I seem not to be understanding the way to implement categories with extra structure. Suppose I want to implement the category Cs of pairs (A,S) where A is a QQ-algebra and S is a linear endomorphism of A with the obvious morphisms. Algebras(QQ) is a full subcategory of Cs by adding the zero endormophism. And we also have the forgetful functor from Cs to Algebras(QQ) which consists to simply forget S.
Now from reading the examples in https://doc.sagemath.org/html/en/reference/categories/index.html or https://doc.sagemath.org/html/en/thematic_tutorials/coercion_and_categories.html I could set up
class Cs(Category)
def super_categories(self):
return[Algebras(QQ)]
And that woul d give me a canonical forgetful functor Cs -> Algebras(QQ). Now my problem is when I want to create a parent of Cs() starting from a parent in Algebras(QQ). That is, I could set up a new parent and elements which will have to implement the methods for Algebras(QQ):
sage: from sage.misc.abstract_method import abstract_methods_of_class
sage: abstract_methods_of_class(Algebras(QQ).parent_class)
{'optional': ['algebra_generators'], 'required': ['__contains__']}
sage: abstract_methods_of_class(Algebras(QQ).element_class)
{'optional': ['_add_', '_mul_'], 'required': ['__nonzero__']}
But instead of implementing those methods I would want to use the methods of the underlying parent of Algebras(QQ). Something like a hypothetical PolynomialRing(QQ, 'x', category=Cs()). In other words I'm looking to implement the functor A -> (A, 0) above.
Finally I have similar concerns about implementing Cs() in the other two possible ways, namely as the subcategory of the category of arrows in Algebras(QQ) with the same source and target, or as a super category of Algebras(QQ) using _subcategory_hook_. Always I get to the point where I don't know how to implement something like PolynomialRing(QQ, category=Cs()). What I am doing now is keeping a copy of the algebra A inside of an instance of Cs.parent_class and then implementing the element and parent mehods of Algebras(QQ) by pointing to the corresponding methods of A but that seems silly. I'd appreciate any pointer.
