* Another question on Grothendieck @ 2010-08-30 14:18 Michael Barr 2010-08-30 18:04 ` Andrew Stacey 2010-08-31 5:50 ` John Baez 0 siblings, 2 replies; 8+ messages in thread From: Michael Barr @ 2010-08-30 14:18 UTC (permalink / raw) To: Categories list Grothendieck introduces, on the top of p. 209 of the Tohoku paper, the notation U_{i_0..i_p} without explanation and uses it again over the next couple pages. Here {U_i} is an open cover of a space X and I have reason to believe that this stands for the intersection of U_{i_j}. Can anyone confirm this? Or give an alternate explanation? The context is that of a claim that (when A is a sheaf) and "every U_{i_0..i_p} is A-acyclic, then"... and that awfully like the definition of a simple cover. Michael [For admin and other information see: http://www.mta.ca/~cat-dist/ ] ^ permalink raw reply [flat|nested] 8+ messages in thread
* Re: Another question on Grothendieck 2010-08-30 14:18 Another question on Grothendieck Michael Barr @ 2010-08-30 18:04 ` Andrew Stacey [not found] ` <E1Or9Si-00031p-Iw@mlist.mta.ca> 2010-08-31 5:50 ` John Baez 1 sibling, 1 reply; 8+ messages in thread From: Andrew Stacey @ 2010-08-30 18:04 UTC (permalink / raw) To: Michael Barr; +Cc: Categories list On Mon, Aug 30, 2010 at 10:18:30AM -0400, Michael Barr wrote: > Grothendieck introduces, on the top of p. 209 of the Tohoku paper, the > notation U_{i_0..i_p} without explanation and uses it again over the next > couple pages. Here {U_i} is an open cover of a space X and I have reason > to believe that this stands for the intersection of U_{i_j}. Can anyone > confirm this? Or give an alternate explanation? > > The context is that of a claim that (when A is a sheaf) and "every > U_{i_0..i_p} is A-acyclic, then"... and that awfully like the definition of > a simple cover. I have absolutely no idea as to what Grothendieck meant, but the notation you describe is quite common in (algebraic) topology and means what you "have reason to believe" that it means. Andrew [For admin and other information see: http://www.mta.ca/~cat-dist/ ] ^ permalink raw reply [flat|nested] 8+ messages in thread
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* Re: Another question on Grothendieck [not found] ` <E1Or9Si-00031p-Iw@mlist.mta.ca> @ 2010-09-03 6:46 ` Vaughan Pratt [not found] ` <4C819AD7.8090403@dm.uba.ar> [not found] ` <E1Ort7T-00087z-SA@mlist.mta.ca> 0 siblings, 2 replies; 8+ messages in thread From: Vaughan Pratt @ 2010-09-03 6:46 UTC (permalink / raw) To: categories list On 9/1/2010 12:21 PM, Eduardo J. Dubuc wrote: > I am wondering, nobody can read the mathematics and come up with what > Grothendieck meant !!! Eduardo raises an excellent point here. Which is more important for a contribution, its meaning or its influence? If the latter, a secondary question is, how was that influence achieved? Improved access to the contribution, e.g. via translation, may help those who understand the mathematics but not the French explain the influence, even if the original meaning remains obscure. Vaughan [For admin and other information see: http://www.mta.ca/~cat-dist/ ] ^ permalink raw reply [flat|nested] 8+ messages in thread
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* Re: Another question on Grothendieck [not found] ` <4C819AD7.8090403@dm.uba.ar> @ 2010-09-04 7:04 ` Vaughan Pratt [not found] ` <4C82A02D.7090703@dm.uba.ar> 0 siblings, 1 reply; 8+ messages in thread From: Vaughan Pratt @ 2010-09-04 7:04 UTC (permalink / raw) To: categories list All I am saying is that one need not read Galois in order to learn Galois theory. When a new idea is introduced, even if it is not explained so clearly that everyone understands it right away, as long as someone understands it and can rephrase it in a helpful way, the impact of the idea has been not only felt but disseminated. Dissemination is not always a single step. Vaughan On 9/3/2010 6:03 PM, Eduardo J. Dubuc wrote: > I confess that I am a little bit confused about what Vaughan is saying. > > This promps me to repeat my posting in other words: > > If a mathematical statement is understood by a reader (the hypotesis, > the conclusion and the proof) > > then the mathematical meaning of any particular notation used should > come up by itself to this reader (that is, it should be clear for him > that only one possible meaning for this particular notation would make > the things work). > > Eduardo > > [For admin and other information see: http://www.mta.ca/~cat-dist/ ] ^ permalink raw reply [flat|nested] 8+ messages in thread
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* Re: Re: Another question on Grothendieck [not found] ` <4C82A02D.7090703@dm.uba.ar> @ 2010-09-05 23:23 ` Vaughan Pratt 0 siblings, 0 replies; 8+ messages in thread From: Vaughan Pratt @ 2010-09-05 23:23 UTC (permalink / raw) To: categories list My apologies to Eduardo. I misread his original post as a complaint about AG, when what he meant was that there was no ambiguity given that the other interpretations made little or no sense. Since complaining about AG makes little sense on a list AG (presumably) doesn't read, when there was another interpretation that would have made more sense had I noticed it, I'm guilty of the very thing Eduardo was complaining about. Vaughan On 9/4/2010 12:38 PM, Eduardo J. Dubuc wrote: > Yes Vaughan, I agree with your point "up to a point", and this is an > interesting topic to discuss (see Point 2) below). > > But I was raising another point, much more simple. > > Point 1) It was about the notation "U_{i_0..i_p}". If you understand the > mathematics, then whether this stands for the intersection, the union, > or any other known construction with the U_i_j, it should be clear which > one is. > > So it seemed to me ridiculous that people in the list, all > mathematicians, start discussing and speculating about possible meanings > of "U_{i_0..i_p}". [For admin and other information see: http://www.mta.ca/~cat-dist/ ] ^ permalink raw reply [flat|nested] 8+ messages in thread
