Categories

AUTHORS:

• David Kohel, William Stein and Nicolas M. Thiery

Every Sage object lies in a category. Categories in Sage are modeled on the mathematical idea of category, and are distinct from Python classes, which are a programming construct.

In most cases, typing x.category() returns the category to which x belongs. If C is a category and x is any object, C(x) tries to make an object in C from x. Checking if x belongs to C is done as usually by x in C.

See Category and sage.categories.primer for more details.

EXAMPLES:

We create a couple of categories:

sage: Sets()
Category of sets
sage: GSets(AbelianGroup([2, 4, 9]))                                                # needs sage.groups
Category of G-sets for Multiplicative Abelian group isomorphic to C2 x C4 x C9
sage: Semigroups()
Category of semigroups
sage: VectorSpaces(FiniteField(11))
Category of vector spaces over Finite Field of size 11
sage: Ideals(IntegerRing())
Category of ring ideals in Integer Ring

Let’s request the category of some objects:

sage: V = VectorSpace(RationalField(), 3)                                           # needs sage.modules
sage: V.category()                                                                  # needs sage.modules
Category of finite dimensional vector spaces with basis
 over (number fields and quotient fields and metric spaces)

sage: G = SymmetricGroup(9)                                                         # needs sage.groups
sage: G.category()                                                                  # needs sage.groups
Join of
 Category of finite enumerated permutation groups and
 Category of finite Weyl groups and
 Category of well generated finite irreducible complex reflection groups

sage: P = PerfectMatchings(3)                                                       # needs sage.combinat
sage: P.category()                                                                  # needs sage.combinat
Category of finite enumerated sets

Let’s check some memberships:

sage: V in VectorSpaces(QQ)                                                         # needs sage.modules
True
sage: V in VectorSpaces(FiniteField(11))                                            # needs sage.modules
False
sage: G in Monoids()                                                                # needs sage.groups
True
sage: P in Rings()                                                                  # needs sage.combinat
False

For parametrized categories one can use the following shorthand:

sage: V in VectorSpaces                                                             # needs sage.modules
True
sage: G in VectorSpaces                                                             # needs sage.groups
False

A parent P is in a category C if P.category() is a subcategory of C.

Note

Any object of a category should be an instance of CategoryObject.

For backward compatibility this is not yet enforced:

sage: class A:
....:   def category(self):
....:       return Fields()
sage: A() in Rings()
True

By default, the category of an element \(x\) of a parent \(P\) is the category of all objects of \(P\) (this is dubious and may be deprecated):

sage: V = VectorSpace(RationalField(), 3)                                       # needs sage.modules
sage: v = V.gen(1)                                                              # needs sage.modules
sage: v.category()                                                              # needs sage.modules
Category of elements of Vector space of dimension 3 over Rational Field

class sage.categories.category.Category

Bases: UniqueRepresentation, SageObject

The base class for modeling mathematical categories, like for example:

• Groups(): the category of groups

• EuclideanDomains(): the category of euclidean rings

• VectorSpaces(QQ): the category of vector spaces over the field of rationals

See sage.categories.primer for an introduction to categories in Sage, their relevance, purpose, and usage. The documentation below will focus on their implementation.

Technically, a category is an instance of the class Category or some of its subclasses. Some categories, like VectorSpaces, are parametrized: VectorSpaces(QQ) is one of many instances of the class VectorSpaces. On the other hand, EuclideanDomains() is the single instance of the class EuclideanDomains.

Recall that an algebraic structure (say, the ring \(\QQ[x]\)) is modelled in Sage by an object which is called a parent. This object belongs to certain categories (here EuclideanDomains() and Algebras()). The elements of the ring are themselves objects.

The class of a category (say EuclideanDomains) can define simultaneously:

• Operations on the category itself (what is its super categories? its category of morphisms? its dual category?).

• Generic operations on parents in this category, like the ring \(\QQ[x]\).

• Generic operations on elements of such parents (e. g., the Euclidean algorithm for computing gcds).

• Generic operations on morphisms of this category.

