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.
