>
> Okay, now I have
>
> class SuperCommutativeAlgebrasWithBasis(SuperModulesCategory):
> # Imitating topological_space.py:
> _base_category_class = (SuperAlgebrasWithBasis,)
>
> def example(self, alphabet = ('a','b','c')):
> from
> sage.categories.examples.super_commutative_algebras_with_basis import
> Example
> return Example(self.base_ring(), alphabet)
>
> Then I have this problem:
>
> sage: SCAB = SuperCommutativeAlgebrasWithBasis(QQ)
> sage: SCAB.example()
> NotImplemented
>
> Why?
>
That is because of this:
sage: type(SCAB)
<class
'sage.categories.super_algebras_with_basis.SuperAlgebrasWithBasis_with_category'>
Now why that is happening, I do not exactly know. It probably has to do
with the fact that you do not have a functor construction, e.g., a
SuperCommutative attribute linking to this class, in
SuperAlgebrasWithBasis. For instance, in Sets(), there is an attribute
Topological that links to TopologicalSpaces. There is also a method
Topological that calls return "TopologicalSpacesCategory.category_of(self)"
in the SubcategoryMethods so all its subcategories can construct the
appropriate subcategory, but that is a bit tangential.
IMO, proper way to implement this is as a CategoryWithAxiom as a part of
SuperAlgebras[WithBasis]. So you add a class
"Commutative(CategoryWithAxiom_over_base_ring)" to the
SuperAlgebras[WithBasis] category and do the normal category implementation
stuff there.
>
> An implementation question: separately from having a category of super
> commutative algebras, I could imagine wanting the option to use a tensor
> product of graded algebras which includes a sign in the product:
>
> (x1 tensor y1) (x2 tensor y2) = (-1)^(deg y1 deg x2) (x1 x2 tensor y1
> y2)
>
> even if neither tensor factor is itself (graded) commutative. How would
> this be implemented? Define a new tensor product? And then somehow label
> the category in which the algebra lives to tell it to use that tensor
> product? Or define an axiom? (Are axioms worth using? I have run into
> problems with inheritance with axioms.)
>
> This is a bit of a tricky issue. Of course, you could define a new
TensorProducts category that implements a product_on_basis with this
version (see AlgebrasWithBasis.TensorProducts.ParentMethods). If the tensor
product parent was in that category (likely just being a
CombinatorialFreeModule), then it should work out of the box. I am not sure
this fits the criteria to be an axiom (but they are definitely worth using
in the appropriate circumstances). The can of worms comes out when you want
a natural way to construct that parent in that category. Plus a good name
will be needed, but I guess the natural place would be part of the
SuperAlgebras.Commutative.TensorProducts category.
Best,
Travis
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