# what is “module” in crossed module?

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.

• 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)... – Praphulla Koushik Sep 14 at 2:55