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I worked a bit on quasicoherent (infinity,1)-sheaf:
I polished the account of the model-category theory presentation by Toen-Vezzosi a little
Then I added the slick general abstract definition in terms of the tangent (oo,1)-category that we once discussed, but which nobody had yet filled into this entry
“have to” in the sentence just before Definition section does not parse.
I do not quite understand the entry. I mean when one has a space $X$, then quasicoherent sheaf is an imprecise abbreviation of “quasicoherent sheaf of O-modules” where O is a distinguished sheaf of rings on $X$. The notion of course depends on the ring what I do not clearly see in the infinite version.
On the other hand, in 1-categorical situation one can consider more generally quasicoherent modules in any fibered category, nothing linear or abelian needed, just the notion of pullback and hence of cartesian sections of the projection of the fibered category. Hence I expect similar generality in infinite context, not only stable case.
The general abstract definition there gives $\mathcal{O}$-modules by the fact and in the sense that the fibers of $T C^{op} \to C^{op}$ over any $A$ play the role of the stable $\infty$-category $A Mod$ of $A$-modules.
Concerning the generalization to other bifibrations: okay, sure, the general abstract formulation that I gave consists of 2 points:
there is a canonical bifibration $Mod$;
a quasicoherent thing of modules on $X$ is just a morphism of $X$ into $Mod$.
So of course one can replace $Mod$ here by some other bifibration.
a quasicoherent thing of modules on X is just a morphism of X into Mod.
Where is the cartesianess ? I mean an arbitrary section is not cartesian, for usual fibered categories one has modules and quasicoherent modules. The fibered category is the same, but the sections are either cartesian or non-cartesian, what makes a difference. “Just a morphism” confuses me. Second in classical case one needs a fibered category, not a bifibered.
Where is the cartesianess ?
I was freely switching perspectives using the Grothendieck construction. For $Mod$ the given fibration, think of it as the corresponding stack. Then morphisms $X \to Mod$ are morphisms of stacks.
We spelled all that out once in detail at quasicoherent sheaf.
Right, Grothendieck construction is wiping out the difference between the quasicoherent and not, everything becomes quasicoherent. But one usually wants to have general O-modules and just a criterium which ones are quasicoherent; investigating this question of quasicoherence in concrete examples is matter of theorems rather than tautology. I am happy with your answer, but I would also like to have the section perspective which can simultaneously accomodate sheaves of modules and sheaves of quasicoherent modules in the same fibered category.
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