Categorical models for path spaces
Abstract
We establish an explicit comparison between two constructions in homotopy theory: the left adjoint of the homotopy coherent nerve functor, also known as the rigidification functor, and the Kan loop groupoid functor. This is achieved by considering localizations of the rigidification functor, unraveling a construction of Hinich, and using a sequence of operators originally introduced by Szczarba in 1961. As a result, we obtain several combinatorial spacelevel models for the path category of a simplicial set. We then pass to the chainlevel and describe a model for the path category, now considered as a category enriched over differential graded (dg) coalgebras, in terms of a suitable algebraic chain model for the underlying space. This is achieved through a localized version of the cobar functor from the categorical Koszul duality theory of Holstein and Lazarev and considering the chains on a simplicial set as a curved coalgebra equipped with higher structure. We obtain a conceptual explanation of a result of Franz stating that the dg algebra quasiisomorphism from the extended cobar construction on the chains of a reduced simplicial set to the chains on its Kan loop group, originally constructed by Hess and Tonks in terms of Szczarba's twisting cochain, is a map of dg bialgebras.
 Publication:

arXiv eprints
 Pub Date:
 January 2022
 arXiv:
 arXiv:2201.03046
 Bibcode:
 2022arXiv220103046M
 Keywords:

 Mathematics  Algebraic Topology;
 Mathematics  Category Theory;
 Mathematics  Quantum Algebra
 EPrint:
 Definition 5.1 was slightly simplified