Timeline for Décalage and the simplicial path object
Current License: CC BY-SA 4.0
7 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Sep 7, 2022 at 14:23 | comment | added | user420620 | thanks, yes, this is clear. I was confused ... | |
| Sep 2, 2022 at 5:28 | comment | added | Jon Pridham | Essentially yes. For instance, since $Y_0 \to Y_0\times_YY^{\Delta^1}$ is a trivial cofibration, you get a weak equivalence $ Y_0\times_YY^{\Delta^1} \to \mathrm{Dec}_+Y$ over $Y$ by the lifting property. | |
| Sep 1, 2022 at 21:02 | comment | added | user420620 | @JonPridham: how do you conclude that "$X\to Y$ factorises though $Y_0$ in the homotopy category iff it lifts to $Dec_+ Y$" ? By a standard diagram chasing argument following directly from the axioms of a model category ? | |
| Sep 1, 2022 at 20:27 | comment | added | Jon Pridham | Writing $\mathrm{Dec}_+:= Y \circ [+1]$ as in Illusie's complexe cotangent (1972), the inclusion $i : Y_0 \to Y$ factorises as $Y_0 \to \mathrm{Dec}_+Y \to Y$. If $Y$ is a Kan complex, the first map is a weak equivalence and the second a Kan fibration. Thus a map $X \to Y$ factorises through $Y_0$ in the homotopy category iff it lifts to $ \mathrm{Dec}_+Y$. Although $Y_0\times_Y Y^{\Delta^1}$ is another space with this property, the final conclusion holds without needing any intermediate comparison; path objects aren't unique. | |
| Sep 1, 2022 at 18:29 | history | edited | LSpice | CC BY-SA 4.0 |
Links
|
| Sep 1, 2022 at 18:27 | history | edited | YCor | CC BY-SA 4.0 |
formatting, added tag
|
| Sep 1, 2022 at 18:14 | history | asked | user420620 | CC BY-SA 4.0 |