Timeline for Free loop space objects and actions
Current License: CC BY-SA 3.0
6 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Sep 26, 2017 at 10:10 | vote | accept | Mike Shulman | ||
| Sep 25, 2017 at 7:32 | answer | added | Pavel Safronov | timeline score: 9 | |
| Sep 20, 2017 at 9:13 | comment | added | Pavel Safronov | Let $\mathcal{D}$ be an $\infty$-category with finite limits. Given a pointed object $A$, $\Omega A$ is a group. Given a morphism $B\rightarrow A$, $\Omega A$ acts on $B\times_A \ast$. You can apply this to $\mathcal{D}=\mathcal{C}_{/X}$, $A=(p_1\colon X\times X\rightarrow X)$ pointed by the diagonal and $B=(p_1\colon X\times Y\rightarrow X)$ with $(\mathrm{id}\times f)\colon X\times Y\rightarrow X\times X$. Unfortunately, I don't know a reference and this might be folklore. | |
| Sep 20, 2017 at 8:08 | comment | added | Mike Shulman | @PavelSafronov Can you supply a citation, and can you define the action as well? | |
| Sep 20, 2017 at 8:00 | comment | added | Pavel Safronov | Consider the map $\ast\coprod\ast\rightarrow \ast$ in pointed spaces (choose an arbitrary pointing on the left). Its Cech conerve is a cogroupoid $\ast\rightrightarrows S^1 ...$ which gives $S^1$ the structure of a cogroup in pointed spaces. Therefore, if $\mathcal{C}$ is cotensored over spaces, $\mathrm{Map}(S^1, X)\rightarrow \mathrm{Map}(\ast, X)$ is a group. | |
| Sep 20, 2017 at 6:47 | history | asked | Mike Shulman | CC BY-SA 3.0 |