AUTHORS:
- Nicolas M. Thiery (2009-2010): initial revision
Bases: sage.categories.covariant_functorial_construction.CovariantFunctorialConstruction
A singleton class for the dual functor
Returns the category of duals of objects of self.
INPUT:
- self – a subcategory of vector spaces over some base ring
The dual of a vector space is the space consisting of all
linear functionals on
(http://en.wikipedia.org/wiki/Dual_space).
Additional structure on
can endow its dual with additional
structure; e.g. if
is an algebra, then its dual is a
coalgebra.
This returns the category of dual of spaces in self endowed with the appropriate additional structure.
See also CovariantFunctorialConstruction.
TODO: add support for graded duals.
EXAMPLES:
sage: VectorSpaces(QQ).DualObjects()
Category of duals of vector spaces over Rational Field
The dual of a vector space is a vector space:
sage: VectorSpaces(QQ).DualObjects().super_categories()
[Category of vector spaces over Rational Field]
The dual of an algebra space is a coalgebra:
sage: Algebras(QQ).DualObjects().super_categories()
[Category of coalgebras over Rational Field, Category of duals of vector spaces over Rational Field]
The dual of a coalgebra space is an algebra:
sage: Coalgebras(QQ).DualObjects().super_categories()
[Category of algebras over Rational Field, Category of duals of vector spaces over Rational Field]
As a shorthand, this category can be accessed with the dual() method:
sage: VectorSpaces(QQ).dual()
Category of duals of vector spaces over Rational Field
TESTS:
sage: C = VectorSpaces(QQ).DualObjects()
sage: C.base_category()
Category of vector spaces over Rational Field
sage: C.super_categories()
[Category of vector spaces over Rational Field]
sage: latex(C)
\mathbf{DualObjects}(\mathbf{VectorSpaces}_{\Bold{Q}})
sage: TestSuite(C).run()
Bases: sage.categories.covariant_functorial_construction.CovariantConstructionCategory
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()