On anticipatory systems: II. The nature of the modelling relation between systems

We embody the concept of a modelling relation between systems, or between descriptions of systems, in terms of a specific mapping between them called an equation of state . We show that the use of models for predictive purposes always involves such an equation of state. After developing some particularly simple examples, we formulate this concept in a variety of contexts in a more general fashion. The basic tool in this development is a commutative diagram expressing a conjugacy relation between mappings or descriptions; depending on the context or the kind of modelling relation under consideration, we may consider conjugacy of two descriptions related by an equation of state, or conjugacy between equations of state governing different systems. We show how all of these ideas may be formulated in an abstract way in terms of category theory.