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 models for the path category of a simplicial set. We then pass to the chain-level 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 simplicial set. This is achieved through a version of the cobar functor inspired by Lazarev and Holstein's categorical Koszul duality. As a consequence, we obtain a conceptual explanation of a result of Franz stating that there is a natural dg bialgebra quasi-isomorphism from the extended cobar construction on the chains of a reduced simplicial set to the chains on its Kan loop group.