T-homotopy and refinement of observation. III. Invariance of the branching and merging homologies (Q851558)

Higher dimensional automata (HDA) model the execution of concurrent processes. The execution trace of a set of concurrent processes is a path in a cube of dimension the number of processes. In modeling deadlocks and unreachable states one is led to the notion of dihomotopy which is a deformation of one trace into an equivalent one. To represent the notion that two such automata are equivalent the author earlier [Homology Homotopy Appl. 5, No. 1, 549-599, electronic only (2003; Zbl 1069.55008)], introduced the notion of a T-homotopy equivalence. The behaviour of an HDA can be seen as a flow. Associated to this flow one can define branching and merging homologies. This paper shows that a generalized T-homotopy equivalence induces isomorphisms of the branching (respectively merging) homology groups. The proof is heavily categorical and relies extensively on model theory. Although part of a series the paper can be read without having read the earlier papers.