Enriched categories and models for spaces of evolving states

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. We model both the ‘space’ and the directed paths by a simplicially enriched category, and show how to adapt some classical constructions to produce a differential graded enrichment.