The theory of diagram spaces with applications to stable homotopy theory and algebraic K-theory

Diagram spaces are functors from geometric or combinatorial diagram categories to the category of topological spaces. The symmetries encoded in diagram spaces allow for constructions useful to the structured homotopical algebra of stable homotopy theory. In this thesis we develop the homotopy theory of diagram spaces.;As a first application we provide models for the infinite loop spaces associated to symmetric spectra, orthogonal spectra, and EKMM S-modules. Each of these categories of structured spectra has a corresponding category of structured spaces that receives the infinite loop space functor O infinity. Diagram spaces provide the appropriate model in the case of diagram spectra. We prove that these models for spaces are Quillen equivalent and that the infinite loop space functors Oinfinity agree. This comparison is then used to show that two different constructions of the spectrum of units gl1R of a commutative ring spectrum R agree.;As a second application, we develop the theory of principal Ainfinity-fibrations and bundles of spectra. We construct classifying spaces for both types of bundle theories and use them to give a geometric description of the cocycles for the algebraic K-theory K(R) of a connective ring spectrum R.