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# Homology, Homotopy and Applications

## Volume 5 (2003)

### Number 2

### Volume of a Workshop at Stanford University

### Some geometric perspectives in concurrency theory

Pages: 95 – 136

DOI: http://dx.doi.org/10.4310/HHA.2003.v5.n2.a5

#### Author

#### Abstract

Concurrency, i.e., the domain in computer science which deals with parallel (asynchronous) computations, has very strong links with algebraic topology; this is what we are developing in this paper, giving a survey of “geometric” models for concurrency. We show that the properties we want to prove on concurrent systems are stable under some form of deformation, which is almost homotopy. In fact, as the “direction” of time matters, we have to allow deformation only as long as we do not reverse the direction of time. This calls for a new homotopy theory: “directed” or di-homotopy. We develop some of the geometric intuition behind this theory and give some hints about the algebraic objects one can associate with it (in particular homology groups). For some historic as well as for some deeper reasons, the theory is at a stage where there is a nice blend between cubical, $\omega$-categorical and topological techniques.

#### Keywords

homology, homotopy, concurrency, cubical sets, di-homotopy

#### 2010 Mathematics Subject Classification

18F20, 54F05, 55P15, 55U10, 68Q55, 68Q60, 68Q85