Prologue: the category of L-spectra
In this prologue, we construct a category whose existence was previously thought
to be impossible by at least two of the authors: a complete and cocomplete
category of spectra, namely the L-spectra, with an associative and commutative
smash product. This contrasts with the category constructed by Lewis and
the fourth author in [48, 38], whose smash product is neither associative nor
commutative (before passage to homotopy categories), and with the category
constructed by the first author in [20], which is neither complete nor cocomplete.
We will also give a function L-spectrum construction that is right adjoint to the
new smash product. The category of L-spectra has all of the properties that we
desire except that its smash product, denoted by Aj^, is not unital. It has a
natural unit map A : S l\% M —• M, which is often an isomorphism and always
a weak equivalence.
The curtain will rise on our real focus of interest in the next chapter, where
we will define an 5-module to be an L-spectrum M such that A : S A& M —• M
is an isomorphism. Restricting A& to S-modules and renaming it As, this will
give us a symmetric monoidal category in which to develop stable topological
1. Background on spectra and the stable homotopy category
We begin by recalling the basic definitions in Lewis and May's approach to
the stable category. We first recall the definition of a coordinate-free spectrum;
see [38, I§2], [20, §2], or [52, Ch.XII] for further details. A coordinate-free
spectrum is a spectrum that takes as its indexing set, instead of the integers, the
set of finite dimensional subspaces of a "universe", namely a real inner product
space U = M°°. Thus, a spectrum E assigns a based space EV to each finite
dimensional subspace V of E/, with (adjoint) structure maps
aViW : EV-^nw-vEW
when V C W. Here W V is the orthogonal complement of V in W and Vtw X is
the space of based maps
X), where
is the one-point compactification
of W. These maps are required to be homeomorphisms and to satisfy an evident
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