Timeline for What are the Čech-local equivalences of (simplicial pre)sheaves?
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| Dec 20, 2013 at 22:43 | comment | added | Zhen Lin | @DavidWhite The Jardine model structure is indeed a left Bousfield localisation of the Čech model structure. | |
| Dec 20, 2013 at 19:59 | comment | added | Dmitri Pavlov | Dugger, Hollander, and Isaksen in their paper “Hypercovers and simplicial presheaves” state before Corollary A.7 that “It would be interesting to know more about sPre(C)_Č, for instance to have an explicit characterization of the weak equivalences.”. So it seems that such an explicit description was an open problem at the time their paper was published (2004). There are, of course, several examples of nonhypercomplete sites, for example, the above paper has one in Example A.10. | |
| Dec 20, 2013 at 18:46 | comment | added | David White | What reference are you using for studying the Cech model structure? For the Jardine? How do you know that not every Jardine weak equivalence is a Cech weak equivalence? What about the converse? I keep thinking the Cech weak equivalences should be the maps which induce isomorphisms on subsheaves or something, but if that was true then there would likely be a Bousfield localization relationship between the Jardine and Cech model structures. | |
| Dec 19, 2013 at 15:50 | comment | added | David White | (continued) It's related to the question of whether $Hom(f^*,Z)$ is a weak equivalence for $Hom$ the internal hom of the monoidal model category vs. whether $Map(f^*,Z)$ is a weak equivalence for the simplicial mapping space. Some details are in my research statement (which I remember you read earlier this fall), and we hope to have a preprint soon. We'll both be at the MSRI conference next month and my goal is to be done by then. | |
| Dec 19, 2013 at 15:49 | comment | added | David White | I attacked a related problem over the summer while visiting Carles Casacuberta in Barcelona. We were also working in motivic spectra, but much of our work applies to motivic spaces as well. Instead of the Cech model structure we were looking at the results of localizing the maps seen to be isomorphisms by various motivic homology theories. This led us to studying the difference between maps which are seen to be isomorphisms by the whole homotopy sheaf vs. just by the homotopy sheaf evaluated at the point. | |
| Dec 19, 2013 at 15:15 | history | asked | Zhen Lin | CC BY-SA 3.0 |