Categorical Theory of Fundamental Dynamical Processes

With the aim of finding a generalization of the principle of general covariance applicable to certain models of emergent quantum mechanics, a categorical theory of dynamics is developed. Given a category, the notion of pre-dynamics is introduced. The dynamics is then defined as the finest pre-dynamics. The categorical recursive 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 in which 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 recursively in terms from the elements of the theory.