Symmetric Monoidal Categories Model All Connective Spectra

The classical infinite loopspace machines in fact induce an equivalence of categories between a localization of the category of symmetric monoidal categories and the stable homotopy category of -1-connective spectra. Introduction Since the early seventies it has been known that the classifying spaces of small symmetric monoidal categories are infinite loop spaces, the zeroth space in a spectrum, a sequence of spaces Xi, i ≥ 0 with given homotopy equivalences to the loops on the succeeding space Xi ∼ −→ ΩXi+1. Indeed, many of the classical examples of infinite loop spaces were found as such classifying spaces ( e.g. [Ma2], [Se]). These infinite loop spaces and spectra are of great interest to topologists. The homotopy category formed by inverting the weak equivalences of spectra is the stable homotopy category, much better behaved than but still closely related to the usual homotopy category of spaces (e.g., [Ad] III ). One has in fact classically a functor Spt from the category of small symmetric monoidal categories to the category of -1-connective spectra, those spectra Xi for which πkXi = 0 when k < i ([Ma2], [Se], [Th2]). Moreover, any two such functors satisfying the condition that the zeroth space of Spt(S) is the “group completion” of the classifying space BS are naturally homotopy equivalent ([Ma4], [Th2]). The aim of this article is to prove the new result (Thm. 5.1) that in fact Spt induces an equivalence of categories between the stable homotopy category of −1-connective spectra and the localization of the category of small symmetric monoidal categories by inverting those morphisms that Spt sends to weak homotopy equivalences. In particular, each −1-connective spectrum is weak equivalent to Spt(S) for some symmetric monoidal category S. Received by the Editors 9 April 1995 Published on 7 July 1995. 1991 Mathematics Subject Classification: Primary: 55P42 Secondary: 18C15, 18D05, 18D10, 19D23 55P47.