We prove the following generalization of a classical result of Adams: for any pointed path-connected
topological space
,
that is not necessarily simply connected, the cobar construction of
the differential graded (dg) coalgebra of normalized singular chains in
with
vertices at
is weakly equivalent as a differential graded associative algebra
(dga) to the singular chains on the Moore based loop space of
at
. We deduce this
statement from several more general categorical results of independent interest. We construct
a functor
from simplicial sets to categories enriched over cubical sets with connections, which,
after triangulation of their mapping spaces, coincides with Lurie’s rigidification functor
from
simplicial sets to simplicial categories. Taking normalized chains of the mapping spaces
of
yields
a functor
from simplicial sets to dg categories which is the left adjoint to the dg nerve functor. For any
simplicial set
with
,
is a dga isomorphic to
, the cobar construction
on the dg coalgebra
of
normalized chains on
. We
use these facts to show that
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.
Keywords
rigidification, cobar construction, based loop space
