Given a cosimplicial space or spectrum $X^\bullet$, there is an associated Bousfield-Kan spectral sequence. This starts out as the bigraded object obtained by taking homotopy groups of each $X^n$ and then taking cohomology of the associated cochain complex (where one may have issues in low degrees if $X^\bullet$ is a cosimplicial space because of non-abelian homotopy groups).
One particularly well studied example of this is the Eilenberg-Moore Spectral sequence obtained from a fiber square. There, we have a cospan $X\to Z\leftarrow Y$ with homotopy pullback $W$. Noticing that the cospan gives $X$ and $Y$ the structure of $Z$-comodules we can build the two-sided cobar construction $C^\bullet(X,Z,Y)$ which is a cosimplicial space with $X\times Z^n \times Y$ in degree $n$ (where the coface maps are given by $X\overset{\Delta}\to X\times X\to X\times Z$, $Y\overset{\Delta}\to Y\times Y\to Y\times Z$ and $Z\overset{\Delta}\to Z\times Z$). The associated BKSS of this cosimplicial space is supposed to give us information then about $W$.
Bousfield has given many conditions that ensure strong convergence of the associated Bousfield-Kan spectral sequence. For instance, for the EMSS described above, if $Z$ is simply connected, the Bousfield-Kan spectral sequence converges strongly.
My questions are the following:
- If the BKSS of a cosimplicial space/spectrum converges strongly, does that mean that the thing it converges to is indeed the homotopy type of the limit of that cosimplicial diagram?
- Is the limit of the cobar construction defining the Eilenberg-Moore spectral sequence always equivalent to the homotopy pullback of that cospan?
- Consider the following example: there is a coaction of $BO$ on $MO$ coming from the Thom diagonal. As such we can consider the cosimplicial spectrum $C^\bullet(MO,\mathbb{S}[BO],\mathbb{S})$ where $\mathbb{S}$ has the trivial $BO$-coaction. The associated spectral sequence is the $MO$-Adams-Novikov spectral sequence and converges to the homotopy of the 2-completed sphere spectrum. Does that mean that the totalization of that cosimplicial object is equivalent to the 2-completed sphere spectrum, or is that just saying that this is all the spectral sequence can see of it?
This last is ultimately a question about Koszul duality. There should be an adjunction between "spectra with an $O$-action" and "spectra with a $BO$-coaction." Constructing $MO$ from the sphere spectrum should be the left adjoint of this equivalence, and I'd like to think that taking a "cotensor product" should be the way of going back. However, it's not clear to me how to check that $\mathbb{S}$ can be recovered in this way from $MO$.
