Hierarchical evolutive systems: A mathematical model for complex systems
In this paper the notion of an evolutive hierarchical system as a mathematical model for complex systems, such as bio-sociological systems, is proposed. This model is based on category theory and presents the structure of the system as it is defined by its internal organization of any level: its components, their interrelations, their degree of complexity, their changing in time. Functional consequences are drawn from the synchronical and diachronical interactions of the components, and from the fact that the behaviour of a complex component will be determined by that of its own components and their links. The state of the system at a given instant is modeled by a category whose objects are its components, the state transition by a functor, a complex object by the limit of a pattern of linked objects, which describe its internal organization. The properties of limits in a category make it possible to ''measure'' the emergence of properties for a complex object with respect to its components, and to reduce the study of a hierarchical system to that of its components of the lowest degree and their links. Categorical constructions describe the formation of a hierarchical evolutive system stepwise, by means of the operations: absorption of external objects, destruction of some components, formation of new complex objects. So this theory combines the structural, relational and functional approaches and it can be used as a means of obtaining a better evaluation of the characteristic relations of the degree of complexity at each level in the hierarchy of the system. The paper contains a number of natural examples which illustrate the general constructions and main principles of the theory.
scientific article; zbMATH DE number 4006039
scientific article