Working in the category $\mathcal{T}$ of based spaces,
we give the basic theory of diagram spaces and
diagram spectra. These are functors
$\mathcal{D}\longrightarrow \mathcal{T}$ for a
suitable small topological category $\mathcal{D}$.
When $\mathcal{D}$ is symmetric monoidal, there is
a smash product that gives the category of
$\mathcal{D}$-spaces a symmetric monoidal structure.
Examples include
\begin{enumerate}
\item[] prespectra, as defined classically,
\item[] symmetric spectra, as defined by Jeff Smith,
\item[] orthogonal spectra, a coordinate-free analogue
of symmetric spectra with symmetric groups replaced by
orthogonal groups in the domain category,
\item[] $\Gamma$-spaces, as defined by Graeme Segal,
\item[] $\mathcal{W}$-spaces, an analogue of
$\Gamma$-spaces with finite sets replaced by
finite CW complexes in the domain category.
\end{enumerate}
We construct and compare model structures on these
categories. With the caveat that $\Gamma$-spaces are
always connective, these categories, and their
simplicial analogues, are Quillen equivalent and their
associated homotopy categories are equivalent to the
classical stable homotopy category.
Monoids in these categories are (strict) ring spectra.
Often the subcategories of ring spectra, module spectra
over a ring spectrum, and commutative ring spectra are
also model categories. When this holds, the respective
categories of ring and module spectra are Quillen
equivalent and thus have equivalent homotopy categories.
This allows interchangeable use of these categories
in applications.
2000 Mathematics Subject Classification:
primary 55P42; secondary 18A25, 18E30, 55U35.
