################

# Parametrized categories whose parent/element class depend only on

# the super categories

##############################################################################

class CategoryWithParameters(Category):

"""

A parametrized category whose parent/element classes depend only on

its super categories.

Many categories in Sage are parametrized, like ``C = Algebras(K)``

which takes a base ring as parameter. In many cases, however, the

operations provided by ``C`` in the parent class and element class

depend only on the super categories of ``C``. For example, the

vector space operations are provided if and only if ``K`` is a

field, since ``VectorSpaces(K)`` is a super category of ``C`` only

in that case. In such cases, and as an optimization (see :trac:`11935`),

we want to use the same parent and element class for all fields.

This is the purpose of this abstract class.

Currently, :class:`~sage.categories.category.JoinCategory`,

:class:`~sage.categories.category_types.Category_over_base` and

:class:`~sage.categories.bimodules.Bimodules` inherit from this

class.

EXAMPLES::

sage: C1 = Algebras(GF(5))

sage: C2 = Algebras(GF(3))

sage: C3 = Algebras(ZZ)

sage: from sage.categories.category import CategoryWithParameters

sage: isinstance(C1, CategoryWithParameters)

True

sage: C1.parent_class is C2.parent_class

True

sage: C1.parent_class is C3.parent_class

False

.. automethod:: _make_named_class

"""

def _make_named_class(self, name, method_provider, cache = False, **options):

"""

Return the parent/element/... class of ``self``.

INPUT:

- ``name`` -- a string; the name of the class as an attribute

of ``self``

- ``method_provider`` -- a string; the name of an attribute of

``self`` that provides methods for the new class (in

addition to what comes from the super categories)

- ``**options`` -- other named options to pass down to

:meth:`Category._make_named_class`.

ASSUMPTION:

It is assumed that this method is only called from a lazy

attribute whose name coincides with the given ``name``.

OUTPUT:

A dynamic class that has the corresponding named classes of

the super categories of ``self`` as bases and contains the

methods provided by ``getattr(self, method_provider)``.

.. NOTE::

This method overrides :meth:`Category._make_named_class`

so that the returned class *only* depends on the

corresponding named classes of the super categories and on

the provided methods. This allows for sharing the named

classes across closely related categories providing the

same code to their parents, elements and so on.

EXAMPLES:

The categories of bimodules over the fields ``CC`` or ``RR``

provide the same methods to their parents and elements::

sage: Bimodules(ZZ,RR).parent_class is Bimodules(ZZ,RDF).parent_class #indirect doctest

True

sage: Bimodules(CC,ZZ).element_class is Bimodules(RR,ZZ).element_class

True

On the other hand, modules over a field have more methods than

modules over a ring::

sage: Modules(GF(3)).parent_class is Modules(ZZ).parent_class

False

sage: Modules(GF(3)).element_class is Modules(ZZ).element_class

False

For a more subtle example, one could possibly share the classes for

``GF(3)`` and ``GF(2^3, 'x')``, but this is not currently the case::

sage: Modules(GF(3)).parent_class is Modules(GF(2^3,'x')).parent_class

False

This is because those two fields do not have the exact same category::

sage: GF(3).category()

Join of Category of finite enumerated fields

and Category of subquotients of monoids

and Category of quotients of semigroups

sage: GF(2^3,'x').category()

Category of finite enumerated fields

Similarly for ``QQ`` and ``RR``::

sage: QQ.category()

Join of Category of number fields

and Category of quotient fields

and Category of metric spaces

sage: RR.category()

Join of Category of fields and Category of complete metric spaces

sage: Modules(QQ).parent_class is Modules(RR).parent_class

False

Some other cases where one could potentially share those classes::

sage: Modules(GF(3),dispatch=False).parent_class is Modules(ZZ).parent_class

False

sage: Modules(GF(3),dispatch=False).element_class is Modules(ZZ).element_class

False

TESTS::

sage: PC = Algebras(QQ).parent_class; PC # indirect doctest

sage: type(PC)

sage: PC.__bases__

(<class 'sage.categories.rings.Rings.parent_class'>,

<class 'sage.categories.associative_algebras.AssociativeAlgebras.parent_class'>,

<class 'sage.categories.unital_algebras.UnitalAlgebras.parent_class'>)

sage: loads(dumps(PC)) is PC

True

"""

cls = self.__class__

if isinstance(cls, DynamicMetaclass):

cls = cls.__base__

key = (cls, name, self._make_named_class_key(name))

try:

return self._make_named_class_cache[key]

except KeyError:

pass

result = Category._make_named_class(self, name, method_provider,

cache=cache, **options)

self._make_named_class_cache[key] = result

return result

@abstract_method

def _make_named_class_key(self, name):

r"""

Return what the element/parent/... class depend on.

