Simplicial models for trace spaces

Directed Algebraic Topology studies topological spaces in which certain directed paths (d-paths) - in general irreversible - are singled out. The main interest concerns the spaces of directed paths between given end points - and how those vary under variation of the end points. The original motivation stems from certain models for concurrent
computation. So far, spaces of d-paths and their topological invariants have only been determined in cases that were elementary to overlook.
In this paper, we develop a systematic approach describing spaces of directed paths - up to homotopy equivalence - as prodsimplicial complexes (with products of simplices as building blocks). This method makes use of certain poset categories of binary matrices and applies to a class of directed spaces that arise from a class of models of computation; still restricted but with a fair amount of generality. In the final section, we outline a generalization to model spaces known as Higher Dimensional Automata.
In particular, we describe algorithms that allow to determine not only the fundamental category of such amodel space, but all homological invariants of spaces of directed paths within it. The prodsimplical complexes and their associated chain complexes are finite, but they will, in general, have a huge number of generators.