### Diagram spaces, diagram spectra, and FSP's, by M. A. Mandell, J. P. May, S. Schwede, and B. Shipley

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 :
(1) Symmetric spectra, as defined by Jeff Smith;
(2) Orthogonal spectra, a coordinate free analogue of symmetric spectra
with symmetric groups replaced by orthogonal groups in the domain
category;
(3) Gamma-spaces, as defined by Graeme Segal; and
(4) W-spaces, an analogue of Gamma-spaces with finite sets replaced by
finite CW complexes in the domain category.

M. A. Mandell <mandell@math.mit.edu>

J. P. May <may@math.uchicago.edu>

S. Schwede <schwede@mathematik.uni-bielefeld.de>

B. Shipley <bshipley@math.purdue.edu>