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* Re: Another question on Grothendieck [not found] ` <E1Ort7T-00087z-SA@mlist.mta.ca> @ 2010-09-05 19:51 ` Steven Vickers 2010-09-06 17:53 ` Eduardo J. Dubuc 0 siblings, 1 reply; 8+ messages in thread From: Steven Vickers @ 2010-09-05 19:51 UTC (permalink / raw) To: Eduardo J. Dubuc; +Cc: categories Dear Eduardo, I have written papers that deliberately have two possible meanings: one classical point-set and one constructive point-free. That is to say, the development in terms of points is done under logical (geometric) constraints that enable it to be interpreted in topos-valid point-free topology (locales), but it can be interpreted directly in point-set topology if one accepts classical logic. I did this for expositional reasons, to help classical topologists understand the topological content of what I was doing. See: "Localic completion of generalized metric spaces I" "The connected Vietoris powelocale" Is this compatible with what you were saying about "only one possible meaning"? Regards, Steve Vickers. On Fri, 03 Sep 2010 22:03:19 -0300, "Eduardo J. Dubuc" <edubuc@dm.uba.ar> wrote: > I confess that I am a little bit confused about what Vaughan is saying. > > This promps me to repeat my posting in other words: > > If a mathematical statement is understood by a reader (the hypotesis, > the conclusion and the proof) > > then the mathematical meaning of any particular notation used should > come up by itself to this reader (that is, it should be clear for him > that only one possible meaning for this particular notation would make > the things work). > > Eduardo [For admin and other information see: http://www.mta.ca/~cat-dist/ ] ^ permalink raw reply [flat|nested] 8+ messages in thread
* Re: Another question on Grothendieck 2010-09-05 19:51 ` Steven Vickers @ 2010-09-06 17:53 ` Eduardo J. Dubuc 0 siblings, 0 replies; 8+ messages in thread From: Eduardo J. Dubuc @ 2010-09-06 17:53 UTC (permalink / raw) To: Steven Vickers; +Cc: categories Dear Steve, I was already aware that my statement "only one possible meaning"? was much too general and I myself speculated (at the time of posting the msage) about many possible exceptions when literally interpreting my statement. But I decided to leave it like that. Luckily it was understood as I meant (private msages). I clarify to you and to those that may rise similar exceptions: The Tohoku paper is just plain old classical mathematics (*), and nothing of the sort of your example is to be found there. I imagine on the other hand that in your papers you do not let the reader stay in the doubt about the meaning of these two possible meanings. (*) where you can of course point out if some reasoning is constructively valid (an exceptional example of this is the chapter on field extensions in the second edition of the classical Van der Waerden book). e.d. Steven Vickers wrote: > Dear Eduardo, > > I have written papers that deliberately have two possible meanings: one > classical point-set and one constructive point-free. > > That is to say, the development in terms of points is done under logical > (geometric) constraints that enable it to be interpreted in topos-valid > point-free topology (locales), but it can be interpreted directly in > point-set topology if one accepts classical logic. > > I did this for expositional reasons, to help classical topologists > understand the topological content of what I was doing. > > See: > > "Localic completion of generalized metric spaces I" > "The connected Vietoris powelocale" > > Is this compatible with what you were saying about "only one possible > meaning"? > > Regards, > > Steve Vickers. > [For admin and other information see: http://www.mta.ca/~cat-dist/ ] ^ permalink raw reply [flat|nested] 8+ messages in thread
* Re: Another question on Grothendieck 2010-08-30 14:18 Another question on Grothendieck Michael Barr 2010-08-30 18:04 ` Andrew Stacey @ 2010-08-31 5:50 ` John Baez 1 sibling, 0 replies; 8+ messages in thread From: John Baez @ 2010-08-31 5:50 UTC (permalink / raw) Cc: Categories list Hi - Grothendieck introduces, on the top of p. 209 of the Tohoku paper, the > notation U_{i_0..i_p} without explanation and uses it again over the next > couple pages. Here {U_i} is an open cover of a space X and I have reason > to believe that this stands for the intersection of U_{i_j}. Can anyone > confirm this? > That's certainly what everyone uses this notation for nowadays, so it sounds like a very good guess. Best, jb [For admin and other information see: http://www.mta.ca/~cat-dist/ ] ^ permalink raw reply [flat|nested] 8+ messages in thread
end of thread, other threads:[~2010-09-06 17:53 UTC | newest] Thread overview: 8+ messages (download: mbox.gz / follow: Atom feed) -- links below jump to the message on this page -- 2010-08-30 14:18 Another question on Grothendieck Michael Barr 2010-08-30 18:04 ` Andrew Stacey [not found] ` <E1Or9Si-00031p-Iw@mlist.mta.ca> 2010-09-03 6:46 ` Vaughan Pratt [not found] ` <4C819AD7.8090403@dm.uba.ar> 2010-09-04 7:04 ` Vaughan Pratt [not found] ` <4C82A02D.7090703@dm.uba.ar> 2010-09-05 23:23 ` Vaughan Pratt [not found] ` <E1Ort7T-00087z-SA@mlist.mta.ca> 2010-09-05 19:51 ` Steven Vickers 2010-09-06 17:53 ` Eduardo J. Dubuc 2010-08-31 5:50 ` John Baez
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