Category Theory/Universal constructions

An important procedure that is made possible by the use of categorical language is the relation of apparently distinct concepts in various categories. Here an example is given, but it should be pointed out that the technique described is of extremely broad application.

Limits
In the category of sets, the notion of Cartesian product is vital. Its description, however, involves mention of the elements of the sets: the Cartesian product of the sets X and Y consists of the ordered pairs (x, y), X x, y Y. In a definition of the product of two objects in an arbitrary category, this definition cannot be imitated because, in the definition of a category, it is not required that the objects should be sets with elements. Therefore, a property of the Cartesian product X ´ Y must be found that characterizes the Cartesian product and that can be expressed in the language of categories. Such a property must be entirely expressible in terms of the data of a category: in terms of objects, morphisms, and composition of morphisms.

For any given set A and functions f: A → X, g: A → Y, there is a unique function h: A → X ´ Y such that equations involving two projections hold (see 355), in which p1: X ´ Y → X, p2: X ´ Y → Y are the projections. This is the clue to the generalization. For two objects X and Y of the category, the triple (Z; p1, p2) is taken to be a product of X and Y if p1: Z → X and p2: Z → Y are morphisms of, and if the universal property holds that, given any object A of and morphisms f: A → X, g: A → Y, there exists a unique morphism h: A → Z such that certain relations hold (see 355).

It is easy to prove that the triple (Z; p1, p2) is, essentially, uniquely determined by the universal property. This means that if (Z¢; p1¢, p2¢) is also a product, then there exists a unique isomorphism w: Z → Z¢ in such that two equalities hold (see 356). Therefore, it is permissible to speak of the product of X and Y in. Of course, X and Y do not necessarily have a product in. If they do have a product in, in the sense defined above, however, then that product is unique. Now the product may be sought in various other categories. For example, in the category of Abelian groups the product, as defined, characterizes the direct sum; in the category of topological spaces it characterizes the topological product. In any category corresponding to a pre-ordered set, the product characterizes the greatest lower bound. Thus, in particular, in the category corresponding to the natural numbers ordered by divisibility, the product is precisely the GCD. It is possible to prove that the product in any category in which it exists is associative in the following sense. Given three objects X1, X2, and X3, then there is an equivalence (see 357). This is, of course, a completely familiar—and trivial—fact for sets: in considering the Cartesian product of three sets, the way in which the sets are to be associated is not relevant. If a proof has been given that is valid in any category, however, it is immediately possible to deduce, for example, that the greatest common divisor satisfies associativity. That is, if a, b, c are any three natural numbers, then a relation involving the GCD holds (see 358). Here there are two points to be made. First, there was no apparent connection between this statement (358) and the statement that the Cartesian product of sets is associative. Second, by proving the statement in sufficient generality—i.e., at the categorical level—it is possible to obtain many more instances of the theorem than could have been obtained by simply being confined to the original category in which the definition of product was carried out; that is, the category of sets.