### Navigation

- index
- modules |
- next |
- previous |
- Category Theory »

# Specific category classes

This is placed in a separate file from categories.py to avoid circular imports (as morphisms must be very low in the hierarchy with the new coercion model).

class `sage.categories.category_types.``AbelianCategory`(s=None)

  Bases: `sage.categories.category.Category`

  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__
      <class '__main__.SemiprimitiveRings_with_category'>

  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 `_repr_object_names()` to customize it.

  `is_abelian`()

    x.__init__(...) initializes x; see help(type(x)) for signature

class `sage.categories.category_types.``Category_ideal`(ambient, name=None)

  Bases: `sage.categories.category_types.Category_in_ambient`

  classmethod `an_instance`()

    Returns an instance of this class

    EXAMPLES:

        sage: AlgebraIdeals.an_instance()
        Category of algebra ideals in Univariate Polynomial Ring in x over Rational Field

  `ring`()

    x.__init__(...) initializes x; see help(type(x)) for signature

class `sage.categories.category_types.``Category_in_ambient`(ambient, name=None)

  Bases: `sage.categories.category.Category`

  `ambient`()

    Return the ambient object in which objects of this category are embedded.

class `sage.categories.category_types.``Category_module`(base, name=None)

  Bases: `sage.categories.category_types.AbelianCategory`, `sage.categories.category_types.Category_over_base_ring`

  Initialize `self`.

  EXAMPLES:

      sage: C = Algebras(GF(2)); C
      Category of algebras over Finite Field of size 2
      sage: TestSuite(C).run()

class `sage.categories.category_types.``Category_over_base`(base, name=None)

  Bases: `sage.categories.category.CategoryWithParameters`

  A base class for categories over some base object

  INPUT:

  - `base` – a category \(C\) or an object of such a category

  Assumption: the classes for the parents, elements, morphisms, of `self` should only depend on \(C\). See trac ticket #11935 for details.

  EXAMPLES:

      sage: Algebras(GF(2)).element_class is Algebras(GF(3)).element_class
      True

      sage: C = GF(2).category()
      sage: Algebras(GF(2)).parent_class is Algebras(C).parent_class
      True

      sage: C = ZZ.category()
      sage: Algebras(ZZ).element_class is Algebras(C).element_class
      True

  classmethod `an_instance`()

    Returns an instance of this class

    EXAMPLES:

        sage: Algebras.an_instance()
        Category of algebras over Rational Field

  `base`()

    Return the base over which elements of this category are defined.

class `sage.categories.category_types.``Category_over_base_ring`(base, name=None)

  Bases: `sage.categories.category_types.Category_over_base`

  Initialize `self`.

  EXAMPLES:

      sage: C = Algebras(GF(2)); C
      Category of algebras over Finite Field of size 2
      sage: TestSuite(C).run()

  `base_ring`()

    Return the base ring over which elements of this category are defined.

    EXAMPLES:

        sage: C = Algebras(GF(2))
        sage: C.base_ring()
        Finite Field of size 2

class `sage.categories.category_types.``ChainComplexes`(base, name=None)

  Bases: `sage.categories.category_types.Category_module`

  The category of all chain complexes over a base ring.

  EXAMPLES:

         sage: ChainComplexes(RationalField())
         Category of chain complexes over Rational Field

         sage: ChainComplexes(Integers(9))
         Category of chain complexes over Ring of integers modulo 9

      TESTS::

         sage: TestSuite(ChainComplexes(RationalField())).run()

  `super_categories`()

    EXAMPLES:

        sage: ChainComplexes(Integers(9)).super_categories()
        [Category of modules with basis over Ring of integers modulo 9]

class `sage.categories.category_types.``Elements`(object)

  Bases: `sage.categories.category.Category`

  The category of all elements of a given parent.

  EXAMPLES:

      sage: a = IntegerRing()(5)
      sage: C = a.category(); C
      Category of elements of Integer Ring
      sage: a in C
      True
      sage: 2/3 in C
      False
      sage: loads(C.dumps()) == C
      True

  classmethod `an_instance`()

    Returns an instance of this class

    EXAMPLES:

        sage: Elements(ZZ)
        Category of elements of Integer Ring

  `object`()

    x.__init__(...) initializes x; see help(type(x)) for signature

  `super_categories`()

    EXAMPLES:

        sage: Elements(ZZ).super_categories()
        [Category of objects]

    TODO:

    check that this is what we want.

class `sage.categories.category_types.``Sequences`(object)

  Bases: `sage.categories.category.Category`

  The category of sequences of elements of a given object.

  This category is deprecated.

  EXAMPLES:

      sage: v = Sequence([1,2,3]); v
      [1, 2, 3]
      sage: C = v.category(); C
      Category of sequences in Integer Ring
      sage: loads(C.dumps()) == C
      True
      sage: Sequences(ZZ) is C
      True

      True
      sage: Sequences(ZZ).category()
      Category of objects

  classmethod `an_instance`()

    Returns an instance of this class

    EXAMPLES:

        sage: Elements(ZZ)
        Category of elements of Integer Ring

  `object`()

    x.__init__(...) initializes x; see help(type(x)) for signature

  `super_categories`()

    EXAMPLES:

        sage: Sequences(ZZ).super_categories()
        [Category of objects]

class `sage.categories.category_types.``SimplicialComplexes`(s=None)

  Bases: `sage.categories.category.Category`

  The category of simplicial complexes.

  EXAMPLES:

      sage: SimplicialComplexes()
      Category of simplicial complexes

  TESTS:

      sage: TestSuite(SimplicialComplexes()).run()

  `super_categories`()

    EXAMPLES:

        sage: SimplicialComplexes().super_categories()
        [Category of objects]

### Navigation

- index
- modules |
- next |
- previous |
- Category Theory »