×

Models and van Kampen theorems for directed homotopy theory. (English) Zbl 1163.55007

The author works with topological spaces equipped with a distinguished set of continuous paths called directed path satisfying natural axioms (Grandis’ notion of \(d\)-space). This kind of objects is useful in the study of concurrency theory using topological models. The directed paths model the execution paths of a concurrent process and the underlying state space is modelled by the underlying topological space. Since these paths are generally not reversible, the directed homotopy classes of directed paths do not assemble into a groupoid, but rather into a small category called the fundamental category. The author studies various full subcategories of the fundamental category, as the fundamental bipartite graph which is the full subcategory generated by the extremal points, i.e the points without past or without future. This full subcategory as well as minimal extremal models are shown to generalize the fundamental group. Various Van Kampen theorems for subcategories, retracts, and models of the fundamental category are also proved. These theorems generalize the Van Kampen theorem proved by Grandis for fundamental categories in [M. Grandis, Cah. Topol. Géom. Différ. Catég. 44, No. 4, 281–316 (2003; Zbl 1059.55009)].

MSC:

55P99 Homotopy theory
68Q85 Models and methods for concurrent and distributed computing (process algebras, bisimulation, transition nets, etc.)
55U99 Applied homological algebra and category theory in algebraic topology

Citations:

Zbl 1059.55009
PDFBibTeX XMLCite
Full Text: DOI arXiv Link