View source for Crossed module

A [[Morphism|morphism]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120290/c1202901.png" /> of groups together with an action of the [[Group|group]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120290/c1202902.png" /> on the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120290/c1202903.png" /> satisfying two conditions which makes the structure a common generalization of:

1) a [[Normal subgroup|normal subgroup]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120290/c1202904.png" />;

2) a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120290/c1202905.png" />-module (cf. also [[Module|Module]]);

3) the inner [[Automorphism|automorphism]] mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120290/c1202906.png" />;

4) an [[Epimorphism|epimorphism]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120290/c1202907.png" /> with central kernel. The two conditions are:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120290/c1202908.png" /></td> </tr></table>

for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120290/c1202909.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120290/c12029010.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120290/c12029011.png" /> is the action of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120290/c12029012.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120290/c12029013.png" />.

The notion is due to J.H.C. Whitehead [[#References|[a9]]] and plays an important role in the theory of algebraic models of homotopy types of spaces.

In topology the structure arises from the second relative [[Homotopy group|homotopy group]] and its boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120290/c12029014.png" /> [[#References|[a5]]], or as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120290/c12029015.png" /> for a [[Fibration|fibration]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120290/c12029016.png" />. The second example has been used in relative [[Algebraic K-theory|algebraic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120290/c12029017.png" />-theory]], to give a crossed module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120290/c12029018.png" /> for an [[Ideal|ideal]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120290/c12029019.png" /> in a [[Ring|ring]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120290/c12029020.png" />.

From the first of these examples, crossed modules can be seen as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120290/c12029022.png" />-dimensional versions of groups, with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120290/c12029023.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120290/c12029024.png" /> as, respectively, the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120290/c12029025.png" />- and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120290/c12029026.png" />-dimensional parts. This impression is confirmed by the existence of a functorial classifying space functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120290/c12029027.png" /> of such a crossed module whose homotopy groups are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120290/c12029028.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120290/c12029029.png" /> in dimensions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120290/c12029030.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120290/c12029031.png" /> and are otherwise <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120290/c12029032.png" />. For any connected pointed [[CW-complex|CW-complex]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120290/c12029033.png" /> there are a crossed module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120290/c12029034.png" /> and a mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120290/c12029035.png" /> inducing an isomorphism in homotopy in dimensions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120290/c12029036.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120290/c12029037.png" />. This results from a homotopy classification of mappings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120290/c12029038.png" />. Thus, crossed modules capture all homotopy <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120290/c12029040.png" />-types. These results are special cases of results on crossed complexes (cf. also [[Crossed complex|Crossed complex]]).

It was proved in [[#References|[a2]]] that the homotopy crossed module functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120290/c12029041.png" /> from pointed pairs of spaces to crossed modules satisfies a generalized Van Kampen theorem, in that it preserves certain colimits. This allows for the determination of certain second relative homotopy groups as crossed modules, thus giving non-trivial non-Abelian information and often determining the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120290/c12029043.png" />-type of a space. Some of the explicit calculations are conveniently done by computer [[#References|[a8]]].

One consequence of the generalized Van Kampen theorem is a result of J.H.C. Whitehead [[#References|[a9]]], giving a certain second relative [[Homotopy group|homotopy group]] as a free crossed module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120290/c12029044.png" />: this is determined by a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120290/c12029045.png" /> from a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120290/c12029046.png" /> to the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120290/c12029047.png" /> and satisfies a universal property. A common situation is when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120290/c12029048.png" /> is the free group on a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120290/c12029049.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120290/c12029050.png" /> is an inclusion: then the free crossed module is determined by the presentation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120290/c12029051.png" /> of the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120290/c12029052.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120290/c12029053.png" /> is known as the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120290/c12029055.png" />-module of identities among relations [[#References|[a4]]], [[#References|[a5]]] for the presentation. This module should be thought of as giving a non-Abelian form of syzygies (cf. also [[Syzygy|Syzygy]]), and as the start of a free crossed resolution of the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120290/c12029056.png" />. Free crossed modules are also conveniently seen as special cases of crossed modules induced from a crossed module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120290/c12029057.png" /> by a morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120290/c12029058.png" /> of groups [[#References|[a2]]], [[#References|[a8]]].

