Configuration spaces and directed paths on the final precubical set

TL;DR

This paper establishes a homotopy equivalence between directed loops on the final precubical set and the total configuration space of points in the plane, with applications to invariants and CW-complex structures.

Contribution

It proves the homotopy equivalence and introduces new invariants, linking directed path spaces to configuration spaces and CW-complexes.

Findings

Directed loop space is homotopy equivalent to total configuration space.

Directed path spaces have CW-complex homotopy types.

Configuration spaces can be represented as nerves of categories.

Abstract

The main goal of this paper is to prove that the space of directed loops on the final precubical set is homotopy equivalent to the "total" configuration space of points on the plane; by "total" we mean that any finite number of points in a configuration is allowed. We also provide several applications: we define new invariants of precubical sets, prove that directed path spaces on any precubical complex have the homotopy types of CW-complexes and construct certain presentations of configuration spaces of points on the plane as nerves of categories.