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In my copy of “Pursuing Stacks” on p40 there is the following passage:
The notion of a ??? here appears as the unifying concept for a synthesis of ??? and ???. This (rather than merely furnishing us with still another description of homotopy types, more convenient for expression of the homotopy groups) seems to me the real “raison d’etre” of the notion of a stack…
The ???s are blacked out. Given the second sentence, I presume that the first ??? should be “stack”. But does anyone know what the other ???s might be?
I don’t know what it was, but I know what it should be :-) it should be “geometry” and “homotopy theory”.
The notion of a stack here appears as the unifying concept for a synthesis of geometry and homotopy theory.
I’ve mentioned this to someone I know on G+ who has an early generation copy (inherited from a student of our Tim Porter). The quality is very good, so the words my still be unredacted.
I can’t read djvu files at the moment, but the French should be clear enough here.
@Urs: that’s one possibility. However, the first ??? is incompletely blacked out, and it looks like it ends with “algebra”. Perhaps “homotopical algebra”?
@David C: That’s the same djvu file as the copy I have.
I can look if you cannot resolve it, but I suggest you contact George Maltsiniotis as he has produced the latex version and will have grappled with such things. (As I said, my copy is in a pile upstairs here and if I pull it out the pile will collapse… it probably will anyway… :-( ) I do not know if Ronnie is reading this he has an early copy, of course.
Is there a latex version available somewhere?
That Guest was me, as I did not realised that I had been timed out.
Thanks very much! I wonder how they came to be blacked out in the djvu version.
Thought! If someone used a highlighter in a colour and then the document was scanned in B&W the scan may not be able to pierce the colour, even though it was only be something like a yellow. I have known such things happen with photocopying.
It would be much more exciting if somehow Grothendieck’s thoughts were considered so high-powered that they must be kept secret, and this is the government redacting key phrases of the document! ;-)
@Tim, that sounds plausible!
Hm, so what is meant by “cohomological algebra” as opposed to “homotopical algebra”?
Does the text go into more detail as to what is meant by this “synthesis”?
Haven't we discussed before the duality between homotopy theory and nonabelian cohomology theory? ETA: Yes, here at the Café.
On the djvu tiff version that I have on my laptop, there are quite a few passages that are black like that.
Haven’t we discussed before the duality between homotopy theory and nonabelian cohomology theory?
But stacks are not what make that duality happen. That’s a duality in homotopy theory, which one can then transport to what I would call the “geometric homotopy theory” of (higher) stacks.
And is that what Grothendieck was referring to, anyway? Are there any futher indications as to which “synthesis” he had in mind?
Perhaps Grothendieck may have meant by “homotopical algebra” to refer to cohomology theories with coefficients in abelian sheaves, which are often calculated by resolving the coefficients, and by “(nonabelian) cohomology theory” to refer to cohomology with coefficients in more geometric objects such as naturally arising stacks of torsors, etc., which are often calculated by resolving the space (Cech covers and hypercovers). These may seem obviously two sides of the same coin to us now, but that may not have been so much the case back then? And the homotopy theory of (higher) stacks is in fact what unifies them, right? injective resolutions of abelian sheaves yield fibrant replacements in model categories for stacks that are generated by localizing at Cech covers.
Without searching in PS I think you are right, Mike. He talks about resolving the ‘space’ or the coefficients and the link between them. I think the two approaches were seen to be equivalent even then but it was not clear why.
I see, so the “synthesis” meant would then be that between abelian sheaf cohomology theory and nonabelian cohomology.
(Maybe it’s more a generalization than a synthesis, as made first fully clear in K. Brown, 73, I suppose. )
(I’d be tempted to argue that “homotopical algebra” is not a good synonym for “abelian sheaf cohomology”, but I guess for various reasons I am not in the position to argue about anything here… ;-)
I suspect that we need to refer back to the exact wording and context of what AG said to see what is a reasonable interpretation of it!
I have found a box file with my copy of PS in it. (As I suspected it WAS right at the bottom of a pile of other boxes!) I will look to see what the discussion near those pages refers to. (NB. As my copy was ’raided’ at certain times for photocopying a missing page or whatever, it is not 100% in order, nor guaranteed to be ’complete’, but if there are queries I may be able to give some help.)
BTW page 40 of the dejavu tif file is page 6 of the typed manuscript, but that does not mean that it is the 6th side as AG often added bits and renumbered.
I recall that the contents pages were numerous, so they probably throw the page count out a fair bit.
I’d be tempted to argue that “homotopical algebra” is not a good synonym for “abelian sheaf cohomology”
I would agree with you, but the point is just that Grothendieck may have been using it that way.
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