Configuration spaces and directed paths on the final precubical set

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.