def DualObjects(self):
r"""
Return the category of spaces constructed as duals of
spaces of ``self``.
The *dual* of a vector space `V` is the space consisting of
all linear functionals on `V` (see :wikipedia:`Dual_space`).
Additional structure on `V` can endow its dual with
additional structure; for example, if `V` is a finite
dimensional algebra, then its dual is a coalgebra.
This returns the category of spaces constructed as dual of
spaces in ``self``, endowed with the appropriate
additional structure.
.. WARNING::
- This semantic of ``dual`` and ``DualObject`` is
imposed on all subcategories, in particular to make
``dual`` a covariant functorial construction.
A subcategory that defines a different notion of
dual needs to use a different name.
- Typically, the category of graded modules should
define a separate ``graded_dual`` construction (see
:trac:`15647`). For now the two constructions are
not distinguished which is an oversimplified model.
.. SEEALSO::
- :class:`.dual.DualObjectsCategory`
- :class:`~.covariant_functorial_construction.CovariantFunctorialConstruction`.
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 is a coalgebra::
sage: sorted(Algebras(QQ).DualObjects().super_categories(), key=str)
[Category of coalgebras over Rational Field,
Category of duals of vector spaces over Rational Field]
The dual of a coalgebra is an algebra::
sage: sorted(Coalgebras(QQ).DualObjects().super_categories(), key=str)
[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
:meth:`~Modules.SubcategoryMethods.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()
"""
return DualObjectsCategory.category_of(self)
def DualObjects(self):
r"""
Return the category of spaces constructed as duals of
spaces of ``self``.
The *dual* of a vector space `V` is the space consisting of
all linear functionals on `V` (see :wikipedia:`Dual_space`).
Additional structure on `V` can endow its dual with
additional structure; for example, if `V` is a finite
dimensional algebra, then its dual is a coalgebra.
This returns the category of spaces constructed as dual of
spaces in ``self``, endowed with the appropriate
additional structure.
.. WARNING::
- This semantic of ``dual`` and ``DualObject`` is
imposed on all subcategories, in particular to make
``dual`` a covariant functorial construction.
A subcategory that defines a different notion of
dual needs to use a different name.
- Typically, the category of graded modules should
define a separate ``graded_dual`` construction (see
:trac:`15647`). For now the two constructions are
not distinguished which is an oversimplified model.
.. SEEALSO::
- :class:`.dual.DualObjectsCategory`
- :class:`~.covariant_functorial_construction.CovariantFunctorialConstruction`.
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 is a coalgebra::
sage: sorted(Algebras(QQ).DualObjects().super_categories(), key=str)
[Category of coalgebras over Rational Field,
Category of duals of vector spaces over Rational Field]
The dual of a coalgebra is an algebra::
sage: sorted(Coalgebras(QQ).DualObjects().super_categories(), key=str)
[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
:meth:`~Modules.SubcategoryMethods.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()
"""
return DualObjectsCategory.category_of(self)