Volume 10, issue 1 (2010)

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Generalized spectral categories, topological Hochschild homology and trace maps

Gonçalo Tabuada

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

Given a monoidal model category C and an object K in C, Hovey constructed the monoidal model category SpΣ(C,K) 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 SpΣ(C,K)–enriched categories. This allow us to construct a (four step) zig-zag of Quillen equivalences comparing dg categories to H–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
References
Publication
Received: 18 September 2008
Revised: 28 July 2009
Accepted: 15 October 2009
Published: 10 January 2010
Authors
Gonçalo Tabuada
Departamento de Matemática e CMA
FCT-UNL
Quinta da Torre
2829-516 Caparica
Portugal