Hammock localization in Waldhausen categories
type: Item
instance of: scholarly article
title: Hammock localization in Waldhausen categories
zbMATH Open document ID: 0938.18007
DOI: 10.1016/S0022-4049(98)00009-7
published in: Journal of Pure and Applied Algebra
publication date: 2000-06-14T00:00:00Z
review text: A Waldhausen category \({\mathcal C}\) is a category with cofibrations and weak equivalences satisfying certain axioms. It is the proper setting to construct and study algebraic K-theory [see \textit{F. Waldhausen}, Lect. Notes Math. 1126, 318-419 (1985; Zbl 0579.18006)]. The study of the algebraic K-theory of a Waldhausen category \({\mathcal C}\) leads to Dwyer-Kan's hammock localization [\textit{W. G. Dwyer} and \textit{D. M. Kan}, J. Pure Appl. Algebra 18, 17-35 (1980; Zbl 0485.18013)] \(L^H({\mathcal C})\) of the category \({\mathcal C}\) with respect to its full subcategory in which morphisms are cofibrations and weak equivalences. Denote by \(\text{mor}^H(D,E)\) the simplicial set of morphisms in \(L^H({\mathcal C})\) between any two objects \(D,E\). The main result in this paper is theorem 2.1 stating that, under certain conditions, if the second variable is fixed, the contravariant functor \(\text{mor}^H(-,E)\) takes cofiber squares to homotopy pullback squares.
Mathematics Subject Classification ID: 18E35
Mathematics Subject Classification ID: 18F25
Mathematics Subject Classification ID: 19D10
zbMATH DE Number: 1308367
zbMATH Keywords: Waldhausen category
zbMATH Keywords: hammock localization
author: Michael S. Weiss
reviewed by: Jianzhong Pan
MaRDI profile type: Publication
cites work: Calculating simplicial localizations
cites work: On c.s.s. Complexes
cites work: Higher algebraic K-theory: I
cites work: The Algebraic Theory of Surgery II. Applications to Topology
cites work: Q3701607
cites work: Duality in Waldhausen Categories
description: scientific article; zbMATH DE number 1308367