Timeline for Groupoid of points, shape and stratified shape of $\operatorname{Sh} (X_\text{pro-ét})$
Current License: CC BY-SA 4.0
10 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Apr 19, 2022 at 21:17 | answer | added | M L | timeline score: 2 | |
| Apr 19, 2022 at 21:02 | comment | added | user40276 | @PiotrAchinger I've edited the question and corrected a mistake. I said profinite étale $\pi_1$ when I've actually meant étale $\pi_1$ for the map with dense image. In order to get the usual SGA1 $\pi_1$ you need to take profinite completion of the shape. | |
| Apr 19, 2022 at 21:00 | comment | added | Michael Hardy | $$\begin{align} & pro-ét \\ {} \\ & \text{pro-ét} \end{align}$$ Above one sees what some parts of this posting looked like before and after the edits by LSpice. In particular, note that a hyphen looks different from a minus sign. | |
| Apr 19, 2022 at 20:58 | history | edited | Michael Hardy | CC BY-SA 4.0 |
added 1 character in body
|
| Apr 19, 2022 at 20:56 | history | edited | LSpice | CC BY-SA 4.0 |
`\DeclareMathOperator`; `\text`; names of references; minor typos; deleted "thanks"
|
| Apr 19, 2022 at 20:53 | comment | added | Piotr Achinger | Sorry, I deleted my comment because I could not edit it. Let me repeat here my comments so that the above makes sense: 1) the map with dense image goes from proetale $\pi_1$ to etale $\pi_1$; in fact there is a third prodiscrete group $\pi_1^{\rm SGA3}$ and maps $\pi_1^{\rm proet}\to \pi_1^{\rm SGA3}\to \pi_1^{\rm et}$ which are respectively the prodiscrete and profinite completion. 2) Artin-Mazur observed that $\pi_1^{\rm SGA3}$ is the $\pi_1$ of the etale homotopy type. Does that imply that $\Pi_\infty(X_{\rm et}$ and $\Pi_\infty(X_{\rm proet})$ are different bc they have different $\pi_1$? | |
| Apr 19, 2022 at 20:48 | history | edited | user40276 | CC BY-SA 4.0 |
deleted 80 characters in body
|
| Apr 19, 2022 at 20:44 | comment | added | user40276 | @PiotrAchinger I'm not well-versed too. I've included the reference (Remark 7.4.12 in BS). What's the étale $\pi_1$ in your comment? I think you mean profinite $\pi_1$. If so, you need to take the profinite completion of the shape or take the finite étale topoi. Am I misunderstanding anything? | |
| Apr 19, 2022 at 20:40 | history | edited | user40276 | CC BY-SA 4.0 |
added 49 characters in body
|
| Apr 19, 2022 at 20:05 | history | asked | user40276 | CC BY-SA 4.0 |