Crossed module
Encyclopedia
In mathematics
, and especially in homotopy theory, a crossed module consists of group
s G and H, where G acts
on H (which we will write on the left), and a homomorphism
of groups
that is equivariant
with respect to the conjugation
action of G on itself:
and also satisfies the so-called Peiffer identity:
'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.
subgroup
of a group G. Then, the inclusion
is a crossed module with the conjugation action of G on N.
For any group G, module
s 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
the onto homomorphism
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
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
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
is a pointed fibration
of spaces, then the induced map of fundamental groups
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.
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.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, and especially in homotopy theory, a crossed module consists of group
Group (mathematics)
In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity...
s G and H, where G acts
Group action
In algebra and geometry, a group action is a way of describing symmetries of objects using groups. The essential elements of the object are described by a set, and the symmetries of the object are described by the symmetry group of this set, which consists of bijective transformations of the set...
on H (which we will write on the left), and a homomorphism
Homomorphism
In abstract algebra, a homomorphism is a structure-preserving map between two algebraic structures . The word homomorphism comes from the Greek language: ὁμός meaning "same" and μορφή meaning "shape".- Definition :The definition of homomorphism depends on the type of algebraic structure under...
of groups
that is equivariant
Equivariant
In mathematics, an equivariant map is a function between two sets that commutes with the action of a group. Specifically, let G be a group and let X and Y be two associated G-sets. A function f : X → Y is said to be equivariant iffor all g ∈ G and all x in X...
with respect to the conjugation
Inner automorphism
In abstract algebra an inner automorphism is a functionwhich, informally, involves a certain operation being applied, then another one performed, and then the initial operation being reversed...
action of G on itself:
and also satisfies the so-called Peiffer identity:
Origin
The first mention of the second identity for a crossed module seems to be in footnote 25 on p. 422 of WhiteheadJ. H. C. Whitehead
John Henry Constantine Whitehead FRS , known as Henry, was a British mathematician and was one of the founders of homotopy theory. He was born in Chennai , in India, and died in Princeton, New Jersey, in 1960....
'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 normalNormal subgroup
In abstract algebra, a normal subgroup is a subgroup which is invariant under conjugation by members of the group. Normal subgroups can be used to construct quotient groups from a given group....
subgroup
Subgroup
In group theory, given a group G under a binary operation *, a subset H of G is called a subgroup of G if H also forms a group under the operation *. More precisely, H is a subgroup of G if the restriction of * to H x H is a group operation on H...
of a group G. Then, the inclusion
is a crossed module with the conjugation action of G on N.
For any group G, module
Module (mathematics)
In abstract algebra, the concept of a module over a ring is a generalization of the notion of vector space, wherein the corresponding scalars are allowed to lie in an arbitrary ring...
s over the group ring
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 one-to-one with the given group. As a ring, its addition law is that of the free...
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
Inner automorphism
In abstract algebra an inner automorphism is a functionwhich, informally, involves a certain operation being applied, then another one performed, and then the initial operation being reversed...
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
the onto homomorphism
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
from the second relative homotopy group to the fundamental group
Fundamental group
In mathematics, more specifically algebraic topology, the fundamental group is a group associated to any given pointed topological space that provides a way of determining when two paths, starting and ending at a fixed base point, can be continuously deformed into each other...
, may be given the structure of crossed module. It is a remarkable fact that this functor
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
is a pointed fibration
Fibration
In topology, a branch of mathematics, a fibration is a generalization of the notion of a fiber bundle. A fiber bundle makes precise the idea of one topological space being "parameterized" by another topological space . A fibration is like a fiber bundle, except that the fibers need not be the same...
of spaces, then the induced map of fundamental groups
may be given the structure of crossed module. This example is useful in algebraic K-theory
Algebraic K-theory
In mathematics, algebraic K-theory is an important part of homological algebra concerned with defining and applying a sequenceof functors from rings to abelian groups, for all integers n....
. 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
Category theory is an area of study in mathematics that examines in an abstract way the properties of particular mathematical concepts, by formalising them as collections of objects and arrows , where these collections satisfy certain basic conditions...
. It can be shown that a crossed module is essentially the same as a categorical group or 2-group
2-group
In mathematics, a 2-group, or 2-dimensional higher group, is a certain combination of group and groupoid. The 2-groups are part of a larger hierarchy of n-groups...
: 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
has a classifying space
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 by a free action of G...
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.
External links
- J. Baez and A. Lauda, Higher-dimensional algebra V: 2-groups
- R. Brown, Groupoids and crossed objects in algebraic topology
- R. Brown, Higher dimensional group theory
- M. Forrester-Barker, Group objects and internal categories
- Behrang Noohi, Notes on 2-groupoids, 2-groups and crossed-modules