Crossed module

In mathematics, specifically in homotopy theory, a crossed module consists of groups Gamma and A, where Gamma acts on A from the left, and a homomorphism of groups

Examples

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

:partial = i : N longrightarrow Gamma

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is a crossed module with the conjugation of Gamma on N.

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• For any group $Gamma$, modules over the group ring are crossed $Gamma$-modules with $partial\; =\; 0$.