Crossed Module - Examples

Examples

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.

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. Thus we have a kind of `automorphism structure' of a group, rather than just a group of automorphisms.

Given any central extension of groups

the onto 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. It is a remarkable fact that this functor

satisfies a form of the van Kampen theorem, in that it preserves certain colimits. See the article on crossed objects in algebraic topology below. The proof involves the concept of homotopy double groupoid of a pointed pair of spaces.

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. While this may sound intimidating, it simply 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 in understanding and using even higher dimensional versions of groups.