Directed algebraic topology and higher dimensional transition system

TL;DR

This paper interprets higher dimensional transition systems using algebraic topology, linking them to labelled precubical sets and process algebras, and provides new categorical insights into concurrency models.

Contribution

It introduces a topological categorical framework for higher dimensional transition systems, connecting them to precubical sets and process algebras like CCS.

Findings

Higher dimensional transition systems form a locally finitely presentable category.

The associated n-cube transition system is the free system generated by an n-dimensional transition.

The approach applies to various process algebras and topological models of concurrency.

Abstract

Cattani-Sassone's notion of higher dimensional transition system is interpreted as a small-orthogonality class of a locally finitely presentable topological category of weak higher dimensional transition systems. In particular, the higher dimensional transition system associated with the labelled n-cube turns out to be the free higher dimensional transition system generated by one n-dimensional transition. As a first application of this construction, it is proved that a localization of the category of higher dimensional transition systems is equivalent to a locally finitely presentable reflective full subcategory of the category of labelled symmetric precubical sets. A second application is to Milner's calculus of communicating systems (CCS): the mapping taking process names in CCS to flows is factorized through the category of higher dimensional transition systems. The method also applies to other process algebras and to topological models of concurrency other than flows.