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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)
I remember you told me once that there was no interesting super version of $ETop \infty Grpd$. Does then $ETopGrpd^{cell} = PSh_{\infty}(Orb)$ prevent any interesting mixture of equivariance with super-extensions?
So $Orb$ is an enhanced version of the full sub-$\infty$-category of $Smooth\infty Grpd$ on the objects of the form $\mathbf{B}G$ for $G$ a compact Lie group. That sub-$\infty$-category genuinely knows smooth structure and does have non-trivial superifications.
Passing to $Orb$ here means replacing external hom-groupoids between these $\mathbf{B}G$s 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…
I see. Yes, curious. I wonder why that’s seen as the natural thing to do.
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:
the external hom space $\mathbf{H}(X,A) \in \infty Grpd$;
the internal one $[X,A]\in \mathbf{H}$ (so far as for any topos);
the realization $\Pi[X,A]\in \infty Grpd$.
It’s the fact that $\Pi$ preserves products which says that $\Pi[X,A]$ gives a secondary enrichment of $\mathbf{H}$ 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 $ETop\infty Grpd$ 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 $SmoothSuper\infty Grpd$ and then make it $G$-equivariant by passing to the $\infty$-topos $[Orb,SmoothSuper\infty Grpd]$.
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