INPUT:

- ``name`` -- a string; the name of the class as an attribute

of ``self``

.. SEEALSO::

- :meth:`_make_named_class`

- :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`

EXAMPLES:

The parent class of an algebra depends only on the category of the base ring::

sage: Algebras(ZZ)._make_named_class_key("parent_class")

Join of Category of euclidean domains

and Category of infinite enumerated sets

and Category of metric spaces

The morphism class of a bimodule depends only on the category

of the left and right base rings::

sage: Bimodules(QQ, ZZ)._make_named_class_key("morphism_class")

(Join of Category of number fields

and Category of quotient fields

and Category of metric spaces,

Join of Category of euclidean domains

and Category of infinite enumerated sets

and Category of metric spaces)

The element class of a join category depends only on the

element class of its super categories::

sage: Category.join([Groups(), Posets()])._make_named_class_key("element_class")

(<class 'sage.categories.groups.Groups.element_class'>,

<class 'sage.categories.posets.Posets.element_class'>)

"""

_make_named_class_cache = dict()

_cmp_key = _cmp_key_named

def _subcategory_hook_(self, C):

"""

A quick but partial test whether ``C`` is a subcategory of ``self``.

INPUT:

- ``C`` -- a category

OUTPUT:

``False``, if ``C.parent_class`` is not a subclass of

``self.parent_class``, and :obj:`~sage.misc.unknown.Unknown`

otherwise.

EXAMPLES::

sage: Bimodules(QQ,QQ)._subcategory_hook_(Modules(QQ))

Unknown

sage: Bimodules(QQ,QQ)._subcategory_hook_(Rings())

False

"""

if not issubclass(C.parent_class, self.parent_class):

return False

return Unknown

#############################################################

# Join of several categories

#############################################################

class JoinCategory(CategoryWithParameters):

"""

A class for joins of several categories. Do not use directly;

see Category.join instead.

EXAMPLES::

sage: from sage.categories.category import JoinCategory

sage: J = JoinCategory((Groups(), CommutativeAdditiveMonoids())); J

Join of Category of groups and Category of commutative additive monoids

sage: J.super_categories()

[Category of groups, Category of commutative additive monoids]

sage: J.all_super_categories(proper=True)

[Category of groups, ..., Category of magmas,

Category of commutative additive monoids, ..., Category of additive magmas,

Category of sets, Category of sets with partial maps, Category of objects]

By :trac:`11935`, join categories and categories over base rings

inherit from :class:`CategoryWithParameters`. This allows for

sharing parent and element classes between similar categories. For

example, since group algebras belong to a join category and since

the underlying implementation is the same for all finite fields,

we have::

sage: G = SymmetricGroup(10)

sage: A3 = G.algebra(GF(3))

sage: A5 = G.algebra(GF(5))

sage: type(A3.category())

sage: type(A3) is type(A5)

True

.. automethod:: _repr_object_names

.. automethod:: _repr_

.. automethod:: _without_axioms

"""

def __init__(self, super_categories, **kwds):

"""

Initializes this JoinCategory

INPUT:

- super_categories -- Categories to join. This category will

consist of objects and morphisms that lie in all of these

categories.

- name -- An optional name for this category.

TESTS::

sage: from sage.categories.category import JoinCategory

sage: C = JoinCategory((Groups(), CommutativeAdditiveMonoids())); C

Join of Category of groups and Category of commutative additive monoids

sage: TestSuite(C).run()

"""

assert(len(super_categories) >= 2)

assert(all(not isinstance(category, JoinCategory) for category in super_categories))

# Use __super_categories to not overwrite the lazy attribute Category._super_categories

# Maybe this would not be needed if the flattening/sorting is does consistently?

self.__super_categories = list(super_categories)

if 'name' in kwds:

Category.__init__(self, kwds['name'])

else:

Category.__init__(self)

def _make_named_class_key(self, name):

r"""

Return what the element/parent/... classes depend on.

