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I will write $\operatorname{Diff}$ to denote a category of generalized smooth spaces e.g. $Sh(\mathsf{CartSp})$. Is there a version of $\operatorname{Diff}$ for which there exists a functor $\mathcal{S^{\infty}}:\mathbb{R-}Sch \to \operatorname{Diff}$ which could legitimately be called the smoothing functor?

I'm not nearly experienced enough in any of the relevant fields to conjecture what such a functor must satisfy so I'm deliberately leaving the vaguenesss in the above question. However, for the sake of completeness I will state some properties I think are relavant here:

  • Sends the affine space to euclidean space

  • Sends smooth integral finite type proper $\mathbb{R}$-schemes to compact (analytic) smooth manifolds.

  • Complexification followed by analytification and then the forgetful functor from complex analytic spaces to $\operatorname{Diff}$ should in some way relate to this smoothing functor. (I'm not even sure that there's a "forgetful" functor from complex analytic spaces to generalized smooth spaces so this is conjectural too).

  • Sends proper maps to proper maps

Already I have the feeling that philosophically this is wrong since it seems to me that the study of real singularities goes very often through complexification. Arguments for why such a functor would be an atrocity are very welcome as well.

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    $\begingroup$ I would think the obvious thing would be to send a finite type affine scheme to its set of real points as a subset of $\mathbb{R}^n$ (with the induced smooth structure), and then to extend to arbitrary schemes by taking colimits. $\endgroup$
    – Eric Wofsey
    Commented Apr 19, 2016 at 23:11
  • $\begingroup$ You will probably lose information of the sort that $C^\infty$ schemes would preserve, and so you might have a factorisation of the sort $Schemes/\mathbb{R} \to C^\infty Schemes \to Diff$ $\endgroup$
    – David Roberts ♦
    Commented Apr 20, 2016 at 6:54
  • $\begingroup$ @EricWofsey I thought as much, and there might not even be a better option here. My question then is what are the properties of such a functor? There are a lot of theorems about the analytification functor and it seems to be not so bad. Can the same be said here? $\endgroup$
    – Saal Hardali
    Commented Apr 20, 2016 at 13:44
  • $\begingroup$ If I understand correctly what you're asking, you might find something here math.boun.edu.tr/instructors/wdgillam/diffspace.pdf (and references) $\endgroup$
    – Mattia Talpo
    Commented Apr 20, 2016 at 17:01
  • $\begingroup$ @DavidRoberts I know almost nothing about $C^{\infty}$-schemes. Is there an obvious functor $C^{\infty}Sch \to Diff$ with good properties? $\endgroup$
    – Saal Hardali
    Commented Apr 23, 2016 at 9:33

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