Timeline for Is there a model structure for S-modules such that cofibrant operad-algebras forget to cofibrant S-modules?
Current License: CC BY-SA 4.0
8 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Nov 29, 2023 at 23:26 | vote | accept | David White | ||
| Jun 28, 2022 at 7:37 | comment | added | David White | @FernandoMuro Example 10.5.2 in the new Hill, Hopkins, and Ravenel book says $e_1: S^{-1}\wedge S^1 \to S^{-0}$ is a weak equivalence (where $S^{-0}$ is the sphere spectrum), but $Sym^n(e_1)$ is not, because $Sym^n$ of the left side is weakly equivalent to the suspension spectrum of the classifying space of principle $\Sigma_n$-bundles with disjoint basepoint. If $S^{-0}$ was cofibrant then you'd need $Sym^n(e_1)$ to be a weak equivalence if you're to have a model structure on commutative monoids. So maybe $Com$ isn't admissible in your Examples 2, 5, and 6. | |
| Jun 27, 2022 at 19:44 | comment | added | David White | Yes, indeed, there are several general theorems that would be true for this new model that are not known in general. Like lifting quillen equivalences between models of spectra to categories of commutative ring spectra or commutative ideals. There are lots of results where you want a commutative O algebra to forget to a commutative underlying object. But this is too far afield for what i need right now | |
| Jun 27, 2022 at 15:46 | comment | added | Tyler Lawson | @DavidWhite That's certainly a possibility. Hopefully then you have some intrinsic interest in the types of O-algebras that can be modeled in your new category! | |
| Jun 27, 2022 at 5:14 | comment | added | David White | Yes, Theorem 3 there does seem to apply (and you already did it, in your Example 6 there). Next is to check that all operads in the new model structure are admissible. We still have the good interval object, but now not all objects are fibrant. It's an exercise. The hard part (to know cof $O$-alg forget to cof obj) is checking good behavior of the functor $(O,A) \to O_A$. The only way I know to do that is to check that, for $X$ with a $\Sigma_n$ action, $X$ cofibrant in $M$, $X \otimes_{\Sigma_n} f^{\Box n}$ is some kind of cofibration, where $f$ is a cofibration. That's really hard. | |
| Jun 26, 2022 at 21:50 | comment | added | Fernando Muro | @DavidWhite that you can do under mild assumptions, in particular for $S$-modules sciencedirect.com/science/article/pii/… I don’t know which of Tyler’s points would fail, though. I suspect that the first one. | |
| Jun 26, 2022 at 17:46 | comment | added | David White | In the application I have in mind, I probably want to force the unit to be cofibrant, and give up that commutative monoids are equivalent to $E_\infty$ things. | |
| Jun 26, 2022 at 15:28 | history | answered | Tyler Lawson | CC BY-SA 4.0 |