View source for Universal property

<!--
u0957101.png
$#A+1 = 30 n = 0
$#C+1 = 30 : ~/encyclopedia/old_files/data/U095/U.0905710 Universal property
Automatically converted into TeX, above some diagnostics.
Please remove this comment and the {{TEX|auto}} line below,
if TeX found to be correct.
-->

{{TEX|auto}}
{{TEX|done}}

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.===

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 [[#References|[a1]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  S. MacLane,   "Duality for groups"  ''Bull. Amer. Math. Soc.'' , '''56'''  (1950)  pp. 485–516</TD></TR></table>