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# Coalgebras with basis

class `sage.categories.coalgebras_with_basis.``CoalgebrasWithBasis`(base_category)

  Bases: `sage.categories.category_with_axiom.CategoryWithAxiom_over_base_ring`

  The category of coalgebras with a distinguished basis.

  EXAMPLES:

      sage: CoalgebrasWithBasis(ZZ)
      Category of coalgebras with basis over Integer Ring
      sage: sorted(CoalgebrasWithBasis(ZZ).super_categories(), key=str)
      [Category of coalgebras over Integer Ring,
       Category of modules with basis over Integer Ring]

  TESTS:

      sage: TestSuite(CoalgebrasWithBasis(ZZ)).run()

  class `ElementMethods`
  class `CoalgebrasWithBasis.``ParentMethods`
    `coproduct`()

      If `coproduct_on_basis()` is available, construct the coproduct morphism from `self` to `self` \(\otimes\) `self` by extending it by linearity. Otherwise, use `coproduct_by_coercion()`, if available.

      EXAMPLES:

          sage: A = HopfAlgebrasWithBasis(QQ).example(); A
          An example of Hopf algebra with basis: the group algebra of the Dihedral group of order 6 as a permutation group over Rational Field
          sage: [a,b] = A.algebra_generators()
          sage: a, A.coproduct(a)
          (B[(1,2,3)], B[(1,2,3)] # B[(1,2,3)])
          sage: b, A.coproduct(b)
          (B[(1,3)], B[(1,3)] # B[(1,3)])

    `coproduct_on_basis`(i)

      The coproduct of the algebra on the basis (optional).

      INPUT:

      - `i` – the indices of an element of the basis of `self`

      Returns the coproduct of the corresponding basis elements If implemented, the coproduct of the algebra is defined from it by linearity.

      EXAMPLES:

          sage: A = HopfAlgebrasWithBasis(QQ).example(); A
          An example of Hopf algebra with basis: the group algebra of the Dihedral group of order 6 as a permutation group over Rational Field
          sage: (a, b) = A._group.gens()
          sage: A.coproduct_on_basis(a)
          B[(1,2,3)] # B[(1,2,3)]

    `counit`()

      If `counit_on_basis()` is available, construct the counit morphism from `self` to `self` \(\otimes\) `self` by extending it by linearity

      EXAMPLES:

          sage: A = HopfAlgebrasWithBasis(QQ).example(); A
          An example of Hopf algebra with basis: the group algebra of the Dihedral group of order 6 as a permutation group over Rational Field
          sage: [a,b] = A.algebra_generators()
          sage: a, A.counit(a)
          (B[(1,2,3)], 1)
          sage: b, A.counit(b)
          (B[(1,3)], 1)

    `counit_on_basis`(i)

      The counit of the algebra on the basis (optional).

      INPUT:

      - `i` – the indices of an element of the basis of `self`

      Returns the counit of the corresponding basis elements If implemented, the counit of the algebra is defined from it by linearity.

      EXAMPLES:

          sage: A = HopfAlgebrasWithBasis(QQ).example(); A
          An example of Hopf algebra with basis: the group algebra of the Dihedral group of order 6 as a permutation group over Rational Field
          sage: (a, b) = A._group.gens()
          sage: A.counit_on_basis(a)
          1

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