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Putting together
$SuperSmooth\infty Grpd \coloneqq Sh_\infty(CartSp \rtimes SuperPoint)$and complex analytic ∞-groupoid, presumably there’s nothing to stop
$Super \mathbb{C} Analytic \infty Grpd \coloneqq Sh_\infty(CplxMfd \rtimes SuperPoint)$to generalise complex supermanifold.
Yes, that should work without problem.
Would we still say that shape is $loc_{\mathbb{R}}$ and that $R$ is $loc_{\mathbb{R}^{0|1}}$ in this complex setting?
So in the complex setting shape would not be localization at $\mathbb{R}$. It might be that it is now localization at $\mathbb{C}$, but I haven’t really thought this through.
But $Rh \simeq loc_{\mathbb{R}^{0|1}}$ 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 $loc_{\mathbb{A}^1}$ and $loc_{\mathbb{R}^{0|1}}$, but isn’t it possible that in some other nontrivial model of solid cohesion that we wouldn’t have $Rh$ as $loc_{\mathbb{R}^{0|1}}$?
Yes, there could be models of solid cohesion where $Rh$ is not $loc_{\mathbb{R}^{0\vert 1}}$. 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 $Spec(\mathbb{R}[\epsilon]/(\epsilon^2))$, but should fairly generally be localization at the collection at all infinitesimally thickened points.
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