The Boardman category of spectra, chain complexes and (co)-localizations
type: Item
instance of: scholarly article
title: The Boardman category of spectra, chain complexes and (co)-localizations
zbMATH Open document ID: 0921.55007
published in: Homology, Homotopy and Applications
publication date: 1999-06-06T00:00:00Z
full work available at URL: https://eudml.org/doc/231572
full work available at URL: volume1-1.htm
review text: A chain functor associates a pair \((X,A)\) of topological spaces to chain complexes \(C(X,A)\) and \(C'(X,A)\). In addition, we require that the complexes \(C(A), C(X), C(X,A)\) and \(C'(X,A)\) share a certain relationship, which ensures that \((X,A)\mapsto H_*C(X,A)\) is a homology theory on the homotopy category of pairs of topological spaces. Singular homology is a special case (with \(C=C'\)). However, the more intricate structural data of ``chain functor'' are designed so that compactly supported homology theories on pairs of CW-spaces defined via chain functors correspond bijectively to such theories defined via spectra (in the sense of Boardman). It follows that, whenever a chain complex construction on chain functors yields again a chain functor, then there is a corresponding construction on spectra. The present article is a survey of the author's recent work in this direction. For example, the tensor product of chain functors is again a chain functor, and so there is a corresponding operation on spectra (which is not the smash product). Notably, this leads to a different view of (co-)localization in the homotopy category of spaces.
Mathematics Subject Classification ID: 55P42
Mathematics Subject Classification ID: 55P60
Mathematics Subject Classification ID: 55N20
Mathematics Subject Classification ID: 55U15
Mathematics Subject Classification ID: 55N07
zbMATH DE Number: 1295632
zbMATH Keywords: chain functors
zbMATH Keywords: Boardman spectra
zbMATH Keywords: tensor products
zbMATH Keywords: localization
zbMATH Keywords: colocalization
author: Friedrich Wilhelm Bauer
reviewed by: George Peschke
MaRDI profile type: Publication
DOI: 10.4310/HHA.1999.V1.N1.A3
description: scientific article; zbMATH DE number 1295632