A friend recently asked me if i had heard anything about a stable Serre Spectral Sequence or one constructed with spectra, has any one else ever heard of this? is there any reason other than historical that we don't think of the Serre SS this way initially?


Any thoughts or references would be great!



Serre spectral sequence with spectra
Since the ordinary Serre spectral sequence is about a fibration of spaces, I don't think you can just talk about a "fibration of spectra" and expect that to be a generalization, since the suspension spectrum functor doesn't preserve fibration sequences. However, there is a version of the Serre spectral sequence involving *parametrized spectra*, which one can think of as a fibration whose base is an ordinary space and whose fibers are spectra. It can be found in section 20.4 of May-Sigurdsson, *Parametrized Homotopy Theory*.


