  Topological spaces - such as classifying spaces, configuration spaces and
spacetimes - often admit extra temporal structure. Qualitative invariants on
such directed spaces often are more informative yet more difficult to calculate
than classical homotopy invariants on underlying spaces 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 temporal structure encoding the orientations of simplices
and 1-cubes. In an attempt to develop calculational tools for directed homotopy
theory, we prove appropriate simplicial and cubical approximation theorems. We
consequently show that geometric realization induces an equivalence between
weak homotopy diagram categories of cubical sets and directed spaces and that
its right adjoint satisfies an excision theorem. Along the way, we give
criteria for two different homotopy relations on directed maps in the
literature to coincide.
