Branching space of precubical set

Using the notion of short natural directed path, we introduce the homotopy branching space of a precubical set. It is unique only up to homotopy equivalence. We prove that, for any precubical set, it is homotopy equivalent to the branching space of any q-realization, any m-realization and any h-realization of the precubical set as a flow. As an application, we deduce the invariance of the homotopy branching space and of the branching homology up to cubical subdivision. By reversing the time direction, the same results are obtained for the merging space and the merging homology of a precubical set.

💡 Research Summary

The paper introduces a homotopy branching space for precubical sets, providing a direct topological invariant that captures the nondeterministic branching behavior of concurrent systems without relying on an indirect flow realization. The authors start by adapting the notion of short directed paths, originally defined in the globular setting, to the cubical context. Because cubical geometry admits a natural length parameter ε∈(0,1), they define the space P⁻(K,ε) of natural directed paths of fixed length ε that start at a vertex of a precubical set K.

A key technical result (Theorem 5.5) shows that the functor P⁻(–,ε):□^{op}Set→Top preserves colimits, meaning that the construction behaves well with respect to all categorical operations on precubical sets (sums, pushouts, etc.). Theorem 6.5 establishes that the homotopy type of P⁻(K,ε) does not depend on the chosen ε: for any ε,ε′∈(0,1) there is a natural homotopy equivalence between the corresponding spaces, and this equivalence is an m‑cofibrant weak equivalence in the model structures considered.

The authors then compare their construction with the branching space of a flow. A flow is a model of directed execution where each n‑cube is sent to a cofibrant replacement of the product poset {0<1}ⁿ. Different realization functors (q‑realization, m‑realization, h‑realization) give rise to possibly different flows, but Corollary 7.8 proves that for any such realization F, the branching space P(F(K)) of the flow is naturally homotopy equivalent to P⁻(K,ε). Thus the homotopy branching space is intrinsic to the precubical set itself, independent of the choice of realization.

The paper’s most striking application is the invariance under cubical subdivision. If f:K↪L is a cubical subdivision (i.e., K is obtained from L by subdividing cubes), then for every original vertex α∈K₀ the branching spaces at α in the realizations of K and L are homotopy equivalent, while any new vertex introduced by the subdivision yields a contractible branching space (Theorem 8.2). Consequently, the branching homology groups H_{‑n}(||K||) are unchanged by subdivision (Corollary 8.4). This mirrors the globular results of Gaucher‑Goubault but in the cubical setting.

By reversing the temporal orientation, the same arguments give analogous results for the merging space and merging homology, capturing nondeterministic merging of execution paths.

Methodologically, the work relies on Δ‑generated spaces, Δ‑kelleyfication, Reedy model structures on cubical diagrams, and the three model structures (q, m, h) on Top. The notion of “homotopy germs of short natural directed paths” (Proposition 6.6) clarifies why the branching space should be viewed as a space of equivalence classes of paths rather than individual paths.

Finally, the authors outline future work: extending the cubical subdivision invariance to a full “generating subdivision” theory for flows, which would allow them to treat transfinite compositions of subdivisions and weak equivalences, thereby strengthening the robustness of branching and merging homology.

Overall, the paper provides a solid, model‑independent definition of branching (and merging) spaces for precubical sets, proves their homotopy invariance under various realizations and subdivisions, and thus equips researchers in directed algebraic topology and concurrency theory with a powerful new tool for analyzing the causal structure of concurrent computations.