crossmodule

A crossed module is an algebraic structure consisting of groups M and P, a group homomorphism μ: M → P, and a left action of P on M by automorphisms, usually denoted p · m. These data must satisfy two compatibility conditions, known as the Peiffer identities.

The Peiffer identities are:

- μ(p · m) = p μ(m) p^{-1} for all p in P and m in M, expressing that μ is compatible with the action.

- μ(m) · n = m n m^{-1} for all m, n in M, expressing that the action of μ(m) on M corresponds to conjugation by m.

Together, these conditions ensure a tight interaction between the homomorphism μ and the P-action on M.

Examples and basic variants:

- A standard example is the inclusion of a normal subgroup N ≤ G, with G acting on N by conjugation. Here μ is the inclusion N ↪ G, and the action is conjugation. The Peiffer identities hold in this setup.

- If P acts trivially on M and μ(M) lies in the center of P, the data form a crossed module as well.

- In many contexts, M and P can be abelian with trivial action, yielding abelian crossed modules that relate to classical module theory.

Morphisms and category:

A morphism of crossed modules (M → P) to (M' → P') consists of group homomorphisms f: M → M' and g: P → P' satisfying μ' ∘ f = g ∘ μ and f(p · m) = g(p) · f(m). This yields a category of crossed modules.

Connections and applications:

Crossed modules model connected homotopy 2-types and are equivalent to strict 2-groups (grouplike 2-categories). They also appear in the study of fundamental crossed modules of topological pairs and in non-abelian cohomology, linking algebraic and homotopical structures.