nLab
group T-complex

Contents

Idea

A group TT-complex is ‘’a simplicial T-complex internal to the category of groups’’. They were first studied in Nick Ashley’s thesis (see references below for the published version)

More precisely:

Definition

A group TT-complex is a pair, (G,T)(G,T), in which GG is a simplicial group and TT is a graded subgroup of GG consisting of thin elements, and which satisfies the conditions:

  • Every degenerate element is thin.
  • Every horn in GG has a unique thin filler.
  • A thin filler of a thin box also has its last face thin.

Results

Lemma

Let D=(D n) n1D = (D_n)_{n\geq 1} be the graded subgroup of GG generated by the images of the degeneracy maps, s i:G nG n+1s_i :G_n \rightarrow G_{n+1}, for all ii and nn, then any box in GG has a filler in DD.

Proof

The algorithmic formulae used when proving that any simplicial group is a Kan complex (cf., entry on simplicial group) give a filler defined as a product of degenerate copies of the faces of the given box, so is in D nD_n.

Proposition

If (G,T)(G,T) is a group TT-complex then T=DT = D.

Proof

To see this, we note that as every degenerate element is this, DTD \subseteq T. Conversely if tT nt \in T_n, then it fills the box made up of (,d 1t,,d nt) ( - , d_1t, \ldots, d_n t ). This, in turn, has a filler, dd, in DD, but, as this filler is also thin, it must be that t=dt = d, since thin fillers are uniquely determined.

This is neat. It says there is basically only one possible group TT-complex structure on a given simplicial group. The next result (again by Ashley) shows that not all simplicial groups carry such a structure.

Proposition

If GG is a simplicial group, then (G,D)(G,D) is a group TT-complex if and only if NGDNG\cap D is the trivial graded subgroup.

Proof

One way around, this is nearly trivial. If (G,D)(G,D) is a group TT-complex and xNG nx\in NG_n, then xx fills a box (,1,,1)(-, 1, \ldots, 1), so if xNG nD nx\in NG_n\cap D_n, xx must itself be the thin filler, however 1 is also a thin filler for this box, so x=1x = 1 as required.

Conversely if NGD={1}NG\cap D = \{1\}, then we must check the other two axioms as the first is trivial. As any box has a standard filler in DD, we only have to check uniqueness, but if xx and yy are in D nD_n, and both fill the same box (with the k thk^{th} face missing) then z=xy 1z = xy^{-1} fills a box with 1s on all faces (and the k thk^{th} face missing).

If k=0k = 0, then as zNG nD nz \in NG_n\cap D_n, we have z=1z = 1 and xx and yy are equal. If k>0k \gt 0, assume that if <k\ell \lt k and zD n iKerd iz \in D_n \cap \bigcap_{i\neq \ell} Ker\, d_i then z=1z = 1, (i.e, that we have uniqueness up to at least the (k1) st(k-1)^{st} case). Consider w=zs k1d kz 1w = zs_{k-1}d_k z^{-1}. This is still in D nD_n and d iw=1d_i w = 1 unless i=k1i = k-1, hence by assumption w=1w = 1. Of course, this implies that z=s k1d kzz = s_{k-1}d_kz, but then d k1z=d kzd_{k-1} z = d_k z. We know that d k1z=1d_{k-1} z = 1, so d kz=1d_k z = 1 and z=1z = 1, i.e., x=yx = y and we have uniqueness at the next stage.

To verify the third axiom, assume that xD n+1x \in D_{n + 1} and each d ixD nd_i x \in D_n for iki \neq k, then we can assume that k=0k = 0, since otherwise we can skew the situation around as before to get that to be true, verify it in that case and ‘skew’ it back again later.

Suppose therefore that d ixD nd_i x \in D_n for all 0<i<n0 \lt i \lt n. As xx must be the degenerate filler given by the standard method, we can calculate xx as follows:

let w n=s n1d nxw_n = s_{n-1}d_n x, w i=w i+1(s i1d iw i+1) 1s i1y iw_i = w_{i+1}(s_{i-1}d_i w_{i+1})^{-1}s_{i-1}y_i for i=1i = 1, then x=w 1x = w_1. We can therefore check that d 0xD nd_0x \in D_n as required.

The group T-complex associated to a simplicial group

This suggests that, given an arbitrary simplicial group, GG, we could form a quotient which would be a group TT-complex, simply by dividing out by the subgroups, NG nD nNG_n\cap D_n. We would need check that the face and degeneracy maps worked correctly, that the result did not somehow generate some new ‘thin’ elements, etc. In fact the idea does not work because of a much more elementary problem: NG nD nNG_n\cap D_n need not be a normal subgroup of G nG_n!

A variant of this does however work. We need the Conduché non-Abelian version of the decomposition theorem for simplicial groups that makes up an important part of the Dold-Kan correspondence. With that we note the fairly obvious point that when we divide out by a graded subgroup in a simplicial group, then it has effects in all dimensions due to the face and degeneracy maps, so if we kill elements in NG nD nNG_n\cap D_n in all dimensions, we must also kill d 0(NG n+1D n+1)d_0(NG_{n+1}\cap D_{n+1}) and all the s k(NG n1D n1)s_k(NG_{n-1}\cap D_{n-1}). The end result of this is a group TT-complex whose Moore complex has a crossed complex structure in a natural way. (The formula is given in the crossed complex entry.)

References

  • N. Ashley, Simplicial T-Complexes: a non abelian version of a theorem of Dold-Kan, Dissertations Math., 165, (1989), 11 – 58.
Revised on March 3, 2010 17:51:07 by Tim Porter (95.147.238.243)