Crossed module

In mathematics, and especially in homotopy theory, a crossed module consists of groups ${\displaystyle G}$ and ${\displaystyle H}$, where ${\displaystyle G}$ acts on ${\displaystyle H}$ by automorphisms (which we will write on the left, ${\displaystyle (g,h)\mapsto g\cdot h}$ , and a homomorphism of groups

${\displaystyle d\colon H\longrightarrow G,}$

that is equivariant with respect to the conjugation action of ${\displaystyle G}$ on itself:

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

and also satisfies the so-called Peiffer identity:

${\displaystyle d(h_{1})\cdot h_{2}=h_{1}h_{2}h_{1}^{-1}}$