Extranatural transformation

<h2 id="definition">Definition</h2>
<p>Let 
  
    
      
        F
        :
        A
        ×
        
          B
          
            
              o
              p
            
          
        
        ×
        B
        →
        D
      
    
    {\displaystyle F:A\times B^{\mathrm {op} }\times B\rightarrow D}
  
 and 
  
    
      
        G
        :
        A
        ×
        
          C
          
            
              o
              p
            
          
        
        ×
        C
        →
        D
      
    
    {\displaystyle G:A\times C^{\mathrm {op} }\times C\rightarrow D}
  
 be two <a href="/facts/Functor/l2u3rIBE">functors</a> of categories.
A family 
  
    
      
        η
        (
        a
        ,
        b
        ,
        c
        )
        :
        F
        (
        a
        ,
        b
        ,
        b
        )
        →
        G
        (
        a
        ,
        c
        ,
        c
        )
      
    
    {\displaystyle \eta (a,b,c):F(a,b,b)\rightarrow G(a,c,c)}
  
 is said to be natural in <i>a</i> and extranatural in <i>b</i> and <i>c</i> if the following holds:
</p>
<ul><li>
  
    
      
        η
        (
        −
        ,
        b
        ,
        c
        )
      
    
    {\displaystyle \eta (-,b,c)}
  
 is a natural transformation (in the usual sense).</li>
<li>(extranaturality in <i>b</i>) 
  
    
      
        ∀
        (
        g
        :
        b
        →
        
          b
          
            ′
          
        
        )
        ∈
        
          M
          o
          r
        
        
        B
      
    
    {\displaystyle \forall (g:b\rightarrow b^{\prime })\in \mathrm {Mor} \,B}
  
, 
  
    
      
        ∀
        a
        ∈
        A
      
    
    {\displaystyle \forall a\in A}
  
, 
  
    
      
        ∀
        c
        ∈
        C
      
    
    {\displaystyle \forall c\in C}
  
 the following <a href="/facts/Commutative_diagram/bdG8xuer">diagram commutes</a></li></ul>

F
                (
                a
                ,
                
                  b
                  ′
                
                ,
                b
                )
              
              
                
                  →
                  
                    F
                    (
                    1
                    ,
                    1
                    ,
                    g
                    )
                  
                
              
              
                F
                (
                a
                ,
                
                  b
                  ′
                
                ,
                
                  b
                  ′
                
                )
              
            
            
              
                
                  
                  
                    F
                    (
                    1
                    ,
                    g
                    ,
                    1
                    )
                  
                
                ↓
                
              
              
              
                
                  
                  
                    η
                    (
                    a
                    ,
                    
                      b
                      ′
                    
                    ,
                    c
                    )
                  
                
                ↓
                
              
            
            
              
                F
                (
                a
                ,
                b
                ,
                b
                )
              
              
                
                  →
                  
                    η
                    (
                    a
                    ,
                    b
                    ,
                    c
                    )
                  
                
              
              
                G
                (
                a
                ,
                c
                ,
                c
                )
              
            
          
        
      
    
    {\displaystyle {\begin{matrix}F(a,b',b)&\xrightarrow {F(1,1,g)} &F(a,b',b')\\_{F(1,g,1)}\downarrow \qquad &&_{\eta (a,b',c)}\downarrow \qquad \\F(a,b,b)&\xrightarrow {\eta (a,b,c)} &G(a,c,c)\end{matrix}}}

<ul><li>(extranaturality in <i>c</i>) 
  
    
      
        ∀
        (
        h
        :
        c
        →
        
          c
          
            ′
          
        
        )
        ∈
        
          M
          o
          r
        
        
        C
      
    
    {\displaystyle \forall (h:c\rightarrow c^{\prime })\in \mathrm {Mor} \,C}
  
, 
  
    
      
        ∀
        a
        ∈
        A
      
    
    {\displaystyle \forall a\in A}
  
, 
  
    
      
        ∀
        b
        ∈
        B
      
    
    {\displaystyle \forall b\in B}
  
 the following diagram commutes</li></ul>

F
                (
                a
                ,
                b
                ,
                b
                )
              
              
                
                  →
                  
                    η
                    (
                    a
                    ,
                    b
                    ,
                    
                      c
                      ′
                    
                    )
                  
                
              
              
                G
                (
                a
                ,
                
                  c
                  ′
                
                ,
                
                  c
                  ′
                
                )
              
            
            
              
                
                  
                  
                    η
                    (
                    a
                    ,
                    b
                    ,
                    c
                    )
                  
                
                ↓
                
              
              
              
                
                  
                  
                    G
                    (
                    1
                    ,
                    h
                    ,
                    1
                    )
                  
                
                ↓
                
              
            
            
              
                G
                (
                a
                ,
                c
                ,
                c
                )
              
              
                
                  →
                  
                    G
                    (
                    1
                    ,
                    1
                    ,
                    h
                    )
                  
                
              
              
                G
                (
                a
                ,
                c
                ,
                
                  c
                  ′
                
                )
              
            
          
        
      
    
    {\displaystyle {\begin{matrix}F(a,b,b)&\xrightarrow {\eta (a,b,c')} &G(a,c',c')\\_{\eta (a,b,c)}\downarrow \qquad &&_{G(1,h,1)}\downarrow \qquad \\G(a,c,c)&\xrightarrow {G(1,1,h)} &G(a,c,c')\end{matrix}}}

<h2 id="properties">Properties</h2>
<p>Extranatural transformations can be used to define wedges and thereby <a href="/facts/End_(category_theory)/KuqcUxzM">ends</a><a class="footnote-ref" id="fnref:2" href="#fn:2"><sup>2</sup></a> (dually co-wedges and co-ends), by setting 
  
    
      
        F
      
    
    {\displaystyle F}
  
 (dually 
  
    
      
        G
      
    
    {\displaystyle G}
  
) constant.
</p><p>Extranatural transformations can be defined in terms of <a href="/facts/Dinatural_transformation/T3w3T7DI">dinatural transformations</a>, of which they are a special case.<a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a>
</p>
<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Dinatural_transformation/T3w3T7DI">Dinatural transformation</a></li></ul>

<h2 id="external-links">External links</h2>
<ul><li><a href="https://ncatlab.org/nlab/show/extranatural+transformation">extranatural+transformation</a> at the <a href="/facts/NLab/lpPvRmgo"><i>n</i>Lab</a></li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1"><p>Eilenberg and Kelly, A generalization of the functorial calculus, J. Algebra 3 366–375 (1966) <a href="/wiki/Samuel_Eilenberg" target="_blank">/wiki/Samuel_Eilenberg</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li>
<li id="fn:2"><p>Fosco Loregian, This is the (co)end, my only (co)friend, arXiv preprint [1] <a href="https://arxiv.org/abs/1501.02503" target="_blank">https://arxiv.org/abs/1501.02503</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li>
<li id="fn:3"><p>Fosco Loregian, This is the (co)end, my only (co)friend, arXiv preprint [1] <a href="https://arxiv.org/abs/1501.02503" target="_blank">https://arxiv.org/abs/1501.02503</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li>
</ol>

Extranatural transformation open-in-new

Extranatural transformation