Directed Algebraic Topology is a recent field, deeply linked with ordinary and higher dimensional Category Theory. A `directed space', e.g. an ordered topological space, has directed homotopies (which are generally non-reversible) and fundamental n-categories (replacing the fundamental n-groupoids of the classical case). Finding a simple model of the latter is a non-trivial problem, whose solution gives relevant information on the given `space'; a problem which is also of interest in general Category Theory, as it requires equivalence relations which are more general than categorical equivalence. Taking on a previous work on `The shape of a category up to directed homotopy', we study now the fundamental 2-category of a directed space. All the notions of 2-category theory used here are explicitly reviewed.
Homology, Homotopy and Applications, Vol. 8 (2006), No. 1, pp.31-70.