Homology, Homotopy and Applications

Volume 11 (2009)

Number 1

Models and van Kampen theorems for directed homotopy theory

Pages: 185 – 202

DOI: https://dx.doi.org/10.4310/HHA.2009.v11.n1.a9


Peter Bubenik (Department of Mathematics, Cleveland State University, Cleveland, Ohio, U.S.A.)


We study topological spaces with a distinguished set of paths, called directed paths. Since these directed paths are generally not reversible, the directed homotopy classes of directed paths do not assemble into a groupoid, and there is no direct analog of the fundamental group. However, they do assemble into a category, called the fundamental category. We define models of the fundamental category, such as the fundamental bipartite graph, and minimal extremal models which are shown to generalize the fundamental group. In addition, we prove van Kampen theorems for subcategories, retracts, and models of the fundamental category.


directed homotopy, fundamental category, van Kampen theorem, $d$-space, reflective subcategory, coreflective subcategory, past retract, future retract, extremal model, fundamental bipartite graph

2010 Mathematics Subject Classification

Primary 55Pxx, 68Q85. Secondary 18A30, 18A40, 55Uxx.

Published 1 January 2009