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 $\left(X,b\right)$, 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 $\mathrm{\Lambda }$ from simplicial sets to dg categories which is the left adjoint to the dg nerve functor. For any simplicial set $S$ with $S_0=\left\{x\right\}$, $\mathrm{\Lambda }\left(S\right)\left(x,x\right)$ is a dga isomorphic to $\mathrm{\Omega }{Q}_{\mathrm{\Delta }}\left(S\right)$, the cobar construction on the dg coalgebra $Q_{\mathrm{\Delta }}\left(S\right)$ of normalized chains on $S$. We use these facts to show that $Q_{\mathrm{\Delta }}$ 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.