Definitions for universal property
uni·ver·sal prop·er·ty

WiktionaryRate this definition:0.0 / 0 votes

1. universal propertynoun

   A definition of a mathematical object, up to isomorphism, in terms of abstract maps between it and other objects of the same category. More specifically, it is either the initial property of an initial object in a coslice-like comma category or the terminal property of a terminal object in a slice-like comma category; these are roughly analogous to the minimum and the maximum (respectively) of a certain lattice, and are used to define the object uniquely up to isomorphism.

WikipediaRate this definition:0.0 / 0 votes

1. Universal property

   In mathematics, more specifically in category theory, a universal property is a property that characterizes up to an isomorphism 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 integers from the natural numbers, of the rational numbers from the integers, of the real numbers from the rational numbers, and of polynomial rings from the field 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 constructions of real numbers are equivalent: it suffices to prove that they satisfy the same universal property. Technically, a universal property is defined in terms of categories and functors by mean of a universal morphism (see § Formal definition, below). Universal morphisms can also be thought more abstractly as initial or terminal objects of a comma category (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 commutative ring R, the field of fractions of the quotient ring of R by a prime ideal p can be identified with the residue field of the localization 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 free objects, direct products and direct sums, free groups, free lattices, Grothendieck group, completion of a metric space, completion of a ring, Dedekind–MacNeille completion, product topologies, Stone–Čech compactification, tensor products, inverse limit and direct limit, kernels and cokernels, quotient groups, quotient vector spaces, and other quotient spaces.

WikidataRate this definition:0.0 / 0 votes

1. Universal property

   In various branches of mathematics, a useful construction is often viewed as the “most efficient solution” to a certain problem. The definition of a universal property uses the language of category theory to make this notion precise and to study it abstractly. This article gives a general treatment of universal properties. To understand the concept, it is useful to study several examples first, of which there are many: all free objects, direct product and direct sum, free group, free lattice, Grothendieck group, product topology, Stone–Čech compactification, tensor product, inverse limit and direct limit, kernel and cokernel, pullback, pushout and equalizer.

How to pronounce universal property?

1. Alex

   US English

   David

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   Mark

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   Daniel

   British

   Libby

   British

   Mia

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   Australian

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   South African

How to say universal property in sign language?

Numerology

1. Chaldean Numerology

   The numerical value of universal property in Chaldean Numerology is: 6

2. Pythagorean Numerology

   The numerical value of universal property in Pythagorean Numerology is: 2

References

1. ^ Wiktionary
   https://en.wiktionary.org/wiki/Universal_Property
2. ^ Wikipedia
   https://en.wikipedia.org/wiki/Universal_Property
3. ^ Wikidata
   https://www.wikidata.org/w/index.php?search=universal property