# Universal property

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In various branches of数学, a useful construction is often viewed as the “most efficient solution” to a certain problem. The definition of auniversal propertyuses the language ofcategory theoryto 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: allfree objectsdirect productdirect sumfree groupfree latticeGrothendieck groupDedekind–MacNeille completionproduct topologyStone–Čech compactificationtensor productinverse limitdirect limitkernelcokernelpullback推出equalizer

## 动力

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 construct 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, thetensor algebraof avector spaceis slightly painful to actually construct, but using its universal property makes it much easier to deal with.
• Universal properties define objects uniquely up to a uniqueisomorphism[1]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 categoryCthen one obtains afunctor打开（放）CFurthermore, this functor is aright or left adjointto the functorUused in the definition of the universal property.[2]
• 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

Suppose thatUDCis afunctorfrom a多类别 Dto a categoryC, and letXbe an object ofCConsider the following二重的(opposite) notions:

Aninitial morphismXUis aninitial objectin the多类别${\displaystyle (X\downarrow U)}$of morphisms fromXUIn other words, it consists of a pair (Φ) whereis an object ofDΦXU() is a morphism inC, such that the followinginitial propertyis satisfied:

• WheneverYis an object ofDfXU(Y) is a morphism inC, then there exists a独特morphismgYsuch that the following diagramcommutes

terminal morphismUXis aterminal objectin thecomma category ${\displaystyle (U\downarrow X)}$of morphisms fromUXIn other words, it consists of a pair (Φ) whereis an object ofDΦU() →Xis a morphism inC, such that the followingterminal propertyis satisfied:

• Whenever Y is an object of D and f : U ( Y ) → X is a morphism in C , then there exists a unique morphism g : Y → A such that the following diagram commutes:

The termuniversal morphism[3]refers either to an initial morphism or a terminal morphism, and the termuniversal propertyrefers either to an initial property or a terminal property. In each definition, the existence of the morphismgintuitively expresses the fact that (Φ) is "general enough", while the uniqueness of the morphism ensures that (Φ) is "not too general".

## Duality

Since the notions of initial and terminal are dual, it is often enough to discuss only one of them, and simply reverse arrows in C for the dual discussion.Alternatively, the word universal is often used in place of both words.

Note: some authors may call only one of these constructions auniversal morphismand the other one aco-universal morphismWhich is which depends on the author, although in order to be consistent with the naming oflimits and colimitsthe latter construction should be named universal and the former couniversal. This article uses the unambiguous terminology of initial and terminal objects.

## 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

LetCbe thecategory of vector spaces K -Vectover a领域 Kand letDbe the category ofalgebras K -Alg结束K(assumed to be单作的associative). Let

U :K -AlgK -Vect

be theforgetful functorwhich assigns to each algebra its underlying vector space.

Given anyvector space V结束Kwe can construct thetensor algebra T(V) ofVThe tensor algebra is characterized by the fact:

“Any linear map fromVto an algebracan be uniquely extended to analgebra homomorphismT(V) to.”

This statement is an initial property of the tensor algebra since it expresses the fact that the pair ( T ( V ), i ), where i  : V → U ( T ( V )) is the inclusion map, is an initial morphism from the vector space V to the functor U .

Since this construction works for any vector spaceV, we conclude thatTis a functor fromK -VectK -AlgThis means thatTleft adjointto the forgetful functorU(see the section below onrelation to adjoint functors).

### 产品

categorical productcan be characterized by a terminal property. For concreteness, one may consider theCartesian product进入配置, thedirect product进入Grp, or theproduct topology进入上衣, where products exist.

Let X and Y be objects of a category D .The product of X and Y is an object X × Y together with two morphisms

π :X×YX
π :X×YY

such that for any other objectZ属于Dand morphismsf :ZXg :ZYthere exists a unique morphismh :ZX×Ysuch thatf= πhg= πh

To understand this characterization as a terminal property we take the categoryCto be the产品类别 D×Dand define thediagonal functor

Δ : D → D × D

by Δ(X) = (XX) and Δ(f :XY) = (ff). Then (X×Y, (π, π)) is a terminal morphism from Δ to the object (XY) ofD×D: If (fg) is any morphism from (ZZ) to (XY), then it must equal a morphism Δ(h :ZX×Y) = (hh) from Δ(Z) = (ZZ) to Δ(X×Y) = (X×YX×Y), followed by (π, π).

