Kan extension

<h2 id="definition">Definition</h2>
A Kan extension proceeds from the data of three categories

A
        
        ,
        
          B
        
        ,
        
          C
        
      
    
    {\displaystyle \mathbf {A} ,\mathbf {B} ,\mathbf {C} }

and two <a href="/facts/Functor/l2u3rIBE">functors</a>

X
 :
 
 A
 
 →
 
 C
 
 ,
 F
 :
 
 A
 
 →
 
 B
 
 
 
 {\displaystyle X:\mathbf {A} \to \mathbf {C} ,F:\mathbf {A} \to \mathbf {B} }
 
,
and comes in two varieties: the "left" Kan extension and the "right" Kan extension of 
 
 
 
 X
 
 
 {\displaystyle X}
 
 along 
 
 
 
 F
 
 
 {\displaystyle F}
 
.
Abstractly, the functor 
 
 
 
 F
 
 
 {\displaystyle F}
 
 gives a pullback map 
 
 
 
 
 F
 
 ∗
 
 
 :
 [
 B
 ,
 C
 ]
 →
 [
 A
 ,
 C
 ]
 
 
 {\displaystyle F^{*}:[B,C]\to [A,C]}
 
. When they exist, the left and right adjoints to 
 
 
 
 
 F
 
 ∗
 
 
 
 
 {\displaystyle F^{*}}
 
 applied to 
 
 
 
 X
 
 
 {\displaystyle X}
 
 gives the left and right kan extensions. Spelling the definition of adjoints out, we get the following definitions;
The right Kan extension amounts to finding the dashed arrow and the <a href="/facts/Natural_transformation/xwPumTHt">natural transformation</a> 
 
 
 
 ϵ
 
 
 {\displaystyle \epsilon }
 
 in the following diagram:

Formally, the right Kan extension of 
 
 
 
 X
 
 
 {\displaystyle X}
 
 along 
 
 
 
 F
 
 
 {\displaystyle F}
 
 consists of a functor 
 
 
 
 R
 :
 
 B
 
 →
 
 C
 
 
 
 {\displaystyle R:\mathbf {B} \to \mathbf {C} }
 
 and a natural transformation 
 
 
 
 ϵ
 :
 R
 F
 →
 X
 
 
 {\displaystyle \epsilon :RF\to X}
 
 that is <a href="/facts/Terminal_object/oeckvXRj">terminal</a> with respect to this specification, in the sense that for any functor 
 
 
 
 M
 :
 
 B
 
 →
 
 C
 
 
 
 {\displaystyle M:\mathbf {B} \to \mathbf {C} }
 
 and natural transformation 
 
 
 
 μ
 :
 M
 F
 →
 X
 
 
 {\displaystyle \mu :MF\to X}
 
, a unique natural transformation 
 
 
 
 δ
 :
 M
 →
 R
 
 
 {\displaystyle \delta :M\to R}
 
 is defined and fits into a commutative diagram:

where 
 
 
 
 
 δ
 
 F
 
 
 
 
 {\displaystyle \delta _{F}}
 
 is the natural transformation with 
 
 
 
 
 δ
 
 F
 
 
 (
 a
 )
 =
 δ
 (
 F
 a
 )
 :
 M
 F
 (
 a
 )
 →
 R
 F
 (
 a
 )
 
 
 {\displaystyle \delta _{F}(a)=\delta (Fa):MF(a)\to RF(a)}
 
 for any <a href="/facts/Object_(category_theory)/7WzxUThf">object</a> 
 
 
 
 a
 
 
 {\displaystyle a}
 
 of 
 
 
 
 
 A
 
 .
 
 
 {\displaystyle \mathbf {A} .}

The functor R is often written 
 
 
 
 
 Ran
 
 F
 
 
 ⁡
 X
 
 
 {\displaystyle \operatorname {Ran} _{F}X}
 
.
As with the other <a href="/facts/Universal_property/5XJBYMNz">universal constructs</a> in <a href="/facts/Category_theory/6zS3DXPa">category theory</a>, the "left" version of the Kan extension is <a href="/facts/Dual_(category_theory)/frtUrhBq">dual</a> to the "right" one and is obtained by replacing all categories by their <a href="/facts/Opposite_category/swDy8NrJ">opposites</a>. 
The effect of this on the description above is merely to reverse the direction of the natural transformations. 

