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.
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。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）C。Furthermore, this functor is aright or left adjointto the functorUused 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.
Suppose thatU：D→Cis afunctorfrom a多类别Dto a categoryC, and letXbe an object ofC。Consider the following二重的(opposite) notions:
Aninitial morphism从X到Uis aninitial objectin the多类别of morphisms fromX到U。In other words, it consists of a pair (一，Φ) where一is an object ofD和Φ：X→U(一) is a morphism inC, such that the followinginitial propertyis satisfied:
WheneverYis an object ofD和f：X→U(Y) is a morphism inC, then there exists a独特morphismg：一→Ysuch that the following diagramcommutes：
一terminal morphism从U到Xis aterminal objectin thecomma categoryof morphisms fromU到X。In other words, it consists of a pair (一，Φ) where一is an object ofD和Φ：U(一) →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 morphismrefers 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 (一，Φ) is "general enough", while the uniqueness of the morphism ensures that (一，Φ) is "not too general".
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 morphism。Which 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.
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 -Vect到K -Alg。This means thatT是left adjointto the forgetful functorU(see the section below onrelation to adjoint functors).
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
such that for any other objectZ属于Dand morphismsf :Z→X和g :Z→Ythere exists a unique morphismh :Z→X×Ysuch thatf= π一∘h和g= π二∘h。
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) = (X，X) and Δ(f :X→Y) = (f，f). Then (X×Y, (π一, π二)) is a terminal morphism from Δ to the object (X，Y) ofD×D: If (f，g) is any morphism from (Z，Z) to (X，Y), then it must equal a morphism Δ(h :Z→X×Y) = (h，h) from Δ(Z) = (Z，Z) to Δ(X×Y) = (X×Y，X×Y), followed by (π一, π二).
Defining a quantity does not guarantee its existence. Given a functorUand an objectXas above, there may or may not exist an initial morphism fromX到U。If, however, an initial morphism (一, φ) does exist then it is essentially unique. Specifically, it is unique高达一独特isomorphism: if (一′, φ′) is another such pair, then there exists a unique isomorphismk：一→一′ such that φ′ =U(k)φ. This is easily seen by substituting (一′, φ′) for (Y，f) 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.
Suppose (一一, φ一) is an initial morphism fromX一到Uand (一二, φ二) is an initial morphism fromX二到U。By the initial property, given any morphismh：X一→X二there exists a unique morphismg：一一→一二such that the following diagram commutes:
If每一个目标XI属于Cadmits an initial morphism toU, then the assignment和defines a functorV从C到D。The maps φIthen define anatural transformationfrom 1C(the identity functor onC) toUV。The functors (V，U) are then a pair ofadjoint functors, withVleft-adjoint toU和Uright-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. LetF和Gbe 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 inC和D：
For each objectX进入C, (F(X), ηX) is an initial morphism fromX到G。That is, for allf：X→G(Y) there exists a uniqueg：F(X) →Yfor which the following diagrams commute.
For each objectY进入D, (G(Y), εY) is a terminal morphism fromF到Y。That is, for allg：F(X) →Ythere exists a uniquef：X→G(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 ).
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.