Timeline for Lemma 2.1.1.4 in Lurie's HTT
Current License: CC BY-SA 3.0
8 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Aug 14, 2021 at 12:53 | comment | added | user12580 | I am confused by your last sentence. Given two homotopic maps from $K\times \Delta^1$ to $X$, is it clear how to prove the induced maps from $K\times \Delta^1$ to $X_{s'}$ are homotopic? | |
| Mar 8, 2015 at 14:52 | comment | added | Edoardo Lanari | Ok, I managed to fix everything, using 2.1.2.10, I am just missing a detail: I want $f_!$ to be independent from the homotopy class of $f$ when seen as an arrow in $\tau_1 S$. Is it true that the condition of being in the same class for arrows $f,f' \in \tau_1 S$ implies them being homotopic as maps $\Delta [1] \to S$? I just need $\Delta[1] \times \Delta[1]/ (\{ 0\} \times \Delta[1]) = \Delta[2]$ | |
| Mar 7, 2015 at 19:01 | comment | added | Dylan Wilson | For the first question: I believe the exact thing I need is the aforementioned corollary or 2.1.2.10, I can try to rewrite this more clearly later. | |
| Mar 7, 2015 at 18:56 | comment | added | Dylan Wilson | For your second question: Applying the last argument for $Y = K$ and $Y = X_s$ shows that in both cases the homotopy class of a lift is determined by $f$, i.e. there is only one homotopy class of lift for $K \times \Delta^1 \rightarrow S$ so it must be the one obtained by composiiton. | |
| Mar 7, 2015 at 14:33 | comment | added | Edoardo Lanari | Secondly, I agree that having a lift for $X_s \times \Delta[1] \to S$ gives me a lift for $K \times \Delta[1] \to S$, but how to obtain the converse in order to establish the equivalence you mentioned? | |
| Mar 7, 2015 at 14:31 | comment | added | Edoardo Lanari | First of all thanks, I just have two questions: I like the argument at the end, but it seems to me that you are using the fact that the pullback of two homotopic maps along a left fibration gives two homotopic maps. Is it true also for left fibrations (I knew it for Kan fibrations)? | |
| Mar 7, 2015 at 13:47 | comment | added | Dylan Wilson | (Alternatively I can quote Corollary 2.1.2.9 of HTT if you don't like that argument at the end.) | |
| Mar 7, 2015 at 13:38 | history | answered | Dylan Wilson | CC BY-SA 3.0 |