Adjunction (category theory)
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
- Steve Awodey: Category Theory (= Oxford Logic Guides . Volume 49 ). Clarendon Press, Oxford 2006, ISBN 978-0-19-856861-2 (9th chapter).
- Martin Brandenburg: Introduction to Category Theory . With detailed explanations and numerous examples. Springer Spektrum, Berlin 2015, ISBN 978-3-662-47067-1 , doi : 10.1007 / 978-3-662-47068-8 (7th chapter).
- Saunders Mac Lane : Categories for the Working Mathematician ( Graduate Texts in Mathematics . Volume 5 ). 2nd Edition. Springer, New York 1998, ISBN 0-387-90035-7 (Chapter IV).
- Bodo Pareigis: Categories and Functors . Teubner, Stuttgart 1969, ISBN 978-3-663-12190-9 , doi : 10.1007 / 978-3-663-12190-9 .
- 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 .
Individual evidence
- ↑ DM Kan: Adjoint functors . In: Transaction American Mathematical Society , 1958, Volume 87, pp. 294-329
- ^ PJ Hilton, U. Stammbach: A Course in Homological Algebra . Springer-Verlag, 1970, ISBN 0-387-90032-2 , Chapter II, Section 7: Adjoint Functors
- ^ 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 .