Symmetric spectra and topological Hochschild homology

ABSTRACT. Usually weak equivalences of spectra are detected by the classical stable homotopy groups defined on the underlying prespectra. The homotopy theory of symmetric spectra is more subtle though and requires a different detection functor. In Symmetric spectra we use a transfinite construction (fibrant replacement) which is useful theoretically but not computationally. Here we define a more concrete countable construction (a homotopy colimit over the category of finite sets and injections) with an associated spectral sequence which leads to a better understanding of the weak equivalences. As an application, the definition of topological Hochschild homology on symmetric ring spectra using the Hochschild complex is shown to agree with Bokstedt's original ad hoc definition. In particular, this shows that Bokstedt's connectivity and convergence conditions are unnecessary.