A combinatorial model for the free loop fibration

We introduce the abstract notion of a closed necklical set in order to describe a functorial combinatorial model of the free loop fibration $\Omega Y\rightarrow \Lambda Y\rightarrow Y$overthegeometricrealization$Y=|X|$ of a path connected simplicial set $X\mathrm{.}$ In particular, to any path connected simplicial set $X$ we associate a closed necklical set $\widehat{\mathbf{\Lambda }}X$ such that its geometric realization $\mathrm{\mid }\widehat{\mathbf{\Lambda }}X\mathrm{\mid }$, a space built out of gluing “freehedrical” and “cubical” cells, is homotopy equivalent to the free loop space $\mathrm{\Lambda }Y$ and the differential graded module of chains $C_{\ast }(\widehat{\mathbf{\Lambda }}X)$ generalizes the coHochschild chain complex of the chain coalgebra $C_{\ast }(X)\mathrm{.}$