Since :trac:`11935`, the element/parent classes of a join

category over base only depend on the element/parent class of

its super categories.

.. SEEALSO::

- :meth:`CategoryWithParameters`

- :meth:`CategoryWithParameters._make_named_class_key`

EXAMPLES::

sage: Modules(ZZ)._make_named_class_key('element_class')

Join of Category of euclidean domains

and Category of infinite enumerated sets

and Category of metric spaces

sage: Modules(QQ)._make_named_class_key('parent_class')

Join of Category of number fields

and Category of quotient fields

and Category of metric spaces

sage: Schemes(Spec(ZZ))._make_named_class_key('parent_class')

Category of schemes

sage: ModularAbelianVarieties(QQ)._make_named_class_key('parent_class')

Join of Category of number fields

and Category of quotient fields

and Category of metric spaces

"""

return tuple(getattr(cat, name) for cat in self._super_categories)

def super_categories(self):

"""

Returns the immediate super categories, as per :meth:`Category.super_categories`.

EXAMPLES::

sage: from sage.categories.category import JoinCategory

sage: JoinCategory((Semigroups(), FiniteEnumeratedSets())).super_categories()

[Category of semigroups, Category of finite enumerated sets]

"""

return self.__super_categories

def additional_structure(self):

r"""

Return ``None``.

Indeed, a join category defines no additional structure.

.. SEEALSO:: :meth:`Category.additional_structure`

EXAMPLES::

sage: Modules(ZZ).additional_structure()

"""

return None

def _subcategory_hook_(self, category):

"""

Returns whether ``category`` is a subcategory of this join category

INPUT:

- ``category`` -- a category.

.. note::

``category`` is a sub-category of this join category if

and only if it is a sub-category of all super categories

of this join category.

EXAMPLES::

sage: base_cat = Category.join([NumberFields(), QuotientFields().Metric()])

sage: cat = Category.join([Rings(), VectorSpaces(base_cat)])

sage: QQ['x'].category().is_subcategory(cat) # indirect doctest

True

"""

return all(category.is_subcategory(X) for X in self._super_categories)

def is_subcategory(self, C):

"""

Check whether this join category is subcategory of another

category ``C``.

EXAMPLES::

sage: Category.join([Rings(),Modules(QQ)]).is_subcategory(Category.join([Rngs(),Bimodules(QQ,QQ)]))

True

"""

if C is self:

return True

hook = C._subcategory_hook_(self)

if hook is Unknown:

return any(X.is_subcategory(C) for X in self._super_categories)

return hook

def _with_axiom(self, axiom):

"""

Return the category obtained by adding an axiom to ``self``.

.. NOTE::

This is just an optimization of

:meth:`Category._with_axiom`; it's not necessarily

actually useful.

EXAMPLES::

sage: C = Category.join([Monoids(), Posets()])

sage: C._with_axioms(["Finite"])

Join of Category of finite monoids and Category of finite posets

TESTS:

Check that axiom categories for a join are reconstructed from

the base categories::

sage: C = Category.join([Monoids(), Magmas().Commutative()])

sage: C._with_axioms(["Finite"])

Category of finite commutative monoids

This helps guaranteeing commutativity of taking axioms::

sage: Monoids().Finite().Commutative() is Monoids().Commutative().Finite()

True

"""

return Category.join([cat._with_axiom(axiom) for cat in self._super_categories])

@cached_method

def _without_axiom(self, axiom):

"""

Return this category with axiom ``axiom`` removed.

OUTPUT:

A category ``C`` which does not have axiom ``axiom`` and such

that either ``C`` is ``self``, or adding back all the

axioms of ``self`` gives back ``self``.

.. SEEALSO:: :meth:`Category._without_axiom`

.. WARNING:: This is not guaranteed to be robust.

