crossed module


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A crossed module (of groups) is:

  • from the nPOV: a convenient way to encode a strict 2-group GG in terms of a morphism of two ordinary groups :G 2G 1\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.


Diagrammatic definition

A crossed module is

  • a pair of groups G 2,G 1G_2, G_1,

  • morphisms of groups

    G 2δG 1 G_2 \stackrel{\delta }{\to}{G_1}


    G 1αAut(G 2) G_1 \stackrel{\alpha}{\to} Aut(G_2)

    (which below we will conceive of as a map α:G 1×G 2G 2\alpha : G_1 \times G_2 \to G_2 analogously to the adjoint action Ad:G×GGAd : G \times G \to G of a group on itself)

  • such that

    G 2×G 2 δ×Id G 1×G 2 Ad α G 2 \array{ G_2 \times G_2 &&\stackrel{\delta \times Id}{\to}&& G_1 \times G_2 \\ & {}_{Ad}\searrow && \swarrow_\alpha \\ && G_2 }


    G 1×G 2 α G 2 Id×δ δ G 1×G 1 Ad G 1 \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 }


We may use the notation (G 2,G 1,δ)(G_2,G_1,\delta), for this if the action is fairly obvious, including an explicit action, (G 2,G 1,δ,α)(G_2,G_1,\delta,\alpha), if there is a risk of confusion.

If one unwraps the definitions in terms of automorphism groups to use merely finite products (for instance, writing the actions as G 1×G 2G 2G_1 \times G_2 \to G_2, together with commuting diagrams encoding the necessary properties), then crossed modules can be defined internal to any cartesian monoidal category CC, namely as a structure involving internal groups in CC. For instance, one might consider Lie crossed modules, which are crossed modules of Lie groups. These are relevant for certain models of the String group.

Alternatively, one can take another tack, and define crossed module objects in categories that support enough structure without using internal groups, the most general case of which, in practice, are semiabelian categories. There one considers the objects to behave ‘like groups’ in the sense that the category they form looks very much like the category of groups. Janelidze (Janelidze 2003) defined the notion of internal crossed module in a semiabelian category (so that in the prototypical example of the category of groups, they reduce to the above notion).

A key result, also due to (Janelidze 2003) and generalising the Brown-Spencer theorem from the case of ordinary crossed modules, is the following:


(Janelidze’s Brown-Spencer theorem). Let CC be a semiabelian category. Then the category XMod(C)XMod(C) of crossed modules in CC is equivalent to the category Gpd(C)Gpd(C) of internal groupoids in CC.

Here the notion of internal groupoid is the usual diagrammatic notion.

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 1G 1g_1\in G_1 on g 2G 2g_2\in G_2 as g 1g 2{}^{g_1}g_2, then the second diagram translates as the equation:

    δ( g 1g 2)=g 1δ(g 2)g 1 1.\delta({}^{g_1}g_2) = g_1\delta(g_2)g_1^{-1}.

    In other words, δ\delta is equivariant for the action of G 1G_1.

  • The first diagram is slightly more subtle. The group G 2G_2 can act on itself in two different ways, (i) by the usual conjugation action, g 2g 2 =g 2g 2 g 2 1{}^{g_2}g^\prime_2=g_2g^\prime_2g_2^{-1} and (ii) by first mapping g 2g_2 down to G 1G_1 and then using the action of that group back on G 2G_2. The first diagram says that the two actions coincide. Equationally this gives:

    δ(g 2)g 2 =g 2g 2 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. Another way to interpret it is to rewrite it slightly:

    δ(g 2)g 2 g 2=g 2g 2 {}^{\delta(g_2)}g^\prime_2 g_2 = g_2g^\prime_2

    The Peiffer rule can thus be seen as a ‘twisted commutativity law’ for G 2G_2.


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

Bf:BGBH \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][G] and [H][H] yields a diagram called a butterfly. See there for more details.


