Just as cat-1-groups (i) give models for connected homotopy 2-types, (ii) are equivalent to crossed modules, or 2-groups,and are an algebraic encoding of internal categories within the category Grp of groups, so it is not surprising that higher dimensional analogues encode higher order homotopy information. This gives one the abstract definition:
A $cat^n$-group is a strict n-fold category internal to Grp.
Regarding a group as a groupoid with a single object, this is the same as an (n+1)-fold groupoid in which in one direction all morphisms are endomorphisms and there is corresponding notion of cat$^n$-groupoid.
As with the cases $n=1$ and 2, there is a neat purely group theoretic definition of these objects.
A cat$^n$-group is a group $G$ together with $2n$ endomorphisms $s_i, t_i, (1 \le i \le n)$ such that
and, for all $i$,
Morphisms of cat$^n$-groups are the obvious things, morphisms of the groups compatible with the endomorphisms.
A cat$^{n}$-group is thus a group with $n$ independent cat$^{1}$-group structures on it.
A $cat^0$-group is a group.
A cat-1-group is a strict 2-group, viewed in a slightly different way.
For simplicity, we describe $\Pi X_{*}$ in a special case, namely when the $n$-cube of spaces $X_{*}$ arises from a pointed $(n + 1)$-ad $(X;X_1,\ldots ,X_n)$ by the rule: $X_{ \langle n \rangle} = X$ and for $A$ properly contained in $\langle n \rangle$, $X_A = \bigcap _{i \not\in A} X_i$, with maps the inclusions. Let $\Phi$ be the space of maps $I^n \to X$ which take the faces of $I^n$ in the $i$th direction into $X_i$. Notice that $\Phi$ has the structure of $n$ compositions derived from the gluing of cubes in each direction. Let $\bullet \in \Phi$ be the constant map at the base point. Then $G = \pi_1(\Phi ,\bullet )$ is certainly a group. Gilbert, 1988, identifies $G$ with Loday’s $\Pi X_{*}$, so that Loday’s results, obtained by methods of simplicial spaces, show that $G$ becomes a cat$^n$-group. It may also be shown that the extra groupoid structures are inherited from the compositions on $\Phi$. It is this cat$^n$-group which is written $\Pi X_*$ and is called the fundamental cat$^n$-group of the $(n + 1)$-ad.
See also crossed n-cube for an alternative description.
Tim Is the first statement above correct? $Cat^n$-groups are examples of strict (n+1)-fold categories, not strict n=categories or am I missing something? (28-09-2010<- corrected)
Ronnie Agreed, and I have corrected that. This is important since an n-category internal to Grp is equivalent to a single vertex crossed complex of length $n+1$.
It is not so clear how to construct a homotopical functor from $n$-cubes of non pointed spaces, and what should be the receiving category.
The original proof of Loday’s result is to be found in
This paper also uses the term n-cat-group, but later use favours the term cat$^n$-group to make it clearer that these were an n-fold category internal to Grp. There are one or two gaps in that proof and various patches and complete proofs were then given. The main one is in
A proof using $cat^n$-groups and a neat detailed analysis of multisimplicial groups and related topics was given in
Porter (1993) gave a simple proof in terms of crossed n-cubes using as little high-powered simplicial techniques as possible in