Representability theorems for presheaves of spectra
type: Item
instance of: scholarly article
title: Representability theorems for presheaves of spectra
zbMATH Open document ID: 1204.55010
published in: Journal of Pure and Applied Algebra
publication date: 2010-10-19T00:00:00Z
review text: The Brown representability theorem gives conditions for the representability of a set-valued contravariant functor defined on the classical stable homotopy category. Let \({\mathrm {s{\mathbf Set}}_*}\) be the category of pointed simplicial sets. Suppose \(\mathcal{M}\) is a closed simplicial model category and \(F:\mathcal{M}^{op}\to {\mathrm {s{\mathbf Set}}_*}\) is a contravariant functor. The main result of this paper gives conditions on \(\mathcal{M}\) and \(F\) which imply that \(F\) is sectionwise weakly equivalent to a representable functor \(hom( \;,Y)\), where \(Y\) is some fibrant object of \(\mathcal{M}\). The conditions on the simplicial model category \(\mathcal{M}\) are abstractions of the general behaviour of the stable model structure of presheaves of spectra. The functor \(F\) should preserve weak equivalences between cofibrant objects and should take homotopy colimits to homotopy inverse limits.
Mathematics Subject Classification ID: 55P65
Mathematics Subject Classification ID: 18F20
Mathematics Subject Classification ID: 55P42
Mathematics Subject Classification ID: 55U35
zbMATH DE Number: 5802621
zbMATH Keywords: Brown representability theorem
zbMATH Keywords: representable functor
author: J. F. Jardine
reviewed by: J. Remedios
MaRDI profile type: Publication
full work available at URL: j.jpaa.2010.04.001
OpenAlex ID: W1982982836
cites work: Failure of Brown representability in derived categories
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cites work: Q4220599
cites work: Motivic symmetric spectra
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cites work: Categorical homotopy theory
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cites work: The Grothendieck duality theorem via Bousfield’s techniques and Brown representability
cites work: Q5715380
DOI: 10.1016/J.JPAA.2010.04.001
description: scientific article; zbMATH DE number 5802621