Cubical Approximation for Directed Topology

Topological spaces such as classifying spaces, configuration spaces and spacetimes often admit extra directionality. Qualitative invariants on such directed spaces often are more informative, yet more difficult, to calculate than classical homotopy invariants because directed spaces rarely decompose as homotopy colimits of simpler directed spaces. Directed spaces often arise as geometric realizations of simplicial sets and cubical sets equipped with ordertheoretic structure encoding the orientations of simplices and 1-cubes. We show that, under definitions of weak equivalences appropriate for the directed setting, geometric realization induces an equivalence between homotopy diagram categories of cubical sets and directed spaces and that its right adjoint satisfies a homotopical analogue of excision. In our directed setting, cubical sets with structure reminiscent of higher categories serve as analogues of Kan complexes. Along the way, we prove appropriate simplicial and cubical approximation theorems and give criteria for two different homotopy relations on directed maps in the literature to coincide.