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