This is achieved as follows:

sage: from sage.categories.category import Category
sage: class EuclideanDomains(Category):
....:     # operations on the category itself
....:     def super_categories(self):
....:         [Rings()]
....:
....:     def dummy(self): # TODO: find some good examples
....:          pass
....:
....:     class ParentMethods: # holds the generic operations on parents
....:          # TODO: find a good example of an operation
....:          pass
....:
....:     class ElementMethods:# holds the generic operations on elements
....:          def gcd(x,y):
....:              # Euclid algorithms
....:              pass
....:
....:     class MorphismMethods: # holds the generic operations on morphisms
....:          # TODO: find a good example of an operation
....:          pass
....:

Note that the nested class ParentMethods is merely a container of operations, and does not inherit from anything. Instead, the hierarchy relation is defined once at the level of the categories, and the actual hierarchy of classes is built in parallel from all the ParentMethods nested classes, and stored in the attributes parent_class. Then, a parent in a category C receives the appropriate operations from all the super categories by usual class inheritance from C.parent_class.

Similarly, two other hierarchies of classes, for elements and morphisms respectively, are built from all the ElementMethods and MorphismMethods nested classes.

EXAMPLES:

We define a hierarchy of four categories As(), Bs(), Cs(), Ds() with a diamond inheritance. Think for example:

• As(): the category of sets

• Bs(): the category of additive groups

• Cs(): the category of multiplicative monoids

• Ds(): the category of rings

sage: from sage.categories.category import Category
sage: from sage.misc.lazy_attribute import lazy_attribute
sage: class As (Category):
....:     def super_categories(self):
....:         return []
....:
....:     class ParentMethods:
....:         def fA(self):
....:             return "A"
....:         f = fA

sage: class Bs (Category):
....:     def super_categories(self):
....:         return [As()]
....:
....:     class ParentMethods:
....:         def fB(self):
....:             return "B"

sage: class Cs (Category):
....:     def super_categories(self):
....:         return [As()]
....:
....:     class ParentMethods:
....:         def fC(self):
....:             return "C"
....:         f = fC

sage: class Ds (Category):
....:     def super_categories(self):
....:         return [Bs(),Cs()]
....:
....:     class ParentMethods:
....:         def fD(self):
....:             return "D"

Categories should always have unique representation; by github issue #12215, this means that it will be kept in cache, but only if there is still some strong reference to it.

We check this before proceeding:

sage: import gc
sage: idAs = id(As())
sage: _ = gc.collect()
sage: n == id(As())
False
sage: a = As()
sage: id(As()) == id(As())
True
sage: As().parent_class == As().parent_class
True

We construct a parent in the category Ds() (that, is an instance of Ds().parent_class), and check that it has access to all the methods provided by all the categories, with the appropriate inheritance order:

sage: D = Ds().parent_class()
sage: [ D.fA(), D.fB(), D.fC(), D.fD() ]
['A', 'B', 'C', 'D']
sage: D.f()
'C'

sage: C = Cs().parent_class()
sage: [ C.fA(), C.fC() ]
['A', 'C']
sage: C.f()
'C'

Here is the parallel hierarchy of classes which has been built automatically, together with the method resolution order (.mro()):

sage: As().parent_class
<class '__main__.As.parent_class'>
sage: As().parent_class.__bases__
(<... 'object'>,)
sage: As().parent_class.mro()
[<class '__main__.As.parent_class'>, <... 'object'>]

sage: Bs().parent_class
<class '__main__.Bs.parent_class'>
sage: Bs().parent_class.__bases__
(<class '__main__.As.parent_class'>,)
sage: Bs().parent_class.mro()
[<class '__main__.Bs.parent_class'>, <class '__main__.As.parent_class'>, <... 'object'>]

sage: Cs().parent_class
<class '__main__.Cs.parent_class'>
sage: Cs().parent_class.__bases__
(<class '__main__.As.parent_class'>,)
sage: Cs().parent_class.__mro__
(<class '__main__.Cs.parent_class'>, <class '__main__.As.parent_class'>, <... 'object'>)

sage: Ds().parent_class
<class '__main__.Ds.parent_class'>
sage: Ds().parent_class.__bases__
(<class '__main__.Cs.parent_class'>, <class '__main__.Bs.parent_class'>)
sage: Ds().parent_class.mro()
[<class '__main__.Ds.parent_class'>, <class '__main__.Cs.parent_class'>,
 <class '__main__.Bs.parent_class'>, <class '__main__.As.parent_class'>, <... 'object'>]

