Adams' cobar construction revisited

  • Manuel Rivera

    Purdue University, 150 N University St., West Lafayette, IN 47906, USA
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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.

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Manuel Rivera, Adams' cobar construction revisited. Doc. Math. 27 (2022), pp. 1213–1223

DOI 10.4171/DM/895