Oh I remember looking at the paper, but not the expression. I should have followed the link.

]]>the notion of vectorial bundle is used by Gomi to model twisted K-theory using geometric objects. Have a look at the entry. These cocycles are not exactly the same as those of Karoubi, but their equivalence classes do model the same K-theory.

]]>Wait a second there is a strange section titled “Virtual vector bundles” in vector bundle – but instead of virtual vector bundles (like in Karoubi’s book: allowing vector bundles of different dimension/rank on different connected components of the base and also allowing negatives of vector bundles), there is a discussion on $Z_2$ gradings which is related but not exactly what I recall from Karoubi (the pairs mean Grothendieck construction so one has negatives, but not the case of only locally constant rank) and there is also strange remark about “vectorial vector bundles” (??) there which I can not relate to virtual v.b.

]]>For usual vector bundles, I put some pain into producing a reference section in vector bundle with main monographs having lots of material on basic of the vector bundles.

]]>I don't know, this is another point of view, equating D-modules with qcoh O-modules for another space, which I was not talking about. I put the reference of Beilinson-Bernstein on proof of Jantzen conjecture which is the main classical reference on D-affinity and algebraic D-geometry. I added some nlab links to D-module and the elementary textbook from Coutinho. I should say that I like your treatment of the interpretation via de Rham space.

]]>by "ordinary" I mean "1-categorical" as opposed to the oo-version. I have now tried to clarify this.

By the way, what's a good reference that we could cite at D-module for the equivalence of D-modules with qcoh sheaves on the deRham space?

]]>There is a wording ordinary quasicoherent sheaves in the entry. As if D-modules were not ordinary quasicoherent sheaves, Full name is quasicoherent sheaves of O-modules in the first case and in the secondc case are the quasicoherent sheaves of D-modules. They are both quasicoherent sheaves in most ordinary sense, the difference is just weather of O-modules or of D-modules. The fact that by abuse of language geometers do not say qcoh sheaves of D-modules but only D-modules and not quasicoherent sheaves of O-modules but only qcoh sheaves or only qcoh modules is a different thing,

One should see if the qcoh 2-sheaves in the sense recently introduced by Toen in the arxiv preprint fits in this framework.

]]>so we can talk of tangent category each time we have a notion of "infinitesimally close objects". however, we know from classical differential geometry that there is another approach to tangent spaces which needs not a notion of infinitesimally close points (even if this naive idea is motivating the construction and actually somehow hidden into it). namely, to construct one considers all (smoth) paths stemming from , i.e. the whole overcategory , and then quotients by a suitable equivalence relation. and, as usual, instead of quotienting we can add an isomorphisms between paths we want to identify. here I stop, cause I don't know exactly how to go on, but now I feel the abstract nonsense construction of as stabilization of overcategories more natural than a few days ago. ]]>

started a subsection on flat oo-vector bundles and D-modules

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<p>Okay. Do we know if <img src="https://nforum.ncatlab.org/extensions//vLaTeX/cache/latex_bab8fa3b10abe063ced5002f58a8b2a4.png" title=" QCOh(X)" style="vertical-align: -20%;" class="tex" alt=" QCOh(X)"/> is <em>exactly</em> what we get when we start with coherent sheaves and then complete under push-forward?</p>
<p>(I have to say I'd be interested, but I don't really care so much. :-) The general nonsense def of QCoh(X) is so good, that I don't see a pressing need to find another universal characterization.)</p>
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I've the doubt we may be unable to distinguish between two stably equivalent modules. but that's my unexperience in the field; I'll try to think more on this.
</blockquote>
<p>Right, yes. I am not sure. Here, too, I don't really care too much. we know the right picture. What we are asking here is to which extent the wrong picture is preserved by the right picture. :-) Would be good to nail it down, but I am not losing sleep over this.</p>
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as far as concerns Mod inside oo-Mod: are we sure of this? I've the doubt we may be unable to distinguish between two stably equivalent modules. but that's my unexperience in the field; I'll try to think more on this. ]]>

they're not, right? already due to the finiteness condition.

]]>do coherent sheaves (not quasi-) form a bifibration? I.e. are they closed under push and pull?

