## Homology, Homotopy and Applications

- Homology Homotopy Appl.
- Volume 11, Number 1 (2009), 185-202.

### Models and van Kampen theorems for directed homotopy theory

#### Abstract

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.

#### Article information

**Source**

Homology Homotopy Appl., Volume 11, Number 1 (2009), 185-202.

**Dates**

First available in Project Euclid: 1 September 2009

**Permanent link to this document**

https://projecteuclid.org/euclid.hha/1251832565

**Mathematical Reviews number (MathSciNet)**

MR2506132

**Zentralblatt MATH identifier**

1163.55007

**Subjects**

Primary: 55P99: None of the above, but in this section 68Q85: Models and methods for concurrent and distributed computing (process algebras, bisimulation, transition nets, etc.)

Secondary: 18A40: Adjoint functors (universal constructions, reflective subcategories, Kan extensions, etc.) 18A30: Limits and colimits (products, sums, directed limits, pushouts, fiber products, equalizers, kernels, ends and coends, etc.) 55U99: None of the above, but in this section

**Keywords**

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

#### Citation

Bubenik, Peter. Models and van Kampen theorems for directed homotopy theory. Homology Homotopy Appl. 11 (2009), no. 1, 185--202. https://projecteuclid.org/euclid.hha/1251832565