Streams, d-Spaces and Their Fundamental Categories

We describe an abstract framework in which the notion of fundamental category can be defined. The structures matching this framework are categories endowed with some additional structure. Provided we have a suitable adjunction between two of them, the fundamental categories defined in both cases can be easily compared. Each of these structures has a “natural” functor to the category of d-spaces [Marco Grandis. Directed homotopy theory, i. the fundamental category. Cahiers de Topologie et Géométrie Différentielle Catégoriques, 44(3):281–316, 2003.] and provide a Van Kampen like theorem. As an application we compare the fundamental categories of streams [Sanjeevi Krishnan. Directed Algebraic Topology and Concurrency. PhD thesis, Chicago University, 2006. Sanjeevi Krishnan. A convenient category of locally preordered spaces. Applied Categorical Structures, 17(5):445–466, 2009.] and d-spaces, actually proving that streams and d-spaces are almost the same notion.