# Tensor Product Functorial Construction¶

AUTHORS:

• Nicolas M. Thiery (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.

The tensor functor is covariant: if A is a subcategory of B, then A.TensorProducts() is a subcategory of B.TensorProducts() (see also CovariantFunctorialConstruction). Hence, the role of Algebras(QQ).TensorProducts() is solely to provide mathematical information and algorithms which are relevant to tensor product of algebras.

Those are implemented in the nested class TensorProducts of Algebras(QQ). This nested class is itself a subclass of TensorProductsCategory.

class sage.categories.tensor.TensorProductsCategory(category, *args)

An abstract base class for all TensorProducts’s categories

TensorProducts()

Returns the category of tensor products of objects of self

By associativity of tensor products, this is self (a tensor product of tensor products of $$Cat$$’s is a tensor product of $$Cat$$’s)

EXAMPLES:

sage: ModulesWithBasis(QQ).TensorProducts().TensorProducts()
Category of tensor products of vector spaces with basis over Rational Field

base()

The base of a tensor product is the base (usually a ring) of the underlying category.

EXAMPLES:

sage: ModulesWithBasis(ZZ).TensorProducts().base()
Integer Ring

sage.categories.tensor.tensor = The tensor functorial construction

The tensor product functorial construction

See TensorProductFunctor for more information

EXAMPLES:

sage: tensor
The tensor functorial construction