Universal property

A property of an object in a category which characterizes it as a representing object for some (covariant or contravariant) set-valued functor defined on the category. More formally, let be a category and a functor (for definiteness, the covariant case is treated here). Then a universal element of is a pair , where is an object of and , such that for every other such pair there is a unique in satisfying . The correspondence between and defines a natural isomorphism between and the functor ; the object is said to be a representing object (or representation) for the functor , and its universal property is the possession of the universal element .

Examples.

1) In any category , the universal property of a (categorical) product is the possession of a pair of projections ; that is, is a universal element for the (contravariant) functor which sends an object to the set of all pairs of morphisms .

2) In the category of modules over a commutative ring , the universal property of a tensor product is the possession of a bilinear mapping ; that is, is a representing object for the covariant functor which sends a module to the set of bilinear mappings .

An object possessing a given universal property is unique up to canonical isomorphism in the appropriate category. The idea of characterizing objects by means of universal properties was first exploited by S. MacLane [a1].

References

[a1]	S. MacLane, "Duality for groups" Bull. Amer. Math. Soc. , 56 (1950) pp. 485–516