  • For HH any group, its automorphism crossed module is

    AUT(H):=(G 2=H,G 1=Aut(H),δ=Ad,α=Id). 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(BH) Aut_{Grpd}(\mathbf{B}H)

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

  • Almost the canonical example of a crossed module is given by a group GG and a normal subgroup NN of GG. We take G 2=NG_2 = N, and G 1=GG_1 = G with the action given by conjugation, whilst δ\delta is the inclusion, inc:NGinc : N \to G. This is ‘almost canonical’, since if we replace the groups by simplicial groups G .G_. and N .N_., then (π 0(G .),π 0(N .),π 0(inc))(\pi_0(G_.),\pi_0(N_.),\pi_0(inc)) is a crossed module, and given any crossed module, (C,P,δ)(C,P,\delta), there is a simplicial group G .G_. and a normal subgroup N .N_., such that the construction above gives the given crossed module up to isomorphism.

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

  • If μ:MP\mu: M \to P is a crossed module with cokernel GG, and MM is abelian, then the operation of PP on MM factors through GG. In fact such crossed modules in which both MM and PP are abelian should not be sneezed at! A good example is μ:C 2×C 2C 4\mu: C_2 \times C_2 \to C_4 where C nC_n denotes the cyclic group of order nn, μ\mu is injective on each factor, and C 4C_4 acts on the product by the twist. This crossed module has a classifying space XX with fundamental and second homotopy groups C 2C_2 and non trivial kk-invariant in H 3(C 2,C 2)H^3(C_2, C_2), so XX 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 kk-invariant. The higher homotopy van Kampen theorem implies that the above XX gives the 2-type of the mapping cone of the map of classifying spaces BC 2BC 4BC_2 \to BC_4.

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

    of pointed spaces, thus pp is a fibration in the topological sense (lifting of paths and homotopies of paths will suffice), F=p 1(b 0)F = p^{-1}(b_0), where b 0b_0 is the basepoint of BB. The fibre FF is pointed at f 0f_0, say, and f 0f_0 is taken as the basepoint of EE as well.

    There is an induced map on homotopy groups

    π 1(F)π 1(i)π 1(E)\pi_1(F) \stackrel{\pi_1(i)}{\longrightarrow} \pi_1(E)

    and if aa is a loop in EE based at f 0f_0, and bb a loop in FF based at f 0f_0, then the composite path corresponding to aba 1a b a^{-1} is homotopic to one wholly within FF. To see this, note that p(aba 1)p(a b a^{-1}) is null homotopic?. Pick a homotopy in BB between it and the constant map, then lift that homotopy back up to EE to one starting at aba 1a b a^{-1}. This homotopy is the required one and its other end gives a well defined element abπ 1(F){}^a b \in \pi_1(F) (abusing notation by confusing paths and their homotopy classes). With this action (π 1(F),π(E),π 1(i))(\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 AXA\to X into a pointed space (X,x 0)(X,x_0), (where we assume x 0Ax_0\in A). We can replace this inclusion by a homotopic fibration, A¯X\overline{A}\to X in ‘the standard way’, and then find that the fundamental group of its fibre is π 2(X,A,x 0)\pi_2(X,A,x_0).

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

δ:π 2(A{e λ 2} λΛ,A,x)π 1(A,x)\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 22-cells. One utility of this is that it enables the expression of nonabelian chains and boundaries ideas in dimensions 11 and 22: thus for the standard picture of a Klein Bottle formed by identifications from a square σ\sigma the formula

δσ=a+ba+b\delta \sigma = a+b-a +b

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

Whitehead’s proof of this theorem used knot theory and transversality. The theorem is also a consequence of the 22-dimensional Seifert-van Kampen Theorem, proved by Brown and Higgins, which states that the functor

Π 2\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 π 2(XCA,X,x)\pi_2(X \cup CA,X,x) as a crossed π 1(X,x)\pi_1(X,x) module, and then Whitehead’s result is the case with A$ is a wedge of circles.


The concept was introduced in

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

see also

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

Further discussion is in

  • Ronnie Brown, Peter Higgins, On the connections between the second relative homotopy groups of some related spaces, Proc. London Math. Soc. (3) 36 (1978) 193-212.

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

  • George Janelidze, Internal crossed modules, Georgian Mathematical Journal 10 (2003) pp 99-114. (EuDML)

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

Revised on January 5, 2018 06:43:43 by David Corfield (