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In his thesis, Jacob Lurie mentioned two work in preparation (by him), namely "Virtual fundamental classes and the motivic sphere" and "Geometric derived stacks". Now that much is written in the DAG series and the two books (Higher Algebra and Higher Topoi), I'm wondering if anybody here know if these two papers are contained in the DAG series already or still yet to be written?

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May I suggest you to write the author? This doesn't look like a mathematical question to me. – Fernando Muro Jun 3 '12 at 21:46
@Fernando: I think it's a reasonable question, and as much a mathematical question as many of the other reference-related questions asked here. I would also recommend that "temp" not write the author unless no answer appears here. Many people receive more emails than they can or do respond to. – Kevin Walker Jun 3 '12 at 23:10
I'd think that the best approach to this problem would be to find the result that you're trying to understand in one of Jacob's books, and then just see what's cited there. It would help if we knew what specifically you were looking for, because I can't seem to find a searchable version of Jacob's thesis itself. – Noah Snyder Jun 4 '12 at 4:54

I'm not sure I understand the question (unless it's intended as general curiosity about the progress of Lurie's work). I can see two legitimate options:

  1. you are interested in a particular result and want to learn about it or apply it, in which case the answer is there are no documents circulating (AFAIK) beyond the vast amount on Jacob's homepage. Are you then looking for help finding a result there, in which case a specific question on MO is likely to provide an answer, or you know what you're looking for isn't there? - in the latter case, sadly I think the answer is you have to wait patiently for future publications like the rest of us.. (This is basically repeating Noah's comment.)

  2. you have your own ideas about a result that was announced as to appear in these manuscripts, and would like to know their status before working hard to complete or publish a result which may become obsolete. In this case I'd say realistically you should either change field or (much better option) not worry about whether your result will be subsumed by his work. Jacob's writing tends to subsume results of many people, it's an obvious danger of working in this (exciting and rapidly developing) field, but the community (judging from my experience) will still very much appreciate other takes, and given the great scope of things that Jacob is working on it's likely you can complete something before it appears.

Of course if you're just starting to think about a problem it might be wiser to find your own questions rather than trying to prove something you saw announced in Jacob's thesis..

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Oh, I was just asking that out of pure curiosity. I want to know what's behind those titles. – temp Jun 7 '12 at 4:20

In December 2005, Jacob Lurie visited Berkeley, and I asked him about the main results of the motivic sphere paper (N.B., his dissertation was from 2004).

He replied, "I cited it, but it does not exist."

It is possible that he wasn't completely serious.

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You mean the citation wasn't serious or his remark here wasn't serious? – temp Jun 5 '12 at 15:35
His ideas on virtual fundamental classes were developed enough to give a talk about it in September 2005: The point at which one can say that a collection of ideas forms a paper seems to vary wildly from person to person. – S. Carnahan Jun 7 '12 at 3:27

Lurie repeatedly cited "Virtual fundamental classes" for seemingly unrelated topics. I don't believe that he has yet returned to any of those topics. He cited "Geometric derived stacks" fewer times and more vaguely, so it is harder to track what happened. I think some of the material appeared in "DAG VIII: Quasi-Coherent Sheaves and Tannaka Duality Theorems," which got folded into the second book. I think he changed his plan and spread out that material, while the first paper has remained intact and just hasn't appeared yet.

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Minor comment: DAG VIII will presumably be folded into the third book (Derived Algebraic Geometry? Higher Geometry??), not the second (Higher Algebra). – David Ben-Zvi Jun 4 '12 at 20:05
oops. I should have checked the table of contents. – penner Jun 4 '12 at 20:55
I looked into this a little and came to the same conclusion as penner. "Virtual fundamental classes" refers to a specific paper that has not yet appeared, while "Geometric derived stacks" no longer refers to one specific paper but the material has mostly appeared in the DAG papers. – Noah Snyder Jun 5 '12 at 4:51
@David, Lurie says he decided to call the third book Spectral Algebraic Geometry. – Omar Antolín-Camarena Jun 5 '12 at 21:04

It's worth noting that Lurie's thesis is not on his webpage, and is not regularly updated like the papers which are on is webpage. I would take this to mean that you're supposed to be reading the current versions of this material in the DAG series, and not reading the 8 year old outline.

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