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A convenient category of locally preordered spaces
 Applied Categorical Structures
, 2008
"... Abstract. As a practical foundation for a homotopy theory of abstract spacetime, ..."
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Abstract. As a practical foundation for a homotopy theory of abstract spacetime,
Context for models of concurrency
 in Preliminary Proceedings of the Workshop on Geometry and Topology in Concurrency and Distributed Computing GETCO 2004, vol NS042 of BRICS Notes
, 2004
"... Abstract. Many categories have been used to model concurrency. Using any of these, the challenge is to reduce a given model to a smaller representation which nevertheless preserves the relevant computerscientific information. That is, one wants to replace a given model with a simpler model with the ..."
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Abstract. Many categories have been used to model concurrency. Using any of these, the challenge is to reduce a given model to a smaller representation which nevertheless preserves the relevant computerscientific information. That is, one wants to replace a given model with a simpler model with the same directed homotopytype. Unfortunately, the obvious definition of directed homotopy equivalence is too coarse. This paper introduces the notion of context to refine this definition. 1.
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
"... 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