Equivalence of Categories of Precubical Sets and Transitional Chu-Spaces, Preserving the Property of Morphisms to be Open

The intention of the paper is to show the applicability of the directed algebraic topology to establish the close categorical relationships between geometrical models of concurrency — precubical sets and transitional Chu-spaces. In particular, we start with introducing categories of the models under consideration. Then, we construct and study the universal di-covering functor from the category of precubical sets to the category of simply di-connected counterpart of precubical sets. Finally, an equivalence of the categories of transitional Chu-spaces and simply di-connected precubical sets is established, preserving an important property of morphisms to be open.