\(E_ n\)-ring categories
type: Item
instance of: scholarly article
title: \(E_ n\)-ring categories
zbMATH Open document ID: 0886.18006
DOI: 10.1016/S0022-4049(96)00015-1
author: Gerald Dunn
published in: Journal of Pure and Applied Algebra
publication date: 1998-04-19T00:00:00Z
review text: 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the more general case where the multiplicative structure is given by a braided cat-operad, defined by Fiedorowicz (unpublished) in the topological case.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.
Mathematics Subject Classification ID: 18D10
Mathematics Subject Classification ID: 55P42
zbMATH DE Number: 1025718
zbMATH Keywords: operad pairs
zbMATH Keywords: homotopy ring structures
zbMATH Keywords: braided monoidal categories
zbMATH Keywords: \(E_ n\)-spaces
zbMATH Keywords: \(E_ n\)-ring spectrum
reviewed by: Rainer Vogt
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1025718