Tensor Product Functorial Construction - Category Framework

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# Tensor Product Functorial Construction

AUTHORS:

- Nicolas M. Thiéry (2008-2010): initial revision and refactorization

class sage.categories.tensor.TensorProductFunctor

  Bases: `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 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 `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`.

  symbol = ' # '
  unicode_symbol = ' ⊗ '

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

  Bases: `CovariantConstructionCategory`

  An abstract base class for all TensorProducts’s categories.

  TensorProducts()

    Return 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

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

    Python

        >>> from sage.all import *
        >>> 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

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

    Python

        >>> from sage.all import *
        >>> 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

      sage: tensor
      The tensor functorial construction

  Python

      >>> from sage.all import *
      >>> tensor
      The tensor functorial construction