# 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. $\mu(h_1) \cdot h_2 = h_1 \cdot h_2 \ \forall \ h_1,h_2 \in H$