A Model Category for the Homotopy Theory of Concurrency

We construct a cofibrantly generated model structure on the category of flows such that any flow is fibrant and such that two cofibrant flows are homotopy equivalent for this model structure if and only if they are S-homotopy equivalent. This result provides an interpretation of the notion of S-homotopy equivalence in the framework of model categories. 1. Geometric models of concurrency Algebraic topological models have been used now for some years in concurrency theory (concurrent database systems and fault-tolerant distributed systems as well) [23]. The earlier models, progress graph (see [6] for instance) have actually appeared in operating systems theory, in particular for describing the problem of “deadly embrace” (as E. W. Dijkstra originally put it in [8], now more usually called deadlock) in “multiprogramming systems”. They are used by J. Gunawardena in [25] as an example of the use of homotopy theory in concurrency theory. Later V. Pratt introduced another geometric approach using strict globular ω-categories in [32]. Some of his ideas would be developed in an homological manner in E. Goubault’s PhD [22], using bicomplexes of modules. The ω-categorical point of view would be developed by the author mainly in [13] [14] [15] [16] using the equivalence of categories between the category of strict globular ω-categories and that of strict cubical ωcategories [1]. The mathematical works of R. Brown et al. [5] [4] and of R. Street [34] play an important role in this approach. The ω-categorical approach also allowed to understand how to deform higher dimensional automata (HDA) modeled by ω-categories without changing their computer-scientific properties (deadlocks, unreachable states, schedules of execution, final and initial points, serializability). The notions of spatial deformation and of temporal deformation of HDA are indeed introduced in [12] in an informal way. Another algebraic topological approach of concurrency is that of local po-space introduced by L. Fajstrup, E. Goubault and M. Raussen. A local po-space is a gluing of topological spaces which are equipped with a closed partial ordering representing the time flow. They are used as a formalization of higher dimensional automata Received August 9, 2003, revised November 30, 2003; published on December 26, 2003. 2000 Mathematics Subject Classification: 55P99, 68Q85.