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I have a memory of reading a quotation from Grothendieck in which he explained why defining “-groupoid” to mean “Kan complex” did not satisfy what he was looking for. Can anyone tell what I may be thinking of and where I could find it?
There is something to this effect fairly early on in Pursuing Stacks. My copy of PS is somewhere in a pile upstairs here, and is in disarray so I will not (at the moment) offer to look for you.
Maltsiniotis quotes this on p. 2 of his
where it says:
In a letter to Tim Porter, Grothendieck clearly explains that this was not the kind of definition he was looking for:
“my main point is that your suggestion that Kan complexes are “the ultimate in lax -groupoids” does not in any way meet with what I am really looking for, and this for a variety of reasons, … “
And he points to
Doh! now where did I put that? I remember seeing it recently and putting it in a ‘safe place’!
I have a scanned copy! it is on another machine, and I have to work out how best to send it!
Mike, look in your e-mail.
A thought: since Maltsiniotis announced the public posting of this letter in “Documents Mathématiques”, maybe it can be made publically available? Say by linking the scan form the nLab? Just a thought.
It is a thought I had had, but I am not yet 100% sure what is the best thing to do. I also have copies of ‘some’ of my letters to AG. They are dated almost exactly 30 years ago. Perhaps to start with a page with a list of some of the known letters to and from AG could be put up. A whole letter might be invasion of his privacy, (or of mine for that matter!!!), but key points of relevance would not. Let me think about it.
To start with (and more will be slow coming as I do not type that fast) I have started a section on my private area which has some thoughts on what to do. (I have added a short mention on the Tim Porter page).
Thanks, Tim! It seems that his “variety of reasons” are (1) he wants composition to be an operation and (2) he wants morphisms to be globular (which he admits are minor/technical/psychological points), and more importantly (3) he regards stacks as very different from sheaves of categories (or simplicial sheaves), even if they are functorially related. His point seems to be that stacks (I guess in the original sense of groupoid- or category-valued pseudofunctors satisfying descent) arise from geometry (e.g. bundles, torsors, and so on, I guess), whereas complexes of sheaves (perhaps nonabelian ones) arise rather from (homotopical or homological) algebra.
This suggests to me that perhaps the final missing phrase in the other thread should be some sort of “geometry”.
Looking at this from a modern perspective, one might still agree or disagree with his technical/psychological points. But as for the third one, we know now that 1-groupoid-stacks and 1-groupoid-sheaves model the same homotopy theory — although they are certainly not identical — but that this is no longer true for -groupoids. Was Grothendieck anticipating the problem of hypercompleteness? (-:
Hypercompleteness aside, it seems as though point (3) could be said to be an objection to strict categories and anything we can do with them that we can’t do with non-strict categories, such as defining strict (rather than pseudo) 2-functors, or a sheaf (rather than stack) condition on objects.
@Mike
Forgive my ignorance; where can I find some information about the homotopy theory of stacks of groupoids vs sheaves of groupoids? The model structure on the latter must be highly non-obvious!
I think, (3) is as much a psychological point as (2). Stacks arise naturally from geometry – for instance, the stack of line bundles on a locally ringed space – whereas sheaves of categories or groupoids are somewhat more contrived – say, the sheaf of non-vanishing functions, which is the sheaf of abelian groups that corresponds to the stack of line bundles. One imagines that there might be some complicated locally-defined data which constitute a stack but which have no obvious algebraic presentation in terms of a sheaf of coefficients or moduli. But if these have the same homotopy theory then I suppose there must always be such an algebraic presentation…
Check out Sharon Hollander’s paper a homotopy theory for stacks.
Great, thanks! This paper finally makes sense to me.
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