Product (category theory)

In category theory, the product of two (or more) objects in a category is a notion designed to capture the essence behind constructions in other areas of mathematics such as the cartesian product of sets, the direct product of groups, the direct product of rings and the product of topological spaces. Essentially, the product of a family of objects is the "most general" object which admits a morphism to each of the given objects.

Let $C$ be a category with some objects $X 1$ and $X 2$ . A product of $X 1$ and $X 2$ is an object $X$ (often denoted $X 1 \times X 2$ ) together with a pair of morphisms $π 1 : X \to X 1$ , $π 2 : X \to X 2$ that satisfy the following universal property:

The unique morphism $f$ is called the product of morphisms $f 1$ and $f 2$ and is denoted $< f 1, f 2 >$ . The morphisms $π 1$ and $π 2$ are called the canonical projections or projection morphisms.

Above we defined the binary product. Instead of two objects we can take an arbitrary family of objects indexed by some set $I$ . Then we obtain the definition of a product.

An object $X$ is the product of a family $(X i) i \in I$ of objects iff there exist morphisms $π i : X \to X i$ , such that for every object $Y$ and a $I$ -indexed family of morphisms $f i : Y \to X i$ there exists a unique morphism $f : Y \to X$ such that the following diagrams commute for all $i \in I$ :

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