Adjoint functors

In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, specifically <a href="/facts/Category_theory/6zS3DXPa">category theory</a>, adjunction is a relationship that two <a href="/facts/Functor/l2u3rIBE">functors</a> may exhibit, intuitively corresponding to a weak form of equivalence between two related <a href="/facts/Category_(mathematics)/7WzxUThf">categories</a>. Two functors that stand in this relationship are known as adjoint functors, one being the left adjoint and the other the right adjoint. Pairs of adjoint functors are ubiquitous in mathematics and often arise from constructions of "optimal solutions" to certain problems (i.e., constructions of objects having a certain <a href="/facts/Universal_property/5XJBYMNz">universal property</a>), such as the construction of a <a href="/facts/Free_group/zPo2NIGb">free group on a set</a> in <a href="/facts/Algebra/nMdqczjo">algebra</a>, or the construction of the <a href="/facts/Stone%25E2%2580%2593%25C4%258Cech_compactification/5smYbEYS">Stone–Čech compactification</a> of a <a href="/facts/Topological_space/v2ut7CbK">topological space</a> in <a href="/facts/Topology/2EqW47cU">topology</a>.
By definition, an adjunction between categories 
 
 
 
 
 
 C
 
 
 
 
 {\displaystyle {\mathcal {C}}}
 
 and 
 
 
 
 
 
 D
 
 
 
 
 {\displaystyle {\mathcal {D}}}
 
 is a pair of functors (assumed to be <a href="/facts/Covariant_functor/l2u3rIBE">covariant</a>)

F
        :
        
          
            D
          
        
        →
        
          
            C
          
        
      
    
    {\displaystyle F:{\mathcal {D}}\rightarrow {\mathcal {C}}}
  
  and  
  
    
      
        G
        :
        
          
            C
          
        
        →
        
          
            D
          
        
      
    
    {\displaystyle G:{\mathcal {C}}\rightarrow {\mathcal {D}}}

and, for all objects 
 
 
 
 c
 
 
 {\displaystyle c}
 
 in 
 
 
 
 
 
 C
 
 
 
 
 {\displaystyle {\mathcal {C}}}
 
 and 
 
 
 
 d
 
 
 {\displaystyle d}
 
 in 
 
 
 
 
 
 D
 
 
 
 
 {\displaystyle {\mathcal {D}}}
 
, a <a href="/facts/Bijection/ayC7YE3z">bijection</a> between the respective morphism sets

h
            o
            m
          
          
            
              C
            
          
        
        (
        F
        d
        ,
        c
        )
        ≅
        
          
            h
            o
            m
          
          
            
              D
            
          
        
        (
        d
        ,
        G
        c
        )
      
    
    {\displaystyle \mathrm {hom} _{\mathcal {C}}(Fd,c)\cong \mathrm {hom} _{\mathcal {D}}(d,Gc)}

such that this family of bijections is <a href="/facts/Natural_transformation/xwPumTHt">natural</a> in 
 
 
 
 c
 
 
 {\displaystyle c}
 
 and 
 
 
 
 d
 
 
 {\displaystyle d}
 
. Naturality here means that there are <a href="/facts/Natural_isomorphism/xwPumTHt">natural isomorphisms</a> between the pair of functors 
 
 
 
 
 
 C
 
 
 (
 F
 −
 ,
 c
 )
 :
 
 
 D
 
 
 →
 
 S
 e
 
 t
 
 op
 
 
 
 
 
 {\displaystyle {\mathcal {C}}(F-,c):{\mathcal {D}}\to \mathrm {Set^{\text{op}}} }
 
 and 
 
 
 
 
 
 D
 
 
 (
 −
 ,
 G
 c
 )
 :
 
 
 D
 
 
 →
 
 S
 e
 
 t
 
 op
 
 
 
 
 
 {\displaystyle {\mathcal {D}}(-,Gc):{\mathcal {D}}\to \mathrm {Set^{\text{op}}} }
 
 for a fixed 
 
 
 
 c
 
 
 {\displaystyle c}
 
 in 
 
 
 
 
 
 C
 
 
 
 
 {\displaystyle {\mathcal {C}}}
 
, and also the pair of functors 
 
 
 
 
 
 C
 
 
 (
 F
 d
 ,
 −
 )
 :
 
 
 C
 
 
 →
 
 S
 e
 t
 
 
 
 {\displaystyle {\mathcal {C}}(Fd,-):{\mathcal {C}}\to \mathrm {Set} }
 
 and 
 
 
 
 
 
 D
 
 
 (
 d
 ,
 G
 −
 )
 :
 
 
 C
 
 
 →
 
 S
 e
 t
 
 
 
 {\displaystyle {\mathcal {D}}(d,G-):{\mathcal {C}}\to \mathrm {Set} }
 
 for a fixed 
 
 
 
 d
 
 
 {\displaystyle d}
 
 in 
 
 
 
 
 
 D
 
 
 
 
 {\displaystyle {\mathcal {D}}}
 
.
The functor 
 
 
 
 F
 
 
 {\displaystyle F}
 
 is called a left adjoint functor or left adjoint to 
 
 
 
 G
 
 
 {\displaystyle G}
 
, while 
 
 
 
 G
 
 
 {\displaystyle G}
 
 is called a right adjoint functor or right adjoint to 
 
 
 
 F
 
 
 {\displaystyle F}
 
. We write 
 
 
 
 F
 ⊣
 G
 
 
 {\displaystyle F\dashv G}
 
.
An adjunction between categories 
 
 
 
 
 
 C
 
 
 
 
 {\displaystyle {\mathcal {C}}}
 
 and 
 
 
 
 
 
 D
 
 
 
 
 {\displaystyle {\mathcal {D}}}
 
 is somewhat akin to a "weak form" of an <a href="/facts/Equivalence_of_categories/k6uzqAcy">equivalence</a> between 
 
 
 
 
 
 C
 
 
 
 
 {\displaystyle {\mathcal {C}}}
 
 and 
 
 
 
 
 
 D
 
 
 
 
 {\displaystyle {\mathcal {D}}}
 
, and indeed every equivalence is an adjunction. In many situations, an adjunction can be "upgraded" to an equivalence, by a suitable natural modification of the involved categories and functors.

Adjoint functors open-in-new

Adjoint functors