Bisimulation and path logic for sheaves:  contextuality and beyond
Giovanni Ciná, Sebastian Enqvist

Abstract:
In the setting of concurrency theory, Joyal, Winskel and Nielsen introduced a general notion of path bisimulation and showed that path bisimulations can be characterized as spans of open maps between presheaves. A modal logic for presheaves, called path logic, was shown to be expressive for such notion of bisimilarity. We consider the special case where the presheaves are defined over a topological space, and in particular where they are sheaves. We illustrate how natural properties of sheaves can be expressed in path logic and show how to encode the key concepts of the sheaf-theoretic treatment of contextuality. We further investigate the associated notion of path bisimulation on sheaves, proving a characterization result for co-spans of open maps. Finally, we introduce a “hybrid path logic” and show that the notion of a sheaf itself can be captured by an axiom of this enriched logic.

Keywords: Bisimulation, Modal logic, Sheaves