Branching spaces of multipointed d-spaces

Abstract

Using the notion of a 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.

arXiv comment25 pages. some material taken from arXiv:2502.11773 ; v2: some reorganization (the numbering of the theorems is modified); v3: fixes and improvements; v4: new introduction; v5: final version