Suppose we have a tower of fibrations of spectra $\{X_k\}_{k\in\mathbb{N}}$ with inverse limit $X_\infty$, and let $F_k$ be the fibre of the map $X_k\to X_{k1}$. There is then a spectral sequence $E^1_{jk}=\pi_j F_k \Longrightarrow \pi_{j+k} X_\infty$. If we instead have a tower of fibrations of based spaces, then we still have something like a spectral sequence except that some of the entries may be nonabelian groups or just pointed sets, and the sense in which $E^{r+1}$ is the homology of $E^r$ must be modified to take account of this. The details are in the book 'Homotopy limits, completions and localizations' by Bousfield and Kan. Now suppose we have a tower of fibrations of unbased spaces. I think I have heard it said that there is still some kind of spectral sequence building up to $\pi_\ast(X_\infty,a_\infty)$, where the basepoint $a_\infty$ is not given in advance but is chosen iteratively by lifting basepoints $a_n\in X_n$ as we work through the spectral sequence. This makes life difficult because $F_{n+1}$ and $\pi_\ast(X_n)$ are not defined until we have chosen $a_n$. Has any theory of this type been worked out in detail?

$\begingroup$ I have the impression that there is a paper of Bousfield that does something like this. $\endgroup$ – Tom Goodwillie Jul 15 '11 at 11:15

3$\begingroup$ Tom, might you be thinking of Bousfield's "Homotopy spectral sequences and obstructions" (Israel J. Math vol. 66)? It does something like this for unbased cosimplicial spaces, rather than a general tower of fibrations. $\endgroup$ – Tyler Lawson Jul 15 '11 at 12:57

$\begingroup$ @Tyler: thanks for the pointer. If you want to promote that comment to an answer, then I will accept it. $\endgroup$ – Neil Strickland Jul 15 '11 at 16:35
As requested, I am reposting this comment as an answer.
Bousfield covers this material in "Homotopy spectral sequences and obstructions," Israel J. Math 66. The discussion is specific to cosimplicial objects (e.g. the discussion of obstruction cocycles in Section 5) and the general method of obtaining "partially" defined spectral sequences without basepoints would have to be extracted. However, I believe that all the necessary content is already there.

9$\begingroup$ Sadly the masters of the library have ordained that I must descend into the bowels of the earth and retrieve a dead tree before I am worthy to study this work in detail. I might get there next week ... $\endgroup$ – Neil Strickland Sep 2 '11 at 15:38

$\begingroup$ Interesting! Am I right if I say that treating the case of cosimplicial spaces is enough since the spectral sequence of a tower of fibrations would be the same as the spectral sequence of the associated cosimplicial space constructed as a kind of AndréQuillen resolution using the nerve of the category $\mathbb N$? $\endgroup$ – Fernando Muro Aug 24 '13 at 22:28

1$\begingroup$ Just answering myself, I wasn't right. $\endgroup$ – Fernando Muro Feb 24 '14 at 12:33