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The paper establishes an equivalence between localizations of (diagrams of)cubical sets and (diagrams of) directed topological spaces by those mapsdefining (natural) cubical homotopy equivalences after application of thedirected singular functor and a directed analogue of fibrant replacement. Thisequivalence both lifts and extends an equivalence between classical homotopycategories of cubical sets and topological spaces. Some simple applicationsinclude combinatorial descriptions and subsequent calculations of directedhomotopy monoids and directed singular 1-cohomology monoids. Anotherapplication is a characterization of isomorphisms between small categories upto zig-zags of natural transformations as directed homotopy equivalencesbetween directed classifying spaces. Cubical sets throughout the paper aretaken to mean presheaves over the minimal symmetric monoidal variant of thecube category. Along the way, the paper characterizes morphisms in this variantas the interval-preserving lattice homomorphisms between finite Boolean latticeand describes some of the test model structure on presheaves over this variant.

Cubical Approximation for Directed Topology II