Grothendieck: more complete URL

Grothendieck: more complete URL

[Michael Barr]

Apparently, some browsers are not recognizing that URL.  Here is a more
complete version:
ftp://ftp.math.mcgill.ca/pub/barr/pdffiles/gk.pdf
My browser automatically prepends the ftp://, but I guess some don't.


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Re: Grothendieck: more complete URL

[Colin McLarty]

Dear Mike,

I would suggest a slightly different translation of the title.  I
think that both AG's writing style at the time, and his view of the
paper are better suited by "Some aspects of homological algebra.".

The title "on certain aspects..." suggest that this is just one
selection of topics, and one could as well have selected others.  But
AG intended this paper to hit the very center of homological algebra:
from now on homological algebra is about derived functors on Abelian
categories with enough injectives -- and he justifies this by proving
(against the general expectation) that all sheaf categories have
enough injectives .   Of course there are other resolutions besides
injective, we will use them too, but they are special cases for
special purposes.

Reading the Serre-Grothendieck correspondence you see they were very
off-hand in their language and consciously casual.  But they were not
at all modest.  So AG remarks on "some aspects" -- but he does not
suggest any limitation to just "certain aspects" as if other aspects
might be equally important.

best, Colin





2010/9/5 Michael Barr <barr@math.mcgill.ca>:
> Apparently, some browsers are not recognizing that URL.  Here is a more
> complete version:
> ftp://ftp.math.mcgill.ca/pub/barr/pdffiles/gk.pdf
> My browser automatically prepends the ftp://, but I guess some don't.

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Re: Grothendieck: more complete URL

[Eduardo J. Dubuc]

I do no see the diference between

"on some aspects ..." and "on certain aspects ..."

On the other hand, Grothendieck writes the "on". I disagree with Colin about
"Some aspects ...", a translator is not an interpreter, and if there was an
"on", he should leave it !!

the reader wants to read Grothendieck, not what the translator thinks he
should have written, or what his view of the paper may have been

I imagine I will distrust any translation made by Colin, sorry Colin

of course, in footnotes the translator can insert any comment he wants.

e.d.



Colin McLarty wrote:
> Dear Mike,
>
> I would suggest a slightly different translation of the title.  I
> think that both AG's writing style at the time, and his view of the
> paper are better suited by "Some aspects of homological algebra.".
>
> The title "on certain aspects..." suggest that this is just one
> selection of topics, and one could as well have selected others.  But
> AG intended this paper to hit the very center of homological algebra:
> from now on homological algebra is about derived functors on Abelian
> categories with enough injectives -- and he justifies this by proving
> (against the general expectation) that all sheaf categories have
> enough injectives .   Of course there are other resolutions besides
> injective, we will use them too, but they are special cases for
> special purposes.
>
> Reading the Serre-Grothendieck correspondence you see they were very
> off-hand in their language and consciously casual.  But they were not
> at all modest.  So AG remarks on "some aspects" -- but he does not
> suggest any limitation to just "certain aspects" as if other aspects
> might be equally important.
>
> best, Colin
>

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Re: Grothendieck: more complete URL

[Steve Vickers]

Dear Colin,

Can I ask a technical question about foundations here?

Does the "enough injectives" result assume a classical base theory? The
classical result for module categories uses choice, and my understanding
is that the result for sheaf categories uses Barr covers to make
available the classical result.

I wonder if there's an unequivocally constructive formulation.

Regards,

Steve.

Colin McLarty wrote:
> AG intended this paper to hit the very center of homological algebra:
> from now on homological algebra is about derived functors on Abelian
> categories with enough injectives -- and he justifies this by proving
> (against the general expectation) that all sheaf categories have
> enough injectives .   Of course there are other resolutions besides
> injective, we will use them too, but they are special cases for
> special purposes.


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Re: Grothendieck: more complete URL

[Michael Barr]

Well Marcia (who is the professional French ---> English translator)
prefers Colin's translation, after discssion.  So that's what it will be.

Michael


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Re: Grothendieck: more complete URL

[Colin McLarty]

I would not be surprised if Grothendieck's Tohoku proof that all AB5
categories have enough injectives is straightforwardly constructive --
with the unhelpful limitation that from a constructive viewpoint
nearly no categories satisfy the AB5 axioms.  Especially the last
axiom, 5, requires completeness in a strong sense which is tailored to
make each step of the classical Baer proof work.

The classical corollary, that all sheaf categories (i.e. sheaves of
modules)  have enough injectives, includes saying that all module
categories have enough injectives.  So it cannot be more constructive
than that.

I have not looked at constructive forms of the result, or relativizing
the whole to any base topos.

best, Colin



2010/9/9 Steve Vickers <s.j.vickers@cs.bham.ac.uk>:
> Dear Colin,
>
> Can I ask a technical question about foundations here?
>
> Does the "enough injectives" result assume a classical base theory? The
> classical result for module categories uses choice, and my understanding
> is that the result for sheaf categories uses Barr covers to make
> available the classical result.
>
> I wonder if there's an unequivocally constructive formulation.
>
> Regards,
>
> Steve.
>

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Re: Grothendieck: more complete URL

[Toby Bartels]

Colin McLarty wrote:

>I would not be surprised if Grothendieck's Tohoku proof that all AB5
>categories have enough injectives is straightforwardly constructive --
>with the unhelpful limitation that from a constructive viewpoint
>nearly no categories satisfy the AB5 axioms.  Especially the last
>axiom, 5, requires completeness in a strong sense which is tailored to
>make each step of the classical Baer proof work.

Does this depend on how the axiom is phrased?
An elementary version says that an Ab5 category
is an cocomplete abelian category such that,
given any object X, subobject A of X,
and directed system (B_i)_i of subobjects of X,
A \cap \sum_i B_i = \sum_i (A \cap B_i).
The obvious proof that modules form an Ab5 category
works in the internal language of any topos with NNO.

It is less clear to me whether it follows from this
that filtered colimits preserve exact sequences,
which is the more abstract form of axiom 5.


--Toby


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Re: Grothendieck: more complete URL

[Colin McLarty]

Certainly it depends on how the axioms are phrased.  And I only said I
would not be surprised!  When I wrote that, I had the idea that in
Tohoku he phrased AB5 for all filtered colimits of objects, but I see
it is in fact phrased for colimits of subobjects of the given one --
which is pretty obvious now that I see it.

I suppose the issue for constructivizing the proof then comes from AB3
on arbitrary sums together  AB4-5 on their properties.   It would
depend sensitively on how "arbitrary sums" are defined.

colin





2010/9/9 Toby Bartels <toby+categories@ugcs.caltech.edu>:
> Colin McLarty wrote:
>
>>I would not be surprised if Grothendieck's Tohoku proof that all AB5
>>categories have enough injectives is straightforwardly constructive --
>>with the unhelpful limitation that from a constructive viewpoint
>>nearly no categories satisfy the AB5 axioms.  Especially the last
>>axiom, 5, requires completeness in a strong sense which is tailored to
>>make each step of the classical Baer proof work.
>
> Does this depend on how the axiom is phrased?
> An elementary version says that an Ab5 category
> is an cocomplete abelian category such that,
> given any object X, subobject A of X,
> and directed system (B_i)_i of subobjects of X,
> A \cap \sum_i B_i = \sum_i (A \cap B_i).
> The obvious proof that modules form an Ab5 category
> works in the internal language of any topos with NNO.
>
> It is less clear to me whether it follows from this
> that filtered colimits preserve exact sequences,
> which is the more abstract form of axiom 5.
>
>
> --Toby
>


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