There are a number of other algebraically defined categories equivalent to the category of crossed modules, namely: group objects in groupoids; groupoid objects internal to the category of groups; group objects internal to the category of groupoids; cat<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120290/c12029059.png" />-groups; simplicial groups whose Moore complex is of length <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120290/c12029060.png" /> (cf. also [[Simplicial complex|Simplicial complex]]); and certain kinds of double groupoids. This last fact is crucial in the proof of the generalized Van Kampen theorem in [[#References|[a2]]], which uses a homotopy double groupoid of a pair of spaces. It is important to note that group objects internal to the category of groups are just Abelian groups: thus, the extension to groupoid objects in groups, or groupoid objects in groupoids, opens up the area of higher-dimensional non-Abelian structures for modeling geometry, and this has been exploited by a number of mathematicians. This seems to answer a dream of the topologists of the early part of the 20th century, of finding a higher-dimensional but still non-commutative analogue of the [[Fundamental group|fundamental group]].

Generalizing the above to the case where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120290/c12029061.png" /> is a [[Groupoid|groupoid]] rather than group yields the notion of crossed module of groupoids. This is important for many applications, and is the convenient format also for the generalized Van Kampen theorem. Such crossed modules are equivalent to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120290/c12029063.png" />-groupoids [[#References|[a3]]].

The notion of groupoid object internal to the category of groups generalizes to groupoid object in other categories, and in many cases (basically when quotients are determined by kernels) one obtains a corresponding notion of crossed module [[#References|[a7]]]. Such a notion is at the root of the area of non-Abelian homological algebra, see for example [[#References|[a1]]], [[#References|[a6]]].

A crossed module of groupoids <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120290/c12029064.png" /> has a classifying space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120290/c12029065.png" />, and this gives a connection with homotopy <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120290/c12029066.png" />-types (see [[#References|[a10]]] for details and references).

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> L. Breen, "Bitorseurs et cohomologie non–Abélienne" P. Cartier (ed.) et al. (ed.) , ''The Grothendieck Festschrift: a Collection of Articles Written in Honour of the 60th Birthday of Alexander Grothendieck'' , '''I''' , Birkhäuser (1990) pp. 401–476 {{MR|1086889}} {{ZBL|0743.14034}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> R. Brown, P.J. Higgins, "On the second relative homotopy groups of some related spaces" ''Proc. London Math. Soc.'' , '''36''' : 3 (1978) pp. 193–212 {{MR|478150}} {{ZBL|0405.55015}} </TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> R. Brown, P.J. Higgins, "The equivalence of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120290/c12029067.png" />-groupoids and crossed complexes" ''Cah. Topol. Géom. Diff.'' , '''22''' (1981) pp. 371–386 {{MR|639048}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top"> R. Brown, J. Huebschmann, "Identities among relations" R. Brown (ed.) T.L. Thickstun (ed.) , ''Low Dimensional Topology'' , ''London Math. Soc. Lecture Notes'' , '''48''' , Cambridge Univ. Press (1982) pp. 153–202 {{MR|0662431}} {{ZBL|0485.57001}} </TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top"> "Two-dimensional homotopy and combinatorial group theory" C. Hog-Angeloni (ed.) W. Metzler (ed.) A.J. Sieradski (ed.) , ''London Math. Soc. Lecture Notes'' , '''197''' , Cambridge Univ. Press (1993) {{MR|1279174}} {{ZBL|0788.00031}} </TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top"> A.S.-T. Lue, "Cohomology of groups relative to a variety" ''J. Algebra'' , '''69''' (1981) pp. 155–174 {{MR|0613866}} {{ZBL|0468.20044}} </TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top"> T. Porter, "Extensions, crossed modules and internal categories in categories of groups with operators" ''Proc. Edinburgh Math. Soc.'' , '''30''' (1987) pp. 373–381</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top"> C.D. Wensley, M. Alp, "XMOD: a GAP share package" , Acad. Press Council (1997)</TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top"> J.H.C. Whitehead, "Combinatorial homotopy II" ''Bull. Amer. Math. Soc.'' , '''55''' (1949) pp. 453–496 {{MR|0030760}} {{ZBL|0040.38801}} </TD></TR><TR><TD valign="top">[a10]</TD> <TD valign="top"> R. Brown, "Groupoids and crossed objects in algebraic topology" ''Homology, Homotopy and Appl.'' , '''1''' (1999) pp. 1–78 {{MR|1691707}} {{ZBL|0920.55002}} </TD></TR></table>