# 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.