(also nonabelian homological algebra)
A crossed module is a bit like a normal subgroup … without being a subgroup. In fact if a crossed module has a boundary map which is a monomorphism then it is isomorphic to the inclusion crossed module of a normal subgroup.
Crossed modules model all connected homotopy 2-types (which by the looping and delooping-theorem means: all 2-groups). Crossed squares model all connected homotopy 3-types (hence all 3-groups) and correspond in much the same way to pairs of normal subgroups.
Suppose $G$ is a group and $M$ and $N$ are normal subgroups of $G$; then of course, so is $M \cap N$. Put these groups in a square, with the inclusion maps between them. Finally note that if $m \in M$ and $n \in N$, then $[m,n]$ is in the intersection $M \cap N$. This gives you a crossed square with $h$-map $h(m,n) = [m,n]$. Removing the condition that the inclusions are inclusions (!) gives the general form.
(The definition that follows is that given by Guin-Valery and Loday in their paper (see references). Another definition can be given that is just the case $n = 2$ of that of crossed n-cube, for which see that entry.
A crossed square
consists of four morphisms of groups $\lambda: L \to M$, $\lambda': L \to N$, $\mu: M \to P$, $\nu: N \to P$, such that $\nu \lambda'= \mu \lambda$ together with actions of the group $P$ on $L, M, N$ on the left, conventionally, (and hence actions of $M$ on $L$ and $N$ via $\mu$ and of $N$ on $L$ and $M$ via $\mu$) and a function $h: M \times N \to L$.
This structure shall satisfy the following axioms:
for all $l\in L, \,m, m'\in M,\, n,n'\in N$ and $p\in P$.
The similarity of these axioms to commutator identities is no accident (see below).
This should be thought of as a crossed module of crossed modules (in either direction!). For instance horizontally:
The image of this morphism is a normal sub-crossed module of $(M,P,\mu)$, so we can form a quotient
and this is a crossed module, as is the kernal crossed module of this (horizontal) morphism.
The classical homotopical example $\Pi(X;A,B)$ is determined by a pointed triad? $(X; A,B)$ where $A,B \subseteq X$, and $P = \pi_1(A \cap B)$, $M = \pi_2(A, A \cap B), N = \pi_2(B, A \cap B)$ and $L=\pi_3(X; A,B)$. The operations of $P$ are the standard ones and $h$ is the generalised Whitehead product. (The conventions may be slightly different from the standard ones in homotopy theory.) This can be generalised to a functor $\Pi$ from squares of pointed spaces to crossed squares.
Ellis uses this construction in
where the fact that that the crossed square associated to a triad is defined directly in terms of certain homotopy classes is important.
The fact that there is a van Kampen type theorem for $\Pi$ implies that one calculates some nonabelian groups. It also implies that one is calculating some (pointed) homotopy 3-types.
The example hinted at above has $P$ a group, $M$ and $N$ normal subgroups, and $L = M \cap N$,
with all maps the evident inclusions, all actions by conjugation, and $h : M \times N \to M\cap N$ given by $h(m,n) = [m,n]$.
If we replace each group in the algebraic example by a simplicial group, we would have a simplicial crossed square, now apply the connected component functor to that and you get a crossed square, and in fact any crossed square can be constructed up to isomorphism in this way.
If we start with a simplicial group, $G$, using the decalage functor we can construct a simplicial group and two normal subgroups and thus get to the previous situation. The result can be interpreted in terms of the Moore complex as follows:
The two morphisms with codomain $G_1$ are inclusions, the other two are induced by $d_0$. The $h$-map can be explicitly given. It can be found in
T. Porter, n-types of simplicial groups and crossed n-cubes, Topology, 32, (1993), 5 – 24,
which also contains the discussion of the generalisation to crossed n-cubes
A crossed module $\mu: M \to P$ determines a $cat^1$-structure on the semidirect product group $M \rtimes P$. Thus to say that the above crossed square is a crossed module of crossed modules suggests that we should ask for $L \rtimes N \to M \rtimes P$ to be a crossed module, so that there is an action which allows the big group $G = (L \rtimes N) \rtimes (M \rtimes P)$ to be a $cat^1$-group. Then $G$ becomes a $cat^2$-group. The $h$-map of the crossed square derives from a commutator in $G$.
This equivalence between crossed squares and $cat^2$-groups confirms the completeness of the axioms for crossed squares. Notice also that to prove a diagram of crossed squares is a colimit diagram, it looks as if you have to make appallingly detailed verifications of axioms. It is much easier to prove the corresponding diagram of $cat^2$-groups is a colimit!
This theme of using two equivalent categories, one for conjecture and proof, the other for calculation and application to traditional invariants, runs through the story of higher homotopy van Kampen theorems.
3-group, 2-crossed module / crossed square, differential 2-crossed module
∞-group, simplicial group, crossed complex, hypercrossed complex