We establish an equivalence between directed homotopy categories of (diagrams of)
cubical sets and (diagrams of) directed topological spaces. 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
-cohomology
monoids. Another application is a characterization of isomorphisms between small
categories up to zigzags 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, we characterize morphisms in this variant
as the interval-preserving lattice homomorphisms between finite Boolean
lattices.
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