all objects are cofibrant. Thomas Kahl proved the existence of a nontrivial model category of pospaces. Yet this structure barely differs from the model structure over topological spaces. In many regards it just consists of forgetting the partial order of the objects. Krzysztof Worytkiewicz uses advanced methods from model category theory (namely localization and completion) to build a model category from the small categories of finite-dimensional directed hypercubes. In fact any attempt to define a model structure over some category of directed spaces has to face the following question: should an inclusion map be a cofibration, a weak equivalence, both (trivial cofibration) or none. For example, if we suppose is a trivial cofibration, then (as a subpospace of the directed plane) is equivalent to a point since the collection of trivial cofibrations is stable under pushout. This fact is prohibitive for computer science application though it is a trivial fact from homotopy theory if we drop the direction feature. Directed coverings ... Software ... References Further reading Directed Algebraic Topology and Concurrency, Lisbeth Fajstrup, Éric Goubault, Samuel Mimram, Emmanuel Haucourt, Martin Raussen Directed homotopy theory, II. Homotopy Constructs, Marco Grandis, Theory and Applications of Categories, Vol. 10, No. 14, 2002, pp. 369–391 A Few Points On Directed Algebraic Topology, Marco Grandis Directed combinatorial homology and noncommutative tori, Marco Grandis, Math. Proc. Cambridge Philos. Soc. 138 (2005), 233-262 Algebraic topology