Timeline for Limits of infinity categories and mapping spaces
Current License: CC BY-SA 4.0
7 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jan 9 at 11:32 | comment | added | user141099 | @danielgratzer How do you construct hom spaces out of taking powers and pullbacks in Cat$_{\infty}$? | |
| Jul 10, 2023 at 10:40 | comment | added | daniel gratzer | @Kim I wouldn't expect this to work with pushouts though: exponentiation with $\Delta^1$ won't commute with colimits. Moreover, if you consider $\Delta^1$ as a category and the pushout converting it into a circle, we go from having every hom-space being subsingleton to having a hom-space with infinitely many connected components. However, colimits commute with taking connected components, so this shouldn't arise if hom spaces commuted with colimits. | |
| Jul 9, 2023 at 14:14 | comment | added | Kim | Thanks for your nice answer! And it seems like the argument also works if we replace colimits with limits. | |
| Jul 9, 2023 at 14:12 | vote | accept | Kim | ||
| Jul 8, 2023 at 11:44 | comment | added | daniel gratzer | Good point @MaximeRamzi! I guess the key point is that we can construct hom spaces out of taking powers and pullbacks in $\mathbf{Cat}_\infty$ and just be done with it. | |
| Jul 8, 2023 at 11:34 | comment | added | Maxime Ramzi | The argument is correct but you don't need to do all this replacing/strictifying to make it work :) if you argue inside the $\infty$-category of $\infty$-categories, you can simply say "limits commute with limits" | |
| Jul 8, 2023 at 11:15 | history | answered | daniel gratzer | CC BY-SA 4.0 |