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.

25 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