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# Tensor Product Functorial Construction

AUTHORS:

> - Nicolas M. Thiery (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 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`.

  TESTS:

      sage: TestSuite(tensor).run()

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

  Bases: `sage.categories.covariant_functorial_construction.CovariantConstructionCategory`

  An abstract base class for all TensorProducts’s categories

  TESTS:

      sage: C = ModulesWithBasis(QQ).TensorProducts()
      sage: C
      Category of tensor products of vector spaces with basis over Rational Field
      sage: C.base_category()
      Category of vector spaces with basis over Rational Field
      sage: latex(C)
      \mathbf{TensorProducts}(\mathbf{WithBasis}_{\Bold{Q}})
      sage: TestSuite(C).run()

  `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 product functorial construction

  See `TensorProductFunctor` for more information

  EXAMPLES:

      sage: tensor
      The tensor functorial construction

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