Categories of Spectra and Infinite Loop Spaces

At the Seattle conference, I presented a calculation of H,(F;Zp) as an algebra, for odd primes p, where F = lim F(n) and F(n) is the topological monoid > of homotopy equivalences of an n-sphere. This computation was meant as a preliminary step towards the computation of H*(BF;Zp). Since then, I have calculated H*(BF;Zp), for all primes p, as a Hopf algebra over the Steenrod and Dyer-Lashof algebras. The calculation, while not difficult, is somewhat lengthy, and I was not able to write up a coherent presentation in time for inclusion in these proceedings. The computation required a systematic study of homology operations on n-fold and infinite loop spaces. As a result of this study, I have also been able to compute H,(2nsnx;Zp), as a Hopf algebra over the Steenrod algebra, for all connected spaces X and prime numbers p. This result, which generalizes those of Dyer and Lashof [3] and Milgram [8], yields explicit descriptions of both An essential first step towards these results was a systematic categorical analysis of the notions of n-fold and infinite loop spaces. The results of this analysis will