ETAS interpretation

ETAS interpretation

0.1

Introduction

ETAS

is the acronym for the

“

Elementary Theory of Abstract Supercategories

”

as defined by the

axioms of metacategories and supercategories

.

The following are simple examples of

supercategories

that are essentially

interpretations

of the eight

ETAC

axioms reported by W. F. Lawvere (1968), with one or several

ETAS

axioms added as indicated in the examples listed. A family, or class, of a specific level (or ’order’)

(

n

+

1

)

of a
supercategory

𝕊

n

+

1

(with

n

being an integer) is defined by the specific ETAS axioms added to the eight ETAC axioms; thus, for

n

=

0

, there are no additional ETAS axioms and the supercategory

𝕊

1

is the limiting, lower type, currently defined as a

category

with only one

composition

law and any standard interpretation of the eight ETAC axioms. Thus, the first level of ’proper’ supercategory

𝕊

2

is defined as an interpretation of ETAS axioms

S1

and

S2

; for

n

=

3

, the supercategory

𝕊

4

is defined as an interpretation of the eight ETAC axioms plus the additional three ETAS axioms:

S2

,

S3

and

S4

. Any (proper)

recursive

formula

or ’function’ can be utilized to generate supercategories at levels

n

higher than

𝕊

4

by adding composition or

consistency

laws to the ETAS axioms

S1

to

S4

, thus allowing a digital

computer

algorithm to generate any finite level supercategory

𝕊

n

syntax, to which one needs then to add

semantic

interpretations (which are

complementary

to the computer generated syntax).

0.2

Simple examples of ETAS interpretation in supercategories

1.

Functor categories

subject only to the eight

ETAC

axioms;

2.

Functor

supercategories

,

ℱ

𝖲

:

𝒜

→

ℬ

,
with both

𝒜

and

ℬ

being ’large’ categories (i.e.,

𝒜

does not need to be small as in the case of

functor categories

);

3.

A

topological groupoid

category

is an example of a particular supercategory
with all

invertible

morphisms

endowed with both a topological and an agebraic

structure

, still subject to all ETAC axioms;

4.

Supergroupoids

(also

definable

as crossed complexes of

groupoids

), and

supergroups

–also definable as crossed modules of groups– seem to be of great

interest

to mathematicians currently

involved in

‘categorified’

mathematical physics

or physical mathematics.)

5.

A

double groupoid

category

is a ‘simple’ example of a higher dimensional supercategory which is useful in

higher dimensional homotopy

theory, especially in

non-Abelian

algebraic topology

;
this

concept

is subject to all eight ETAC axioms, plus additional axioms related to the

definition

of the double groupoid (generally non-Abelian) structures;

6.

An example of ‘standard’ supercategories was recently introduced in mathematical (or more specifically ‘categorified’) physics, on the web’s

http://golem.ph.utexas.edu/category/2007/07/supercategories.html

n-Category

café’s web site under

“Supercategories”

. This is a rather ‘simple’ example of supercategories, albeit in a much more restricted sense as it still involves only the standard

categorical

homo-morphisms, homo-functors, and so on; it begins with a somewhat standard definiton of super-categories, or ‘super categories’ from

category theory

, but then it becomes more interesting as it is being tailored to

supersymmetry

and

extensions

of ‘Lie’ superalgebras, or superalgebroids, which are sometimes called graded ‘Lie’

algebras

that are thought to be relevant to

quantum gravity

(

[

6

]

and references cited therein). The following is an almost exact quote from the above n-Category cafe’ s website posted mainly by Dr. Urs Schreiber:
A

supercategory

is a

diagram

of the form:

⋄

⋄

I

d

C

⋄

𝐂

⋄

⋄

s

in

Cat

–the

category of categories

and (homo-) functors between categories– such that:

⋄

⋄

𝐼𝑑

⋄

⋄

I

d

C

⋄

𝐂

⋄

𝐂

⋄

⋄

s

⋄

⋄

s

=

⋄

⋄

I

d

C

⋄

I

d

C

⋄

⋄

𝐼𝑑

,

(where the ‘

diamond

’ symbol should be replaced by the symbol ‘square’, as in the original Dr. Urs Schreiber’s postings.)

