satisfying some stability properties and whose elements both preserve the past of their source and the future of their target. Then is defined as the quotient which is proven to be equivalent to the localization of a category . The category of components of a PV program P is then defined as where is the geometric model of P. As an interesting property, the category of components of any PV program is finite. Topics Higher order directed homotopy The higher order directed homotopy theory can be developed through cylinder functor and path functor, all constructions and properties being expressed in the setting of categorical algebra. This approach emphasizes the combinatorial role of cubical sets in directed algebraic topology. The model category approach Philippe Gaucher proposed an alternative formalisation of the notion of directed space which is, roughly speaking, based on the category of directed graphs enriched in topological spaces i.e. the collection of arrows from x to y is endowed with a topology. This approach gives rise to the so-called category of Flows, which admits a nontrivial model category structure. He introduced a topological version (here a topological category means a category equipped with a topological forgetful functor towards the category of sets) using a variant of Marco Grandis' d-spaces, the multipointed d-spaces. In recent papers, he constructed similar model category structures on cubical higher-dimensional transition systems (whose a reflective subcategory is the one of Cattani-Sassone higher-dimensional transition systems) and on labelled symmetric precubical sets. The common points of all these model category structures is 1) the presence of the cofibration {0,1}→{0} identifying two states, 2) the non-contractibility of the directed segment, 3) the strong relationship with the computer-scientific notion of bisimulation. The cylinders of the category of flows and of the category of multipointed d-spaces make the globes oscillate by keeping the set of states constant. All objects of the model categories of flows and multipointed d-spaces are fibrant. It can be checked that the cylinders of these model categories satisfy the homotopy exchange property introduced by Lafont-Métayer-Worytkiewicz in their work about globular omega-categories. The cylinders of the category of cubical transition systems and of labelled symmetric precubical sets make the cubes oscillate by keeping the set of states constant as well. These latter model category structures are constructed using M. Olschok's PhD which generalizes Cisinski's work on the homotopy theory of toposes. In these latter model category structures,