Opened 8 years ago

# Make a proper distinction in the categories between dual and graded dual — at Version 1

Reported by: Owned by: darij major sage-6.4 categories nthiery, sage-combinat N/A #10963

Tangent off the #10963 discussion...

```+ @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`.
+
+
+ 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]
```

I know this is not a big issue since the `dual()` of an algebra *is* a coalgebra in probably all cases in which `dual()` is implemented (not least because in the infinite-dimensional cases it usually means the graded dual). But at some point it probably *will* become an issue (maybe with the introduction of non-graded bases for graded algebras?), and I'm unhappy with the docstring lying in my face. And Nicolas suggests that "we need to clean up the distinction between dual and graded dual; this is not completely obvious to set the things up so that we can still share some code between the two".

At this occasion, DualObjectsCategory? should be made into a category over base ring, if not just to inherit from the an_instance method. Compare:

```sage: sage: ModulesWithBasis.Graded.an_instance()
Category of graded modules with basis over Rational Field
sage: ModulesWithBasis.DualObjects.an_instance()
Type error ...
```

### comment:1 Changed 8 years ago by nthiery

• Description modified (diff)
• Summary changed from Docstring and probably also code is dangerously naive about duality to Make a proper distinction in the categories between dual and graded dual
• Type changed from task to defect
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