Spectral Sequences from Sequences of Spectra: Towards the Spectrum of the Category of Spectra

As is well known, it is our manifest destiny as 21st century algebraic topologists to compute homotopy groups of spheres. This noble venture began even before the notion of homotopy was around. In 1931, Hopf was thinking about a map he had encountered in geometry from S to S and wondered whether or not it was essential. He proved that it was by considering the linking of the fibers. After Hurewicz developed the notion of higher homotopy groups this gave the first example, aside from the self-maps of spheres, of a non-trivial higher homotopy group. Hopf classified maps from S to S and found they were given in a manner similar to degree, generated by the Hopf map, so that