In mathematics, and especially in homotopy theory, a crossed module consists of groups
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 J. H. C. Whitehead's 1941 paper cited below, while the term 'crossed module' is introduced in his 1946 paper cited below.
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.
For any group H, the homomorphism from H to Aut(H) sending any element of H to the corresponding inner automorphism is a crossed module.
Given any central extension of groups the surjective homomorphism together with the action of
Conversely, a crossed module with surjective boundary defines a central extension.
The functor satisfies a form of the van Kampen theorem, in that it preserves certain colimits.
The result on the crossed module of a pair can also be phrased as: if is a pointed fibration of spaces, then the induced map of fundamental groups may be given the structure of crossed module.
These examples suggest that crossed modules may be thought of as "2-dimensional groups".
This allows one to prove that (pointed, weak) homotopy 2-types are completely described by crossed modules.