Bundles of spectra and algebraic K-theory

TL;DR

This paper develops a bundle-theoretic approach to parametrized spectra, linking algebraic K-theory of ring spectra with classification theorems for parametrized R-module spectra and principal G fibrations.

Contribution

It introduces a geometric bundle perspective on parametrized spectra and establishes classification theorems connecting them to homotopy classes of maps into classifying spaces.

Findings

Parametrized R-module spectra classify cohomology theories via algebraic K-theory.

Classification of parametrized spectra over X by homotopy classes of maps to BAut_R(M).

Extension of classification results to principal G fibrations with G an A_infinity space.

Abstract

A parametrized spectrum E is a family of spectra E_x continuously parametrized by the points x of a topological space X. We take the point of view that a parametrized spectrum is a bundle-theoretic geometric object. When R is a ring spectrum, we consider parametrized R-module spectra and show that they give cocycles for the cohomology theory determined by the algebraic K-theory K(R) of R in a manner analogous to the description of topological K-theory K^0(X) as the Grothendieck group of vector bundles over X. We prove a classification theorem for parametrized spectra, showing that parametrized spectra over X whose fibers are equivalent to a fixed R-module M are classified by homotopy classes of maps from X to the classifying space BAut_R(M) of the A_\infty space of R-module equivalences from M to M. In proving the classification theorem for parametrized spectra, we define of the notion of a principal G fibration where G is an A_\infty space and prove a similar classification theorem for principal G fibrations.