This specific

instance

is that of a supercategory which has only

one object

– the above quoted superdiagram of diamonds, an arbitrary

abstract category

C

(subject to all ETAC axioms), and the standard category

identity

(homo-) functor; it can be further specialized to the previously introduced concepts of

supergroupoids

(also definable as crossed complexes of groupoids), and

supergroups

(also definable as crossed modules of groups), which seem to be of great interest to mathematicians involved in ‘Categorified’ mathematical physics or physical mathematics.) This was then continued with the following interesting example. “What, in this sense, is a

braided

monoidal

supercategory ?

”. Dr. Urs Schreiber, suggested the following answer: “like an ordinary braided monoidal catgeory is a 3-category which in lowest degrees looks like the

trivial

2-group, a braided monoidal supercategory is a 3-category which in lowest degree looks like the strict 2-group that comes from
the crossed module

G

⁢

(

2

)

=

(

⋄

2

⋄

𝐼𝑑

⋄

2

)

”. Urs called this

generalization

of stabilization of n-categories,

G

⁢

(

2

)

-

stabilization

. Therefore, the claim would be that ‘braided monoidal supercategories come from

G

⁢

(

2

)

-stabilized 3-categories, with

G

⁢

(

2

)

the above strict 2-group’;

7.

An

organismic set

of order

n

can be regarded either as a category of algebraic
theories representing organismic sets of different orders

o

≤

n

or as a

discrete topology

organismic supercategory

of algebraic theories (or supercategory only with discrete topology, e.g. , a

class

of objects);

8.

Any ‘standard’ topos with a (

commutative

) Heyting

logic algebra

as a

subobject classifier

is an example of
a commutative (and

distributive

) supercategory with the additional axioms to ETAC being those that
define the Heyting logic algebra;

9.

The generalized

L

⁢

M

n

(Łukasiewicz–Moisil) toposes are supercatgeories of

non-commutative

, algebraic

n

-valued logic diagrams that are subject to the axioms of

L

⁢

M

n

algebras of

n

-valued logics

;

10.

n

-categories are supercategories restricted to interpretations of the ETAC axioms;

11.

An

organismic supercategory

is defined as a supercategory subject to the ETAC axioms
and also subject to the ETAS axiom of

complete

self–reproduction involving

π

–entities (

viz

. Löfgren, 1968;

[

1

]

); its objects are classes representing organisms
in terms of morphism (super) diagrams or equivalently as

heterofunctors

of organismic classes
with

variable

topological structure;

Definition 0.1

.

Organismic Supercategories (

[

1

]

)

An example of a class of supercategories interpreting such ETAS axioms as those stated above
was previously defined for organismic structures with different levels of complexity (

[

1

]

);

organismic supercategories

were thus defined as

superstructure

interpretations of ETAS

(including ETAC, as appropriate) in terms of

triplets

𝐊

=

(

𝐶

,

Π

,

𝑁

)

, where

C

is an arbitrary category (interpretation of ETAC axioms, formulas, etc.),

Π

is a category of complete self–reproducing entities,

π

, (

[

4

]

) subject to the

negation

of the axiom of

restriction

(for elements of sets):

∃

S

:

(

S

≠

⊘

)

a

n

d

∀

u

:

[

u

∈

S

)

⇒

∃

v

:

(

v

∈

u

)

a

n

d

(

v

∈

S

)

]

, (which is known to be

independent

from the ordinary logico-mathematical and biological reasoning),
and

N

is a category of non-atomic

expressions

, defined as follows.

Definition 0.2

.

