Directed Algebraic Topology

This is the first authored book to be dedicated to the new field of directed algebraic topology that arose in the 1990s, in homotopy theory and in the theory of concurrent processes. Its general aim can be stated as modelling non-reversible phenomena and its domain should be distinguished from that of classical algebraic topology by the principle that directed spaces have privileged directions and directed paths therein need not be reversible. Its homotopical tools (corresponding in the classical case to ordinary homotopies, fundamental group and fundamental groupoid) should be similarly non-reversible: directed homotopies, fundamental monoid and fundamental category. Homotopy constructions occur here in a directed version, which gives rise to new shapes, like directed cones and directed spheres. Applications will deal with domains where privileged directions appear, including rewrite systems, traffic networks and biological systems. The most developed examples can be found in the area of concurrency.

Spis treści:
Introduction
Part I. First Order Directed Homotopy and Homology
1. Directed structures and first order homotopy properties
2. Directed homology and noncommutative geometry
3. Modelling the fundamental category
Part II. Higher Directed Homotopy Theory
4. Settings for higher order homotopy
5. Categories of functors and algebras, relative settings
6. Elements of weighted algebraic topology
Appendix A. Some points of category theory
References
Index of symbols
General index.