# 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

${G}_{2}\stackrel{\delta }{\to }{G}_{1}$G_2 \stackrel{\delta }{\to}{G_1}

and

${G}_{1}\stackrel{\alpha }{\to }\mathrm{Aut}\left({G}_{2}\right)$G_1 \stackrel{\alpha}{\to} Aut(G_2)

(which below we will conceive of as a map $\alpha :{G}_{1}×{G}_{2}\to {G}_{2}$ analogous the adjoint action $\mathrm{Ad}:G×G\to G$ of a group on itself)

• such that

$\begin{array}{ccccc}{G}_{2}×{G}_{2}& & \stackrel{\delta ×\mathrm{Id}}{\to }& & {G}_{1}×{G}_{2}\\ & {}_{\mathrm{Ad}}↘& & {↙}_{\alpha }\\ & & {G}_{2}\end{array}$\array{ G_2 \times G_2 &&\stackrel{\delta \times Id}{\to}&& G_1 \times G_2 \\ & {}_{Ad}\searrow && \swarrow_\alpha \\ && G_2 }

and

$\begin{array}{ccc}{G}_{1}×{G}_{2}& \stackrel{\alpha }{\to }& {G}_{2}\\ {↓}^{\mathrm{Id}×\delta }& & {↓}^{\delta }\\ {G}_{1}×{G}_{1}& \stackrel{\mathrm{Ad}}{\to }& {G}_{1}\end{array}$\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 $\left({G}_{2},{G}_{1},\delta \right)$, for this if the action is fairly obvious, including an explicit action, $\left({G}_{2},{G}_{1},\delta ,\alpha \right)$, 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:

$\delta \left({}^{{g}_{1}}{g}_{2}\right)={g}_{1}\delta \left({g}_{2}\right){g}_{1}^{-1}.$\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, ${}^{{g}_{2}}{g}_{2}^{\prime }={g}_{2}{g}_{2}^{\prime }{g}_{2}^{-1}$ 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:

${}^{\delta \left({g}_{2}\right)}{g}_{2}^{\prime }={g}_{2}{g}_{2}^{\prime }{g}_{2}^{-1}.${}^{\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 $\left[G\right]$ and $H$, a morphism of strict 2-groups $f:G\to H$, and hence a morphism of crossed modules $\left[f\right]:\left[G\right]\to H$ is a 2-functor

$Bf:BG\to BH$\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 $\left[G\right]$ and $\left[H\right]$ yields a diagram called a butterfly. See there for more details.

## Examples

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

$\mathrm{AUT}\left(H\right):=\left({G}_{2}=H,{G}_{1}=\mathrm{Aut}\left(H\right),\delta =\mathrm{Ad},\alpha =\mathrm{Id}\right)\phantom{\rule{thinmathspace}{0ex}}.$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

${\mathrm{Aut}}_{\mathrm{Grpd}}\left(BH\right)$Aut_{Grpd}(\mathbf{B}H)

of automorphisms in the category Grpd of groupoids on the one-object delooping groupoid $BH$ 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, $\mathrm{inc}:N\to G$. This is ‘almost canonical’, since if we replace the groups by simplicial groups ${G}_{.}$ and ${N}_{.}$, then $\left({\pi }_{0}\left({G}_{.}\right),{\pi }_{0}\left({N}_{.}\right),{\pi }_{0}\left(\mathrm{inc}\right)\right)$ is a crossed module, and given any crossed module, $\left(C,P,\delta \right)$, 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 $\mu :{C}_{2}×{C}_{2}\to {C}_{4}$ 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 ${H}^{3}\left({C}_{2},{C}_{2}\right)$, 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 ${\mathrm{BC}}_{2}\to {\mathrm{BC}}_{4}$.

• 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), $F={p}^{-1}\left({b}_{0}\right)$, 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

${\pi }_{1}\left(F\right)\stackrel{{\pi }_{1}\left(i\right)}{⟶}{\pi }_{1}\left(E\right)$\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 $ab{a}^{-1}$ is homotopic to one wholly within $F$. To see this, note that $p\left(ab{a}^{-1}\right)$ 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 $ab{a}^{-1}$. This homotopy is the required one and its other end gives a well defined element ${}^{a}b\in {\pi }_{1}\left(F\right)$ (abusing notation by confusing paths and their homotopy classes). With this action $\left({\pi }_{1}\left(F\right),\pi \left(E\right),{\pi }_{1}\left(i\right)\right)$ 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 $\left(X,{x}_{0}\right)$, (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 ${\pi }_{2}\left(X,A,{x}_{0}\right)$.

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

$\delta :{\pi }_{2}\left(A\cup \left\{{e}_{\lambda }^{2}{\right\}}_{\lambda \in \Lambda },A,x\right)\to {\pi }_{1}\left(A,x\right)$\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

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

## 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.

Revised on July 21, 2012 21:11:42 by Ronnie Brown (86.154.163.18)