Note 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__
<class '__main__.myparent_with_category'>
sage: D.__class__.__bases__
(<class '__main__.myparent'>, <class '__main__.Ds.parent_class'>)
sage: D.__class__.mro()
[<class '__main__.myparent_with_category'>,
<class '__main__.myparent'>,
<class 'sage.structure.parent.Parent'>,
<class 'sage.structure.category_object.CategoryObject'>,
<class '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
<class '__main__.myparent_with_category.element_class'>
sage: D.element_class.mro()
[<class '__main__.myparent_with_category.element_class'>,
<class ...__main__....Element...>,
<class '__main__.Ds.element_class'>,
<class '__main__.Cs.element_class'>,
<class '__main__.Bs.element_class'>,
<class '__main__.As.element_class'>,
<... 'object'>]

_super_categories()

The immediate super categories of this category.

This lazy attribute caches the result of the mandatory method 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 super_categories().

Note

This attribute is likely to eventually become a tuple. When this happens, we might as well use Category._sort(), if not Category._sort_uniq().

EXAMPLES:

sage: Rings()._super_categories
[Category of rngs, Category of semirings]

_super_categories_for_classes()

The super categories of this category used for building classes.

This is a close variant of _super_categories() used for constructing the list of the bases for parent_class(), element_class(), and friends. The purpose is ensure that Python will find a proper Method Resolution Order for those classes. For background, see sage.misc.c3_controlled.

See also

_cmp_key().

Note

This attribute is calculated as a by-product of computing _all_super_categories().

EXAMPLES:

sage: Rings()._super_categories_for_classes
[Category of rngs, Category of semirings]

_all_super_categories()

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

Since github issue #11943, the order of super categories is determined by Python’s method resolution order C3 algorithm.

See also

all_super_categories()

Note

this attribute is likely to eventually become a tuple.

Note

this sets _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]

_all_super_categories_proper()

All the proper super categories of this category.

Since github issue #11943, the order of super categories is determined by Python’s method resolution order C3 algorithm.

See also

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]

_set_of_super_categories()

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

Note

this is used for speeding up category containment tests.

See also

all_super_categories()

EXAMPLES:

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]
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]

_make_named_class(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).

• 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 _super_categories_for_classes(). This is to guarantee that Python will build consistent method resolution orders. For background, see sage.misc.c3_controlled.

See also

CategoryWithParameters._make_named_class()

EXAMPLES:

sage: PC = Rings()._make_named_class("parent_class", "ParentMethods"); PC
<class 'sage.categories.rings.Rings.parent_class'>
sage: type(PC)
<class 'sage.structure.dynamic_class.DynamicMetaclass'>
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
<class 'sage.categories.rings.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

_repr_()

Return the print representation of this category.

EXAMPLES:

sage: Sets() # indirect doctest
Category of sets

_repr_object_names()

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'

_test_category(**options)

Run generic tests on this category

See also

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()

_with_axiom(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())
<class 'sage.categories.category.JoinCategory_with_category'>
sage: Magmas().Finite().Commutative().super_categories()
[Category of commutative magmas, Category of finite sets]
sage: C = Algebras(QQ).WithBasis().Commutative()
sage: C 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.

_with_axiom_as_tuple(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 _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 vector spaces over Rational Field)

_without_axioms(named=False)

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 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

static _sort(categories)

Return the categories after sorting them decreasingly according to their comparison key.

See also

_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 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)

static _sort_uniq(categories)

Return the categories after sorting them and removing redundant categories.

Redundant categories include duplicates and categories which are super categories of other categories in the input.

INPUT:

• categories – a list (or iterable) of categories

OUTPUT: a sorted tuple of mutually incomparable categories

EXAMPLES:

sage: Category._sort_uniq([Rings(), Monoids(), Coalgebras(QQ)])
(Category of rings, Category of coalgebras over Rational Field)

Note that, in the above example, Monoids() does not appear in the result because it is a super category of Rings().

static __classcall__(*args, **options)

Input mangling for unique representation.

Let C = Cs(...) be a category. Since github issue #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.

