# Crossed module

Crossed module

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

## Origin

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.

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

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. Conversely, a crossed module with surjective boundary defines a central extension.

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. It is a remarkable fact that this functor $\Pi \colon (\text{pairs of pointed spaces}) \rightarrow (\text{crossed modules})$

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 $F \rightarrow E \rightarrow B \!$

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. 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.

## Classifying space

Any crossed module $M= (d\colon H \longrightarrow G) \!$

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 conveniently 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.

Wikimedia Foundation. 2010.

### Look at other dictionaries:

• Crossed-field amplifier — A crossed field amplifier (CFA) is a specialized vacuum tube, first introduced in the mid 1950s and frequently used as a microwave amplifier in very high power transmitters. Raytheon engineer William C. Brown s work to adapt magnetron principles… …   Wikipedia

• List of mathematics articles (C) — NOTOC C C closed subgroup C minimal theory C normal subgroup C number C semiring C space C symmetry C* algebra C0 semigroup CA group Cabal (set theory) Cabibbo Kobayashi Maskawa matrix Cabinet projection Cable knot Cabri Geometry Cabtaxi number… …   Wikipedia

• Symmetry — For other uses, see Symmetry (disambiguation) …   Wikipedia

• J. H. C. Whitehead — John Henry Constantine Whitehead Naissance 11 novembre 1904 Chennai (  Inde …   Wikipédia en Français

• J. H. C. Whitehead — Infobox Scientist name = J. H. C. Whitehead image width = 150px caption = John Henry Constantine Whitehead birth date = birth date|1904|11|11|df=y birth place = Madras (Chennai), India death date = death date and age|1960|5|8|1904|11|11|df=y… …   Wikipedia

• Gear — For the gear like device used to drive a roller chain, see Sprocket. This article is about mechanical gears. For other uses, see Gear (disambiguation). Two meshing gears transmitting rotational motion. Note that the smaller gear is rotating… …   Wikipedia

• List of gear nomenclature — Gears have a wide range of unique terminology known as gear nomenclature. Many of the terms defined cite the same reference work. Contents 1 Addendum 2 Addendum angle 3 Addendum circle …   Wikipedia

• Von Neumann algebra — In mathematics, a von Neumann algebra or W* algebra is a * algebra of bounded operators on a Hilbert space that is closed in the weak operator topology and contains the identity operator. They were originally introduced by John von Neumann,… …   Wikipedia

• Physical Sciences — ▪ 2009 Introduction Scientists discovered a new family of superconducting materials and obtained unique images of individual hydrogen atoms and of a multiple exoplanet system. Europe completed the Large Hadron Collider, and China and India took… …   Universalium

• Group cohomology — This article is about homology and cohomology of a group. For homology or cohomology groups of a space or other object, see Homology (mathematics). In abstract algebra, homological algebra, algebraic topology and algebraic number theory, as well… …   Wikipedia