\(E_{\infty}\)-spaces and injective \(\Gamma\)-spaces
type: Item
instance of: scholarly article
title: \(E_{\infty}\)-spaces and injective \(\Gamma\)-spaces
zbMATH Open document ID: 0649.57018
DOI: 10.1007/BF01259329
published in: Manuscripta Mathematica
publication date: 1988-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/155311
review text: The work of \textit{J. M. Boardman} and the second author [Homotopy invariant algebraic structures on topological spaces (Lect. Notes Math. 347) (1973; Zbl 0285.55012)], \textit{G. Segal} [Topology 13, 293-312 (1974; Zbl 0284.55016)] and \textit{J. P. May} \([E_{\infty}\) ring spaces and \(E_{\infty}\) ring spectra (Lect. Notes Math. 577) (1977; Zbl 0345.55007)] shed a lot of light on the properties of homotopy invariant algebraic structures. More recently \textit{M. Bökstedt} [Habilitationsschrift, Bielefeld, 1986], by combining these ideas with work of Waldhausen on the algebraic K-theory of topological spaces, has shown how an algebraic analysis of homotopy coherent algebraic structures implies new results in algebra and algebraic number theory. A problem in this area is always that the different models of homotopy coherence are good at doing different things, hence one needs means of comparing the models, of translating between the different ``languages'' being used. In this paper, the authors attack this problem with regards to homotopy monoid structures. They do this using \(\Gamma\)-spaces (cf. Segal loc. cit.) in which certain of the structure maps are closed cofibrations, the injective \(\Gamma\)-spaces of the title. The homotopy categories of \(\Gamma\)-spaces and injective \(\Gamma\)-spaces are equivalent. The authors then construct an equivalence between the latter category and the homotopy category \(Ho(E_{\infty}\)-Top) of \(E_{\infty}\)-spaces.
reviewed by: Timothy Porter
Mathematics Subject Classification ID: 57P99
Mathematics Subject Classification ID: 55P47
Mathematics Subject Classification ID: 18F25
Mathematics Subject Classification ID: 55U35
zbMATH DE Number: 4060321
zbMATH Keywords: homotopy monoid structures
zbMATH Keywords: \(\Gamma \)-spaces
zbMATH Keywords: closed cofibrations
zbMATH Keywords: injective \(\Gamma \)-spaces
zbMATH Keywords: \(E_{\infty }\)-spaces
Wikidata QID: Q114233974
author: Roland Schwänzl
author: Rainer Vogt
MaRDI profile type: Publication
cites work: Homotopy-everything 𝐻-spaces
cites work: Homotopy invariant algebraic structures on topological spaces
cites work: Tensor products of theories, application to infinite loop spaces
cites work: An adjunction theorem for locally equiconnected spaces
cites work: Multiplicative infinite loop space theory
cites work: The uniqueness of infinite loop space machines
cites work: Categories and cohomology theories
cites work: Infinite loop structures on the algebraicK-theory of spaces
cites work: Uniqueness of delooping machines
cites work: Convenient categories of topological spaces for homotopy theory
cites work: Q3048986
cites work: Q3868473
description: scientific article; zbMATH DE number 4060321