Adjunction (category theory)

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Adjunction is a term from the mathematical branch of category theory . Two functors and between categories and are called adjoint if they convey a certain relationship between sets of morphisms . This term was introduced by DM Kan .

definition

Two functors and between categories and form an adjoint functor pair , if the functors

and

from being naturally equivalent to the category of sets set . (Together with the two categories and the two functors, the natural equivalence forms an adjunction .)

is called right adjoint to , is called left adjoint to .

Unity and co-unity of adjunction

If the natural equivalence is , then the natural transformations are called

and

Unit or co-unit of adjunction .

Unit and co-unit have the property that the two induced transformations

and

the identity revealed. Conversely, one can show that two such natural transformations determine an adjunction.

properties

  • If and are quasi-inverse to one another, then right and left adjoint to .
  • Right adjoint functors get limits (they are left exact ), left adjoint functors get colimites (they are right exact ).
  • If is right adjoint to , the unity, and the counity of the adjunction, then with is a monad in .

Examples

  • The functor “ free Abelian group over a set” is left adjoint to the forgetful functor Ab  →  Set .
  • The functor "equip a set with the discrete topology" is left adjoint to the forgetting functor Top  →  Set .
  • The functor "equip a set with the trivial topology" is right adjoint to the forgetting functor Top  →  Set .
  • The functor “disjoint union with a one-point space” is left adjoint to the forgetting functor Top *  →  Top .
  • The functor " Stone-Čech compactification " is left adjoint to the forget functor from the category of compact Hausdorff spaces to the category of all topological spaces .
  • The functor “ completion ” is left adjoint to the forget functor from the category of complete metric spaces to the category of all metric spaces .
  • The reduced hanging is left adjoint to the loop space ; Both categories are the dotted topological spaces with the homotopy classes of dotted mappings as morphisms.
  • In a Cartesian closed category , the functor is left adjoint to the functor for each object . The monad resulting from these functors, for which the object mapping is, is precisely the state monad with state object .
  • If one understands functions as special relations , a forgetting functor results , with for sets and for functions . The right adjoint functor assigns sets to their power sets and relations to the function . The component of the unit of adjunction,, is . The component of the co-unit of the adjunction,, is precisely the element relation that is restricted to it .

literature

Individual evidence

  1. DM Kan: Adjoint functors . In: Transaction American Mathematical Society , 1958, Volume 87, pp. 294-329
  2. ^ PJ Hilton, U. Stammbach: A Course in Homological Algebra . Springer-Verlag, 1970, ISBN 0-387-90032-2 , Chapter II, Section 7: Adjoint Functors
  3. ^ H. Schubert: Categories II (=  Heidelberger Taschenbuch . Volume 66 ). Springer, Berlin 1970, ISBN 978-3-540-04866-4 , doi : 10.1007 / 978-3-642-95156-5 .