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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.
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.
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.
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.
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
, 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.
RESEARCH SUMMARY
"... I am active in three areas of research: computational algebraic topology and data analysis, directed homotopy theory and concurrent computing, and homotopy theory, differential graded algebra and toric topology. Together with my collaborator Peter T. Kim, I am combining topological and statistical m ..."
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I am active in three areas of research: computational algebraic topology and data analysis, directed homotopy theory and concurrent computing, and homotopy theory, differential graded algebra and toric topology. Together with my collaborator Peter T. Kim, I am combining topological and statistical methods to aid practitioners in analyzing large, highdimensional data sets [11, 7]. Independently and with various collaborators I am developing a directed version of homotopy theory for the purpose of modeling concurrent (parallel) computing [13, 6, 3, 9, 12]. My research background is in homotopy theory, in which I have made contributions to the classical question of how the attachment of a cell affects invariants such as the loop space homology [2] and the homotopy Lie algebra [4]. Currently I am working with Leah Gold to combine these topological techniques with algebraic techniques to the new field of toric topology [8]. In addition, I have published work with John Holbrook on constructing statistical samples [10], I have provided analytic support for George Bubenik’s work on Moose hearing [1], and I have contributed to Zhiming Luo’s work in homotopical algebra [29]. 1. Computational Algebraic Topology and Data Analysis