Timeline for Homotopy pullback of $\mathbb{A}^1$-projections in the Nisnevich localization
Current License: CC BY-SA 4.0
8 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Aug 30, 2019 at 10:53 | vote | accept | L. Xie | ||
| Aug 20, 2019 at 16:18 | comment | added | Marc Hoyois | @Alexis The homotopy pullback of $A^1\times X\to X$ along any map $F\to X$ is $A^1\times F\to F$, which is still an $A^1$-weak equivalence, so indeed the localization does not change. On the other hand, the class of all $A^1$-weak equivalences in $L_{Nis}sPre$ is not closed under homotopy pullbacks, so if you localize at the closure you will get a different localization (which still may or may not be an ∞-topos!). To actually get a left exact localization, you have to localize at the simultaneous closure of $A^1$-weak equivalences under homotopy colimits and homotopy pullbacks. | |
| Aug 20, 2019 at 16:16 | comment | added | L. Xie | The homotopy closure I mean is in Rezk's notes section 5.5 faculty.math.illinois.edu/~rezk/homotopy-topos-sketch.pdf . This should be the quivalent to that quoted in Raptis and Strunk. I ask it because I wonder if there is a reason not to localize with repect to the homotopy pullback closure to get a model topos. | |
| Aug 20, 2019 at 15:28 | comment | added | David White | I think we have different definitions of what it means to be "closed under homotopy pullbacks" - in my answer, it means that if you have a map of diagrams that is an objectwise weak equivalence then it induces a weak equivalence on homotopy limits. Googling your comment about "model topos" I find this paper by Raptis and Strunk, where it closure seems to mean something else arxiv.org/pdf/1704.08467.pdf | |
| Aug 20, 2019 at 14:44 | comment | added | L. Xie | If just localizing with respect to $\bar{S}$ once, I expect them not to be the same. $L_{\bar{S}}$ is left exact. But it's a fact that motivic homotopy theory is not a model topos. What's the problem here? | |
| Aug 20, 2019 at 13:20 | comment | added | David White | You would get the same result. In general, if you localize a model category with respect to a class of maps contained in the weak equivalences, nothing changes. So, if you first localize with respect to S, then with respect to the homotopy pullback closure of S (or just do both together, with this one localization), you get the same as if you just did it with respect to S. | |
| Aug 20, 2019 at 10:40 | comment | added | L. Xie | The $A_1$-localization is for the class $S$ of all maps $A_1\times X\to X$. What will happen if localizing $L_{Nis}sPre$ at the homotopy pullback closure of $S$? | |
| Aug 19, 2019 at 22:51 | history | answered | David White | CC BY-SA 4.0 |