2006/08/02 by Stefan Schwede · 2 citations
Mathematics · #Algebra over a field #Algebraic Geometry and Number Theory #Algebraic structures and combinatorial models #Cofibration #Homotopy #Homotopy and Cohomology in Algebraic Topology #Homotopy category #Homotopy sphere #Mathematics #Model category #Morphism #Pure mathematics #Regular homotopy #math.AT #msc:55P42 #n-connected
paper · pdf · doi:10.2140/gt.2008.12.1313
published as Geom. Topol. 12 (2008) 1313-1344 · 25 pages
arxiv created 2006/08/02 · openalex publication_date 2008/06/03 · arxiv updated 2014/11/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
Symmetric spectra are an easy-to-define and convenient model for the stable homotopy category with a nice smash product. Symmetric ring spectra first showed up under the name ‘FSP on spheres ‘ in the context of algebraic K-theory and topological Hochschild homology. Around 1993, Jeff Smith made the crucial observation that the ‘FSP on spheres ‘ are the monoids in a category of ‘symmetric spectra ‘ with respect to an associative and commutative smash prodct, and he suspected compatible model category structures so that one obtains as homotopy categories ‘the ‘ stable homotopy category (for symmetric spectra), the homotopy category of A ∞ ring spectra (for symmetric ring spectra), respectively the homotopy category of E ∞ ring spectra (for commutative symmetric ring spectra). The details of various model structures were worked out by Hovey, Shipley and Smith in [HSS]. Maybe the only tricky point with symmetric spectra is that the stable equivalences can not be defined by looking at stable homotopy groups (defined as the classical sequential colimit of the unstable homotopy groups of the terms in a symmetric spectrum). Formally inverting the π∗-isomorphisms, i.e., those morphisms which induce isomorphisms