An

atomically self–reproducing entity

is a unit class

relation

u

such that

π

⁢

π

⁢

⟨

π

⟩

, which means
“

π

stands in the relation

π

to

π

”,

π

⁢

π

⁢

⟨

π

,

π

⟩

, etc.

An expression that does not contain any such atomically self–reproducing entity is called a

non-atomic expression

.

References

1

See references [13] to [26] in the

Bibliography for Category Theory

and Algebraic Topology (

http://planetmath.org/CategoricalOntologyABibliographyOfCategoryTheory

)

2

W.F. Lawvere: 1963. Functorial Semantics of Algebraic Theories.,

Proc. Natl. Acad. Sci. USA

,

50

: 869–872.

3

W. F. Lawvere: 1966. The Category of Categories as a

Foundation

for Mathematics. , In

Proc. Conf.

Categorical Algebra

–La Jolla

, 1965, Eilenberg, S et al., eds. Springer –Verlag: Berlin, Heidelberg and New York, pp. 1–20.

4

L. Löfgren: 1968. On Axiomatic Explanation of Complete Self–Reproduction.

Bull. Math. Biophysics

,

30

: 317–348.

5

R. Brown R, P.J. Higgins, and R. Sivera.:

“Non–

Abelian

Algebraic Topology”

(2008).

http://www.bangor.ac.uk/mas010/nonab--t/partI010604.pdf

PDF file

6

R. Brown, J. F. Glazebrook and I. C. Baianu: A categorical and

higher dimensional algebra

framework

for

complex systems

and spacetime structures,

Axiomathes

17

:409–493.
(2007).

7

R. Brown and G. H. Mosa: Double algebroids and crossed modules of algebroids, University of Wales–Bangor, Maths Preprint, 1986.

8

R. Brown and C.B. Spencer: Double groupoids and crossed modules,

Cahiers Top. Géom.Diff.

17

(1976), 343–362.

9

I.C. Baianu: Łukasiewicz–Topos Models of Neural Networks, Cell Genome and Interactome Nonlinear Dynamics). CERN Preprint EXT-2004-059.

Health Physics and Radiation Effects

(June 29, 2004).

10

I.C. Baianu, Brown R., J. F. Glazebrook, and Georgescu G.: 2006, Complex Nonlinear Biodynamics in
Categories, Higher Dimensional Algebra and Łukasiewicz–Moisil Topos:

Transformations

of
Neuronal, Genetic and Neoplastic

networks

,

Axiomathes

16

Nos. 1–2, 65–122.

11

I.C. Baianu and M. Marinescu: 1974, A Functorial Construction of

(M,R)

– Systems.

Revue Roumaine de Mathematiques Pures et Appliquees

19

: 388–391.

12

I.C. Baianu: 1977, A Logical Model of Genetic Activities in Łukasiewicz Algebras: The Non–linear Theory.

Bulletin of

Mathematical Biophysics

,

39

: 249–258.

13

I.C. Baianu: 1987a, Computer Models and Automata Theory in Biology and Medicine., in M. Witten (ed.),

Mathematical Models in Medicine

, vol. 7., Pergamon Press, New York, 1513–1577;

http://doc.cern.ch//archive/electronic/other/ext/ext-2004-072.pdf

CERN Preprint No. EXT-2004-072.

Title

ETAS interpretation

Canonical

name

ETASInterpretation

Date of creation

2013-03-22 18:16:04

Last modified on

2013-03-22 18:16:04

Owner

bci1 (20947)

Last modified by

bci1 (20947)

Numerical id

80

Author

bci1 (20947)

Entry type

Topic

Classification

msc 81-00

Classification

msc 92B05

Classification

msc 03G30

Classification

msc 18-00

Synonym

elementary theory of abstract supercategories

Synonym

ETAS

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ETA

Defines

axioms of metacategories and supercategories

Defines

examples of supercategories and metacategories

Defines

ETAS interpretation

Defines

ETAS axiom

Defines

ETAS

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