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In
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, and especially in
homotopy theory In mathematics, homotopy theory is a systematic study of situations in which maps can come with homotopies between them. It originated as a topic in algebraic topology but nowadays is studied as an independent discipline. Besides algebraic topolo ...
, a crossed module consists of
groups A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic ide ...
G and H, where G
acts The Acts of the Apostles ( grc-koi, Πράξεις Ἀποστόλων, ''Práxeis Apostólōn''; la, Actūs Apostolōrum) is the fifth book of the New Testament; it tells of the founding of the Christian Church and the spread of its message ...
on H by
automorphisms In mathematics, an automorphism is an isomorphism from a mathematical object to itself. It is, in some sense, a symmetry of the object, and a way of mapping the object to itself while preserving all of its structure. The set of all automorphisms ...
(which we will write on the left, (g,h) \mapsto g \cdot h , and a
homomorphism In algebra, a homomorphism is a morphism, structure-preserving map (mathematics), map between two algebraic structures of the same type (such as two group (mathematics), groups, two ring (mathematics), rings, or two vector spaces). The word ''homo ...
of groups : d\colon H \longrightarrow G, that is equivariant with respect to the
conjugation Conjugation or conjugate may refer to: Linguistics *Grammatical conjugation, the modification of a verb from its basic form * Emotive conjugation or Russell's conjugation, the use of loaded language Mathematics *Complex conjugation, the change ...
action of G on itself: : d(g \cdot h) = gd(h)g^ and also satisfies the so-called Peiffer identity: : d(h_) \cdot h_ = h_h_h_^


Origin

The first mention of the second identity for a crossed module seems to be in footnote 25 on p. 422 of
J. H. C. Whitehead John Henry Constantine Whitehead FRS (11 November 1904 – 8 May 1960), known as Henry, was a British mathematician and was one of the founders of homotopy theory. He was born in Chennai (then known as Madras), in India, and died in Princeton, ...
'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. Whitehead's ideas on crossed modules and their applications are developed and explained in the book by Brown, Higgins, Sivera listed below. Some generalisations of the idea of crossed module are explained in the paper of Janelidze.


Examples

Let N be a
normal subgroup In abstract algebra, a normal subgroup (also known as an invariant subgroup or self-conjugate subgroup) is a subgroup that is invariant under conjugation by members of the group of which it is a part. In other words, a subgroup N of the group G ...
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'',
module Module, modular and modularity may refer to the concept of modularity. They may also refer to: Computing and engineering * Modular design, the engineering discipline of designing complex devices using separately designed sub-components * Modul ...
s over the
group ring In algebra, a group ring is a free module and at the same time a ring, constructed in a natural way from any given ring and any given group. As a free module, its ring of scalars is the given ring, and its basis is the set of elements of the giv ...
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 In abstract algebra an inner automorphism is an automorphism of a group, ring, or algebra given by the conjugation action of a fixed element, called the ''conjugating element''. They can be realized via simple operations from within the group itse ...
is a crossed module. Given any central extension of groups : 1 \to A \to H \to G \to 1 \! the surjective 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 In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called point ...
(i.e. A is a subspace of X, and x is a point in A), then the homotopy boundary : d\colon \pi_(X,A,x) \rightarrow \pi_(A,x) \! from the second relative homotopy group to the
fundamental group In the mathematical field of algebraic topology, the fundamental group of a topological space is the group of the equivalence classes under homotopy of the loops contained in the space. It records information about the basic shape, or holes, o ...
, may be given the structure of crossed module. The functor : \Pi \colon (\text) \rightarrow (\text) satisfies a form of the
van Kampen theorem A van is a type of road vehicle used for transporting goods or people. Depending on the type of van, it can be bigger or smaller than a pickup truck and SUV, and bigger than a common car. There is some varying in the scope of the word across th ...
, in that it preserves certain colimits. The result on the crossed module of a pair can also be phrased as: if : F \rightarrow E \rightarrow B \! is a pointed
fibration The notion of a fibration generalizes the notion of a fiber bundle and plays an important role in algebraic topology, a branch of mathematics. Fibrations are used, for example, in postnikov-systems or obstruction theory. In this article, all ma ...
of spaces, then the induced map of fundamental groups : d\colon \pi_(F) \rightarrow \pi_(E) \! may be given the structure of crossed module. This example is useful in
algebraic K-theory Algebraic ''K''-theory is a subject area in mathematics with connections to geometry, topology, ring theory, and number theory. Geometric, algebraic, and arithmetic objects are assigned objects called ''K''-groups. These are groups in the sense ...
. 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 Category theory is a general theory of mathematical structures and their relations that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Nowadays, ca ...
. It can be shown that a crossed module is essentially the same as a
categorical group Categorical may refer to: * Categorical imperative, a concept in philosophy developed by Immanuel Kant * Categorical theory, in mathematical logic * Morley's categoricity theorem, a mathematical theorem in model theory * Categorical data analysis ...
or 2-group: that is, a group object in the category of categories, or equivalently a category object in the category of groups. This 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 for higher-dimensional versions of groups.


Classifying space

Any crossed module : M= (d\colon H \longrightarrow G) \! has a ''
classifying space In mathematics, specifically in homotopy theory, a classifying space ''BG'' of a topological group ''G'' is the quotient of a weakly contractible space ''EG'' (i.e. a topological space all of whose homotopy groups are trivial) by a proper free ac ...
BM '' with the property that its homotopy groups are Coker d, in dimension 1, Ker d in dimension 2, and 0 in dimensions above 2. It is possible to describe 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.


External links

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References

* * * * {{refend Group actions (mathematics) Algebraic topology