categorical dynamics

categorical dynamics

0.1

Introduction

Categorical dynamics

is a relatively recent area (1958- ) of applied

algebraic topology

/

category theory

and

higher dimensional algebra

concerned with

system dynamics

that utilizes

concepts

such as:

categories

,

functors

,

natural transformations

, higher dimensional categories and

supercategories

(

http://planetmath.org/Supercategory

) 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

(

http://planetmath.org/RosettaGroupoids

), 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 of dynamic systems and their

transformations

.

0.2

Basic definitions in categorical dynamics

Definition 0.1

.

An

ultra-complex system,

U

C

⁢

S

is defined as an object

representation

in the following non-commutative
diagram of dynamic systems and dynamic system

morphisms

or dynamic transformations:

\xymatrix

@

C

=

5

p

c

[

S

U

P

E

R

-

C

O

M

P

L

E

X

]

\ar

[

r

]

(

Higher
Dim

)

\ar

[

d

]

Λ

&

(

U

C

⁢

S

=

U

L

T

R

A

-

C

O

M

P

L

E

X

)

\ar

[

d

]

o

⁢

n

⁢

t

⁢

o

C

O

M

P

L

E

X

&

\ar

[

l

]

(

Generic Map

)

[

S

I

M

P

L

E

]

One notes that the above diagram is indeed not ‘natural’ (that is, 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. 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 functions

or dynamic laws, denoted by

δ

for either classical or chaotic physical systems and
by a class of transition functions:

{

λ

τ

}

τ

∈

T

,

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 topology

. Therefore, in the latter two cases, highly

complex systems

are in fact represented, respectively, by

functor categories

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 categories

; 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

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