Cubical rigidification, the cobar construction, and the based loop space

TL;DR

This paper generalizes a classical result relating the cobar construction of chains on a space to the chains on its based loop space, extending it to non-simply connected spaces using categorical and dg algebra techniques.

Contribution

It introduces a new functor from simplicial sets to dg categories that links the cobar construction with chains on loop spaces, generalizing Adams' classical theorem.

Findings

Cobar construction of chains is weakly equivalent to chains on the Moore loop space.

Constructs a functor linking simplicial sets to dg categories via cubical sets.

Shows that certain dg coalgebra maps induce quasi-isomorphisms under the cobar functor.

Abstract

We prove the following generalization of a classical result of Adams: for any pointed and 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 $\mathfrak{C}_{\square_c}$ from simplicial sets to categories enriched over cubical sets with connections which, after triangulation of their mapping spaces, coincides with Lurie's rigidification functor $\mathfrak{C}$ from simplicial sets to simplicial categories. Taking normalized chains of the mapping spaces of $\mathfrak{C}_{\square_c}$ yields a functor $\Lambda$ from simplicial sets to dg categories which is the left adjoint to the dg nerve functor. For any simplicial set $S$ with $S_0=\{x\}$, $\Lambda(S)(x,x)$ is a dga isomorphic to $\Omega Q_{\Delta}(S)$, the cobar construction on the dg coalgebra $Q_{\Delta}(S)$ of normalized chains on $S$. We use these facts to show that $Q_{\Delta}$ sends categorical equivalences between simplicial sets to maps of connected dg coalgebras which induce quasi-isomorphisms of dga's under the cobar functor.