A combinatorial model for the free loop fibration

Abstract

We introduce the abstract notion of a closed necklical set in order to describe a functorial combinatorial model of the free loop fibration ΩY → ΛY → Y over the geometric realization Y = |X| of a path-connected simplicial set X. In particular, to any path-connected simplicial set X we associate a closed necklical set ΛX such that its geometric realization |ΛX|, a space built out of glueing ‘freehedrical’ and ‘cubical’ cells, is homotopy equivalent to the free loop space ΛY and the differential graded module of chains C*(ΛX) generalizes the coHochschild chain complex of the chain coalgebra C*(X).