Crossed module

Definition

Suppose $G$ and $H$ are groups. A crossed module structure for $H$ over $G$ (i.e., with $H$ being the crossed module over $G$) is the following data:

• A homomorphism of groups $\mu :H\to G$
• A group action of $G$ on $H$, i.e., a homomorphism of groups $\alpha :G\to \operatorname {Aut} (H)$

satisfying the following conditions:

1. The map $\alpha$ pushes forward via $\mu$ to the conjugation action of $G$ on itself: $\mu (\alpha (g)(h))=g\mu (h)g^{-1}\ \forall \ g\in G,h\in H$
2. The map $\alpha$ pulls back via $\mu$ to the conjugation action of $H$ on itself: $\alpha (\mu (h_{1}))(h_{2})=h_{1}h_{2}h_{1}^{-1}\ \forall h_{1},h_{2}\in H$

The conditions are more easily stated if we use $\cdot$ to denote all the conjugation actions within a group and to denote the action $\alpha$, so that $\alpha (g)(h)$ becomes $g\cdot h$. In that case, the conditions become:

1. $\mu (g\cdot h)=g\cdot \mu (h)\ \forall \ g\in G,h\in H$
2. $\mu (h_{1})\cdot h_{2}=h_{1}\cdot h_{2}\ \forall \ h_{1},h_{2}\in H$

Related notions

• Crossed module over a Lie ring

Facts

• Crossed module defines a compatible pair of actions

References

• Combinatorial Homotopy II. by J. H. C. Whitehead, Volume 55, Page 453 - 496(Year 1949): ^{Official PDF (ungated)More info}

Other uses

• Van Kampen theorems for diagrams of spaces by Ronald Brown and Jean-Louis Loday, Topology, Volume 26,Number 3, Page 311 - 335(Year 1987): ^{ungated copy (PDF)More info}, Section 1 beginning