The deduction of qualitative properties of a complex system from the calculation of homotopy invariants on an associated state space dates back to the origins of modern algebraic topology. However, state spaces typically have some directed structure finer than mere topology. While generalizations of (co)homology, fundamental groups, and higher homotopy capture some of the directed structure in a directed space, all but the latter have been explicitly studied in the literature.

In this talk, we construct directed homotopy monoids, discuss their many uses (solving word problems on monoids, classifying higher dimensional flows, detecting obstructions to causality in a spacetime), give a higher categorical interpretation of them, and give a subsequent calculation of directed homotopy monoids of directed spheres. We then speculate/sketch some potential applications in dynamics and classical homotopy theory.