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 $ {\mathcal C} $ be a category and $ F: {\mathcal C} \rightarrow \mathop{\rm Set} $ a functor (for definiteness, the covariant case is treated here). Then a universal element of $ F $ is a pair $ ( A, x) $, where $ A $ is an object of $ {\mathcal C} $ and $ x \in F( A) $, such that for every other such pair $ ( B, y) $ there is a unique $ f: A \rightarrow B $ in $ {\mathcal C} $ satisfying $ F( f )( x)= y $. The correspondence between $ y $ and $ f $ defines a natural isomorphism between $ F $ and the functor $ \mathop{\rm Hom} _ {\mathcal C} ( A, -) $; the object $ A $ is said to be a representing object (or representation) for the functor $ F $, and its universal property is the possession of the universal element $ x $.

Examples.

1) In any category $ {\mathcal C} $, the universal property of a (categorical) product $ A \times B $ is the possession of a pair of projections $ ( p: A \times B \rightarrow A, q : A \times B \rightarrow B) $; that is, $ ( A \times B, ( p, q)) $ is a universal element for the (contravariant) functor which sends an object $ C $ to the set of all pairs of morphisms $ ( f: C \rightarrow A, g: C \rightarrow B) $.

2) In the category of modules over a commutative ring $ R $, the universal property of a tensor product $ M \otimes _ {R} N $ is the possession of a bilinear mapping $ M \times N \rightarrow M \otimes _ {R} N $; that is, $ M \otimes _ {R} N $ is a representing object for the covariant functor which sends a module $ P $ to the set of bilinear mappings $ M \times N \rightarrow P $.

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