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I have been working on some new material for the Menagerie (and the notes for an excellent Luminy conference, which finished a week ago). Some of this has been put into the Lab at higher generation by subgroups which is an account of a paper by Abels and Holz. One consequence of this is that I have been looking at the group action on the nerve of the covering by cosets from a family of subgroups. (Please see that nLab entry if you want more details.) I am linking this with complexes of groups, which are a type of pseudo functor from a small category (without loops) to the category of groups and in this Abels and Holz nerve case this leads to a simplex of groups, and a sort of fundamental domain simplex for the action.
My query is whether anyone knows of a similar idea but with a n-categorical context. Here the group is an ordinary group and the nerve is actually a simplicial complex (and if you replace it by a simplicial set life seems to get complicated). We have talked of 2-group actions and I remember looking at Jim Dolan’s notes on 2-geometries, but never finished them and the ideas have slipped. Any pointers (or help?) would be appreciated.
We have talked of 2-group actions and I remember looking at Jim Dolan’s notes on 2-geometries, but never finished them and the ideas have slipped. Any pointers (or help?) would be appreciated.
One elegant way to talk about actions $\rho$ of an $n$-group $G$ (in a suitable ambient context) on an object $V$ is to identify them with homotopy fiber sequences of the form
$V \to Q \stackrel{\rho}{\to} \mathbf{B}G \,.$Here the middle piece $Q := V//G$ is the corresponding $n$-action groupoid.
I know of that, but what I do not see is the ’geometric’ context, as the situations that I have been stumbling on seem to reply on direct small models of objects (simplicial complex or small category without loops) where explicit calculations are sort of more evident. The homotopy fibre method is used but can tend to smudge out all that geometry, hiding it under a weight of homotopy structure that partially obscures what is going on. (The usual way of dealing with the Volodin models of K-theory is exactly a case in point, where Suslin uses the homotopy fibre method early on but crushes together the algebraic structure. I think however that using the n-action groupoid may be much better as it does retain more of the action information near the surface of the structure. I will have another look at that later on. Thanks.
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