Simplicial models of trace spaces (Q986682)

Directed algebraic topology is a field of research between concurrency and algebraic topology involving the use of a space of states equipped with an additional structure modeling time. In this kind of model, there is always an underlying state space equipped with a set of privileged continuous paths playing the role of execution paths. In general, the reverse of an execution path is not an execution path. Calculating the homotopy type of the space of execution paths is the main problem in this theory. For example, calculating the fundamental category means calculating the path-connected components of this space [\textit{M. Grandis}, Directed algebraic topology. Models of non-reversible worlds. New Mathematical Monographs 13. Cambridge: Cambridge University Press. (2009; Zbl 1176.55001)]. In other approaches like the reviewer's approach of directed algebraic topology [\textit{P. Gaucher}, Homology Homotopy Appl. 5, No.~1, 549--599, electronic only (2003; Zbl 1069.55008), Theory Appl. Categ. 22, 588--621 (2009; Zbl 1191.55013)], the homotopy type of the space of execution paths also plays a fundamental role, e.g. in the definition of the weak equivalences of model category structures. In this paper, the author develops a systematic approach describing spaces of execution paths up to homotopy equivalence as finite prodsimplicial complexes [\textit{D. Kozlov}, Combinatorial algebraic topology. Algorithms and Computation in Mathematics 21. Berlin: Springer. (2008; Zbl 1130.55001)]. This approach makes feasible (at least for some class of models of computation) the calculations of some of their algebraic topological invariants. The author describes algorithms to determine not only the fundamental category of such a model space, but all homological invariants of spaces of execution paths within it.