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    • CommentRowNumber1.
    • CommentAuthorMike Shulman
    • CommentTimeDec 9th 2010

    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.

    • CommentRowNumber2.
    • CommentAuthorTim_Porter
    • CommentTimeDec 9th 2010

    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.

    • CommentRowNumber3.
    • CommentAuthorDavidRoberts
    • CommentTimeDec 10th 2010

    The correct notion of a 2-transitive action of the 2-group GG on the groupoid XX is of course ’the restricted action functor χ:*×GX×GX\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)Stab(x) of the chosen point x:*Xx:*\to X - it is the essential fibre if χ\chi over xx. Note that Stab(x)GStab(x) \to G is faithful, so is a sub-2-group. Then XX (if I remember the calculations I did last year correctly) should be a weak quotient G//Stab(x)G//Stab(x). It is obvious that XX is the coproduct of the full subgroupoids consisting of orbits, though I wonder if there is another way of writing a general GG-groupoid as a colimit over transitive bits (not necessarily full subgroupoids) that isn’t just a coproduct.

    • CommentRowNumber4.
    • CommentAuthorMike Shulman
    • CommentTimeDec 10th 2010

    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).

    • CommentRowNumber5.
    • CommentAuthorDavidRoberts
    • CommentTimeDec 10th 2010

    Yes - identifying 2-groups with pointed connected 2-types, we can always cover a sub-2-group of GG by one that is 1-connected. There is then only a set of 1-connected sub-2-groups of a given 2-group

    • CommentRowNumber6.
    • CommentAuthorTim_Porter
    • CommentTimeDec 10th 2010

    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 GO_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 GO_G is not clear to me however.

    • CommentRowNumber7.
    • CommentAuthorMike Shulman
    • CommentTimeDec 11th 2010

    Well, I think for a 1-group G the category of G-sets is the free extensive completion of O GO_G, aka Fam(O G)Fam(O_G); is that the sort of thing you mean?

    • CommentRowNumber8.
    • CommentAuthorTim_Porter
    • CommentTimeDec 11th 2010

    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.