Tensor Product Functorial Construction¶
Nicolas M. Thiéry (2008-2010): initial revision and refactorization
- class sage.categories.tensor.TensorProductFunctor¶
A singleton class for the tensor functor.
This functor takes a collection of vector spaces (or modules with basis), and constructs the tensor product of those vector spaces. If this vector space is in a subcategory, say that of
Algebras(QQ), it is automatically endowed with its natural algebra structure, thanks to the category
Algebras(QQ).TensorProducts()of tensor products of algebras. For elements, it constructs the natural tensor product element in the corresponding tensor product of their parents.
The tensor functor is covariant: if
Ais a subcategory of
A.TensorProducts()is a subcategory of
CovariantFunctorialConstruction). Hence, the role of
Algebras(QQ).TensorProducts()is solely to provide mathematical information and algorithms which are relevant to tensor product of algebras.
- class sage.categories.tensor.TensorProductsCategory(category, *args)¶
An abstract base class for all TensorProducts’s categories
Returns the category of tensor products of objects of
By associativity of tensor products, this is
self(a tensor product of tensor products of \(Cat\)’s is a tensor product of \(Cat\)’s)
sage: ModulesWithBasis(QQ).TensorProducts().TensorProducts() Category of tensor products of vector spaces with basis over Rational Field
The base of a tensor product is the base (usually a ring) of the underlying category.
sage: ModulesWithBasis(ZZ).TensorProducts().base() Integer Ring