A Categorial Theory of Objects as Observed Processes

~ e semantic domain for object-oriented languages and systems outlined in this paper is an amalgamation of two approaches: the objects-as-sheaves approach of the second author published nearly two decades ago, and the more recent ideas of the first and third authors on objects as processes endowed with observation. The basic insight is that objects in the latter sense correspond to object morphisms in the former sense. After an informal introduction into both approaches, we first elaborate on the sheaf model, using the term "behaviour" for objects in this sense, and avoiding concepts from topology. Behaviours and behaviour morphisms are shown to form a complete category where parallel composition is reflected by limits. Objects are defined to be behaviour morphisms, giving rise to a cocompletc category of objects where colimits reflect object aggregation. Object morphisms reflect different forms of inheritance, and also object reifieation (implementing objects over objects) is conveniently expressed in this framework.