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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,
RELATIVE DIRECTED HOMOTOPY THEORY OF PARTIALLY ORDERED SPACES
"... Algebraic topological methods have been used successfully in concurrency theory, the domain of theoretical computer science that deals with parallel computing. L. Fajstrup, E. Goubault, and M. Raussen have introduced partially ordered spaces (pospaces) as a model for concurrent systems. In this pape ..."
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Algebraic topological methods have been used successfully in concurrency theory, the domain of theoretical computer science that deals with parallel computing. L. Fajstrup, E. Goubault, and M. Raussen have introduced partially ordered spaces (pospaces) as a model for concurrent systems. In this paper it is shown that the category of pospaces under a fixed pospace is both a fibration and a cofibration category in the sense of H. Baues. The homotopy notion in this fibration and cofibration category is relative directed homotopy. It is also
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,
unknown title
, 2008
"... Covers of a (nice) topological space X are classified by the fundamental group π1X or, if we wish to avoid assuming that X is connected and has a basepoint, ..."
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Covers of a (nice) topological space X are classified by the fundamental group π1X or, if we wish to avoid assuming that X is connected and has a basepoint,
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
KRZYSZTOF WORYTKIEWICZ
, 808
"... Abstract. We study the homotopy theory of locally ordered spaces, that is manifolds with boundary whose charts are partially ordered in a compatible way. Their category is not particularly wellbehaved with respect to colimits. However, this category turns out to be a certain full subcategory of a t ..."
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Abstract. We study the homotopy theory of locally ordered spaces, that is manifolds with boundary whose charts are partially ordered in a compatible way. Their category is not particularly wellbehaved with respect to colimits. However, this category turns out to be a certain full subcategory of a topos of sheaves over a simpler site. A precise characterisation of this subcategory is provided. The ambient topos makes available some general homotopical machinery. 1.