Introduction
The last thirty years have seen the importation of more and more algebraic
techniques into stable homotopy theory. Throughout this period, most work in
stable homotopy theory has taken place in Boardman's stable homotopy category
[6], or in Adams' variant of it [2], or, more recently, in Lewis and May's variant
[38]. That category is analogous to the derived category obtained from the
category of chain complexes over a commutative ring k by inverting the quasi-
isomorphisms. The sphere spectrum 5 plays the role of fc, the smash product
A plays the role of the tensor product, and weak equivalences play the role of
quasi-isomorphisms. A fundamental difference between the two situations is that
the smash product on the underlying category of spectra is not associative and
commutative, whereas the tensor product between chain complexes of fc-modules
is associative and commutative. For this reason, topologists generally work with
rings and modules in the stable homotopy category, with their products and
actions defined only up to homotopy. In contrast, of course, algebraists generally
work with differential graded /c-algebras that have associative point-set level
multiplications.
We here introduce a new approach to stable homotopy theory that allows one
to do point-set level algebra. We construct a new category Ms of 5-modules
that has an associative, commutative, and unital smash product A5. Its derived
category @s is obtained by inverting the weak equivalences; @s is equivalent
to the classical stable homotopy category, and the equivalence preserves smash
products. This allows us to rethink all of stable homotopy theory: all previous
work in the subject might as well have been done in ^ 5 . Working on the point-
set level, in Ms, we define an 5-algebra to be an 5-module R with an associative
and unital product RAsR R; if the product is also commutative, we call R
a commutative 5-algebra. Although the definitions are now very simple, these
are not new notions: they are refinements of the ^IQO and £"00 ring spectra that
were introduced over twenty years ago by May, Quinn, and Ray [48]. In general,
the latter need not satisfy the precise unital property that is enjoyed by our new
5-algebras, but it is a simple matter to construct a weakly equivalent 5-algebra
from an AQQ ring spectrum and a weakly equivalent commutative 5-algebra from
an EQO ring spectrum.
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