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 . ${\ displaystyle F: {\ mathcal {C}} \ rightarrow {\ mathcal {D}}}$${\ displaystyle G: {\ mathcal {D}} \ rightarrow {\ mathcal {C}}}$${\ displaystyle {\ mathcal {C}}}$${\ displaystyle {\ mathcal {D}}}$

## definition

Two functors and between categories and form an adjoint functor pair , if the functors ${\ displaystyle F \ colon {\ mathcal {C}} \ rightarrow {\ mathcal {D}}}$${\ displaystyle G \ colon {\ mathcal {D}} \ rightarrow {\ mathcal {C}}}$${\ displaystyle {\ mathcal {C}}}$${\ displaystyle {\ mathcal {D}}}$

${\ displaystyle (X, Y) \ mapsto \ operatorname {Mor} _ {\ mathcal {D}} (X, FY)}$

and

${\ displaystyle (X, Y) \ mapsto \ operatorname {Mor} _ {\ mathcal {C}} (GX, Y)}$

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 .) ${\ displaystyle {\ mathcal {D}} ^ {\ operatorname {op}} \ times {\ mathcal {C}}}$

${\ displaystyle F}$is called right adjoint to , is called left adjoint to . ${\ displaystyle G}$${\ displaystyle G}$${\ displaystyle F}$

## Unity and co-unity of adjunction

If the natural equivalence is , then the natural transformations are called${\ displaystyle t}$${\ displaystyle \ operatorname {Mor} _ {\ mathcal {D}} (\ cdot _ {1}, F (\ cdot _ {2})) \ to \ operatorname {Mor} _ {\ mathcal {C}} ( G (\ cdot _ {1}), \ cdot _ {2})}$

${\ displaystyle \ eta \ colon \ operatorname {id} _ {\ mathcal {D}} \ to FG}$
${\ displaystyle X \ mapsto t _ {(X, GX)} ^ {- 1} (\ operatorname {id} _ {GX})}$

and

${\ displaystyle \ varepsilon \ colon GF \ to \ operatorname {id} _ {\ mathcal {C}}}$
${\ displaystyle Y \ mapsto t _ {(FY, Y)} (\ operatorname {id} _ {FY})}$

Unit or co-unit of adjunction .

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

${\ displaystyle F \ rightarrow FGF \ rightarrow F}$

and

${\ displaystyle G \ rightarrow GFG \ rightarrow G}$

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 .${\ displaystyle F}$${\ displaystyle G}$ ${\ displaystyle F}$${\ displaystyle G}$
• 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 .${\ displaystyle F}$${\ displaystyle G}$${\ displaystyle \ eta \ colon \ mathrm {id} _ {\ mathcal {D}} \ to FG}$${\ displaystyle \ varepsilon \ colon GF \ to \ mathrm {id} _ {\ mathcal {C}}}$${\ displaystyle (FG, \ eta, \ mu)}$${\ displaystyle \ mu _ {X}: = F (\ varepsilon _ {GX})}$${\ displaystyle {\ mathcal {D}}}$

## 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 .${\ displaystyle {\ mathcal {C}}}$${\ displaystyle S}$${\ displaystyle (-) \ times S \ colon {\ mathcal {C}} \ to {\ mathcal {C}}}$${\ displaystyle (-) ^ {S} \ colon {\ mathcal {C}} \ to {\ mathcal {C}}}$${\ displaystyle A \ mapsto (A \ times S) ^ {S}}$${\ displaystyle S}$
• If one understands functions as special relations , a forgetting functor results , with for sets and for functions . The right adjoint functor assigns sets to their power sets and relations to the function . The component of the unit of adjunction,, is . The component of the co-unit of the adjunction,, is precisely the element relation that is restricted to it .${\ displaystyle G \ colon \ mathbf {Set} \ to \ mathbf {Rel}}$${\ displaystyle G (X) = X}$${\ displaystyle X}$${\ displaystyle G (f) = \ {(x, f (x)) \ mid x \ in X \} \ subseteq X \ times Y}$${\ displaystyle f \ colon X \ to Y}$${\ displaystyle G}$${\ displaystyle F \ colon \ mathbf {Rel} \ to \ mathbf {Set}}$${\ displaystyle r \ subseteq X \ times Y}$${\ displaystyle M \ mapsto \ {y \ mid (x, y) \ in r, x \ in M ​​\}}$${\ displaystyle X}$${\ displaystyle \ eta _ {X} \ colon X \ to {\ mathcal {P}} X}$${\ displaystyle x \ mapsto \ {x \}}$${\ displaystyle Y}$${\ displaystyle \ varepsilon _ {Y} \ subseteq {\ mathcal {P}} Y \ times Y}$${\ displaystyle {\ mathcal {P}} Y}$

## 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 .