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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeJul 18th 2014
   • (edited Jul 18th 2014)
   • PermaLink
   Author: Urs
   Format: MarkdownItexI am _still_ trying to see that in non-archimedean analytic geometry the polydiscs are Artin-Mazur-etale contractible. One strategy would be this: show that the analytic spectrum functor from analytic domains to topological spaces is close enough to preserving split hypercovers. If it is close enough, then given any split hypercover by polydiscs (affinoid domains) of a polydisc, the corresponding etale homotopy type would be that of the image of that simplicial object under the [[analytic spectrum]] functor.That would hopefully be close to a split hypercover itself, and since this then would be a hypercover of the space underlying a polydisc, which is contractible, the etale homotopy type would be contractible. That's the strategy. Now to make it work. The first would seem to show that forming analytic spectra preserves fiber products along admissible maps ([[affinoid domain embeddings]]), so that intersections go to intersections. That seems straightforward, but I need to convince myself of this in detail. Then next a problem is that analytic spectra of affinoid covers are _closed covers_. So a priori we end up not with a hyper-cover like simplicial object of open topological sets but of closed sets. I suppose one will want to argue that passing to interiors everywhere is okay. This, too might be straightforward, but I need to convince myself of the details. So it seems easy enough. But some care is required, I suppose.

   I am still trying to see that in non-archimedean analytic geometry the polydiscs are Artin-Mazur-etale contractible.

   One strategy would be this: show that the analytic spectrum functor from analytic domains to topological spaces is close enough to preserving split hypercovers.

   If it is close enough, then given any split hypercover by polydiscs (affinoid domains) of a polydisc, the corresponding etale homotopy type would be that of the image of that simplicial object under the analytic spectrum functor.That would hopefully be close to a split hypercover itself, and since this then would be a hypercover of the space underlying a polydisc, which is contractible, the etale homotopy type would be contractible.

   That’s the strategy. Now to make it work.

   The first would seem to show that forming analytic spectra preserves fiber products along admissible maps (affinoid domain embeddings), so that intersections go to intersections. That seems straightforward, but I need to convince myself of this in detail.

   Then next a problem is that analytic spectra of affinoid covers are closed covers. So a priori we end up not with a hyper-cover like simplicial object of open topological sets but of closed sets. I suppose one will want to argue that passing to interiors everywhere is okay. This, too might be straightforward, but I need to convince myself of the details.

   So it seems easy enough. But some care is required, I suppose.

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeJul 19th 2014
   • (edited Jul 19th 2014)
   • PermaLink
   Author: Urs
   Format: MarkdownItexAh, the Huber affinoids are open (as oppsed to the Berkovich ones, which are closed). So using Huber spaces instead would deal with the second issue above.

   Ah, the Huber affinoids are open (as oppsed to the Berkovich ones, which are closed). So using Huber spaces instead would deal with the second issue above.