Universal property

A universal property is a method of mathematics, and there in particular abstract algebra, to gain a desired structure without specifying a concrete construction. Here is defined for objects of a certain category, eg the category of abstract algebra, a property, for example, that there is an injective mapping in the algebra of a vector space V.

The universal design now is to assert the existence of a "smallest" element U of the category which satisfies the property. Being in the example this would be the tensor algebra TV by V. " Smallest " means that there is for each object W the category that satisfies the required property, a uniquely determined morphism f: U → W is, which is compatible with the property in the example commutes with the embedding of V.

The "smallest" element need not be uniquely determined, but all the "smallest" elements, where they exist, are isomorphic. As a proof of existence a concrete construction is usually specified, but the details of construction of the theory of the structure are mostly insignificant.

Examples

  • The tensor algebra, see above.
  • The core of a linear map.
  • The linear span of a subset of a vector space as the smallest subspace that contains this amount.
  • The affine hull of a subset of an affine space.
  • The convex hull of a subset of an affine space.
  • The topological degree of a subset of a topological space.
  • The interior of a subset of a topological space as the largest open set contained in the subset.

Motivation

What are universal properties well? Once one has recognized that a certain construction satisfies a universal property, so one gains from this

  • Universal properties define objects up to unique isomorphisms; to show that two objects satisfy the same universal property, hence is a possible strategy to show their isomorphism.
  • The exact details of the given structure may be complex and highly technical in nature, but thanks to the universal property, you can forget all these details: Everything you need to know about the construct is already included in the universal property. If you use the universal property instead of the specific details, this makes a proof usually short and elegant.
  • If the universal design can be performed for each object in a category, we obtain a functor in the target category.
  • This functor is on top of that right or linksadjungiert at a given functor. But such functors exchange principle with colimits or Limites. In this way, for example, follows immediately that the core of the direct product of two linear maps equal to the product of the nuclei ( canonically isomorphic ).

Formal definition

Be a functor from the category in the category and was an object of. A universal morphism from to consists of a pair, one object and one morphism is in, so that the following universal property is satisfied:

  • For each object and each morphism, there is exactly one morphism, so that is true, that is, such that the following diagram commutes:

Intuitively, the existence of that is " general enough ", while the uniqueness ensures that not " too general " is. You can reverse this definition also all arrows, that is, consider the category theoretical dual. A universal morphism from to is a pair, one object and one morphism is in, so that the following universal property is satisfied:

  • For each object and each morphism, there is exactly one morphism, so that is true, that is, such that the following diagram commutes:

Properties

Existence and uniqueness

The mere definition still does not guarantee existence. For a functor and an object as above, a universal morphism from to exist or not. If, however, a universal morphism exists, it is unique up to unique isomorphism. So is another such pair, then there's a unique isomorphism. This is easily seen by applying the definition of the universal property to.

Equivalent formulations

Defining a universal morphism be formulated in various ways. With a functor and an object are the following statements are equivalent:

  • Is a universal morphism from to
  • Is an initial object in the comma category
  • Is an illustration of the functor

Accordingly, the dual statements are equivalent:

  • Is a universal morphism from to
  • Is a final object of the comma category
  • Is an illustration of the functor

Relationship to adjoint functors

Be a universal morphism from to and from to. Due to the universal property is available for every morphism exactly one morphism.

Is there even any object of the category a universal morphism after so defines the mapping, a functor. The morphisms form by a natural transformation of ( the Identitätsfunktor on ). Then a pair of adjoint functors, and that is the left - adjoint to and right - adjoint to.

The same applies, mutatis mutandis, in the dual case.

History

Universal properties were introduced in connection with various topological structures in 1948 by Pierre Samuel. Later Nicolas Bourbaki used them extensively. The closely related concept of Adjungiertheit of functors introduced independently of this 1958 Daniel Kan.

Algebra over a field Tensoralgebra Initial and terminal objects Free group Monad (category theory)
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