Crossed module
1977281
205093757
2008-04-12T08:49:16Z
RonnieBrown
417456
/* Examples */
In [[mathematics]], and especially in [[homotopy theory]], a '''crossed module''' consists of [[group (mathematics)|group]]s <i>G</i> and <i>H</i>, where <i>G</i> [[group action|acts]] on <i>H</i> (which we will write on the left), and a [[homomorphism]] of groups
: <math> d\colon H \longrightarrow G, \! </math>
that is [[equivariant]] with respect to the [[inner automorphism|conjugation]] action of <i>G</i> on itself:
: <math> d(gh) = gd(h)g^{-1} \! </math>
and also satisfies the so-called [[Peiffer identity]]:
: <math> d(h_{1})h_{2} = h_{1}h_{2}h_{1}^{-1} \! </math>
== Examples ==
Let <i>N</i> be a [[normal subgroup|normal]] [[subgroup]] of a group <i>G</i>. Then, the inclusion
: <math> d\colon N \longrightarrow G \! </math>
is a crossed module with the conjugation action of <i>G</i> on <i>N</i>.
For any group <i>G</i>, [[module (mathematics)|module]]s over the [[group ring]] are crossed <i>G</i>-modules with <i>d</i> = 0.
For any group <i>H</i>, the homomorphism from <i>H</i> to Aut(<i>H</i>) sending any element of <i>H</i> to the corresponding [[inner automorphism]] can be given the structure of 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
: <math> 1 \to A \to H \to G \to 1 \! </math>
the onto homomorphism
: <math> d\colon H \to G \! </math>
together with the action of <i>G</i> on <i>H</i> 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 (<i>X</i>,<i>A</i>,<i>x</i>) is a pointed pair of [[topological spaces]], then the homotopy boundary
: <math> d\colon \pi_{2}(X,A,x) \rightarrow \pi_{1}(A,x) \! </math>
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
:<math> \Pi \colon (\text{pairs of pointed spaces}) \rightarrow (\text{crossed modules}) </math>
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
: <math> F \rightarrow E \rightarrow B \! </math>
is a pointed [[fibration]] of spaces, then the induced map of fundamental groups
: <math> d\colon \pi_{1}(F) \rightarrow \pi_{1}(E) \! </math>
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 <i>n</i>-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
: <math> M= (d\colon H \longrightarrow G) \! </math>
has a <i> classifying space BM </i> 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 <i>BM</i>. This allows one to prove that (pointed, weak) homotopy 2-types are completely described by crossed modules.
== External links ==
* J. Baez and A. Lauda, [http://arxiv.org/abs/math.QA/0307200 Higher-dimensional algebra V: 2-groups]
* R. Brown, [http://intlpress.com/HHA/v1/n1/a1/ Groupoids and crossed objects in algebraic topology]
* R. Brown, [http://www.bangor.ac.uk/r.brown/hdaweb2.htm Higher dimensional group theory]
* M. Forrester-Barker, [http://arxiv.org/abs/math.CT/02212065 Group objects and internal categories]
* Behrang Noohi, [http://arxiv.org/pdf/math.CT/0512106 Notes on 2-groupoids, 2-groups and crossed-modules]
[[Category:Group actions]]
[[Category:Algebraic topology]]