product

• cartisian product of two sets \(S_{1}\) and \(S_{2}\) is denoted as \(S_{1} \times S_{2}\), which is a set whose elements are ordered pair of elements in \(S_{1}\) and \(S_{2}\) respectively

• The abstract concept of such products generalizes from Set to any other category , only that in general products of any given objects may or may not actually exist in that category.

  • it's a special case of a limit

• definition of product of objects in a category \(\mathcal{C}\)

  • for objects \(\{X_{i}\}_{i \in I}\), the product (if exists) is denoted as \(\Pi_{i \in I} X_{i} \in \mathcal{C}\)
  • and equipped with morphisms \(p_{i} : (\Pi_{i \in I} X_{i}) \to X_{i}\) (projections)
  • the following universal property is satisfied
    • for any other object \(Q \in \mathcal{C}\) with morphism \(Q \to X_{i}\)
    • there's an unique morphism \(Q \to (\Pi_{i \in I} X_{i})\) such that the diagram commutes