Binary Cartesian Product

A binary cartesian product in a Category 𝒞 𝒞 \mathcal{C} of two objects X , Y : 𝒞 : 𝑋 𝑌 𝒞 X,Y:\mathcal{C} is an object X × Y 𝑋 𝑌 X\times Y equipped with a pair of projections π 1 : X × Y X : subscript 𝜋 1 𝑋 𝑌 𝑋 \pi_{1}:X\times Y\to X , π 2 : X × Y Y : subscript 𝜋 2 𝑋 𝑌 𝑌 \pi_{2}:X\times Y\to Y that satisfies the following universal property:

Properties

Global Definition

A category 𝒞 𝒞 \mathcal{C} has all binary cartesian products if and only if the functor Δ : 𝒞 𝒞 × 𝒞 : Δ 𝒞 𝒞 𝒞 \Delta:\mathcal{C}\to\mathcal{C}\times\mathcal{C} that sends an object X 𝑋 X to ( X , X ) 𝑋 𝑋 (X,X) has a Right Adjoint.