(Recall that a <a href="/facts/Natural_transformation/xwPumTHt">natural transformation</a> 
 
 
 
 τ
 
 
 {\displaystyle \tau }
 
 between the functors 
 
 
 
 F
 ,
 G
 :
 
 C
 
 →
 
 D
 
 
 
 {\displaystyle F,G:\mathbf {C} \to \mathbf {D} }
 
 consists of having an <a href="/facts/Morphism/Kdpuyz3S">arrow</a> 
 
 
 
 τ
 (
 a
 )
 :
 F
 (
 a
 )
 →
 G
 (
 a
 )
 
 
 {\displaystyle \tau (a):F(a)\to G(a)}
 
 for every object 
 
 
 
 a
 
 
 {\displaystyle a}
 
 of 
 
 
 
 
 C
 
 
 
 {\displaystyle \mathbf {C} }
 
, satisfying a "naturality" property. When we pass to the opposite categories, the source and target of 
 
 
 
 τ
 (
 a
 )
 
 
 {\displaystyle \tau (a)}
 
 are swapped, causing 
 
 
 
 τ
 
 
 {\displaystyle \tau }
 
 to act in the opposite direction).
This gives rise to the alternate description: the left Kan extension of 
 
 
 
 X
 
 
 {\displaystyle X}
 
 along 
 
 
 
 F
 
 
 {\displaystyle F}
 
 consists of a functor 
 
 
 
 L
 :
 
 B
 
 →
 
 C
 
 
 
 {\displaystyle L:\mathbf {B} \to \mathbf {C} }
 
 and a natural transformation 
 
 
 
 η
 :
 X
 →
 L
 F
 
 
 {\displaystyle \eta :X\to LF}
 
 that is <a href="/facts/Initial_object/oeckvXRj">initial</a> with respect to this specification, in the sense that for any other functor 
 
 
 
 M
 :
 
 B
 
 →
 
 C
 
 
 
 {\displaystyle M:\mathbf {B} \to \mathbf {C} }
 
 and natural transformation 
 
 
 
 α
 :
 X
 →
 M
 F
 
 
 {\displaystyle \alpha :X\to MF}
 
, a unique natural transformation 
 
 
 
 σ
 :
 L
 →
 M
 
 
 {\displaystyle \sigma :L\to M}
 
 exists and fits into a commutative diagram:

 
where 
 
 
 
 
 σ
 
 F
 
 
 
 
 {\displaystyle \sigma _{F}}
 
 is the natural transformation with 
 
 
 
 
 σ
 
 F
 
 
 (
 a
 )
 =
 σ
 (
 F
 a
 )
 :
 L
 F
 (
 a
 )
 →
 M
 F
 (
 a
 )
 
 
 {\displaystyle \sigma _{F}(a)=\sigma (Fa):LF(a)\to MF(a)}
 
 for any object 
 
 
 
 a
 
 
 {\displaystyle a}
 
 of 
 
 
 
 
 A
 
 
 
 {\displaystyle \mathbf {A} }
 
.
The functor L is often written 
 
 
 
 
 Lan
 
 F
 
 
 ⁡
 X
 
 
 {\displaystyle \operatorname {Lan} _{F}X}
 
.
The use of the word "the" (as in "the left Kan extension") is justified by the fact that, as with all universal constructions, if the object defined exists, then it is unique <a href="/facts/Up_to/ydz4ztzX">up to</a> unique <a href="/facts/Isomorphism/pj8KgkxU">isomorphism</a>. In this case, that means that (for left Kan extensions) if 
 
 
 
 L
 ,
 M
 
 
 {\displaystyle L,M}
 
 are two left Kan extensions of 
 
 
 
 X
 
 
 {\displaystyle X}
 
 along 
 
 
 
 F
 
 
 {\displaystyle F}
 
, and 
 
 
 
 η
 ,
 α
 
 
 {\displaystyle \eta ,\alpha }
 
 are the corresponding transformations, then there exists a unique isomorphism of functors 
 
 
 
 σ
 :
 L
 →
 M
 
 
 {\displaystyle \sigma :L\to M}
 
 such that the second diagram above commutes. Likewise for right Kan extensions.

<h2 id="properties">Properties</h2>
<h3>Kan extensions as (co)limits</h3>
Further information: <a href="/facts/Limit_(category_theory)/JoLugLs3">limit (category theory)</a>
Suppose 
 
 
 
 X
 :
 
 A
 
 →
 
 C
 
 
 
 {\displaystyle X:\mathbf {A} \to \mathbf {C} }
 
 and 
 
 
 
 F
 :
 
 A
 
 →
 
 B
 
 
 
 {\displaystyle F:\mathbf {A} \to \mathbf {B} }
 
 are two functors. If A is <a href="/facts/Small_category/7WzxUThf">small</a> and C is cocomplete, then there exists a left Kan extension 
 
 
 
