# PlanetPhysics/ETAS Interpretation

### IntroductionEdit

*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') of a supercategory (with being an integer) is defined by the specific ETAS axioms added to the eight ETAC axioms; thus, for , there are no additional ETAS axioms and the supercategory 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 is defined as an interpretation of ETAS axioms **S1** and **S2** ; for , the supercategory 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 higher than 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 syntax, to which one needs then to add semantic interpretations (which are complementary to the computer generated syntax).

### Simple examples of ETAS interpretation in supercategoriesEdit

*functor categories*subject only to the eight*ETAC*axioms;*\htmladdnormallink{functor*{http://planetphysics.us/encyclopedia/TrivialGroupoid.html} supercategories},**Failed to parse (unknown function "\F"): {\displaystyle \mathsf{\F_S}: \mathcal{A} \to \mathcal{B}}**,

with both and being 'large' categories (i.e.,
does not need to be small as in the case of *functor categories* );

- A
*\htmladdnormallink{topological groupoid*{http://planetphysics.us/encyclopedia/GroupoidHomomorphism2.html} 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; *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.)- A
*\htmladdnormallink{double groupoid*{http://planetphysics.us/encyclopedia/ThinEquivalence.html} 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;

- An example of `standard' supercategories was recently introduced in mathematical (or more specifically `categorified') physics, on the web's n-Category caf\'e'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 (^{[1]}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:
in **Cat** --the category of categories and (homo-) functors between categories-- such that:
**Failed to parse (unknown function "\textsl"): {\displaystyle \diamond \diamond \textsl{Id} \diamond \diamond Id_C \diamond '''C''' \diamond '''C''' \diamond \diamond s \diamond \diamond s = \diamond \diamond Id_C \diamond Id_C \diamond \diamond \textsl{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 **Failed to parse (unknown function "\textsl"): {\displaystyle G(2)=(\diamond 2 \diamond \textsl{Id} \diamond 2)}**
. Urs called this generalization of stabilization of n-categories, -*stabilization* . Therefore, the claim would be that `braided monoidal supercategories come from -stabilized 3-categories, with the above strict 2-group';

- An
*organismic set*of order can be regarded either as a category of algebraic theories representing organismic sets of different orders or as a*discrete topology*organismic supercategory of algebraic theories (or supercategory only with discrete topology, e.g. , a*class*of objects); - 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;

- The generalized (\L{}ukasiewicz- Moisil) toposes are supercatgeories of

*non-commutative* , algebraic -valued logic diagrams that are subject to the axioms of \emph{ algebras of
-valued logics};

- -categories are supercategories restricted to interpretations of the ETAC axioms;
- 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
-entities (*viz* . L\"ofgren, 1968; ^{[2]}); its objects are classes representing organisms
in terms of morphism (super) diagrams or equivalently as heterofunctors of organismic classes
with variable topological structure;

*Organismic Supercategories ( ^{[2]})*
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 (

^{[2]});

*organismic supercategories*were thus defined as

*superstructure interpretations of ETAS*(including ETAC, as appropriate) in terms of triples , where

*C*is an arbitrary category (interpretation of ETAC axioms, formulas, etc.), is a category of complete self--reproducing entities, , (

^{[3]}) subject to the negation of the axiom of restriction (for elements of sets): , (which is known to be independent from the ordinary logico-mathematical and biological reasoning), and is a category of non-atomic expressions, defined as follows.

An *atomically self--reproducing entity* is a unit class relation such that , which means
" stands in the relation to ", , etc.

An expression that does not contain any such atomically self--reproducing entity is called a *non-atomic expression* .

## All SourcesEdit

^{[2]}^{[4]}^{[5]}^{[3]}^{[6]}^{[1]}^{[7]}^{[8]}^{[9]}^{[10]}^{[11]}^{[12]}^{[13]}^{[14]}

## ReferencesEdit

- ↑
^{1.0}^{1.1}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). - ↑
^{2.0}^{2.1}^{2.2}^{2.3}See references [13] to [26] in the Bibliography for Category Theory and Algebraic Topology - ↑
^{3.0}^{3.1}L. L\"ofgren: 1968. On Axiomatic Explanation of Complete Self--Reproduction.*Bull. Math. Biophysics*,**30**: 317--348. - ↑
W.F. Lawvere: 1963. Functorial Semantics of Algebraic Theories.,
*Proc. Natl. Acad. Sci. USA*,**50**: 869--872. - ↑
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. - ↑
R. Brown R, P.J. Higgins, and R. Sivera.:
*"Non--Abelian Algebraic Topology"*(2008). PDF file - ↑ R. Brown and G. H. Mosa: Double algebroids and crossed modules of algebroids, University of Wales--Bangor, Maths Preprint, 1986.
- ↑
R. Brown and C.B. Spencer: Double groupoids and crossed modules,
*Cahiers Top. G\'eom.Diff.***17**(1976), 343-362. - ↑
I.C. Baianu: \L ukasiewicz-Topos Models of Neural Networks, Cell Genome and Interactome Nonlinear Dynamics). CERN Preprint EXT-2004-059.
*Health Physics and Radiation Effects*(June 29, 2004). - ↑
I.C. Baianu, Brown R., J. F. Glazebrook, and Georgescu G.: 2006, Complex Nonlinear Biodynamics in
Categories, Higher Dimensional Algebra and \L ukasiewicz--Moisil Topos: Transformations of
Neuronal, Genetic and Neoplastic networks,
*Axiomathes***16**Nos. 1--2, 65--122. - ↑
I.C. Baianu and M. Marinescu: 1974, A Functorial Construction of
-- Systems.*(M,R)**Revue Roumaine de Mathematiques Pures et Appliquees***19**: 388-391. - ↑
I.C. Baianu: 1977, A Logical Model of Genetic Activities in \L ukasiewicz Algebras: The Non-linear Theory.
*Bulletin of Mathematical Biophysics*,**39**: 249-258. - ↑
I.C. Baianu: 1980, Natural Transformations of Organismic Structures. \emph{Bulletin of Mathematical
Biophysics}
**42**: 431-446. - ↑
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; CERN Preprint No. EXT-2004-072.