ii iii Abstract. Let S be the sphere spectrum. We construct an associative, commutative, and unital smash product in a complete and cocomplete category M S of "S-modules" whose derived category D S is equivalent to the classical stable homotopy category. This allows a simple and algebraically manageable definition of "S-algebras" and "commutative S-algebras" in terms of associative, or associative and commutative, products R ∧ S R −→ R. These notions are essentially equivalent to the earlier notions of A ∞ and E ∞ ring spectra, and the older notions feed naturally into the new framework to provide plentiful examples. There is an equally simple definition of R-modules in terms of maps R ∧ S M −→ M .When R is commutative, the category M R of R-modules also has an associative, commutative, and unital smash product, and its derived category D R has properties just like the stable homotopy category.Working in the derived category D R , we construct spectral sequences that specialize to give generalized universal coefficient and Künneth spectral sequences. Classical torsion products and Ext groups are obtained by specializing our constructions to Eilenberg-Mac Lane spectra and passing to homotopy groups, and the derived category of a discrete ring R is equivalent to the derived category of its associated Eilenberg-Mac Lane S-algebra.We also develop a homotopical theory of R-ring spectra in D R , analogous to the classical theory of ring spectra in the stable homotopy category, and we use it to give new constructions as MU-ring spectra of a host of fundamentally important spectra whose earlier constructions were both more difficult and less precise.Working in the module category M R , we show that the category of finite cell modules over an S-algebra R gives rise to an associated algebraic K-theory spectrum KR. Specialized to the Eilenberg-Mac Lane spectra of discrete rings, this recovers Quillen's algebraic K-theory of rings. Specialized to suspension spectra Σ ∞ (ΩX) + of loop spaces, it recovers Waldhausen's algebraic K-theory of spaces.Replacing our ground ring S by a commutative S-algebra R, we define Ralgebras and commutative R-algebras in terms of maps A ∧ R A −→ A, and we show that the categories of R-modules, R-algebras, and commutative R-algebras are all topological model categories. We use the model structures to study Bousfield localizations of R-modules and R-algebras. In particular, we prove that KO and KU are commutative ko and ku-algebras and therefore commutative S-algebras.We define the topological Hochschild homology R-module T HH R (A; M ) of A with coefficients in an (A, A)-bimodule M and give spectral sequences for the calculation of its homotopy and homology groups. Again, classical Hochschild homology and cohomology groups are obtained by specializing the constructions to Eilenberg-Mac Lane spectra and passing to homotopy groups. iv

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, 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 Smith's category of symmetric spectra, with an intermediate stop at a new category that may be of interest in its own right, whose objects we call symmetric functors.Comment: 59 pages, 1 figur

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Let G be a compact Lie group. A canonical method is given for constructing a C-space from homotopy theoretic information about its fixed point sets. The construction is a special case of the categorical bar construction. Applications include easy constructions of certain classifying spaces, as well as C-Eilenberg-Mac Lane spaces and Postnikov towers. 0. Introduction. Let G be a compact Lie group and X a G-space. The equivariant homotopy theory of X is reflected to a remarkable extent in its system of fixed point sets, defined as a functor from a certain category 0G to Top, the category of topological spaces. (Our spaces will be compactly generated weak Hausdorff; they may or may not be equipped with a basepoint, depending on the context.) These functors, or systems, have considerable technical advantages over G-spaces; it is easy to apply most homotopy theoretic constructions to them, whereas in many cases it is unclear how to proceed for G-spaces. It is the purpose of this paper to present a canonical way of recovering from any system a G-space which preserves all the homotopy theoretic structure of the system. This allows us to give easy equivariant versions of some standard topological constructions such as Eilenberg-Mac Lane spaces and Postnikov towers, and to simplify other equivariant constructions.1 1. Statements of the main theorems. Throughout, G is a fixed compact Lie group. Definitions. The category of canonical orbits, written 0G, is a topological category with discrete object space \0G\ = (G/77: 77 a closed subgroup of G} and morphisms the G-maps, topologized by requiring the natural bijection (•) Hoxn0ciG/H,G/K)^{G/K)H to be a homeomorphism. By an Oc-space we shall mean a continuous contravariant functor from 0G to Top; these functors form the objects of a topological category in the usual manner. We will also consider GG-rings, Oc-groups, etc., defined similarly.

We show that the K -theory construction of our paper [7], which preserves multiplicative structure, extends to a symmetric monoidal closed bicomplete source category, with the multiplicative structure still preserved. The source category of [7], whose objects are permutative categories, maps fully and faithfully to the new source category, whose objects are (based) multicategories.