Enriched categories and models for spaces of dipaths

Partially ordered sets, causets, partially ordered spaces and their local counterparts are now often used to model systems in computer science and theoretical physics. The order models ‘time’ which is often not globally given. In this setting directed paths are important objects of study as they correspond to an evolving state or particle traversing the system. Many physical problems rely on the analysis of models of the path space of a space-time manifold. Many problems in concurrent systems use ‘spaces’ of paths in a system. We review some ideas from algebraic topology and discrete differential geometry that suggest how to model the dipath space of a pospace by an enriched category. Much of the earlier material is ‘well known’, but, coming from different areas, is dispersed in the literature.