See also

subcategory_class()

EXAMPLES:

sage: A = Algebras(QQ)
sage: A.__class__
<class 'sage.categories.algebras.Algebras_with_category'>
sage: A is Algebras(QQ)
True
sage: A is A.__class__(QQ)
True

__init__()

Initialize 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__
<class '__main__.SemiprimitiveRings_with_category'>

Note

If the default name of the category (built from the name of the class) is not adequate, please implement _repr_object_names() to customize it.

Realizations()

Return the category of realizations of the parent self or of objects of the category self

INPUT:

• self – a parent or a concrete category

Note

this function is actually inserted as a method in the class Category (see Realizations()). It is defined here for code locality reasons.

EXAMPLES:

The category of realizations of some algebra:

sage: Algebras(QQ).Realizations()
Join of Category of algebras over Rational Field
    and Category of realizations of unital magmas

The category of realizations of a given algebra:

sage: A = Sets().WithRealizations().example(); A                                # needs sage.modules
The subset algebra of {1, 2, 3} over Rational Field
sage: A.Realizations()                                                          # needs sage.modules
Category of realizations of
 The subset algebra of {1, 2, 3} over Rational Field

sage: C = GradedHopfAlgebrasWithBasis(QQ).Realizations(); C
Join of Category of graded Hopf algebras with basis over Rational Field
    and Category of realizations of Hopf algebras over Rational Field
sage: C.super_categories()
[Category of graded Hopf algebras with basis over Rational Field,
 Category of realizations of Hopf algebras over Rational Field]

sage: TestSuite(C).run()

See also

• Sets().WithRealizations

• ClasscallMetaclass

Todo

Add an optional argument to allow for:

sage: Realizations(A, category=Blahs()) # todo: not implemented

WithRealizations()

Return the category of parents in self endowed with multiple realizations.

INPUT:

• self – a category

See also

• The documentation and code (sage.categories.examples.with_realizations) of Sets().WithRealizations().example() for more on how to use and implement a parent with several realizations.

• Various use cases:

  • SymmetricFunctions

  • QuasiSymmetricFunctions

  • NonCommutativeSymmetricFunctions

  • SymmetricFunctionsNonCommutingVariables

  • DescentAlgebra

  • algebras.Moebius

  • IwahoriHeckeAlgebra

  • ExtendedAffineWeylGroup

• The Implementing Algebraic Structures thematic tutorial.

• sage.categories.realizations

Note

this function is actually inserted as a method in the class Category (see WithRealizations()). It is defined here for code locality reasons.

EXAMPLES:

sage: Sets().WithRealizations()
Category of sets with realizations

Parent with realizations

Let us now explain the concept of realizations. A parent with realizations is a facade parent (see Sets.Facade) admitting multiple concrete realizations where its elements are represented. Consider for example an algebra \(A\) which admits several natural bases:

sage: A = Sets().WithRealizations().example(); A                                # needs sage.modules
The subset algebra of {1, 2, 3} over Rational Field

For each such basis \(B\) one implements a parent \(P_B\) which realizes \(A\) with its elements represented by expanding them on the basis \(B\):

sage: # needs sage.modules
sage: A.F()
The subset algebra of {1, 2, 3} over Rational Field in the Fundamental basis
sage: A.Out()
The subset algebra of {1, 2, 3} over Rational Field in the Out basis
sage: A.In()
The subset algebra of {1, 2, 3} over Rational Field in the In basis
sage: A.an_element()
F[{}] + 2*F[{1}] + 3*F[{2}] + F[{1, 2}]

If \(B\) and \(B'\) are two bases, then the change of basis from \(B\) to \(B'\) is implemented by a canonical coercion between \(P_B\) and \(P_{B'}\):

sage: # needs sage.combinat sage.modules
sage: F = A.F(); In = A.In(); Out = A.Out()
sage: i = In.an_element(); i
In[{}] + 2*In[{1}] + 3*In[{2}] + In[{1, 2}]
sage: F(i)
7*F[{}] + 3*F[{1}] + 4*F[{2}] + F[{1, 2}]
sage: F.coerce_map_from(Out)
Generic morphism:
  From: The subset algebra of {1, 2, 3} over Rational Field in the Out basis
  To:   The subset algebra of {1, 2, 3} over Rational Field in the Fundamental basis

allowing for mixed arithmetic:

sage: (1 + Out.from_set(1)) * In.from_set(2,3)                                  # needs sage.combinat sage.modules
Out[{}] + 2*Out[{1}] + 2*Out[{2}] + 2*Out[{3}] + 2*Out[{1, 2}]
+ 2*Out[{1, 3}] + 4*Out[{2, 3}] + 4*Out[{1, 2, 3}]

