Another question on Grothendieck

Another question on Grothendieck

[Michael Barr]

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


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Re: Another question on Grothendieck

[Andrew Stacey]

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

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Re: Another question on Grothendieck

[John Baez]

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


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Re: Another question on Grothendieck

[Vaughan Pratt]

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


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Re: Another question on Grothendieck

[Vaughan Pratt]

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
>
>

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Re: Another question on Grothendieck

[Steven Vickers]

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

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Re: Re: Another question on Grothendieck

[Vaughan Pratt]

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}".


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Re: Another question on Grothendieck

[Eduardo J. Dubuc]

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.
>

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