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If G is a (discrete 1-)group, then any G-set decomposes uniquely as a coproduct of transitive G-sets, each of which is the quotient of G by a subgroup. Is something analogous true for actions of 2-groups on groupoids, or for n-groups on (n-1)-groupoids? I remember there was a lot of Cafe discussion about higher Klein geometry, subgroups and homogeneous spaces of higher groups, but I don’t remember whether something like this came up.
Some of the stuff in Magnus Forrester-Barker’s thesis may be relevant to this. He looked at representations of crossed modules on the naive 2-vector spaces, and did quite a few calculations.It is here in case it is of use.
The correct notion of a 2-transitive action of the 2-group $G$ on the groupoid $X$ is of course ’the restricted action functor $\chi:*\times G \to X \times G \to X$ is essentially surjective’. This map can then be used to describe the weak stabiliser $Stab(x)$ of the chosen point $x:*\to X$ - it is the essential fibre if $\chi$ over $x$. Note that $Stab(x) \to G$ is faithful, so is a sub-2-group. Then $X$ (if I remember the calculations I did last year correctly) should be a weak quotient $G//Stab(x)$. It is obvious that $X$ is the coproduct of the full subgroupoids consisting of orbits, though I wonder if there is another way of writing a general $G$-groupoid as a colimit over transitive bits (not necessarily full subgroupoids) that isn’t just a coproduct.
Of course; thanks. I guess the only disconcerting bit (to me) is that a 2-group generally has a proper class of “subgroups.” Which means that unlike for 1-groups, the transitive actions don’t form a small site that presents the classifying (2,1)-topos of a 2-group. I suppose they are an “essentially small site” in the usual sense, though (having a small family of objects that cover all the rest).
Yes - identifying 2-groups with pointed connected 2-types, we can always cover a sub-2-group of $G$ by one that is 1-connected. There is then only a set of 1-connected sub-2-groups of a given 2-group
Surely the point is to categorify the statement about ’decompositions into coproducts’ and so to state it as a property of the category of G-sets. What is the 2-cat. version of that property. That has never been completely clear to me. I have had thoughts about it and there is the category, $O_G$, of orbits of a group (which Ronnie, Marek Golasinski, Andy Tonks and myself used in two papers on equivariant stuff), see also Ieke Moedijk’s papers with Svenson. What the decomposition theorem says about $O_G$ is not clear to me however.
Well, I think for a 1-group G the category of G-sets is the free extensive completion of $O_G$, aka $Fam(O_G)$; is that the sort of thing you mean?
Exactly. The 2-category of G-groupoids (G a 2-group) should have some similar property (suitably categorified). That suggests that the arguments for decomposing G-sets as coproducts of ’simpler’ one should generalise ??? I am getting lost. The other idea (related probably) is that the action of a 2-groupoid on its underling groupoid needs studying.
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