Timeline for Understanding how group cohomology classifies extensions using the derived functor point of view
Current License: CC BY-SA 4.0
10 events
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| Jul 8 at 9:49 | comment | added | DevVorb | @QiaochuYuan I was anyway going to ask for a homotopical interpretation of my problem if the derived functor interpretation wasn't satisfactory. This response is perfect, thank you very much | |
| Jul 5 at 22:57 | comment | added | Qiaochu Yuan | Actually the classification can be stated very conceptually in this language: every extension is given by an action of $G$ on $A$, except that "actions" have to be understood in a homotopical sense, as a morphism of $2$-groups $G \to \textbf{Aut}(A)$. This means $g$ and $h$ don't act exactly the same way but can differ by a conjugation in a way that satisfies a coherence condition. All of this can be written down without any explicitly homotopical or even $2$-categorical language, I think. | |
| Jul 5 at 22:55 | comment | added | Qiaochu Yuan | @Steve: if $A$ is nonabelian then $B^2 A$ has to be replaced with a more complicated space, namely what John Baez calls $B \textbf{Aut}(A)$. This is the classifying space of the automorphism $2$-group of $A$, so roughly speaking it is the "moduli space of $A$'s," and it has $\pi_1 \cong \text{Out}(A), \pi_2 \cong Z(A)$. Then you can write down a classification of homotopy classes of maps into this thing in terms of $\pi_1$ and $\pi_2$ and you get the more typical version of the classification of extensions. | |
| Jul 5 at 21:05 | comment | added | Steve D | This is such an interesting perspective that I've never seen before. I want to spend some time thinking about what this says in the case the "kernel group" $A$ is non-Abelian (in which case you consider $H^n(G,Z(A))$ for $n=2,3$). | |
| Jul 5 at 20:16 | history | edited | Qiaochu Yuan | CC BY-SA 4.0 |
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| Jul 5 at 17:29 | vote | accept | DevVorb | ||
| Jul 5 at 16:21 | history | edited | Qiaochu Yuan | CC BY-SA 4.0 |
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| Jul 5 at 16:08 | history | edited | Qiaochu Yuan | CC BY-SA 4.0 |
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| Jul 5 at 14:26 | history | edited | Qiaochu Yuan | CC BY-SA 4.0 |
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| Jul 5 at 14:20 | history | answered | Qiaochu Yuan | CC BY-SA 4.0 |