Marco Grandis

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Directed Algebraic Topology studies phenomena where privileged directions appear, derived from the analysis of concurrency, traffic networks, space-time models, etc. This is the sequel of a paper, ‘Directed homotopy theory, I. The fundamental category’, where we introduced directed spaces, their non reversible homotopies and their fundamental category. Here(More)
Résumé. La Topologie Algébrique Dirigée est en train d'émerger, à partir de plusieurs applications. La structure de base que nous étudions ici, un espace dirigé ou d-éspace, est un éspace topologique muni d'une famille convenable de chemins dirigés. Dans ce cadre, les homotopies dirigées, généralement non réversibles, sont répresentées par des foncteurs(More)
Extended cubical sets (with connections and interchanges) are presheaves on a ground category, the extended cubical site K, corresponding to the (augmented) simplicial site, the category of finite ordinals. We prove here that K has characterisations similar to the classical ones for the simplicial analogue, by generators and relations, or by the existence(More)
In order to facilitate a natural choice for morphisms created by the (left or right) lifting property as used in the definition of weak factorization systems, the notion of natural weak factorization system in the category K is introduced, as a pair (comonad, monad) over K. The link with existing notions in terms of morphism classes is given via the(More)
This work is a contribution to a recent field, Directed Algebraic Topology. Categories which appear as fundamental categories of ‘directed structures’, e.g. ordered topological spaces, have to be studied up to appropriate notions of directed homotopy equivalence, which are more general than ordinary equivalence of categories. Here we introduce past and(More)
Cubical sets have a directed homology, studied in a previous paper and consisting of preordered abelian groups, with a positive cone generated by the structural cubes. By this additional information, cubical sets can provide a sort of 'noncommutative topology', agreeing with some results of noncommutative geometry but lacking the metric aspects of(More)
This is a brief study of the homology of cubical sets, with two main purposes. First, this combinatorial structure is viewed as representing directed spaces, breaking the intrinsic symmetries of topological spaces. Cubical sets have a directed homology, consisting of preordered abelian groups where the positive cone comes from the structural cubes. But(More)
A reparametrization (of a continuous path) is given by a surjective weakly increasing self-map of the unit interval. We show that the monoid of reparametrizations (with respect to compositions) can be understood via “stop-maps” that allow to investigate compositions and factorizations, and we compare it to the distributive lattice of countable subsets of(More)