Generalized spectral categories, topological Hochschild homology and trace maps

Abstract

Given a monoidal model category $\mathcal{C}$ and an object $K$ in $\mathcal{C}$, Hovey constructed the monoidal model category ${\mathsf{Sp}}^{\Sigma }\left(\mathcal{C},K\right)$ of $K$–symmetric spectra over $\mathcal{C}$. In this paper we describe how to lift a model structure on the category of $\mathcal{C}$–enriched categories to the category of ${\mathsf{Sp}}^{\Sigma }\left(\mathcal{C},K\right)$–enriched categories. This allow us to construct a (four step) zig-zag of Quillen equivalences comparing dg categories to $H\mathbb{Z}$–categories. As an application we obtain: (1) the invariance under weak equivalences of the topological Hochschild homology (THH) and topological cyclic homology (TC) of dg categories; (2) non-trivial natural transformations from algebraic $K$–theory to THH.