A combinatorial model for the path fibration (Q2002818)

In this article the authors construct a combinatorial model of the path space fibration over the geometric realization of a path-connected simplicial set. To this end they define necklical sets as contravariant functors from the category of necklaces [\textit{D. Dugger} and \textit{D. Spivak}, Algebr. Geom. Topol. 11, No. 1, 225--261 (2011; Zbl 1213.55015)] to the category of sets, as well as their geometric realizations, which are cubical complexes. The loop space and the path space of the geometric realization of a path-connected simplicial set are then modeled by necklical sets. In the last section of the article, the relationship between the necklical model of the loop space and Adams' cobar construction [\textit{J. Adams}, Proc. Natl. Acad. Sci. USA 42, 409--412 (1956; Zbl 0071.16404)] is described.