Extension of the multiplication operation in \(E_{\infty }\)-algebras to an \(A_{\infty }\)-morphism of \(E_{\infty }\)-algebras and Cartan objects in the category of May algebras
type: Item
instance of: scholarly article
title: Extension of the multiplication operation in \(E_{\infty }\)-algebras to an \(A_{\infty }\)-morphism of \(E_{\infty }\)-algebras and Cartan objects in the category of May algebras
zbMATH Open document ID: 1232.18011
DOI: 10.1134/S0001434611050087
author: Juan-Miguel Gracia
published in: Mathematical Notes
publication date: 2011-11-25T00:00:00Z
review text: The homotopy theory of \(E_\infty\)-coalgebras has been developed by \textit{V. A.\ Smirnov} [Izv. Akad. Nauk SSSR, Ser. Mat. 49, No. 6, 1302--1321 (1985; Zbl 0595.55008)]. The author shows that the multiplication operation \(\cup : X\otimes X\to X\) on an arbitrary \(E_\infty\)-algebra can be extended to an \(A_\infty\)-morphism of the \(E_\infty\)-algebras \(X\otimes X\) and \(X\). As a corollary, an excellent result is derived: Every May algebra [\textit{J. P. May}, Steenrod Algebra Appl., Lect. Notes Math. 168, 153--231 (1970 (1970; Zbl 0242.55023)] originating from some \(E_\infty\) algebra is a Cartan object in the category of May algebras.
reviewed by: Marek Golasiński
Mathematics Subject Classification ID: 18D50
Mathematics Subject Classification ID: 55P43
zbMATH DE Number: 5980815
zbMATH Keywords: \(A_\infty\)-morphism, Cartan object
zbMATH Keywords: cohomology of a topological space
zbMATH Keywords: differential module
zbMATH Keywords: \(E_\infty\)-algebra
zbMATH Keywords: May algebra
zbMATH Keywords: Steenrod operations
MaRDI profile type: Publication
cites work: Q5653373
cites work: Q3726984
cites work: HOMOLOGY OFB-CONSTRUCTIONS AND CO-B-CONSTRUCTIONS
cites work: Homotopy theories of algebras over operads
full work available at URL: s0001434611050087
OpenAlex ID: W2042932018
description: scientific article; zbMATH DE number 5980815