In our example, there are three realizations:

sage: A.realizations()                                                          # needs sage.modules
[The subset algebra of {1, 2, 3} over Rational Field in the Fundamental basis,
 The subset algebra of {1, 2, 3} over Rational Field in the In basis,
 The subset algebra of {1, 2, 3} over Rational Field in the Out basis]

Instead of manually defining the shorthands F, In, and Out, as above one can just do:

sage: A.inject_shorthands()                                                     # needs sage.combinat sage.modules
Defining F as shorthand for
 The subset algebra of {1, 2, 3} over Rational Field in the Fundamental basis
Defining In as shorthand for
 The subset algebra of {1, 2, 3} over Rational Field in the In basis
Defining Out as shorthand for
 The subset algebra of {1, 2, 3} over Rational Field in the Out basis

Rationale

Besides some goodies described below, the role of \(A\) is threefold:

• To provide, as illustrated above, a single entry point for the algebra as a whole: documentation, access to its properties and different realizations, etc.

• To provide a natural location for the initialization of the bases and the coercions between, and other methods that are common to all bases.

• To let other objects refer to \(A\) while allowing elements to be represented in any of the realizations.

We now illustrate this second point by defining the polynomial ring with coefficients in \(A\):

sage: P = A['x']; P                                                             # needs sage.modules
Univariate Polynomial Ring in x over
 The subset algebra of {1, 2, 3} over Rational Field
sage: x = P.gen()                                                               # needs sage.modules

In the following examples, the coefficients turn out to be all represented in the \(F\) basis:

sage: P.one()                                                                   # needs sage.modules
F[{}]
sage: (P.an_element() + 1)^2                                                    # needs sage.modules
F[{}]*x^2 + 2*F[{}]*x + F[{}]

However we can create a polynomial with mixed coefficients, and compute with it:

sage: p = P([1, In[{1}], Out[{2}] ]); p                                         # needs sage.combinat sage.modules
Out[{2}]*x^2 + In[{1}]*x + F[{}]
sage: p^2                                                                       # needs sage.combinat sage.modules
Out[{2}]*x^4
+ (-8*In[{}] + 4*In[{1}] + 8*In[{2}] + 4*In[{3}]
   - 4*In[{1, 2}] - 2*In[{1, 3}] - 4*In[{2, 3}] + 2*In[{1, 2, 3}])*x^3
+ (F[{}] + 3*F[{1}] + 2*F[{2}] - 2*F[{1, 2}] - 2*F[{2, 3}] + 2*F[{1, 2, 3}])*x^2
+ (2*F[{}] + 2*F[{1}])*x
+ F[{}]

Note how each coefficient involves a single basis which need not be that of the other coefficients. Which basis is used depends on how coercion happened during mixed arithmetic and needs not be deterministic.

One can easily coerce all coefficient to a given basis with:

sage: p.map_coefficients(In)                                                    # needs sage.combinat sage.modules
(-4*In[{}] + 2*In[{1}] + 4*In[{2}] + 2*In[{3}]
 - 2*In[{1, 2}] - In[{1, 3}] - 2*In[{2, 3}] + In[{1, 2, 3}])*x^2
+ In[{1}]*x + In[{}]

Alas, the natural notation for constructing such polynomials does not yet work:

sage: In[{1}] * x                                                               # needs sage.combinat sage.modules
Traceback (most recent call last):
...
TypeError: unsupported operand parent(s) for *:
'The subset algebra of {1, 2, 3} over Rational Field in the In basis'
and 'Univariate Polynomial Ring in x over
The subset algebra of {1, 2, 3} over Rational Field'

The category of realizations of \(A\)

The set of all realizations of \(A\), together with the coercion morphisms is a category (whose class inherits from Category_realization_of_parent):

sage: A.Realizations()                                                          # needs sage.modules
Category of realizations of
 The subset algebra of {1, 2, 3} over Rational Field

The various parent realizing \(A\) belong to this category:

sage: A.F() in A.Realizations()                                                 # needs sage.modules
True

\(A\) itself is in the category of algebras with realizations:

sage: A in Algebras(QQ).WithRealizations()                                      # needs sage.modules
True

The (mostly technical) WithRealizations categories are the analogs of the *WithSeveralBases categories in MuPAD-Combinat. They provide support tools for handling the different realizations and the morphisms between them.

