Categorical Theory of Fundamental Dynamical Processes

With the aim of finding a generalization of the principle of general covariance applicable to certain models for a {\it fundamental dynamical}, a categorical approach to dynamics is developed. Given a category, the notion of pre-dynamics is introduced. The dynamics is then defined as a {\it finest pre-dynamics}. The {\it categorical recurrence principle}, which constitutes a categorical generalization of the principle of general covariance, is formulated. Natural consequences of the theory are the deterministic character of the fundamental dynamics and the evolving character of the physical dynamical law itself. One main problem considered is the way how time parameters can be defined in terms of the elements of the category and set theory, as a part of the implementation of the categorical recurrence principle. Aimed to address this question, a construction is presented showing how time parameters are defined from the elements of the theory recursively.