### Navigation

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

# Graded Hopf algebras with basis

class `sage.categories.graded_hopf_algebras_with_basis.``GradedHopfAlgebrasWithBasis`(base_category)

  Bases: `sage.categories.graded_modules.GradedModulesCategory`

  The category of graded Hopf algebras with a distinguished basis.

  EXAMPLES:

      sage: C = GradedHopfAlgebrasWithBasis(ZZ); C
      Category of graded hopf algebras with basis over Integer Ring
      sage: C.super_categories()
      [Category of hopf algebras with basis over Integer Ring,
       Category of graded algebras with basis over Integer Ring]

      sage: C is HopfAlgebras(ZZ).WithBasis().Graded()
      True
      sage: C is HopfAlgebras(ZZ).Graded().WithBasis()
      False

  TESTS:

      sage: TestSuite(C).run()

  class `ElementMethods`
  class `GradedHopfAlgebrasWithBasis.``ParentMethods`
  class `GradedHopfAlgebrasWithBasis.``WithRealizations`(category, *args)

    Bases: `sage.categories.with_realizations.WithRealizationsCategory`

    TESTS:

        sage: from sage.categories.covariant_functorial_construction import CovariantConstructionCategory
        sage: class FooBars(CovariantConstructionCategory):
        ...       _functor_category = "FooBars"
        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()

    `super_categories`()

      EXAMPLES:

          sage: GradedHopfAlgebrasWithBasis(QQ).WithRealizations().super_categories()
          [Join of Category of hopf algebras over Rational Field
               and Category of graded algebras over Rational Field]

      TESTS:

          sage: TestSuite(GradedHopfAlgebrasWithBasis(QQ).WithRealizations()).run()

### Navigation

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