↑ arvau

Wikipedia / 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 means of a universal morphism (see , below). Universal morphisms can also be thought more abstractly as initial or terminal objects of a comma category (see , 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 , the field of fractions of the quotient ring of by a prime ideal can be identified with the residue field of the localization of at (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, kernelss and cokernels, quotient groups, quotient vector spaces, and other quotient spacess.

Motivation

Before giving a formal definition of universal properties, we offer some motivation for studying such constructions.

  • The concrete details of a given construction may be messy, but if the construction satisfies a universal property, one can forget all those details: all there is to know about the construction is already contained in the universal property. Proofs often become short and elegant if the universal property is used rather than the concrete details. For example, the tensor algebra of a vector space is slightly complicated to construct, but much easier to deal with by its universal property.

  • Universal properties define objects uniquely up to a unique isomorphism. Therefore, one strategy to prove that two objects are isomorphic is to show that they satisfy the same universal property.

  • Universal constructions are functorial in nature: if one can carry out the construction for every object in a category C then one obtains a functor on C. Furthermore, this functor is a right or left adjoint to the functor U used in the definition of the universal property.

  • Universal properties occur everywhere in mathematics. By understanding their abstract properties, one obtains information about all these constructions and can avoid repeating the same analysis for each individual instance.

Formal definition

To understand the definition of a universal construction, it is important to look at examples. Universal constructions were not defined out of thin air, but were rather defined after mathematicians began noticing a pattern in many mathematical constructions (see below). Hence, the definition may not make sense at first, but will become clear when one reconciles it with concrete examples.

.

.

A which has the following property, commonly referred to as a universal property:

, there exists a unique such that the following diagram commutes:

We can dualize this categorical concept. A that satisfies the following universal property:

, there exists a unique such that the following diagram commutes:

which behaves as above satisfies a universal property.

Connection with comma categories

Universal morphisms can be described more concisely as initial and terminal objects in a comma category (i.e. one where morphisms are seen as objects in their own right).

is the category where

  • such that the diagram commutes:

such that the following diagram commutes.

Note that the equality here simply means the diagrams are the same. Also note that the diagram on the right side of the equality is the exact same as the one offered in defining a .

is the category where

  • such that the diagram commutes:

such that the following diagrams commute.

The diagram on the right side of the equality is the same diagram pictured when defining a .

Examples

Below are a few examples, to highlight the general idea. The reader can construct numerous other examples by consulting the articles mentioned in the introduction.

Tensor algebras

be the category of vector spaces -Vect over a field be the category of algebras -Alg (assumed to be unital and associative be the forgetful functor which assigns to each algebra its underlying vector space.

Given any vector space we can construct the tensor algebra. The tensor algebra is characterized by the fact:

  1. can be uniquely extended to an algebra homomorphism.”

.

is a functor from -Vect to -Alg is left adjoint (see the section below on relation to adjoint functors).

Products

A categorical product can be characterized by a universal construction. For concreteness, one may consider the Cartesian product in Set, the direct product in Grp, or the product topology in Top, where products exist.

together with two morphisms

.

to be the product category and define the diagonal functor

. As a commutative diagram: For the example of the Cartesian product in Set.

Limits and colimits

Categorical products are a particular kind of limit in category theory. One can generalize the above example to arbitrary limits and colimits.

a small index category be the corresponding functor category. The diagonal functor

.

), the limit. Dually, the colimit.

Properties

Existence and uniqueness

does exist, then it is essentially unique. Specifically, it is unique up to a unique isomorphism in the definition of a universal morphism.

is also a universal morphism.

Equivalent formulations

. Then the following statements are equivalent:

The dual statements are also equivalent:

Relation to adjoint functors

such that the following diagram commutes:

If every then define a natural transformation are then a pair of adjoint functors.

).

(see the article on adjoint functors:

  • for which the following diagrams commute.

  • for which the following diagrams commute.

).

History

Universal properties of various topological constructions were presented by Pierre Samuel in 1948. They were later used extensively by Bourbaki. The closely related concept of adjoint functors was introduced independently by Daniel Kan in 1958.

See also

Notes

References

  • Paul Cohn, Universal Algebra (1981), D.Reidel Publishing, Holland. .

  • Borceux, F. Handbook of Categorical Algebra: vol 1 Basic category theory (1994) Cambridge University Press, (Encyclopedia of Mathematics and its Applications)

  • N. Bourbaki, Livre II : Algèbre (1970), Hermann, .

  • Milies, César Polcino; Sehgal, Sudarshan K.. An introduction to group rings. Algebras and applications, Volume 1. Springer, 2002.

  • Jacobson. Basic Algebra II. Dover. 2009.