Difference between revisions of "Universal property"
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| − | A property of an object in a [[Category|category]] which characterizes it as a representing object for some (covariant or contravariant) set-valued [[Functor|functor]] defined on the category. More formally, let | + | <!-- |
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| + | A property of an object in a [[Category|category]] which characterizes it as a representing object for some (covariant or contravariant) set-valued [[Functor|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.=== | ===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 $ | |
| − | 2) In the category of modules over a commutative ring | + | 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 [[#References|[a1]]]. | 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 [[#References|[a1]]]. | ||
Revision as of 08:27, 6 June 2020
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 |
Universal property. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Universal_property&oldid=49093