Directed algebraic topology studies topological spaces in which certain directed paths (d-paths) are singled out; in most cases of interest, the reverse path of a d-path is no longer a d-path. We are mainly concerned with 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, homotopy types of 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 finite prodsimplicial complexes, ie with products of simplices as building blocks. This method makes use of a certain poset category of binary matrices related to a given model space. It applies to a class of directed spaces that arise from a certain 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 us to determine not only the fundamental category of such a model 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 cells and generators.
"Simplicial models of trace spaces." Algebr. Geom. Topol. 10 (3) 1683 - 1714, 2010. https://doi.org/10.2140/agt.2010.10.1683