and
nonabelian homological algebra
A crossed module (of groups) is:
From other points of view it is:
Historically they were the first example of higher dimensional algebra to be studied.
A crossed module is
a pair of groups $G_2, G_1$,
morphisms of groups
and
(which below we will conceive of as a map $\alpha : G_1 \times G_2 \to G_2$ analogously to the adjoint action $Ad : G \times G \to G$ of a group on itself)
such that
and
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.
If one unwraps the definitions in terms of automorphism groups to use merely finite products (for instance, writing the actions as $G_1 \times G_2 \to G_2$, together with commuting diagrams encoding the necessary properties), then crossed modules can be defined internal to any cartesian monoidal category $C$, namely as a structure involving internal groups in $C$. 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 $C$ be a semiabelian category. Then the category $XMod(C)$ of crossed modules in $C$ is equivalent to the category $Gpd(C)$ of internal groupoids in $C$.
Here the notion of internal groupoid is the usual diagrammatic notion.
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:
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:
This equation is known as the Peiffer rule in the literature.
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
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.
For $H$ any group, its automorphism crossed module is
Under the equivalence of crossed modules with strict 2-groups this corresponds to the automorphism 2-group
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
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
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
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.
2-group, crossed module, differential crossed module
3-group, 2-crossed module / crossed square, differential 2-crossed module
∞-group, simplicial group, crossed complex, hypercrossed complex
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.
George Janelidze, Internal crossed modules, Georgian Mathematical Journal 10 (2003) pp 99-114. (EuDML)
J. H. C. Whitehead, Combinatorial Homotopy II, Bull. Amer. Math. Soc., 55 (1949), 453–496.