categorical dynamics

0.1 Introduction

Categorical dynamics is a relatively recent area (1958- ) of applied algebraic topology/category theoryMathworldPlanetmathPlanetmathPlanetmathPlanetmath and higher dimensional algebraPlanetmathPlanetmath concerned with system dynamicsPlanetmathPlanetmath that utilizes conceptsMathworldPlanetmath such as: categoriesMathworldPlanetmath, functorsMathworldPlanetmath, natural transformations, higher dimensional categories and supercategoriesPlanetmathPlanetmath ( to study motion and dynamic processes in classical/ quantum systems, as well as complex or super-complex systems (biodynamics).

A type of categorical dynamics was first introduced and studied by William F. Lawvere for classical systems. Subsequently, a complex class of categorical, dynamic (M,R)–systems representing the categorical dynamics involved in metabolic–replication processes in terms of categories of sets and ODE’s was reported by Robert Rosen in 1970.

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 dimensionsPlanetmathPlanetmath of human neural network dynamics and the simple algebra of physical dynamics, as shown in the following, essentially non-commutative categorical diagram of dynamic systems and their transformationsPlanetmathPlanetmath.

0.2 Basic definitions in categorical dynamics

Definition 0.1.

An ultra-complex system, UCS is defined as an object representation in the following non-commutative diagram of dynamic systems and dynamic system morphismsMathworldPlanetmath or dynamic transformations:

\xymatrix@C=5pc[SUPER-COMPLEX]\ar[r](Higher Dim)\ar[d]Λ&(UCS=ULTRA-COMPLEX)\ar[d]ontoCOMPLEX&\ar[l](Generic Map)[SIMPLE]

One notes that the above diagram is indeed not ‘natural’ (that is, it is not commutativePlanetmathPlanetmathPlanetmath) 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. Moreover, each type of dynamic system shown in the above diagram is in its turn represented by a distinct diagram representing its dynamics in terms of transitions occurring in a state space S according to one or several transition functionsMathworldPlanetmath or dynamic laws, denoted by δ for either classical or chaotic physical systems and by a class of transition functions:


where T is an index class consisting of dynamic parameters τ that label the transformation stages of either a super-complex or an ultra-complex system, thus keeping track of the switches that occur between dynamic laws in highly complex dynamic systems with variable topologyPlanetmathPlanetmath. Therefore, in the latter two cases, highly complex systems are in fact represented, respectively, by functor categoriesPlanetmathPlanetmath and supercategories of diagrams because categorical diagrams can be defined as functors. An important class of the simpler dynamic systems can be represented by algebraic categoriesPlanetmathPlanetmathPlanetmath; an example of such class of simple dynamic systems is that endowed with monadic dynamics represented by the category of Eilenberg-Moore algebras.

Title categorical dynamics
Canonical name CategoricalDynamics
Date of creation 2013-03-22 18:26:59
Last modified on 2013-03-22 18:26:59
Owner bci1 (20947)
Last modified by bci1 (20947)
Numerical id 18
Author bci1 (20947)
Entry type Topic
Classification msc 55M05
Classification msc 18E05
Classification msc 18-00
Synonym Dynamics in categories
Synonym organismic supercategories
Related topic CategoriesInPhysics
Related topic HighlyComplexSystems
Related topic Supercategory
Related topic Supercategory3
Related topic DynamicalSystem
Related topic QuantumStateSpace
Related topic FunctionalBiology
Related topic QuantumTopos
Related topic WilliamFrancisLawvere