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Towards a theory of natural directed paths

Philippe Gaucher

TL;DR

The paper develops a unified theory of natural directed paths by framing various cube-like models as presheaves on thick categories of cubes. It demonstrates a common directed metric and homotopy theory across models, extends Raussen's natural $d$-path and Ziemiański's cube chains to this abstract setting, and proves that cube-chains faithfully capture the homotopy type of tame natural $d$-paths. An application to process-algebra synchronization using coskeleton in Symmetric Transverse Sets shows the framework yields the correct homotopy type of paths, validating the approach as a general axiomatization for directed-path homotopy in concurrency.

Abstract

We introduce the abstract setting of presheaf category on a thick category of cubes. Precubical sets, symmetric transverse sets, symmetric precubical sets and the new category of (non-symmetric) transverse sets are examples of this structure. All these presheaf categories share the same metric and homotopical properties from a directed homotopy point of view. This enables us to extend Raussen's notion of natural $d$-path for each of them. Finally, we adapt Ziemiański's notion of cube chain to this abstract setting and we prove that it has the expected behavior on precubical sets. As an application, we verify that the formalization of the parallel composition with synchronization of process algebra using the coskeleton functor of the category of symmetric transverse sets has a category of cube chains with the correct homotopy type.

Towards a theory of natural directed paths

TL;DR

The paper develops a unified theory of natural directed paths by framing various cube-like models as presheaves on thick categories of cubes. It demonstrates a common directed metric and homotopy theory across models, extends Raussen's natural -path and Ziemiański's cube chains to this abstract setting, and proves that cube-chains faithfully capture the homotopy type of tame natural -paths. An application to process-algebra synchronization using coskeleton in Symmetric Transverse Sets shows the framework yields the correct homotopy type of paths, validating the approach as a general axiomatization for directed-path homotopy in concurrency.

Abstract

We introduce the abstract setting of presheaf category on a thick category of cubes. Precubical sets, symmetric transverse sets, symmetric precubical sets and the new category of (non-symmetric) transverse sets are examples of this structure. All these presheaf categories share the same metric and homotopical properties from a directed homotopy point of view. This enables us to extend Raussen's notion of natural -path for each of them. Finally, we adapt Ziemiański's notion of cube chain to this abstract setting and we prove that it has the expected behavior on precubical sets. As an application, we verify that the formalization of the parallel composition with synchronization of process algebra using the coskeleton functor of the category of symmetric transverse sets has a category of cube chains with the correct homotopy type.
Paper Structure (5 sections, 25 theorems, 69 equations, 1 figure)

This paper contains 5 sections, 25 theorems, 69 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Thick category of cubes
  3. Metric and homotopical study
  4. Ziemiański's cube chain
  5. Application

Key Result

Theorem 1

(Proposition Orem, Theorem thick-cube and Theorem greatest) There exists a greatest thick category of cubes for the inclusion not containing the symmetry maps.

Figures (1)

  • Figure 1: Construction of $h_1$

Theorems & Definitions (61)

  • Theorem
  • Theorem
  • Theorem
  • Theorem
  • Definition 2.1
  • Definition 2.3
  • Definition 2.4
  • Proposition 2.5
  • Proposition 2.6
  • proof
  • ...and 51 more