 
 Lan
 
 F
 
 
 ⁡
 X
 
 
 {\displaystyle \operatorname {Lan} _{F}X}
 
 of 
 
 
 
 X
 
 
 {\displaystyle X}
 
 along 
 
 
 
 F
 
 
 {\displaystyle F}
 
, defined at each object b of B by

(
        
          Lan
          
            F
          
        
        ⁡
        X
        )
        (
        b
        )
        =
        
          
            
              lim
              →
            
          
          
            f
            :
            F
            a
            →
            b
          
        
        ⁡
        X
        (
        a
        )
      
    
    {\displaystyle (\operatorname {Lan} _{F}X)(b)=\varinjlim _{f:Fa\to b}X(a)}

where the colimit is taken over the <a href="/facts/Comma_category/wTa3tURp">comma category</a> 
 
 
 
 (
 F
 ↓
 
 const
 
 b
 
 
 )
 
 
 {\displaystyle (F\downarrow \operatorname {const} _{b})}
 
, where 
 
 
 
 
 const
 
 b
 
 
 :
 ∗
 →
 
 B
 
 ,
 ∗
 ↦
 b
 
 
 {\displaystyle \operatorname {const} _{b}\colon \ast \to \mathbf {B} ,\ast \mapsto b}
 
 is the constant functor. Dually, if A is small and C is complete, then right Kan extensions along 
 
 
 
 F
 
 
 {\displaystyle F}
 
 exist, and can be computed as the limit

(
        
          Ran
          
            F
          
        
        ⁡
        X
        )
        (
        b
        )
        =
        
          
            
              lim
              ←
            
          
          
            F
            a
            ←
            b
          
        
        ⁡
        X
        (
        a
        )
      
    
    {\displaystyle (\operatorname {Ran} _{F}X)(b)=\varprojlim _{Fa\leftarrow b}X(a)}

over the comma category 
 
 
 
 (
 
 const
 
 b
 
 
 ↓
 F
 )
 
 
 {\displaystyle (\operatorname {const} _{b}\downarrow F)}
 
.

<h3>Kan extensions as (co)ends</h3>
Further information: <a href="/facts/Coend_(category_theory)/KuqcUxzM">coend (category theory)</a>
Suppose 
 
 
 
 X
 :
 
 A
 
 →
 
 C
 
 
 
 {\displaystyle X:\mathbf {A} \to \mathbf {C} }
 
 and 
 
 
 
 F
 :
 
 A
 
 →
 
 B
 
 
 
 {\displaystyle F:\mathbf {A} \to \mathbf {B} }
 
 are two functors such that for all objects a and a′ of A and all objects b of B, the copowers 
 
 
 
 
 B
 
 (
 F
 
 a
 ′
 
 ,
 b
 )
 ⋅
 X
 a
 
 
 {\displaystyle \mathbf {B} (Fa',b)\cdot Xa}
 
 exist in C. Then the functor X has a left Kan extension 
 
 
 
 
 Lan
 
 F
 
 
 ⁡
 X
 
 
 {\displaystyle \operatorname {Lan} _{F}X}
 
 along F, which is such that, for every object b of B,

(
        
          Lan
          
            F
          
        
        ⁡
        X
        )
        b
        =
        
          ∫
          
            a
          
        
        
          B
        
        (
        F
        a
        ,
        b
        )
        ⋅
        X
        a
      
    
    {\displaystyle (\operatorname {Lan} _{F}X)b=\int ^{a}\mathbf {B} (Fa,b)\cdot Xa}

when the above <a href="/facts/Coend_(category_theory)/KuqcUxzM">coend</a> exists for every object b of B.
Dually, right Kan extensions can be computed by the <a href="/facts/End_(category_theory)/KuqcUxzM">end</a> formula

(
        
          Ran
          
            F
          
        
        ⁡
        X
        )
        b
        =
        
          ∫
          
            a
          
        
        X
        
          a
          
            
              B
            
            (
            b
            ,
            F
            a
            )
          
        
        .
      
