Comparing equivalences on precubical sets and spaces

We study equivalences of concurrent processes represented by objects of algebraic topology. We use methods of category theory and consider precubical sets (analogs of semisimplicial sets) and precubical spaces (analogs of cell complexes). In particular, we consider categories of these objects and construct subcategories of path-objects. We define open morphisms with respect to these subcategories and formulate criteria for a morphism to be open. We prove that the equivalence of precubical sets (spaces) based on open morphisms coincides with a behavioral equivalence of concurrent processes.