Typically, VectorSpaces(QQ).FiniteDimensional().WithRealizations() will eventually be in charge, whenever a coercion \(\phi: A\mapsto B\) is registered, to register \(\phi^{-1}\) as coercion \(B \mapsto A\) if there is none defined yet. To achieve this, FiniteDimensionalVectorSpaces would provide a nested class WithRealizations implementing the appropriate logic.

WithRealizations is a regressive covariant functorial construction. On our example, this simply means that \(A\) is automatically in the category of rings with realizations (covariance):

sage: A in Rings().WithRealizations()                                           # needs sage.modules
True

and in the category of algebras (regressiveness):

sage: A in Algebras(QQ)                                                         # needs sage.modules
True

Note

For C a category, C.WithRealizations() in fact calls sage.categories.with_realizations.WithRealizations(C). The later is responsible for building the hierarchy of the categories with realizations in parallel to that of their base categories, optimizing away those categories that do not provide a WithRealizations nested class. See sage.categories.covariant_functorial_construction for the technical details.

Note

Design question: currently WithRealizations is a regressive construction. That is self.WithRealizations() is a subcategory of self by default:

sage: Algebras(QQ).WithRealizations().super_categories()
[Category of algebras over Rational Field,
 Category of monoids with realizations,
 Category of additive unital additive magmas with realizations]

Is this always desirable? For example, AlgebrasWithBasis(QQ).WithRealizations() should certainly be a subcategory of Algebras(QQ), but not of AlgebrasWithBasis(QQ). This is because AlgebrasWithBasis(QQ) is specifying something about the concrete realization.

additional_structure()

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 category with axiom.

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 categories with axiom 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, functorial construction categories 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()
  

See also

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 structure()), and to test whether a category is a full subcategory of some other category (see 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 github issue #16340.

all_super_categories(proper=False)

Returns the list of all super categories of this category.

INPUT:

• proper – a boolean (default: False); whether to exclude this category.

Since github issue #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 _all_super_categories(), _all_super_categories_proper(), or _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

classmethod an_instance()

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()                                               # needs sage.rings.real_mpfr
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

axioms()

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'})

category()

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

category_graph()

Returns the graph of all super categories of this category

EXAMPLES:

sage: C = Algebras(QQ)
sage: G = C.category_graph()                                                # needs sage.graphs
sage: G.is_directed_acyclic()                                               # needs sage.graphs
True

The girth of a directed acyclic graph is infinite, however, the girth of the underlying undirected graph is 4 in this case:

sage: Graph(G).girth()                                                      # needs sage.graphs
4

element_class()

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.

See also

• parent_class(), morphism_class()

• Category for details

EXAMPLES:

sage: C = Algebras(QQ).element_class; C
<class 'sage.categories.algebras.Algebras.element_class'>
sage: type(C)
<class 'sage.structure.dynamic_class.DynamicMetaclass'>

By github issue #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 instances may not have a __dict__:

sage: E = FiniteEnumeratedSets().element_class
sage: E.__dictoffset__
0

See also

parent_class()

example(*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 instead.

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

full_super_categories()

Return the immediate full super categories of self.

See also

• super_categories()

• 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 cannot 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]

is_abelian()

Return 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

is_full_subcategory(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 structure() from the result of 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 full_super_categories() for a discussion.

See also

• is_subcategory()

• 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 EuclideanDomains currently being a structure category. Is this what we want?

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

is_subcategory(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'))                                # needs sage.rings.finite_rings
sage: M3.is_subcategory(M9)                                                 # needs sage.rings.finite_rings
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

static 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

See also

__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 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 meaningful 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

static meet(categories)

Returns the meet of a list of categories

INPUT:

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

See also

__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

morphism_class()

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.

