  The paper establishes an equivalence between localizations of (diagrams of)
cubical sets and (diagrams of) directed topological spaces by those maps
defining (natural) cubical homotopy equivalences after application of the
directed singular functor and a directed analogue of fibrant replacement. This
equivalence both lifts and extends an equivalence between classical homotopy
categories of cubical sets and topological spaces. Some simple applications
include combinatorial descriptions and subsequent calculations of directed
homotopy monoids and directed singular 1-cohomology monoids. Another
application is a characterization of isomorphisms between small categories up
to zig-zags of natural transformations as directed homotopy equivalences
between directed classifying spaces. Cubical sets throughout the paper are
taken to mean presheaves over the minimal symmetric monoidal variant of the
cube category. Along the way, the paper characterizes morphisms in this variant
as the interval-preserving lattice homomorphisms between finite Boolean lattice
and describes some of the test model structure on presheaves over this variant.
