There are many models of concurrent computing, which are inherently geometry. Higher Dimensional Automata are (pre) cubical complexes, the PV-model as (subsets of) products of directed graphs – in both cases, executions are directed paths respecting a local partial order – time. Executions are equivalent if the corresponding paths are homotopic in a directed sense. Fundamentally, understanding the spaces of directed paths supported by a given geometry is the key. We will give the background for the interest in these spaces, give combinatorial models of them, obstructions to these being trivial and present a recent result about collapsing.