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# Homology, Homotopy and Applications

## Volume 10 (2008)

### Number 1

### Towards a homotopy theory of process algebra

Pages: 353 – 388

DOI: https://dx.doi.org/10.4310/HHA.2008.v10.n1.a16

#### Author

#### Abstract

This paper proves that labelled flows are expressive enough to contain all process algebras which are a standard model for concurrency. More precisely, we construct the space of execution paths and of higher dimensional homotopies between them for every process name of every process algebra with any synchronization algebra using a notion of labelled flow. This interpretation of process algebra satisfies the paradigm of higher dimensional automata (HDA): one non-degenerate full $n$-dimensional cube (no more no less) in the underlying space of the time flow corresponding to the concurrent execution of $n$ actions. This result will enable us in future papers to develop a homotopical approach of process algebras. Indeed, several homological constructions related to the causal structure of time flow are possible only in the framework of flows.

#### Keywords

homotopy, model category, precubical set, concurrency, process algebra

#### 2010 Mathematics Subject Classification

55Pxx, 68Q85

Published 1 January 2008