@SaalHardali That's a great question! I actually don't have any evidence that they're inequivalent as commutative algebras, and in fact my guess would be that they are at least equivalent as E_k-algebras with k growing with the prime
One reason to think that is that establishing this would be a first step in the proof of showing a multiplicative version of the equivalence between the homotopy category of E-local spectra and differential E_*E-comodules
And I think the latter is a reasonable guess because the asymptotic equivalence of Barthel, Schlank and Stapleton is in fact symmetric monoidal
@skd based on unstable adams charts, it looks like the non-divided alpha elements are available in pi_* S^3, but their p-divisions don’t all become available until you further stabilize
I'm certainly not the first to have done so, but I've written the classical theory of G-spectra from a completely ∞-categorical point of view in these notes
And that's even before we throw in spectral Mackey functors
@HarryGindi I did not write down the details for the Lie group version when you want all closed subgroups because we did not need them for that seminar, but it works almost verbatim with the addition of a bunch of annoying shifts
By the way, if someone knows of a version for general topological groups when you consider more than just the finite subgroups I'd love to hear it
Does Bar-Cobar adjunction for a pair of koszul dual operads interchange free algebras with trivial coalgebras for the koszul dual co-operad and vice versa? (in nice circumstances). Mainly looking for the "rule of thumb" not the most general strongest statement
@skd actually, it looks like the element in the 31-stem (idk if that's the smallest example, just happened to be where i looked on the page) isn't there at all 'til S^5
at any rate, i'd try to understand this through the EHPSS & i'd try to find the relevant fact in the mahowald paper w/ the approximate title the image of j in the EHPSS
