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I apologize if this question is slightly vague but I don't know how to ask it non-vaguely. Moreover, my question is about an ideal situation. If there's a close answer which doesn't satisfy all the conditions it will still worth a lot!

Let $\mathcal{C}$ be the category of smooth (/analytic) manifolds.

The question is about existence of certain definitions which admit certain properties. Here are the definitions I require:

  1. A grothendieck topology $J$ on $\mathcal{C}$
  2. A closed symmetric-monoidal stable $\infty$-category $\mathcal{V}$ (or triangulated if you prefer, though descent issues should then be handled carefully).

Here are the properties I'd like them to satisfy:

  1. The cohomology theory of $\mathcal{V}$-valued $J$-sheaves on $\mathcal{C}$ has all six functors.

This means that for every map $f: X \to Y$, we have four functors: $f_!,f^!,f^*,f_*$ on the corresponding categories of sheaves with the familiar adjunctions between them. Moreover there are the functors $\otimes, \mathcal{Hom(-,-)}$.

  1. There's a natural way to assign to any differential complex of vector bundles $E_\bullet$ on a manifold $X$ its corresponding complex of sheaf of (compactly supported-) sections (smooth/analytic/distributional/sobolev etc...) as an object in $Sh(X,\mathcal{V})$

In other words for every manifold $X$ there exist (well behaved) sheaves of section functors $Diff(X) \overset{\Gamma^{\#}}{\to} Sh(X,\mathcal{V})$ with $\# = \text{smooth, analytic, distributional, continuous, sobolev etc...}$. Where $Diff(X)$ is the $dg$-category of complexes of differential operators between vector bundles and morphism complexes are the complexes of differential operators between shifts.

  1. Verdier duality $Sh(X,\mathcal{V})$ is compatible with schwartz kernels when they exist.

This means that in "favorable situations" we'd want sections of $\mathbb{D}\mathcal{E} \otimes \mathcal{F}$ to correspond to certain kernels of integral operators living in $\mathcal{D}(X \times X, E^*\otimes Dens_X \otimes F)$ (with some regularity in the $F$ direction).

This is not the most complete list one could write, and it's terribly imprecise. I've avoided being too precise because i'm not sure what really one should ask for. What I wrote was only for the purpose of figuring out whether this sort of question has some answers in the literature.

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  • $\begingroup$ It might be worth pointing out that the category of (finite-dimensional, paracompact, hausdorff) manifolds is a full subcategory of R-algebras, by taking M to $C^\infty(M)$. Actually, there is a definition of smooth R-algebra, and commutative smooth R-algebras are precisely those coming from manifolds (if I remember correctly). The reason I mention it is that this might help for point 3. $\endgroup$
    – Yosemite Sam
    Commented Oct 13, 2017 at 2:17
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    $\begingroup$ @YosemiteSam: There is a counterexample to your claim here: mathoverflow.net/questions/86426/… $\endgroup$
    – Dmitri Pavlov
    Commented Oct 14, 2017 at 21:04
  • $\begingroup$ @DmitriPavlov you mean I should have said second-countable instead of paracompact? $\endgroup$
    – Yosemite Sam
    Commented Oct 14, 2017 at 23:02
  • $\begingroup$ @YosemiteSam: Either second-countable, or C^∞-rings instead of R-algebras. $\endgroup$
    – Dmitri Pavlov
    Commented Oct 15, 2017 at 4:30
  • $\begingroup$ @DmitriPavlov, thank you for spotting this. I happily retract/edit my original statement (which I incorrectly copied from Nestruev's book). I'm not sure I even care about non-second countable manifolds! $\endgroup$
    – Yosemite Sam
    Commented Oct 15, 2017 at 7:16

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