The category of abelian groups, i.e. additive abelian monoids where each element has an inverse.

EXAMPLES:

sage: C = CommutativeAdditiveGroups(); C
sage: C.super_categories()
sage: sorted(C.axioms())
True
True


Note

This category is currently empty. It’s left there for backward compatibility and because it is likely to grow in the future.

TESTS:

sage: TestSuite(CommutativeAdditiveGroups()).run()


The empty covariant functorial construction category classes CartesianProducts and Algebras are left here for the sake of nicer output since this is a commonly used category:

sage: CommutativeAdditiveGroups().CartesianProducts()
Category of Cartesian products of commutative additive groups
Category of commutative additive group algebras over Rational Field


Also, it’s likely that some code will end up there at some point.

class Algebras(category, *args)

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()


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()


#### Previous topic

Coalgebras with basis