Branching space of multipointed d-space

TL;DR

This paper introduces the branching space of multipointed d-spaces using short directed paths, proving its topological invariance and relating it to flow categorization, with applications to homology invariance under subdivision.

Contribution

It defines the branching space for multipointed d-spaces, proves its homeomorphism with flow categorization for q-cofibrant spaces, and establishes topological invariance of branching homology.

Findings

Branching space is homeomorphic to flow categorization for q-cofibrant spaces.

Topological invariance of branching space and homology under globular subdivision.

Reversal of time yields similar results for merging space and homology.

Abstract

Using the notion of short directed path, we introduce the branching space of a multipointed $d$-space. We prove that for any q-cofibrant multipointed $d$-space, it is homeomorphic to the branching space of the q-cofibrant flow obtained by applying the categorization functor. As an application, we deduce a purely topological proof of the invariance of the branching space and of the branching homology of cellular multipointed $d$-spaces up to globular subdivision. By reversing the time direction, the same results are obtained for the merging space and the merging homology.