Generalized spectral categories, topological Hochschild homology and trace maps

Tabuada, Gonçalo

doi:10.2140/agt.2010.10.137

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Generalized spectral categories, topological Hochschild homology and trace maps

Gonçalo Tabuada

Algebraic & Geometric Topology 10 (2010) 137–213

DOI: 10.2140/agt.2010.10.137

Abstract

Given a monoidal model category $C$ and an object $K$ in $C$ , Hovey constructed the monoidal model category ${S p}^{Σ} (C, K)$ 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 ${S p}^{Σ} (C, K)$ –enriched categories. This allow us to construct a (four step) zig-zag of Quillen equivalences comparing dg categories to $H ℤ$ –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.

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

Volume 10, issue 1 (2010)