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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeApr 14th 2014

    created equivariant homotopy theory – table displaying the various cohesive \infty-toposes and their bases \infty-toposes (for inclusion in “Related entries” at the relevant entries)

    • CommentRowNumber2.
    • CommentAuthorDavid_Corfield
    • CommentTimeApr 14th 2014

    I remember you told me once that there was no interesting super version of ETopGrpdETop \infty Grpd. Does then ETopGrpd cell=PSh (Orb)ETopGrpd^{cell} = PSh_{\infty}(Orb) prevent any interesting mixture of equivariance with super-extensions?

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeApr 14th 2014
    • (edited Apr 14th 2014)

    So OrbOrb is an enhanced version of the full sub-\infty-category of SmoothGrpdSmooth\infty Grpd on the objects of the form BG\mathbf{B}G for GG a compact Lie group. That sub-\infty-category genuinely knows smooth structure and does have non-trivial superifications.

    Passing to OrbOrb here means replacing external hom-groupoids between these BG\mathbf{B}Gs with geometic realization of internal hom-groupoids. That is a curious step which somehow governs the whole theory, but which remains a bit subtle, to my mind. I am not sure if superification goes along with this. I really wish I had a better concrete feeling for this step…

    • CommentRowNumber4.
    • CommentAuthorDavid_Corfield
    • CommentTimeApr 14th 2014

    I see. Yes, curious. I wonder why that’s seen as the natural thing to do.

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeApr 14th 2014
    • (edited Apr 14th 2014)

    I suppose one way to think of this is that this is the natural way to have the geometry enter the hom-spaces, so that it is in a way “more geoemtric” than not doing it.

    As I remarked elswhere once when we discussed this, this is inded one of the very things that Lawvere pointed out cohesion is good for back in “axiomatic cohesion”: given a cohesive topos (or \infty-topos, same story): then it has three kinds of hom-objects:

    1. the external hom space H(X,A)Grpd\mathbf{H}(X,A) \in \infty Grpd;

    2. the internal one [X,A]H[X,A]\in \mathbf{H} (so far as for any topos);

    3. the realization Π[X,A]Grpd\Pi[X,A]\in \infty Grpd.

    It’s the fact that Π\Pi preserves products which says that Π[X,A]\Pi[X,A] gives a secondary enrichment of H\mathbf{H} in \infty-groupoids. The self-enrichment in point 2 contains “more information” than both 1 and 3, but leads us to enriched \infty-category theory. The third point retains as much of the geometric information in 2 as possible while still staying non-trivially enriched.

    So that’s what the global equivariant indexing category is: the full sub-\infty-category of that secondary enrichment of ETopGrpdETop\infty Grpd on the deloopings of compact Lie groups.

    From that perspective it is quite natural to consider this.

    But even if conceptually naturally, I still find it hard (harder) to reason about this, to work with it.

    On the other hand, depending on what you have in mind with supergeometry here:

    you can take the \infty-topos SmoothSuperGrpdSmoothSuper\infty Grpd and then make it GG-equivariant by passing to the \infty-topos [Orb,SmoothSuperGrpd][Orb,SmoothSuper\infty Grpd].