Crossed module

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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:

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

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