Timeline for Precise definition of the $\infty$-category of spaces, continuous maps, homotopies, homotopies between homotopies, and so on
Current License: CC BY-SA 4.0
11 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Dec 4, 2021 at 17:03 | vote | accept | user997814 | ||
| Dec 4, 2021 at 1:09 | answer | added | CoconesFromTwistedArrows | timeline score: 2 | |
| Dec 4, 2021 at 0:28 | answer | added | Emily | timeline score: 7 | |
| Dec 3, 2021 at 19:40 | comment | added | Fernando Muro | A homotopy is just a map from a cylinder. A homotopy between homotopies is just a homotopy between such maps, relative to the boundary of the cylinder. And so on. This is not very simplicial so you can't make a quasi category out of it. | |
| Dec 3, 2021 at 13:32 | comment | added | user997814 | Thanks, Zhen Lin! | |
| Dec 3, 2021 at 13:16 | answer | added | Tim Porter | timeline score: 5 | |
| Dec 3, 2021 at 13:14 | comment | added | Zhen Lin | I'm not sure what you are expecting there. In some sense the answer is yes – the homotopy-coherent nerve functor sends Dwyer–Kan equivalences of Kan-enriched categories to equivalences of quasicategories, so it suffices to verify the claim at the level of Kan-enriched categories, and I'm sure this can be extracted from the proof of the Quillen equivalence between topological spaces (CW complexes) and simplicial sets (Kan complexes). | |
| Dec 3, 2021 at 13:06 | comment | added | user997814 | Thanks! Can you prove the homotopy hypothesis with that definition? | |
| Dec 3, 2021 at 12:41 | comment | added | Zhen Lin | It should be possible to do something like this: define a Kan-enriched category $\underline{\textbf{Top}}$ where the simplicial set $\underline{\textbf{Top}} (X, Y)$ has as its $n$-simplices the continuous maps $\Delta^n \times X \to Y$. Then you can take the homotopy-coherent nerve to get a quasicategory. This has the property that the homotopy category is the naïve homotopy category. | |
| S Dec 3, 2021 at 12:02 | review | First questions | |||
| Dec 3, 2021 at 12:07 | |||||
| S Dec 3, 2021 at 12:02 | history | asked | user997814 | CC BY-SA 4.0 |