×

An infinite loop space machine for theories with noncontractible multiplication. (English) Zbl 0802.55005

The free loop space functor \(Q:X \mapsto \Omega^ \infty \Sigma^ \infty X\) is a monad on topological spaces. \(Q\) has a finitary approximation \(D_{\vartheta_ Q}\) [G. Segal, Topology 13, 293- 312 (1974; Zbl 0284.55016)], which is a theory in the sense of [J. M. Boardman and R. M. Vogt, Homotopy invariant algebraic structures on topological spaces, Lect. Notes Math. 347 (1973; Zbl 0285.55012)]. (A special case of) Theorem 1.3 asserts, that the algebras of \(D_{\vartheta_ Q}\) are also canonically (weakly) equivalent to \(\Gamma\)-spaces in the sense of Segal. (Theorem 1.3 is a general criterion for theories to produce \(\Gamma\)-spaces). \(D_{\vartheta_ Q}\) is augmented over the theory of abelian groups – a categorical version of the usual Hurewicz map. In the language of theories this is saying to encode algebraic operations by products of components of \(Q(S^ 0)\). The operation “aditive inverse” for instance is given by the degree –1 maps.
This is called coherent encoding in the paper. A different notion of coherence had been used in [the reviewer and R. M. Vogt, Topology 28, No. 4, 481-484 (1989; Zbl 0688.55009)].

MSC:

55P47 Infinite loop spaces
PDFBibTeX XMLCite
Full Text: DOI