Crossed module
In mathematics, and especially in homotopy theory, a crossed module consists of groups G {\displaystyle G} and H {\displaystyle H} , where G {\displaystyle G} acts on H {\displaystyle H} by automorphisms (which we will write on the left), ( g , h ) ↦ g ⋅ h {\displaystyle (g,h)\mapsto g\cdot h} , and a homomorphism of groups d : H ⟶ G , {\displaystyle d\colon H\longrightarrow G,} that is equivariant with respect to the conjugation 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}} == Origin == The first mention of the second identity for a crossed module seems to be in footnote 25 on p. 422 of J. H. C. Whitehead's 1941 paper cited below, while the term 'crossed module' is introduced in his 1946 paper cited below.