Timeline for Density Theorem for $\infty$-Categories (HTT, Lemma 5.1.5.3)
Current License: CC BY-SA 4.0
10 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Oct 4, 2022 at 9:42 | comment | added | Maxime Ramzi | @Ken : my pleasure ! :) sorry for the earlier mistake :D | |
| Oct 3, 2022 at 10:56 | vote | accept | Ken | ||
| Oct 3, 2022 at 10:56 | comment | added | Ken | @MaximeRamzi Ahhh, that's so clever. Thank you very much for your patient instruction! | |
| Oct 3, 2022 at 7:49 | comment | added | Maxime Ramzi | @Ken : sorry, my previous comment was silly, what I really meant to write was that $(\mathcal C_{/X})^\triangleright$ was equivalent, over $\mathcal P(S)$, to $\mathcal D_{/X}$, where $\mathcal D$ is the full subcategory spanned by $\mathcal C$ and $X$. So now $\mathcal E$ is $\mathcal D_{/X} \times_\mathcal D \mathcal D_{j(s)/}$ and so you can do the exact same argument as for $\mathcal E^0$ | |
| Oct 2, 2022 at 20:33 | comment | added | Ken | @MaximeRamzi No, nor do I understand why that particular fact is relevant. Could you elaborate on that approach? | |
| Oct 2, 2022 at 13:20 | comment | added | Maxime Ramzi | @Ken : have you tried using the fact that if $D$ has a terminal object (e.g. $\mathcal C_{/X}$), then $D^\triangleright$ deformation retracts onto $D$ ? | |
| Oct 2, 2022 at 7:49 | comment | added | Z. M | @Ken I just want to point out that there is no essential difference between the slice and the "fat" one denoted by a superscript — there is a canonical categorical equivalence between them, thus also a weak equivalence. | |
| Oct 2, 2022 at 7:46 | comment | added | Ken | Your argument works perfectly in the case of $\mathcal{E}^0$, but I am not sure how to proceed in the case of $\mathcal{E}$; I was able to construct homotopies $\mathcal{E}\times \Delta^1\to \mathcal{P}(S)_{j(s)/}$ and $\mathcal{E}\times \Delta^1 \to \mathcal{C}_{/ X}$, (In Lurie's notation, these should be the "upper slices", but let's put this aside for now) but they don't seem to produce the same map upon postcomposing the maps into $\mathcal{P}(S)$. Could you expand on that part when you have time? | |
| Oct 1, 2022 at 9:58 | comment | added | Ken | Thanks, Maxime! This looks very promising. Your answer contains exactly what I had been missing, as you correctly guessed. Give me some time to work out the details before I accept your answer. I also appreciate the comment on the alternative proof. | |
| Oct 1, 2022 at 9:24 | history | answered | Maxime Ramzi | CC BY-SA 4.0 |