Endofunctors modelling higher-order behaviours

In this paper we show how the abstract behaviours of higher-order systems can be modelled as final coalgebras of suitable behavioural functors. These functors have the challenging peculiarity to be circularly defined with their own final coalgebra. Our main contribution is a general construction for defining these functors, solving this circularity which is the essence of higher-order behaviours. This characterisation is syntax agnostic. To achieve this property, we shift from term passing to behaviour passing: in the former higher-order is expressed by passing around syntactic objects (such as terms or processes) as representations of behaviours whereas the former ditches the syntactic encoding altogether and works directly with behaviours i.e. semantic objects. From this perspective, the former can be seen as syntactic higher-order whereas the later as semantic higher-order