Branching space of precubical set

TL;DR

This paper introduces the homotopy branching space of a precubical set using short natural directed paths, establishing its invariance and homotopy equivalence across various realizations and subdivisions, with applications to merging spaces.

Contribution

It defines the homotopy branching space for precubical sets and proves its invariance and equivalence across different realizations and subdivisions.

Findings

Homotopy branching space is unique up to homotopy equivalence.

Homotopy branching space is invariant under cubical subdivision.

Results extend to merging spaces and homology via time reversal.

Abstract

Using the notion of short natural directed path, we introduce the homotopy branching space of a precubical set. It is unique only up to homotopy equivalence. We prove that, for any precubical set, it is homotopy equivalent to the branching space of any q-realization, any m-realization and any h-realization of the precubical set as a flow. As an application, we deduce the invariance of the homotopy branching space and of the branching homology up to cubical subdivision. By reversing the time direction, the same results are obtained for the merging space and the merging homology of a precubical set.