#### Volume 10, issue 1 (2010)

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Generalized spectral categories, topological Hochschild homology and trace maps

Algebraic & Geometric Topology 10 (2010) 137–213
##### Abstract

Given a monoidal model category $\mathsc{C}$ and an object $K$ in $\mathsc{C}$, Hovey constructed the monoidal model category ${Sp}^{\Sigma }\left(\mathsc{C},K\right)$ of $K$–symmetric spectra over $\mathsc{C}$. In this paper we describe how to lift a model structure on the category of $\mathsc{C}$–enriched categories to the category of ${Sp}^{\Sigma }\left(\mathsc{C},K\right)$–enriched categories. This allow us to construct a (four step) zig-zag of Quillen equivalences comparing dg categories to $Hℤ$–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.

##### Keywords
symmetric spectra, Eilenberg–Mac Lane spectra, spectral categories, Dg categories, Quillen model structure, Bousfield localization, topological Hochschild homology, topological cyclic homology, trace maps
##### Mathematical Subject Classification 2000
Primary: 55P42, 18D20, 18G55
Secondary: 19D55