### Limits and colimits

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

LetJCbe categories withJ index categoryand letCJbe the corresponding函子范畴这个diagonal functor

Δ :CCJ

is the functor that maps each object N in C to the constant functor Δ( N ): J → C to N (i.e. Δ( N )( X ) = N for each X in J ).

Given a functorF :JC(thought of as an object inCJ), the限制属于F, if it exists, is nothing but a terminal morphism from Δ toFDually, the上极限属于Fis an initial morphism fromFto Δ.

## 性能

### Existence and uniqueness

Defining a quantity does not guarantee its existence. Given a functorUand an objectXas above, there may or may not exist an initial morphism fromXUIf, however, an initial morphism (, φ) does exist then it is essentially unique. Specifically, it is unique高达独特 isomorphism: if (′, φ′) is another such pair, then there exists a unique isomorphismk′ such that φ′ =U(k)φ. This is easily seen by substituting (′, φ′) for (Yf) in the definition of the initial property.

It is the pair ( A , φ) which is essentially unique in this fashion.The object A itself is only unique up to isomorphism.Indeed, if ( A , φ) is an initial morphism and k : A → A ′ is any isomorphism then the pair ( A ′, φ′), where φ′ = U ( k )φ, is also an initial morphism.

### Equivalent formulations

The definition of a universal morphism can be rephrased in a variety of ways.Let U be a functor from D to C , and let X be an object of C .Then the following statements are equivalent:

The dual statements are also equivalent:

• ( A , φ) is a terminal morphism from U to X
• (, φ) is aterminal objectof the comma category (UX)
• (, φ) is a representation of HomC(U—，X)

### Relation to adjoint functors

Suppose (, φ) is an initial morphism fromXUand (, φ) is an initial morphism fromXUBy the initial property, given any morphismhXXthere exists a unique morphismgsuch that the following diagram commutes:

If每一个目标XI属于Cadmits an initial morphism toU, then the assignment${\displaystyle X_{i}\mapsto A_{i}}$${\displaystyle h\mapsto g}$defines a functorVCDThe maps φIthen define anatural transformationfrom 1C(the identity functor onC) toUVThe functors (VU) are then a pair ofadjoint functors, withVleft-adjoint toUUright-adjoint toV

Similar statements apply to the dual situation of terminal morphisms from U .If such morphisms exist for every X in C one obtains a functor V : C → D which is right-adjoint to U (so U is left-adjoint to V ).

Indeed, all pairs of adjoint functors arise from universal constructions in this manner. LetFGbe a pair of adjoint functors with unit η and co-unit ε (see the article onadjoint functorsfor the definitions). Then we have a universal morphism for each object inCD

• For each objectX进入C, (F(X), ηX) is an initial morphism fromXGThat is, for allfXG(Y) there exists a uniquegF(X) →Yfor which the following diagrams commute.
• For each objectY进入D, (G(Y), εY) is a terminal morphism fromFYThat is, for allgF(X) →Ythere exists a uniquefXG(Y) for which the following diagrams commute.

Universal constructions are more general than adjoint functor pairs: a universal construction is like an optimization problem; it gives rise to an adjoint pair if and only if this problem has a solution for every object of C (equivalently, every object of D ).

## 历史

Universal properties of various topological constructions were presented byPierre Samuelin 1948. They were later used extensively by布尔巴基The closely related concept of adjoint functors was introduced independently by丹尼尔·阚in 1958.

## 笔记

1. ^ Jacobson (2009), Proposition 1.6, p. 44.
2. ^ See for example, Polcino & Sehgal (2002), p. 133. exercise 1, about the universal property ofgroup rings
3. ^ It is also called a universal arrow; for example in (Mac Lane 1998, Ch. III, § 1.)

## 推荐信

• 鲍尔·科恩Universal Algebra(1981), D.Reidel Publishing, Holland.ISBN 90-277-1213-1
• Mac Lane, Saunders数学工作者的范畴2nd ed. (1998), Graduate Texts in Mathematics 5. Springer.ISBN 0-387-98403-8
• Borceux, F.Handbook of Categorical Algebra: vol 1 Basic category theory(1994) Cambridge University Press, (Encyclopedia of Mathematics and its Applications)ISBN 0-521-44178-1
• N. Bourbaki,Livre II : Algèbre(1970), Hermann,ISBN 0-201-00639-1
• Milies, César Polcino; Sehgal, Sudarshan K..An introduction to group ringsAlgebras and applications, Volume 1. Springer, 2002.ISBN 978-1-4020-0238-0
• Jacobson. Basic Algebra II. 多弗2009.ISBN 0-486-47187-X