Product (Category Theory)¶
In Category Theory, the product is the limit of a diagram inexed by a discrete category \(J\) with only identity morphisms.
The Limit Cone and Universal Property¶
For a diagram \(F: J \to C\) for such a category \(J\), a cone over \(F\) is a \(J\)-indexed family of morphisms \(\lambda_j: c \to Fj\). The limit is denoted \(\prod_{j \in J} Fj\), and the legs of the limit cone are maps \(\pi_k: \prod_{j \in J} Fj \to Fk\) for \(k \in J\), called projection maps.
The universal property is that composition with the product projection defines a natural isomorphism:
Topological Spaces¶
Let \(X\) and \(Y\) be topological spaces. The product \(X \times Y\) has the continuous projection functions \(\pi_X: X \times Y \to X\) and \(\pi_Y: X \times Y \to Y\), satisfying the universal property that for any space \(Z\) with continuous maps \(f: Z \to X\) and \(g: Z \to Y\), there exists a unique continuous function \(h: Z \to X \times Y\) such that composing \(h\) with the projection maps is equivalent to applying \(f\) or \(g\).
The Product Topology¶
Suppose \(Z\) is the single point \(*\), which represents the forgetful functor \(U: \text{Top} \to \text{Set}\). We have the bijection \(\text{Top}(*, X \times Y) \cong \text{Top}(*, X) \times \text{Top}(*, Y)\), implying that the points in \(X \times Y\) are in the Cartesian product of the underlying sets.
If instead we consider \(Z = X \times Y\), then we have \(\text{Top}(X \times Y, X \times Y) \cong \text{Top}(X \times Y, X) \times \text{Top}(X \times Y, Y)\), and so the topology on \(X \times Y\) is the coarsest topology such that the projection functions are continuous. This is indeed the correct notion of a product topology.