Timeline for Homotopy coherent action on differential graded vector space
Current License: CC BY-SA 4.0
9 events
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| Feb 6, 2019 at 12:25 | comment | added | Denis Nardin | @NajibIdrissi I probably interpreted "homotopy coherent action" in a different way than you: I interpreted it as a map of $E_1$-spaces $G→\mathrm{Map}(Y,Y)$ where $\mathrm{Map}$ is the derived mapping space of $Y$. Then since $\mathrm{Map}(Y,Y)\cong \mathrm{Map}(X,X)$ as $E_1$-spaces we are done. | |
| Feb 6, 2019 at 12:21 | comment | added | Najib Idrissi | @Lukas Yes, if an algebra $A$ is Koszul then the operad $(0,A,0,0,\dots)$ is Koszul if I'm not mistaken. The HTT is probably overkill here, but I think that if you write down the details of the strategy from your question, you essentially end up reproving HTT for $A$-modules, no? | |
| Feb 6, 2019 at 12:20 | comment | added | Najib Idrissi | @Denis Okay, but given chain complexes $X \simeq Y$ and a $G$-action on $X$, how do you find a $G$-action on $Y$? | |
| Feb 6, 2019 at 12:15 | comment | added | Phil Tosteson | @Denis Also over the integers. | |
| Feb 6, 2019 at 11:41 | comment | added | Lukas Woike | Thank you, Naijib. Just to be sure that I am following your argument correctly: You mention that the group algebra is Koszul because that implies that the corresponding operad (being non-trivial only in arity 1) is Koszul such that you can apply that HT theorem? Another small question: I was also briefly thinking about applying HTT here, but it seemed a bit of an overkill. In the question I sketched I very naive way of transferring which should work in this special case. Do you think that would also work? | |
| Feb 6, 2019 at 11:31 | vote | accept | Lukas Woike | ||
| Feb 6, 2019 at 11:10 | comment | added | rori | @DenisNardin it appears that the question itself speaks of "vector spaces" so presumably it is not answerer's fault that they assume that everything is over a field | |
| Feb 6, 2019 at 11:06 | comment | added | Denis Nardin | Over a field the chain complex is quasi-isomorphic to its cohomology... | |
| Feb 6, 2019 at 10:47 | history | answered | Najib Idrissi | CC BY-SA 4.0 |