Adams' cobar construction revisited

TL;DR

This paper provides a simplified proof that the cobar construction of normalized singular chains on a path-connected space is quasi-isomorphic to the chains on the loop space, extending Adams' classical theorem.

Contribution

It offers a streamlined proof of a recent generalization of Adams' theorem relating cobar constructions to loop space chains.

Findings

Cobar construction is quasi-isomorphic to loop space chains for general spaces.

The proof relates the cobar functor to the homotopy coherent nerve's left adjoint.

Extends classical results from simply connected to all path-connected spaces.

Abstract

We give a short and streamlined proof of the following statement recently proven by the author and M. Zeinalian: the cobar construction of the dg coassociative coalgebra of normalized singular chains on a path-connected pointed space is naturally quasi-isomorphic as a dg associative algebra to the singular chains on the based loop space. This extends a classical theorem of F. Adams originally proven for simply connected spaces. Our proof is based on relating the cobar functor to the left adjoint of the homotopy coherent nerve functor.