Quotient models of a category up to directed homotopy

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 a fundamental category (replacing the fundamental groupoid of the classical case).Finding a simple -possibly finite -model of the latter is a non-trivial problem, whose solution gives relevant information on the given 'space'; a problem which is of interest for applications as well as in general Category Theory.Here we continue the work "The shape of a category up to directed homotopy", with a deeper analysis of 'surjective models', motivated by studying the singularities of 3dimensional ordered spaces.