Branching space of multipointed d-space

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.

💡 Research Summary

The paper “Branching space of multipointed d‑space” develops a purely topological construction of the branching space and branching homology for multipointed d‑spaces, bypassing the traditional reliance on the category of flows. The authors first introduce the notion of a short directed path, which is a minimal execution path that starts at a given state and immediately proceeds to another state. By taking germs of such short paths at each state α, they define the branching space G⁻_α(X) of a multipointed d‑space X, equipped with the Δ‑Kelley topology.

The central result (Theorem 6.3) states that if X is q‑cofibrant—a condition satisfied by cellular multipointed d‑spaces—then for every state α there is a homeomorphism between G⁻_α(X) and the branching space P⁻_α(cat X) of the flow obtained by applying the categorization functor cat : MdTop → Flow. Consequently, the branching homology H⁻_n(X), originally defined as the homology of the flow L cat(X), can be computed directly from the topological data of X without invoking cat. This is made explicit in Corollary 6.4, where a chain complex built from the germs of short paths yields the same homology groups as the flow‑based definition.

The paper then addresses invariance under globular subdivision, a refinement operation that replaces a globular cell by a collection of smaller globular cells while preserving causal structure. Theorem 7.8 proves that a globular subdivision f : X → Y induces a homeomorphism G⁻α(X) ≅ G⁻{f(α)}(Y) for every state α. As a direct consequence, Corollary 7.9 shows that the branching homology groups are invariant under such subdivisions: H⁻_n(X) ≅ H⁻_n(Y) for all n. This provides a purely topological proof of a result that previously required the flow framework and the left derived functor of cat (as in