Algebraic & Geometric Topology

Generalized spectral categories, topological Hochschild homology and trace maps

Gonçalo Tabuada

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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.

Article information

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

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

Permanent link to this document
https://projecteuclid.org/euclid.agt/1513882310

Digital Object Identifier
doi:10.2140/agt.2010.10.137

Mathematical Reviews number (MathSciNet)
MR2580431

Zentralblatt MATH identifier
1206.55012

Subjects
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]

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

Citation

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. https://projecteuclid.org/euclid.agt/1513882310


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