Grothendieck: more complete URL
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. [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
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.
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]
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
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]
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.
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]
participants (4)
-
Colin McLarty -
Eduardo J. Dubuc -
Michael Barr -
Steve Vickers