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Branching space of precubical set

Philippe Gaucher

TL;DR

This work develops a cubical analogue of topological branching by introducing the homotopy branching space for precubical sets via short natural directed paths of fixed length $0<psilon<1$, denoted $ P^-(K,psilon)$. It proves the construction is colimit-preserving, independent of $psilon$ up to homotopy, and naturally compatible with multiple flow-realization functors, yielding invariance under cubical subdivisions and enabling a cubical branching theory parallel to the globular setting. By time-reversing, the same framework applies to merging spaces and merging homology, providing a robust invariant for nondeterministic branching in concurrent systems modeled by flows. The results integrate with the directed-topology toolkit, showing that branching invariants are stable under subdivision and can be computed via homotopy germs of short directed paths, while connecting to the associated branching homology. Overall, the paper delivers a self-contained, subdivision-invariant cubical branching theory with applications to directed spaces and flow models of concurrency.

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.

Branching space of precubical set

TL;DR

This work develops a cubical analogue of topological branching by introducing the homotopy branching space for precubical sets via short natural directed paths of fixed length , denoted . It proves the construction is colimit-preserving, independent of up to homotopy, and naturally compatible with multiple flow-realization functors, yielding invariance under cubical subdivisions and enabling a cubical branching theory parallel to the globular setting. By time-reversing, the same framework applies to merging spaces and merging homology, providing a robust invariant for nondeterministic branching in concurrent systems modeled by flows. The results integrate with the directed-topology toolkit, showing that branching invariants are stable under subdivision and can be computed via homotopy germs of short directed paths, while connecting to the associated branching homology. Overall, the paper delivers a self-contained, subdivision-invariant cubical branching theory with applications to directed spaces and flow models of concurrency.

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.

Paper Structure

This paper contains 8 sections, 30 theorems, 31 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Topological and homotopical reminders
  3. From precubical set to directed space
  4. Homotopy branching space of a precubical set
  5. Categorical properties of the homotopy branching space
  6. Homotopical properties of the homotopy branching space
  7. Homotopy branching space of a flow
  8. Homotopy branching space and cubical subdivision

Key Result

Theorem 1

(Theorem thm:G-colimit-preserving) The functor $\mathcal{P}^-(-,\epsilon):\square^{op}{\rm{\mathbf{Set}}} \to {\mathbf{Top}}$ from precubical sets to topological spaces is colimit-preserving.

Figures (3)

  • Figure 1: The natural directed path $\gamma_h$ from $(0,0)$ to $(0,h)$ and from $(0,h)$ to $(\frac{\epsilon-h}{2}\,\frac{\epsilon+h}{2})$ of natural length $\epsilon$ in the $2$-cube $[0,1]\times [0,1]$
  • Figure 2: Cubical subdivision $\mathop{\mathrm{Sub}}\nolimits_3(\square[2])$
  • Figure 3: An execution path of $\mathbb{P}_{(0,0),(1,1)}||\square[2]||\backslash \mathbb{P}_{(0,0),(1,1)}||\mathop{\mathrm{Sub}}\nolimits_3(\square[2])||$

Theorems & Definitions (60)

  • Theorem
  • Theorem
  • Theorem
  • Theorem
  • Theorem
  • Theorem 2.1
  • Definition 2.2
  • Theorem 2.3
  • proof
  • Definition 3.2
  • ...and 50 more