Crossed module

In mathematics, specifically in homotopy theory, a crossed module consists of group (mathematics)s Gamma and A, where Gamma group action on A from the left, and a homomorphism of groups

:partial : A longrightarrow Gamma,

that is equivariant with respect to the inner automorphism action of Gamma on itself such that aba^{-1}=(partial a) b.

= Examples =

Let N be a normal subgroup subgroup of a group Gamma. Then, the inclusion

:partial = i : N longrightarrow Gamma is a crossed module with the conjugation of Gamma on N.

For any group Gamma, module (mathematics) over the group ring are crossed Gamma-modules with partial = 0.