Adams' cobar construction as a monoidal $E_{\infty}$-coalgebra model of the based loop space

TL;DR

This paper proves that Adams' cobar construction on singular chains is a monoidal $E_$-coalgebra quasi-isomorphism to the chains on the based loop space, extending Baues' result.

Contribution

It establishes that Adams' cobar construction preserves monoidal $E_$-coalgebra structures explicitly, extending previous results.

Findings

Adams' map is a quasi-isomorphism.

The map preserves monoidal $E_$-coalgebra structures.

Extension of Baues' result on structure preservation.

Abstract

We prove that the classical map comparing Adams' cobar construction on the singular chains of a pointed space and the singular cubical chains on its based loop space is a quasi-isomorphism preserving explicitly defined monoidal $E_\infty$-coalgebra structures. This contribution extends to its ultimate conclusion a result of Baues, stating that Adams' map preserves monoidal coalgebra structures.