A group -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)
A group -complex is a pair, , in which is a simplicial group and is a graded subgroup of consisting of thin elements, and which satisfies the conditions:
Let be the graded subgroup of generated by the images of the degeneracy maps, , for all and , then any box in has a filler in .
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 .
If is a group -complex then .
To see this, we note that as every degenerate element is this, . Conversely if , then it fills the box made up of . This, in turn, has a filler, , in , but, as this filler is also thin, it must be that , since thin fillers are uniquely determined.
This is neat. It says there is basically only one possible group -complex structure on a given simplicial group. The next result (again by Ashley) shows that not all simplicial groups carry such a structure.
If is a simplicial group, then is a group -complex if and only if is the trivial graded subgroup.
One way around, this is nearly trivial. If is a group -complex and , then fills a box , so if , must itself be the thin filler, however 1 is also a thin filler for this box, so as required.
Conversely if , then we must check the other two axioms as the first is trivial. As any box has a standard filler in , we only have to check uniqueness, but if and are in , and both fill the same box (with the face missing) then fills a box with 1s on all faces (and the face missing).
If , then as , we have and and are equal. If , assume that if and then , (i.e, that we have uniqueness up to at least the case). Consider . This is still in and unless , hence by assumption . Of course, this implies that , but then . We know that , so and , i.e., and we have uniqueness at the next stage.
To verify the third axiom, assume that and each for , then we can assume that , 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 for all . As must be the degenerate filler given by the standard method, we can calculate as follows:
let , for , then . We can therefore check that as required.
This suggests that, given an arbitrary simplicial group, , we could form a quotient which would be a group -complex, simply by dividing out by the subgroups, . 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: need not be a normal subgroup of !
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 in all dimensions, we must also kill and all the . The end result of this is a group -complex whose Moore complex has a crossed complex structure in a natural way. (The formula is given in the crossed complex entry.)