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:

  1. $\tau(\alpha(h))(h')=hh'h^{-1}$ for all $h,h'\in H$.
  2. $\alpha(\tau(g)(h))=g\alpha(h)g^{-1}$ 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.

  • 1
    $\begingroup$ I do not know how I missed that "abelian" $H$ condition... Yes, a $G$-module is an abelian group $M$ with a homomorphism of groups $G\rightarrow \text{Aut}(M)$... For any one who needs reference, it is defined in page $186$ of Peter J. Hilton, Urs Stammbach - A Course in Homological Algebra (second edition)... $\endgroup$ – Praphulla Koushik Sep 14 at 2:55

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