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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeDec 13th 2010
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   Author: Urs
   Format: MarkdownItexI 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

   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

2. • CommentRowNumber2.
   • CommentAuthorzskoda
   • CommentTimeDec 14th 2010
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   Author: zskoda
   Format: MarkdownItex"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.

   “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.

3. • CommentRowNumber3.
   • CommentAuthorUrs
   • CommentTimeDec 14th 2010
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   Author: Urs
   Format: MarkdownItexThe 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: 1. there is a canonical bifibration $Mod$; 1. 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.

   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:

   1. there is a canonical bifibration $Mod$;

   2. 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.

4. • CommentRowNumber4.
   • CommentAuthorzskoda
   • CommentTimeDec 14th 2010
   • (edited Dec 14th 2010)
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   Author: zskoda
   Format: MarkdownItex> 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.

   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.

5. • CommentRowNumber5.
   • CommentAuthorUrs
   • CommentTimeDec 15th 2010
   • (edited Dec 15th 2010)
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   Author: Urs
   Format: MarkdownItex> 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]].

   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.

6. • CommentRowNumber6.
   • CommentAuthorzskoda
   • CommentTimeDec 15th 2010
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   Author: zskoda
   Format: MarkdownItexRight, 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.

   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.