Crossed module

In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, and especially in <a href="/facts/Homotopy_theory/MJwYyRny">homotopy theory</a>, a crossed module consists of <a href="/facts/Group_(mathematics)/IQTYqINt">groups</a> 
 
 
 
 G
 
 
 {\displaystyle G}
 
 and 
 
 
 
 H
 
 
 {\displaystyle H}
 
, where 
 
 
 
 G
 
 
 {\displaystyle G}
 
 <a href="/facts/Group_action_(mathematics)/Vh9VtUUw">acts</a> on 
 
 
 
 H
 
 
 {\displaystyle H}
 
 by <a href="/facts/Automorphism/WnyxNFkQ">automorphisms</a> (which we will write on the left, 
 
 
 
 (
 g
 ,
 h
 )
 ↦
 g
 ⋅
 h
 
 
 {\displaystyle (g,h)\mapsto g\cdot h}
 
 , and a <a href="/facts/Homomorphism/0gyqdEse">homomorphism</a> of groups

d
        :
        H
        ⟶
        G
        ,
      
    
    {\displaystyle d\colon H\longrightarrow G,}

that is <a href="/facts/Equivariant/KkIkDwIH">equivariant</a> with respect to the <a href="/facts/Inner_automorphism/awJPzhDa">conjugation</a> action of 
 
 
 
 G
 
 
 {\displaystyle G}
 
 on itself:

d
        (
        g
        ⋅
        h
        )
        =
        g
        d
        (
        h
        )
        
          g
          
            −
            1
          
        
      
    
    {\displaystyle d(g\cdot h)=gd(h)g^{-1}}

and also satisfies the so-called Peiffer identity:

d
        (
        
          h
          
            1
          
        
        )
        ⋅
        
          h
          
            2
          
        
        =
        
          h
          
            1
          
        
        
          h
          
            2
          
        
        
          h
          
            1
          
          
            −
            1
          
        
      
    
    {\displaystyle d(h_{1})\cdot h_{2}=h_{1}h_{2}h_{1}^{-1}}

Crossed module open-in-new

Crossed module