Crossed module

In mathematics, and especially in homotopy theory, a crossed module consists of groups G and H, where G acts on H by automorphisms (which we will write on the left, ${\displaystyle (g,h)\mapsto g\cdot h}$ , and a homomorphism of groups

that is equivariant with respect to the conjugation action of G on itself:

and also satisfies the so-called Peiffer identity:

The first mention of the second identity for a crossed module seems to be in footnote 25 on p. 422 of Whitehead's 1941 paper cited below, while the term 'crossed module' is introduced in his 1946 paper cited below. These ideas were well worked up in his 1949 paper 'Combinatorial homotopy II', which also introduced the important idea of a free crossed module. Whitehead's ideas on crossed modules and their applications are developed and explained in the book by Brown, Higgins, Sivera listed below. Some generalisations of the idea of crossed module are explained in the paper of Janelidze.

Let N be a normal subgroup of a group G. Then, the inclusion

is a crossed module with the conjugation action of G on N.

For any group G, modules over the group ring are crossed G-modules with d = 0.

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