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    • CommentRowNumber1.
    • CommentAuthorDavid_Corfield
    • CommentTimeMay 5th 2016
    • (edited May 5th 2016)

    Given the HoTT activity we’ve just seen at regular action, is there anything similarly interesting to say at transitive action for \infty-actions?

    We have a result there

    Let ρ:G×XX\rho : G \times X \to X be a transitive action and suppose that XX is inhabited. Then ρ\rho is equivalent to the action of GG by multiplication on a coset space G/HG/H, where the subgroup HH is taken as the stabilizer subgroup H=G x={gGgx=x}H = G_x = \{ g \in G \mid g x = x \} of some arbitrary element xXx \in X. In particular, the transitivity of ρ\rho guarantees that the GG-equivariant map G/HXG/H \to X defined by gHgxg H \mapsto g x is a bijection.

    I guess Urs has done everything at stabilizer+group#ForInfinityGroupActions, but with a transitive action BStab ρ(x)\mathbf{B}Stab_\rho(x) becomes X/GX/G.

    Given that transitivity concerns connectness of quotient, there could be a generalization to kk-connectness. There’s also nn-transitivity to consider.

    • CommentRowNumber2.
    • CommentAuthorDavidRoberts
    • CommentTimeMay 5th 2016

    An action would be transitive if, I guess, x yx=y\prod_x \prod_y x=y is inhabited, for x,y:X//Gx,y:X//G

    • CommentRowNumber3.
    • CommentAuthorDavid_Corfield
    • CommentTimeMay 5th 2016
    • (edited May 5th 2016)

    That would allow empty XX. Should we do that? The main definition speaks of a single orbit, yet later we’re supposing that XX is inhabited.

    If we opt for isContr(||X//G|| 0)isContr(||X//G||_0), that would have to have XX inhabited.

    • CommentRowNumber4.
    • CommentAuthorDavid_Corfield
    • CommentTimeMay 5th 2016

    Looking about, the consensus is that XX is non-empty. So then we should stop after ’action’ in

    Let ρ:G×XX\rho : G \times X \to X be a transitive action and suppose that XX is inhabited,

    or is there something subtle going on with ’inhabited’?

    I see Mike and Toby were thrashing out what it meant at inhabited object.

    • CommentRowNumber5.
    • CommentAuthorDavidRoberts
    • CommentTimeMay 5th 2016
    • (edited May 5th 2016)

    David #3 that’s probably what I was thinking. I guess ’exactly one orbit’ is the thing that is used.

  1. In terms of whether the definition of transitive action should require the set to be inhabited, I’ve seen it both ways and like the more general definition. Something in the background for this particular fact is that really there is a 1-to-1 correspondence between pointed transitive G-actions and subgroups of G, by the stabilizer and coset action constructions. I do not know whether this has been explained nicely somewhere in terms of an equivalence of categories–I first learned about it from Samuel Vidal’s thesis and later in a nice series of blog posts by Qiaochu Yuan. I am kind of offline for a few days, but it would be great to incorporate this into the article at some point.

    • CommentRowNumber7.
    • CommentAuthorDavid_Corfield
    • CommentTimeMay 5th 2016

    As I said in #1, all ingredients for the full \infty-action story are at stabilizer group.

    • CommentRowNumber8.
    • CommentAuthorDavid_Corfield
    • CommentTimeNov 30th 2016

    We never got round to settling which version (inhabited or possibly empty) to adopt. What we had was inconsistent, so I added to make it at least consistent. But then now the discussion of kk-transitivity needs attention. Do people want a 3-transitive action on XX to require at least 3 elements?