The directed homotopy hypothesis

The homotopy hypothesis was originally stated by Grothendieck [13]: topological spaces should be "equivalent" to (weak) 1-groupoids, which give algebraic representatives of homotopy types. Much later, several authors developed geometrizations of computational models, e.g. for rewriting, distributed systems, (homotopy) type theory etc. But an essential feature in the work set up in concurrency theory, is that time should be considered irreversible, giving rise to the field of directed algebraic topology. Following the path proposed by Porter, we state here a directed homotopy hypothesis: Grandis' directed topological spaces should be "equivalent" to a weak form of topologically enriched categories, still very close to (1,1)-categories. We develop, as in ordinary algebraic topology, a directed homotopy equivalence and a weak equivalence, and show invariance of a form of directed homology.