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
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.
scientific article; zbMATH DE number 5980815