Timeline for $E_\infty$-maps of diagrams
Current License: CC BY-SA 4.0
6 events
| when toggle format | what | by | license | comment | |
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| Apr 8, 2020 at 14:40 | comment | added | Phil Tosteson | This is the sort of question that obstruction theory is meant to answer. Dennis's obstruction is in lifting the 1 skeleton, but there should be higher ones also. | |
| Apr 8, 2020 at 12:12 | comment | added | Maxime Ramzi | @DenisNardin : ah you're right, that was obvious (although I'll have to think of an example). However this doesn't seem to generalize immediately to $\Delta^{op}$ (but I do think there will be counterexamples there too, I just don't see them), so I'm still interested in that. | |
| Apr 8, 2020 at 11:09 | comment | added | Denis Nardin | You can get a counterexample by taking a pair of map of algebras that are not homotopic to each other but they become so when you forget the algebra structure (by taking $I=\Delta^1$, and the two diagrams just the constant diagram on the source and target respectively). It should be possible to cook up such an example... | |
| Apr 8, 2020 at 9:57 | comment | added | Maxime Ramzi | I think theorem 8.19 in Lurie's DAG III might be relevant : it says that for a nice enough simplicial symmetric monoïdal model category $A$, the model structure on commutative algebras in $A$ has as underlying $\infty$-category precisely the $\infty$-category of $E_\infty$-algebras in the underlying $\infty$-category of $A$. It's also mentioned in the same paper that $Sp^\Sigma$ (symmetric spectra) are such a nice model category. I'm not sure how to use that yet, but it might be useful | |
| Apr 8, 2020 at 8:52 | history | edited | Maxime Ramzi | CC BY-SA 4.0 |
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| Apr 8, 2020 at 8:21 | history | asked | Maxime Ramzi | CC BY-SA 4.0 |