In general terms this project is devoted to simplifying and making more usable several powerful, complicated, and subtle algebraic techniques of topology. In the 1960's, Daniel Quillen described how to define the homology of simplical objects over many different categories, including sets, algebras over a ring, and unstable algebras over the Steenrod algebra. Many of the calculations in homology theory can be cast in this light, among them the computation of the E2-term of certain "unstable Adams spectral sequences" such as the Bousfield-Kan spectral sequence. Past experience has indicated the strength of this approach; the purpose of this project is to explore this idea in depth. A particular ramification would be the exploration of the homology of function complexes using tools of Jean Lannes and A.K. Bousfield.
