Adjoint functors

Adjunction is a term from the mathematical branch of category theory. Two functors between categories and hot adjoint if they convey a certain relationship between morphism. This term was introduced by D. M. Kan.

Definition

Two functors between categories and form a pair of adjoint functors if the functors

And

Of the category of sets kit are of course equivalent. ( The natural equivalence is part of the structure " adjoint Funktorpaar ". )

Ie rechtsadjungiert to say to linksadjungiert.

Unit and Koeinheit the adjunction

Is the natural equivalence, so hot the natural transformations

And

Unit or Koeinheit the adjunction.

Koeinheit unit and have the property that the two induced transformation

And

The identity arise. Conversely, one can show that two such natural transformations define an adjunction.

Properties

  • Are and quasi- inverse to each other, as is right and linksadjungiert to.
  • Rechtsadjungierte functors obtained Limites ( ie are left exact), linksadjungierte functors preserve colimits ( they are quite exact).
  • Is rechtsadjungiert to that unit, and the Koeinheit the adjunction, then with a monad.

Examples

  • The functor " free abelian group over a set " is linksadjungiert for forgetful From → Set.
  • The functor " a lot of equipped with the discrete topology " is linksadjungiert for forgetful Top → Set.
  • The functor ' disjoint union with a one-point space " is linksadjungiert for forgetful Top * → Top.
  • The functor "Stone - Čech compactification " is linksadjungiert for forgetful of the category of compact Hausdorff spaces in the category of all topological spaces.
  • The functor " completion " is linksadjungiert for forgetful of the category of complete metric spaces in the category of all metric spaces.
  • The reduced device to attach to the loop space is linksadjungiert; both categories are the dotted topological spaces with the homotopy classes of maps as morphisms dotted.
  • In a cartesian closed category is for each object of the functor to the functor linksadjungiert. The resultant by these functors monad, in which the object image is just the Zustandsmonade with state object.
  • Summarizing functions as special relations on, the result is a forgetful, for with quantities and functions. The functor assigns to rechtsadjungierte amounts of their power set and relations to the function. The component of the unit of the adjunction, is. The component of the Koeinheit the adjunction, is precisely the element limited to relation.
Monad (category theory) Element (mathematics)
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