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: $$\alpha(\mu(h_1))(h_2) = h_1h_2h_1^{-1} \ \forall h_1,h_2 \in H$$

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

Other uses

 * , Section 1 beginning