Spaces of directed paths and traces in directed algebraic topology

 

In directed algebraic topology, one studies topological spaces with fewer admissible "directed" paths. In particular, it is usually not possible to invert a directed path.  Execution paths in a model of concurrent computation and causal curves in relativity theory occur among the intended applications.

In ordinary algebraic topology, properties of paths and of path spaces are essentially equivalent to those of loops (start and end point coincide) and loop spaces, at least within a given path component. In directed algebraic topology, it is necessary to consider spaces of directed paths from any start point to any end point and to investigate whether and how much these path spaces vary under variation of the two end points. It is often advisable to work with traces instead of directed paths: Two paths represent the same trace if they only differ up to increasing reparametrizations.

If the directed topological space considered has a combinatorial model, like a pre-cubical complex glued out of (hyper-)cubes of  various dimensions, then the trace spaces in question are not too complicated: They are locally compact, locally contractible and metrizable, and they have the homotopy type of a CW-complex.
 
It is the aim of the project to provide systematic information about algebraic topological invariants (e.g. homology) of these trace spaces using suitable decompositions of the original space (or rather of the pairs of points in that space). One can hope for a best possible decomposition of that space into "components" with the following properties: If start and end point vary only within a component, then trace spaces are essentially (homotopy) equivalent.  In many cases it is possible to use combinatorial and order information to give an essentially finite description of the path spaces involved as a simplicial complex. Using that complex, one may calculate algebraic topological invariants of paths spaces, e.g., their homology groups. For applications, it is already important to know the path components (corresponding to equivalence classes of executions) of these trace spaces.  Actual computations with realistic execution spaces can be huge and have to be performed on a computer.

(Rev. 01.01.2010)