Generalized spectral categories, topological Hochschild homology and trace maps

Given a monoidal model category C and an object K in C, Hovey constructed in [22] the monoidal model category Sp_C of K–symmetric spectra over C . In this paper we describe how to lift a model structure on the category of C-enriched categories to the category of Sp_C–enriched categories. This allow us to construct a (fourstep) zig-zag of Quillen equivalences comparing dg categories to HZ–categories. As an application we obtain: (1) the invariance under weak equivalences of thetopological Hochschild homology (THH) and topological cyclic homology (TC) of dg categories; (2) non-trivial natural transformations from algebraic K–theory toTHH.