Timeline for Motives and homotopy theories of algebraic varieties
Current License: CC BY-SA 4.0
17 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| S May 17, 2019 at 21:06 | history | bounty ended | CommunityBot | ||
| S May 17, 2019 at 21:06 | history | notice removed | user138661 | ||
| May 17, 2019 at 21:06 | vote | accept | CommunityBot | ||
| May 16, 2019 at 12:51 | answer | added | Will Sawin | timeline score: 2 | |
| May 16, 2019 at 12:24 | comment | added | François Brunault | The category $\mathrm{DM}(S,\Lambda)$ over any base scheme $S$ and any coefficient ring $\Lambda$ has been constructed by Cisinski-Déglise in their book. | |
| May 16, 2019 at 12:12 | history | edited | user138661 | CC BY-SA 4.0 |
added 177 characters in body
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| May 16, 2019 at 12:06 | answer | added | user25309 | timeline score: 0 | |
| May 16, 2019 at 9:54 | answer | added | François Brunault | timeline score: 1 | |
| S May 14, 2019 at 16:19 | history | bounty started | CommunityBot | ||
| S May 14, 2019 at 16:19 | history | notice added | user138661 | Draw attention | |
| May 7, 2019 at 13:08 | comment | added | Jon Pridham | Starting from categories of dualisable motives over $X$, a point in $X(k)$ gives you a fibre functor to $k$-motives, and Tannakian formalism then gives a form of homotopy type in $k$-motives. A Weil cohomology theory would give a fibre functor down to complexes, and the resulting thing would then be an arithmetic, rather than a geometric, homotopy type. For these homotopy types to behave, you want a $t$-structure (so the standard conjectures would help). For some thoughts about this, see arxiv.org/abs/1309.0637 | |
| May 7, 2019 at 10:47 | comment | added | François Brunault | See Dundas, Levine, Østvær, Röndigs, Voevodsky, Motivic homotopy theory. Lectures from the Summer School held in Nordfjordeid, August 2002. Universitext. MR2334212 | |
| May 6, 2019 at 17:47 | comment | added | Denis Nardin | Relevant: arxiv.org/abs/0712.3291 | |
| May 6, 2019 at 17:21 | comment | added | François Brunault | It seems the category you want would map into the category DM and not the other way round, so I'm not sure my comment will be useful for your question. You could also look for the notion of motivic fundamental group, I don't know if this can be made into a functor from some natural category (just as motivic cohomology can be seen as a functor from DM). | |
| May 6, 2019 at 17:14 | comment | added | user138661 | @FrançoisBrunault but does every "Weil homotopy theory" factor through it, like every Weil cohomology theory supposedly factors through the category of motives? Assuming we know what a Weil homotopy theory is. | |
| May 6, 2019 at 17:07 | comment | added | François Brunault | There is the $\mathbb{A}^1$-homotopy theory of schemes (Morel, Voevodsky). Voevodsky's triangulated category of motives $DM$ embeds into the $\mathbb{A}^1$-derived category $D_{\mathbb{A}^1}$ where the objects have "explicit" descriptions (essentially, complexes of presheaves on the category of smooth schemes). Is this in the spirit of what you are looking for? | |
| May 6, 2019 at 15:23 | history | asked | user138661 | CC BY-SA 4.0 |