1 Categorical Dynamics

A relatively recent area (1958- ) of category theory, higher dimensional algebra and mathematical physics developing applications of categories, functors, natural transformations, higher dimensional categories and supercategories to dynamics in classical, quantum, complex and super-complex systems.

One can represent in square categorical diagrams the emergence of ultra-complex dynamics from the super-complex dynamics of human organisms coupled via social interactions in characteristic patterns represented by Rosetta biogroupoids, together with the complex–albeit inanimate–systems with chaos. With the emergence of the ultra-complex system of the human mind– based on the super-complex human organism– there is always an associated progression towards higher dimensional algebras from the lower dimensions of human neural network dynamics and the simple algebra of physical dynamics, as shown in the following, essentially non-commutative categorical diagram.

Definition 1.1. An ultra-complex system, U_CS is defined as an object representation in the following non-commutative diagram of systems and dynamic system morphisms or ‘dynamic transformations’:

Note that the above diagram is indeed not ‘natural’ (i.e. it is not commutative) for reasons related to the emergence of the higher dimensions of the super–complex (biological/organismic) and/or ultra–complex (psychological/neural network dynamic) levels in comparison with the low dimensions of either simple (physical/classical) or complex (chaotic) dynamic systems.