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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 cylindre et(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 2. The link with existing notions in terms of morphism classes is given via the(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)
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)
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 cate-gory', where we introduced directed spaces, their non reversible homotopies and their fundamental category.(More)
On introduit une théorie d'homotopie combinatoire pour le topos des ensembles simpliciaux symétriques (préfaisceaux sur les cardinaux finis positifs), en étendant une théorie développée pour les complexes simpliciaux [11]; comme avantage essentiel de cette extension, le groupoïde fondamental devient l'adjoint à gauche d'un foncteur nerf symétrique et(More)
The category of finite cardinals (or, equivalently, of finite sets) is the symmetric analogue of the category of finite ordinals, and the ground category of a relevant category of presheaves, the augmented symmetric simplicial sets. We prove here that this ground category has characterisations similar to the classical ones for the category of finite(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)