Homotopy theory of Γ-spaces, spectra, and bisimplicial sets

Abstract

in [Segal I], Graeme Segal introduced the concept of a F-space and proved that a certain homotopy category of F-spaces is equivalent to the usual homotopy category of connective spectra. Our main purpose is to show that there is a full-fledged homotopy theory of r-spaces underlying Segal's homotopy category. We do this by giving F-spaces the structure of a closed model category, i.e. defining "fibrations," "cofibrations," and "weak equivalences" for r-spaces so that Quillen's theory of homotopical algebra can be applied. Actuall~ we give two such structures (3.5, 5.2) leading to a "strict" and a "stable" homotopy theory of F-spaces. The former has had applications, cf. [Friedlander], but the latter is more closely related to the usual homotopy theory of spectra. In our work on F-spaces, we have adopted the "chain functor" viewpoint of [Anderson]. However, we do not require our F-spaces to be "special," cf. §4, because "special" F-spaces are not closed under direct limit constructions. We have included in § §4,5 an exposition, and slight generalization, of the Anderson-Segal results on the construction of homology theories from r-spaces, and on the equivalence of the homotopy categories of F-spaces and connective spectra. To set the stage for our work on F-spaces, we have given in §2 an exposition of spectra from the standpoint of homotopical algebra. We have also included an appendix (§B) on bislmplicial sets, where we outline some well-kno~n basic results needed in this paper and prove a rather strong fibratlon theorem (B.~) for diagonals of bisimplicial sets. We apply B.4 to prove a generalization of S~pl~orted in part by NSF Grants