Diagram spaces, diagram spectra, and FSP's
M.A. Mandell
MIT
mandell@math.mit.edu
J.P. May
University of Chicago
may@math.uchicago.edu
S. Schwede
Universitet Bielefeld, Germany
schwede@mathematik.uni-bielefeld.de
B. Shipley
Purdue University and University of Chicago
bshipley@math.purdue.edu
Working in the category $T$ of based spaces, we give the basic theory
of diagram spaces, diagram spectra, and functors with smash product.
For a small topological category $D$, a $D$-space is just a continuous
functor $D >--> T$. There is an external smash product that takes a
pair of $D$-spaces to a $(D x D)$-space. If $D$ is symmetric monoidal,
there is an internalization of this smash product that makes the
category $DT$ of $D$-spaces a symmetric monoidal category. This allows
the definition of monoids R in $DT$, modules over monoids R, and,
when R is commutative, monoids in the category of R-modules. These
structures are defined in terms of the internal smash product, but
they all have more elementary descriptions in terms of the external
smash product. A monoid R is a symmetric monoidal functor $D >--> T$,
and the external version of an R-module is a $D$-spectrum over R. We
show that there is a new category $D_R$ such that a $D_R$-space has the
same structure as a $D$-spectrum over R. When R is commutative, the
external version of a monoid in the category of R-modules is a $D$-FSP
(functor with smash product) over R. We are especially interested in
functors relating categories such as these as $D$ varies. With R taken
as a canonical sphere diagram space, examples include
Symmetric spectra, as defined by Jeff Smith.
Orthogonal spectra, a coordinate free analogue of symmetric spectra
with symmetric groups replaced by orthogonal groups in the domain
category.
Gamma-spaces, as defined by Graeme Segal.
$W$-spaces, an analogue of Gamma-spaces with finite sets replaced by
finite CW complexes in the domain category.