Cubical rigidification, the cobar construction and the based loop space

Abstract

We prove the following generalization of a classical result of Adams: for any pointed path-connected topological space (X, b), that is not necessarily simply connected, the cobar construction of the differential graded (dg) coalgebra of normalized singular chains in X with vertices at b is weakly equivalent as a differential graded associative algebra (dga) to the singular chains on the Moore based loop space of X at b. We deduce this statement from several more general categorical results of independent interest. We construct a functor (formula presented) from simplicial sets to categories enriched over cubical sets with connections, which, after triangulation of their mapping spaces, coincides with Lurie’s rigidification functor (formula presented) from simplicial sets to simplicial categories. Taking normalized chains of the mapping spaces of (formula presented) yields a functor ⋀ from simplicial sets to dg categories which is the left adjoint to the dg nerve functor. For any simplicial set S with (formula presented) is a dga isomorphic to ΩQΔ(S), the cobar construction on the dg coalgebra QΔ(S) of normalized chains on S. We use these facts to show that QΔ sends categorical equivalences between simplicial sets to maps of connected dg coalgebras which induce quasi-isomorphisms of dgas under the cobar functor, which is strictly stronger than saying the resulting dg coalgebras are quasi-isomorphic.