Universal property

目录

动力 [ 编辑 ]

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, the tensor algebra of a vector space is 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 unique isomorphism 。 ^{[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 category C then one obtains a functor 打开（放） C 。 Furthermore, this functor is a right or left adjoint to the functor U used 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 [ 编辑 ]

Whenever Y is an object of D 和 f ： X → U ( Y ) is a morphism in C , then there exists a 独特 morphism g ： 一 → Y such that the following diagram commutes ：

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:

^{[3]}

Duality [ 编辑 ]

Examples [ 编辑 ]

Tensor algebras [ 编辑 ]

U : K -Alg → K -Vect

“Any linear map from V to an algebra 一 can be uniquely extended to an algebra homomorphism 从 T ( V ) to 一 .”

产品 [ 编辑 ]

π _{一}: X × Y → X π _{二}: X × Y → Y

_{一}_{二}

Δ : D → D × D

_{一}_{二}_{一}_{二}

Limits and colimits [ 编辑 ]

^{J}

Δ : C → C ^{J}

^{J}

性能 [ 编辑 ]

Existence and uniqueness [ 编辑 ]

Equivalent formulations [ 编辑 ]

( A , φ) is an initial morphism from X to U ( 一 , φ) is an initial object of the comma category ( X ↓ U ) ( 一 , φ) is a representation of Hom _{C}( X ， U —)

( A , φ) is a terminal morphism from U to X ( 一 , φ) is a terminal object of the comma category ( U ↓ X ) ( 一 , φ) is a representation of Hom _{C}( U —， X )

Relation to adjoint functors [ 编辑 ]

_{一}_{一}_{一}_{二}_{二}_{二}_{一}_{二}_{一}_{二}

_{I}_{I}_{C}

For each object X 进入 C , ( F ( X ), η _{X}) is an initial morphism from X 到 G 。 That is, for all f ： X → G ( Y ) there exists a unique g ： F ( X ) → Y for which the following diagrams commute. For each object Y 进入 D , ( G ( Y ), ε _{Y}) is a terminal morphism from F 到 Y 。 That is, for all g ： F ( X ) → Y there exists a unique f ： X → G ( Y ) for which the following diagrams commute.

历史 [ 编辑 ]

See also [ 编辑 ]

Free object Natural transformation Adjoint functor Monad (category theory) Variety of algebras Cartesian closed category

笔记 [ 编辑 ]

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

推荐信 [ 编辑 ]

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 rings 。 Algebras and applications, Volume 1. Springer, 2002. ISBN 978-1-4020-0238-0 Jacobson. Basic Algebra II. 多弗 2009. ISBN 0-486-47187-X

External links [ 编辑 ]

nLab , a wiki project on mathematics, physics and philosophy with emphasis on the N -categorical point of view André Joyal ， CatLab , a wiki project dedicated to the exposition of categorical mathematics Hillman, Chris. " A Categorical Primer". CiteSeerX 10.1.1.24.3264 ： Missing or empty |url= ( 帮助 ) formal introduction to category theory. J. Adamek, H. Herrlich, G. Stecker, Abstract and Concrete Categories-The Joy of Cats 斯坦福哲学百科全书 : " Category Theory "—by Jean-Pierre Marquis. Extensive bibliography. List of academic conferences on category theory Baez, John, 1996," The Tale of n -categories. " An informal introduction to higher order categories. WildCats is a category theory package for 数学软件 。 Manipulation and visualization of objects, 态射 , categories, 仿函数 ， 自然变换 ， universal properties 。 The catsters , a YouTube channel about category theory. "Category Theory" 。 PlanetMath 。 Video archive of recorded talks relevant to categories, logic and the foundations of physics. Interactive Web page which generates examples of categorical constructions in the category of finite sets.