Abstract
The geometric models of concurrency  Dijkstra's PVmodels 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 dhomotopy (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 postructure 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 postructure.
BibTeX  Entry
@InProceedings{fajstrup:DSP:2005:132,
author = {Lisbeth Fajstrup},
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},
year = {2005},
editor = {Ralph Kopperman and Michael B. Smyth and Dieter Spreen and Julian Webster},
number = {04351},
series = {Dagstuhl Seminar Proceedings},
ISSN = {18624405},
publisher = {Internationales Begegnungs und Forschungszentrum f{\"u}r Informatik (IBFI), Schloss Dagstuhl, Germany},
address = {Dagstuhl, Germany},
URL = {http://drops.dagstuhl.de/opus/volltexte/2005/132},
annote = {Keywords: Cubical Complex , Higher Dimensional Automaton , Ditopology}
}
Keywords: 

Cubical Complex , Higher Dimensional Automaton , Ditopology 
Seminar: 

04351  Spatial Representation: Discrete vs. Continuous Computational Models 
Issue Date: 

2005 
Date of publication: 

22.04.2005 