Timeline for Weakly contractible cover in étale homotopy theory
Current License: CC BY-SA 4.0
9 events
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| Oct 7, 2020 at 16:37 | comment | added | user30211 | @QiaochuYuan I thought about your example with $\text{String}(3)$. There are, as you know, several cartesian closed categories that manifolds fit into, e.g. in synthetic differential geometry. Would $\text{String}(3)$ fit into such a category as that? If so, it might still be reasonable for a cover to fit into sheaves on some big site here. | |
| Sep 22, 2020 at 16:10 | comment | added | user30211 | @MarcHoyois. I guess that would be analogous to a map $\mathbb{R} \rightarrow S^3$ factoring through an inclusion $S^1 \rightarrow S^3$ through a covering $\mathbb{R} \rightarrow S^1$. Analogously, the map $\text{Spec}(\mathbb{F}_p^{sep}) \rightarrow \text{Spec}(\mathbb{Z})$ factors through $\text{Spec}(\mathbb{F}_p) \rightarrow \text{Spec}(\mathbb{Z})$ by a covering map $\text{Spec}(\mathbb{F}_p^{sep}) \rightarrow \text{Spec}(\mathbb{F}_p)$. The relevant homotopy groups are the same in each example. In each case we lose track of $\pi_3$ of the base space. | |
| Sep 22, 2020 at 5:58 | comment | added | Marc Hoyois | You can take $E$ to be a separable closure of a generic point of $B$... But it is not possible to achieve this with $E\to B$ surjective. | |
| Sep 21, 2020 at 20:23 | comment | added | user30211 | Thanks Qiaochu, you're a legend | |
| Sep 21, 2020 at 20:07 | comment | added | Qiaochu Yuan | Maybe you want a loop space that shifts all the etale homotopy up one degree though; I don't think this construction will have that property but I have no idea. | |
| Sep 21, 2020 at 20:04 | comment | added | Qiaochu Yuan | Depending on what you want out of a loop space you can do the following instead. You can construct the based loop space $\Omega X$ of a based space from its free loop space $LX$ by fixing a point on the circle, considering the corresponding projection $LX \to X$, and taking a (homotopy) fiber. There's an analogous construction in AG: the (derived) free loop space $LX$ of a scheme (or stack, derived stack, etc.) is the (derived) fiber product $X \times_{X \times X} X$, where the two maps $X \to X \times X$ are both the diagonal $\Delta$. This is a global version of Hochschild homology. | |
| Sep 21, 2020 at 19:59 | comment | added | user30211 | @QiaochuYuan. Could it still be a higher topos? My goal was to construct the loop space of a scheme by taking $E \times_B E$, but I did not foresee this problem. | |
| Sep 21, 2020 at 19:51 | comment | added | Qiaochu Yuan | What sort of an object are you expecting $E$ to be? In the topological setting already the $3$-connected cover of $S^3$ isn't a manifold (it's the string group $\text{String}(3)$) so I imagine it's too much to expect $E$ to be a scheme (as opposed to an ind-scheme or something stranger). | |
| Sep 21, 2020 at 18:10 | history | asked | user30211 | CC BY-SA 4.0 |