Timeline for Is there a higher categorical structure which models the (higher) conjugation actions of a group acting on itself?
Current License: CC BY-SA 4.0
7 events
| when toggle format | what | by | license | comment | |
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| May 11, 2021 at 13:47 | vote | accept | Chris H | ||
| May 11, 2021 at 11:52 | comment | added | Tim Porter | As the technicalities of the simplicial question would not really fit here, I have gone over to contacting people who may be interested by e-mail. (If there are several people who would like to join in, we could start a discussion on the Cateogy zulip site, or you can contact me directly.) | |
| May 11, 2021 at 11:00 | comment | added | Tim Porter | After a bit of ferreting, and perhaps thinking, one could try and replace the $cat^n$-group by an equivalent simplicial group, thought of as an $\infty$-group, $G$,now take the auto-equivalences of that $G$, this gives an $\infty$-groupoid analogue of the automorphism group of the original object, ... but that merely says there should be something there it does not produce it for you as a construction on the original object. Working with simplicial groups however may give a slightly different perspective on the original question. | |
| May 11, 2021 at 10:34 | comment | added | Tim Porter | To be honest, I'm not sure. It may be that one gets a $cat^{2n}$-group. I know that there are abstract versions of the $G\to Aut(G)$ construction in certain types of category which include the category of crossed modules, but I cannot recall the details. | |
| May 10, 2021 at 9:20 | comment | added | Fernando Muro | Hi, Tim! I meant, given a $cat^n$-group $G$, is it possible to define an automorphism $cat^n$-group $Aut(G)$ and merge them to an `inner automorphisms' $cat^{n+1}$-group? Like with $G\to Aut(G)$ when $G$ is a group. We can take it one step further using Norrie's paper since $G\to Aut(G)$ is a crossed module. I was giving for granted this had already been done. | |
| May 9, 2021 at 10:53 | history | edited | Tim Porter | CC BY-SA 4.0 |
Added a different perspective that might help.
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| May 9, 2021 at 10:48 | history | answered | Tim Porter | CC BY-SA 4.0 |