# nLab crossed module

### Context

#### Higher category theory

higher category theory

## 1-categorical presentations

#### Homological algebra

homological algebra

and

nonabelian homological algebra

diagram chasing

# Content

## Idea

A crossed module (of groups) is:

• from the nPOV: a convenient way to encode a strict 2-group $G$ in terms of a morphism of two ordinary groups $\partial : G_2 \to G_1$.

From other points of view it is:

Historically they were the first example of higher dimensional algebra to be studied.

## Definition

### Diagrammatic definition

A crossed module is

• a pair of groups $G_2, G_1$,

• morphisms of groups

$G_2 \stackrel{\delta }{\to}{G_1}$

and

$G_1 \stackrel{\alpha}{\to} Aut(G_2)$

(which below we will conceive of as a map $\alpha : G_1 \times G_2 \to G_2$ analogous the adjoint action $Ad : G \times G \to G$ of a group on itself)

• such that

$\array{ G_2 \times G_2 &&\stackrel{\delta \times Id}{\to}&& G_1 \times G_2 \\ & {}_{Ad}\searrow && \swarrow_\alpha \\ && G_2 }$

and

$\array{ G_1 \times G_2 &\stackrel{\alpha}{\to}& G_2 \\ \downarrow^{Id \times \delta} && \downarrow^{\delta} \\ G_1 \times G_1 &\stackrel{Ad}{\to}& G_1 }$

commute.

We may use the notation $(G_2,G_1,\delta)$, for this if the action is fairly obvious, including an explicit action, $(G_2,G_1,\delta,\alpha)$, if there is a risk of confusion.

### Definition in terms of equations

The two diagrams can be translated into equations, which may often be helpful.

• If we write the effect of acting with $g_1\in G_1$ on $g_2\in G_2$ as ${}^{g_1}g_2$, then the second diagram translates as the equation:

$\delta({}^{g_1}g_2) = g_1\delta(g_2)g_1^{-1}.$

In other words, $\delta$ is equivariant for the action of $G_1$.

• The first diagram is slightly more subtle. The group $G_2$ can act on itself in two different ways, (i) by the usual conjugation action, ${}^{g_2}g^\prime_2=g_2g^\prime_2g_2^{-1}$ and (ii) by first mapping $g_2$ down to $G_1$ and then using the action of that group back on $G_2$. The first diagram says that the two actions coincide. Equationally this gives:

${}^{\delta(g_2)}g^\prime_2 = g_2g^\prime_2g_2^{-1}.$

This equation is known as the Peiffer rule in the literature.

### Morphisms

For $G$ and $H$ two strict 2-groups coming from crossed modules $[G]$ and $H$, a morphism of strict 2-groups $f : G \to H$, and hence a morphism of crossed modules $[f] : [G] \to H$ is a 2-functor

$\mathbf{B}f : \mathbf{B}G \to \mathbf{B}H$

between the corresponding delooped 2-groupoids. Expressing this in terms of a diagram of the ordinary groups appearing in $[G]$ and $[H]$ yields a diagram called a butterfly. See there for more details.

## Examples

• For $H$ any group, its automorphism crossed module is

$AUT(H) := (G_2 = H, G_1 = Aut(H), \delta = Ad, \alpha = Id) \,.$

Under the equivalence of crossed modules with strict 2-groups this corresponds to the automorphism 2-group

$Aut_{Grpd}(\mathbf{B}H)$

of automorphisms in the category Grpd of groupoids on the one-object delooping groupoid $\mathbf{B}H$ of $H$.

• Almost the canonical example of a crossed module is given by a group $G$ and a normal subgroup $N$ of $G$. We take $G_2 = N$, and $G_1 = G$ with the action given by conjugation, whilst $\delta$ is the inclusion, $inc : N \to G$. This is ‘almost canonical’, since if we replace the groups by simplicial groups $G_.$ and $N_.$, then $(\pi_0(G_.),\pi_0(N_.),\pi_0(inc))$ is a crossed module, and given any crossed module, $(C,P,\delta)$, there is a simplicial group $G_.$ and a normal subgroup $N_.$, such that the construction above gives the given crossed module up to isomorphism.

• Another standard example of a crossed module is $M \to ^0 P$ where $P$ is a group and $M$ is a $P$-module. Thus the category of modules over groups embeds in the category of crossed modules.

