Algebraic & Geometric Topology

Generalized spectral categories, topological Hochschild homology and trace maps

Gonçalo Tabuada

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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.

Article information

Algebr. Geom. Topol., Volume 10, Number 1 (2010), 137-213.

Received: 18 September 2008
Revised: 28 July 2009
Accepted: 15 October 2009
First available in Project Euclid: 21 December 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 55P42: Stable homotopy theory, spectra 18D20: Enriched categories (over closed or monoidal categories) 18G55: Homotopical algebra
Secondary: 19D55: $K$-theory and homology; cyclic homology and cohomology [See also 18G60]

symmetric spectra Eilenberg–Mac Lane spectra spectral categories Dg categories Quillen model structure Bousfield localization topological Hochschild homology topological cyclic homology trace maps


Tabuada, Gonçalo. Generalized spectral categories, topological Hochschild homology and trace maps. Algebr. Geom. Topol. 10 (2010), no. 1, 137--213. doi:10.2140/agt.2010.10.137.

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