@Drew in fact, I think that somehow this mysterious project jack has me working on is connected to all of that. which is what i wanted all along. he's just being very mysterious about it.
I sort of get all the pieces, I just still can't really see his big picture. Or rather, his/Kathryn Hess'/John Rognes'/etc.'s big picture.
But I know for sure that that's a party I'll pay just about anything to get into.
@EricPeterson is this a very recent phenomenon? or are you talking about like, over a long period of time?
BTW @EricPeterson I bookmarked a conversation from here, do you have any clue how to access it again? It's the one we were having about chromatic stuf.
I'll have to tell the first years at JHU about MO. There are at least one or two interested in topology. Maybe I can get them to stir up some shit.
i mean, u know this is based on an actual paper on the arxiv right?
Jesus. "This reformulates the Einstein-Hilbert action as the Lagrangian for a conformally invariant physical theory involving a unimodular pseudometric..."
I'm gonna need Craig's flight of stairs again.
The quotes in the beginning of the sections are pretty amazing though.
like, given a map of "objects" (either schemes or spaces, maybe ring spectra in RngSpectra^{op} or something...)
$X\to Y$
I guess a map of sites.
then we can talk about descent.
in the correct homotopical generality, descent should be controlled by (I think!) a cosimplicial dude. something like, and well, here's my first point of confusion, either $X\otimes^L_Y X$ or $Y\otimes^L_X Y$
and i guess this depends on your variance. that is, are we using "rings" or "spaces"
but yeah Hopkins says that for a faithfully flat map R-->S you form that descent hopf algebroid (S,S\otimes_{R} S) and there is an equivalence of categories between comodules over that and the category of descent data
I guess ultimately my question is about the fact that $MU\wedge^{\mathbb{L}}_{\mathbb{S}}MU$ is actually (I think???) the same as $THH_{\mathbb{S}}(MU)$, i.e. it's simultaneously the thing that controls deformation data "at" the "point" MU and it's the coring which controls descent from MU to $\mathbb{S}$ along the unit.
Yeah, but this whole line of thought was triggered by Aaron Royer talking about the mod-p ASS sort of... converging to things in a formal neighborhood of $p$ in Spec(S)
But this bugs me, because, in Lurie's language, the only formal neighborhoods that Spec(S) has are about arithmetic points.
And we all know that's, at least morally, unsatisfying.
So, if we think about spectra as sheaves over a good notion of Spec(S) (i.e. something that looks like the Bousfield lattice) we can think of the various E ANSS's as telling me about, I'm not sure, something like the value of that sheaf "on" an infinitesimal neighborhood of E "in" Spec(S)
But we can also think of them as being something like.... I dunno, Tor(THH_S(E),X), I think...
Okay, well then there's this other thing: there is a coring interpretation of formal groups (a la Demazure), where comodules are like formal schemes over a given formal group.
it's a little confusing. there's this weird relationship between simplicial and cosimplicial things that makes some of these constructions look the same
Yeah, Demazure gives this equivalence (for a field k) between k-corings and k-formal-schemes....
So, a coring over k "is" a formal scheme, so, in particular, if there is a k-algebra A, then A\otimes_k A is a particular formal scheme over k
but it's ALSO the descent coring associated to descending from A to k?
And Eric tells me that comodules over this coring correspond to formal curves on the formal scheme (oh and also I think algebras in the coring category are formal groups). Anyway, a space X gives me an MU\wedge MU comodule, i.e. a curve on the formal scheme associated to MU over S?
No, there's no specific situation. I just happened to notice that there is a coring associated to descent, but corings are the same thing as formal schemes, in an ideal situation.
the ground object for that coring is R, not S (or "is A, not k" using your letters from earlier), which makes the corings <~~~> formal schemes correspondence less perfect. i'll bet there is still something you can say, though; maybe start by writing down what a T-point of the coring is for T an R-algebra and seeing if you find something familiar
the T-points of a formal scheme presented as a coring C are something else; they're elements of C (x) T (thought of as Hom(C^*, T)) satisfying some properties (thought of as the conditions to be a Hom)
Yeah, an $R$-algebra $T$ is not naturally an $R\otimes_S R$ comodule, I don't think. But let's see, $R\otimes_S T\cong R\otimes_S S\otimes_S T\to R\otimes_S R\otimes_S T$
Some journals published by learned societies or national academies require that “communications” be presented (or sponsored) by a member of the society. This was the case, for example, of the PNAS (Proceedings of the National Academy of Sciences) until July 2010; the top of an article looked like this:
@EricPeterson well, I have a map from H_*(BSp) to H_*(BU) from an old Sem. Cartan.
given in terms of duals of characteristic classes.
Jack has recommended that I work with the homology of SO and Sp instead, since the exterior algebras make things easier, but then how will I ever determine the homology of MSO\wedge_{MSp} MSO from the homology of SO\wedge_{Sp} SO???
