Sheaf semantics for concurrent interacting objects

Abstract

This paper uses concepts from sheaf theory to explain phenomena in concurrent systems, including object, inheritance, deadlock, and non-interference, as used in computer security. The approach is very; general, and applies not only to concurrent object oriented systems, but also to systems of differential equations, electrical circuits, hardware description languages, and much more. Time can be discrete or continuous, linear or branching, and distribution is allowed over space as well as time. Concepts from categpru theory help to achieve this generality: objects are modelled by sheaves; inheritance by sheaf morphisms; systems by diagrams; and interconnection by diagrams of diagrams. In addition, behaviour is given by limit, and the result of interconnection by colimit. The approach is illustrated with many examples, including a semantics for a simple concurrent object-based programming language.