#10963: More functorial constructions
-------------------------------------+-------------------------------------
Reporter: nthiery | Owner: stumpc5
Type: enhancement | Status: needs_review
Priority: major | Milestone: sage-6.1
Component: categories | Resolution:
Keywords: days54 | Merged in:
Authors: Nicolas M. Thiéry | Reviewers: Simon King, Frédéric
Report Upstream: N/A | Chapoton
Branch: | Work issues:
public/ticket/10963 | Commit:
Dependencies: #11224, #8327, | 478de48553d203516cddb47e0cb89c34ccc210ee
#10193, #12895, #14516, #14722, | Stopgaps:
#13589, #14471, #15069, #15094, |
#11688, #13394, #15150, #15506 |
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Comment (by darij):
{{{
+ @cached_method
+ def DualObjects(self):
+ r"""
+ Return the category of duals of objects 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; e.g. if `V` 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.
+
+ .. SEEALSO::
+
+ - :class:`.dual.DualObjectsCategory`
+ -
:class:`~.covariant_functorial_construction.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 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]
}}}
I know this is not a big issue since the `dual()` of an algebra *is* a
coalgebra in all cases in which `dual()` is implemented (not least because
in the infinite-dimensional cases it usually means the graded dual). But
I'm still unhappy with the docstring lying in my face. Can anyone write a
reasonably worded .. WARNING about this `dual()` not being the actual
vector-space dual?
--
Ticket URL: <http://trac.sagemath.org/ticket/10963#comment:324>
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