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There are two parts to my question:

Question #1: First a sanity-check: Am I right in that the category of quasi-coherent sheaves (over e.g. an affine scheme) is not a topos?

My reasoning is thus:

It is well-known that there is no non-trivial category that is both a topos and an abelian category -- since the topos is cartesian closed, it cannot have a 0 object without becoming trivial (see here: Can a topos ever be an abelian category?).

However, by the affine Serre theorem, it is well-known that the category of $R$-modules is equivalent to $QCoh(Spec(R))$, or the category of quasi-coherent sheaves on the space $Spec(R)$. Since the category of $R$-modules is abelian, it follows that the category $QCoh(Spec(R))$ is not a topos despite the fact that it is some kind of category of sheaves (satisfying some extra property). Am I correct?

Question #2) If this is true, can someone give me a conceptual understanding of why one would wish to present the category of $R$-modules as the category of quasi-coherent sheaves, even if it's not a topos?

I get the rough idea that we're constructing the category of $R$-modules from the categories of modules over local rings $R_{\mathfrak{p}}$ -- but I'm not sure I understand this very clearly. Can someone explain this a bit more clearly? Where does quasi-coherence figure in this picture?

Further, suppose we have a category of sheaves over the site $Spec(R)$ endowed with the Zariski topology, which is a topos (right?). Is the internal category of $R$-modules within this topos equivalent to the category of quasi-coherent sheaves? Or is that not the right way to think about the category of quasi-coherent sheaves? (Perhaps the answer is sitting in Ingo's PhD thesis somewhere, but I haven't found it...)

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    $\begingroup$ One simple reason to represent $R$-modules as some kind of sheaves over $\operatorname{Spec}R$ is that you want to find a definition that works also for non-affine schemes... $\endgroup$ Apr 10, 2021 at 18:05
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    $\begingroup$ If you study algebraic geometry, you will for one reason or another probably be interested in sheaves on your spaces anyway. Quasicoherent sheaves are those which, at least locally, behave well under localization. Alternatively there is the (classical) description as sheaves which locally admit presentations. These categories end up being equivalent to categories of modules, and generalize nicely to other schemes, despite the fact they aren't topoi. $\endgroup$
    – Wojowu
    Apr 10, 2021 at 18:21
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    $\begingroup$ As a matter of fact $\mathrm{Qco}(X)$ is a Grothendieck category, i.e. an abelian category with a set of generators and that satisfies AB5, directed colimits are exact. These categories may be considered as the additive analogous of a topos (as the category of abelian objects in a topos with enough points is). $\endgroup$
    – Leo Alonso
    Apr 10, 2021 at 22:30
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    $\begingroup$ Toposes are categories of sheaves of sets. As Leo Alonso says, the abelian analogue of a topos is a Grothendieck abelian category. $\endgroup$
    – Zhen Lin
    Apr 10, 2021 at 23:19
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    $\begingroup$ To amplify: the only abelian category which is a topos is the zero category. In fact, the only additive category which is a topos is the zero category. In fact, the only category with a zero object which is a topos is the zero category. In a topos which is not the zero category, there is no morphism $1 \to \emptyset$ where $1$ is the terminal object and $\emptyset$ is the initial object. Whereas in a category with zero object, these two objects are isomorphic. $\endgroup$
    – Tim Campion
    Apr 11, 2021 at 2:19

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