    
    {\displaystyle (\operatorname {Ran} _{F}X)b=\int _{a}Xa^{\mathbf {B} (b,Fa)}.}

<h3>Limits as Kan extensions</h3>
The <a href="/facts/Limit_(category_theory)/JoLugLs3">limit</a> of a functor 
 
 
 
 F
 :
 
 C
 
 →
 
 D
 
 
 
 {\displaystyle F:\mathbf {C} \to \mathbf {D} }
 
 can be expressed as a Kan extension by

lim
        F
        =
        
          Ran
          
            E
          
        
        ⁡
        F
      
    
    {\displaystyle \lim F=\operatorname {Ran} _{E}F}

where 
 
 
 
 E
 
 
 {\displaystyle E}
 
 is the unique functor from 
 
 
 
 
 C
 
 
 
 {\displaystyle \mathbf {C} }
 
 to 
 
 
 
 
 1
 
 
 
 {\displaystyle \mathbf {1} }
 
 (the <a href="/facts/Category_(mathematics)/7WzxUThf">category</a> with one object and one arrow, a <a href="/facts/Terminal_object/oeckvXRj">terminal object</a> in 
 
 
 
 
 C
 a
 t
 
 
 
 {\displaystyle \mathbf {Cat} }
 
). The colimit of 
 
 
 
 F
 
 
 {\displaystyle F}
 
 can be expressed similarly by

colim
        ⁡
        F
        =
        
          Lan
          
            E
          
        
        ⁡
        F
        .
      
    
    {\displaystyle \operatorname {colim} F=\operatorname {Lan} _{E}F.}

<h3>Adjoints as Kan extensions</h3>
A functor 
 
 
 
 F
 :
 
 C
 
 →
 
 D
 
 
 
 {\displaystyle F:\mathbf {C} \to \mathbf {D} }
 
 possesses a <a href="/facts/Left_adjoint/bJ1P7AN2">left adjoint</a> <a href="/facts/If_and_only_if/bYSxGJ66">if and only if</a> the right Kan extension of 
 
 
 
 Id
 :
 
 C
 
 →
 
 C
 
 
 
 {\displaystyle \operatorname {Id} :\mathbf {C} \to \mathbf {C} }
 
 along 
 
 
 
 F
 
 
 {\displaystyle F}
 
 exists and is preserved by 
 
 
 
 F
 
 
 {\displaystyle F}
 
. In this case, a left adjoint is given by 
 
 
 
 
 Ran
 
 F
 
 
 ⁡
 Id
 
 
 {\displaystyle \operatorname {Ran} _{F}\operatorname {Id} }
 
 and this Kan extension is even preserved by any functor 
 
 
 
 
 C
 
 →
 
 E
 
 
 
 {\displaystyle \mathbf {C} \to \mathbf {E} }
 
 whatsoever, i.e. is an absolute Kan extension.
Dually, a right adjoint exists if and only if the left Kan extension of the identity along 
 
 
 
 F
 
 
 {\displaystyle F}
 
 exists and is preserved by 
 
 
 
 F
 
 
 {\displaystyle F}
 
.

<h2 id="applications">Applications</h2>
The <a href="/facts/Codensity_monad/F190gfQr">codensity monad</a> of a functor 
 
 
 
 G
 :
 
 D
 
 →
 
 C
 
 
 
 {\displaystyle G:\mathbf {D} \to \mathbf {C} }
 
 is a right Kan extension of G along itself.

<ul><li><a href="/facts/Henri_Cartan/IGrXDS4a">Cartan, Henri</a>; <a href="/facts/Samuel_Eilenberg/vs6aksKX">Eilenberg, Samuel</a> (1956). Homological algebra. Princeton Mathematical Series. Vol. 19. Princeton, New Jersey: <a href="/facts/Princeton_University_Press/kwvFlKEj">Princeton University Press</a>. <a href="/facts/Zbl_(identifier)/P6rFxKKx">Zbl</a> <a href="https://zbmath.org/?format=complete&q=an:0075.24305">0075.24305</a>.</li>
<li><a href="/facts/Saunders_Mac_Lane/rjgUqfkA">Mac Lane, Saunders</a> (1998). <a href="/facts/Categories_for_the_Working_Mathematician/SLQDat9B">Categories for the Working Mathematician</a>. <a href="/facts/Graduate_Texts_in_Mathematics/VTIfY29j">Graduate Texts in Mathematics</a>. Vol. 5 (2nd ed.). New York, NY: <a href="/facts/Springer-Verlag/nAesf6nT">Springer-Verlag</a>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-387-98403-8. <a href="/facts/Zbl_(identifier)/P6rFxKKx">Zbl</a> <a href="https://zbmath.org/?format=complete&q=an:0906.18001">0906.18001</a>.</li></ul>
<h2 id="external-links">External links</h2>
<ul><li><a href="https://mathoverflow.net/q/223722">Model independent proof of colimit formula for left Kan extensions</a></li>
<li><a href="https://ncatlab.org/nlab/show/Kan+extension">Kan extension</a> at the <a href="/facts/NLab/lpPvRmgo">nLab</a></li>
<li><a href="https://math.stackexchange.com/q/2031247">Kan extension as a limit: an example</a></li></ul>

Kan extension open-in-new

Kan extension