Rings, modules, and algebras in infinite loop space theory

A. D. Elmendorf and M. A. Mandell

Subject classes: Primary 19D23; Secondary 55P43, 18D10

xxx-LANL identifier: math.KT/0403403

Addresses:
A. D. Elmendorf
Dept. of Mathematics
Purdue University Calumet
Hammond, IN 46323
aelmendo@calumet.purdue.edu

M. A. Mandell (current)
DPMMS
CMS
University of Cambridge
Cambridge CB3 0WB
England
M.A.Mandell@dpmms.cam.ac.uk

M. A. Mandell (effective Fall 2005)
Department of Mathematics
Indiana University
Bloomington, IN 47405
mmandell@indiana.edu

This is a major revision of a previous submission of the same name.
We have completely rewritten sections 5 -- 7, giving a new
construction of the first part of our functor.  The main abstract
is as follows:

We give a new construction of the algebraic $K$-theory
of small permutative categories that preserves multiplicative
structure, and therefore allows us to give a unified treatment of
rings, modules, and algebras in both the input and output.  This
requires us to define multiplicative structure on the category of
small permutative categories.  The framework we use is the concept
of multicategory (elsewhere also called colored operad), a generalization of
symmetric monoidal category
that precisely captures the multiplicative structure we have
present at all stages of the construction. Our method ends up in
the Hovey-Shipley-Smith category of symmetric spectra, with an intermediate stop
at a category of functors out of a particular wreath product.