# A combinatorial model for the path fibration

## TL;DR

This paper introduces a new combinatorial model called necklical sets to represent the path fibration over simplicial sets, providing a homotopy equivalent space to the based loop space with an algebraic structure extending Adams' cobar construction.

## Contribution

It defines necklical sets as a novel combinatorial framework for modeling path fibrations and loop spaces, linking geometric realizations with algebraic structures.

## Key findings

- Homotopy equivalence between the geometric realization of the necklical set and the based loop space.
- The chain complex forms a differential graded associative algebra extending Adams' cobar construction.
- A functorial combinatorial model for the path fibration over simplicial sets.

## Abstract

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 $|\widehat{\mathbf{\Omega}}X|$, a space built out of gluing cubical cells, is homotopy equivalent to the based loop space on $|X|$ and the differential graded module of chains $C_*(\widehat{\mathbf{\Omega}}X)$ is a differential graded associative algebra generalizing Adams' cobar construction.