Dihomotopy Classes of Dipaths in the Geometric Realization of a Cubical Set: from Discrete to Continuous and back again

Author Lisbeth Fajstrup



PDF
Thumbnail PDF

File

DagSemProc.04351.13.pdf
  • Filesize: 79 kB
  • 3 pages

Document Identifiers

Author Details

Lisbeth Fajstrup

Cite AsGet BibTex

Lisbeth Fajstrup. Dihomotopy Classes of Dipaths in the Geometric Realization of a Cubical Set: from Discrete to Continuous and back again. In Spatial Representation: Discrete vs. Continuous Computational Models. Dagstuhl Seminar Proceedings, Volume 4351, pp. 1-3, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2005)
https://doi.org/10.4230/DagSemProc.04351.13

Abstract

The geometric models of concurrency - Dijkstra's PV-models and V. Pratt's Higher Dimensional Automata - rely on a translation of discrete or algebraic information to geometry. In both these cases, the translation is the geometric realisation of a semi cubical complex, which is then a locally partially ordered space, an lpo space. The aim is to use the algebraic topology machinery, suitably adapted to the fact that there is a preferred time direction. Then the results - for instance dihomotopy classes of dipaths, which model the number of inequivalent computations should be used on the discrete model and give the corresponding discrete objects. We prove that this is in fact the case for the models considered: Each dipath is dihomottopic to a combinatorial dipath and if two combinatorial dipaths are dihomotopic, then they are combinatorially equivalent. Moreover, the notions of dihomotopy (LF., E. Goubault, M. Raussen) and d-homotopy (M. Grandis) are proven to be equivalent for these models - hence the Van Kampen theorem is available for dihomotopy. Finally we give an idea of how many spaces have a local po-structure given by cubes. The answer is, that any cubicalized space has such a structure after at most one subdivision. In particular, all triangulable spaces have a cubical local po-structure.
Keywords
  • Cubical Complex
  • Higher Dimensional Automaton
  • Ditopology

Metrics

  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    0
    PDF Downloads
Questions / Remarks / Feedback
X

Feedback for Dagstuhl Publishing


Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail