[Paper Review] Relative directed homotopy theory of partially ordered spaces

Thomas Kahl|ArXiv.org|Jan 4, 2006

Homotopy and Cohomology in Algebraic Topology14 references8 citations

TL;DR

This paper establishes that the category of partially ordered spaces (pospaces) admits a closed model category structure where weak equivalences are dihomotopy equivalences, fibrations are difibrations, and cofibrations are dicofibrations. It further shows that relative directed homotopy theory in pospaces is well-behaved, with the category of pospaces under a fixed pospace forming both a fibration and cofibration category, enabling a robust homotopical framework for modeling concurrent systems in computer science.

ABSTRACT

Algebraic topological methods have been used successfully in concurrency theory, the domain of theoretical computer science that deals with distributed computing. L. Fajstrup, E. Goubault, and M. Raussen have introduced partially ordered spaces (pospaces) as a model for concurrent systems. In this paper it is shown that the category of pospaces under a fixed pospace is both a fibration and a cofibration category in the sense of H. Baues. The homotopy notion in this fibration and cofibration category is relative directed homotopy. It is also shown that the category of pospaces is a closed model category such that the homotopy notion is directed homotopy.

Motivation & Objective

• To develop a homotopy-theoretic framework for partially ordered spaces (pospaces) used in concurrency theory.
• To define and formalize relative directed homotopy (dihomotopy) in the context of pospaces under a fixed base pospace.
• To show that the category of pospaces is a closed model category with directed homotopy as the homotopy notion.
• To prove that all pospaces are both fibrant and cofibrant, ensuring good homotopical behavior.
• To establish that dihomotopy in the closed model category sense coincides with the standard notion of dihomotopy.

Proposed method

• Constructs the category of pospaces under a fixed pospace, using the comma category construction to formalize relative homotopy.
• Defines dimaps (directed continuous maps) that preserve the partial order, modeling time-ordered system transitions.
• Introduces relative dihomotopy via a homotopy H: (X,≤) × (I,Δ) → (Y,≤) that fixes the base pospace C throughout the deformation.
• Applies Quillen's closed model category axioms to the category of pospaces, defining weak equivalences as dihomotopy equivalences.
• Uses mapping path factorizations and the construction of a subpospace E ⊆ (X×Y Y^I) × I to factor morphisms into a trivial dicofibration followed by a difibration.
• Proves that the inclusion i: X → E is a trivial dicofibration by constructing a deformation retraction via a dimap H that preserves order and fixes the base.

Experimental results

Research questions

• RQ1 Can the category of pospaces be endowed with a closed model category structure that reflects directed homotopy?
• RQ2 Is relative directed homotopy (dihomotopy) well-behaved in the context of pospaces under a fixed base pospace?
• RQ3 Do the standard homotopical structures (fibrations, cofibrations, weak equivalences) in pospaces align with the directed notions from concurrency theory?
• RQ4 Are all pospaces both fibrant and cofibrant in this model structure?
• RQ5 Does the homotopy in the closed model category sense coincide with the standard notion of dihomotopy?

Key findings

• The category of pospaces is a closed model category with weak equivalences as dihomotopy equivalences, fibrations as difibrations, and cofibrations as dicofibrations.
• Every pospace is both fibrant and cofibrant, ensuring that all objects are well-behaved in homotopical constructions.
• The homotopy relation in the closed model category sense coincides exactly with the standard notion of dihomotopy between dimaps.
• The relative dihomotopy relation—where paths are equivalent if they are homotopic through a deformation fixing the initial and final states—is captured precisely by the model structure.
• The mapping path factorization construction yields a factorization of any dimap into a trivial dicofibration followed by a difibration, satisfying Quillen's factorization axioms.
• The inclusion of the image of a pospace into the constructed path space E is a trivial dicofibration, which is essential for proving the model category axioms.