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    • CommentRowNumber1.
    • CommentAuthorDavid_Corfield
    • CommentTimeMay 7th 2015

    Putting together

    SuperSmoothGrpdSh (CartSpSuperPoint) SuperSmooth\infty Grpd \coloneqq Sh_\infty(CartSp \rtimes SuperPoint)

    and complex analytic ∞-groupoid, presumably there’s nothing to stop

    SuperAnalyticGrpdSh (CplxMfdSuperPoint) Super \mathbb{C} Analytic \infty Grpd \coloneqq Sh_\infty(CplxMfd \rtimes SuperPoint)

    to generalise complex supermanifold.

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeMay 7th 2015

    Yes, that should work without problem.

    • CommentRowNumber3.
    • CommentAuthorDavid_Corfield
    • CommentTimeMay 13th 2015
    • (edited May 13th 2015)

    Would we still say that shape is loc loc_{\mathbb{R}} and that RR is loc 0|1loc_{\mathbb{R}^{0|1}} in this complex setting?

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeMay 13th 2015

    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 Rhloc 0|1Rh \simeq loc_{\mathbb{R}^{0|1}} would still hold, and I would still write it this way, yes.

    • CommentRowNumber5.
    • CommentAuthorDavid_Corfield
    • CommentTimeMay 13th 2015
    • (edited May 13th 2015)

    Ah, so I wonder how things should be said in a model independent way.

    We could have loc 𝔸 1loc_{\mathbb{A}^1} and loc 0|1loc_{\mathbb{R}^{0|1}}, but isn’t it possible that in some other nontrivial model of solid cohesion that we wouldn’t have RhRh as loc 0|1loc_{\mathbb{R}^{0|1}}?

    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeMay 13th 2015
    • (edited May 13th 2015)

    Yes, there could be models of solid cohesion where RhRh is not loc 0|1loc_{\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([ε]/(ε 2))Spec(\mathbb{R}[\epsilon]/(\epsilon^2)), but should fairly generally be localization at the collection at all infinitesimally thickened points.