A crossed module consists of groups $G$ and $H$ with maps $\alpha:H\rightarrow G$ and $\tau:G\rightarrow \text{Aut}(H)$ satisfying following conditions:
- $\tau(\alpha(h))(h')=hh'h^{-1}$ for all $h,h'\in H$.
- $\alpha(\tau(g)(h))=g\alpha(h)g^{-1}$ for $g\in G, h\in H$.
Questions : Does the word "module" in crossed module has anything to do with the standard notion of an $R$-module for a commutative ring $R$?
In case of $R$-module $M$, we have an action map $R\times M\rightarrow M$ satisfying some conditions. Here we have an action map (??) $G\times H\rightarrow H$ (along with $H\rightarrow G$) satisfying some condtions.
Is this the only relevance between crossed module and the standard notion of $R$-module? Is there anything more to this?
If $\alpha:H\to G$ is a crossed module such that $\alpha(h)=1_G$ for all $h\in H$, then the Peiffer condition implies that $$hh'h^{-1}=\tau(\alpha(h))(h')=h',$$ thus $H$ is abelian, and is thus a $\mathbb{Z}[G]$-module. Thus a module over a group is a particular case of a crossed module.