A remark on \(K\)-theory and \(S\)-categories
type: Item
instance of: scholarly article
title: A remark on \(K\)-theory and \(S\)-categories
zbMATH Open document ID: 1054.55004
author: Gabriele Vezzosi
author: Bertrand Toën
published in: Topology
publication date: 2004-08-16T00:00:00Z
full work available at URL: https://arxiv.org/abs/math/0210125
review text: \textit{M. Schlichting} [Invent. Math. 150, 111--116 (2002; Zbl 1037.18007)] has shown that there exist two Waldhausen categories having equivalent (triangulated) homotopy categories and non-weakly equivalent associated (Waldhausen) \(K\)-theory spectra. The authors' starting point is this fact which implies that the \(K\)-theory spectrum \(K(C)\) of a Waldhausen category \(C\) depends on strictly more than its homotopy category. First the authors introduce a notion of a good Waldhausen category which is built up in Section 2. By a good Waldhausen category the authors mean a Waldhausen category being embeddable in the category of fibrant objects of a pointed model category. One finds that given a Waldhausen category there is a good Waldhausen category with the same \(K\)-theory spectrum up to homotopy. Moreover by an \(S\)-category one means a category enriched over the category of simplicial sets. Then the main purpose of this paper is to prove that if \(C\) and \(D\) are good Waldhausen categories, and if their simplicial localizations \(L^HC\) and \(L^HD\) are equivalent in the homotopy category of \(S\)-categories then the \(K\)-theory spectra \(K(C)\) and \(K(D)\) are isomorphic in the homotopy category of \(K\)-spectra (Corollary 6.6). The key to the authors' approach is defining a \(K\)-theory spectrum \(K(T)\) for an \(S\)-category \(T\) satisfying some properties which depends (up to weak equivalences) on the \(S\)-equivalence class of \(T\) (Section 5). By applying this construction to the simplicial localization \(L^HC\) of a good Waldhausen category \(C\) one can obtain a weak equivalence \(K(C)\simeq K(L^HC)\) (Theorem 6.1) from which the above result follows immediately. The basic results needed for the proof of this theorem are provided in Sections 3 and 4. In addition this theorem yields the following interesting consequence (Corollary 7.1): There exist two stable model categories \(M_1\) and \(M_2\) such that \(L^HM_1\) and \(L^HM_2\) cannot be equi…
reviewed by: Haruo Minami
Mathematics Subject Classification ID: 55N15
Mathematics Subject Classification ID: 19D10
Mathematics Subject Classification ID: 18E30
Mathematics Subject Classification ID: 18G55
Mathematics Subject Classification ID: 18D20
zbMATH DE Number: 2091693
zbMATH Keywords: K-theory
zbMATH Keywords: simplicial categories
zbMATH Keywords: derived categories
MaRDI profile type: Publication
OpenAlex ID: W1966878416
arXiv ID: math/0210125
DOI: 10.1016/J.TOP.2003.10.008
description: scientific article