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am starting an entry smooth spectrum (in the sense of smooth infinity-groupoid). But nothing much there yet.
And are there parameterized smooth spectra forming the tangent category to $Smooth \infty Grpd$, as parameterized spectra do to $\infty Grpd$?
Yes, that’s the story already discussed at tangent cohesion: the tangent topos of smooth $\infty$-groupoids is the “extension” of that by the smooth spectra. I just thought that eventually this should not be hidden in the general discussion at “tangent cohesion” but get its own entry here. I hope to be expanding that entry soon. Just need to get a bunch of other items out of the way right now…
added a Properties-section From chain complexes of smooth modules, so far containing the proposition asserting that presheaves of $\mathbf{R} \coloneqq C^\infty(-,\mathbb{R})$-modules are already $(\infty,1)$-sheaves of spectra (without further $\infty$-stackification).
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