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.
is a crossed module with the conjugation action of G on N.
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 G on H defines a crossed module. Thus, central extensions can be seen as special crossed modules. Conversely, a crossed module with surjective boundary defines a central extension.
If (X,A,x) is a pointed pair of topological spaces (i.e. A is a subspace of X, and x is a point in A), then the homotopy boundary
from the second relative homotopy group to the fundamental group, may be given the structure of crossed module. 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. This example is useful in algebraic K-theory. There are higher-dimensional versions of this fact using n-cubes of spaces.
These examples suggest that crossed modules may be thought of as "2-dimensional groups". In fact, this idea can be made precise using category theory. It can be shown that a crossed module is essentially the same as a categorical group or 2-group: that is, a group object in the category of categories, or equivalently a category object in the category of groups. This means that the concept of crossed module is one version of the result of blending the concepts of "group" and "category". This equivalence is important for higher-dimensional versions of groups.
Any crossed module
has a classifying space BM with the property that its homotopy groups are Coker d, in dimension 1, Ker d in dimension 2, and 0 above 2. It is possible to describe the homotopy classes of maps from a CW-complex to BM. This allows one to prove that (pointed, weak) homotopy 2-types are completely described by crossed modules.
- J. Baez and A. Lauda, Higher-dimensional algebra V: 2-groups
- R. Brown, Groupoids and crossed objects in algebraic topology
- R. Brown, Higher dimensional group theory
- R. Brown, P.J. Higgins, R. Sivera, Nonabelian algebraic topology: filtered spaces, crossed complexes, cubical homotopy groupoids, EMS Tracts in Mathematics Vol. 15, 703 pages. (August 2011).
- M. Forrester-Barker, Group objects and internal categories
- Behrang Noohi, Notes on 2-groupoids, 2-groups and crossed-modules
- Whitehead, J. H. C., On adding relations to homotopy groups, Ann. of Math. (2) 42 (1941) 409–428.
- Whitehead, J. H. C., Note on a previous paper entitled "On adding relations to homotopy groups", Ann. of Math. (2) 47 (1946) 806–810.
- Whitehead, J. H. C., Combinatorial homotopy. II, Bull. Amer. Math. Soc. 55 (1949) 453–496.
- Janelidze, G. Internal crossed modules. Georgian Math. J. 10 (2003), no. 1, 99–114.