A crossed module is the ‘algebraic core’ of a $\cat^1$-group, in as much as within a $\cat^1$-group we can find a crossed module in a simple way from which the whole of the $\cat^1$-group in all its glory can be reconstructed.
A crossed square is similarly the ‘algebraic core’ of a $\cat^2$group.
A crossed $n$-cube should be the ‘algebraic core’ of a $\cat^n$-group.
A crossed square in the Guin-Valery Loday specification has quite a few axioms, many more than those for a crossed module. Does that mean that crossed $n$-cubes will be defined with a very large number of axioms (perhaps dependent on $n$)? No (unless you think that 11 is large!)
Finally a particular example of a crossed square is given by a group with two normal subgroups, and their intersection together with the commutator map. It would be appropriate if a group together with $n$ normal subgroups, all the intersections and all the commutator maps, formed an example of a crossed $n$-cube. It does. The axioms below govern the $h$-maps, by using analogues of the relationships between commutator maps in such a $n$-cube of inclusions of intersections.
The result (definition due to Graham Ellis and Richard Steiner, reference below) is a set of 11 axioms. That is all, and these objects model all homotopy $(n+1)$-types.
We denote by $\langle n \rangle$ the set $\{1,2,\ldots, n\}$.
A crossed $n$-cube, $M$, is a family of groups, $\{M_A: A \subseteq \langle n \rangle\}$, together with homomorphisms, $\mu_i :M_A \rightarrow M_{A-\{i\}}$, for $i \in \langle n \rangle , A \subseteq \langle n \rangle$, and functions, $h:M_{A} \times M_{B} \rightarrow M_{A\cup B}$, for $A, B \subseteq \langle n \rangle$, such that if $^{a}b$ denotes $h(a,b)b$ for $a \in M_{A}$ and $b \in M_{B}$ with $A \subseteq B$, then for $a, a^\prime \in M_{A},\, b, b^\prime \in M_{B},\, c \in M_{C}$ and $i, j \in \langle n \rangle$, the following axioms hold:
$\mu_i a = a$ if $i \notin A$
$\mu_i\mu_j a = \mu_j\mu_i a$
$\mu_i h(a,b) = h(\mu_i a,\mu_i b)$
$h(a,b) = h(\mu_i a,b) = h(a,\mu_i b)$ if $i \in A \cap B$
$h(a,a^\prime ) = [a,a^\prime ]$
$h(a,b) = h(b,a)^{-1}$
$h(a,b) = 1$ if $a = 1$ or $b = 1$
$h(aa^\prime ,b) = {}^{a}h(a^\prime ,b)h(a,b)$
$h(a,bb^\prime ) = h(a,b){ }^b h(a,b^\prime )$
${ }^{a}h(h(a^{-1},b),c)^{c}h(h(c^{-1},a),b)^{b}h(h(b^{-1},c),a) = 1$
${ }^{a}h(b,c) = h(^{a}b,^{a} c)$ if $A \subseteq B \cap C$.
A morphism of crossed $n$-cubes
is a family of homomorphisms, $\{f_{A}: M_{A} \rightarrow M^{\prime}_A \,|\, A \subseteq \langle n \rangle\}$, which commute with the maps, $\mu_i$, and the functions, $h$.
This gives us a category, $Crs^{n}$, equivalent to that of $\cat^n$-groups.
The fundamental crossed $n$-cube of groups functor $\Pi '$ is defined from $n$-cubes of pointed spaces to crossed $n$-cubes of groups: $\Pi 'X_{*}$ is simply the crossed $n$-cube of groups equivalent to the cat$^n$-group $\Pi X_{*}$. It is easier to identify $\Pi '$ in classical terms in the case $X_{*}$ is the $n$-cube of spaces arising from a pointed $(n + 1)$-ad $\mathcal{X} = (X;X_1,\ldots ,X_n)$. That is, let $X_{ \langle n \rangle } = X$ and for $A$ properly contained in $\langle n \rangle$ let $X_A = \bigcap _{i \not\in A} X_i$. Then $M = \Pi '\mathcal{X}$ is given as follows (Ellis and Steiner, 1987): $M_{\emptyset} = \pi_1(X_\emptyset )$; if $A = {i_1,\ldots ,i_r}$, in the right order, then $M$ is the homotopy $(r + 1)$-ad group $\pi _{r+1}(X _A;X_A \cap X_{i_1} ,\ldots ,X_A \cap X_{i_r} )$; the maps $\mu$ are given by the usual boundary maps; the $h$-functions are given by generalised Whitehead products.
Note that whereas these separate elements of structure had all been considered previously, the aim of this theory is to consider the whole structure, despite its apparent complications. The equivalence of categories is a convincing reason for supposing that the axioms for a crossed $n$-cube of groups are a complete axiomatisation of this homotopical structure, as was not previously known.
Let us put a bit of flesh on the example given in the introduction. Suppose that $G$ is a group and $N_1, N_2, .., N_n$ are $n$-normal subgroups of $G$ (there may be repeats).
Now define for $A \subseteq \langle n \rangle$, $M_A = \bigcap\{N_i\mid i \in A\}$, and for $A\in M_A$, $b\in M_B$, $h(a,b) = [a,b]$, the commutator of $a$ and $b$ in $G$. The result is a crossed $n$-cube (sometimes called the inclusion crossed $n$-cube determined by the normal $n$-ad of subgroups).
If instead of a space you start with a simplicial group $G$, as model for a connected homotopy type, then there is a crossed $n$-cube generalising the crossed square given in terms of the Moore complex.
G. J. Ellis and R. Steiner, Higher dimensional crossed modules and the homotopy groups of
(n + 1)-ads_, J. Pure Appl. Alg., 46, (1987), 117–136.
R. BrownComputing homotopy types using crossed n-cubes of groups, Adams Memorial Symposium on
Algebraic Topology}, Vol 1, edited N. Ray and G Walker, Cambridge University Press, 1992, 187-210. Here is a link to a revised version with hyperref
R. BrownHomotopical excision (work to be done!) Presentation at Higher Structures Lisbon, July 2017. RB’s preprints
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