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<p>right, associating vector bundles is just composition of cocycles, which is just composition of morphisms in the oo-topos, yes.</p>
<p>Concerning the other aspect you mention: it is true that with the tangent oo-category approach we don't get <em>just</em> the classical theory. But we get something into which the classical theory embeds.</p>
<p>When we do this over <img src="https://nforum.ncatlab.org/extensions//vLaTeX/cache/latex_34d0a67079d60aa8f0cd6ea8ad81abfa.png" title=" C = SRing^{op} " style="vertical-align: -20%;" class="tex" alt=" C = SRing^{op} "/> then, as the example 8.6 in "Stable oo-Categories" shows, <img src="https://nforum.ncatlab.org/extensions//vLaTeX/cache/latex_03cf21e0dd38b8f3d0170d22802c53f4.png" title=" \infty Mod(R) " style="vertical-align: -20%;" class="tex" alt=" \infty Mod(R) "/> over an ordinary ring <img src="https://nforum.ncatlab.org/extensions//vLaTeX/cache/latex_07473a8bf5ab2af6f9d299667342c3db.png" title=" R" style="vertical-align: -20%;" class="tex" alt=" R"/> is the oo-category whose homotopy category is the derived category of <img src="https://nforum.ncatlab.org/extensions//vLaTeX/cache/latex_07473a8bf5ab2af6f9d299667342c3db.png" title=" R" style="vertical-align: -20%;" class="tex" alt=" R"/> -modules, yes. But ordinary modules still sit inside there.</p>
<blockquote>
here one can come back to one of the original issues between urs and zoran: what is Qcoh(X) to Vect(X)? we know that it is not abelianization, since Coh(X) is enough.
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<p>right, we need to say this precisely, eventually.</p>
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on the other hand I'm still unsure of how recover the classical theory (do we want to recover it) from the infinity-stable one. according to Lurie, in infinity-stable categories, looking at commutative rings as to simplicial commutative rings, what one ends up with over a ring R (more precisely R has to be a -algebra)

is not the category of R-modules but something stable which knows about the derived category of R-modules.

so, in going from an affine scheme to an arbitrary scheme $X$, it's likely that the tangent category approach does not really see but rather its stabilization or, maybe better, the stabilization of the category of complexes of quasicoherent sheaves on X.

here one can come back to one of the original issues between urs and zoran: what is Qcoh(X) to Vect(X)? we know that it is not abelianization, since Coh(X) is enough. but it may happen that Vect(X), Coh(X) and Qcoh(X) are stably equivalent, and so from a stable point of view there would be no difference between them. ]]>

I have forwarded this discussion to the blog, here

]]>this morning I have worked on the entry and basically rewritten it, taking into account all the discussion we had, and also polishing this and that. I'd be very grateful if you could have another look

There are various immedie things to be done from here on. I indicate one in the new section on Associated oo-vector bundles.

]]>The strictification indirect way: take any category over a category. The category opposite to the category of it sections is the category of modules in it. Take any section and from it make a Grothendieck construction. Then take a section of the latter, what you obtain is a quasicoherent section. Quasicoherent modules are objects of the opposite category. They can be also expressed via cartesian sections from a trivial fibration.

]]>Thanks, Zoran and Domenico, for fixing things!

]]>I'll also try to think to how to recover the classical case from the tangent (infinity,1)-category approach. ]]>

I will change the notation. I mean calling f* is fit to inverse image functor, not the map of rings. Map in the category of rings is just opposite to the map in the category of spectra. So I would either call f map of rings and Spec f map of spaces, or call f map of rings and f circ or f op the map of spectra.

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<p>Thanks again for all the comments. I went offline yesterday before I could reply to them.</p>
<blockquote>
I keep telling you this for last two years but it never develops into conversation, as you forget that there is first step and want to make both steps in one, and then I can not help you as I do not see them as one.
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<p>My apologies, Zoran, for being dense. I suppose I never understood what you were telling me.</p>
<p>So let's sort this out in detail for plain QC sheaves first, before we continue talking about the oo-categorification further.</p>
<p>I just made an attempt to add most of what I understand about QC sheaves to the entry <a href="http://ncatlab.org/nlab/show/quasicoherent+sheaf">quasicoherent sheaf</a>. Please see the announcement of the additions I made at <a href="http://www.math.ntnu.no/~stacey/Vanilla/nForum/comments.php?DiscussionID=608&page=1#Item_1">quasicoherent sheaf (forum)</a>. Let me know what you think.</p>
<p>Also Domenico, please have a look at the entry on QC sheaves now. It tries to make very explicit the way QC sheaves may be thought of from the <a href="http://ncatlab.org/nlab/show/nPOV">nPOV</a> as simply homs into the the stack of modules.</p>
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