We introduce the abstract setting of presheaf category on a thick category of cubes. Precubical sets, symmetric transverse sets, symmetric precubical sets and the new category of (non-symmetric) transverse sets are examples of this structure. All these presheaf categories share the same metric and homotopical properties from a directed homotopy point of view. This enables us to extend Raussen's notion of natural $d$-path for each of them. Finally, we adapt Ziemiański's notion of cube chain to this abstract setting and we prove that it has the expected behavior on precubical sets. As an application, we verify that the formalization of the parallel composition with synchronization of process algebra using the coskeleton functor of the category of symmetric transverse sets has a category of cube chains with the correct homotopy type.

翻译：我们引入了厚立方体范畴上预层范畴的抽象框架。预立方集、对称横贯集、对称预立方集以及新范畴（非对称）横贯集均为该结构的实例。从有向同伦观点来看，所有这些预层范畴均具有相同的度量与同伦性质。这使我们能够为其中每个范畴扩展Raussen的自然$d$-路径概念。最后，我们将Ziemiański的立方链概念适配至此抽象框架，并证明其在预立方集上具有预期性质。作为应用，我们验证了使用对称横贯集范畴的余骨架函子对进程代数同步并行组合的形式化，其立方链范畴具有正确的同伦类型。