@TylerLawson that's a clear explanation, thanks

@CharlesRezk passing to connective covers feels like cheating though

i'm confused about ando-hopkins-rezk again :( i'm trying to understand condition 7.10.ii, about the bernoulli denominators. this is a necessary condition, coming from translating a null homotopy of spin --> gl_1 S --> gl_1 KO_p to a map bspin --> Q (x) gl_1 KO_p and requiring that it agree with the miller invariant on postcomposition to Q/Z (x) gl_1 KO_p. that's fine

their citation for what that postcomposition does is 5.18, which says that the miller invariant can be calculated by any integral orientation, so pick the ABS orientation and use it. that seems circular if we're trying to construct the ABS orientation? is the point that if there are E_infty orientations, then they would restrict to A_infty orientations, so we could calculate their unstable miller invariant instead, and the ABS orientation is somehow otherwise known to be A_infty?

i <3 asking a question & watching 5 people quit the room within a minute later

@JoeBerner Yes there are two model catgory construction. One due to Fabien Morel "Ensembles profinis simpliciaux et interprétation géométrique du foncteur T" and one due to Isaksen "Completion of pro-spaces". There is also an infinity-categorical version due to Lurie. Together with Ilan Barnea and Yonatan Harpaz, we have written a paper (geoffroy.horel.org/Pro-categories.pdf) in which we compare these three constructions.

ok, i'm pretty sure that's what's happening: you use the A_infty-ness of the orientation MU --> KU, then use the effects of the maps BU --> BO and KO --> KU on homotopy. that's cool, and i guess that's what's written, without the emphasis on A_infty vs E_infty. follow-up question: how do you amplify this into something that works for the unstable miller invariant of (L_K(1)) tmf? do you have, idk, BU<6> --> BString, tmf --> K^Tate, and something nice about the sigma-orientation of K^Tate?

@GeoffroyHorel seeing you in the chat room was actually a partial motivation for asking this, since I've used that paper in what I'm working on. The constructions you're mentioned due respectively to Quick, Morel, and Isaksen (his thesis specifically), model the profinite spaces, p-profinite spaces, and protruncated spaces of Lurie(I have a short argument for this, but you can probably imagine it).

I want to localize profinite spaces away from p, which I am concerned is different. I think I have an example of a map of spaces that is an equivalence at each \ell different from p but is not a profinite equivalence "away from p". My issue is that \ell-profinite spaces can only see nilpotent quotients of the fundamental group, but being a "profinite equivalence away from p" shouldn't worry about nilpotence

so I've been sent a preprint, and in fact some people are thinking about this.

@CharlesRezk yeah, i think i finally caught up to that, which is good. still puzzled about computing even the homotopy associative version of the miller invariant for (L_K(1)) tmf, though

Yeah I'm not sure either. But definitely the idea is that we have many orientations for various elliptic spectra, e.g., from Ando-Hopkins-Strickland, and this can be used to give us information about miller invariants for things related to tmf.

I am confused my something seems simple: If we apply (equivariant) Anderson dual to a spectrum X twice, do we get X back? If not, what's a nice example of seeing it?