Crossed module

From Wikipedia, the free encyclopedia

In mathematics, and especially in homotopy theory, a crossed module consists of groups G and H, where G acts on H (which we will write on the left), and a homomorphism of groups

$d\colon H \longrightarrow G, \!$

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

$d(gh) = gd(h)g^{-1} \!$

and also satisfies the so-called Peiffer identity:

$d(h_{1})h_{2} = h_{1}h_{2}h_{1}^{-1} \!$

[edit] Examples

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

$d\colon N \longrightarrow G \!$

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 can be given the structure of a crossed module.

Given any central extension of groups

$1 \to A \to H \to G \to 1 \!$

the onto homomorphism

$d\colon H \to G \!$

together with the action of G on H defines a crossed module. Thus, central extensions can be seen as special crossed modules.

If (X,A,x) is a pointed pair of topological spaces, then the homotopy boundary

$d\colon \pi_{2}(X,A,x) \rightarrow \pi_{1}(A,x) \!$

from the second relative homotopy group to the fundamental group, may be given the structure of crossed module.

The previous result can also be phrased as: if

$F \rightarrow E \rightarrow B \!$

is a pointed fibration of spaces, then the induced map of fundamental groups

$d\colon \pi_{1}(F) \rightarrow \pi_{1}(E) \!$

may be given the structure of crossed module. This example is useful in algebraic K-theory.

These examples suggest that crossed modules may be thought of a "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 a somewhat disguised version of the result of blending the concepts of "group" and "category".