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Published online by Cambridge University Press: 23 October 2009
Abstract. In 1991, Gaunce Lewis published a theorem showing that a quite minimal list of desiderata for an “ideal” category of spectra was inconsistent; see [4]. This result requires any category modeling stable homotopy theory to make some compromises in its formal structure. This short paper describes the compromises present in wS, the category of S-modules developed in [2], together with the amusing consequence that wS contains a copy of the (unstable!) category of topological spaces.
At this point we have several categories of spectra that are symmetric monoidal, with their smash products descending to the smash product in the stable category; let me mention in particular the S-modules of [2] and the symmetric spectra of [3]. These categories are much more nicely behaved than any of their predecessors, but their behavior is not absolutely ideal, because it can't be. This is a theorem of Gaunce Lewis's, whose paper [4] was published before any of the current batch of symmetric monoidal categories of spectra were developed. Suppose we have a candidate for a “good” category of spectra, which we ambiguously call S. Lewis sets out the following pretty minimal list of properties for S, all of which are devoutly to be desired:
The category S has a symmetric monoidal product, which we call smash and write Λ, as usual.
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