Universal property

In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, more specifically in <a href="/facts/Category_theory/6zS3DXPa">category theory</a>, a universal property is a property that characterizes <a href="/facts/Up_to/ydz4ztzX">up to</a> an <a href="/facts/Isomorphism/pj8KgkxU">isomorphism</a> the result of some constructions. Thus, universal properties can be used for defining some objects independently from the method chosen for constructing them. For example, the definitions of the <a href="/facts/Integer/PUwwrawx">integers</a> from the <a href="/facts/Natural_number/u65og8JE">natural numbers</a>, of the <a href="/facts/Rational_number/VJQmyY05">rational numbers</a> from the integers, of the <a href="/facts/Real_number/R02gw5Pb">real numbers</a> from the rational numbers, and of <a href="/facts/Polynomial_ring/ua3vk8Ih">polynomial rings</a> from the <a href="/facts/Field_(mathematics)/xAjAS4ko">field</a> of their coefficients can all be done in terms of universal properties. In particular, the concept of universal property allows a simple proof that all <a href="/facts/Constructions_of_real_numbers/rxoyDRav">constructions of real numbers</a> are equivalent: it suffices to prove that they satisfy the same universal property. 
Technically, a universal property is defined in terms of <a href="/facts/Category_(mathematics)/7WzxUThf">categories</a> and <a href="/facts/Functor/l2u3rIBE">functors</a> by means of a universal morphism (see § Formal definition, below). Universal morphisms can also be thought more abstractly as <a href="/facts/Initial_and_terminal_objects/oeckvXRj">initial or terminal objects</a> of a <a href="/facts/Comma_category/wTa3tURp">comma category</a> (see § Connection with comma categories, below). 
Universal properties occur almost everywhere in mathematics, and the use of the concept allows the use of general properties of universal properties for easily proving some properties that would need boring verifications otherwise. For example, given a <a href="/facts/Commutative_ring/QS1r2ovR">commutative ring</a> R, the <a href="/facts/Field_of_fractions/cmlJrDSS">field of fractions</a> of the <a href="/facts/Quotient_ring/X2WkXyEw">quotient ring</a> of R by a <a href="/facts/Prime_ideal/pmrGf6DA">prime ideal</a> p can be identified with the <a href="/facts/Residue_field/e6nflEmB">residue field</a> of the <a href="/facts/Localization_(commutative_algebra)/rjevzBJf">localization</a> of R at p; that is 
 
 
 
 
 R
 
 p
 
 
 
 /
 
 p
 
 R
 
 p
 
 
 ≅
 Frac
 ⁡
 (
 R
 
 /
 
 p
 )
 
 
 {\displaystyle R_{p}/pR_{p}\cong \operatorname {Frac} (R/p)}
 
 (all these constructions can be defined by universal properties).
Other objects that can be defined by universal properties include: all <a href="/facts/Free_object/VxL28y8B">free objects</a>, <a href="/facts/Direct_product/GehGQmNV">direct products</a> and <a href="/facts/Direct_sum/x3ljVsP6">direct sums</a>, <a href="/facts/Free_group/zPo2NIGb">free groups</a>, <a href="/facts/Free_lattice/p0vMu8nP">free lattices</a>, <a href="/facts/Grothendieck_group/HSLHgFvh">Grothendieck group</a>, <a href="/facts/Completion_of_a_metric_space/lsckmU8G">completion of a metric space</a>, <a href="/facts/Completion_of_a_ring/Q0vjA9JI">completion of a ring</a>, <a href="/facts/Dedekind%25E2%2580%2593MacNeille_completion/yCa3eCAL">Dedekind–MacNeille completion</a>, <a href="/facts/Product_topology/0hg02IaX">product topologies</a>, <a href="/facts/Stone%25E2%2580%2593%25C4%258Cech_compactification/5smYbEYS">Stone–Čech compactification</a>, <a href="/facts/Tensor_product/7K3pR1ap">tensor products</a>, <a href="/facts/Inverse_limit/f6QecRUu">inverse limit</a> and <a href="/facts/Direct_limit/V3JuHnYK">direct limit</a>, <a href="/facts/Kernel_(linear_algebra)/adhGUYa0">kernels</a> and <a href="/facts/Cokernel/vYFQRPoh">cokernels</a>, <a href="/facts/Quotient_group/HlwpHtjb">quotient groups</a>, <a href="/facts/Quotient_vector_space/WmQ3lqdI">quotient vector spaces</a>, and other <a href="/facts/Quotient_space_(disambiguation)/xgIQVnIY">quotient spaces</a>.

Universal property open-in-new

Universal property