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QUOTIENT MODELS OF A CATEGORY UP TO DIRECTED HOMOTOPY Abstract.
"... Directed Algebraic Topology is a recent field, deeply linked with ordinary and higher dimensional Category Theory. A ‘directed space’, e.g. an ordered topological space, has directed homotopies (which are generally non reversible) and a fundamental category (replacing the fundamental groupoid of the ..."
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Directed Algebraic Topology is a recent field, deeply linked with ordinary and higher dimensional Category Theory. A ‘directed space’, e.g. an ordered topological space, has directed homotopies (which are generally non reversible) and a fundamental category (replacing the fundamental groupoid of the classical case). Finding a simple possibly finite model of the latter is a nontrivial problem, whose solution gives relevant information on the given ‘space’; a problem which is of interest for applications as well as in general Category Theory. Here we continue the work “The shape of a category up to directed homotopy”, with a deeper analysis of ‘surjective models’, motivated by studying the singularities of 3dimensional ordered spaces.
COVERING SPACE THEORY FOR DIRECTED TOPOLOGY
, 812
"... Abstract. The state space of a machine admits the structure of time. For example, the geometric realization of a precubical set, a generalization of an unlabeled asynchronous transition system, admits a “local preorder ” encoding control flow. In the case where time does not loop, the “locally preor ..."
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Abstract. The state space of a machine admits the structure of time. For example, the geometric realization of a precubical set, a generalization of an unlabeled asynchronous transition system, admits a “local preorder ” encoding control flow. In the case where time does not loop, the “locally preordered” state space splits into causally distinct components. The set of such components often gives a computable invariant of machine behavior. In the general case, no such meaningful partition could exist. However, as we show in this note, the locally preordered geometric realization of a precubical set admits a “locally monotone ” covering from a state space in which time does not loop. Thus we hope to extend geometric techniques in static program analysis to looping processes. 1.
MODELS AND VAN KAMPEN THEOREMS FOR DIRECTED HOMOTOPY THEORY
, 810
"... Abstract. We study topological spaces with a distinguished set of paths, called directed paths. Since these directed paths are generally not reversible, the directed homotopy classes of directed paths do not assemble into a groupoid, and there is no direct analog of the fundamental group. However, t ..."
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Abstract. We study topological spaces with a distinguished set of paths, called directed paths. Since these directed paths are generally not reversible, the directed homotopy classes of directed paths do not assemble into a groupoid, and there is no direct analog of the fundamental group. However, they do assemble into a category,
COVERING SPACE THEORY FOR DIRECTED TOPOLOGY
"... Abstract. The state space of a machine admits the structure of time. For example, the geometric realization of a precubical set, a generalization of an unlabeled asynchronous transition system, admits a “local preorder ” encoding control flow. In the case where time does not loop, the “locally preor ..."
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Abstract. The state space of a machine admits the structure of time. For example, the geometric realization of a precubical set, a generalization of an unlabeled asynchronous transition system, admits a “local preorder ” encoding control flow. In the case where time does not loop, the “locally preordered ” state space splits into causally distinct components. The set of such components often gives a computable invariant of machine behavior. In the general case, no such meaningful partition could exist. However, as we show in this note, the locally preordered geometric realization of a precubical set admits a “locally monotone ” covering from a state space in which time does not loop. Thus we hope to extend geometric techniques in static program analysis to looping processes. 1.
A FEW POINTS ON DIRECTED ALGEBRAIC TOPOLOGY by Marco GRANDIS ( *)
"... Dedicated to Charles Ehresmann, on the centennial of his birth Abstract. Directed Algebraic Topology is a recent field, deeply linked with Category Theory. A 'directed space ' has directed homotopies (generally non reversible), directed homology groups (enriched with a preorder) and fundam ..."
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Dedicated to Charles Ehresmann, on the centennial of his birth Abstract. Directed Algebraic Topology is a recent field, deeply linked with Category Theory. A 'directed space ' has directed homotopies (generally non reversible), directed homology groups (enriched with a preorder) and fundamental ncategories (replacing the fundamental ngroupoids of the classical case). Applications have been mostly developed in the theory of concurrency. Unexpected links with noncommutative geometry and the modelling of biological systems have emerged.
ABSTRACT. ABSOLUTE LAX 2CATEGORIES
"... We have introduced, in a previous paper, the fundamental lax 2category of a ‘directed space ’ X. Here we show that, when X has a T1topology, this structure can be embedded into a larger one, with the same objects (the points of X), the same arrows (the directed paths) and the same cells (based on ..."
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We have introduced, in a previous paper, the fundamental lax 2category of a ‘directed space ’ X. Here we show that, when X has a T1topology, this structure can be embedded into a larger one, with the same objects (the points of X), the same arrows (the directed paths) and the same cells (based on directed homotopies of paths), but a larger system of comparison cells. The new comparison cells are absolute, in the sense that they only depend on the arrows themselves rather than on their syntactic expression, as in the usual settings of lax or weak structures. It follows that, in the original structure, all the diagrams of comparison cells commute, even if not constructed in a natural way and even if the composed cells need not stay within the old system.
DIRECTED ALGEBRAIC TOPOLOGY, CATEGORIES AND HIGHER CATEGORIES
"... Abstract. Directed Algebraic Topology is a recent field, deeply linked with Category Theory. A ‘directed space ’ has directed homotopies (generally non reversible), directed homology groups (enriched with a preorder) and fundamental ncategories (replacing the fundamental ngroupoids of the classical ..."
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Abstract. Directed Algebraic Topology is a recent field, deeply linked with Category Theory. A ‘directed space ’ has directed homotopies (generally non reversible), directed homology groups (enriched with a preorder) and fundamental ncategories (replacing the fundamental ngroupoids of the classical case). On the other hand, directed homotopy can give geometric models for lax higher categories. Applications have been mostly developed in the theory of concurrency. Unexpected links with noncommutative geometry and the modelling of biological systems have emerged. 1.
COVERING
"... Abstract. The state space of a machine admits the structure of time. For example, the geometric realization of a precubical set, a generalization of an unlabeled asynchronous transition system, admits a “local preorder ” encoding control flow. In the case where time does not loop, the “locally preor ..."
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Abstract. The state space of a machine admits the structure of time. For example, the geometric realization of a precubical set, a generalization of an unlabeled asynchronous transition system, admits a “local preorder ” encoding control flow. In the case where time does not loop, the “locally preordered ” state space splits into causally distinct components. The set of such components often gives a computable invariant of machine behavior. In the general case, no such meaningful partition could exist. However, as we show in this note, the locally preordered geometric realization of a precubical set admits a “locally monotone ” covering from a state space in which time does not loop. Thus we hope to extend geometric techniques in static program analysis to looping processes. 1.
Abstract. KAN EXTENSIONS IN DOUBLE CATEGORIES (ON WEAK DOUBLE CATEGORIES, PART III)
"... are closely related to the orthogonal adjunctions introduced in a previous paper. The pointwise case is treated by introducing internal comma objects, which can be defined in an arbitrary double category. ..."
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are closely related to the orthogonal adjunctions introduced in a previous paper. The pointwise case is treated by introducing internal comma objects, which can be defined in an arbitrary double category.