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# Cartesian Product Functorial Construction

AUTHORS:

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

class `sage.categories.cartesian_product.``CartesianProductFunctor`

  Bases: `sage.categories.covariant_functorial_construction.CovariantFunctorialConstruction`

  A singleton class for the Cartesian product functor.

  EXAMPLES:

      sage: cartesian_product
      The cartesian_product functorial construction

  `cartesian_product` takes a finite collection of sets, and constructs the Cartesian product of those sets:

      sage: A = FiniteEnumeratedSet(['a','b','c'])
      sage: B = FiniteEnumeratedSet([1,2])
      sage: C = cartesian_product([A, B]); C
      The cartesian product of ({'a', 'b', 'c'}, {1, 2})
      sage: C.an_element()
      ('a', 1)
      sage: C.list()         # todo: not implemented
      [['a', 1], ['a', 2], ['b', 1], ['b', 2], ['c', 1], ['c', 2]]

  If those sets are endowed with more structure, say they are monoids (hence in the category \(Monoids()\)), then the result is automatically endowed with its natural monoid structure:

      sage: M = Monoids().example()
      sage: M
      An example of a monoid: the free monoid generated by ('a', 'b', 'c', 'd')
      sage: M.rename('M')
      sage: C = cartesian_product([M, ZZ, QQ])
      sage: C
      The cartesian product of (M, Integer Ring, Rational Field)
      sage: C.an_element()
      ('abcd', 1, 1/2)
      sage: C.an_element()^2
      ('abcdabcd', 1, 1/4)
      sage: C.category()
      Category of Cartesian products of monoids

      sage: Monoids().CartesianProducts()
      Category of Cartesian products of monoids

  The Cartesian product functor is covariant: if `A` is a subcategory of `B`, then `A.CartesianProducts()` is a subcategory of `B.CartesianProducts()` (see also `CovariantFunctorialConstruction`):

      sage: C.categories()
      [Category of Cartesian products of monoids,
       Category of monoids,
       Category of Cartesian products of semigroups,
       Category of semigroups,
       Category of Cartesian products of unital magmas,
       Category of Cartesian products of magmas,
       Category of unital magmas,
       Category of magmas,
       Category of Cartesian products of sets,
       Category of sets, ...]

      [Category of Cartesian products of monoids,
       Category of monoids,
       Category of Cartesian products of semigroups,
       Category of semigroups,
       Category of Cartesian products of magmas,
       Category of unital magmas,
       Category of magmas,
       Category of Cartesian products of sets,
       Category of sets,
       Category of sets with partial maps,
       Category of objects]

  Hence, the role of `Monoids().CartesianProducts()` is solely to provide mathematical information and algorithms which are relevant to Cartesian product of monoids. For example, it specifies that the result is again a monoid, and that its multiplicative unit is the cartesian product of the units of the underlying sets:

      sage: C.one()
      ('', 1, 1)

  Those are implemented in the nested class `Monoids.CartesianProducts` of `Monoids(QQ)`. This nested class is itself a subclass of `CartesianProductsCategory`.

class `sage.categories.cartesian_product.``CartesianProductsCategory`(category, *args)

  Bases: `sage.categories.covariant_functorial_construction.CovariantConstructionCategory`

  An abstract base class for all `CartesianProducts` categories.

  TESTS:

      sage: C = Sets().CartesianProducts()
      sage: C
      Category of Cartesian products of sets
      sage: C.base_category()
      Category of sets
      sage: latex(C)
      \mathbf{CartesianProducts}(\mathbf{Sets})

  `CartesianProducts`()

    Return the category of (finite) Cartesian products of objects of `self`.

    By associativity of Cartesian products, this is `self` (a Cartesian product of Cartesian products of \(A\)‘s is a Cartesian product of \(A\)‘s).

    EXAMPLES:

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

  `base_ring`()

    The base ring of a cartesian product is the base ring of the underlying category.

    EXAMPLES:

        sage: Algebras(ZZ).CartesianProducts().base_ring()
        Integer Ring

`sage.categories.cartesian_product.``cartesian_product`

  The cartesian product functorial construction.

  See `CartesianProductFunctor` for more information.

  EXAMPLES:

      sage: cartesian_product
      The cartesian_product functorial construction

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