# 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.