Branching space of multipointed d-space
Philippe Gaucher
TL;DR
The work develops a explicit geometric description of the branching space and branching homology for cellular multipointed $d$-spaces in the globular setting, avoiding reliance on the categorization functor. It proves a central equivalence, $\mathcal{G}^-_\alpha(X) \cong \mathbb{P}^-_\alpha{\mathrm{cat}}(X)$, for all states $\alpha$ when $X$ is $q$-cofibrant, linking globular and flow perspectives. Moreover, it provides a purely topological proof of the invariance of branching homology under globular subdivision (and the analogous merging results via time reversal), and it demonstrates the preservation of branching structures under globular subdivisions, establishing invariance of branching homology and clarifying the relationship to cubical approaches. Together with the companion cubical-branching work, the paper clarifies how globular and cubical formalisms capture the causal structure of concurrent computations and lays groundwork for unified treatments of branching phenomena in directed topology.
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.