\(E_ n\)-ring categories
The notion of an operad pair introduced by \textit{J. P. May} [``\(E_\infty\) ring spaces and \(E_\infty\) ring spectra'', Lect. Notes Math. 577 (1977; Zbl 0345.55007)] can be extended in the obvious way to operad pairs \(({\mathcal C}, {\mathcal G})\) where each \({\mathcal C} (n)\) and \({\mathcal G} (n)\) is a topological category and the structure maps are continuous functors. Since a topological space can be considered as a topological category with only identity morphisms this allows a unifying treatment of homotopy ring structures on spaces and on categories such as bipermutative categories. \textit{G. Dunn} treats the more general case where the multiplicative structure is given by a braided cat-operad, defined by Fiedorowicz (unpublished) in the topological case. The paper starts with the definition of homotopy ring structures on topological categories with a special emphasis on \(E_n\) braided ring structures. The second section discusses the passage from categories with homotopy ring structures to ring spectra and corrects mistakes made by others in the passage from homotopy ring spaces to ring spectra. The remaining sections discuss various examples of \(E_n\)-ring categories and their associated categories of algebras.
scientific article; zbMATH DE number 1025718