See also

• parent_class(), element_class()

• Category for details

EXAMPLES:

sage: C = Algebras(QQ).morphism_class; C
<class 'sage.categories.algebras.Algebras.morphism_class'>
sage: type(C)
<class 'sage.structure.dynamic_class.DynamicMetaclass'>

or_subcategory(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

parent_class()

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.

See also

• element_class(), morphism_class()

• Category for details

EXAMPLES:

sage: C = Algebras(QQ).parent_class; C
<class 'sage.categories.algebras.Algebras.parent_class'>
sage: type(C)
<class 'sage.structure.dynamic_class.DynamicMetaclass'>

By github issue #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 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 Bimodules, Category_over_base and sage.categories.category.JoinCategory.

required_methods()

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

structure()

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 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 additional_structure() on the super categories of self.

subcategory_class()

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.

See also

• github issue #12895

• parent_class()

• element_class()

• _make_named_class()

EXAMPLES:

sage: cls = Rings().subcategory_class; cls
<class 'sage.categories.rings.Rings.subcategory_class'>
sage: type(cls)
<class 'sage.structure.dynamic_class.DynamicMetaclass'>

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

sage: isinstance(Rings(), cls)
True
sage: isinstance(AlgebrasWithBasis(QQ), cls)
True

super_categories()

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 github issue #10963, the order of the categories in the result is irrelevant. For details, see On the order of super categories.

Note

Whenever speed matters, developers are advised to use the lazy attribute _super_categories() instead of calling this method.

class sage.categories.category.CategoryWithParameters

Bases: 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 github issue #11935), we want to use the same parent and element class for all fields. This is the purpose of this abstract class.

Currently, JoinCategory, Category_over_base and 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

Category._make_named_class(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).

• 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 _super_categories_for_classes(). This is to guarantee that Python will build consistent method resolution orders. For background, see sage.misc.c3_controlled.

See also

CategoryWithParameters._make_named_class()

EXAMPLES:

sage: PC = Rings()._make_named_class("parent_class", "ParentMethods"); PC
<class 'sage.categories.rings.Rings.parent_class'>
sage: type(PC)
<class 'sage.structure.dynamic_class.DynamicMetaclass'>
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
<class 'sage.categories.rings.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

class sage.categories.category.JoinCategory(super_categories, **kwds)

Bases: 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 github issue #11935, join categories and categories over base rings inherit from 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: # needs sage.groups sage.rings.finite_rings
sage: G = SymmetricGroup(10)
sage: A3 = G.algebra(GF(3))
sage: A5 = G.algebra(GF(5))
sage: type(A3.category())
<class 'sage.categories.category.JoinCategory_with_category'>
sage: type(A3) is type(A5)
True

Category._repr_object_names()

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'

Category._repr_()

Return the print representation of this category.

EXAMPLES:

sage: Sets() # indirect doctest
Category of sets

Category._without_axioms(named=False)

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 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

additional_structure()

Return None.

Indeed, a join category defines no additional structure.

See also

Category.additional_structure()

EXAMPLES:

sage: Modules(ZZ).additional_structure()

is_subcategory(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

super_categories()

Returns the immediate super categories, as per 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]

sage.categories.category.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 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 github issue #15801).

EXAMPLES:

sage: G = sage.categories.category.category_graph(categories=[Groups()])        # needs sage.graphs
sage: G.vertices(sort=True)                                                     # needs sage.graphs
['groups', 'inverse unital magmas', 'magmas', 'monoids', 'objects',
 'semigroups', 'sets', 'sets with partial maps', 'unital magmas']
sage: G.plot()                                                                  # needs sage.graphs sage.plot
Graphics object consisting of 20 graphics primitives

sage: sage.categories.category.category_graph().plot()                          # needs sage.graphs sage.plot
Graphics object consisting of ... graphics primitives

sage.categories.category.category_sample()

Return a sample of categories.

It is constructed by looking for all concrete category classes declared in sage.categories.all, calling 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)                                        # needs sage.groups
[Category of Coxeter groups,
 Category of Dedekind domains,
 Category of G-sets for Symmetric group of order 8! as a permutation group,
 Category of Hecke modules over Rational Field,
 Category of Hopf algebras over Rational Field,
 Category of Hopf algebras with basis over Rational Field,
 Category of Lie algebras over Rational Field,
 Category of Weyl groups,
 Category of abelian varieties 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, ...

sage.categories.category.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