New York Journal of Mathematics
Volume 16 (2010) 409-461


Philippe Gaucher

Directed algebraic topology and higher dimensional transition systems

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Published: November 15, 2010
Keywords: labelled symmetric precubical set, higher dimensional transition system, locally presentable category, topological category, small-orthogonality class, directed homotopy, model category, process algebra
Subject: 18C35, 18A40, 18A25, 18F20, 18G55, 68Q85

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.

Author information

Laboratoire PPS (CNRS UMR 7126), Universit{é} Paris 7-Denis Diderot, Site Chevaleret, Case 7014, 75205 PARIS Cedex 13, France