A combinatorial model for the path fibration

We introduce the abstract notion of a necklical set in order to describe a functorial combinatorial model of the path fibration over the geometric realization of a path connected simplicial set. In particular, to any path connected simplicial set $X$ we associate a necklical set $\widehat{\mathbf{\Omega }}X$ such that its geometric realization $\mathrm{\mid }\widehat{\mathbf{\Omega }}X\mathrm{\mid }$, a space built out of gluing cubical cells, is homotopy equivalent to the based loop space on $\mathrm{\mid }X\mathrm{\mid }$ and the differential graded module of chains $C_{\ast }(\widehat{\mathbf{\Omega }}X)$ is a differential graded associative algebra generalizing Adams’ cobar construction.