Vector Spaces¶
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class
sage.categories.vector_spaces.
VectorSpaces
(K)¶ Bases:
sage.categories.category_types.Category_module
The category of (abstract) vector spaces over a given field
??? with an embedding in an ambient vector space ???
EXAMPLES:
sage: VectorSpaces(QQ) Category of vector spaces over Rational Field sage: VectorSpaces(QQ).super_categories() [Category of modules over Rational Field]
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class
CartesianProducts
(category, *args)¶ Bases:
sage.categories.cartesian_product.CartesianProductsCategory
TESTS:
sage: from sage.categories.covariant_functorial_construction import CovariantConstructionCategory sage: class FooBars(CovariantConstructionCategory): ....: _functor_category = "FooBars" ....: _base_category_class = (Category,) sage: Category.FooBars = lambda self: FooBars.category_of(self) sage: C = FooBars(ModulesWithBasis(ZZ)) sage: C Category of foo bars of modules with basis over Integer Ring sage: C.base_category() Category of modules with basis over Integer Ring sage: latex(C) \mathbf{FooBars}(\mathbf{ModulesWithBasis}_{\Bold{Z}}) sage: import __main__; __main__.FooBars = FooBars # Fake FooBars being defined in a python module sage: TestSuite(C).run()
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extra_super_categories
()¶ The category of vector spaces is closed under Cartesian products:
sage: C = VectorSpaces(QQ) sage: C.CartesianProducts() Category of Cartesian products of vector spaces over Rational Field sage: C in C.CartesianProducts().super_categories() True
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class
VectorSpaces.
DualObjects
(category, *args)¶ Bases:
sage.categories.dual.DualObjectsCategory
TESTS:
sage: from sage.categories.covariant_functorial_construction import CovariantConstructionCategory sage: class FooBars(CovariantConstructionCategory): ....: _functor_category = "FooBars" ....: _base_category_class = (Category,) sage: Category.FooBars = lambda self: FooBars.category_of(self) sage: C = FooBars(ModulesWithBasis(ZZ)) sage: C Category of foo bars of modules with basis over Integer Ring sage: C.base_category() Category of modules with basis over Integer Ring sage: latex(C) \mathbf{FooBars}(\mathbf{ModulesWithBasis}_{\Bold{Z}}) sage: import __main__; __main__.FooBars = FooBars # Fake FooBars being defined in a python module sage: TestSuite(C).run()
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extra_super_categories
()¶ Returns the dual category
EXAMPLES:
The category of algebras over the Rational Field is dual to the category of coalgebras over the same field:
sage: C = VectorSpaces(QQ) sage: C.dual() Category of duals of vector spaces over Rational Field sage: C.dual().super_categories() # indirect doctest [Category of vector spaces over Rational Field]
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class
VectorSpaces.
ElementMethods
¶
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class
VectorSpaces.
ParentMethods
¶
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class
VectorSpaces.
TensorProducts
(category, *args)¶ Bases:
sage.categories.tensor.TensorProductsCategory
TESTS:
sage: from sage.categories.covariant_functorial_construction import CovariantConstructionCategory sage: class FooBars(CovariantConstructionCategory): ....: _functor_category = "FooBars" ....: _base_category_class = (Category,) sage: Category.FooBars = lambda self: FooBars.category_of(self) sage: C = FooBars(ModulesWithBasis(ZZ)) sage: C Category of foo bars of modules with basis over Integer Ring sage: C.base_category() Category of modules with basis over Integer Ring sage: latex(C) \mathbf{FooBars}(\mathbf{ModulesWithBasis}_{\Bold{Z}}) sage: import __main__; __main__.FooBars = FooBars # Fake FooBars being defined in a python module sage: TestSuite(C).run()
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extra_super_categories
()¶ The category of vector spaces is closed under tensor products:
sage: C = VectorSpaces(QQ) sage: C.TensorProducts() Category of tensor products of vector spaces over Rational Field sage: C in C.TensorProducts().super_categories() True
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class
VectorSpaces.
WithBasis
(base_category)¶ Bases:
sage.categories.category_with_axiom.CategoryWithAxiom_over_base_ring
TESTS:
sage: C = Modules(ZZ).FiniteDimensional(); C Category of finite dimensional modules over Integer Ring sage: type(C) <class 'sage.categories.modules.Modules.FiniteDimensional_with_category'> sage: type(C).__base__.__base__ <class 'sage.categories.category_with_axiom.CategoryWithAxiom_over_base_ring'> sage: TestSuite(C).run()
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class
CartesianProducts
(category, *args)¶ Bases:
sage.categories.cartesian_product.CartesianProductsCategory
TESTS:
sage: from sage.categories.covariant_functorial_construction import CovariantConstructionCategory sage: class FooBars(CovariantConstructionCategory): ....: _functor_category = "FooBars" ....: _base_category_class = (Category,) sage: Category.FooBars = lambda self: FooBars.category_of(self) sage: C = FooBars(ModulesWithBasis(ZZ)) sage: C Category of foo bars of modules with basis over Integer Ring sage: C.base_category() Category of modules with basis over Integer Ring sage: latex(C) \mathbf{FooBars}(\mathbf{ModulesWithBasis}_{\Bold{Z}}) sage: import __main__; __main__.FooBars = FooBars # Fake FooBars being defined in a python module sage: TestSuite(C).run()
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extra_super_categories
()¶ The category of vector spaces with basis is closed under Cartesian products:
sage: C = VectorSpaces(QQ).WithBasis() sage: C.CartesianProducts() Category of Cartesian products of vector spaces with basis over Rational Field sage: C in C.CartesianProducts().super_categories() True
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class
VectorSpaces.WithBasis.
TensorProducts
(category, *args)¶ Bases:
sage.categories.tensor.TensorProductsCategory
TESTS:
sage: from sage.categories.covariant_functorial_construction import CovariantConstructionCategory sage: class FooBars(CovariantConstructionCategory): ....: _functor_category = "FooBars" ....: _base_category_class = (Category,) sage: Category.FooBars = lambda self: FooBars.category_of(self) sage: C = FooBars(ModulesWithBasis(ZZ)) sage: C Category of foo bars of modules with basis over Integer Ring sage: C.base_category() Category of modules with basis over Integer Ring sage: latex(C) \mathbf{FooBars}(\mathbf{ModulesWithBasis}_{\Bold{Z}}) sage: import __main__; __main__.FooBars = FooBars # Fake FooBars being defined in a python module sage: TestSuite(C).run()
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extra_super_categories
()¶ The category of vector spaces with basis is closed under tensor products:
sage: C = VectorSpaces(QQ).WithBasis() sage: C.TensorProducts() Category of tensor products of vector spaces with basis over Rational Field sage: C in C.TensorProducts().super_categories() True
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VectorSpaces.WithBasis.
is_abelian
()¶ Return whether this category is abelian.
This is always
True
since the base ring is a field.EXAMPLES:
sage: VectorSpaces(QQ).WithBasis().is_abelian() True
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class
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VectorSpaces.
additional_structure
()¶ Return
None
.Indeed, the category of vector spaces defines no additional structure: a bimodule morphism between two vector spaces is a vector space morphism.
See also
Todo
Should this category be a
CategoryWithAxiom
?EXAMPLES:
sage: VectorSpaces(QQ).additional_structure()
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VectorSpaces.
base_field
()¶ Returns the base field over which the vector spaces of this category are all defined.
EXAMPLES:
sage: VectorSpaces(QQ).base_field() Rational Field
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VectorSpaces.
super_categories
()¶ EXAMPLES:
sage: VectorSpaces(QQ).super_categories() [Category of modules over Rational Field]
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class