EXAMPLES::

sage: C = Posets() & FiniteEnumeratedSets() & Sets().Facade(); C

Category of facade finite enumerated posets

sage: C._without_axiom("Facade")

Category of finite enumerated posets

sage: C = Sets().Finite().Facade()

sage: type(C)

sage: C._without_axiom("Facade")

Category of finite sets

"""

result = Category.join(C._without_axiom(axiom) for C in self.super_categories())

assert axiom not in result.axioms()

assert result._with_axioms(self.axioms()) is self

return result

def _without_axioms(self, named=False):

"""

When adjoining axioms to a category, one often gets a join

category; this method tries to recover the original

category from this join category.

INPUT:

- ``named`` -- a boolean (default: ``False``)

See :meth:`Category._without_axioms` for the description

of the ``named`` parameter.

EXAMPLES::

sage: C = Category.join([Monoids(), Posets()]).Finite()

sage: C._repr_(as_join=True)

'Join of Category of finite monoids and Category of finite posets'

sage: C._without_axioms()

Traceback (most recent call last):

...

ValueError: This join category isn't built by adding axioms to a single category

sage: C = Monoids().Infinite()

sage: C._repr_(as_join=True)

'Join of Category of monoids and Category of infinite sets'

sage: C._without_axioms()

Category of magmas

sage: C._without_axioms(named=True)

Category of monoids

TESTS:

``C`` is in fact a join category::

sage: from sage.categories.category import JoinCategory

sage: isinstance(C, JoinCategory)

True

"""

axioms = self.axioms()

for category in self._super_categories:

if category._with_axioms(axioms) is self:

return category._without_axioms(named=named)

raise ValueError("This join category isn't built by adding axioms"

" to a single category")

def _cmp_key(self):

"""

Return a comparison key for ``self``.

See :meth:`Category._cmp_key` for the specifications.

EXAMPLES:

This raises an error since ``_cmp_key`` should not be called

on join categories::

sage: (Magmas() & CommutativeAdditiveSemigroups())._cmp_key()

Traceback (most recent call last):

...

ValueError: _cmp_key should not be called on join categories

"""

raise ValueError("_cmp_key should not be called on join categories")

def _repr_object_names(self):

"""

Return the name of the objects of this category.

.. SEEALSO:: :meth:`Category._repr_object_names`, :meth:`_repr_`, :meth:`._without_axioms`

EXAMPLES::

sage: Groups().Finite().Commutative()._repr_(as_join=True)

'Join of Category of finite groups and Category of commutative groups'

sage: Groups().Finite().Commutative()._repr_object_names()

'finite commutative groups'

This uses :meth:`._without_axioms` which may fail if this

category is not obtained by adjoining axioms to some super

categories::

sage: Category.join((Groups(), CommutativeAdditiveMonoids()))._repr_object_names()

Traceback (most recent call last):

...

ValueError: This join category isn't built by adding axioms to a single category

"""

from sage.categories.category_with_axiom import CategoryWithAxiom

return CategoryWithAxiom._repr_object_names_static(self._without_axioms(named=True), self.axioms())

def _repr_(self, as_join = False):

"""

Print representation.

INPUT:

- ``as_join`` -- a boolean (default: False)

EXAMPLES::

sage: Category.join((Groups(), CommutativeAdditiveMonoids())) #indirect doctest

Join of Category of groups and Category of commutative additive monoids

By default, when a join category is built from category by

adjoining axioms, a nice name is printed out::

sage: Groups().Facade().Finite()

Category of facade finite groups

But this is in fact really a join category::

sage: Groups().Facade().Finite()._repr_(as_join = True)

'Join of Category of finite groups and Category of facade sets'

The rationale is to make it more readable, and hide the

technical details of how this category is constructed

internally, especially since this construction is likely to

change over time when new axiom categories are implemented.

This join category may possibly be obtained by adding axioms

to different categories; so the result is not guaranteed to be

unique; when this is not the case the first found is used.

.. SEEALSO:: :meth:`Category._repr_`, :meth:`_repr_object_names`

TESTS::

sage: Category.join((Sets().Facade(), Groups()))

Category of facade groups

"""

if not as_join:

try:

return super(JoinCategory, self)._repr_()

except ValueError:

pass

return "Join of " + " and ".join(str(cat) for cat in self._super_categories)

Coverage for local/lib/python2.7/site-packages/sage/categories/category.py : 73%

378 statements 276 run 102 missing 0 excluded