منابع مشابه
Directed Homology
We introduce a new notion of directed homology for cubical sets with connections and transpositions. We show that it respects directed homotopy and is functorial, and that it reduces to the usual homology in case of reversible cubical sets. However it has an undesired cancellation property, which we propose to remedy by refining the dihomology relation.
متن کاملDirected combinatorial homology and noncommutative geometry
We will present 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 cubical sets can also express topo...
متن کاملInequilogical spaces, directed homology and noncommutative geometry (*)
We introduce a preordered version of D. Scott's equilogical spaces [Sc], called inequilogical spaces, as a possible setting for Directed Algebraic Topology. The new structure can also express 'formal quotients' of spaces, which are not topological spaces and are of interest in noncommutative geometry, with finer results than the ones obtained with equilogical spaces, in a previous paper. This s...
متن کاملDirected Homology Theories and Eilenberg-Steenrod Axioms
In this paper, we define and study a homology theory, that we call “natural homology”, which associates a natural system of abelian groups to every space in a large class of directed spaces and precubical sets. We show that this homology theory enjoys many important properties, as an invariant for directed homotopy. Among its properties, we show that subdivided precubical sets have the same hom...
متن کاملPersistent Homology Over Directed Acyclic Graphs
We define persistent homology groups over any set of spaces which have inclusions defined so that the underlying graph between the spaces is directed and acyclic. This method simultaneously generalizes standard persistent homology, zigzag persistence and multidimensional persistence to arbitrary directed acyclic graphs, and it also allows the study of arbitrary families of topological spaces or...
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ژورنال
عنوان ژورنال: Electronic Notes in Theoretical Computer Science
سال: 2004
ISSN: 1571-0661
DOI: 10.1016/j.entcs.2004.09.005