Cubical Approximation for Directed Topology I

Abstract

Topological spaces—such as classifying spaces, configuration spaces and spacetimes—often admit 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 order-theoretic structure encoding the orientations of simplices and 1-cubes. We prove dual simplicial and cubical approximation theorems appropriate for the directed setting and give criteria for two different homotopy relations on directed maps in the literature to coincide.