Model categories of diagram spectra

with Mike Mandell, Peter May, and Stefan Schwede

ABSTRACT. We first give a unifying definition for some of the new symmetric monoidal categories of spectra: symmetric spectra, gamma spaces, orthogonal spectra, and W-spaces; they are all examples of categories of diagram spectra. The category of ring spectra in each case is a generalization of functors with smash product, an older notion of ring spectra which had not been recognized as coming from a symmetric monoidal category of spectra. Then we give comparisons of these categories that combine with other known equivalences to show that all of the known homotopy theories of highly structured ring and module spectra are equivalent. So a result in any such approach can be transported to any other, and one is free to work with the category which best suits a problem.