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Putting together
SuperSmooth∞Grpd≔Sh∞(CartSp⋊SuperPoint)and complex analytic ∞-groupoid, presumably there’s nothing to stop
SuperℂAnalytic∞Grpd≔Sh∞(CplxMfd⋊SuperPoint)to generalise complex supermanifold.
Yes, that should work without problem.
Would we still say that shape is locℝ and that R is locℝ0|1 in this complex setting?
So in the complex setting shape would not be localization at ℝ. It might be that it is now localization at ℂ, but I haven’t really thought this through.
But Rh≃locℝ0|1 would still hold, and I would still write it this way, yes.
Ah, so I wonder how things should be said in a model independent way.
We could have loc𝔸1 and locℝ0|1, but isn’t it possible that in some other nontrivial model of solid cohesion that we wouldn’t have Rh as locℝ0|1?
Yes, there could be models of solid cohesion where Rh is not locℝ0|1. We discussed this before, that it seems unclear and in fact unlikely that solid cohesion as presently set up is sufficient to fully characterize supergeometry.
But slight variants of this localization business will work more generally. For instance I suppose it follows readily that in complex analytic cohesion the shape modality is given by localization at the collection of all all Stein spaces.
Similarly, reduction will maybe not in general be localization at Spec(ℝ[ε]/(ε2)), but should fairly generally be localization at the collection at all infinitesimally thickened points.
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