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Stated the hypothèse inspiratrice, mentioned a proof of a lift of it to derivators by Cisinski, mentioned my conjecture that the original statement is independent of ZFC (feel free to remove this if felt inappropriate), and listed all references to it that I am aware of by Grothendieck in Pursuing Stacks.
Regarding your conjecture, it’s certainly not inappropriate in general to include conjectures on nLab pages, even conjectures of our own. However, it feels weird to me to hear a conjecture like this without any explantion of why one might believe it. I certainly don’t see any reason to expect this statement to be independent of ZFC, not any more so than any other unsolved problem. Are you thinking that it might have to do with some kind of large cardinal hypothesis, similar to how certain category-theoretic statements like “there is an inaccessible left exact endofunctor of $Set$” or “$Set$ has a small codense subcategory” depend on the (non)existence of measurable cardinals?
added hyperlinks to a few terms in the first paragraph, notably to Pursuing Stacks (!), also italicized that.
concretified “homotopy category” by classical homotopy category
@Richard: You say on that page: … blacked out (leaving the original illegible) statement …. For what it is worth I think the problem was that someone used a highlighter (probably yellow) and that photocopies black!
Re #5 - and it’s not visible in the Scrivener’s version of PS?
I have just added a link to a scanned version of that page. (Scanned from my copy.)
Added proof of the analogous statement for Set, and explained how an analogous proof can be given for the hypothèse inspiratrice if one ’lifts’.
Tried to give some justification for my conjecture that the original hypothèse inspiratrice is independent of ZFC.
Tweaked part of the section with quotes from Pursuing Stacks in the light of Tim’s contributions.
A couple of minor things in addition.
Thanks very much Tim! That you have made available the scan is enough to make me feel that it was worthwhile creating this page! I’ve incorporated your remark about highlighting, because I think it is a useful/important point for those interested in the history (there is a big difference between someone highlighting something to indicate its significance, and Grothendieck wishing to delete something!).
Thanks very much too, Urs! I lacked a bit of energy to do this myself yesterday; it is rewarding to see you help out! I completely agree regarding the references; I look forward to the day I have time to implement a new referencing mechanism to make this easier and more easily standardised.
Thanks very much for the comments/thoughts, Mike!
However, it feels weird to me to hear a conjecture like this without any explantion of why one might believe it.
I completely agree. I thought about adding something, but felt I would not be able to do so satisfactorily. The conjecture is not something new to me, I first came to believe this conjecture maybe 10 years ago, possibly even before I was aware of HoTT! I have mentioned it to a few people over the years; David Ayala I think was one. Yet I cannot really give any strong evidence for it. I have tried now to make some kind of remarks in the entry. If people feel that these are not adequate, I am completely fine with just removing that section from the page.
Are you thinking that it might have to do with some kind of large cardinal hypothesis, similar to how certain category-theoretic statements like “there is an inaccessible left exact endofunctor of Set” or “Set has a small codense subcategory” depend on the (non)existence of measurable cardinals?
I think there may be some kind of analogy with this kind of situation, yes. Somehow I feel one will have to step ’outside’ of set theory, to a natively homotopy theoretic foundations, to be able to have a model for the homotopy category for which one can prove the statement. See what is currently in the entry for a little more.
Thanks Richard! I don’t find the remark convincing, though; just pointing out why a conjecture is hard, or why tools that work for a modified version of it don’t seem to apply to it directly, doesn’t to me give any reason to believe it to be independent of ZFC, at least not any more than any other hard unsolved problem. So I’d be in favor of removing that section, but if others feel differently please speak up.
No problem, that’s fine with me. Let’s see if anybody else has an opinion; otherwise I’ll remove the section.
Thanks, Tim! That addresses one of the issues with highlighting listed here. If you could see your way to looking up the small number of other missing transcribed pieces, I for one would be very grateful. I’m glad your copy of PS has emerged from the depths of storage.
… depths of storage’ now means ‘on top of the pile of books and files in the corner in front of me’! When that earlier discussion was started I searched for some of those references and was not lucky. They had been highlighted and were unreadable. I will check them again if I have time … and remember.
Thanks again, Tim :-)
Yes, thanks, Tim! I’ve now incorporated this and put up new pdfs on github.
Thanks very much, I found this very nice, and the distinction between the directed and undirected cases to be insightful!
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