• If $\mu: M \to P$ is a crossed module with cokernel $G$, and $M$ is abelian, then the operation of $P$ on $M$ factors through $G$. In fact such crossed modules in which both $M$ and $P$ are abelian should not be sneezed at! A good example is $\mu: C_2 \times C_2 \to C_4$ where $C_n$ denotes the cyclic group of order $n$, $\mu$ is injective on each factor, and $C_4$ acts on the product by the twist. This crossed module has a classifying space $X$ with fundamental and second homotopy groups $C_2$ and non trivial $k$-invariant in $H^3(C_2, C_2)$, so $X$ is not a product of Eilenberg-MacLane spaces. However the crossed module is an algebraic model and so one one can do algebraic constructions with it. It gives in some ways a better feel for the space than the $k$-invariant. The higher homotopy van Kampen theorem implies that the above $X$ gives the 2-type of the mapping cone of the map of classifying spaces $BC_2 \to BC_4$.

• Suppose $F\stackrel{i}{\to}E\stackrel{p}{\to}B$ is a fibration sequence

of pointed spaces, thus $p$ is a fibration in the topological sense (lifting of paths and homotopies of paths will suffice), $F = p^{-1}(b_0)$, where $b_0$ is the basepoint of $B$. The fibre $F$ is pointed at $f_0$, say, and $f_0$ is taken as the basepoint of $E$ as well.

There is an induced map on homotopy groups

$\pi_1(F) \stackrel{\pi_1(i)}{\longrightarrow} \pi_1(E)$

and if $a$ is a loop in $E$ based at $f_0$, and $b$ a loop in $F$ based at $f_0$, then the composite path corresponding to $a b a^{-1}$ is homotopic to one wholly within $F$. To see this, note that $p(a b a^{-1})$ is null homotopic?. Pick a homotopy in $B$ between it and the constant map, then lift that homotopy back up to $E$ to one starting at $a b a^{-1}$. This homotopy is the required one and its other end gives a well defined element ${}^a b \in \pi_1(F)$ (abusing notation by confusing paths and their homotopy classes). With this action $(\pi_1(F), \pi(E), \pi_1(i))$ is a crossed module. This will not be proved here, but is not that difficult. (Of course, secretly, this example is ‘really’ the same as the previous one since a fibration of simplicial groups is just morphism that is an epimorphism in each degree, and the fibre is thus just a normal simplicial subgroup. What is fun is that this generalises to ‘higher dimensions’.)

• A particular case of this last example can be obtained from the inclusion of a subspace $A\to X$ into a pointed space $(X,x_0)$, (where we assume $x_0\in A$). We can replace this inclusion by a homotopic fibration, $\overline{A}\to X$ in ‘the standard way’, and then find that the fundamental group of its fibre is $\pi_2(X,A,x_0)$.

A deep theorem of J.H.C. Whitehead is that the crossed module

$\delta: \pi_2(A \cup \{e^2_\lambda\}_{\lambda \in \Lambda},A,x) \to \pi_1(A,x)$

is the free crossed module on the characteristic maps of the $2$-cells. One utility of this is that it enables the expression of nonabelian chains and boundaries ideas in dimensions $1$ and $2$: thus for the standard picture of a Klein Bottle formed by identifications from a square $\sigma$ the formula

$\delta \sigma = a+b-a +b$

makes sense with $\sigma$ a generator of a free crossed module; in the usual abelian chain theory we can write only $\partial \sigma =2b$, thus losing information.

Whitehad’s proof of this theorem used knot theory and transversality. The theorem is also a consequence of the $2$-dimensional Seifert-van Kampen Theorem, proved by Brown and Higgins, which states that the functor

$\Pi_2$: (pairs of pointed spaces) $\to$ (crossed modules)

preserves certain colimits (see reference below).

This last example was one of the first investigated by Whitehead and his proof appears also in a little book by Hilton; see also Nonabelian algebraic topology, however the more general result of Brown and Higgins determines also the group $\pi_2(X \cup CA,X,x)$ as a crossed $\pi_1(X,x)$ module, and then Whitehead’s result is the case with A\$ is a wedge of circles.

## References

• R. Brown, “Groupoids and crossed objects in algebraic topology”, Homology, Homotopy and Applications, 1 (1999) 1-78.

• R. Brown and P.J. Higgins, “On the connections between the second relative homotopy groups of some related spaces”, Proc. London Math. Soc. (3) 36 (1978) 193-212.

• R. Brown, P. J. Higgins, and R. Sivera, Nonabelian Algebraic Topology: Filtered spaces, Crossed Complexes, Cubical Homotopy Groupoids, EMS Tracts in Mathematics, Vol. 15, (Autumn 2010).

• Peter J. Hilton, 1953, An Introduction to Homotopy Theory, Cambridge University Press.

• J. H. C. Whitehead, Combinatorial Homotopy II, Bull. Amer. Math. Soc., 55 (1949), 453–496.

Revised on March 3, 2014 08:50:08 by JCMc Keown (174.116.173.185)