Finite sets¶
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class
sage.categories.finite_sets.
FiniteSets
(base_category)¶ Bases:
sage.categories.category_with_axiom.CategoryWithAxiom_singleton
The category of finite sets.
EXAMPLES:
sage: C = FiniteSets(); C Category of finite sets sage: C.super_categories() [Category of sets] sage: C.all_super_categories() [Category of finite sets, Category of sets, Category of sets with partial maps, Category of objects] sage: C.example() NotImplemented
TESTS:
sage: TestSuite(C).run() sage: C is Sets().Finite() True
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class
Algebras
(category, *args)¶ Bases:
sage.categories.algebra_functor.AlgebrasCategory
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
()¶ EXAMPLES:
sage: FiniteSets().Algebras(QQ).extra_super_categories() [Category of finite dimensional vector spaces with basis over Rational Field]
This implements the fact that the algebra of a finite set is finite dimensional:
sage: FiniteMonoids().Algebras(QQ).is_subcategory(AlgebrasWithBasis(QQ).FiniteDimensional()) True
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class
FiniteSets.
ParentMethods
¶ -
is_finite
()¶ Return
True
sinceself
is finite.EXAMPLES:
sage: C = FiniteEnumeratedSets().example() sage: C.is_finite() True
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class
FiniteSets.
Subquotients
(category, *args)¶ Bases:
sage.categories.subquotients.SubquotientsCategory
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
()¶ EXAMPLES:
sage: FiniteSets().Subquotients().extra_super_categories() [Category of finite sets]
This implements the fact that a subquotient (and therefore a quotient or subobject) of a finite set is finite:
sage: FiniteSets().Subquotients().is_subcategory(FiniteSets()) True sage: FiniteSets().Quotients ().is_subcategory(FiniteSets()) True sage: FiniteSets().Subobjects ().is_subcategory(FiniteSets()) True
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class