Directed lifting properties

A directed space is a geometric object for which certain paths are singled out as the directed paths. A circle with only paths running clockwise is a simple example. Such geometries are used as models of relativity and of concurrent computing. In the latter case, directed paths are executions of programs and direction is given by local time.

In algebraic topology, spaces are studied and compared via maps between them - and via the properties of these maps. Lifting properties are examples of such properties: Given a map f:Y -> X, is every path in X the image of a path in Y; if so, is the correspondence unique - given y in Y does each path in X starting at f(y) correspond to precisely one path in Y starting at y; via the map f. These properties are called path lifting and unique path lifting. Other lifting properties can be seen as a correspondence between interesting geometric shapes other than paths in X and in Y via the map f. Lifting properties are a tool for calculations of invariants - knowing invariants of Y and calculating invariants of X and vice versa.

In the directed setting, we encounter many complications compared to the well known non-directed cases. In particular, lifting properties for directed paths do not imply liftings of other geometric shapes. The topology of X and Y may be very different even if all directed paths lift. Hence it is essential to apply methods of modern category theory in order to arrive at results corresponding to the non-directed setting.

In concurrency, lifting of directed paths implies a comparison of executions in two programs, hence a map with sufficient directed lifting properties can be considered an equivalence of programs.

The aim of this project is to study lifting properties in the directed setting; and to compare directed invariants via maps with lifting properties. Moreover, we aim at classifying all maps f:Y -> X satisfying given lifting properties for a given space X.