nLab
cat-n-group

Idea

A ${\mathrm{cat}}^n$-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.

As with the cases $n=1$ and 2, there is a neat purely group theoretic definition of these objects.

Algebraic definition

A cat${}^n$-group is a group $G$ together with $2n$ endomorphisms $s_i,t_i,(1\le i\le n)$ such that

$$s_i{t}_i=t_i,\mathrm{and}{t}_i{s}_i=s_i\mathrm{for}\mathrm{all}i,$$s_i t_i = t_i, and t_i s_i = s_i for all i,

$$s_i{s}_j=s_j{s}_i,t_i{t}_j=t_j{t}_i,s_i{t}_j=t_j{s}_i\mathrm{for}i\ne j$$s_i s_j = s_j s_i, t_i t_j = t_j t_i, s_i t_j = t_j s_i for i\neq j

and, for all $i$,

$$[\mathrm{Ker}{s}_i,\mathrm{Ker}{t}_i]=1.$$[ Ker\, s_i, Ker\, t_i] = 1.

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.

Special cases

• A ${\mathrm{cat}}^0$-group is a group.

• A cat-1-group is a strict 2-group, viewed in a slightly different way.

Homotopical example

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 $(X;X_1,\dots ,X_n)$ by the rule: $X_{⟨n⟩}=X$ and for $A$ properly contained in $⟨n⟩$, $X_A={\bigcap }_{i\neg \in A}{X}_i$, 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 $•\in \Phi$ be the constant map at the base point. Then $G={\pi }_1(\Phi ,•)$ 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.

Remarks

• Even though ${\mathrm{cat}}^n$-groups are examples of strict (n+1)-fold categories, Loday has shown that the homotopy category of ${\mathrm{cat}}^{n-1}$-groups is equivalent to that of spaces which are pointed connected homotopy n-types. Hence for ${\mathrm{cat}}^n$-groups (thought of as $(n+1)$-fold groupoids) the homotopy hypothesis is true in this sense. See there for more details.

Tim Is the first statement above correct? ${\mathrm{Cat}}^n$-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.

References

The original proof of Loday’s result is to be found in

• J.-L. Loday, Spaces with finitely many nontrivial homotopy groups, J.Pure Appl. Alg., 24, (1982), 179–202.

This paper also uses the term n-cat-group, but we later used 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

• R. Steiner, Resolutions of spaces by cubes of fibrations. J. London Math. Soc. (2) 34 (1986), 169–176.

A proof using ${\mathrm{cat}}^n$-groups and a neat detailed analysis of multisimplicial groups and related topics was given in

• M. Bullejos?, A. M. Cegarra, and J. Duskin, On ${\mathrm{cat}}^n$ -groups and homotopy types, J. Pure Appl. Alg., 86, (1993), 135–154.

Porter (1993) gave a simple proof in terms of crossed n-cubes using as little high-powered simplicial techniques as possible in

• T. Porter, n-types of simplicial groups and crossed n-cubes, Topology, 32, (1993), 5–24.