Tensor Product Functorial Construction¶
AUTHORS:
Nicolas M. Thiéry (2008-2010): initial revision and refactorization
- class sage.categories.tensor.TensorProductFunctor¶
Bases:
sage.categories.covariant_functorial_construction.CovariantFunctorialConstruction
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 categoryAlgebras(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
A
is a subcategory ofB
, thenA.TensorProducts()
is a subcategory ofB.TensorProducts()
(see alsoCovariantFunctorialConstruction
). Hence, the role ofAlgebras(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
ofAlgebras(QQ)
. This nested class is itself a subclass ofTensorProductsCategory
.
- class sage.categories.tensor.TensorProductsCategory(category, *args)¶
Bases:
sage.categories.covariant_functorial_construction.CovariantConstructionCategory
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 informationEXAMPLES:
sage: tensor The tensor functorial construction