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I finally gave spectral super-scheme an entry, briefly stating the idea.
This goes back to the observation highlighted in Rezk 09, section 2. There is some further support for the idea that a good definition of supergeometry in the spectrally derived/E∞ context is nothing but E∞-geometry over even periodic ring spectra. I might add some of them later.
Thanks to Charles Rezk for discussion (already a while back).
Does this relate to any grading by 𝕊, or augmentation or any other such structure?
That ℤ-grading on π•(E), that’s π0(𝕊)-grading.
Indeed, by Sagave-Schlichtkrull, theorem 1.7-1.8 every connective E∞-ring is canonically 𝕊-graded.
So E∞-geometry already is the higher analog of super-geometry, but of ℤ-graded supergeometry, not of the proper ℤ/2-graded supergeometry.
The remaining problem is to turn this categorified/homotopified ℤ-graded supergeometry into genuine ℤ/2-graded supergeometry. This is achieved by requiring an even periodiv ground ∞-ring
I have added a brief mentioning of the corresponding concept of the spectral superpoint.
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