Timeline for Conjectures in Grothendieck's "Pursuing stacks"
Current License: CC BY-SA 4.0
16 events
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| Jul 11, 2022 at 6:49 | history | edited | Martin Sleziak | CC BY-SA 4.0 |
a minor typo
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| Feb 22, 2015 at 7:56 | comment | added | Tim Porter | Just to record the fact, I read PS as it was being written and sent to Bangor. This can be seen from occasional references in it to the feedback that I gave to AG. I understand Jean Benabou's reaction to pursuing further the weak n-category line. When Ronnie and I went on with trying to get research funding for ideas related to PS we were consistently shot down and derided by some of the referees and hence got no funding. (Note Bangor maths dept. no longer exists partially because of this.) Finally referring to specific results in PS is a bit difficult as the manuscript was evolving. | |
| Jan 28, 2015 at 0:32 | comment | added | David Roberts♦ | On the more recent side of things, Carlos Simpson's book Homotopy theory of higher categories (hal.archives-ouvertes.fr/hal-00449826/document) specifically cites PS: "One of the main inspirations for the recent interest in higher categories came from Grothendieck’s manuscript Pursuing stacks. He set out a wide vision of the possible developments and applications of the theory of n-categories going up to n = ω. Many of his remarks continue to provide important research directions, and many others remain untouched." | |
| Jan 28, 2015 at 0:23 | comment | added | David Roberts♦ | This is a late response, but Voevodsky has said fairly recently that in the late 80s some of the Russian school (Drinfel'd I guess the most prominent) was taking Grothendieck's work of the 80s very seriously. VV's (joint) work on higher categories published around 1990 grew out of this, but had an error - this is discussed at length by him in recent talks dealing with his decision to move to formal proof assistants. VV's work on motivic homotopy theory is a natural progression from the bare homotopy types/higher categories to the 'schematic' side of things. | |
| Dec 11, 2012 at 9:51 | comment | added | Jonathan Chiche | (ctd) written anything interesting from a mathematical point of view at this period. Even Quillen did not answer to the letter! It is also true that, from what I have been told, it was almost impossible to "sell" the issues raised by Grothendieck at the time. For instance, Bénabou had discussed the question of higher analogs of bicategories (that is, weak n-categories) with Grothendieck around 1966 but he told me that basically he thought he could not have been "forgiven" to work on such a topic, and that only Grothendieck could be forgiven. I am unsure he has been. | |
| Dec 11, 2012 at 9:41 | comment | added | Jonathan Chiche | David, thanks for your comments. I hope you did not understand I was claiming nobody had read "Pursuing Stacks" and I am glad you have. But how many precise references to this text could one point out in the literature? It is true that some people present (I am not qualified to judge whether they are right or not) their work as solutions to questions raised by Grothendieck, but what has been written about the solutions brought by Grothendieck himself? This is something which I find very sad, but for many years most people seem to have been thinking that Grothendieck could not have (ctd) | |
| Dec 11, 2012 at 1:11 | comment | added | David Roberts♦ | However, before I read it, Larry Breen told me that after the first section, it was all a bit messy, and only the first part (weak $n$- and $\infty$-groupoids) was worth reading in his opinion, due to the 'research diary' nature of the manuscript (at least, that was the impression I got from a brief conversation at a conference). | |
| Dec 11, 2012 at 1:09 | comment | added | David Roberts♦ | I should say I have read the entirety of (the electronically available copy of) Pursuing Stacks, so apart from possibly faulty memory, I know what's in there. Toën mentions here ens.math.univ-montp2.fr/~toen/msri2002-2.pdf (on page 1) that what he describes is his understanding of the problem of schematisation of homotopy types as presented in PS. This note by Toën is just an outline of other work he has done. Maltsiniotis and Cisinski have both certainly read PS. I don't know if Lurie has, but he is in effect 'second generation' after Toën-Vezzosi. | |
| Dec 10, 2012 at 19:15 | comment | added | Todd Trimble | @Jonathan: You can bet your last dollar that Ronnie Brown has read Pursuing Stacks. I believe Tim Porter has as well. | |
| Dec 10, 2012 at 15:49 | history | edited | Jonathan Chiche | CC BY-SA 3.0 |
Added one paragraph at the end.
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| Dec 6, 2012 at 18:28 | comment | added | Tim Porter | I second Todd's point. | |
| Dec 6, 2012 at 16:23 | comment | added | Todd Trimble | I would like to remind Jonathan (or anyone else) that the nLab (ncatlab.org/nlab/show/HomePage) is a wiki, and that you should feel encouraged to modify anything that you find wrong or objectionable. (But) speaking on behalf of nLab regulars, I would like to ask those who work on nLab entries to make a note of their efforts at the nForum nforum.mathforge.org/discussions/?CategoryID=0, which also serves as a discussion board. Alternatively, you can initiate a discussion at the nForum of proposed changes to nLab entries, before you actually make those changes. | |
| Dec 6, 2012 at 15:47 | comment | added | Tim Porter | I should add that Grothendieck's conjectures are often informally stated, and he sometimes suggests that deciding what the theory should look like, and therefore what the conjectures are, is the greater part of the battle for the development of any theory. | |
| Dec 6, 2012 at 15:45 | comment | added | Tim Porter | I have just gone into the n-Lab and have tried to change the wording to reflect Jonathan's very valid points. I have only added a little and changed a little, but would suggest that the entry does need some more work. Does any one have typed out a table of contents of PS? Then linked comments with the future development would of the main themes might be useful. | |
| Dec 6, 2012 at 11:28 | comment | added | David Roberts♦ | I agree with this. While PS is a working diary of sorts and there are backtrackings aplenty, it is hard, especially a mathematician of Grothendieck's stature, to write 600 pages and not prove a theorem. I also agree that Grothendieck wanted an algebraic definition of higher groupoid, and this is what he gave. | |
| Dec 6, 2012 at 9:36 | history | answered | Jonathan Chiche | CC BY-SA 3.0 |