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This is quite a broad question regarding constructions of categories of sheaves in geometry.

Let $\textbf{Sch}$ denote the category of schemes. Let $\textbf{SchAff}$ denote the full subcategory of affine schemes. Recall that this category is canonically isomorphic to the opposite of the category of commutative rings, which we denote by $\textbf{ComR}$. To any $X\in \textbf{Sch}$, we have the Abelian category of quasi-coherent $\mathcal{O}_X$-modules, which we denote by $\mathbf{QCoh(X)}.$ For an affine scheme $Spec(R)$ this is just the category of $R$-modules. Pullback allows us to arrange all this data into a 2-functor:

$$f^*: \textbf{Sch}^{op} \rightarrow \textbf{Ab},$$

where $\textbf{Ab}$ denotes the 2-category of Abelian categories. If we restrict this functor to $\textbf{ComR}$ it is particularly easy to understand: it sends a ring to its category of modules and pullback is just tensor product.

This type of construction occurs all the time in geometry. For example, instead of quasi-coherent sheaves, we could take etale, or $\ell$-adic sheaves. If we restricted to smooth, complex schemes over $Spec(\mathbb{C})$ we could take $D$-modules. If we restrict to schemes over a finite field we could take arithmetic $D$-modules.

My first question is the following:

After strictifying, are all of these pullback functors right Kan-extensions from their restriction to commutative rings? If we took pushforward instead of pullback would they be left Kan-extensions?

We could also beef things up to the derived world. For example, in the quasi-coherent case, pullback would give the $\infty$-functor:

$$f_{\infty}^*: N(\textbf{Sch}^{op}) \rightarrow \textbf{Stab}_{\infty},$$ $$\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; X \rightarrow D(X),$$ where $\textbf{Stab}_{\infty}$ is the $\infty$-category of stable $\infty$-categories, $N(\textbf{Sch}^{op})$ is the nerve, and $D(X)$ is the ($\infty$-categorical) derived category of quasi-coherent sheaves on $X$.

As above, this basic construction can be carried out in many different settings.

My second question is the following:

Are all of these pullback $\infty$-functors right Kan-extensions (in the $\infty$-categorical sense) from their restriction to $N(\textbf{ComR})$?

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    $\begingroup$ Kan extensions of what? Are you asking if we can determine $\mathbf{Qcoh}(X)$ for a scheme $X$ from just knowing the case where $X$ is affine? The answer there is yes: this is because $\mathbf{Qcoh}$ is a stack for the Zariski (indeed, fpqc) topology. $\endgroup$ – Zhen Lin Aug 15 '13 at 19:10
  • $\begingroup$ I suppose I'm asking about other cases too, like $\ell$-adic sheaves, D-modules or indeed the conjectural category of mixed motives. $\endgroup$ – Alexander Paulin Aug 15 '13 at 20:25
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    $\begingroup$ When you unwind the definitions, doesn't the right Kan property just boil down to the fact that you can glue quasi-coherent sheaves over a Zarski (hyper-) cover? So it should hold for any Zariski local category, like $\ell $-adic sheaves or D-modules. Or does strictifying mess things up? The argument doesn't dualise, though, so I don't know about the left Kan property for pushforward. I also don't know enough about $\infty $ categories to be sure, but presumably in this case it will again just boil down to the fact that the $\infty$-categorical derived caregory is Zariski local? $\endgroup$ – ChrisLazda Aug 16 '13 at 9:25
  • $\begingroup$ Is the point really that it should be the case as long as your theory is local for some (not necessarily the Zariski) Grothendieck topology on $\textbf{SchAff}$. $\endgroup$ – Alexander Paulin Aug 16 '13 at 11:04
  • $\begingroup$ Yes, exactly, for $f^*$ at least. $\endgroup$ – ChrisLazda Aug 16 '13 at 14:25

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