Timeline for Homotopy fixed points of involutive automorphisms of discrete groups
Current License: CC BY-SA 4.0
8 events
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| Dec 19, 2021 at 23:01 | answer | added | rvk | timeline score: 3 | |
| Oct 13, 2021 at 13:41 | history | edited | rvk | CC BY-SA 4.0 |
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| Oct 7, 2021 at 11:28 | comment | added | rvk | @BertramArnold: Your explanation is a bit over my head. Is there a more pedestrain way to say this? Something like: the internal mapping functor on groupoids commutes with the nerve construction. Further, the nerve of a groupoid is Kan. So I can compute $map(B\Gamma, B(\Gamma \ltimes G)$ in groupoids, and then similarly for the space of sections (writing it as a fiber product)? But I am a bit out of my depth here. | |
| Oct 7, 2021 at 7:29 | comment | added | Bertram Arnold | Any category $C$ with a $\Gamma$-action, i.e. (2-)functor $B\Gamma\to\operatorname{Cat}$, can be equivalently encoded via its Grothendieck construction, the (co-)cartesian fibration $C_{h\Gamma}\to B\Gamma$. Homotopy fixed points are sections of this functor. In your example, the homotopy orbits $BG_{h\Gamma}$ are equivalent to the classifying space of the semidirect product $\Gamma\ltimes G$, and the space of sections works out to be your description - one way to see this is that 1-types are closed under limits and equivalent to groupoids, so you can compute the limit in either (2-)category. | |
| Oct 7, 2021 at 6:32 | history | edited | rvk | CC BY-SA 4.0 |
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| Oct 7, 2021 at 6:23 | history | edited | rvk | CC BY-SA 4.0 |
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| Oct 7, 2021 at 6:09 | history | edited | YCor | CC BY-SA 4.0 |
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| Oct 7, 2021 at 1:28 | history | asked | rvk | CC BY-SA 4.0 |