\(E_{\infty}\)-spaces and injective \(\Gamma\)-spaces. (English) Zbl 0649.57018

The work of J. M. Boardman and the second author [Homotopy invariant algebraic structures on topological spaces (Lect. Notes Math. 347) (1973; Zbl 0285.55012)], G. Segal [Topology 13, 293-312 (1974; Zbl 0284.55016)] and 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 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.
Reviewer: T.Porter


57P99 Generalized manifolds
55P47 Infinite loop spaces
18F25 Algebraic \(K\)-theory and \(L\)-theory (category-theoretic aspects)
55U35 Abstract and axiomatic homotopy theory in algebraic topology
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