Homotopy limits of triples
type: Item
instance of: scholarly article
title: Homotopy limits of triples
zbMATH Open document ID: 1032.55008
DOI: 10.1016/S0166-8641(02)00217-1
author: Assaf Libman
published in: Topology and its Applications
publication date: 2003-05-27T00:00:00Z
review text: Let \(\mathbf E\) be a unitary ring spectrum, and define a functor \(E:{\mathcal S}_* \to {\mathcal S}_*\) from the category of pointed spaces to itself by the rule \(E:X\mapsto \Omega^\infty(\Sigma^\infty X\wedge {\mathbf E})\). If \(E\) is an \(S\)-algebra or a symmetric spectrum, then \(E\) is a triple on \({\mathcal S}_*\) and is also a continuous functor in the sense that the natural map \[ \text{ map}_*(X,Y) \to \text{ map}_*(EX,EY) \] of spaces is continuous. There is then a cosimplical resolution \(E^\bullet X\) and partial terms \(E^{\bullet \leq n} = {\text{ Tot}}_n(E^\bullet_{\text{ fib}} X)\), (where \(E^\bullet_{\text{ fib}} X\) comes with a Reedy fibrant approximation). The main result shows that \(E_n\) is a homotopy functor, and that the induced functor on the homotopy category is a triple.
Mathematics Subject Classification ID: 55P60
Mathematics Subject Classification ID: 55P65
Mathematics Subject Classification ID: 18A25
zbMATH DE Number: 1917850
zbMATH Keywords: homotopy limits
zbMATH Keywords: triples
zbMATH Keywords: completions
reviewed by: Jeffrey Strom
MaRDI profile type: Publication
cites work: The unstable Adams spectral sequence for generalized homology
cites work: Cosimplicial resolutions and homotopy spectral sequences in model categories
cites work: Homotopy limits, completions and localizations
cites work: Cellular spaces, null spaces and homotopy localization
cites work: Q3808872
cites work: Q3036295
cites work: Q4896542
cites work: Q4220599
cites work: Symmetric spectra
cites work: Q3934579
cites work: Universal spaces for homotopy limits of modules over coaugmented functors. I
cites work: Q4693151
description: scientific article; zbMATH DE number 1917850