Branching space of precubical set
Abstract
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.
Cite
@article{arxiv.2508.14839,
title = {Branching space of precubical set},
author = {Philippe Gaucher},
journal= {arXiv preprint arXiv:2508.14839},
year = {2026}
}
Comments
23 pages. sequel of arXiv:2502.11773 and cubical analogue of arXiv:2507.08377; v2: finiteness hypothesis removed from Theorem 8.2 and Corollary 8.4; v3: additional fact added to Theorem 8.2 to deduce Corollary 8.4 whose proof is fully expanded; v4: new introduction