Bases: sage.categories.category.Category
A category with tensor product is a category endowed with a tensor product functor (operation on its parents and on its elements).
Technically, let CClass be a class inheriting from
CategoryWithTensorProduct. An instance of CClass is a category.
CClass must implement a method
tensor_product_category() which returns the category of
tensor products of parents in . With the default implementation
of tensor_product_category(), it is sufficient to provide a
class CClass.TensorProductCategory whose constructor takes as
parameter the category
and returns the desired category.
If is a subcategory of another category with tensor product
, C.tensor_product_category() is automatically considered
as a subcategory of
.
See also CovariantFunctorialConstruction.
TESTS:
sage: TestSuite(CategoryWithTensorProduct()).run() # mostly to silence sage -coverage on this abstract class
Returns the tensor product of its arguments, as an element of the tensor product of the parents of those elements.
EXAMPLES:
sage: C = AlgebrasWithBasis(QQ)
sage: A = C.example()
sage: (a,b,c) = A.algebra_generators()
sage: a.tensor(b, c)
B[word: a] # B[word: b] # B[word: c]
FIXME: is this a policy that we want to enforce on all parents?
Returns the tensor product of the parents
EXAMPLES:
sage: C = AlgebrasWithBasis(QQ)
sage: A = C.example(); A.rename("A")
sage: A.tensor(A,A)
A # A # A
The category of tensor products of parents in self
EXAMPLES:
sage: ModulesWithBasis(QQ).tensor_category()
Category of tensor products of modules with basis over Rational Field
Bases: sage.categories.tensor.CategoryWithTensorProduct
An abstract base class for all TensorCategory’s defined in CategoryWithTensorProduct’s.
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 A’s is a tensor product of A’s)
EXAMPLES:
sage: ModulesWithBasis(QQ).tensor_category().tensor_category()
Category of tensor products of modules with basis over Rational Field
Bases: sage.categories.category.CovariantFunctorialConstruction
A singleton class for the tensor functor