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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 G on the groupoid X is of course ’the restricted action functor χ:*×GX×GX is essentially surjective’. This map can then be used to describe the weak stabiliser Stab(x) of the chosen point x:*X - it is the essential fibre if χ over x. Note that Stab(x)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.

    • 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 G 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, OG, 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 OG 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 OG, aka Fam(OG); 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.