Working in the category of based spaces, we give the basic theory of diagram spaces and diagram spectra. These are functors for a suitable small topological category . When is symmetric monoidal, there is a smash product that gives the category of -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[] -spaces, as defined by Graeme Segal, \item[] -spaces, an analogue of -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 -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.

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Model categories of diagram spectra

Nilpotence and stable homotopy theory II

A bstract . Let E be a homotopy commutative ring spectrum, and suppose the ring of cooperations E ∗E is flat over E ∗. We wish to address the following question: given a commutative E ∗-algebra A in E∗E -comodules, is there an E ∞ -ring spectrum X with E∗X ≅ A as comodule algebras? We will formulate this as a moduli problem, and give a way – suggested by work of Dwyer, Kan, and Stover – of dissecting the resulting moduli space as a tower with layers governed by appropriate Andre-Quillen cohomology groups. A special case is A = E∗E itself. The final section applies this to discuss the Lubin-Tate or Morava spectra E n . Some years ago, Alan Robinson developed an obstruction theory based on Hochschild cohomology to decide whether or not a homotopy associative ring spectrum actually has the homotopy type of an A ∞ -ring spectrum. In his original paper on the subject [35] he used this technique to show that the Morava K -theory spectra K(n) can be realized as an A ∞-ring spectrum; subsequently, in [3], Andrew Baker used these techniques to show that a completed version of the Johnson-Wilson spectrum E(n) can also be given such a structure. Then, in the mid-90s, the second author and Haynes Miller showed that the entire theory of universal deformations offinite height formal group laws over fields of non-zero characteristic can be lifted to A ∞-ring spectra in an essentially unique way.

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Structured Ring Spectra: Moduli spaces of commutative ring spectra

Let d ≥ 1 and X a pointed topological space. We denote as usual by π d (X) the d-th homotopy group of X. One of the starting point in homotopy theory is the following result: Theorem 1.1.1. Let n > 0 be any integer. 1) If d n then π d (S N ) is a finite group unless n is even and d = 2n - 1 in which case it a direct sum of ℤ and a finite group.

On the Motivic π0 of the Sphere Spectrum

Given a small category C, we show that there is a universal way of expanding C into a model category, essentially by formally adjoining homotopy colimits. The technique of localization becomes a method for imposing `relations' into these universal gadgets. The paper develops this formalism and discusses various applications, for instance to the study of homotopy colimits, the Dwyer-Kan theory of framings, and to the homotopy theory of schemes.

M. G. Barratt

Papers

Geometric Applications of Homotopy Theory II