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1. • CommentRowNumber1.
   • CommentAuthorzskoda
   • CommentTimeJul 6th 2010
   • (edited Jul 6th 2010)
   • PermaLink
   Author: zskoda
   Format: MarkdownItexNew entry [[cohomological descent]].

   New entry cohomological descent.

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeJul 6th 2010
   • PermaLink
   Author: Urs
   Format: MarkdownItexThanks, Zoran. We also once wrote [[higher monadic descent]]. I added a link there to cohomological descent. Would be nice if we could eventually discuss the relation more.

   Thanks, Zoran.

   We also once wrote higher monadic descent. I added a link there to cohomological descent. Would be nice if we could eventually discuss the relation more.

3. • CommentRowNumber3.
   • CommentAuthorzskoda
   • CommentTimeJul 6th 2010
   • (edited Jul 6th 2010)
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   Author: zskoda
   Format: MarkdownItexIt is not the same! The cohomological descent is not about equivalence of categories but about fully faithful at the total derived level. Second it is at the level of triangulated categories. Of course it is related in good cases. Finally it is always taken with repsect to hypercovers to suit the needs in algebraic geometry.

   It is not the same! The cohomological descent is not about equivalence of categories but about fully faithful at the total derived level. Second it is at the level of triangulated categories. Of course it is related in good cases. Finally it is always taken with repsect to hypercovers to suit the needs in algebraic geometry.

4. • CommentRowNumber4.
   • CommentAuthorUrs
   • CommentTimeJul 6th 2010
   • PermaLink
   Author: Urs
   Format: MarkdownItex> It is not the same! [...] Of course it is related in good cases. Right, so that's why I said it would be good to discuss the relation, eventually! :-)

   It is not the same! […] Of course it is related in good cases.

   Right, so that’s why I said it would be good to discuss the relation, eventually! :-)

5. • CommentRowNumber5.
   • CommentAuthorzskoda
   • CommentTimeJul 6th 2010
   • PermaLink
   Author: zskoda
   Format: MarkdownItexIn original booklet of Giraud from 1962 (not the later book) also for usual descent the fully faithful case is discussed and called 1-descent while what we call descent he called 2-descent. In cohomological descent one thus has some sort of cohomological version of 1-descent. To get the true derived descent I was suggesting to Rosenberg to look at cohomological descent first; after few days they had (with Kontsevich) also the A-infinity version. But I myself never apart from the ideas fully understood the relation between the derived 1-descent and the cohomological descent for total derived functors.

   In original booklet of Giraud from 1962 (not the later book) also for usual descent the fully faithful case is discussed and called 1-descent while what we call descent he called 2-descent. In cohomological descent one thus has some sort of cohomological version of 1-descent. To get the true derived descent I was suggesting to Rosenberg to look at cohomological descent first; after few days they had (with Kontsevich) also the A-infinity version. But I myself never apart from the ideas fully understood the relation between the derived 1-descent and the cohomological descent for total derived functors.