Abstract
Algebr. Geom. Topol. 18 (2018) 3789-3820 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.
