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Idea

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:

Abstract definition

A cat ncat^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 catn^n-groupoid.

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

Algebraic definition

A catn^n-group is a group GG together with 2n2n endomorphisms s i,t i,(1in)s_i, t_i, (1 \le i \le n) such that

s it i=t i,andt is i=s iforalli,s_i t_i = t_i, and t_i s_i = s_i for all i,
s is j=s js i,t it j=t jt i,s it j=t js iforij 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 ii,

[Kers i,Kert i]=1.[ Ker\, s_i, Ker\, t_i] = 1.

Morphisms of catn^n-groups are the obvious things, morphisms of the groups compatible with the endomorphisms.

A catn^{n}-group is thus a group with nn independent cat1^{1}-group structures on it.

Special cases

  • A cat 0cat^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 ΠX *\Pi X_{*} in a special case, namely when the nn-cube of spaces X *X_{*} arises from a pointed (n+1)(n + 1)-ad (X;X 1,,X n) (X;X_1,\ldots ,X_n) by the rule: X n=XX_{ \langle n \rangle} = X and for AA properly contained in n\langle n \rangle, X A= i¬AX iX_A = \bigcap _{i \not\in A} X_i, with maps the inclusions. Let Φ\Phi be the space of maps I nXI^n \to X which take the faces of I nI^n in the iith direction into X iX_i. Notice that Φ\Phi has the structure of nn compositions derived from the gluing of cubes in each direction. Let Φ\bullet \in \Phi be the constant map at the base point. Then G=π 1(Φ,)G = \pi_1(\Phi ,\bullet ) is certainly a group. Gilbert, 1988, identifies GG with Loday’s ΠX *\Pi X_{*}, so that Loday’s results, obtained by methods of simplicial spaces, show that GG becomes a catn^n-group. It may also be shown that the extra groupoid structures are inherited from the compositions on Φ\Phi. It is this catn^n-group which is written ΠX *\Pi X_* and is called the fundamental catn^n-group of the (n+1)(n + 1)-ad.

See also crossed n-cube for an alternative description.

Remarks

  • Even though cat ncat^n-groups are examples of strict (n+1)-fold categories, Loday has shown that the homotopy category of cat n1cat^{n-1}-groups is equivalent to that of spaces which are pointed connected homotopy n-types. Hence for cat ncat^n-groups (thought of as (n+1)(n+1)-fold groupoids) the homotopy hypothesis is true in this sense. See there for more details.

Tim Is the first statement above correct? Cat nCat^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+1n+1.

It is not so clear how to construct a homotopical functor from nn-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 later use favours the term catn^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 cat ncat^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

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

Revised on November 20, 2013 05:34:54 by Tim Porter (2.26.40.238)