product (category theory)

The categorical product abstracts the idea of a Cartesian product — pairing two things together — into pure arrow language. The key insight is that an ordered pair (a, b) is the same data as a pair of separate pieces of information: an a and a b. So the product of two objects should be the universal object out of which you can read off a coordinate in each factor, and into which a map is exactly a map to each factor.

Formally, a product of objects A and B is an object A × B together with two projection morphisms p : A × B -> A and q : A × B -> B, satisfying the universal property: for every object X with morphisms f : X -> A and g : X -> B, there exists a unique morphism ⟨f, g⟩ : X -> A × B such that p ∘ ⟨f, g⟩ = f and q ∘ ⟨f, g⟩ = g. The product is thus the universal way to map into A and B simultaneously.

Products need not exist in every category, but when they do they are unique up to unique isomorphism. The construction recovers familiar examples: the product in Set is the Cartesian product, in Grp the direct product of groups, in Top the product space with the product topology. A product is precisely a limit over a two-object diagram with no arrows, which is why it generalizes to arbitrary products of families and beyond.