Abstract
In this paper, we consider 2-dimensional precubical sets, which can be used to model systems of two concurrently executing processes. From the point of view of concurrency theory, two precubical sets can be considered equivalent if their geometric realizations have the same directed homotopy type relative to the extremal elements in the sense of P. Bubenik. We give easily verifiable conditions under which it is possible to reduce a 2-dimensional precubical set to an equivalent smaller one by collapsing an edge or eliminating a square and one or two free faces. We also look at some simple standard examples in order to illustrate how our results can be used to construct small models of 2-dimensional precubical sets.
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Communicated by Martin Raussen.
This research has been supported by FEDER funds through “Programa Operacional Factores de Competitividade-COMPETE” and by FCT-Fundação para a Ciência e a Tecnologia through projects Est-C/MAT/UI0013/2011 and PTDC/MAT/0938317/2008.
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Kahl, T. Some collapsing operations for 2-dimensional precubical sets. J. Homotopy Relat. Struct. 7, 281–298 (2012). https://doi.org/10.1007/s40062-012-0010-7
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DOI: https://doi.org/10.1007/s40062-012-0010-7
Keywords
- Cubical sets
- d-spaces
- Fundamental bipartite graph
- Fundamental category
- Trace spaces
- Directed homotopy theory
- Concurrency theory