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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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095710/u0957101.png" /> be a category and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095710/u0957102.png" /> a functor (for definiteness, the covariant case is treated here). Then a universal element of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095710/u0957103.png" /> is a pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095710/u0957104.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095710/u0957105.png" /> is an object of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095710/u0957106.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095710/u0957107.png" />, such that for every other such pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095710/u0957108.png" /> there is a unique <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095710/u0957109.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095710/u09571010.png" /> satisfying <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095710/u09571011.png" />. The correspondence between <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095710/u09571012.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095710/u09571013.png" /> defines a natural isomorphism between <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095710/u09571014.png" /> and the functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095710/u09571015.png" />; the object <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095710/u09571016.png" /> is said to be a representing object (or representation) for the functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095710/u09571017.png" />, and its universal property is the possession of the universal element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095710/u09571018.png" />.
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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) $.
  
1) In any category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095710/u09571019.png" />, the universal property of a (categorical) product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095710/u09571020.png" /> is the possession of a pair of projections <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095710/u09571021.png" />; that is, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095710/u09571022.png" /> is a universal element for the (contravariant) functor which sends an object <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095710/u09571023.png" /> to the set of all pairs of morphisms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095710/u09571024.png" />.
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2) In the category of modules over a commutative ring $  R $,  
 
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the universal property of a tensor product $  M \otimes _ {R} N $
2) In the category of modules over a commutative ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095710/u09571025.png" />, the universal property of a tensor product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095710/u09571026.png" /> is the possession of a bilinear mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095710/u09571027.png" />; that is, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095710/u09571028.png" /> is a representing object for the covariant functor which sends a module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095710/u09571029.png" /> to the set of bilinear mappings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095710/u09571030.png" />.
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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]]].

Latest 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
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