CHAPTER I

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

algebra.

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

F(SW,

X), where

Sw

is the one-point compactification

of W. These maps are required to be homeomorphisms and to satisfy an evident

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http://dx.doi.org/10.1090/surv/047/01