# Universal property

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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
How to Cite This Entry:
Universal property. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Universal_property&oldid=17411
This article was adapted from an original article by P.T. Johnstone (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article