Timeline for Group cohomology as homotopy groups
Current License: CC BY-SA 4.0
6 events
| when toggle format | what | by | license | comment | |
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| Jul 15, 2020 at 17:52 | comment | added | Aaron Mazel-Gee | In paragraph 5, I think you may be looking for the notion of a parametrized spectrum over $BG$. You can think of such a thing as a functor $BG \to Spectra$, or equivalently as an infinite loop object in $Spaces_{/BG}$. More generally, for a base space $X$, if $X$ has an $E_n$-monoidal structure (e.g. an $n$-fold loopspace) you can contemplate $E_n$-maps $X \to Spectra$. This sort of situation (in the case where the functor is valued in invertible spectra) is studied in Antolin-Camarena--Barthel arXiv:1411.7988. | |
| Jul 15, 2020 at 6:21 | comment | added | Denis Nardin | Can you give a reference for the definition $H^i(G,A)$ when for a group extension? Or do you just mean normal group cohomology and the extension is just an extension with abelian first term? | |
| Jul 12, 2020 at 22:44 | comment | added | Zhen Lin | I mean second. I'm not familiar with $H^2$ for group extensions. | |
| Jul 12, 2020 at 22:39 | comment | added | David Corwin | Do you mean in my fourth paragraph? | |
| Jul 12, 2020 at 22:27 | comment | added | Zhen Lin | Isn't the observation in your second paragraph actually a theorem? Specifically, for a space $X$ and a sheaf $A$ of abelian groups on $X$, $H^n (X, A) \cong \pi_{m - n} \mathbf{R} \Gamma (X, K(A, m))$, where $K(A, m)$ is the simplicial sheaf version of the usual thing. Group cohomology comes about when you have $X = B G$. The existence of $H^1 (G, A)$ for non-abelian $A$ comes about because we can define $K (A, 1) = B A$ for non-abelian $A$ as well, etc. | |
| Jul 12, 2020 at 21:04 | history | asked | David Corwin | CC BY-SA 4.0 |