License: Creative Commons Attribution 4.0 International license (CC BY 4.0)
When quoting this document, please refer to the following
DOI: 10.4230/DagSemProc.04351.13
URN: urn:nbn:de:0030-drops-1328
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2005/132/
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Fajstrup, Lisbeth

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

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04351.FajstrupLisbeth.ExtAbstract.132.pdf (0.08 MB)

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.

BibTeX - Entry

@InProceedings{fajstrup:DagSemProc.04351.13,
  author =	{Fajstrup, Lisbeth},
  title =	{{Dihomotopy Classes of Dipaths in the Geometric Realization of a Cubical Set: from Discrete to Continuous and back again}},
  booktitle =	{Spatial Representation: Discrete vs. Continuous Computational Models},
  pages =	{1--3},
  series =	{Dagstuhl Seminar Proceedings (DagSemProc)},
  ISSN =	{1862-4405},
  year =	{2005},
  volume =	{4351},
  editor =	{Ralph Kopperman and Michael B. Smyth and Dieter Spreen and Julian Webster},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/opus/volltexte/2005/132},
  URN =		{urn:nbn:de:0030-drops-1328},
  doi =		{10.4230/DagSemProc.04351.13},
  annote =	{Keywords: Cubical Complex , Higher Dimensional Automaton , Ditopology}
}

Keywords: Cubical Complex , Higher Dimensional Automaton , Ditopology
Collection: 04351 - Spatial Representation: Discrete vs. Continuous Computational Models
Issue Date: 2005
Date of publication: 22.04.2005

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