#### Volume 18, issue 7 (2018)

 Recent Issues
 The Journal About the Journal Editorial Board Editorial Interests Subscriptions Submission Guidelines Submission Page Policies for Authors Ethics Statement ISSN (electronic): 1472-2739 ISSN (print): 1472-2747 Author Index To Appear Other MSP Journals
Cubical rigidification, the cobar construction and the based loop space

### Manuel Rivera and Mahmoud Zeinalian

Algebraic & Geometric Topology 18 (2018) 3789–3820
##### 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 $\Lambda$ from simplicial sets to dg categories which is the left adjoint to the dg nerve functor. For any simplicial set $S$ with ${S}_{0}=\left\{x\right\}$, $\Lambda \left(S\right)\left(x,x\right)$ is a dga isomorphic to $\Omega {Q}_{\Delta }\left(S\right)$, the cobar construction on the dg coalgebra ${Q}_{\Delta }\left(S\right)$ of normalized chains on $S\phantom{\rule{0.3em}{0ex}}$. 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 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
##### Mathematical Subject Classification 2010
Primary: 18G30, 55P35, 55U10, 57T30
Secondary: 18D20, 55U35, 55U40