Much work in the semantics of concurrent systems has been based on notions of observations of system behaviours. A fundamental property of concurrency theory is the great variety of possible observations. The large variety of possible observations depends on the fact that at present there is no widely accepted metalanguage to describe concurrent systems and their behaviours in an abstract setting, and that different aspects of system behaviour are important in different applications.
The definition of a uniform metalanguage to describe concurrent systems and their behaviours is the main concern of this paper. The basic idea is that the observation of the possible behaviours of an independent component of a system (called object) can be formalized as a function to some set of observable attributes. This key idea leads to the notion of sheaf: a sheaf is a collection of the possible behaviours of an object. The main feature of concurrent systems is that they consist of independent components (objects) interacting with each other along their computations. Using concepts of sheaf theory, the author shows that interacting concurrent systems can be described as diagrams of sheaves (connections among conponents are obtained by elegant categorical constructions), and that computations of a concurrent system are described, categorically, by taking the limit of the diagram. The main features and the insights which could be gained by using the sheaf approach are illustrated with several interesting examples.