Invariance of directed spaces and persistence. Lecture #13178. Workshop on Topological Methods in Combinatorics, Computational Geometry and the Study of Algorithms. Venue: Mathematical Sciences Research Instutue, Berkeley, California, USA. Time: October 05, 2006 - 10:30 AM to 12:00 PM

With motivations arising from concurrency theory within Computer Science, a new field of research, directed algebraic topology, has emerged. The main characteristic is, that it involves spaces of "directed paths'' (or timed paths, executions) in a "directed space''; these directed paths can be concatenated, but in general /not/ reversed; time is not reversable. The combinatorics of spaces of directed paths with fixed endpoints can be studied via (homotopy or homology) functors from the preorder category of the directed space to a category like /Ho-Top/ or /Ab/; this point of view allows to generalize previous work on the fundamental category of such a space. Birth and death of homology classes indicate structural changes at various levels. We give a definition of a directed homotopy equivalence (which is not the obvious generalization of the classical notion), and we describe "smaller" models for the homotopy and homology functors mentioned above.